This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
% prirce denotes the subgroup of
K*/
A*
K*
generated by
p.
The
can be calculated explicitly. The result, which
will not be used in the sequel, is as follows. 2 Write A = f »A 0 as in sec. 1, and let k
be the number of
137 factors
p
in
f.
The character
x
is
as
i
n
sec. 6, and
X Q is
the corresponding character for A Q . If k = 0 we have K*/
A* = 7L =0 = 7L/27L Next let k > 0.
if
X (P)
= 1,
if
X (P)
= -1,
if
X (P)
= 0.
In most cases we have
K*/
A* = 7L © (Z/(p - l)p k ~ X7L) k
]
= Z/(p + l)p " 7L
if
X Q (P) = 1,
if
x Q (p) - " I ,
k
= (7L/27L) © (Z/p Z)
if
xo(P> - 0.
The precise list of exceptions is as follows. The group
K*/
A*
is isomorphic to 7L © (Z/2Z) © (2L/2k~22L)
if p = 2, k > 2 and
x Q (2) = 1;
(Z/2Z) © (2/3«2 " 2Z)
if p = 2, k > 2 and
XQ
Z/4Z
if p = 2, k = l and
A Q =-4 mod 16;
if p = 2, k > 2 and
A Q s - 4 mod 32;
k
2
(2Z/42) © ( Z / 2 k " 1 2 ) k
X
(7L/37L) © (2Z/2-3 ~ 7L) Combining this description of
if p = 3, k > l and K*/
A*
we obtain an algebraic description of that
F
( 2 ) = -1;
A =-3 mod 9.
with (9.2), (9.1) and (9.3) F.
In particular, we see
is the direct sum of a finite group and a free abelian
group of countably infinite rank. The natural action of sec. 1) on
F
is given by
a(F) = F
a (see
, for F e F.
10. The topological structure of F From this point onward we assume that case of negative
A
A
is positive. The
is similar, but will not be needed in the
sequel. The group homomorphism coset
(M, aQ*p.)K* ^
F -> C
defined in sec. 8 maps the
to the ideal class of M. We denote by G
the kernel of this homomorphism. The coset of G
if and only if M = A3 G = {(A, YQ* 0 )K* > 0 :
For
Y
j,Y2€K*
only if
we have
for some Y
(M, aQ*,.) belongs to
3 e K* -., so, dividing by 3:
e K*}. (A, Y , ^ ) ^ = (A, Y 2 Q* 0 )K* > 0
Y J Q * Q = ^ Y O ^ Q for some
£ e A* with
this it follows that the map d: G -> H / R Z d((A, Y Q * 0 ) K * > Q ) = (JloglY/cj(Y)l mod R)
if and
N(c) = +1 . From
138 is a well defined group homomorphism; here A,
defined in sec. 6. The map
d
"distance" defined by Shanks [24]. We have E
R
is the regulator of
is a small modification of the ker(d) = {1, E } ,
with
as defined in sec. 9. It follows that the map G -> (1R/RZ) © {±1}
obtained by combining
d
with the map
ip
from sec. 9 is an
infective group homomorphism. For cardinality reasons it is not surjective. However, its image is dense in follows from the fact that
G
(3R/RZ) © {±1};
this
is infinite (sec. 9 ) , and it can also
be seen directly. We conclude that
G
may be considered as a dense subgroup of
the product of a circle group of
f
order two. The
in
h
cosets of
G
circumference*
cosets of such a subgroup. A coset of cycle of
F,
and
G
F
R
and a group of
may be considered as the G
in
F
will be called a
itself is the principal cycle. The agreement
with the terminology introduced in sec. 5 is intentional, and will be justified in sec. 11. Every cycle consists of two circles, a positive
and a negative circle, containing forms with positive and
negative leading coefficients, respectively; cf. figure 1. If F.
F , F? e F
belong to the same cycle, the distance from -1 d(F 2 F ) , which is a real number
to
F2
is defined to be
modulo
R.
The distance is zero if and only if
F. = F«
or
F. =
F ? *E. If F , F 2 e F do not belong to the same cycle, the distance 2 from F to * 2 is not defined. 1 wv/ Replacing G by the full group (1R/RZ) © {±1}, and similarly with the cosets, we obtain an embedding of
F
as a dense
139 subset in a compact topological space
F.
see that the group multiplication of
F
It is not difficult to can be extended to
F,
making it into a topological group. This can be done using fibred sums, or by defining
F = (I © ((K ®
It)*/m* ))/K*
interest to notice that the group
F
certain group of idele classes of
K,
.
It is of
can also be described as a as follows. For background,
see [ 2 ] . Let
A = lim A/nA
be the profinite completion of
ranging over the positive integers. We may consider of the restricted product places of A*
K
and
K
TTf K ,
with
v
with
n
as a subring
ranging over the finite
denoting the completion of 1
may be considered as a subgroup of
A
A,
T7 K*;
K
at
v.
Hence
for example, if
A =
An (see sec. 1) then A* = TT U , where U consists of the 0 v v v local units at v. Adding lfs at the infinite places, we may consider
A*
as a subgroup of the group
J
of ideles of
K
K satisfying the product formula. Now we have
This group is very similar to the group
J /(K*»TT U ) , K. v v
the
compactness of which is equivalent to the conjunction of the Dirichlet unit theorem and the finiteness of the class number. The isomorphism (10.1), which will not be used in the sequel, indicates what is the right generalization of
F
for algebraic number fields
of higher degrees. 11. Reduced forms in Since no two forms in may consider
R
F R
as a subset of
quoted in sec. 5, the cycles of
are in the same orbit under F. R
By the fundamental theorem are precisely the intersections
of the cycles of Figure 2.
r, we
F
particular, we have
with
R;
in
P = G n R.
In
fact, the cyclical structure of each cycle of
R
is reflected by the way
it is sitting in the corresponding cycle of
F,
as suggested by fig. 2.
More precisely, if
F e R,
is the first element of
R
then
p(F)
that is
140 encountered if the two circles are simultaneously traversed in the positive direction, starting from F; this fixes the sense that for no it is automatic that
G e R F
one also has
and
p(F)
p(F)
G*E e R;
uniquely in and, finally,
are on different circles. The
last statement reflects the fact that the sign of the leading coefficient is changed if
p
is applied.
The proof of these statements can most conveniently be given by interpreting
R
and
p
in terms of lattice points on the
boundary of the convex hull of the totally positive part of a 2 lattice in 1R . We do not go into the details. The fundamental theorem quoted in sec. 5 is a consequence of the above results. We calculate the distance from F in
correspond to the coset M
F = (a, b) e R
(M, aQ* ) K * Q .
Choosing
to a
p(F). Let primitive
we then have M = 2Za + Z $ , aX
2
Applying
+ bXY + cY p
($a(a) - aa(3))//A>0, 2
= N(Xa + Y$)/N(M).
means first applying the element
and next an element of
T.
~
of
SL«(2Z)
The latter element does not change the
F-orbit, so we only have to investigate the effect of
~|.
changes the form into
N(XB - Ya)/W(M),
(M, 3Q* 0 )K* >0 .
(M, BQ*Q)(M, a Q ^ ) " 1 = (A, (3/a)Q*Q)
Since
3/a = (b + /A)/(2a),
This
corresponding to the coset
we find that the distance from
F
to
and p(F)
is given by d(p(F)F"1) = ilog taken modulo
6/a a(B/a)
b + /A
- Hog b - /A
R.
It is of interest to determine upper and lower bounds for this quantity. Since
0 < b < /A
b + /A ilog b - /A Using that
for a reduced form
(b + /A) 4ac
(a, b ) , we have
< |log A.
b > 1 one can prove the lower bound
(11.0 A 5,
but this is
useless. A more satisfactory lower bound is obtained by considering the distance traversed if p is applied twice, i.e. from F to 2 p (F). Let, with the notation as before, p map the coset of (M, aQ* ) (M, Y Q * Q ) •
to the coset of
(M, 3Q* Q ),
and similarly
(M, 3Q*Q )
Using the geometrical interpretation with convex hulls
that we suppressed it is quite easy to see that
|yl > 2[ ct | and
to
y/a > log 2. This gives the a(y/a) following lower bound for the distance traversed if p is applied
141
|a(y)I < Jla(a)I, so
twice: b» + b + /A (11.2) > log 2, b 2 °g b f - A p((a, b)) = (c, b 1 ) . A heuristic argument suggests that the
ilog where
average of
Jlogl (b +/A)/(b -/A) | T
somewhere near Levy s constant
over all reduced forms should be 2 TT /(12-log 2) = 1.18656911... .
Since the circumference of the whole cycle is
R,
we have
b + / A
R = I
(11.3)
b - /A '
the sum ranging over the reduced forms cycle. If there are
I
(a, b)
belonging to a fixed
reduced forms in the cycle, the above
inequalities yield H'log 2 < R < JJl-log A. Therefore, if two cycles of R
(11.4) contain
I.
and
£«
forms,
respectively, we have
This is an explicit version of a theorem of Skubenko, asserting that £j/£ 2 = O(logA), see [27; 15, pp. 558, 586]. I am indebted to A. Schinzel for mentioning this theorem to me.
12.
Reduction in
F
The reduction algorithm of sec. 4 can be formulated as follows. Extend the map
where we assumed that shows that applying distance of |b| < /A, R
p: R -> R
to a map
b, c)) = (c, b T ) ,
p((a,
p: F -> F
b f = -b mod 2c,
by bf e J ,
b e J . As in the previous section, one p
comes down to moving along the cycle over a
|log|(b + /A)/(b - / A ) |
= log|(b + /A)//|4ac| |;
also, if
one changes to the companion circle. The reduction map
p~(F) p (F)> (F), where k is the least P Q (F) = P k non-negative integer for which p (F) is reduced. Clearly, p~ P0: F
is defined by
the identity on The map a form in it.
R
R. p~
assigns to every form in
that is "not too far away" from
More precisely, let
F~, F., F~
consecutive forms on a cycle of F
0
= F
2^'
and
F
let
F €
^
be
^
n tne
R
be three (possibly
interval
142 between that
Fn
and
F~
that is opposite to
P Q ( F ) is one of
F Q , F., ¥„•
that the distance from
F
Then it can be shown
By (11.1) it follows from this,
P Q ( F ) ^S
to
F..
at
most
log A
in absolute
value. A more detailed analysis shows, in fact, that |d(p()(F)F"*1) | < Jlog(l + e/A) where
6 = (1 + /5)/2
and
for all
This is usually very small with respect to
R,
The multiplication F
*
on
R
for
(12.1) x e H/RZ.
the circumference of 5
the cycle, which may have order of magnitude
multiplication in
F e F,
|x| = min{|y|: y e x}
A .
defined in sec. 5 is just
followed by the reduction map
p~.
This
remark, and the inequalities (11.2) and (12.1), easily imply the approximate associative law (5.1), with
|n| < 1 + 41og( 1 + 6/A) /log 2.
We leave the pleasure of investigating the properties of
m(k,£)
in (5.2) to the reader.
13. The algorithm for positive discriminants We shall mainly be concerned with the calculation of the regulator
R,
which is the circumference of each circle. It can be
determined by applying the powers of p to a fixed form F € R, £ until we find p (F) = F, and then using (11.3). This is essentially the classical algorithm, which is often phrased in terms 0(A 5
of continued fractions. It has running time
£
)
for every
e > 0. We describe two more efficient methods, which make use of the function
d
defined in sec. 10. The calculations are all done in
the principal cycle
G,
and mostly in
P = G n R.
is not only specified by its coefficients parameter to read of
6
6 6
which is such that
a, b,
A form
F e G
but also by a real
d(F) = (6 mod R ) .
It is not easy
directly from the coefficients, but one can keep track
under all operations built up from
and inversion in
F,
1 = (1, b Q ) when applying
p
and multiplication
by the following rules:
has p
6=0; to
when multiplying in when inverting in
F, F,
11
(a, b ) , add add up both
~"
b - /A 6*s;
change the sign of
In particular, we can keep track of
6
to
6;
6.
under the composition
143 *: P x ? -* V
from sec. 5.
The inequality
R < /A»log A
(see (6.2)) and the baby step -
giant step technique now lead to an 0(A R,
)-determination of
as follows. Starting from the unit form
(1, b Q ) we build up a
stock of forms by successive applications of
p
("baby steps"),
until one of two things happens. It may happen, that (i) a form (a, b) (* (1, b Q )) is encountered that is its own inverse, i.e. for which
a
divides
b;
in that case,
and we stop. But for most large finds a form with after at most
A
R
is twice the current
6,
it happens sooner, that (ii) one
$ > 6Q = (/A*log A ) 5 . By (11.2), this happens '+ e )
1 + (26Q/log 2) = 0(A^
applications of p.
At this moment, we have a stock of forms that, together with their inverses, cover an interval of length
^ 26 Q along the principal
cycle. Now we start taking "giant steps", with step length a little bit less than
26^. More precisely, by
and applying a small power of whose
6
p
*-squaring the current form,
, one determines a form
F e P
satisfies
- Jlog(l + 6/A) - £log A < 6 < 26 Q - JlogO + 6 A ) . *1 *2 The giant steps are taken by calculating F = F, F = F * F, 26
..., F
= F * (F ) , . . . .
Our inequalities guarantee that
the "step length", i.e. the distance from for all
i between
6Q
and
F*1
to F*^ 1 +
26 Q . Hence after
, is
0(R/6Q) = 0(A* + E)
giant steps we have traversed the entire cycle, and we will discover F
among our "baby" forms and their inverses. Then we have two
values of
6
for the same form, and the difference of these values
is the regulator. The above algorithm calculates the regulator to any prescribed e
precision in 0(A
R
steps. The fundamental unit + £
(u + v/A)/2 since
)
cannot be calculated in 0(A^ ' '
(« number of decimal digits of u
order of magnitude
5
)
steps; in fact,
and v)
A , one cannot even write down
n = e =
n
is often of in time less
than that, let alone calculate it. It is, however, possible to calculate
u
and v
modulo any fixed positive integer
m
in time
0(A
) , the implied constant depending on m, by a procedure
similar to the above one, cf. [9]. The same remarks apply to the algorithm described below.
144 If the generalized Riemann hypothesis is assumed, we can give an
0(A
)-algorithm for the calculation of
R.
is analogous to the determination of the order of A < 0,
The procedure
F
in the case
see sec. 7, so we only sketch the main points. Using the
class number formula, we find a number
R = /A-TTp
0(A° 0(A ^(1/5) 6~ « A
being
P,
x « A1'5,
„ fl - A I E I ) " 1
.
prime, p < X ^ P that is close to an integer multiple
hR
of
R,
' .
Next, by repeated squarings and multiplications in
steps from this
).
The baby forms are now made as above, but with
F,
F
whose
6
is close to
Taking giant
6!s for the same form,
and the difference
R#
regulator; here
is supposedly not far from
h
R.
in both directions, we encounter a form that is
already in the "baby" stock. That gives two
),
the difference
e
we jump to a form
(:> A
(13.1)
is an unknown integer multiple h.
If
hR
of the
h
is large
this is discovered by finding another match after taking
some more giant steps. The remaining cases by looking if the unit form from itself, for
(1, b^)
1 < m ^ A
.
h ^ A
are checked
is found at distance
—R#
We notice that the latter
technique can also be applied in the case
A < 0,
to avoid
factoring. This finishes our sketchy description of the algorithm to determine
R.
We notice that the Riemann hypothesis is only needed
to guarantee the efficiency of the algorithm; once the answer is found, its correctness does not depend on any unproved assumptions. The determination of the class number in the case
A < 0,
with
subgroup generated by sufficiently large,
P
F, h
Otherwise, select a form
and
R
h
now runs exactly as
playing the role of the
in sec. 7, and its order. If
R
is
is determined by the class number formula. G e R9
and determine its order in
In this fashion one proceeds until a large enough subgroup of has been determined to fix
h
F/G. F/G
uniquely.
In this procedure one needs an algorithm that tests if a given reduced form belongs to the principal cycle. By the baby step giant step technique this can be done in
O(R 5 A )
steps. In
particular, equivalence of two reduced forms can be tested in 0(A(1/4) + e)
steps.
145 The conclusion is exactly as in the case Riemann hypotheses,
h
can be determined in
A < 0.
Modulo the
0(A
but the structure of the class group may take
)
0(A
steps, )
steps.
We have only considered the regulator, class number and class Rf,
group in the strict sense. To obtain the regulator h1
number
Cf
and class group
look halfway the principal cycle, i.e. at distance unit form
class
in the ordinary sense, one has to
(1, b ~ ) . If at this point the form
|R
from the
(-1, b~)
is found,
then Rf = ^R,
h f = h,
C ? = C.
Otherwise, one finds halfway
P
and
is a non-trivial factor of
b = 0 mod a.
Then
|a|
a form
F = (a, b)
with
|a| > 1 A,
and
one has RT = R, where
h 1 = ^h,
C, c c
of the form
Cf = C/C Q
is the subgroup of order two generated by the class (-1, b Q ) .
The distance of two reduced froms an integer multiple of
Rf
if and only if
This implies that the role of f
played by
R .
(a, b)
R
|a| = |a f |
is
b = bf.
and
in the above algorithm can also be
In particular, we can replace
close to the integer multiple
(a 1 , b T )
and
hfRf
of
R1.
R
by
^R,
which is
I am indebted to
R. Tijdeman for this observation.
14. A numerical example The algorithms described in sections 7 and 13 have been programmed in Amsterdam by R.J. Schoof on the CDC Cyber 750 computer system, for discriminants of up to
28
digits
[21],
Using only a hand held calculator like the HP67 one can deal with discriminants of up to - up to
6
10
digits. For much smaller discriminants
digits, roughly - it is often faster to apply the
classical algorithm (see sec. 13). We give an example which was calculated using an HP67. Let A = 40919537. cycle
P
In table 1 one finds forms lying on the principal
belonging to this discriminant. The first column gives an
identification number to each form. In the text below, form indicated by
F .
#n
is
The second column shows how the form is obtained
146 from previous forms in the table. Here are as in sec. 5, and
p
and the multiplication
*
-s- is multiplication with the inverse. The
next two columns contain the coefficients
a, b
final column gives
F.
6,
the distance from
of the form. The to the form, rounded
to five decimals from the value given by the calculator.
Table 1. #
def.
A = 40919537. a
1 = unit
1 6395
2 = p(D 3 = p(2) 4 = p(3)
tt def.
b 0
27 = 26*26
2654 2391
1234.67199
-5878 5361
4.42393
28 = 27*27
-364 6159
2469.19812
518 6035
5.63858
29 = 28*28
-137 6371
4936.94461
-2171
5 = p(4)
3904 5159
7.84756
30 = 29*29
-512 5671
9873, 63784
6 = p(5)
-916 5833
8.96447
31 = 30-22 -3584 4647
9822, 13330
7 = p(6)
1882 5459
10.50290
32 = 31-22
1586 3695
9770, 79649
8 = p(7)
-1477 6357
11.77140
33 = 32-22
-614 6129
9719, 95084
9 = p(8)
86 6371
14.65578
34 = 33-22
2294 3371
9668, 63890
10 = p(9)
-959 5137
17.75720
35 = 34v22
2857 3553
9616, 67814
11 =p(10)
3788 2439
18.86435
36 = 35-22
562 5345
9566, 02209
-2308 2177
19.26591
37 = 36-22
3934 1973
9514. 51755
3919 5661
19.62037
14 = p(13) -566 5659
21.01860
38 = 30*22-3584 5671
9925.14238
15 = p(14) 3929 2199 16 = p(15) -2296 2393 17 = p(16) 3832 5271
22.41539
39 = 38*22-3581
1479
9976.97826
22.77375
40 = 39*22
86 6371
10027.06848
18 = p(17) -857 5013 19 = p(18) 4606 4199
24.33607
41 = 29*22
-959 6371
4988.44915
25.39088
42 = 28*21
-842 5735
2496.66310
20 = p(19) -1264 5913 21 =p(20) 1178 5867
26.17737
43 = 42*3
794 5003
2502.05241
27.79557
44 = p(43) -5003 5003
2503.10318
45 = 36-27-1477 5459
8331.90585
13 = p(12)
23.16692
22=19*19
7 6385
51.50454
23 = 22*22
49 6385
103.00908
24=23*23
2401 2465 206.01816
46 = 37-22
-56 6343
9461.41380
25 = 23*24
-157 6151
47 = 46-22
-8 6391
9409.90926
26 = 25*25
-172 6113 617.15922
308.07526
147 Taking
X = 100
are taken from 6 « R.
F.
in (13.1) we find
to
F^,.
steps:
F 4 Q = Fg.
(F.«
Therefore
10012.41270 = hR,
to R
(F~ Q F,~)
Looking halfway
R#
6371 m -5137 mod 2-959.
4
Since
±842
does not divide
3
F , Q , we must have
so
h < 20.
h, A
F, . = F..,
F,«
with
6
since
Hence
close to
h
|R#
is not in the baby list, this means and that exactly at
£R#
a non-
will be found. Looking there, out of
divides
F,~ = F~ .
to
6(41) + 6(10) = JR*.
curiosity, we find the ambiguous form
the match
we find no baby
we find another match: Notice that
trivial factorization of
To test if
which has
R# - 6(40) - 6(9) =
F~ 7
is even. Looking again halfway we find
that
OQ> F~ 7 )
say. Since no other baby form, or inverse baby
R > 10012.41270 - 6(37) + 6(21) > 525,
a = -842.
to
Baby steps
we find one after three
divides
form, is found in the interval from
and
F
Then we jump to
Taking giant steps backward
form, but going forward
R = 9839.22.
h,
F,,,
yielding
we look near
Therefore
6
divides
A = 5003*8179.
(5/6)R#
and find
and
h = 6 or 18.
h,
We exclude the latter possibility by taking one more giant step (F, /)
to improve the above upper bound to
have now proved that
R > 578,
h < 18.
R = (1/6)R# = 1668.73545.
The most likely value for the strict class number h = 6. h
We show that in any case
is even. To see that
obviously a cube: e.g., F,
the cube of
We have
3
6
divides
divides
h
F, 7 = F . ^ v F ^
F = (-2, 6395)
h.
s
and it is, in
so if
F
3
or raising it to the
in the class group, and that
If one checks that difficult to prove that
5003
).
were on
Multiplying
and
h s 2 mod 4.
F
so by
11-th power, we derive in
6
8179
divides
F
has
h.
are primes, it is not
So if
and
"p prime, p > .00 0 - * £ * • ) " ' >3.05, which is very unlikely.
¥^F
6 s (-602.50344)/3 mod R/3,
each of the three cases a contradiction. We conclude that order
h =
By sec. 13, end,
a = -8,
6 « -200.83448, 355.41067 or 911.65582 mod R. F^,,
is
we search for a form that is na
6(47) = 9409.90926 s -602.50344 mod R,
or by
h
(we could also have used
the principal cycle it would have
F«/
We
h * 6
then
h > 18,
148 We leave to the reader the pleasure to find out how multiplicative relations between the
a f s can be exploited to
shorten the above calculations.
15. Concluding remarks (i) The algorithms described in this lecture can be used for an experimental approach to Gauss T s class number problems [4, sees 302-307]. Thus, they have been employed in the search for fields with irregular class groups, see [20] for references. It would also be interesting to investigate the decreasing density of fields with class number one among the real quadratic fields with prime discriminants, cf. [25, sec. 5; 12; 16, sec. 1 ] . (ii) The connection between the factorizations of the discriminant and the elements of order two in the class group gives rise to interesting factorization algorithms. Using negative discriminants, as Shanks does in [23], one obtains an algorithm factoring any positive integer
n
in
0(n
)
steps, if we
assume the Riemann hypotheses. Positive discriminants can be used in several ways. We can look halfway the principal cycle (cf. the end of sec. 13), for discriminants that are small multiples of
n.
Modulo the Riemann hypotheses it can be shown that this also leads to an
0(n
)-algorithm. A second factoring method employing
positive discriminants will be described by Shanks [26], cf. [28; 17]. This method has expected running time composite
n.
0(n
),
for
It is so simple that it can be programmed for a
pocket calculator like the HP67 for numbers of up to twenty digits. (iii) As Shanks suggested in [25, sec. 1; 29, sec. 4.4], it should be possible to adapt his techniques for number fields of higher degrees, like complex cubic fields. From sec. 10 we know that the "right" group to consider is a group whose "size" is essentially the product of the class number and the regulator. The main complication is that the circles are replaced by higher dimensional tori.
149 References 1. 2.
3. 4. 5. 6. 7. 8.
9.
10.
11.
12. 13. 14. 15. 16.
17. 18.
19. 20. 21.
22.
Z.I. Borevic", I.R. Safarevic", Teorija cisel, Moscow 1964. Translated into German, English and French. J.W.S. Cassels, Global fields, pp. 42-84 in: J.W.S. Cassels, A. Frohlich (eds), Algebraic number theory, Academic Press, London 1967. J.W.S. Cassels, Rational quadratic forms, Academic Press, London 1978. C.F. Gauss, Disquisitiones arithmeticae, Fleischer, Leipzig 1801. L.-K. Hua, On the least solution of Pellfs equation, Bull. Amer. Math. Soc. 4£ (1942),731-735. I. Kaplansky, Composition of binary quadratic forms, Studia Math. 2L (1968)> 523-530. J.C. Lagarias, Worst-case complexity bounds for algorithms in the theory of integral quadratic forms, J. Algorithms 1 (1980) 142-186. J.C. Lagarias, On the computational complexity of determining the solvability or unsolvability of the equation X 2 - D Y 2 = - 1 , Trans. Amer. Math. Soc. _26£ (1980), 485-508. J.C. Lagarias, Succinct certificates for the solvability of binary quadratic diophantine equations, Proc. 20th IEEE Symp. foundations comp. sci., 1979, 47-56. J.C. Lagarias, H.L. Montgomery, A.M. Odlyzko, A bound for the least prime ideal in the Chebotarev density theorem, Inventiones math. 54 (1979), 271-296. J.C. Lagarias, A.M. Odlyzko, Effective versions of the Chebotarev density theorem, pp. 409-464 in: A. Frohlich (ed.), Algebraic number fields, Academic Press, London 1977. R.B. Lakein, Computation of the ideal class group of certain complex quartic fields, II, Math. Comp. _29 (1975), 137-144. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2 Bande, Teubner, Leipzig 1909; 2nd ed., Chelsea, New York 1953. P.G. Lejeune Dirichlet, R. Dedekind, Vorlesungen liber Zahlentheorie, Braunschweig 1893^; reprint, New York 1968. A.V. Malyshev, Yu.V. Linnik?s ergodic method in number theory, Acta Arith. 27_ (1975), 555-598. J.M. Masley, Where are number fields with small class number?, pp. 221-242 in: M.B. Nathanson (ed.), Number Theory Carbondale 1979, Lecture Notes in Mathematics 751, Springer, Berlin 1979. L. Monier, Algorithmes de factorisation d'entiers, These de 3° cycle, Orsay 1980. J. Oesterle, Versions effectives du theoreme de Chebotarev sous l'hypothese de Riemann generalised, pp. 165-167 in: Asterisque 61 (Journees arithmetiques de Luminy), Soc. Math, de France 1979. J.M. Pollard, Theorems on factorization and primality testing, Proc. Cambridge Philos. Soc. 76» (1974), 521-528. R.J. Schoof, Quadratic fields and factorization, in: Number theory and computers, Mathematisch Centrum, Amsterdam, to appear. R.J. Schoof, Two algorithms for determining class groups of quadratic fields, Mathematisch Instituut, Universiteit van Amsterdam, to appear. I. Schur, Einige Bemerkungen zu der vorstehenden Arbeit des Herrn G. Polya: Uber die Verteilung der quadratischen Reste und Nichtreste, Nachr. Kon. Ges. Wiss. Gottingen, Math.-phys. Kl.
150
23.
24. 25.
26. 27.
28. 29.
(1918), 30-36; pp. 239-245 in: Gesammelte Abhandlungen, vol. II, Springer, Berlin 1973. D. Shanks, Class number, a theory of factorization, and genera, pp. 415-440 in: Proc. Symp. Pure Math. _20 (1969 Institute on number theory), Amer. Math. S o c , Providence 1971. D. Shanks, The infrastructure of a real quadratic field and its applications, Proc. 1972 number theory conference, Boulder, 1972. D. Shanks, A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view), pp. 15-40 in: Congressus Numerantium YJ_ (Proc. 7th S-E Conf. combinatorics, graph theory, and computing, Baton Rouge 1976), Utilitas Mathematica, Winnipeg 1976. D. Shanks, Square-form factorization, a simple 0(N1/1+) algorithm, unpublished manuscript. B.F. Skubenko, The asymptotic distribution of integers on a hyperboloid of one sheet and ergodic theorems (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 26^ (1962), 721-752. S.S. Wagstaff, Jr., M.C. Wunderlich, A comparison of two factorization methods, to appear. H.C. Williams, D. Shanks, A note on class number one in pure cubic fields, Math. Comp. 33 (1979), 1317-1320.
H.W. Lenstra, Jr. Mathematisch Instituut Universiteit van Amsterdam Roetersstraat 15 1018 WB
Amsterdam
Netherlands.
PETITS DISCRIMINANTS DES CORPS DE NOMBRES Jacques MARTINET *
Soit K un corps de nombres, de degre n et de signature (r, s)
(i.e.,
K possede
r places reelles et s places imaginaires)
Les travaux recents de Odlyzko, Poitou et Serre, reposant sur des me*thodes analytiques, ont conduit a des minorations de la valeur absolue du discriminant de K bien meilleuresque celles qui avaient e*te obtenues jusqu'a present a l'aide des me*thodes geometriques issues des travaux de Minkowski. C'est d'abord de ces travaux que nous rendrons compte. Les lettres n , r , s , eventuellement indexe*es par K ont le sens ci-dessus dans la suite ; on note d
le
discriminant du corps K : c'est la valeur du determinant det(Tr(uu. 0).)) 1
j
ou u)-,...,ou 1
n
de*signe une base de K sur Q et Tr
d^signe la trace de K sur Q . Etant donne'e une extension L./K , on dispose e*galement d'une notion de discriminant relatif, qui est un ide*al de K . Pour d'autres notions de discriminant, voir l'appendice 1. Laboratoire associ^ au C.N. R.S. n° 226
152 Le probleme fondamental concernant les discriminants est le suivant : etant donnes un couple ( r , s ) n = r + 2 s > 0 ) , et un entier de nombres
d'entiers
( r > 0 , s>0 ,
d , a quelle condition existe-t-il un corps
K de signature
(r, s)
tel que d
= d ? L'entier d
rv
est soumis a. un certain nombre de contraintes : (i)
le signe de d est lie au couple (r, s) par l'egalite jd| = ( - i ) s d ,
(ii)
l'entier
d verifie la congruence de Stickelberger :
d = 0 ou 1 mod 4 (voir l'appendice (iii)
2) ,
pour chaque nombre premier p , l'exposant
v (d)
de p
dans d ne peut prendre qu'un nombre fini de valeurs (voir l'appendice 3) , (iv)
enfin, et c'est la l'essentiel,on dispose de minorations de |d|
en fonction du couple (r, s) .
La fagon dont ces minorations s'obtiennent par une etude des fonctions zeta des corps de nombres est expliquee dans le paragraphe 1. On trouvera en fin d'article, grace a l'obligeance de Odlyzko d'une part, de Diaz y Diaz et de Poitou d'autre part, des tables de minorations.
Le paragraphe 2 est consacre a. des applications, essentiellement a des problemes de nombres de classes, des minorations exposees dans le paragraphe 1 . Enfin, dans le paragraphe 3, nous etudions le probleme de construire des corps de discriminants (en valeur absolue) petits, c'est-a-dire aussi voisins que possible des bornes inferieures provenant du paragraphe 1. L'expe*rience montre que la recherche de petits discriminants est liee a la recherche de corps euclidiens par
153 la me'thode de Lenstra ; pour cette raison, nous exposons avec quelques details en quoi consiste cette methode. Deux tables illustrent le paragraphe 3 : une table de petits discriminants de corps de degre S 8 et une table de petits discriminants de corps totalement imaginaires.
1. - Minorations 1 . 1 . - Formules explicites. - Serre s'est rendu compte que les premieres minorations obtenues par Odlyzko s'interpretaient comme cas particuliers des "formules explicites" dues a. Weil [_W1, W2] ; nous renvoyons le lecteur a l'expose [ P I ] de Poitou pour l'aspet historique de la question. Nous utilisons les formules explicites sous la forme donnee par Poitou dans QP 2] (les formules enoncees et demontrees dans [P 2] concernent les fonctions zeta des corps de nombres, ce qui suffit pour les applications que nous avons en vue ; l'extension aux fonctions
L ne presente pas
de difficult^ particuliere) . On considere un corps de nombres K , de degre n=r+2s , et une fonction reelle paire
F , verifiant certaines conditions de
regularite et assez rapidement decroissante a l'infini. Voici les conditions donnees par Poitou : (i) Les fonctions
x *—> F(x)
et x»—> (F(0 )- F(x)) /x
sont a
variation borne*e sur toute la droite, et leur valeur en chaque point est la moyenne de leurs limites a gauche et a droite. (ii) A l'infini, la fonction
F est O(e"^ + )' X I)
On considere la fonction <$ definie pour nant a la bande critique 0
pour un 6 >0 .
s complexe apparte-
par la formule :
154
$(s)=J
F(x) exp((s-£)x)dx
-00
et les trois nombres re*els A_, , B
, C
p+0° F(0 ) - F(x) F"J 2ch(x/2)
d^finis par les integrales :
=
_ p+0° F(0 ) - F(x) F"J0 2sh(x/2) 0 = 4 /
*¥(*)
X
X
ch(x/2)dx.
0 THEOREME. - Quelle que soit la fonction F paire ve*rifiant les conditions (i) €>t (ii) , on a l'e'galite : F(0)[log|d|-n(Y+log8n)-r^]+rAF+nBF+CF = (
lim
Z
*(p))+ 2 2
Dans le membre de gauche,
;
y
^ f r / 9 F(m log N p ) .
d est le discriminant d'un corps
de nombre K de degr^ n possedant complexes
lQg
r places reelles et s places
designe la constante d'Euler
( y = 0, 577 215... ) .
Le membre de droite est la somme de deux termes. Dans le premier, la sommation est ^tendue aux ze*ros de la fonction z§ta de K de partie re*elle positive ; dans le second, la sommation est ^tendue aux entiers m>0
et aux ide*aux premiers p
de K .
Les convergences des series et integrales qui interviennent ( N designe la norme)
sont clairement a s s u r e s par les hypotheses
faites sur la fonction F .
1 . 2 . - Principe d'obtention des inegalites. - Pour tirer du the*oreme ci-dessus des ine*galite*s sur les discriminants, on choisit F de fagon que le second membre de l'egalite soit > 0 , en prenant
une fonction F positive dont la transformed de Mellin $
155 prend des
valeurs de parties replies positives aux ze*ros de la fonction zeta de K (la partie re*elle suffit par symetrie) . Faute de pouvoir localiser ces ze*ros avec precision, le mieux que l!on puisse faire dans l'e'tat actuel de nos connaissances est de s'arranger pour que $ soit de partie re*elle positive dans la bande critique 0 < R e ( s ) < l . Si l'on veut bien admettre l'hypothese de Riemann ge*ne*ralisee (GRH) , il suffit de prendre la partie reelle de $ droite
s = -^ ; alors, lorsque l'on pose
positive sur la
s = ^ + i t , <1> devient une
fonction cp de la variable re*elle t , de*finie par la formule +» cp(t)=J F(x) e
dx .
-00
On reconnaft la transformed de Fourier de F , et il suffit alors d'utiliser des fonctions
F positives dont la partie re*elle de
la transformee de Fourier (transformee de Fourier en coainus) soit e*galement positive.
1 . 3 . - Minorations asymptotiques. - On se donne un reel P€ [0>l]>
e
t l ! on considere une suite de corps K dont les degres
n tendent vers l'infini pour laquelle le rapport
r/n tend vers p ;
il s'agit de minorer la limite infe*rieure de la suite
a) Minoration sous
| d^ |
GRH (Serre)
On considere une fonction F positive a transformee de Fourier positive ; un exemple de telle fonction est donne par la figure de gauche ci-apres, les fonctions de ce type etant les carres de convolution des fonctions represente*es par la figure de droite.
156
On considere ensuite une suite de fonctions de la forme x -* aF(x/b)
dans laquelle on ajuste les nombres reels
a et b
de fagon a obtenir la convergence de la suite (au sens des distributions) vers la mesure de Dirac en 0 . Apres division par la valeur en 0 cles fonctions, on constate que les contributions dues aux termes contenant A
, B-^ et C
tendent vers 0 . On en deduit
la minoration asymptotique r I , | l / n « Y+fTI" / 2 ) 0 l i m inf | d | > oTTe Numeriquement, on obtient les inegalites liminf | d^ |
>44,7
(resp. liminf
|cL^|
^215,3)
pour une
suite de corps totalement imaginaires (resp. totalement reels) . b) Mino rations inconditionnelles(Odlyzko) Le procede ci-dessus n'assure pas la positivite de <> |
dans
toute la bande critique. Odlyzko s'est apergu que l'on y parvenait en remplagant les fonctions x«—• a F(x/b) /ch(x/2).
x ^ a F(x/b)
par les fonctions
La contribution des termes en A
et
B
intervient alors dans le resultat, et l'on trouve la minoration asymptotique liminf | d K | l / n > 4 n e Y + P
.
Par rapport a. la minoration trouvee par Serre sous on a perdu un facteur croissant en fonction de 0
GRH,
de 2 dans le cas
, totalement imaginaire a
2e
157 ' " = 3,539...
dans le tas totale-
ment reel.
1.4. - Minorations pour une signature donnee. aux minorations donnees de 1. 3, que ce soit sous
Par rapport
GRH
ou incon-
ditionnellement, les termes
A , B et C introduisent une F F F ^ / correction affaiblissant les minorations de dT_| . II est posJS,
sible de construire des tables ; la qualite des resultats depend du choix de la fonction
F . Grace a l'obligeance de Odlyzko (resp.
Poitou et Diaz y Diaz) , on trouvera de telles tables a la fin de cet article sous GRH
(resp. inconditionnelles) .
Ces tables partagent les deux proprietes suivantes : (a) pour un rapport
r/n
donne\ la minoration de
jd |
est une fonction croissante de n ; (b)
es
pour un degre donne, la minoration de |dT^I
t
u n e
fonction croissante de r/n . Outre les tables citees ci-dessus, Poitou a donne diverses formules faciles a programmer sur une calculatrice de poche, concernant le
cas "inconditionnel". Nous reproduisons avec son
autorisation la formule suivante, qui est un bon compromis eritre la simplicite et la qualite du re*sultat ; elle conduit a une minoration de
— log |d|
en fonction homographique de r/n
:
n Dans cette formule, b
et c D
y
est la constante d'Euler. Les coefficients
, qui dependent de r / n ,
cL
entier positif, on pose
sont definis ainsi : pour k
158 X(k) r)(k)
= l +l/3
k
k
+ l / 5 + ...
=1-l/2
k
+ l/3k-l/4k+...
2k+1 C
2
=
(12b5)/(35b3)
Voici les valeurs numeriques utiles : X(3)
= 1,0 51 799 790 . . .
r\(2) = 0,822 467 0 33 . . .
X(5)
= 1,004 523 762...
r)(4) = 0 , 947 0 32 829...
Y + log ^
4TT
3
= 3,108 239 911 . . .
= 11, 243 134 665...
Exemple. - Pour les corps totalement imaginaires, on peut utiliser la formule ^•log |d|^3,108 - 6 , 8 6 l / ( n 2 / + 0 , 9 3 6 )
.
Pour n > 30 , cette formule donne des resultats meilleurs que les tables inconditionnelles de Odlyzko (qui ne sont pas reproduites ici, celles de Diaz y Diaz et Poitou etant un petit peu meilleures) .
1 . 5 . - Le second membre des formules explicites. - Dans les paragraphes precedents, le second membre des formules explicites n'intervenait que par la condition qu'il soit >0 . La contribution du premier terme semble difficile a estimer. On peut en revanche tenir compte du second terme, et cela est particulierement facile a faire loreque l'on utilise des fonctions nulles en dehors d'un compact, ce qui est le cas des fonctions utilisees par Odlyzko pour constiruire ses tables . Voici un exemple d'application de*rons le corps de degre
: consi-
8 , extension cyclique de degre 4 de
D(i)
de conducteur un ide*al premier au-dessus de 17 . Son dis8 3 criminant est 2 .17 ; c'est le plus petit discriminant connu
parmi les discriminants de corps totalement imaginaires de degre
8 ,
et ,
dans ce corps ,
2
est le carre d'un ideal
159 premier de degre 4 . En tenant compte du second terme, on peut montrer inconditionnellement que, si un corps totalement imagi8 3 naire de degre* 8 a un discriminant au plus egal a 2 . 1 7 , alors la decomposition de 2 dans K est de l'une des trois formes suivantes :
2 est inerte,
(2)= £
ou (2) = ? $' ,
ou ?
et ?'
designent des ideaux premiers de degre* 4 . De fagon generale, si le discriminant d'un corps est proche de la limite inferieure donnee par les tables, ce corps ne contient pas d'ideaux premiers de petite norme. En outre, dans ces conditions, le premier terme du second membre des formules explicites est lui aussi petit.
1.6.- Grands discriminants. - Fixons un degre n et une signature
(r, s) . Les corps soumis a ces conditions peuvent avoir
des discriminants arbitrairement grands. Au premier membre de la formule explicite, le corps n'intervient que par le terme en ~ l o g | d | . Le second terme du second membre est borne : une borne superieure se determine en calculant la somme m
£
——&-—v-l ]r( m iogp)
dans la situation (impossible en fait a
'*> (Npf
cause du theoreme de Tchebotarev) ou la decomposition des nombres premiers est la plus defavorable. II s'en suit que, pour une suite de corps dont les discriminants tendent vers l'infini, le terme lim ( L .^(P)) T-*« |lm(p)FST
tend vers l'infini. Cela traduit une tendance des
zeros de la fonction zeta a se rapprocher de l'axe re*el. II y a peutetre la (remarque de Poitou) une possibility d'obtenir des renseignements sur la repartition statistique des zeros des fonctions zeta en fonction des discriminants en utilisant des families convenables de fonctions
F .
160 1.7.- Peut-on faire mieux ? - La question posee est de savoir si l'on peut esperer ameliorer les minorations ci-dessus. Nous verrons au paragraphe 3 que pour de petits degres, on connait des exemples de corps
K pour lesquels
Id I
est proche
des minorations en question. II n'en est pas de meme quand on considere de grands degres, mais cela est sans doute du au fait que l'on ne connait pas de bonnes methodes pour decrire des corps de grands degres. A defaut de disposer d'exemples, on peut tenter d'analyser de plus pres la methode elle meme. Faute de renseignements precis sur la localisation des zeros des fonctions zeta, et si l'on ne desire pas introduire de conditions sur la decomposition des nombres p r e m i e r s , on ne peut qje considerer des fonctions
F posi-
tives telles que <> | soit positive dans la bande critique (ou simplement sur la droite
s = 1/2
si l'on accepte d'utiliser G R H ) .
Lors-
que l'on recherche des minorations asymptotiques, les minorations de Serre (sous G R H ) et de Odlyzko (sans hypothese) apparaissent etre les meilleures possibles ; Odlyzko m'a ecrit qu'il avait etudie la situation ge*nerale, et qu'il disposait de resultats montrant que l'on ne pouvait pas esperer obtenir d'ameliorations substantielles des minorations en s'imposant les regies ci-dessus sur
F
et § .
1 . 8 . - Classes d'ideaux. - Dans les methodes geometriques (Minkowski et ses continuateurs, jusqu'a
Mulholland), les mino-
rations des discriminants apparaissent comme corollaire a un r e sultat prouvant la finitude du nombre de classes d'ideaux des corps de nombres : il existe une constante d'ideaux contient un ideal 51 la norme de 21 par
c
r, s de norme S c
telle que toute classe
J\ dl ; en minorant r, s 1 , on obtient une minoration de |d| .
161 L'emploi des formules explicites ne conduit pas a des resultats de ce type. II convient ne*anmoins de signaler que Zimmert [Z] a obtenu recemment des resultats de la forme ci-dessus par des me*thodes analytiques, desquels il est possible de deduire la minoration asymptotique trouvee par Odlyzko.
2. - Applications des minorations des discriminants 2. 1. - Extensions non ramifiees. - Soit K un corps de nombres de degre n et soit L une extension non ramifiee de K . On a entre les discriminants de K et L la relation O].
En cons,quence,
si { ^ ^
est
inferieur a la minoration asymptotique du paragraphe 1, l'extension maximale non ramifiee de K est de degre fini ; si
|d|
est meme inferieur a la minoration correspondant au degre 2n , le corps K ne possede aucune extension non ramifiee autre que lui-meme. Les resultats de Minkowski permettaient deja de prouver cette proprie'te pour certains corps K , en particulier pour K = (Q , permettant a Minkowski de re*soudre un des problemes ouverts celebres de la theorie des nombres ; les progres recents des minorations des discriminants ont considerablement augmente le champ d'application de la methode. Bien entendu, nous devons faire des demonstrations, et par consequent utiliser les minorations inconditionnelles. Ceux qui croient en la validite de l'hypothese de Riemann generalisee en conclueront que le procede ci-dessus ne s'applique qu'aun nombre fini de corps. 2. 2. - Corps de nombre
de classes 1 . - On note h__ le K nombre de classes du corps de nombres K . La theorie du corps de classes montre qu'un corps de nombres K a un nombre de
162 1 si et seulement si K ne possede pas d'extension abelien-
classes
ne non ramifiee autre que lui-meme. Le procede indique en 2 . 1 . fournit done des exemples de corps dont le nombre de classes est 1 . L'egalite
^
= 1 equivaut aussi a l'absence d'extension resoluble
non ramifiee de K autre que K lui-meme. Mais rien n'empSche qu'un corps de nombres avec h
=1 possede des extensions non
ramifiees a groupe de Galois simple non abelien ; nous n'avons pas d'interpretation de l'existence de telles extensions . Comme l'a remarque* Masley (cf [Ms] et sa bibliographie) , on peut, en utilisant l'action des groupes de Galois sur les groupes de c l a s s e s , prouver l'egalite
h
=1 dans le cas de corps pour l e s -
quels les minorations des discriminants n'entrament pas immediatement l'absence d'extension non ramifiee. Le principe est le suivant : soit
L une extension galoisienne non ramifiee d'un corps K , de
groupe de Galois ses
CL
G ; le groupe
G opere sur le groupe des clas-
de L . Si l'on peut prouver que l'action de G est fidele,
on a l'inegalite* h non ramifiee
M/K
> [ L : K] , d'ou. l'existence d'une extension de K de degre relatif > [ L : K ]
, ce qui peut
etre contradictoire avec les minorations des discriminants pour le degre
[M : Qj . Voici comment on peut utiliser cette remarque dans
le cas ou. G est cyclique d'ordre premier
t . Soit p un nombre
p r e m i e r , et soit H le sous-groupe de Cl forme des classes P L verifiant l'egalite sur
xp = 1 ; H
fournit une representation de G
IF . Si p ^ {, , cette representation est semi-simple ; par
consequent, si H
est non trivial, son ordre est divisible par
p
ou m
est le plus petit entier pour lequel IF m contient une racine m P d'ordre I de l'unite, i . e . p = 1 m o d i . Si p = £ , e t s i H P est non trivial, le sous-groupe de H fixe par G est egalement
non trivial ; des formules classiques de classes invariantes r e montant a Furtwangler, Takagi et Chevalley fournissent un calcul
,
163 explicite du nombre de classes invariantes permettant eventuellement de prouver que h
est d'ordre premier a. £ . Un cas par-
ticulier important est le suivant : si h une unique place se ramifie dans
est premier a. £ , et si
L/K , alors h
est egalement
premier a. t . 2. 3. - Exemples 2. 3.1. - Masley a prouve* que,pour n S 50 , le sous-corps reel maximal du corps des racines n-emes de l'unite* a pour nombre de classes 1 . Ces re*sultats ont e*te etendus par van der Linden [vL] a un grand nombre de corps abeliens reels. 2. 3. 2. - On peut a l'aide de ces methodes, et en utilisant les re*sultats connus sur les classes des corps de degre* 3 ou 4 , etudier les extensions maximales non ramifiees des corps quadratiques imaginaires ; pour
jd|8<4TTe , i.e.
|d| ^ 499 , les corps qua-
dratiques imaginaires de discriminant d ont une extension maximale non ramifie*e de degre fini. Pour
|d| S 250 , on trouve que
1'extension maximale non ramifiee d'un corps quadratique de discriminant d est son corps de classes de Hilbert,. aux sept exceptions pres suivantes, pour lesquelles nous donnons la structure du groupe de Galois relatif de l'extension maximale non ramifiee, le symbole D
(resp. H ) designant un groupe diedral (resp. qua-
ternionien) d'ordre n d = -115 :
D
d = -184 :
H
6
:
d = -120 : H g
;
d = -155
:H
12
:
d = -195
;
d = -235
:D
io
12 d = -248 : H .
:H
16
Voici deux autres exemples interessants : d = -283 : l'extension maximale non ramifiee est de degre 48 ; son groupe de Galois relatif (resp. absolu) est le groupe A .
164 (resp. S 4 ) , obtenu comme image reciproque dans le revetement S —• S 0 (IR)
d'un sous-groupe de SO (1R)
isomorphe a
A.
(resp. S4 ) . d = -420 :
le groupe des classes du corps quadratique est de type
(2,2,2),
et c'est le seul exemple dans lequel le groupe des clas-
ses contient un groupe a trois generateurs pour lequel on sache montrer que l'extension maximale non ramifiee est de degre fini. On montre sans difficult^ que l'extension maximale non ramifiee K a un degre sur
Q multiple de 64 ; si l'on accepte l'hypothese
de Riemann generalised, il est immediat que [K : Q] = 64 ; demontrer ce resultat sans utiliser l'hypothese de Riemann generalisee est beaucoup plus difficile ; une demonstration m'a ete communiquee par Rene Schoof pendant la conference d'Exeter [Sch] . 2. 3. 3. - On trouve dans [Mr 2] un exemple de corps totalement imaginaire de degre* 116 et de nombres de classes
1 ;
c'est le plus grand degre connu pour un coips avec hT = 1 . Un autre exemple interessant, signale par Lenstra, est celui du corps de classe de Hilbert du corps des racines degre
23-emes de l'unite, de
66 .
3. - Recherche de petits discriminants Dans ce paragraphe, nous cherchons des exemples de corps ayant un petit discriminant relativement aux minorations exposees dans le paragraphe 1 . Nous avons systematiquement compare, pour un corps
K de degre n , |d |
par Odlyzko sous
GRH
avec la minoration donnee
pour la signature de K ; il semble en ef-
fet plus raisonnable d'admettre
GRH
pour avoir une idee de ce
que l'on peut esperer ; de toute fagon, pour les degres assez petits, les minorations inconditionnelles sont proches des minorations
165 obtenues sous GRH. Les corps de discriminant petit
ont pour nombre de classes
1 . Souvent, on peut montrer qu'ils sont euclidiens. Inversement, la recherche de corps euclidiens par la methode de Lenstia conduit a. des exemples de petits discriminants ; pour cette raison, nous avons donne quelques details sur cette methode.
3. 1. - Tours de corps de classes. - Dans ce sous-paragraphe, nous cherchons a tester les minorations asymptotiques de Serre. On a vu (cf
§ . 1 . 3 ) , que pour tout reel p £ [0 , 1] , on avait pour
une suite de corps K de degre* n tendant vers l'infini pour lesquels
r/n
|
tend vers p
> 8rre
*'
la minoration asymptotique
(resp. 4n e
)
selon que l'on accepte ou non
d'utiliser l'hypothese de Riemann generalisee. Un resultat important de Golod et Safarevic est que, pour tout p
rationnel £ [0, 1] ,
il existe une suite de corps de degre tendant vers l'infini avec lim de
— = p et lim |d | < + « (autrement dit, la limite inferieure 1/ |cL-| pour n -»« et ~—• p est finie) . Le principe est
d'utiliser des tours de corps de classes, c'est-a-dire "d'empiler" des extensions abeliennes non ramifiees ; on ne sait pas ce qui se passe si
lim ~~ = p n'est pas un nombre rationnel. Voici brieven ment comment on procede. a) On se fixe un nombre premier groupe d1 (G)
p . Etant donne un pro-p-
G , on sait definir le nombre minimum de generateurs de G et le nombre de relations
d (G)
entre ces genera-
teurs ; une formulation de ces definitions est d.(G) = dirrr d(A)
H (G , ^ / p ^ ) . Pour un groupe abelien fini A , on note
le p-rang de A : d(A) = dim
Z/pZ® A . P b) Soit K un corps de nombres, et soit L/K une extension
finie, non ramifiee, galoisienne, dont le groupe de Galois
G est
166 un p-groupe
; soit E
(resp. L ) . Alors, si
c) Soit
(resp. p
E ) le groupe des unites de K
ne divise pas
G un p-groupe fini
h
, on a l'egalite (cftSe]):
; Golod et Safarevic (cf [Ro])
ont montre* l'inegalite* d2(G)>d1(G)2/4 . d) II re*sulte de b) et c) que si le p-rang
t du groupe des
classes de K est assez grand, alors la p-tour de corps de classes de K est infinie : en effet, soit
G le groupe de Galois de la
p-extension maximale non ramifie*e de K ; la theorie du corps de classes montre que l'on a l'e'galite'
d. (G) = t ; en utilisant
b)
et
le theorem e de Dirichlet sur les unite's, on voit tout de suite que, si G est fini, on a l'ine*galite* d9(G) - d1 (G) S r+s . Cette ine*galite* est contradictoire avec
c)
si t depasse
2 4- 2A/ r+s+1 .
e) Soit alors K un corps de nombre de degre n = r +2s r & ' o o o o avec r /n = p . Un calcul de classes invariantes montre que le 2-rang du groupe des classes du corps
K (^/rn) , m
entier positif,
tend vers l'infini avec le nombre de facteurs premiers de m . Pour un choix convenable de m , le corps
K= K
(\/m)
2-tour de corps de classes infinie, et le rapport discriminant eleve a la puissance pectivement a
p
eta
Jd j
l/n
a done une r/n
ainsi que le
sont constants, egaux r e s -
, lorsque l'on considere les dif-
f^rents Stages de la tour de corps de classes,
C . Q . F . D.
Les exempies de tours avec une valeur pas trop grande pour ont e*te obtenues en rendant d) le meilleur exemple connu (cf [Mr 1])
explicite
. Pour
p =0 ,
est celui du corps
K= D (cos 2TT/11 , V-46) , de degre 10 , pour lequel i, ,1/10 --4/5 o3/2 OQl/2 n o o , o Jd I = 11 . 2 ' . 23 ' - 92, 368 . . . , ce qui depasse consi-
167 de*rablement la borne inferieure asymptotique
8lTeY = 44, 763 ... .
II est extrSmement probable qu'il existe des tours donnant de meilleurs resultats. Ainsi, conside'rons un corps quadratique imaginaire K avec cinq nombres premiers ramifies, si bien que le 2-rang du groupe des classes de K est £gal a. 4 . Si la 2-tour de corps de classes de K est finie, alors d'apres b) et c) , le groupe G (en tant que pro-2-groupe) peut etre defini par quatre generateurs et cinq relations. On ignore s'il existe de tels 2-groupes ; les specialistes conjecturent qu'un tel groupe n'existe pas. Une demonstration de ce fait permettrait alors de remplacer par
2. (3. 5. 7. 13)
2
92, 368 . . .
=73,891... , comme on le voit en considerant
le corps Q(v/-3. 5.7.13). A defaut d'ameliorer le theoreme de Golod et Safarevic, ce qui est probablement difficile, on peut essayer d'utiliser d'autres proprietes des groupes de Galois. L'exemple suivant d\X a Koch et Venkov ([K - V]) estimations de
est encourageant, bien que n'ameliorant pas les |dr<-|
impair, 1'eVentualite
:
c e s
deux auteurs ont montre que pour p
d (G) = d (G) = 3 etait impossible pour le
groupe de Galois (suppose fini) de la tour de corps de classes d'un corps quadratique imaginaire, bien qu'il existe des p-groupes finis ayant ces valeurs pour d1
et d9
JL
Ld
; leur demonstration est en fait
un theoreme de th^orie des groupes, que l'on applique aux corps quadratiques imaginaires en utilisant I1 existence de l'automorphisme d'ordre
2 induit par la conjugaison complexe.
A mon avis, l'obtention de
resultats significativement meil-
leurs passe par des progres en theorie des groupes.
3.2.- Corps totalement imaginaires. - La theorie du corps de classes permet de decrire les extensions abeliennes d'un corps de nombres k a partir de donnees provenant exclusivement de k :
168 groupes des classes d'ideaux et des unites de k , et decomposition dans k des nombres p r e m i e r s . L'iteration du procede est difficile, car la theorie du corps de classes, si elle fournit la decomposition des ideaux p r e m i e r s dans les extensions abeliennes de k , n'indique rien au sujet des unites ni des classes d'ideaux de ces extensions. On n'a alors pas d'autre recours que la determination explicite de polynomes, ce qui conduit a des calculs d'une complexite croissant rapidement avec le degre. L'utilisation de la theorie du corps de classes est d'autant plus facile que le groupe des unites est de rang petit. La table I, construite d'apres des calculs faits dans [Mr 2 ] , donne, pour tiers
n compris entre
imaginaires
22 en-
2 et 80 , des exemples de corps totalement
K de degre n pour lesquels
|d
|
n'excede que
de 3 % au plus la minoration obtenue sous GRH par Odlyzko ; tous les corps sont des corps de classes de rayon sur un corps de degre 2 , 3 ou 4 possedant au plus une unite fondamentale ; ils sont tous de nombre de classes
1.
3. 3. - Corps euclidiens. - L'etude des corps de nombres euclidiens (pour la norme) a ete completement renouvelee par Lenstra. Introduisons, pour tout corps K , la constante L(K) , borne inferieure des normes des ideaux premiers de K , et la "constante de Lenstra" M(K) , borne superieure des entiers £ existe une suite
CAK , . . . , au
d'elements de K dont les differences
sont des unites (on a clairement
tels qu'il
pour toute suite du type
; en outre, on peut toujours
uu = 1 , d'ou les inegalites
2 <M(K)^ L(K)S 2* 2*1)1) .
Dans [Ln] , Lenstra determine des constantes a si K est un corps de nombres possedant
telles que, r, s r places reelles et s
places complexes, l'inegalite M(K) >cc
vldr^l r, s
.Lv
entraine que K
169 est euclidien. En cherchant parmi les corps de petits discriminants, on a done de bonnes chances de decouvrir des corps euclidiens. II se trouve que, reciproquement, on trouve des corps de petits discriminants en cherchant des corps euclidiens par la methode de Lenstra : e'est en effet un fait d'experience que les corps pour lesquels
l
es
t tres proche des minorations que l'on obtient a
l'aide des formules explicites ont une grande constante M(K) . Par exemple, pour le corps K de la table 2 , on a M = 9 , 4, 3 alors que l'inegalite triviale M > 2
suffit a prouver que ce corps
est euclidien. Un autre exemple est le corps K
pour lequel on
sait que M vaut 10 ou 11 alors que l'inegalite facile
M>3
suffit a prouver qu'il est euclidien. On peut noter a ce propos que Lenstra etablit dans [Ln] une minoration de M(K) L(K) . Or, il se trouve qu'un corps pour lequel
en fonction de
j c3__ |
est proche
des minorations obtenues par les formules explicites a necessairement une grande constante
L(K) : cela se voit immediatement en
examinant le deuxieme terme du second membre des formules explicites. II y a peut etre dans cette remarque une explication au lien qui semble exister entre les constantes M et les discriminants. 3. 4. - Petits discriminants jusqu'au degre 8". - On trouve dans la table II pour chaque couple (n , r + s)
avec 2 ^ n ^ 8
n-eme du discriminant d'un corps, note K n , r+s
la racine
; ces discrimi-
nants sont en valeur absolue les plus petits que je connaisse dans leur categoric Voici comment ont ete choisis les
23 corps de. la
table II : a) pour
13 d'entre eux, il a ete prouve qu'ils realisent le
minimum en valeur absolue des discriminants des corps correspondant a ce couple (n , r+s) ; ce sont les corps avec n < 5 , n = 6 et r + s = 3 ou 6 , et enfin
n = r+s = 7 . Le resultat etait con-
170 nus de Gauss pour les degre*s 2 et 3 en termes de minima de formes quadratiques ou cubiques.
Les resultats pour n = 4 sont dus a
Mayer ([My], 1929) et ceux pour n=5 a Hunter ([H], 1957).
Les
trois autres re*sultats sont beaucoup plus recents, et ont ete obtenus par Kaur (CKa],1970) pour n= r+s = 6 , ([L-Z],1977) pour
pour
par Liang et Zassenhaus
n= 6 , r + s = 3 et par Pohst ([Po], a paraitre)
n = r+s = 7 .
b) Les autres corps de la table de degre totalement imaginaire de degre* 8
(6 corps)
6 ou 7 et le corps ont e*te choisis a
cause de la valeur de leur constante M . Pour prouver que les corps K si
sont euclidiens il suffit de prouver l'inegalite
(p,q)= (6,4) , (7,4) ou (8,4)
et l'inegalite
M^6
M>5
si
(p, q) = (6, 5) . Des recherches tres e*tendues de corps verifiant l'une des ine*galites M> 5 ou M> 6 ayant et^ faites, par Lenstra en particulier, on peut conjecturer, compte tenu des remarques du paragraphe precedent, que ces quatre corps fournissent les plus petits discriminants parmi ceux correspondant a l'un des couples (6,4), (6,5), (7,4)
et (8,4) ; les deux corps de degre
6 sont du
reste bien connus des specialistes. On ne peut pas etre aussi affirmatif en ce qui concerne les deux autres corps, faute d1 experimentation suffisante. K
Le corps
a ete* trouve lors d'une recherche de corps verifiant
M>7
7, 5 en conside"rant les polynSmes unitaires f(0), f(l), f ( - l ) , f(i), f(-j) l'in^galit^
M>7
f pour lesquels
2
(i +l =0 , j 2 + j + l = 0 )
sont des unites,
se voyant alors sur la suite 0 , 1 , x ,
( x - l ) / x , x+1 , x +1 ou x designe une racine de f . ( [ L t ] , cf [Mr 3]) a montre* que l ! on avait l'inegalite
Leutbecher M>7
les corps de*finis par une racine d'un polyn6me unitaire f(0), f(l), f ( - l ) , f(2), f(0)
2
(e -0-1=0)
l/(l-x),
pour
f tel que
sont des unites, a l'aide
171 de la suite 0 , 1 , x, x+1 , x , x / ( x - l ) , l/(2-x) . C'est cette suite que nous avons utilisee pour trouver le corps K determine* les polyndmes verifiant
,
: nous avons
f(-l) = f(2) = 1 , f(0) = 1 ,
f(l) = f(0)= -1, ce qui assure l'existence dfau moins 5 racines reelles pour f ; on trouve une famille de polynQmes dependant d'un parametre, que l'on choisit pour que les coefficients de f ne soient pas trop grands.
Le premier essai a conduit au corps K , . 7, o
c) Enfin, les quatre corps non totalement imaginaires de degre* 8 ont e*te cherche*s comme extensions quadratiques de corps de degre 4 au moyen de la theorie du corps de classes K
; le corps
m'a e*te signale il y a longtemps par Lenstra.
o, o
Tous les corps de la table II sont de nombre de classes on sait que dix-huit d'entre eux sont euclidiens,
1 ;
et cela peut se de*-
montrer a l'aide de minorations de M ; nous en dressons la liste ci-dessous en indiquant l'ine*galite* a. demontrer pour M en fonction du couple (n , r+s) : (2,1) ,(2,2) ,(3, 2) , (3, 3) , (4,2) , (4,3) . M>2
M>3
(4,4) ,(5,3) ,(6,3)
M>4
(5,4)
M>5
(5,5) ,(6,4) ,(7,4) , (8,4)
M>6
(6,5)
M>8
(7,5) (6,6) ,(8,5) .
A cette liste, on pourrait adjoindre le corps totament imaginaire de degre 10 de la table I ; il est euclidien, l'inegalite M>1 5 ayant et£ demontree par Mestre ([Me]) en utilisant la multiplication complexe
(M > 10 suffit) .
172 4. - Tables Nous expliquons dans ce paragraphe l'usage des tables . Table I. - La table I decrit des corps totalement imaginaires K . La premiere colonne contient le degre* du corps, obtenu comme corps de classes de rayon sur un corps k dont le degre (2 , 3 ou 4) est donne dans la colonne 2 et le discriminant dans la colonne 3 ; la colonne 4 contient le conducteur de K sur k , la notation ^ P (resp. p ) designant un ideal premier convenable de k au-dessus du nombre premier
p , de degre 1
et 6 contiennent la valeur de
(resp. 2 ) ; les colonnes 5
|d j
, arrondie aux trois decimales
les plus proches dans la colonne 6 ; la colonne 7 contient la minoration Odl
pour le degre n obtenue sous GRH par Odlyzko
(cf. table 3) ; enfin, la colonne 8 contient l'expression /Odl ) - 1 , exprimee en pourcentage et arrondie aux deux decimales les plus proches. Table II. - A 1'intersection de la colonne n et de la ligne r + s figurent trois nombres, relatifs a un corps K de degre n / n,r+s et de signature (r, s) : |d | / , arrondi aux trois decimales les plus proches, Odl
minoration de Odlyzko sous
GRH pour le degre n et la signature sion (|d^|
/Odl
)-l
(r, s) , et enfin
l'expres-
exprimee en pourcentage, le resultat
xv n, r T s etant arrondi aux deux decimales les plus proches. Les corps K , sont decrits ci-dessous au moyen de l'un des procedes 3 n, r+s ^ suivants : a) Si K
est abelien sur Q , on le definit par un gene-
rateur mis sous forme trigonometrique. b) Si K
n'est pas abelien sur C mais est abelien sur
un sous-corps k , on le definit par k et son conducteur, la notation $
designant un ideal p r e m i e r convenable de degre
1 de k
173 au-dessus du nombre premier
p .
c) Lorsque les procedes ci-dessus ne sont pas applicables, on definit K par le polynSme minimal de l'un de ses elements n, r+s r ^ J K K K K
2,1 2,2 3,2 3,3
K 4,2 K 4,3 K 4,4 K 5,3 K 5,4 K 5, 5 K 6,3 K 6,4 K 6,5 K 6,6 K 7,4 K 7,5 K 7,6 K 7,7 K 8,4 K 8,5 K 8,6 K 8,7 K 8,8
d=
-
3 , premier
d:
+
5 , premier
d=
- 2 3 , premier
d=
+ 49 =
72
d=
+ 117 =
2
d=
-
d=
+ 725 =
3 .13
275 = + 52.29
d=
+
1 609, premier
d=
- 4 5 1 1 , premier
d
+ 14 641 =
d
-
9 747 =
-33.192 23 2 .53
Q(2cos 2n/5) X3-X-l ;K. . 3,^ Q(2cos 2n/7) conducteur
i3
SUr
K
SUr
K
2,l
conducteur il 2,2 conducteur $^^ sur K 2 J 2 5 3 2 29 X*- X\X 5
3
+X-1 2
X -2X +X -1 Q ( 2 c o s 2TT/11) conducteur $., _ s u r K^ . 19 2, 1 conducteur $r _ sur K. ,2 53 X6+X5-2X4-3X3-X2+2X+1
d
+
28 037 =
d
-
92779, premier
d
+ 300 125 =
53.74
C(2cos 2n/5 + 2cos 2n/7)
d
- 184 60 7, premier
x 7 -x 6 -x 5 +x 3 +x 2 -x-i
d
- 612233 =
X7-2X6+3X5-2X4-2X3+2X2-3X+1
d = -
71. 8623
2 306 599 =-107. 21 557
X ? -3 X 6 -X 5 +9X 4 -3X 3 -7 X2f2X+l
d = + 20 134 393 = 71. 283 583 d :+ 1 257 728 = 2 8 . 1 7 3
conducteur
d = -
conducteur
4461875 =-5 4 .11 2 .59
d :+ 15243125 = 5 4 . 2 9 3 d = - 68856875 =- 5 4 29 2 131 d : + 282300 416 =2 1 2 4 1 3
sur Q(i)
conducteur
^59 8 U r P 2 9 sur
conducteur
P
conducteur
131 S U r P 4 1 sur
K
4, 3 K22
K
4,4 Q(/2)
174 Table HI. - Cette table est extraite de tables multigraphiees dues a Odlyzko, datees du 29 novembre 1976 ([Odl]) . Elle donne, sous l'hypothese de Riemann ge*neralisee, une minoration de la valeur absolue du discriminant gre* n et de la signature
d d'un corps
K en fonction du de-
(r, s) .
La premiere partie de la table concerne les corps totalement re*els ou totalement imaginaires ; une minoration de |dj
s'ob-
tient par lecture directe en fonction de n ; l'usage du parametre b figurant dans les colonnes 2 et 4 est explique
ci-dessous.
La seconde partie de la table concerne des corps de signature (r, s)
arbitraire ; des fonctions
A, B et E du parametre
b
donne* dans la premiere colonne sont calculees, et l'on a, quel que it r 2s -E soit le choix de b , | d | > A B e . II faut optimiser b ; la valeur a chercher est comprise entre les deux valeurs donnees dans la premiere partie de la table qui concerne les cas extremes r =0
et r = n . Si l'on tient compte du deuxieme terme du second membre des
formules explicites, on obtient des mino rations de B
2S
f
e "
E
|d|
de la forme
Z (log(Np) /Mpf ^) F(log N(p) m ) p ,m (cf. § . 1) , pour un choix convenable de la fonction F . Le choix a |d|>A
r
,
1
ou f=2
faire est le suivant : on prend F(x) = G(x/b) , ou G est la fonction paire, nulle en dehors de l'intervalle 0^ x ^ 2,
[-2, +2] , et de'finie, pour
par la formule G(x) = (l-x/2) cos(rr/2)x+(l/n)sin(n/2)x .
Odlyzko, en serrant de plus pres les minorations, peut ame*liorer quelque peu les minorations de la table III. II m'a signal^ les trois exemples suivants concernant des corps totalement imaginaires : on peut remplacer 5,734 par
5,743
en degr^
1,721
par
1,725
8 et 15,225 par
en degre* 2 , 15,238 en degre 4 8 .
175 Les pourcentages indique*s dans la table I seraient alors 0 , 4 1 % au lieu de 0,64% gre
8 et 1,54%
en degre
2,
0,76%
au lieu de 1,62%
au lieu de 0,92%
ende-
en degre* 48.
Table IV. - Nous donnons des extraits des tables calcule*es par Diaz y Diaz ([Dy D]) , donnant, pour un corps K de degre* n , une minoration de
|d |
pendante de GRH
en fonction de la signature du corps, inde*(minorations inconditionnelles).
Nous donnons les re*sultats jusqu'au degre* 10 pour toutes les signatures possibles, et de larges extraits des tables concernant les corps totalement reels ou totalement imaginaires. Nous n'avons r e produit que quatre des huit de*cimales calculees par Diaz y Diaz.
176
TABLE I n
K//n
l^ K|l/n
Odl
%
1, 732
1, 721
0, 64
3, 289
3, 263
0, 79
31/2.19 1 / 3
4, 622
4, 592
0, 65
2.17 3 / 8
5, 787
5, 734
0,92
J'\ 312/5
6, 841
6, 726
1, 70
7, 687
7, 598
1,17
f
m
31/2
2
2
-3
4
2
-3
6
2
-3
8
2
-22
10
2
-3
12
4
3 2 13
14
2
-71
16
4
3 2 13
18
3
-23
20
4
22
2
-7
24
4
3 2 13
32
2
-15
36
4
229
40
4
3 2 17 3 (1)
31
44
4
3 3 .7
48
4
5 2 19 2 >2»2
52
4
56
4
257
60
4
3 2 37
72
4
28
80
4
257
28
(1) *13 *19 »17 »31
1
^ . l ^ . U S
(1)
7ll/2
8,426
8, 371
0, 66
3l/ 2 .13l/ 4 .241 3 / 1 6
9, 198
9,068
1,43
72/9.l95/l823l/3
9,929
9,697
2,40
10,438
10, 270
1, 64
11,003
10,797
1,91
3l/2.13l/ 4 .397 5 / 2 4
11,441
11,283
1,40
21/2. 3 3 / 4 . 5 7 / 8
13, 181
12,912
2,08
229l/ 4 . 3072/9
13, 889
13, 581
2, 26
14, 501
14,183
2, 24
3 3 / 4 .7l / 4 .463 5 /22
14,960
14,728
1, 57
22/3 J/Z
15,472
15, 225
1,62
53/4.9113/]3
16,114
15, 680
2, 77
21/4.2lll3/56.257l/4
16,493
16,097
2,46
3l/2.37l/ 4 .19 7 / 1 5
16, 880
16,482
2,41
22.577 17 / 72
17,948
17,497
2, 57
25 7 1/ 4 .641 1 9 / 8 0
18,583
18,073
2,82
*241
Vl9 »11
/
6
*163
2 2 .n^ 5
*23
*397
^307
*463
53
*2*211
/2
.17 3 / 4
19 !/2
"19
*577 »641
177 TABLE II
\ . n r+sX 1
2
3
4
5
6
7
8
2
3
1,732 1,721 0 , 64 % 2,844 2, 236 2, 225 2, 820 0, 50 % 0 , 8 5 % 3, 659 3,639 0, 56%
4
3,289 3, 263 0, 79 % 4,072 4,036 0,90 % 5,189 5,124 1,27%
5
4, 378 4, 345 0, 77% 5,381 5, 322 1,11% 6,80 9 6, 640 2,55%
6
4,622 4,592 0,65% 5,512 5,484 0,51% 6,728 6,638 1,36% 8,182 8,143 0,48%
7
8
5,654 5,619 0,61% 6,710 6,653 0,85% 8,110 7,960 1,88% 11,051 9,611 14,99%
5, 787 5,734 0,92% 6,779 6,675 1,56% 7,905 7,834 0,90 % 9,544 9,266 3,00% 11,385 11,036 3,16%
178
TABLE III - Minorations sous GRH Premiere Partie corps totalemenit imaginaires
corps totalement reels n
b 1 2 3 4 5
6 7 8
9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 24 25
26 28 30 32 34 36 38 40 42 44 46 48 50
0. 340 0. 700 1.0 50 1. 350 1. 550 1. 750 1.900 2.0 50 2. 200 2. 300 2.400 2. 500 2. 550 2. 650 2. 700 2. 800 2.850 2.900 2. 950 3.000 3. 100 3.100 3. 100 3. 200 3. 200 3. 300 3. 300 3.400 3.500 3. 500 3. 600 3. 700 3. 700 3.800 3. 800 3. 800 3.900 3.900
idi 1 /" 0.996
2.225 3.630 5.124 6.640 8.143 9.611 11.036 12.410 13. 736 15.012 16.238 17.422 18.559 19.657 20.711 21.734 22. 720 23.672 24.594 25.474 26.351 27.178 28.001 28.787 29.554 31.0 20 32.425 33.750 35.005 36.219 37.356 38.471 39.514 40.542 41.504 42.456 43.356
b
!dl 1 / n
0. 300 0.580 0.800 1.0 50 1. 200 1. 350 1.500 1.600 1.700 1.800 1.900 2.000 2.050 2.100 2. 200 2. 250 2. 300 2.400 2.450 2.500 2. 550 2.550 2. 600 2.650 2. 700 2.750 2.800 2.850 2.950 3.000 3.000 3.100 3.100 3. 200 3.200 3. 300 3. 300 3.400
0. 874 1.721 2.519 3. 263 3.954 4.592 5. 185 5. 734 6. 247 6.726 7. 176 7.598 7.997 8. 371 8. 730 9.068 9. 390 9.697 9.990 10. 270 10.539 10.797 11.045 11.283 11. 512 11.733 12. 153 12. 545 12.912 13. 258 13. 581 13.894 14.183 14.465 14.728 14.984 15. 225 15.456
179 TABLE III
-
Minorations sous GRH
Premiere Partie
corps totalemenit imaginaires
corps totalement re*els n 52 56 60 64 68 72 76 80 84 88 92
96 100 110 120 130 140 150 160 170 180 190 200 220 240 260 280 300 320 340 360 380 400 480 600 720 840 960
b 4.000 4.000 4.100 4. 200 4.200 4. 300 4.400 4.400 4. 500 4. 500 4. 500 4. 600 4. 600 4. 700 4.800 4.900 5.000 5.000 5. 100 5. 200 5. 200 5. 300 5. 300 5.400 5. 500 5. 600 5. 700 5. 700 5.800 5.900 5.900 6.000 6.000 6. 200 6.400 6. 600 6. 800 6.900
(suite)
idi1/* 44.230 45.884 47.452 48.913 50.285 51.601 52.822 54.014 55.119 56.204 57.214 58.20 5 59.141 61.335 63. 335 65.169 66.853 68.426 69.897 71.255 72.553 73.760 74.90 9 77.026 78.943 80.689 82.283 83.775 85.155 86.424 87.642 88.760 89.833 93.555 97.979 101.488 104. 361 106.815
b 3.400 3.500 3.500 3.600 3.700 3.700 3.800 3.800 3.900 3.900 4.000 4.000 4.000 4.100 4. 200 4. 300 4.400 4.400 4.500 4.600 4.600 4.700 4. 700 4.800 4.900 5.000 5.100 5.100 5.200 5.200 5. 300 5.400 5.400 5.600 5.800 6.000 6.100 6. 300
ld| l/n 15.680 16.097 16.482 16.846 17.180 17.497 17.793 18.073 18. 338 18.589 18.826 19.055 19. 268 19. 770 20 . 221 20. 631 21.00 3 21.345 21.666 21.959 22. 236 22.493 22. 735 23.178 23. 575 23.934 24. 258 24. 560 24.838 25.091 25.332 25.552 25.763 26.485 27.328 27.984 28. 515 28.961
180 TABLE III - Minorations sous GRH P r e m i e r e Partie corps totalement re*els n 1 000 1 200 1 332 2 400 4 800 4 840 8 862 10 000 31 970 100 000 254 228 1 000 000 2 391 978 10 000 000
b 6.900 7. 200 7. 200 7. 800 8.400 8. 600 9. 200 9. 200 10.400 11. 600 12. 500 14.000 14. 500 16.000
(suite)
corps totalement imaginaires
1
Mi /" 10 7. 548 110.728 112. 575 122. 112 132.020 132. 126 139. 766 141.218 153. 252 162. 651 168.971 176. 415 180. 319 185. 655
b 6. 300 6.500 6.600 7. 200 7.800 7.800 8.400 8.600 9.800 10.800 11.800 13.000 14.000 15.000
idi1/11 29.094 29.673 29.992 31.645 33.298 33. 315 34. 541 34. 768 36. 613 37.994 38.895 39.923 40.458 41.122
181 Table III - Minorations sous GRH Deuxieme Partie
b 0. 300 0. 320 0. 340 0. 360 0. 380 0.400 0.420 0. 580 0. 600 0. 625 0. 650 0. 675 0. 700 0. 750 0. 800 0.850 0.900 0.950 1.000 1.0 50 1. 100 1.150 1. 200 1. 250 1. 300 1. 350 1.400 1.450 1. 500 1. 550 1. 600 1. 650 1. 700 1. 750 1. 800 1.850 1.900 1.950
A 2. 623 2.820 3.020 3. 222 3.427 3.636 3.847 5.640 5.878 6.179 6.485 6.795 7.110 7. 753 8.416 9.096 9.795 10. 513 11.248 12.001 12. 772 13.561 14.366 15.189 16.028 16.883 17. 754 18.641 19.543 20.459 21. 390 22.335 23.293 24.264 25.248 26.244 27.251 28.269
B 2. 324 2.478 2.633 2. 787 2.941 3.095 3. 249 4.475 4.628 4.818 5.008 5.198 5. 387 5.765 6.142 6.517 6.891 7. 262 7. 633 8.001 8. 368 8.732 9.095 9.455 9.814 10 . 170 10.524 10.876 11.225 11.572 11.916 12.258 12.598 12.935 13.269 13.600 13.929 14.256
E 0.9769 1.0426 1.10 85 1.1745 1. 240 6 1. 30 68 1.3732 1.9107 1.9788 2.0643 2.1501 2.2363 2.3229 2.4973 2.6735 2.8517 3.0319 3. 2143 3.3991 3.5863 3.7761 3.9686 4.1641 4.3626 4.5643 4.7693 4.9779 5.1902 5.40 62 5.6264 5.850 7 6.0793 6.3126 6.550 5 6.7935 7.0415 7.2949 7.5539
TABLE III - Minorations sous GRH D euxieme Partie
(suite)
b
A
B
2.000 2.0 50 2. 100 2. 200 2. 250 2. 300 2. 350 2.400 2.450 2. 500 2. 550 2. 600 2. 650 2. 700 2. 750 2. 800 2. 850 2.900 2.950 3.000 3. 100 3. 200 3. 300 3.400 3. 500 3. 600 3. 700 3.800 3.900 4.000 4.100 4.200 4. 300 4.400 4. 500 4. 600 4. 700 4. 800 4.900
29. 298 30. 338 31.386 33.511 34.585 35.667 36.757 37.853 38.955 40.063 41.176 42.295 43.417 44.543 45.673 46.80 6 47.941 49.079 50. 218 51.359 53.643 55.928 58.211 60.490 62.762 65.024 67.275 69.513 71.735 73.940 76.126 78.292 80.436 82.558 84.656 86.730 88.778
14.579 14.900 15.218 15.845 16.154 16.461 16. 764 17.065 17.363 17.658 17.950 18.239 18.525 18.808 19.089 19.366 19.641 19.912 20.181 20.446 20.969 21.480 21.980 22.469 22.946 23.413 23.868 24.313 24.747 25.171 25.585 25.988 26.382
90.799 92.794
26.766 27.141 27. 50 7 27.863 28.211 28.550
E
7.8187 8.0894 8.3664 8.9400 9.2371 9.5414 9.8532 10.173 10.501 10.837 11.181 11.535 11.898 12.270 12. 653 13.045 13.448 13.863 14.289 14.726 15.638 16. 603 17.624 18.706 19.851 21.066 22.355 23.723 25.176 26.719 28.360 30 . 10 5 31.962 33.939 36.044 38.286 40.676 43.224 45.941
183 TABLE III - Minorations sous GRH Deuxieme Partie
b 5.000 5. 100 5. 200 5. 300 5.400 5. 500 5. 600 5. 700 5.800 5.900 6.000 6. 100 6. 200 6. 300 6.400 6. 500 6. 600 6. 700 6. 800 6.900 7.000 7. 200 7.400 7. 600 7.800 8.000 8. 200 8.400 8. 600 8. 800 9.000 9. 200 9.400 9. 600 9. 800 10.000 10. 200 10.400 10. 600
A 94.761 96. 701 98. 612 100.495 10 2.349 104. 174 10 5. 970 10 7. 737 109.475 111. 184 112.863 114.514 116.137 117.731 119.296 120. 834 122. 344 123. 826 125.282 126. 710 128. 112 130.839 133.464 135. 991 138.423 140.764 143.015 145. 182 147.266 149.272 151. 201 153.058 154.845 156. 565 158. 220 159. 814 161.348 162.826 164. 250
(suite)
B 28.881 29. 20 3 29.518 29.825 30.123 30.415 30 . 699 30.976 31.247 31.510 31.767 32.018 32.262 32.501 32.733 32.960 33.181 33.397 33.608 33.813 34.013 34.400 34.768 35.119 35.453 35.772 36.076 36.366 36.643 36.908 37.161 37.40 3 37.634 37.855 38.067 38.270 38.464 38.650 38.828
E 48.840 51.934 55.237 58.764 62.532 66.559 70.863 75.465 80. 387 85.652 91.287 97.319 103.78 110 . 70 118.11 126.05 134.56 143.68 153.47 163.96 175.22 200 . 26 229.13 262.43 300 . 88 345.31 396.69 456.15 525.04 604.89 697.52 80 5.0 6 929.98 1.0753 1.2442 1. 440 9 1.6700 1.9371 2.2485
e e e e e e
03 03 03 03 03 03
184
TABLE III - Minorations sous GRH Deuxieme Partie (suite)
b
10.800 11.000 11. 200 11.400 11.600 11.800 12.000 12.500 13.000 13.500 14.000 14.500 15.000 16.000 17.000 18.000 19.000 20.000 22.000 24.000 26.000 28.000 30.000 32. 500 35.000 37.500 40.000 42. 500 45.000 47.500 50.000 52. 500 55.000 57.500 60.000 65.000 70.000
A
165. 622 166. 944 168. 219 169.447 170. 633 1.71. 776 172. 879 175.472 177. 848 180.0 31 182.037 183.886 185.592 188.628 191.237 193.493 195.455 197. 170 200.00 9 20 2. 240 20 4.0 24 20 5.471 20 6. 660 20 7.868 20 8.842 20 9. 639 210. 298 210.849 211.315 211.712 212.053 212. 348 212. 60 5 212. 830 213.029 213. 361 213.626
B
E
39.000 39.164 39.322 39.473 39.619 39.759 39.893 40 . 20 7 40 . 494 40.754 40.993 41.211 41.412 41.766 42.069 42.328 42.553 42.748 43.069 43.320 43. 520 43.681 43.813 43.946 44.054 44.141 44.214 44.274 44.325 44.369 44.40 6 44.438 44.466 44.491 44.512 44.549 44.577
2.6120 3.0365 3.5324 4.1122 4.790 3 5. 5840 6.5134 9. 5972 1.4194 2.1065 3.1366 4.6851 7.0186 1. 5882 3. 6298 8.3717 1.9467 4.560 6 2. 5532 1.4625 8.5416 5.0734 3.0577 2.9393 2.8743 2.8528 2.8685 2.9175 2.9977 3.1082 3.2493 3.4220 3.6281 3.870 2 4. 1518 4.850 3 5.7572
e 03 e 03 eO3 e03 e 03 e 03 e03 e 03 e 04 e 04 e 04 e 04 e 04 e 05 e 05 e 05 e 06 e 06 e 07 e 08 e 08 e 09 e 10 e 11 e 12 e 13 e 14 e 15 e 16 e 17 e 18 e 19 e 20 e 21 e 22 e 24 e 26
185 TABLE IV - Minorations inconditionnelles Premiere Partie
: nS 10
Dans chaque ligne, on lit, pour un degre* n donne, la suite des minorations de |cL-l [(n+l)/2]
pour r+s croissant de
a n ; ces minorations tiennent compte de la correction
la plus de*favorable que donne le second membre des formules explicites en fonction de la decomposition dans K des petits nombres premiers
; pour n = l , le re*sultat avec ses 8 decimales est
0,99 999 524. n= 2 :
1.7297 ;
2.2280
n = 3 : 2.8185 ;
3. 6128
n = 4 : 3.2584 ;
4.0143 ;
5.0674
n = 5 : 4.31.75 ;
5. 2638 ;
6.5240
n = 6 : 4.5577 ;
5.4194 ;
6.5239
;
7.9424
n=7 :
5.5485 ;
6.5355 ;
7. 7664
;
9.30 27
n=8 :
5.6593 ;
6. 5540 ;
7.6448
;
8.9749 ;
10 .5972
n=9 :
6.5763 ;
7. 5583 ;
8.7337
; 10. 1404 ;
11 .8242
n = 10 :
6.6003 ;
7.4952 ;
8. 550 2 ;
12.9853 .
9.7934 ;
11 .2585
;
186
TABLE IV - Minorations inconditionnelles Deuxieme Partie Corps totalement re*els
N 1 2 3 4 5
6 7 8
9 10 11 12 13 14
15 16 17 18
19 20 21 22 23 24 25
26 28 30 32 34
36
Minoration 0.9979 2. 2234 3. 6108 5.0 670 6.5235 7.9414 9. 3017 10.5964 11.8238 12.9850 14.0831 15.1217 16. 1047 17.0359 17.9192 18. 7580 19.5555 20. 3148 21.0386 21.7294 22.3896 23.0212 23.6261 24.20 61 24.7628 25.2976 26. 30 71 27.2440 28.1165 28.9315 29.6948
N
Minoration
N
38 40 42 44
30.4117 31.0865 31.7232 32. 3252 32. 8954 33.4365 33. 950 8 34.440 5 35. 3532 36. 1874 36.9536 37. 660 5 38. 3151 38.9235 39. 490 8 40.0 213 40.5188 40.9865 41.4272 41. 8434 42. 7899 43. 6232 44.3638 45.0273 45. 6260
240 260 280 300 320 340 360 380 400 450 480 500 600 700 720 800 840
46 48 50 52 56 60 64 68 72 76 80 84 88 92
96 100 110 120 130 140 150 160 170 180
190 200 220
46.1696 46. 6658 47.1211 47. 540 6 47.9287 48.6246
900 960 1 000 1 100 1 200 1 300 1 332 1 400 1 500 1 600 1 700 1 800 1 900 2 000
Minoration 49.2319 49.7675 50. 2442 50. 6717 51.0579 51.4087 51.7292 52.0233 52.2945 52.8888 53.1982 53. 3882 54. 1850 54.7962 54.9021 55. 2829 55.4513 55. 6813 55. 8880 56.0147 56. 2986 56.5438 56. 7581 56. 8212 56.9474 57. 1161 57.2674 57.4042 57. 5285 57. 6421 57.7464
187 TABLE IV - Minorations inconditionnelles Deuxieme Partie Corps totalement imaginaires
Minoration
N 2 4
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52
56 60 64 68
1.7221 3.2545 4. 5570 5.6593 6. 600 3 7.4128 8. 1224 8.7484 9.3056 9-80 57 10.2575 10. 6683 11.0 438 11.3889 11. 70 72 12.0022 12. 2764 12. 5322 12. 7715 12.9960 13. 20 71 13.40 61 13.5941 1.3. 7721 13.9409 14. 1013 14.3993 14.6707 14.9193 15.1479
N 72 76 80 84 88 92
96 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 320 340
Minoration 15. 3591 15. 5549 15. 7371 15. 9071 16.0663 16. 2158 16. 3563 16.4889 16. 7898 17.0539 17. 2880 17.4974 17. 6859 17.8568 18.0126 18.1553 18. 2867 18.4081 18. 520 7 18.6254 18.7232 18. 8148 18.900 7 18.9815 19.0577 19. 1297 19.1979 19.2625 19.3823 19.4911
N 360 380 400 480 500 600 700 720 800 840 900 960 1 000 1 100 1 200 1 300 1 332 1 400 1 500 1 600 1 700 1 800 1 900 2 000 2 200 2 400 2 600 2 800 3 000 4 000
Minoration 19. 590 3 19.6813 19.7652 20.0443 20. 10 29 20. 3483 20. 5363 20. 5688 20.6858 20. 7375 20. 80 81 20. 8715 20 . 910 3 20.9973 21.0 724 21.1380 21.1573 21.1959 21.2475 21.2937 21.3355 21.3735 21.4082 21.4401 21.4966 21.5453 21.5877 21.6252 21.6585 21.7825
188 APPENDICE I Diverses notions de discriminant
Classiquement, on de*finit, pour toute extension K/k de n o m b r e s , un discriminant
6
/
de corps
qui est un ide*al entier de k .
Lorsque k = Q , cela ne definit le discriminant habituel qu'au signe p r e s . En considerant des bases de K sur k , on definit naturellement un discriminant k = Q,
d^ /
€ S t u n
element de k /k*
la connaissance du discriminant usuel &
naissance de
6
R / Q
.
Lorsque
e*quivaut a la con-
et d K / Q .
Le discriminant
d__ /,
partage avec le discriminant usuel
lorsque k = Q les deux proprietes suivantes : (i)
# #2 Ce discriminant met en bijection les elements de k /k
avec les extensions quadratiques de k , a condition de conside*rer k lui-mSme comme une extension quadratique de k . (ii)
II permet d'associer a. toute extension K/k
quadratique canonique K'
de k caracte'rise'e par l'egalite*
Dans cette situation, soit x£K de Galois
une extension
tel que k(x) = K . Le groupe
G de la clSture galoisienne de K/k
g^brique donnee k de k opere sur l'ensemble
dans une clSture a l R des racines du
polyndme minimal de x sur k , permettant d'identifier
G a un
sous-groupe du groupe syme'trique S(R) . Alors, l'extension
K'/k
correspond a. l'intersection de G et du sous-groupe alterne* de S(R) si
; en particulier,
d
,
est trivial dans k /k
si et seulement
G se plonge dans le sous-groupe alterne de S(R) . Soit p
un ide*al p r e m i e r non nul de k . La clSture integrale
dans K de l'anneau local de k en p
est un (0 ) -module libre,
189 auquel on peut associer un discriminant dans (0, ) de*finit le discriminant
dL./, • En rempla^ant
(0 )
\(&. )
> cjui
par son comple-
te* en p , et en introduisant les places a l'infini, on obtient le discriminant ide*lique introduit par Frohlich (Discriminants of algebraic number fields, Math. Z. 74 (I960), 18-28 ; Ideals in an extension field as modules over the algebraic integers in a finite number field, Math. Z. 74 (I960), 29-38) ; c'estun element de J^/U^
(ideles
modulo les Carre's des ideles unite's) , qui appartient plus pre'cise'ment a
K* J__ /U__ . xv K
#2 La consideration des discriminants de*finis dans (&.) \ (0, ) permet de verifier pour & /, une ^galit^ de la forme 2 K/k 6 / =6 , o , ou 0 est un ide*al entier de k , ce qui entraine un re*sultat analogue pour le discriminant idelique.
APPENDICE II La congruence de Stickelberger
La congruence classique est le fait que, pour tout corps de nombre K , on a d = 0
ou 1 mod. 4 ; une de*montration tres
simple de ce theoreme de Stickelberger a 6t6 donnee par Schur (Elementarer Beweis eines Satzes von L. Stickelberger, Math. Z 29 (1929), 464-465,et oeuvres, tome 3, p. 87-88). Comme l'a montre* FrOhlich (cf. appendice I) , la demonstration de Schur permet de de*montrer un theoreme analogue pour le discriminant id^lique. Voici une generalisation du the*oreme de Stickelberger aux discriminants de nombres
M
ide*aux" 6 ; soit
r
,
: soit K/k une extension finie de corps
le nombre de places complexes de K au-
dessus d'une place reelle de k ; alors,
N, / ^ de*signant la norme
190 absolue, on a la congruence : N
K/D(6K/k)S°
° U (- 1 " 2
m
°
d 4
-
Je n'ai pas de re*fe*rence a. proposer pour cet e'nonce' ; on peut le de'duire de la congruence "idelique" de FrBhlich. On peut e*galement le de*montre r en le ve*rifiant pour l e s extensions quadratiques, puis en passant au cas ge'ne'ral au moyen de la formule 6 i - 6 , 0
£nonce*e dans l'appendice I.
APPENDICE III Valeurs de v
Soit k un corps de nombres (ou une extension finie d'un corps p-adique Q ) , soit n un entier et soit p un ide*al premier non nul de k . L'exposant v (6
, ) de p dans le discriminant re-
latif d'une extension K/k de degre* n ne peut prendre qu'un nombre fini de valeurs, dont on peut dresser la liste en fonction de la caracte*ristique re*siduelle p de p , du degre* n de K/k et de l'indice absolu de ramification
e
de p . Cela resulte des travaux de Ore
(Existenzbeweise flir algebraische Korper mit vorgeschriebenen Eigenschaften, Math. Z. 25 (1926), 474-489) et de Thomson (On the possible forms of discriminants of algebraic fields, Amer. J. Math. 53(1931), 81-90
et 55(1933), 111-118).
Voici le principe de la me'thode de Ore ; on commence par e*tudier le cas local totalement ramifie. Une uniform is ante II K est racine d'un polynSme d'Eisenstein f £ k [ X ] n
f(X) = X + a
n
X " + . . . + a X+n
ou a.£p
et rr
de
: est une unifor-
misante de k . La valuation dans k du discriminant de K/k
est
191 e*gale a la valuation dans K de la differente de K/k , qui est engendre*e par fl(n) = n l l n "
+(n-l)a
II*1" + . . . + a
; les differents
t e r m e s de cette somme ont des valuations dans K deux a deux incongrues modulo n , si bien que la valuation cherche'e est la valeur minimum des valuations des differents t e r m e s ci-dessus trouve ainsi une liste de valeurs possibles pour v (6
; on
, ) , qui se
pre*sentent toutes effectivement. La plus petite est n , ou n-1, que l'on obtient par exemple pour le polyndme X +TTX + TT , et la plus grande est n v (n) + n - l , que l'on obtient par exemple pour le polynSme
On passe dela au cas local ge*ne*ral en e*crivant K comme extension totalement ramifiee d'une extension de k non ramifie*e, puis on globalise en approchant par le the'oreme d'approximation un produit de polynSmes de k [X]
par un polynDme de k[X]
dont
P on peut a s s u r e r l'irre'ductibilite en lui imposant d'etre un polynSme d'Eisenstein en un ide*al p r e m i e r de k autre que p ; le the'oreme d'approximation montre e*galement que l'on peut imposer le nombre de places replies de k ramifiees dans K . Voici les resultats pour le discriminant d'un corps de nombres de degre* n (e = 1 ) . On ecrit le developpement p-adique de n : n= £ a.p i=0 1
,
avec 0 S a . < p . La valeur maximum de v (d) est 1 P 1 N n,p i = 0
ou r de*signe le nombre de coefficients
a. non nuls.
Les valeurs possibles pour v (d) sont celles de l'intervalle P [0 ,N ] a l'exclusion des suivantes : 1 n,pJ (i) (ii) (iii)
ip-1
sin =p
1
,i>l
X
i p - ! si n = p +l , i^ 2 1
pour
p =2
(cf. appendice II) .
192 BIBLIOGRAPHIE
[Dy D]
F . DIAZ Y DIAZ.- Tables minor ant la racine n-ieme du discriminant d'un corps de degre n , Publications Mathematiques d'Orsay, a paraftre.
CH]
J. HUNTER. - The minimum discriminant of quintic fields, Proc. Glasgow Math. Association 3 (1957), 57-67.
[Ka]
G. KA.UR. - The minimum discriminant of sixth degree totally real algebraic number fields, J. Indian Math. Soc. 34 (1970), 123-134.
[K-V]
H. KOCH, B . B . VENKOV.- Uber den p-KlassenkSrperturm eines imaginar-quadratischen Zahlko'rpers, Asterisque 24-25 (1975), 57-67.
[Ln]
H. W. LENSTRA. - Euclidean number fields of large degree, Invent. Math. 36 (1977), 237-254.
[Lt]
A. LEUTBECHER. - Communication privee .
[L-Z]
J. LIANG, H. ZASSENHAUSS. - The minimum Discriminant of Sixth Degree Totally Complex Algebraic Number Fields, J. Number Theory 9 (1977), 16-35.
Cv L]
F . J . van der LINDEN.- Class Numbers of real abelian number fields of small conductors, expose aux Jour nee s Arithmetiques d'Exeter.
[Mr l]
J. MARTINET.- Tours de corps de classes et estimation de discriminants, Invent. Math. 44 (1978), 65-73.
[Mr 2]
J. MARTINET.- Petits Discriminants, Ann. Inst. Fourier 29, 1 (1979), 159-17(h
[Mr 3]
J. MARTINET.- Sur la constantede Lenstra des corps de nombres, Seminaire de Theorie des Nombres, Bordeaux, 1979-80, exp. n°17.
[Ms]
J . M . MASLEY.- Where are number fields with small class number ? in Number Theory, Carbondale 1979, Lecture Notes in Mathematics, n°751, Berlin-HeidelbergNew-York; Springer, 1979.
193 [My]
J. MAYER. - Die Absolut kleinsten Discriminanten der biquadratischen Zahlkflrper, S.B. Akad. Wiss. Wien, II a, 138 (1929), 733-742.
[Me]
J.-F. MESTRE.- Corps euclidiens, unites exceptionnelles et courbes elliptiques, J. Number Theory (a paraitre).
fodl]
A. ODLYZKO.- Discriminant Bounds, tables multigr a phiees datees du 29 Novembre 1976.
[Po]
M. POHST. - The minimum discriminant of seventh degree totally real algebraic number field, J. Number Theory (a paraitre).
[Pi]
G. POITOU.- Minorations de discriminants (d'apres A.M. Odlyzko), Seminaire Bourbaki, expose 479, Fevrier 1976 ; Lecture Notes in Maths, n°567, BerlinHeidelberg-New-York ; Springer, 1977.
[P 2]
G. PQITOU.- Sur les petits discriminants, Seminaire D . P . P . , Paris, expose n°6, 1976-77.
CRo]
P . ROQUETTE.- On class field t o w e r s , i n J . W . S . Cassels and A. FrChlich, Algebraic Number Theory, p . 231-249 ; New-York, London, Academic P r e s s , 1967.
[Sch]
R. SCHOOF.- Communication p r i v e e .
[Se]
J . - P . SERRE. - Cohomologie galoisienne, 4 e m e e d . , Lecture Notes in Maths, n°5 ; Berlin - Heidelberg New-York ; Springer, 1973.
[Wl]
A. WEIL.- Sur les'formules explicites'de la theorie des nombres premiers, Comm. Sem. Math. Lund (1952), 252-265.
[W 2]
A. WEIL. - Sur les for mules explicites de la theorie des nombres, Izvestia Akad. Nauk. S.S.S.R., Ser. Math., 36 (1972), 3-18.
[Z]
R. ZIMMERT. - Ideale kleiner Norm in Idealklassen und eine Regulatorabschatzung, a paraitre.
STICKELBERGER RELATIONS IN CLASS GROUPS AND GALOIS NODULE STRUCTURE Leon R. McCulloh*
Hilbert's [H] proof of the Stickelberger relations for the cyclotomic field tensions of
Q
iQKy..)
is based on the fact that tame cyclic ex-
of degree
£
(prime) have normal integral bases.
This connects two apparently distinct aspects of Galois module structure: a) the structure of the ideal class group C (=(Z/£Z) X )
tion of the Galois group
b) the structure of rings of integers L/Q
of degree
Cl(Z[y.])
of 0.
U(y £ )/Q
under the acand
in tame cyclic extensions
£ under the action of their Galois groups
G (~Z/£Z) .
The connection is made by the resolvents: (v|x) =
£ aCv)x(a) aeG
for
v e L , x
the rings
C
and
Hom(G,yJ x
which belong to the composite fields Galois groups
€
L(y.) , are acted on by both
G , and embody the Galois module structure of
0.
Using the same bridge, Frohlich [F] was able to prove the Stickelberger relations for an arbitrary cyclotomic field
IQ) (y_p) ,
avoiding Stickelberger's explicit factorization of Gauss sums. This approach can also be used to obtain relations on class groups associated to
ZG
for certain abelian groups
G .
These re-
lations arise by applying general results on the Galois module structure of tame extensions G
to the ground field
K = QJ
L/K
with Galois group isomorphic to
where all such extensions have normal
integral bases. With an arbitrary number field
K
as ground field, the situ-
ation becomes more complicated in two ways: 1)
L/K
2)
even if it does and is tame (or even unramified), it may have no
may have no (relative) integral basis and
(relative) normal integral basis. (In fact,
K
(Example:
Q ( /-"5, /^T) /Q ( /^S) .)
has tame quadratic extensions without normal integral
bases if and only if the ray class group (mod (2)) of
K
is non-
* This research was made possible by the enlightened sabbatical policy of the University of Illinois. The author wishes to express his gratitude to Universitat Regensburg and to King's College, London for the generous hospitality extended to him while the major part of this work was being done.
195 trivial.)
To deal with these difficulties, we introduce the class
group. Let K = an algebraic number field, G = a finite abelian group, and 0 = 0.,=
the ring of integers in
Definition
The class group
K .
Cl(oG)
of the group ring
oG
Cl(oG) = I(oG)/((KG)*) where and
I(oG) ((KG) ) If
the class lows: some
is the group of invertible fractional oG-ideals in
L/K
is a tame extension with Galois group
(0 L )
of
0.
is an element of
Cl(oG)
Gal(L/K) ? G ,
defined as fol-
By the normal basis theorem of Galois theory, v e L .
Then
Definition Evidently, for some
v' e L
Notice that Gal(L/K) * G
KG
is the group of principal invertible oG-ideals.
0. = m»v (0 L )
is the image of
(0 L ) = 1 (i.e., (0, )
for some
iff L/K
m
L = KG«v
for
m e I(oG) . m
in
Cl(oG) . 0 L = OG v1
is principal iff
has a normal integral basis).
depends on the choice of the isomorphism
and is unique only up to the action of
AutG
on
Cl(oG) . Definition
The set of 'realizable classes' is defined as
R(oG) = {(0 L ) | L/K tame, Gal(L/K) = G} c Cl(oG) and is a union of
AutG
orbits in
Cl(oG) .
(Note:
, Cl(oG)
is a
Z[AutG]-module.) One easily sees, in fact, that R(oG) c Cl'(oG) = Kernel(Cl(oG)^->Cl(o) ) where oG
e »o .
is the map on class groups induced by the augmentation The reason, in essence, is that if
Tr(0, ) = o , the principal class in
Cl(o) .
L/K
is tame, then
Beyond this, for
196 arbitrary ground field
K ,
R(oG)
has been described
precisely
only for elementary abelian groups. Let
G
be elementary abelian of order
I
.
Then
AutG = G1,(F O ) . If we identify G with the additive group of the x finite field F ^ , then the multiplicative group C = C. = (F ^) acts (by multiplication) on group of
AutG .
Thus,
that there is an ideal
%
.
becomes a
R(oG)
(We write the group
ZC
ZC
module.
We can show
in the following sense:
R(oG) = Cl'(oG) J
THEOREM 1. action of
and is embedded as a (Cartan) sub-
J c ZC , analogous to the classical Stickel-
berger ideal which describes
order
G
Cl(oG)
for
G
Cl(oG)
elementary
abelian of
multiplicatively
and the
exponentially.
The ideal
J
is a relative of one introduced by Kubert and
Lang in their series on units in modular function fields. fined as follows: map
Tr: F ^
Let
» F
negative residue
»Z
It is de-
be the composite of the trace
with the canonical lifting to the least nonF_
Definition
tr: C
^ [0,£) n Z .
Then
The Stickelberger element and ideal are defined,
respectively as 0 = 0, = k
T tr(6)6~ 1 6 ZC 6eC
and
J = J, = ZC« (0/£) n ZC K Remarks 1)
If
C 1 = F* = (Z/£Z) X = Gal(Q(yJl)/Q)
k = 1 ,
and
J1
is
the classical Stickelberger ideal. 2)
If
k = 1
and
the above theorem gives that
Cl'(oG)
I = 2
then
C1 = 1 ,
R(oG) = Cl'(oG) .
ZC,, = J ^ = Z
and
One can also compute
is isomorphic to the ray class group (mod (2)) of
K
from which it follows that the vanishing of this ray class group is necessary and sufficient for all tame quadratic extensions
L/K
to
have normal integral bases. 3) Moreover,
If
k = 1
and
K = Q , then
Cl(ZG) = Cl(Z[y 0 ])
serves the
C.
action.
o = Z
so
Cl'(ZG) = C K Z G ) .
(see [R]) and the isomorphism pre-
So our theorem says
R(ZG) = Cl(Z[y.])
' .
This points up explicitly the fact, implicit in Hilbert's proof, that the Stickelberger relations for
y(y«)
are equivalent to the
normal integral basis theorem for tame cyclic extensions of prime degree
I .
Q
of
197 The theorem was obtained first [M] in the special case where k = 1
and
k = 1
was able to obtain the inclusion
assuming
y0 c K .
In 1977, L.N. Childs, [C], still in the case 7 R(oG) 2 Cl'(oG)
relations from the normal basis theorem. found in the case the hypothesis (s) group
without
v u s K , which is sufficient to recover the Stickelberger
C^
k = 2
In 1978, C. Glass [G]
an explicit description of
R(oG)
under
y 0 c K , based on the diagonal (split Cartan) sub-
of
Gl^CF.) .
This work of Glass, in preliminary ver-
sion, was instrumental in drawing my attention to the elementary abelian case . Now, just as in Remark 3) above, if we put theorem with group of
in the
ZG :
Corollary order
K = UJ
k > 1 , we obtain Stickelberger relations on the class CHZG)
= 1
for
G
elementary abelian of
£ This suggested looking for an analogue of Iwasawa's class
number formula [I].
Let
x: G
element to its inverse. symmetric part
A
THEOREM 2.
*G
, with respect to For
be the involution sending each
This allows us to consider the skew-
G
x , of a
ZC
module
elementary abelian of order
|C1(ZG)~| = [ZC~:J~]
if
I
is odd,
|C1(ZG) | = [ZG :J ]
if
I = 2 .
I
A . ,
and
The proof of this theorem consists of computing both sides and comparing the results.
Let
M
be the maximal order of ttJG .
Iwasawa's class number formula allows us to express index of the ideal of ideal
D(ZG)
Cl(jl|)
as the
generated by the classical Stickelberger
J ^ c ZC,, s ZC , that is,
ing by get
ZC
|C1(M)~| = [ZC~ : (ZC • J ^ )" ] .
the kernel of the surjection
CKZG)
Denot-
>C1 CM) > we
|C1(ZG)~| = |C1(M)~|•|D(ZG)"| . Frohlich CF 1 ] has computed
|D(ZG) I
for arbitrary abelian il-groups and Kubert and Lang [KL]
have developed techniques which can be applied to computing [CZC«J 1 )":J~] .
Details appear in [N'].
The proof of Theorem 1 follows the spirit of Hilbert's proof discussed at the beginning of this article.
One attempts to connect
two situations: a) the structure of the class group of the group
C
of automorphisms of
KG
CKO.G) over
K
under the action and
198 b) the Galois module structure of rings of integers tame extensions
L/K
with Galois group isomorphic to
0.
in
G .
The connection is made by the 'resolvends': v =
][ o{\/)a
for
v £ L ,
which are elements of the group ring characters
x
groups,
and
C
°^
are
G
LG
G , act on
action on the elements of
LG . G .
LG
X^v) = Cv | x) .
The action of
The action of
tion on the coefficients which lie in act on
and whose images under the
the resolvents:
L .
C
G
Both
is through its
is through its ac-
Of course,
G
may also
by left multiplication, but that action does not pre-
serve the ring structure.
However, both actions of
G
agree on the
resolvends, and indeed, the resolvends are characterized by that property.
From that fact, one deduces an important result on the
behaviour of resolvends under the action of the annihilator of a e ZC ,
L
c KG
G
regarded as a
if and only if
ZC
C :
a e A .
make sense in this relation, we regard
L
LG .)
A = Ann-^pG ,
Then, for
(For the exponent to as an element of the
multiplicative group of all rank one, free which are generated by units of
Let
module.
KG
modules in
LG
The annihilator ideal
A
is
also closely related to the Stickelberger ideal
J .
J = A(0/£)
be a normal basis
and that
generator of the class
A + J - ZC .
L/K , and
0. = m # v
(0, ) £ Cl(oG) .
v
- w e KG .
propriate sense,
0.
v
m e I(oG)
and represents
Raising the module of integral resolvends
to the £th power (noting that where
Now, let where
One shows that
£ e A ) we find that
0, = m «w
The crucial step is to observe that, in an apis an integral &th power free ideal of
oG
and can be represented in the form 0* - a 9 where
a
is a square-free integral ideal of
of its C-conjugates. every
Then, one shows that
a £ A , and using the fact that
(0. ) e Cl(oG)
.
v
generator at all prime divisors of , the
o'G
for
to be a normal integral basis £ , and then to replace
ideal which it generates, where
class group can be regarded as a quotient of o'G
e Cl(oG)
A + J = ZC , one obtains
(The details are actually somewhat more involved.
It is more convenient to choose m1
oG , distinct from all (0. )
m
o 1 = o[1/£] .
by The
I(o'G) , and since
is a maximal o'-order, the notions of £th power free and square
free are meaningful.) The most difficult part of the proof is to construct, for each
199 class in
Cl'(oG)
for which
(0, )
, a tame extension
L/K
is the given class.
with Galois group
This is done by constructing,
in effect, a 'Kummer extension 1 of the group algebra elements of
G
G
KG
with the
playing the role of the group of roots of unity.
This algebra is then shown to be of form abelian extension
L/K
L »,, KG
for a suitable
which turns out to be the desired extension.
From the construction, one sees that there are infinitely many such extensions, and that the discriminant may be chosen relatively to any preassigned ideal of
prime
K .
It is natural to look for generalizations of these theorems to other abelian groups. (& ,...,& ) , rank of
k .
Aut G = Gl k (Z/£ n Z)
Now let
G
be an abelian group of type
Again we consider a certain Cartan subgroup .
Specifically, if
tegers in the unique unramified extension of then
(R/£nR)+
= G
and we take
by multiplication. of the trace
> Z/£ Z
least non-negative residue
is the ring of in(QL
of degree
C = C, = (R/£nR)X K, n
As before, let
Tr: C
R
tr: C
»Z
k ,
, acting on
G
denote the composite
with the canonical lifting to the mod & n .
Then
Definition 0 = 9k J = J, K
In case
=
,n
I
tr(6)6~ 1 e ZC
= ZC(0/£ n ) n ZC
k = 1 ,
G = Z/£ n Z
and
.
and
J^
n
is again classical.
How-
ever, [ZC
i,n:
J
i , n ] = IClCZC^nin - h n
which, in general is much smaller than n
| C K Z G D " | = | DCZG) " | n h " r . a
r 1
^
To generalize the results, we need an analogue to the ring of cyclotomic integers
Z[y
n]
for
k > 1 .
Let H be the elementary abelian subgroup of G of rank k . Then the group algebra d|G decomposes as a product IIJ(G/H) x A^ n where A, is a product of copies of 5J(y n ) • In fact, the algebra A, is a Galois extension of dj under the action of the K, n Galois group C, . For increasing n , these extensions can be K, n fitted into a tower of Galois extensions producing an infinite Galois extension with Galois group
lim C. =" R ^ K, n
Denote by A. K
,n
200 the
image
of
ZG
in
A,
and
y
'n
fk1
k,n
by
( k) n
the image
of
G
.
(Then
*
^ THEOREM 3. i) CKA,K , n) k ' n = 1 . ii)
If
I
is odd,
[ZC~ K
where
Remark analogous enlarged and
E = (k-1 ) U
We can e n l a r g e
to the S t i c k e l b e r g e r i d e a l , we obtain
the class
J,
to an ideal
ideal
used
equality
number without
The proof method
( n
: J" ] = £ E |C1CA, )"| , n K,n K., n 1 )k ~ - 1)/2 .
of ii) p a r a l l e l s
for computing
between
the extra
|D(ZG)
I
S,
[S].
the index
p o w e r of
£
that of T h e o r e m extends
c ZC,
by Sinnott
of the
ideal
appearing
2.
also to
For the
in i i ) .
Frohlich's
ID C A,
) I . The , n needed is that the units of finite o r d e r in ( k) ±1 • y . The proof also shows that the m a j o r K
only A,
a d d i t i o n a l fact are p r e c i s e l y
ingredient h
of the equality
is the
nn
£ discriminant Taken A. K , n
fact that
i i ) , in addition
the o r d e r
k,n to the characters
respect
of i) comes
by a d a p t i n g
Stickelberger
relations
in c y c l o t o m i c
been p o s s i b l e
to extend
Theorem
n > 1 , and
obtain
case w h e r e
The
I
c Cl
G
in the
'' n
of o r d e r
, by m e t h o d s
case
o = Z
that D(ZG)
is a r e g u l a r p r i m e , the
relatively
$L = 3
prime
to
I,
index
whereas
it is a n o n - t r i v i a l when
of
As y e t , it has
of such
ft
G
in the
an e x t e n s i o n .
one could
not
case In
conceivably
the
show
s i m i l a r to the proof of T h e o r e m
is, however, false.
in p a r t i c u l a r that
shows
for
Frohlich's proof [F] of the
fields.
1 to the group
i) as a corollary
is cyclic
(oG)
converse
imply and
formula
a conductor-
n u m b e r of w a y s .
The proof
R(oG)
to Iwasawa'a
satisfies
C, K, n t o g e t h e r , these results show that the (non-maximal) orders g e n e r a l i z e the rings of integers in cyclotomic fields in a
significant
f o r m u l a with
A.
F o r , if it were Cl(ZG)
is a n n i h i l a t e d
is a n n i h i l a t e d of
J.
in
Frohlich's
p o w e r of
I
when
by ZC.
J,
by .
J. But
• (1-T)/2
computation n > 2
1.
t r u e , it would
of
if is
|D(ZG) |
(except
for
n = 2 ) .
REFERENCES [C] prime
C h i l d s , L.N., S t i c k e l b e r g e r relations and d e g r e e , I l l i n o i s 3.Math, (to a p p e a r ) .
tame e x t e n s i o n s
of
[F] Frb'hlich, A., S t i c k e l b e r g e r w i t h o u t Gauss s u m s , in A l g e b r a i c N u m b e r F i e l d s , P r o c . Durham S y m p o s i u m , A c a d e m i c P r e s s , L o n d o n , 1 9 7 7 . [F'] F r o h l i c h , A., On the c l a s s g r o u p finite a b e l i a n groups I I , N a t h e m a t i k a
of integral g r o u p r i n g s 19 ( 1 9 7 2 ) , 5 1 - 5 6 .
of
201 [G] Glass, C.A., Realizable classes in the class groups of integral group rings, Thesis, London University, 1980. [H] Hilbert, D., Die Theorie der algebraischen Zahlkb'rper, in Gesammelte Abhandlungen, Chelsea, New York, 1965. [I] Iwasawa, K., A class number formula for cyclotomic fields, Ann.Nath. T6_ (1962), 171-179. [KL] Kubert, D. and Lang, S., Cartan-Bernoulli numbers as values of L-series, Nath.Ann. 2^_0 (1979), 21-26. [N] NcCulloh, L.R., A Stickelberger condition on Galois module structure for Kummer extensions of prime degree, in Algebraic Number Fields, Proc. Durham Symposium, Academic Press, London, 1977. [N 1 ] McCulloh, L.R., A class number formula for elementary-abeliangroup rings, J. Algebra _68_ (1981), 443-452. [R] Rim, D.S., Nodules over finite groups. Ann.Nath. 69 (1959), 700-712. CS] Sinnott, W., On the Stickelberger ideal and the circular units of a cyclotomic field, Ann.Hath. _1_CH3 (1978), 107-134.
UNIFORM DISTRIBUTION OF SEQUENCES OF INTEGERS W. Narkiewicz Mathematical Institute Wroclaw University PL-50-384 Wroclaw, Plac Grunwaldzki 2/4 Poland 1.
In this report we shall be concerned with uniform distri-
bution of sequences of integers in residue classes and shall deal with two related notions:
uniform distribution in all residue
classes with respect to a given modulus by
UD (mod N)
which are prime to bution
(mod N)
(mod N ) ,
N , which we shall call weakly uniform distri-
and denote by
Thus a sequence UD (mod N)
N , which we shall denote
and uniform distribution in residue classes
a*,a-,...
provided for all W{n < x : a
WUD (mod N) . j
of integers is said to be one has
= j(mod N)}
.
EL
lim
I
m
x+°° and it is said to be
WUD (mod N) , provided the set
{n : (an,N) = 1} is infinite and moreover for all W{n < x : a n =j(mod N)} llm W{n < x : (a ,N) = 1}
x-*°°
j =
prime to ^ JUTJ
N
one has (2)
n
Although the first paper in which
UD (mod N)
was considered
from a general point of view appeared in 1961 (I. Niven [27]), many results concerning this notion were known much earlier.
The oldest
dealt with uniform distribution of values of polynomials with integer coefficients in residue classes with respect to a prime.
Dickson
called them permutational polynomials and there is a huge literature concerning them.
Here we want to point out the highlights of this
theory. Let
f
be a given polynomial over
integers and denote by the sequence
M(f)
the set of all integers
f(1),f(2),...
is
in 1926 the conjecture, that if primes, then the degree of
Z , the ring of rational
f
UD (mod N) . M(f)
such that
contains infinitely many
is prime to
6
and moreover
be written as a composition of cyclic polynomials Tchebycheff polynomials
N
I. Schur [30] stated
ax
+ b
f and
can
203 T n (x) = 2" 1 " n {(x + [x 2 + 4] 2 l ) n + (x - Cx 2+ 4]^) n } . Schur himself proved its truth in the case, when the degree of
f
was equal to an odd prime and later V.A. Kurbatov [18] extended his result and covered the case when the degree of
f
is either divis-
ible by two primes or is squarefree with at most four prime divisors. The long-awaited proof of Schur's conjecture was finally given in 1970 by M. Fried [10]. The following question seems to be still unanswered: Problem 1
Let
be a prime number.
f
be a given polynomial over
For any
j = 0,1,...,p-1
Z
and let
denote by
d.(f,p)
p
the number of solutions of the congruence fCX) = j (mod p) , and denote by
A Cf)
the sequence
d o (f,p),d 1 (f,p),...,d p _ 1 (f,p) . Let finally
f
be a fixed polynomial over
ible to characterize all those polynomials that for infinitely many primes
Apm
p
f
Z .
Is it poss-
with the property
one has
- Aptfo)
in a way similar to the conjecture of Schur, which covers the case f Q [X) = X ? 2.
Another conjecture concerning permutational polynomials
was put forward by L.E. Dickson. of
f
equals
4 , then
N(f)
and if the degree equals to
N(f)
is
11 .
8
He noted namely that if the degree
cannot contain any prime exceeding
7
then the biggest prime which may belong
These observations lead to the conjecture that
for polynomials of even degree
M(f)
is finite.
This follows obvi-
ously from Fried's theorem but was established earlier by H. Davenport and D.J. Lewis [4], and D.R. Hayes [11]. It follows from the Chinese Remainder Theorem that if then the product and
b
belong to
prime power
p°
ab
lies in
N(f)
if and only if both factors
a
and the usual lifting argument gives that a
(c > 2)
and the derivative of
M(f)
(a,b) = 1
f
lies in
N(f)
has no zeros
if and only if (mod p) .
p e N(f)
This shows in
2 particular that if p e N(f) then all powers of p lie in M(f) . Denote by Ptf) the set of all primes in N(f) and by S(f) the
204 set of all primes
p
such that
p
e M(f) .
Then
N(f)
consists
of all integers of the form a
where the lie in
mination of [28]).
•••
p.'s
S(f)
a m %
1
?1 ••• p m q 1
are different and lie in
and the exponents MCf)
a.
P(f)\S(f) , the
are arbitrary.
reduces to that of
P(f)
and
q.'s
Thus the deter-
S(f) (W. Nobauer
An interesting result concerning these sets was obtained in
W. Nobauer [29]:
if
S c P
are two finite sets of primes, then
there exists an infinite sequence the degree of
f
f^f-,...
of polynomials with
tending to infinity and such that for every
n
one has
S(f ) = S and P(f ) = P . n n The following question is still open: Problem 2
Determine all pairs
S c P
of subsets of the set
of all primes such that a)
there exists a polynomial
f
with
S(f) = S
and
P(f) = P , b)
there exists infinitely many such polynomials with arbi-
trarily large degree. 3.
Uniform distribution of arithmetical functions
f
which
are not polynomials was considered in its full generality by I. Niven [27], although certain particular functions were studied earlier (e.g. the
UD
property for
u)(n)
ago) Niven gave a necessary condition for
and ft( n) UD (mod N)
was known long and later
S. Uchiyama [34] gave the appropriate Weyl-type condition which is both necessary and sufficient and runs as follows: A sequence h = 1, . . . ,N-1
f(1),f(2),...
is
UD (mod N)
if and only if for
one has
I exp{~p- hf(k)} = o(T) . N k=1 Of course this is a special case of the known criterion for uniform distribution in compact abelian groups.
See also A. Dijksma, H.G.
Neijer [8], L. Kuipers, S. Uchiyama [17], S. Uchiyama [35] and N. Uchiyama, S. Uchiyama [33], For certain classes of functions ditions for
UD (mod N)
stated above.
f(n)
one can state con-
which are easier to apply than the one
In the case of integer-valued additive functions this
was done by H. Delange [5] who showed that an additive function is
UD (mod N)
if and only if either for
m = 1,2,...,N-1
and
f
205 r = 1,2,3,...
the ratio
2mf(2 r )/N is integral and odd, or the series such that
N/fmf(p)
Zp
taken over all primes
p
diverges.
A similar result ([6]) holds for systems of additive functions, provided one extends appropriately the notion of uniform distribution.
Examples of multiplicative functions which are
UD (mod N)
for all
N
were given by H. Delange.
The question for which integers sequence of the second order is Bumby [1]. a is
n n+2
powers
a given linear recurrence
UD (mod N)
was solved by R.T.
He showed that such a sequence, defined by = Aa
A n+1
UD (mod N)
m > 1
N
p
+ Ba
n
if and only if it is
which divide exactly
if and only if it is
N
UD (mod p m ) and it is
UD (mod p)
for all prime
UD (mod p )
for
and one of the following
cases holds: (i)
p > 5
(ii)
p = 3 ,
A 2 + B £ 0 (mod 9)
(iii)
p = 2 ,
A = 2 (mod 4) ,
Finally the sequence have in case odd of
p = 2 ,
p ,
p|A
{a } 2
is
B = 3 (mod 4) .
UD (mod p)
if and only if we
n
+ 4B , p/A , p/2a^ - Aa.
and in the case
2/A , 2/TBa2 - a,, .
This condition for primes was obtained also by M.B. Nathanson [24] and for prime powers by P. Bundschuh, 3.S. Shiue [3] and W.A. Webb, C.T. Long [36]. Earlier the special cases of the Fibonacci and Lucas sequences were settled (P. Bundschuh [2], L. Kuipers, J.S. Shiue [13], [15], H. Niederreiter [26]) . If, as before, we define N
for which the sequence
can ask which subsets as
M(f)
N
M(f)
the set of all those integers
f(1),f(2),... f .
then one
An answer to that question was
obtained by A. Zame [38] who proved that M
UD (mod N)
of the set of positive integers can serve
for a suitably chosen
and only if
is
M
has this property if
contains all divisors of its elements.
He obtained
this as a corollary to a more general result concerning groups. One considered also joint distribution of values of a finite set of arithmetic functions in residue classes.
In this respect see
206 H. Delange [6], L. Kuipers, J.S. Shiue [16], L.Kuipers, H. Niederreiter [12] and N.B. Nathanson [25].
4.
Now we turn to
WUD (mod N) , as defined in (2) .
Par-
ticular functions with this property were studied long ago, in fact, the quantitative form of the Dirichlet's prime number theorem expresses the fact that the sequence of all primes is for all
N .
WUD (mod N)
One can easily formulate the Weyl-type conditions for
WUD , namely for all nonprincipal characters
X mod N
one should
have
I
X ( a n ) = oC I
n<x where
X(a))
n<x
X
denotes the principal character
(mod N) .
cases however one may obtain much simpler criteria.
In certain
Let
f(n)
be
an integer-valued multiplicative function, which is polynomial-like, i.e. for every
j = 1,2,...
such that for all primes
there exists a polynomial i
p
V. e Z[x] ^
f(p J ) = V . ( x ) .
one has
(This con-
dition may be weakened without affecting most results but we do not want to complicate matters.)
Denote by
R.
the set
{V.(x): (xV. (x),N) = 1} and let
A.
be the subgroup of
reduced residue classes assume that not all sets
R^R-,...
the smallest index
k
f (1 ) , f ( 2 ) , . . .
will be
principal character exists a prime
p
G(N) , the multiplicative group of
(mod N) , generated by with
R,
If we now
N
will denote
non-empty, then the sequence
WUD (mod N)
X (mod N)
R. .
are empty, and
if and only if for every non-
which is trivial on
^
there
such that (W. Narkiewicz [19])
- a. This implies that if
A^ = G(N) , then our sequence is
which was already observed in the case
N = 1
Using this criterion one can find all integers
WUD (mod N)
by E. Wirsing [37]. N
for which the
Euler function, the divisor function or the sum of the divisors are WUD (mod N) values of
(W. Narkiewicz [19], J . Sliwa [32]). cj) (n)
the values of
are
a(n)
the divisor function are
WUD (mod N)
divide
N
WUD (mod N) are d(n)
Thus e.g. the
if and only if. (6,N) = 1
WUD (mod N)
if and only if
6/N
.
and For
things are more complicated - its values
if and only if the smallest prime which does not
is a primitive root
(mod N) .
207 One can reformulate this criterion so that it could be applied to arbitrary multiplicative integer-valued functions, not necessarily polynomia1-like.
Let
that there exists an integer
k
f
be such a function and assume
such that the series
I P (f(pk),N)=1 diverges.
Let the smallest such
be the subgroup of
G(N)
k
be denoted by
generated by those
M .
Now let
A
j (mod N) , (j,N) = 1
for which the series
I P f(p n )Ej(mod N) diverges. place of
Now we can repeat the condition involving (3) with
A in
An .
This condition makes sense for arbitrary multiplicative functions but it is not clear whether it is equivalent with
WUD (mod N)
except in two cases: 1) primes
when for each p
Re s > 1
such th that
j (mod N) , (j,N) - 1
f(p ) =j(mod N)
the set
A. .of
is regular, i.e. for
one has
I
-I- = c log ^ L
+
f (S)
J J peA . p J where c. are nonnegative integers and g.(s) J J ular in the closed half-plane Re s > 1 , and 2) when the series
I
is a function reg-
~
P converges (H. Delange [7], W. Narkiewicz [20]). In the general case it is equivalent with the following condition, which may be called for all
j
with
WUD (mod N)
(j,N) = 1
I i,,m M
n f(n)Ej(mod N) r "S n (f(n),N)=1
=
1 (J> (N)
in the sense of Dirichlet:
208 (W. Narkiewicz, J. Sliwa [23]).
To obtain
WUD
from this one needs
tauberian theorems and it seems that one should be able to construct a multiplicative function for which the last condition is satisfied for a certain 5.
N
If
f(p) = V(p)
but nevertheless it is not
f(n)
is a multiplicative function satisfying
with a non-constant polynomial
sume that it is not of the form and
WUD (mod N) .
V = cW
V , about which we as-
where
W
is a polynomial
k > 2 , then it is possible to utilize evaluations of character
sums resulting from A. Weil's proof of Riemann hypothesis for curves to deduce that there is an integer in terms of
V
such that if
D , which can be given explicitly
(N,D) = 1 , then the values of
WUD (mod N)
(W. Narkiewicz [21]).
WUD (mod p)
for all sufficiently large primes
f
are
In particular they are p .
This answers a
question of P. Erdos asked at one of the meetings of the Dberwolfach Number Theory Conference. An analogous result holds also for systems of multiplicative functions, provided one adapts appropriately the notion of
WUD .
This was recently applied to the study of joint distribution of values of
(}>(n)
and
a(n)
(W. Narkiewicz [22]).
This paper brings
also an explicit procedure which in most cases leads to the determination of the set of all
N's
for which a given polynomial-like
multiplicative function has its values No analogue of Zame's result for known for from
WUD .
WUD (mod M)
WUD (mod N) . UD
quoted in section 3 is
In fact there are difficulties in determining when one can deduce
WUD (mod N) .
This question was
studied by E.J. Scourfield [31], who obtained certain sufficient conditions.
So we have:
Problem 3
Prove an analogue of Zame's result for
Problem 4
Characterize all pairs of integers
WUD (mod M)
implies
M,N
WUD . such that
WUD (mod N) .
Naybe these questions would be easier if one would consider only polynomial-like multiplicative functions. REFERENCES 1. Bumby, R.T., A distribution property for linear recurrence of the second order, Proc.Amer.riath.Soc. E>£ (1975), 101-106. 2. Bundschuh, P., On the distribution of Fibonacci numbers, Tamchang J.Math. 5 (1974), 75-79. 3. Bundschuh, P., Shiue, J.S., Solution of a problem on the uniform distribution of integers, Atti Accad.Naz. Lincei, Rend.Cl.Sci.
209 Fis.Mat.Nat. (8), 5^ (1973), 172-177. 4. Davenport, H. and Lewis, D.J., Notes on congruences I, Quart. J.Math., Oxford Ser. J_4 (1963), 51-60. 5. Delange, H., On integral-valued additive -Functions, J. Number Theory _1_ (1969) , 419-430. 6. Delange, H., On integra1-valued additive functions II, ibidem 6_ (1974) , 161-170. 7. Delange, H., Sur les fonctions multiplicatives a valeurs entieres, C.R.Acad.Sci . Paris 283 (1976), A1065-A1067. 8. Dijksma, A. and Meijer, H.G., Note on uniformly distributed sequences of integers, Nieuw Arch.Wisk. Y7_ (1969), 210-213. 9. Dowidar, A.F., Summability methods and distribution of sequences on integers, J . Nat. Sci . Math . VZ_ (1972), 337-341. 10. Fried, M., On a conjecture of Schur, Michigan Math.J. j_7 (.1970), 41-55. 11. Hayes, D.R., A geometric approach to permutation polynomials over a finite field, Duke Math.J. _34 (1967), 293-305. 12. Kuipers, L. and Niederreiter, H., Asymptotic distribution (.mod m) and independence of sequence of integers, I, II, Proc. Japan Acad.Sci. 5_0 (1974), 256-260, 261-265. 13. Kuipers, L. and Shiue, J.S., A distribution property of the sequence of Fibonacci numbers, Fibonacci Quart. 10 (1972), 375-376, 392. 14. Kuipers, L. and Shiue, J.S., A distribution property of a linear recurrence of the second order, Atti Accad.Naz.Lincei, Rend. Cl.Sci.Fis.Mat.Nat. (8), 5_2 (1972), 6-10. 15. Kuipers, L. and Shiue, J.S., A distribution property of the sequence of Lucas numbers, Elem.Math. 2_7 (1972), 10-11. 16. Kuipers, L. and Shiue, J.S., Asymptotic distribution modulo m of sequences of integers and the notion of independence, Atti Accad. Naz. Lincei, Mem.Cl.Sci.Fis.Mat.Nat. (8), JJ_ (1972), 63-90. 17. Kuipers, L. and Uchiyama, S., Notes on the uniform distribution of sequences of integers, Proc.Japan.Acad. _44 (1968), 608-613. 18. Kurbatov, V.A., on the monodromy group of an algebraic function (in Russian), Mat. Sbornik_25 (.1949), 51-94. 19. Narkiewicz, W., On distribution of values of multiplicative functions in residue classes, Acta Arith. YZ_ (1966-67), 269-279. 20. Narkiewicz, W., Values of integer-valued multiplicative functions in residue classes, ibidem ^2 (.1977), 179-182. 21. Narkiewicz, W., On a kind of uniform distribution for systems of multiplicative functions, Litovskij Mat. Sbornik, to appear. 22. Narkiewicz, W,, Euler function and the sum of divisors, 3. Reine Angew.Math. 323 (1981), 200-212. 23. Narkiewicz, W. and Sliwa, J., On a kind of uniform distribution of values of multiplicative functions in residue classes, Acta Arith. 3_1 (1976), 291-294. 24. Nathanson, M.B., Linear recurrences and uniform distribution, Proc. Amer. Math. Soc. 4_8 (.1975), 289-291. 25. Nathanson, M.B., Asymptotic distribution and asymptotic independence of sequences of integers, Acta Math.Hungar. 2J3 C1977), 207-218. 26. Niederreiter, H., Distribution of Fibonacci numbers mod 5 , Fibonacci Quart. 10 (1972), 373-374.
210 27. Niven, I., Uniform distribution of sequences of integers, Trans. Amer. Math. Soc. 9J3 (1961), 52-61. 28. Nb'bauer, W., Uber Permutationspolynome und Permutationsfunktionen fur Primzahlpotenzen, Monatsh . f . Math . _6_9 (1965), 230-238. 29. Nobauer, W., Polynome, welche fur gegebene Zahlen Permutationspolynome sind, Acta Arith. V\_ (1966), 437-442. 30. Schur, I., Uber den Zusammenhang zwischen einem Problem der Zahlentheorie und einem Satz uber algebraische Funktionen, SBer. Preuss.Akad.Wiss. (1923), 123-134. 31. Scourfield, E.J., On polynomial-like multiplicative functions weakly uniformly distributed (mod N ) , J. London Nath.Soc. 9_ (1974), 245-260. 32. Sliwa, J., On distribution of values of a(n) in residue classes, Colloq.Math. 27 (1973), 283-291, 332. 33. Uchiyama, N. and Uchiyama, S., A characterization of uniformly distributed sequences of integers, J.Fac.Sci. Hokkaido JJ5 (1962), 238-248. 34. Uchiyama, S., On the uniform distribution of sequences of integers, Proc.Japan.Acad. 37_ (1961), 605-609. 35. Uchiyama, S., A note on the uniform distribution of sequences of integers, J.Fac.Sci. Shinshu Univ. 3 (1968), 163-169. 36. Webb, W.A. and Long, C.T., Distribution modulo p of the general linear second order recurrence, Atti Accad.Naz. Lincei, Rend. Cl.Sci.Fis.Nat.Nat. (8), _5I3 (1975), 92-100. 37. Wirsing, E., Das asymptotische Verhalten von Summen uber multiplikativen Funktionen, Hath. Annalen 143 (1967), 75-102. 38. Zame, A., On a problem of Narkiewicz concerning uniform distributions of sequences of integers, Colloq.Nath. _24 (1972), 271-273.
DIOPHANTINE EQUATIONS WITH PARAMETERS A. Schinzel
Introduction The starting point of the investigations to be presented here is: Hilbert's Irreducibility Theorem (1892) [7].
Let
F.(xi,...,x , ti,...,t ) (1 < j < k) be polynomials irreducible Then for every polynomial G 6 (j Ltl»#'*>tJ> G ^ 0, there
over Q.
exist integers ti*,...,t *
such that G(ti*,...,t *) ^ 0 and for all
j ^ k the polynomials F.(xx,...,x , ti*,...,t *) are irreducible over Q.
In what follows, we shall use the abbreviated vector notation (xi,...,x ) = x and (ti,...,t ) = t; so that, for example, s r Where r = 1 we write ti = t, but there should be no risk of confusion.
Hilbert's theorem easily implies: THEOREM I. Let F € Q [x,t], and suppose that, for all t* €- 7LX , there exists x £ Q
such that F(x,t*) = 0.
Then there exists
s
x 6
For the proof it suffices to decompose F over Q into irreducible factors F(x,t) = c
I I F.(x,t) j = 1 J
J
and then in Hilbert's Irreducibility Theorem take G to be the product of leading coefficients of F.(x,t) with respect to x.
212 Closely related to Theorem I is:
THEOREM II (Kojima (1915) [s\ - Skolem (1921) [l8]). F eQ x 6 2
Let
[x,t], and suppose that, for all t* 6 7L , there exists S
such that F(x,t*) = 0.
Then there exists X G Q [t]S such
that F(X(t),t) = 0.
Let us denote by I
the set of all polynomials in ^[t ,...,t 1 that
take integer values for all t* € 2Z .
COROLLARY 1. Let G 6 I
,
Theorem II implies:
Suppose that, for all t* G 2 r , there
r
k exists x e 7L such that x = G(t*).
Then there exists X 6 I such
k
that X(t) = G(t).
COROLLARY 2. Let G 6 7L [t]. exists x e 2
such that x
Suppose that for all t* 6 2 r , there
= G(t*) .
Then there exists X € 2Z [t]
k
such that X(t) = G(t) .
For r = 1, both Corollaries have been proved by many authors: Franel [4], Grosch |6| (only for k = 2), Fried and Suranyi £>] , Lovasz [10] and see also Shapiro [l7].
For r > 1 an extension
to algebraic number fields has been given by Ribenboim JJ2] .
Theorems I and II and Corollaries 1 and 2 suggest four types of statements to be investigated for Diophantine equations F(xi,...,x , ti,...,t ) = 0, with s unknowns and r parameters. Namely the following four statements. (i) If, for all t* € ZZr,
there exists x € QS such that F(x,t*) = 0,
then there exists x € Q(t) S such that F(x(t),t) = 0. (ii) If, for all t* € 2 r , there exists x € 2 s S
such that
F(x,t*) = 0, then there exists X € Q[t] such that F(X(t),t) = 0.
213 r
(iii) If, for all t* €• Z , there exists x € 7L F(x,t*) = 0, then there exists X G I
S
S
such that
such that F(X(t),t) = 0.
(iv) If for all t* € 2Zr, there exists x e 7L S such that F(x,t*) = 0, then there exists X € TL [t] S such that F(X(t),t) = 0.
In addition to the examples afforded by Corollaries 1 and 2, the statements (iii) and (iv) occur in the following theorems and examples.
THEOREM III. (Skolem (1937), [l9]). Statement (iii) holds if s F = A (t) + E A.(t) x., where the polynomials A.(t) (1 < i < s) have no common zero.
Example 1. (Skolem (1940) [l9j).
Statement (iii) fails if
F = t x 2 + (t!2 + t 2 2 ) X! + t x t 2 x 2 . THEOREM IV. (Davenport, Lewis and Schinzel (1964) [2] - Chowla (1966) [l]).
Statement (iv) holds if F = xx2 + x 2 2 - C(t), where
C(t) 6 2Z [t].
The results and conjectures given in the sequel have been obtained or proposed during the last ten years and many of them are not yet published.
In their formulation capital letters (except Q
and TL ) denote polynomials with integral coefficients.
1. Concerning Statement (i)
THEOREM 1. Statement (i) holds if s = 2 and F(x,t) = 0 represents a finite union of curves of genus 0 over the algebraic closure of Q(t).
The crucial case when F is quadratic over Q(t) was settled in [3] for r = 1 and in [9j f or r > 1.
The case of one curve of genus
0 reduces to the former due to a theorem of Poincare* (see f2l], p 71) as has been pointed out by M.Fried.
214 CONJECTURE 1. Statement (i) fails for F = x ^ - x 2
2
- (8 t
2
2
+ 5) .
It is shown in [9] how via a result of Stephens [22}, the conjecture follows from the so-called Selmer's conjecture in the theory of rational points on elliptic curves.
2. Concerning statement (ii)
THEOREM 2. Statement (ii) holds if either: 1) F = G(xi,x2) - C(t), where G is a quadratic form; or r = 1 and F satisfies one of the conditions: 2) F = 0 represents a parabola over Q(t), 3) F = L(xi,t) - M(t) x 2 , where L is a polynomial of degree at most 4 in x j , 4) F = A(t) xin + B(t) - M(t) x 2 , where n i 0 (mod 8 ) . The proof of (ii) in case 1) is implicit in [2] for r = 1 and in Q2J for r > 1.
The proof in cases 2) to 4) will appear in [l5].
The assumption in 1) that G is a quadratic form seems to be due to the imperfections of the method.
CONJECTURE 2. (cf [l4]).
In fact we quote:
Statement (ii) holds for
F = G(xi,x2) - C(t), where G is an arbitrary form, C an arbitrary polynomial.
On the other hand, assumption 2) to 4) are natural, as is shown by the following examples.
Example 2 (Ql5J).
Statement (ii) fails for F = x j 2 - (4t 2 +l) 3 x 2 2 +l.
Example 3 ([l5]).
Statement (ii) fails for
F =
(Xl2
+ 3)(Xl
3
Example 4 ([l5j). Example 5 ([l5]). F = xi
2
+ 3) - (3t + 1) x 2 . Statement (ii) fails for F = x x 8 - 16 - (2t +l)x2. Statement (ii) fails for 2
+ 1 - ((4t! + I ) 2 + t 2 2 ) x 2 .
215 In particular, Example 5 shows that in the assumptions 2) to 4) one parameter t cannot be replaced by two parameters.
Nevertheless we
have the following: THEOREM 2a. ([l6]).
Let F = L( Xl ,x,t) - M(T,t)x2, where L is of
degree at most 4 in xi or L(xi,T,t) = A(x,t)x1 A,B,M are arbitrary and n j£ 0 (mod 8 ) . r
+ B(x,t), where
Suppose that, for all
and all t* <=• 2 , there exists x G 7L1 such that
T* € 7L ~
F(x,x*,t*) = 0.
Then there exists x € Q(x)[t] 2 such that
F(x,x,t) = 0.
3. Concerning statement (iii)
THEOREM 3 ( [l 5] ) . F = A(t)x 1
n
Statement (iii) holds if r = 1,
+ B(t) - M(t)x2 and n t 0 (mod 8 ) .
Example 6 ([3]). Example 7 ([l5]).
Statement (iii) fails for F = (2x1 + t) (2xl + t + 1) Statement (iii) fails for
F = (2xx + l)(3x! + 1) - (5t + l)x2. 4. Concerning statement (iv)
THEOREM 4 ([l4]).
Statement (iv) holds if F = G(xx,x2) - C(t),
C ۥ 2Z [t] and either G is an integral quadratic form with fundamental discriminant equivalent (properly or improperly) to every form in its genus or G = xj
x 2 ^ and the greatest common
divisor of the values of C (the fixed divisor of C) equals the greatest common divisor of the coefficients of C (the content of C ) . Theorem 4 suggests the following:
CONJECTURE 3. Statement (iv) holds if F = G(x x ,x 2 ) - C(t), G is an arbitrary integral form and the following conditions are satisfied.
1) G(x x ,x 2 ) = H(a 1 1 x 1 + a 1 2 x 2 , a 2 1 x x + a 2 2 x 2 ) , H G 7L [x,y] , a ^ G 7L implies det ( a . . ) = ± 1. 2) the fixed divisor of C equals the content of C.
216 The following examples show that neither condition can be dispensed with. Example 8 ([l l] , see also [23]). F = xx
2
+ 3x 2
2
2
Statement (iv) fails for 2
- (t x + t x t 2 + t 2 ) .
Example 9 ([l4]).
Statement (iv) fails for F = x 2 2 x 2 3 - 2t2(t+l)2.
5. The case s > 2.
Theorems 1 to 4 all referred to the case s = 2.
Besides Theorem
III there are a few affirmative results concerning the case s > 2. Some of the simpler ones are quoted below.
THEOREM 5 ( [l 3]).
Statement (i) holds if F = A(t) NK(x) + B(t),
where N (x) is the norm form of a field K of prime degree s.
THEOREM 6 ([2] for r = 1, [l3] for r > 1).
Statement (ii) holds
if F = Nj N (x) + C(t), where N (x) is the norm form of a cyclic is.
is.
field K.
The assumptions about the field K made in Theorems 5 and 6 are essential, as the following example shows. Example 10 ([2]).
Statements (i) and (ii) fail
for
F = NK(x) - t2, where K =
References 1. Chowla S, Some problems of elementary number theory, J.Reine Angew.Math.222 (1966), pp 71-74. 2. Davenport H., Lewis D.J. and Schinzel A., Polynomials of certain special types, Acta Arith.9 (1964) pp 107-116. 3. Davenport H., Lewis D.J. and Schinzel A., Quadratic diophantine equations with a parameter, Acta Arith.11 (1966) pp 353-358 4. Franel J., SixieTne reponse a la question 37, Intermed. Math.2 (1985) pp94-96 5. Fried E. and Suranyi J., Neuer Beweis eines zahlentheoretischen
Satzes liber Polynome (Hungarian), Math.Lapok 11 (1960) pp 75-84. 6. Grosch W., LBsung zu Aufgabe 402, Arch.Math.Phys. (3)21 (1913) pp 368-369. 7. Hilbert D. Uber die IrreduzibilitMt ganzer rationalen Functionen mit ganzzahligen Coeffizienten, J.Reine Angew.Math.110(1892) pp 104-129 Ges.Abh.Bd.II, Springer 1970 pp 264-286.
217 8. Kojima T. Note on Number-theoretic properties of algebraic functions, Tohoku Math.J.8 (1915) pp 24-37. 9. Lewis D.J. and Schinzel A., Quadratic diophantine equations with parameters, Acta Arith. 37 (1980) pp 133-141. 10. Lovasz L., Connections between number theoretic properties of polynomials and their substitutional values (Hungarian), Mat.Lapok 20 (1969) pp 129-132. 11. Perlis R. and Schinzel A., Zeta functions and the equivalence of integral forms, J.Reine Angew.Math. 309(1979) pp 176-182. 12. Ribenboim P., Polynomials whose values are powers. J.Reine Agnew. Math. 168/169 (1974) pp 34-40. 13. Schinzel A., On a theorem of Bauer and some of its applications II, Acta Arith. 22 (1972) pp 221-231. 14. Schinzel A., On the relation between two conjectures on polynomials, Acta Arith. 38 (to appear). 15. Schinzel A., Families of curves having each an integer point, Acta.Arith. (1980) pp 285-322. 16. Schinzel A., An application of Hilbert's irreducibility theorem to Diophantine equations, Acta Arith.41 (to appear). 17. Shapiro H.S., The range of an integer-valued polynomial, Amer. Math.Monthly 64(1957) pp 424-425. 18. Skolem T., Untersuchungen liber die mBglichen Verteilungen ganzzahliger Lflsungen gewisser Gleichungen, Kristiania Vid. Selskab. Skrifter I, 1921 No.17. 19. Skolem T., Uber die Lflsbarkeit gewisser linearer Gleichungen im Bereiche der ganzwertigen Polynome, Kong,Norske Vid. Selskab Forh. 9 (1937) No.3420. Skolem T., Einige SMtze liber Polynome, Avh.Norske Vid.Akad. Oslo I 1940 No. 4. 21. Skolem T., Diophantische Gleichungen (Chelsea reprint, New York 1950) • 22. Stephens N.M., Congruence properties of congruent numbers, Bull. London Math.Soc. 7 (1975) pp 182-185. 23. Watson G.L., Determination of a binary quadratic form by its values at integer points, Mathematika 26 (1979) pp 12-15.
GALOIS MODULE STRUCTURE OF RINGS OF INTEGERS M.J. Taylor Queen Nary College London
Let
E/K
be a Galois extension of number fields, let
the ring of integers of
E
and let
T = Gal(E/K) .
as a (right) module over the integral group ring
(Dp
We consider
ZF .
be (Dp
The funda-
mental problem in the theory of Galois module structure of rings of integers is to determine whether or not (i.e. whether or not
IDE
{aT}y e F , i = 1 . . . [K : Q ] ) . to the
{a.}
(Dp
is a free ZF-module
possesses a Z-basis of the form If
(DE
is
ZF free, then we refer
as a normal integral basis.
In Section 1 we describe the history of the subject up to the appearance of the article [F1].
Then in the second section we de-
scribe the 'general methods' introduced by Frb'hlich in [F1].
Lastly
in section 3 we outline the main ideas involved in the proof of Frohlich's conjecture. 1 . HISTORY The first known result in the subject is due to Hilbert (cf. [H] ) , who showed: (1.1) the order of
If
K = U ,
F
is abelian and the prime divisors of
F , |F| , are non-ramified
in
E/IEJ , then
(Dp
is a
free (rank one) 2.T-module. The condition that the 'prime divisors of ramified in E/Q
|F|
be non-
E/0 ' was subsequently slackened to 'the extension
be tame, i.e. at most tamely ramified'.
The credit for this
generalisation is usually given to Speiser - although there appears to be no evidence to support this claim. Nore recently
(cf. [T1]) this result has been extended by the
author to arbitrary basefield, i.e. (1.2) then
(DE
If the extension
is a free ZT-module
E/K
is tame and if
(of rank
V
is abelian,
[K : BJ ] ) .
Returning to chronological order, the next important result after the Hilbert-Speiser Theorem is due to E. Noether who considered the local structure of
ID,-
over
(D^F .
In fact, she too
219 only considered extensions where the divisors of ramified (cf. [N]).
|F|
are non-
However, it is usual to call the following
Noether's Theorem. (1.3)
(DE
is locally free over
if, and only if,
p
ID^F
at a prime
is at most tamely ramified in
p
of
K
E .
For an outline of the proof of this result see page 21 of [CF]. In the sequel we shall always assume the extension
E/K
to
be tame. The first person to obtain good global results on the existence of normal integral bases for non-abelian Galois groups was 3. Martinet.
In [N1] he showed
(1.4) prime),
If
then
K = UJ
(Dp
and
F
is dihedral of order
21
(I an odd
is TT-free.
However, when he applied his techniques to Galois groups which were quaternion groups of order extensions where
(Dp
8 , he discovered that there exist
is not ZF-free (cf. [M3]).
It was Frohlich's
interpretation of Martinet's apparently negative result which is really the richness of the whole subject.
In order to describe his
interpretation it is necessary to introduce some notation. Let
F
be an arbitrary finite group, and let
Grothendieck group of virtual characters of denote the 'extended
1
where
A(S,X)
= W(X)A(1-S,X)
be the x
x
e
^r- *
by
we
A(S,X)
satisfies a functional equation
,
i-s the complex conjugate of
x
R_
For
Artin L-function associated to
(as defined in [N2]).
A(S,X)
F .
known as the Artin root number of
x •
x •
The constant
W(x)
is
From the general theory of
Artin root numbers (cf. [Tel for instance) it is known that for real valued characters Now let
x >
W(x) = ±1 •
K = U , let
F
be a quaternion group of order
and let
x
Then
is a symplectic and hence real valued.
X
be the unique non-abelian irreducible character of
(Dr
is ZT-free
if, and only if,
F .
In [F2] Frohlich
showed that (1.5)
8
W (X) - 1 .
220 2. GENERAL P1ETHODS So far we have only considered special types of Galois group. We now introduce a number of results which will enable us to tackle the general problem. Let K ( Z D be the Grothendieck group of locally free ZFmodules. The rank homomorphism yields a surjection K ( Z D -> Z . Let M be a locally free ZF-module of rank m . We denote by (PI) the stable isomorphism class of PI in K ( Z D minus the stable isomorphism class of the direct sum of m copies of ZF . Thus (PI) e C K Z D , and we refer to (PI) as the class of PI . Let M be a maximal order of QF which contains ZF . Then extension of scalars yields a surjection Cl(ZF) -> C1(M) ; we denote the kernel of this homomorphism by D ( Z D . Jacobinski has shown that the sub-group D(ZF) is independent of the choice of maximal order M . The reason for our introducing D ( Z D is that Frohlich, following a conjecture of Martinet, showed (cf. Theorem 11 of [F1]) ((DE) e D(ZF)
.
(2.1)
The main aim of this section is to firstly give a description of D ( Z D , and then secondly, using this description, to give a representative of the class ((Dp) . Description of D(ZF) Let F be a 'large enough 1 and which is Galois over Q contain the |r| roots of unity and all we introduce. For any number field PI c F For a prime number I we put
and more generally, if S Uo =
S
II
LJ „
.
Then
LL
is
number field which is . In particular F is to other number fields which , we write ft^ = Gal(F/Pl) .
is a finite set of primes, we write an £L,-module i n the n a t u r a l way, and,
if
££s * S y x € F is an S-unit, we view x as an element of U« via the diagonal embedding. Homo (R r ,Ik) is the group of homomorphisms from R r to Ug which commute with ft^-action. (We view ft-, as acting on R^ valuewise) . Homn+ is the group of homomorphisms from R r to QJJ 4 (R p , CD * D TQ) r F I r which commute with ft^ and which are totally positive on all symplectic characters. Note that any fL.-homomorphism necessarily takes real values on symplectic characters since such characters are, of course,
221 real valued. We now wish to introduce a third group of homomorphisms on Let
z € Z 0 F* .
Let
x
De
We want to describe a homomorphism
a character of
r
R_ .
Det(z) : R r -*U 0 .
which is afforded by a representation
T : r -»• GL ((Dp) .
We extend
T : Z^r + M^CDp )
and we define
T
to a homomorphism of algebras
D e t ( z H x ) = det(T(z)) . Because the character
x
determines
left hand side is well-defined. of
Rp
by Z-linearity.
Det(z)
T
up to conjugacy, the
Now we extend
Det(z)
to the whole
In fact from Appendix I of [F1] we know that
is an £1^.-homomorphism.
We write
DetCZ F*)
for the group of
all such homomorphisms, and, more generally, we put Det(Z c T*) = b
n Det(Z o r*) z JUS
.
By II.2 in [F1] we have an isomorphism s) (2
(R(D*) where we take
S
Remark 1 D(ZF)
to be the set of prime divisors of
'2)
|r|
In (2.2) we have written the group operation in
multiplicatively instead of additively. Remark 2
The two essential ideas to understand in this des-
cription are firstly that the class of a module is represented by a homomorphism, and secondly, if we wish to find out whether the class of the module is trivial or not, then we have to develop methods for distinguishing whether a given homomorphism lies in the denominator or not. Example (due to Swan and Uilom) to
|r|
and let
(r,Z)
Let
r
be an integer prime
be the two sided ZT-ideal
Swan (cf. [S]) showed that
(r,Z)
is a locally free
rZT + Z Z y . yeT IV module.
Ullom (cf. (2.4) of [U]) showed that its class lies in
D(ZT)
and
that this class is represented under. (2.2) by the homomorphism
for
x ^ Rp .
Here
e
is the identity character of
is the standard inner product of character theory.
r
and
(, )
222 Class of
(Dp
The remainder of this section is devoted to
describing a representative of the class I.
We define the homomorphism
W
(GDp)
under (2.2).
: Rp -*• ±1
by stipulating
that W(x)
if
X
i-s irreducible and symplectic,
1
if
x
i-s irreducible and non-symplectic.
W'( X ) =
From Theorem 9 of [F1] we know that
W
is an £1™-homomorphism.
Following an idea of Philippe Cassou-Nogues, we define the class represented by II. T(X)
•
For
x
e
Rp *
W we
t(W)
to be
under (2.2). denote the Galois Gauss sum of
x
The reader is referred to [F12] for the definition of
D
y
T(X) .
However, it is worth pointing out that T ( X ) is equal to 1 /2 W(x)N f (x) multiplied by a certain fourth root of unity :y which is ,1/2 determined by the behaviour of the infinite primes. (Here N f is the positive square root of the absolute norm of the Artin conductor of
x • )
III.
By Noether's Theorem (1.3), we can choose
that the index
(Q)._ : a . L D
is prime to
|r| .
a e L
so
We define
A = n ( I aYa.y"1) a
yeT
where the product is taken (in any order) over (arbitary) extensions to into
F
a e ftp which are
of a complete set of embeddings of
K
F . We define
u : Rp -*• F*
to be the homomorphism
T
.W'.Det(A) .
In Section 9 of [F1] Frohlich showed that (a)
u
is an fto-homomorphism.
(b)
u
is S-unit valued.
(c)
Viewing
the class
(Q)E)
Remark
u
as an element of
under (2.2)
Horn
(Rr,U ) , u
represents
M
That (a), (b) and (c) hold clearly represents a very
deep relationship between the Galois Gauss sum and the Galois module structure of
QJ^ .
Indeed, in [F-T], the (tame) Galois Gauss sum is
actually characterised completely in terms of module invariants. 3. FROHLICH'S CONJECTURE One immediate application of Frb'hlich's general methods was to permit the calculation of F .
((Dp)
for many new types of Galois group
Following calculations for quaternion groups of order
4£
223 ( I an odd prime) in [F3], for groups of square free order (and more recently cube free order) in [C], and for Jl-groups in [T2], Frohlich conjectured the following: Conjecture
(Q)E) = t(W)
So that in particular
(Dp
(a)
(Q)E)2 = 1
(b)
the only obstructions
to the vanishing of the class of
are the signs of the Artin root numbers of the irreducible
plectic characters
of
sym-
V .
It is worth pointing out that (1.2),
(1.4),
(1.5) together
with certain self-duality results (cf. [T3] and [F4]) would follow very easily from the above conjecture. The remainder of this article is concerned with describing the main ideas contained in the proof of the following result (for details see [T4]) . (3.1) E/Q , then such
If the prime divisors of ((Dp) = t(W)
\T\
(i.e. Frohlich's
are non-ramified
conjecture
in
is true for
extensions). Remark
While it is clearly desirable to avoid all restric-
tions on the extension
K/Q , it is worth emphasising that the
above result is genuinely relative in that
K/y
can be wildly
ramified. The proof of this result naturally breaks up into three steps. Step 1
Using an idea of Deligne (cf. §5 of [D]) we adjust
the usual Galois Gauss sum by multiplying by a certain root of unity valued, ClQ-homomorphism. Thus we see that
We denote this adjusted Gauss sum by
(QJp)t(W)
x* .
is represented by the homomorphism
x*~ 1 Det(A) . Step 2
Next one shows that there is a number field
taining the normal closure of at
S
M
con-
E/Q , which is non-ramified over
Q
and which has the property that x* € DetdD M r*)
.
(A)
"s The proof of this result involves developing a method which enables us to decide whether a given
QM
homomorphism from
Rr
to
224 LL
lies
problem
in in
Step is
the
theory
3
By
non-ramified,
under that
the the
Let v = Det(z)
i.e. i .e
.
€
can
R
p ^
we
(quite
Then,
, but
e ty™ ^p
because
be
v
the
integral
.
. by
fundamental
group
rings.
Moreover,
shown
*•>• CDJVJ F
some
e
really
A e (D^F
represented
Det((D|v| F*)
w
over
know
(D^F
i in
is
easily)
is
situation ' lee"t t
This
classgroups
embedding
in
this x
of
((D^]t(W)
lies
consider
or n o t .
definition it
diagonal class
s a y ) which We
Det(CD(v| F*)
that
since
Consequently
we
a homomorphism
which which is an
S
A e IDJVJ F*
ft^
see (v ,
homomorphism.
detail.
aa nn dd
choos choose
is an
z e CDJVJ F*
so
that
ft0-homomorphism
Det(z)(x W ) = (Det(z)(x)) W = Det((z w )(x^))
and so
Det(z) =
Det Equivalently, stipulating
that
for
if
we x
D e t ( X ) -a) = D e t ( x W )
view
e (D^ F * ,
DetCffljv] a) e
T*)
as
an
ft^-module
.
Then clearly the above work translates as saying an ft^-fixed point. fi
by
ft^
However, by [T5], because
S
Det((Dn F*) ° = Detfffl^ F* ^) = Det(Z s F*)
.
Det(z)
is
is non-ramified
MAP
Consequently the representative homomorphism (CDp)t(W) Remark
(B) v
of the class
lies in the denominator of (2.2), as we require. It is interesting to note that the main idea in both
the proof of (A) and (B) is the method of integral logarithms which was developed in [T5],
REFERENCES [CF] J.W.S. Cassels and A. Frohlich, Algebraic number theory, Academic Press, New York and London, 1967. [C] Ph. Cassou-Nogues, Quelques theoremes de base normale d'entiers, Ann.Inst. Fourier, Grenoble, 2jB, 3 (1978), 1-33. [D] P. Deligne, Les constantes des equations fonctionnelles des fonctions L, Modular forms in one variable II, 1973, Lecture Notes in Mathematics 349, 501-597. [F1] A. Frohlich, Arithmetic and Galois module structure for tame extensions, J. reine angew.Math. 286/7 (1976), 380-440. [F2] A. Frohlich, Artin root numbers and normal integral bases for quaternion fields, Invent.Math. V7_ (1972), 143-166. [F3] A. Frohlich, Module invariants and root numbers for quaternion fields of degree 4£r, Proc . Camb . Phi 1. Soc . 7j3_ (1974), 393-399.
225 [F4]
A. Frohlich, to appear in the Springer Ergebnisse series.
[F-T] A. Frohlich and M.J. Taylor, The arithmetic theory of local Galois Gauss sums for tame characters, to appear in the Phil. Trans. Royal Soc. [H] D. Hilbert, Die Theorie der algebraischen Zahlkorper, Satz 132, Jahr.ber.d.d.Math.Ver (4) (1897), 175-546, or Ges.Abh. I, New York, 63-367. [M1] J. Martinet, Sur 1'arithmetique des extensions galoisiennes h groupe de Galois diedral d'ordre 2p, Ann.Inst. Fourier _1_9 (1969), 1-80. [M2] J. Martinet, Character theory and Artin L-functions, Algebraic number fields (ed. A. Frohlich), Academic Press, London (1977). [M3] J. Martinet, Nodules sur l'alg^bre du groupe quaternonien, Ann.Sci. Ecole Norm. Sup. 4 (1971), 229-308. [N] E. Noether, Normalbasis bei Korpern ohne hohere Verzweigung, J. reine angew.Math. r§7_ (1932), 147-152. [S] R. Swan, Periodic resolutions for finite groups, Ann. of Math. _72_ (1960) , 267-291 . [Te] J. Tate, Local constants, Algebraic number fields (ed. A. Frohlich), Academic Press, London (1977). [T1] M.J. Taylor, Galois module structure of relative abelian extensions, 3. reine angew.Math. 303/4 (1978), 97-101. [T2] M.J. Taylor, Adams operations, local root numbers and Galois module structure of rings of integers, Proc.L.M.S. (3), 39_ (1979), 147-175. [T3] M.J. Taylor, On the self-duality of a ring of integers as a Galois module, Invent.Math. ^6 (1978), 173-177. [T4] M.J. Taylor, On Frohlich's conjecture for rings of integers of tame extensions, to appear. [T5] M.J. Taylor, A logarithmic approach to classgroups of integral group rings, to appear in the J.Alg. [U] S.V. Ullom, Non-trivial lower bounds for classgroups, Illinois 3.Math. 20 (1976), 361-367.
ON THE FRACTIONAL PARTS OF an 3 , 3n2 AND
R.C. BAKER
1.Introduction integer.
Let
We denote by
|| ...|| the distance to the nearest
e be an arbitrary positive number.
In the present note
we prove a theorem announced in [1]. THEOREM
Let
a , $
and
y
———
numbers with
Then f o r
be real numbers.
Let
— — ______ _______________
______
r\ _L
n 0 > nq
9
_.
o
b£
—~"~*
0 < n. < 1 >
N > c (e)
t h e r e i s a n a t u r a l number
< nx ,
||ar/||
||Pn2||
< n2
n <_ N
and
having
|| y n ||
< n3 .
We mention some related results obtained since the appearance of the survey paper let
[1] .
k-1 K = 2 . Let
Then for
Let
k
a.,..., a,
be a natural number, k >_2 , and be real numbers where
h <_ K/2 .
N > c. (k , e) we have k
II mm l < n < N
The upper bound on
max l < i < _ h
h
|| a . n
II ||
, T -(l/hK) + e <
N
X
is larger than that of [2].
This inequality
is due to myself and G. Harman (to appear). Let
Q ,..., Q,
be quadratic forms in
s >^ c. (h,e) . Then for all zero, having
N >_ 1
s
variables where
there are integers
x.,...a x
|x. | <_N ,
l<_i£h
||Q.(x.,..., x )|| < 1 J S -
The exponent announced in
[l] was
N"(2/h)
(-1/h) + e .
+ £
.
not
227 The new exponent, due to myself and G. Harman, is sharp (this is explained in [1]). Finally, my student G-. Harman has proved min || a n 2 || < N " ( l / 2 ) 1 £ n <_ N
+ £
(Heilbronn's theorem) by elementary means.
That is, he uses no
integrals or infinite series. e(x) = e 27riX .
In what follows let
2.
Preliminary lemmas.
LEMMA 1.
Suppose that
Suppose that
Let
and
K
k x + a,
f(x) = a
N > c 5 (k , e).
k
be as above.
x
k—1
+ ... + a x .
Now if
N | Z e(f(x)) | >. B , x=l
(1)
where BiN
l-(l/K)
(e/2\
+
then there exists a natural number
"*K K + £ r < B N 7u\th
and
Proof
|| V
]
r || <_B~K N K " k + 1 +
This is a special case of Lemma 11A of
LEMMA 2. holds.
(2)
Let
f(x) and
N
suppose further that
. (3)
[6] .
be as in Lemma 1 and suppose that
Suppose that there is a natural number
e
m
having
(1)
228 B~k Nk
+ £
(6)
and
||t cull < B" k N k " j Proof.
This is Lemma 4
+£
(j=l,...,k).(7)
of [3].
Perhaps a few words of explanation would help. that given a large exponential sum as in
It is classical
(1), one obtains a ^ood*
rational approximation to the leading coefficient of
f.
Good
rational approximation was extended to the leading pair of coefficients by Schmidt [6] . The idea in Lemma 2 is that given a good rational approximation to all but the lowest coefficient, one can use the large exponential sum to improve the rational approximation and at the same time extend it to the lowest coefficient.
We now apply this
to particular cubic polynomials. 3. Proof of the theorem. finite set
S.
PROPOSITION.
We write
| S | for the cardinality of a
The theorem is the case Let
S
| S | = 3 of {1,2,3}.
be a nonempty subset of
———
Let
N > cAS9e)
——•
Q
n M < jeS 3 For each n £ N
j _in S , let
0.
known
Then there is a natural number
with ||n^e.|| < 3
Proof.
be real.
By induction on
MT 1 D
for all
| S |. For
j
in
| S | =1
[6] . In our inductive step let
S.
(9)
the result is well
2 <_ | S | <_ 3 . Suppose
that the simultaneous inequalities (9) have no solution
n <_ N.
By
229 standard arguments T
of
S
[4] it follows that there is a nonempty subset
such that
N I | I e ( Z m. 6. x 3 ) | > c (k,e)N , m.(jeT) x=l jeT 3 3
where
c« > 0
having
and the outer sum is over sets of integers
0 < |m.| < M. N '
for all
We see that there is one such set of N . | E e ( Z m. 6. x3 )| ^ N 1 x=l jeT 3 3 Define
B
(1) holds. Write
£/4
( n M.) jeT 3
.
The inequality
a natural number
§
3
k = max {j : j e T }
We distinguish two cases.
r
m. (j e T) having
= N 1 " 6 7 4 '( n M 3. ) " 1 jeT
Lemma 1.
j eT.
by B
then
m.(j e T)
and
Suppose first that
(2) follows from
K = 2k~1 . T ± {1}. We apply
(8), so there exists
r with
£B~KNK+£
<_ ( n M . ) K N 2 £ jeT 3
<_ N 1 " 8 ,
and
llrm.e.H < B N for all
i > 1
in
^ ^ V
T . Notice that
jeT by
3
(8) ; similarly
||rn,.e.|| for all
i > 1
in
T.
< N-i + 1 " e By Lemma 2, then, there is a natural number t
k
t < B~ N
k+e
3
<_ ( n M.) N jeT 3
230 (10)
2e
having || tin. 6. || < B" k N k " i + £
< ( IT M . ) V i + 2 e ~" jeT 3
1 1
for all
i
in
T .
Suppose now that
T = {1}. Then the inequalities
hold for some natural number the end of Lemma 4 Write
(11)
s =
t
s
of
t.
This is proved by arguing as at
[4] . Thus
(10), (11) hold in both cases.
H |m.| ; then , from jeT ^ < ( It M . ) 4 " jeT 3
(10) , (11)
(10) and (11) ,
N3£
(12)
and 8 6. || < ( n 1
for all
i
in
jeT T.
3
1+
^£
(13)
x
By induction (or trivially if
there is a natural number v
M.1 N
M.)
T = {1,2,3}
v with
<_ ( IT M . ) 4 j^T ^
N£
(14)
and || v i s 1 0 i | | < Let
n = sv
n
while if
M*1
for all
; then , from
i { T .
(15)
(12), (13) and (14),
£ ( IJ M. ) 4 N ^ £ N ,
(16)
i e T,
II n i 6 i || £ N1"*1 v H s 6 . l l < N 1 " 1 ( n M . A n M . ) 4 MT1 N~1+4e j^T 1
M"1
.
D
jeT
3
x
(17)
231 Combining (15) , (16) and (17) we obtain a solution
n <^ N
of (9).
This is a contradiction, and the inductive step is complete.
References. 1.
R.C. Baker,
'Recent results on fractional parts of polynomials1.
Number Theory, Carbondale 1979, 10-18.
Lecture Notes in
Mathematics no. 751 (Springer, Berlin). 2.
R.C. Baker, 'Fractional parts of several polynomials III1. Quart. JT. Math. Oxford (2), 31 (1980), 19-36.
3.
R.C. Baker, 'On the distribution modulo 1 of the sequence an
4.
3
+ 3n
2
+ yn' , to appear, Acta Arith.
R.C. Baker and J. Gajraj , "Some non-linear Diophantine approximatior Acta Arith. 31 (1976), 325-341.
5.
R.C. Baker and G. Karman, 'Small fractional parts of quadratic and additive forms'.
6.
To appear.
W. Schmidt, Small fractional parts of polynomials.
Regional
conference series no. 32, American Math. Soc. , Providence 1977.
Royal Holloway College, Egham, Surrey
IRREGULARITIES O F POINT DISTRIBUTION IN UNIT CUBES W. W. L. CHEN
Let U n = [0,1) and U, = ( 0 , 1 ] . distribution P(k,N) of N points in U positive integer.
P(k,N)
denote
which
+
, where k is a k+ 1 -) in U 1 , let
For any x_ = (x ,...,x 1 —
Z[P(k,N);xJ
Suppose we have a
1
K. + 1
t h e number o f p o i n t s
l i e in t h e box 0 < y i < xi
x
=
1
(•¥ \> • • ' >Y\c+0
(i = l , . . . , k + l ) ,
°^ and
wri te D[P(k,N);x] We are interested
= Z[P(k,N);x]
-
N x ^ . a ^ ^
in measuring the irregularity of the
distrib ition P(k,N) by considering, for 0 < W < «> t |D[P(k,N)]|| w
ff
!
^r"
-1 | W '
we shall also consider ^) ]||
Roth
=
sup |D[P(k,N) ;x.] | . k+1 =e 1
[7] proved in 1954 that there exists a positive
constant c,(k), depending only on k, such that for every P(k,N),
(1)
l|D[P(k,N)]||2 >
C l (k)(log
N)^k.
It follows easily from (1) that there exists a positive constant c 2 ( k ) , depending only on k, such that for every
(2)
Jk
1| D [ P C k , N ) ] | | _ > c 2 ( k ) ( l o g N )
233
For the special case k = 1, a sharp lower bound (see later) was obtained in 1972 by Schmidt [11]. He showed that there exists a positive absolute constant c
(3)
such that for every
||D[PC1,N) IH^ > c 3 l o g N .
Recently, an alternative proof of (3) was obtained by Halasz [3]. To obtain his estimate (1), Roth constructed an auxiliary function F (xj such that, writing D(,x) for D[P(k,N);^], (4)
1F(x)D(x)dx
f
k*
> c (k)(log N ) k ,
and (5)
f
F 2 ( x ) d x < c (k)(log N ) k .
These, together with Schwarz's inequality, give (1). A few years ago, Schmidt [12] showed that this auxiliary function F(£) also satisfies (6)
||F||r < c6(k,r)(log N ) J K
(r > 0 ) .
He did this by showing that F 2m (x)dx : < c (k,m)(log N ) m k
(m = 1,2,...)
(4) and (6), together with Holder's inequality, give
234 Theorem 1.
For every W > 1, there exists a positive
number c R (k,W), depending only on k and W, such that for every P(k,N) , ||D[P(k,N)]||w > c8(k,W)(log N ) * k . Schmidt [12] also showed that for some positive number Cg(k), depending only on k, and for large N,
||D[P(k,N)]||
> c (k)
1Og
l0g
N
.
log log log N Recently, Halasz [3] has improved this to (7)
|| D[P(k,N) ]|| _ > c in (k)(log N ) ^ . 1
1U
That (3) is essentially best possible was established by Lerch [6] in 1904, and later by van der Corput [1] using a different method.
In 1960, Halton [4] showed that for a
suitable number c..,(k), depending only on k, there exists, corresponding to every natural number N > 2, a distribution P(k,N) such that
(8)
ilDfPCk^)]!^ < c n ( k ) ( l o g N ) k .
However, for k > 2, there remains a gap between (2) and (8). On the other hand, Roth's lower bound (1) has been shown to be sharp, apart from the value of the constants.
This was
established in the cases k = 1 and k = 2 by Davenport [2] and Roth [9] respectively and more recently for general k by Roth [10].
Meanwhile, alternative proofs for the case k = 1 were
given by Vilenkin [13], Halton and Zaremba [5] and Roth [8].
235 Recently, the author was able to show that Theorem 1 is also sharp, apart from the value of the constants. Theorem 2.
Let W > 0.
For a suitable number c
depending only on k and W, there exists, corresponding
(k,W), to
every natural number N > 2, a distribution P(k,N) such that l|D[P(k,N)]||w < c 1 2 (k,W)(log N ) * k " A detailed proof is too long to be included here, and will be published in Mathematika.
References 1.
J.G. van der Corput.
Verteilungs funktionen, II, Proc.
Kon. Ned. Akad. v. Wetensch., 38 (1935), 1058-1066. 2.
H. Davenport.
Note on irregularities of distribution,
Mathematika, 3 (1956), 131-135. 3.
On Rothfs method in the theory of
G. Halasz.
irregularities of point distributions, to appear in Proc. Conf. Analytic Number Theory at Durham (1979). 4.
J.H. Halton.
On the efficiency of certain quasirandom
sequences of points in evaluating multi-dimensional integrals, Num. Math., 2 (1960), 84-90. 5.
J.H. Halton and S.K. Zaremba.
The extreme and L
discrepancies of some plane sets, Monatsh. fur Math., 73 (1969), 316-328. 6.
M. Lerch.
Question 1547, LfIntermediaire Math.,
11 (1904), 144-145.
236 7.
K.F. Roth,
On irregularities of distribution,
Mathematika, 1 (1954), 73-79. 8.
K.F. Roth.
On irregularities of distribution, II,
Communications on Pure and Applied Math., 29 (1976), 749-754. 9.
K.F. Roth.
On irregularities of distribution, III,
Acta Arith., 35 (1979), 373-384. 10.
K.F. Roth.
On irregularities of distribution, IV,
to appear in Acta Arith. 11.
W.M. Schmidt.
Irregularities of distribution, VII,
Acta Arith., 21 (1972), 45-50. 12.
W.M. Schmidt.
Irregularities of distribution, X,
Number theory and algebra, pp. 311-329, Academic Press, New York (1977) . 13.
I.V. Vilenkin.
Plane nets of integration (Russian),
Z. Vycisl. Mat, i Mat. Fiz., 7 (1967), 189-196; English translation in U.S.S.R. Comp. Math, and Math. Phys., 7(1) (1967), 258-267.
Imperial College, London, England.
THE HASSE PRINCIPLE FOR PAIRS OF QUADRATIC FORMS D.F. Coray* Universite de GenSve, Section de mathematiques 2-4, rue du Lievre CH-1211, Geneve 24, Switzerland
1. MOTIVATIONS The arithmetic study of rational surfaces was initiated by B. Segre [13], who considered the case of cubic surfaces in some detail, over a field which was mostly
Q
or
R .
This study was
continued by Manin [9] and Iskovskih [7], who obtained a complete birational classification over an arbitrary perfect field
k .
Without going into the details of this classification, one may mention the subdivision into two main types: (a) Del Pezzo surfaces,
which include in particular all
smooth quadric and cubic surfaces in projective space 4 smooth intersections of two quadrics in P, .
P. , and all
K
(b) Conic bundle surfaces,
which can be described as fibra-
tions over a rational curve with general fibre a conic. example is the surface defined in affine space
A,
A standard
by the equation:
y 2 + d z 2 = PCx) where
d e k*
and
(1.1)
PCx)
is a polynomial with coefficients in k . 1 The fibration is given by (x,y,z) w- x e A. j above each point 1 2 2 x e A, , there lies a conic, with equation y + dz = PCx ) . O
OK
Del Pezzo surfaces have been extensively studied with regard to unirationality
([10], chap. 4 ) , a problem which is wide open for
the surfaces of type (b):
it is not even known whether (1.1) can
have a rational solution without having infinitely many.
On the
other hand, there are some results on rational equivalence for conic bundle surfaces ([3];[1], lecture 7 ) , whose analogues have not been fully investigated for the surfaces of type (a).
Concerning the
Hasse principle, the following example is due to Iskovskih [6]:
*This is a report on joint work with J-L. Colliot-Thelene and I-.]. Sansuc l"4i.
238 Example
For
let
c e Z
equation z
2
= (t c - x
2
2 W Jlx -c
V c
•5
be the surface with
1)
This is a special case of equation (1.1).
(1 .2) For almost all
x
e h{
the fibre above
x_ is irreducible. The only exceptional points o = ± /c and above which the fibre is a union of two lines: (y + i z H y - i z ) = 0
As Iskovskih showed in [6], the Hasse principle fails for this variety whenever
c
is positive and congruent to
3 modulo 4 .
other words, (1.2) has no solution in rational numbers although it can be solved p-adically for every prime over
U .
Actually the proof is fairly simple:
characterization of sums of two squares in (1.2) can be solved over
Q
Z
In
(x,y,z) , p
and also
the well-known implies readily that
if and only if the following system
can :
u
v,j = c - x
[1 .3)
2 2 V- = X - C
For
c E 3 (4) , this system has no 2-adic solution.
Iskovskih was interested in (1.2) because he knew how to get 4 from it a (singular) intersection of two quadrics in P which violates the Hasse principle. Our interest in [1.2) stems rather '1
239 is an intersection of two quadrics in V.
has no point with coordinates in
satisfying the Hasse principle!
Ay •
If
c E 3 (4) , then
y« , which is an easy way of
As a matter of fact, our main re-
sult (§2) implies that the Hasse principle holds for (1.3) for every value of
c .
As a consequence we get:
Proposition
Let
c e Z
surface defined by (1.2). only if:
c < 0
cases, the set V
is even
or
and
Then
c E 3 (4)
V(y)
V c A
V
be the conic bundle
has no rational point if and c = 4 n (8m+7) . In all other
or
of ^-rational
points of
V
is infinite:
Q-unirational.
The beauty of this result is that
V , in general, does not
satisfy the Hasse principle, as we saw above. precisely for what values of
c
And yet we can say
the equation (1.2) has a solution!
In fact, this is only a special case of a general procedure, explored systematically by Manin ([10], chap.6), for bringing to light possible obstructions to the Hasse principle over a number field For instance, let us consider the variety
V
k.
defined by the
equation : y 2 + d z 2 = P/|(x)P2(x) ... P R (x) where
-d e k*
(1.4)
is not a square and the polynomials
are irreducible over
k(/-d)
and coprime in pairs.
let us assume also that the degree of each
P.
P.(x) e k[x] For simplicity
is even.
Suppose
now that (1.4) has solutions everywhere locally, which we write: V(k^) * 0
for every place
of varieties of descent
V.
v .
Then one can produce a finite set
such that the following alternative
holds : Either: V.(k ) = 0 .
(a) for all
i , there is a place
place
(b) there exists an
v .
also for
Vi
for which
This condition is equivalent to the Manin
attached to the Brauer group Or:
v
obstruction
Br V , and it implies that i
such that
V(k) = 0 .
V.(k ) * 0
for every
In this case, Nanin's obstruction is empty for .
And if
V^k) * 0
then
V , and
V(k) * 0 .
It is not known whether Nanin's obstruction is the only obstruction to the Hasse principle for rational surfaces.
As the
above alternative implies, it is worth while to investigate the Hasse principle for the very special varieties can be described by a system of relations:
Vi c ^ k
.
They
240 (1.5)
{0 * u? + dv? = a i p i ^ n i ! s 1 ^ ! ^ n with
n II a. = 1 . i=1 1 Since Manin's obstruction is guaranteed to be empty for these
varieties, it is natural to ask whether the Hasse principle holds for them, in which case one has an explicit procedure for deciding whether or not the conic bundle surface (1.4) has a k-rational point.
The results described in the forthcoming section furnish a
positive answer to that question for the case n = 2 , deg P. = deg Py
=
case:
indeed, for
2 .
A positive answer may also be expected in the general k = Q , Colliot-Thelene and Sansuc have shown
that the Hasse principle for (1.5) can be derived from Schinzel's conjecture
H
(see [5]).
But this hypothesis is very much stronger
than the twin prime conjecture.
The results discussed below suggest
that the Hasse principle question is less inaccessible.
On the
other hand, more sceptical mathematicians may find it easier to produce a counter-example to the Hasse principle for a system like (1.5) than to disprove Schinzel's conjecture
H
by a direct attack!
2. THE CLEAN HASSE PRINCIPLE The main result of [4] can be stated as follows: THEOREM. degenerate V c P
Let
k
be a number field,
, <j>2
binary quadratic forms with coefficients
the 3-dimensional be th
in
three nonk .
Let
variety defined by the following pair
of equations: ix,y) (2.1 )
Assume, moreover,
that
$,,
the 'Clean Hasse Principle* point defined over
k
every proper model of In fact,
V
(j)^ are not both isotropic.
holds for
Scholium.
V :
if
, for every completion V
V
(hence
k
V(k)
Then
contains a smooth of
k , then
contains a point with coordinates
is even k-unirational
is
in
k .
infinite).
In order to verify the conclusion, it suffices to
find one model of present case,
and
V
V
on which there is a smooth k-point.
itself has this property.
In the
For details concerning i
the birational invariance of the Clean Hasse Principle, see [4], §3'. This notion clears away a number of pathological situations, like affine plane curves with rational points only at infinity, or
241 projective curves whose only rational points are a few accidental singularities.
A typical flaw in the usual definition of the Hasse
principle is illustrated by the following example, which is due to W. Ellison: 2 - 2 2 2 q u. - 3v1 = 923x + y 2 2 2 2 U 2 ~ 2 = ~ ^^x + y ^
(2
This is a special case of (2.1), in which §x . U
<J>-
-2)
is proportional to
It is a counter-example to the ordinary Hasse principle over
(add the two equations;
3
is not a sum of two squares).
None
the less, the Clean Hasse Principle does hold, as there are only finitely many solutions with coordinates in are all singular.
If
$*
and
<J>2
Q9
or
Q_
and they
are coprime, then the Clean
Hasse Principle reduces to the ordinary one. Remark. false.
If
<j>.
and
$~
are
D
°th isotropic, the theorem is
This can be seen on the following example: u1 - 5v1 = 2xy ' ' u 2 - 5v 2 = 2(x+20y)(x+25y)
(2.3)
for which Manin's obstruction is non-empty, as a local analysis reveals.
Furthermore, a fine study of the invariant
H (k,PicV)
shows the Hasse principle of the theorem to be new, in the sense that no birational transformation can carry the variety
V
into one
for which the Hasse principle is well-known to hold (quadric, Severi-Brauer variety, etc.). 3. SOME WORDS ABOUT THE PROOF (a) The first idea is to replace the pair of equations (2.1) by one quadratic form over
k(t) , where
t
is an indeterminate.
The following result of Brumer therefore plays a crucial role in the argument: THEOREM [2]. ficients in
k .
trivially over over
k
Let
$^ , $ 2
The system
be two quadratic forms with coef-
$. = $ 9 = 0
if and only if the form
can be solved non$,. + t $ 9
is isotropic
k(.t) . Thus it is equivalent to solve (2.1) over ((((u^vj + tcf)(u9,v9) = 4>1(x,y) + t(|)9(x,y)
k
or the equation (3.1)
242 over
k(t) .
There is no strong Hasse principle for quadratic forms
over
k(t) .
So it is not immediately clear what has been gained
from this reformulation. (b) Without loss of generality we may assume that the form
+ dv
with
d e k* .
tive form (also called a Pfister form). (3.1) turns out to be
<}> is of
This is a multiplica-
Now the left-hand side of
$ & t$ , which is also multiplicative, by
general theory (cf. [8], chap. 10). In [12], Pfister proved a kind of Hasse principle over for equations of the type k-form and
f
$ = f , where
$
k(t)
is a multiplicative
a polynomial with coefficients in
fore tempting to substitute suitable elements of
k . k[t]
It is therefor
x
y , so that the right-hand side of (3.1) becomes a polynomial
and f ,
and to ask whether a convenient type of Hasse principle applies to the equation
$ 9 t<|> = f .
If
f
is irreducible, Pfister's method
leads to the consideration of the equation is the class of
t
in the number field
| $ if = 0 , where
k[t]/(f) .
x
This equation
can be handled using the local-to-global principle for number fields. Example. 2
Written homogeneously, (1.3) becomes:
2 2 2 + v^ = cy - x
J^
[3.2) J2
+
v 2 = x 2 - (c-1)y 2
It is of no use choosing
x
and
us back to the original problem. x = x,j + tx 2 where the later.
x.
,
y
in
Q :
this choice would lead
The next simplest possibility is:
y = y^ + ty 2
and the
y.
(3.3)
are suitable integers, to be determined
The problem is then reduced to that of solving: (u 2 + v 2 ) + T ( U 2 + v 2 ) = 0
in
Q[t]/(f) , where
T
(3.4)
is a root of the cubic polynomial
f .
Let
us take a look at that polynomial: f = (x 2 - (c-1)y 2 )t 3 + ... + (cy2 - x 2 )
(3.5)
Notice that if the leading and the constant coefficients of f
are units, then
T
is a unit of
Q(x)
and (3.4) can be solved
243 everywhere locally, except possibly above the prime places at infinity.
and at the
Note also that the places at infinity are harm-
less if all the coefficients of T
2
f
happen to be positive, for then
is negative for every real embedding of
JJ(T) .
The problem is
then reduced to a 2-adic computation, in which the product formula is useful. The main difficulty we are thus faced with amounts to solving the two simultaneous equations: x^ - (c-1)y* = 1
(3.6)
cy^ - x^ = 1
(3.7)
But it should strike any one that these two equations can be solved independently, since the variables are disjoint.
In other words,
Brumer's theorem, together with the transformation effect of separating
the variables1.
(3.3), have the
This is a very important fea-
ture of the general proof. (3.6) is nothing else than a Pell equation, and it has nontrivial solutions whenever
c-1
is positive and not a square.
Of
course, (3.7) does not always admit a solution (it is a mock Pell equation, with a
-1
instead of a
+1 ) .
But the argument can be
slightly modified, so as to replace (3.7) by cy1 - x 1 = p where
p
(3.8)
is a suitable sum of two squares.
the coefficients of
f
For showing that all
can be made positive, one has to use the
fact that (.3.6) has infinitely many solutions.
It goes without
saying that such a proof, based on the properties of the Pell equation, does not generalize easily to an arbitrary number field or to the case of arbitrary forms
$,, , <j>2 -
The forthcoming points
list some of the main tools that are needed for this generalization. (c) Although there is no strong Hasse principle over
k(.t) ,
there are two weak Hasse principles: Proposition
Let
k
be a number field.
The natural homo-
morphism of Witt rings W(k(t) ) where
v
•*
n W(k v (t) ) , v
runs through the set of places of
k , is injective.
244 Hence, if two k(t)-forms are equivalent over place
v , they are equivalent over
class of valuations in ideals
k(t) .
k (t)
for every
But there is another
k(t) , namely those corresponding to prime
(TT) in the principal ideal domain
k[t] .
The following
proposition, due to Harder and Milnor, is more general than Pfister's results mentioned in (b): Proposition [11]
Let
k
zlent over all completions equivalent over
be any field. k(t)
If two kit)-forms are
, then they are equivalent
k(t) Cd) For Pfister forms, a weak Hasse principle is just as good
as a strong one.
This is because a multiplicative K-forrn
resents an element over <J>
K .
f e K*
For suppose
is equivalent to
if and only if
$
f$
represents
f
$
$
rep-
is equivalent to
f$
everywhere locally.
Then
everywhere locally, hence also globally if
the weak Hasse principle holds.
It follows that
$
represents
f
globally. (e) Pell equations are replaced by a powerful theorem, which basically goes back to Hecke.
Such a result deserves to be used
much more widely than has been the case so far in the literature. See C4J, §2, for a more comprehensive version than is given here. THEOREM (Hecke). integers, Suppose
\\)
ty
ideal
k
be a number field,
is anisotropic and primitive.
elements of prime to
Let
A
m .
p c A
A
its ring of
a binary quadratic form with coefficients and
m c A
Let
x
,y
a non-zero ideal such that
Then there exist elements
x ,y
in
A
in
A .
be given i/>(x ,y )
is
and a prime
such that
x = x° mod m
,
y = y° mod m
(3.9)
and (i|/(x,y)) = p In fact
x ,y
and
further proximity y
(3.10) p
to Jbe arbitrarily
archimedean
can be chosen in infinitely many ways and
conditions
can be imposed:
we may require
close to specified elements
valuations
v
is real, the pair
(x,y)
angular region of
kv
except one,
say.
,y
x
, via the embedding 0
and
for all
And, assuming
may also be required to lie in a given
x kv 0
v
x
k *••• k v O
v
245
k
v
(3.10), with ty = (J)^ , should be compared with equation This is also where the assumption that comes into the proof.
(J>.
or
4>2
(3.8).
is anisotropic
The choice of an angular region can be made
so as to ensure, whenever necessary, that the coefficients of the polynomial
f
are all positive.
(f) In (3.8), arily a prime. a prime
p
was a sum of two squares, and not necess-
Now, in the situation of the example, (3.10) yields
p , which - a priori - is not a sum of two squares.
The
argument is therefore wound up by showing that, as a matter of fact, it is!
More generally one shows that the prime ideal
pears in (.3.10) splits completely in the extension call, from Cb), that
<J)(u,v) = u
+ dv ) .
p
that ap"
kt/^Hl/k
(.re-
This is made possible
by a series of consistent choices (in particular the ideal
m
in
Hecke's theorem) and by a final application of the product formula. (g) Once we know that
V
contains a smooth point
ordinates in
k , it is easy to show, that
deed, let
be a k-rational hyperplane section of
through
W P .
Since
V
W
W
such a surface is k-unirational if V(k) .
through
P .
W.
V
In-
passing
4 , and it is known that
Wtk) * 0 .
The k-unirationality
by the same argument, in which we replace the generic element
with co-
can be chosen non-singular.
is a del Pezzo surface of degree
finite, a fortiori
P
is infinite.
has only finitely many singular points, a
theorem of Bertini guarantees that Then
V(k)
k
by
Hence of
W(k) V
k(t)
is in-
is obtained and
of a pencil of hyperplane sections
W
by
246 REFERENCES 1. S. Bloch, Lectures on algebraic cycles, Duke University Mathematics Series IV, Durham, N.C., 1980. 2. A. Brumer, Remarques sur les couples de formes quadratiques; C.R. Acad.Sci. Paris, 286A (1978), 679-681. 3. J-L. Colliot-Thelene and D. Coray, L'equivalence rationnelle sur les points fermes des surfaces rationnelles fibrees en coniques, Compositio Mathematica, 3SI (1979), 301-332. 4. J-L. Colliot-Thelene, D. Coray and J-J. Sansuc, Descente et principe de Hasse pour certaines varietes rationnelles% 3. Crelle, 320 (1980), 150-191 . 5. J-L. Colliot-Thelene and 3-3. Sansuc, Sur le principe de Hasse et l'approximation faible, et sur une hypothese de Schinzel,- to appear in Acta Arithmetica. 6. V.A. Iskovskih, A counter-example to the Hasse principle for a system of two quadratic forms in five variables, Nat. Zametki, W_ (1971), 253-257 (transl. Math. Notes, 1_0 (1971), 575-577). 7. V.A. Iskovskih, Minimal models of rational surfaces over arbitrary fields (in Russian), Izv.Akad. Nauk S.S.S.R. 4_3 (1979), 19-43. 8. T.Y. Lam, The algebraic theory of quadratic forms, Benjamin, Reading, 1973. 9. Ju.I. Manin, Rational surfaces over perfect fields, Publ.Math. I.H.E.S. ^3(3 (1966), 55-113 (transl. Amer. Math. Soc .Transl . 84 (1969), 137-186) . 10. Ju.I. Manin, Cubi'c forms, Nauka, Moscow, 1972 (transl. NorthHolland, Amsterdam, 1974). 11. J. Milnor, Algebraic K-theory and quadratic forms, Inventiones Math. ^ (1970), 318-344. 12. A. Pfister, Sums of squares in the function field R(x,y)j in: Computers in Number Theory, ed. by Atkin & Birch, Academic Press, London, 1971, pp.77-81. 13. B. Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae Univ. Rosario V\_ (1951), 1-68.
ALGORITHME D'APPROXIMATION
DIOPHANTIENNE
Eugene Dubois Departement de Nathematiques Universite de Caen Esplanade de la Paix 14032 CAEN CEDEX RESUME Nous introduisons une notion de meilleure approximation de zero par une forme lineaire une fonction le cas
F
de
n = 2 ,
F
p
+ P/jW. + . . . + p w
(p ,p.,...,p ) .
, relativement a
Dans (II) nous avons etudie
etant une forme quadratique.
Le cas
n = 1
completement resolu par l'algorithme des fractions continues. donnons ici une condition suffisante sur
F
est
Nous
pour que les meilleures
approximations soient de 'bonnes' approximations et nous donnons dans le cas
n = 2
et
F(p ,p<.,p«) = Max(p,.,p~)
un algorithme pour
construire la suite des meilleures approximations
(definition habit-
uelle) . I. DEFINITION ET PROPRIETIES D'APPROXIMATION Soient
1,w.,...,w
des nombres reels superieurs a
lineairement independants, d'entiers. Rn+1
P = (p ,p,.,...,p )
Considerons la fonction
L
1,1}-
un (n + 1)-uple
definie sur
Z
ou sur
par: L(P) • p o • p i W i
+ ... • P n w n
et une fonction distance, F , definie sur pour tout
X ,Y
dans
R
F(X+Y) < F(X) + F(Y) , V e > 0 ,
3C
,
(c'est-a-dire que t
on a
F(X) > 0 ,
F(tX) = |tjf(X) ) et telle que
VC > C
o
R
et tout nombre reel
,
0 < # { P e Z
n + 1
, 0 < L(P)
< e,F(P)
< C} < «
Cette condition est, en particulier, verifiee lorsque la racine d'une forme quadratique positive de rang est une fonction distance ne dependant que de Definition 1 de zero par
(1)
o
P
dans
Z
L , relativement a
n
n
F
est
et lorsque
parametres.
est dit meilleure approximation F
(F - M.A. de 1, w,.,...,w )
0 < L(P) < 1 V P ' e Z n + 1 , P ' * P , 0 < L ( P ' ) < L ( P ) => F(P')>F(P)
si
F
248 Puisque tout
e>0
Soit
Q
1,w>l,...,w
l'ensemble
sont U~ lineairement independants, pour
{PeZ
/ 0
un point de cet ensemble.
0 < L(P) < e , une suite
F(P) < F(Q)}
^k^k>n ' '"^k^
n'est jamais vide. {P e Z n +
D'apre's (1)
est fini.
,
Les F.N.A. forment done
decroit vers zero et
F(P, )
croit vers
l'infini. Graphiquement, en representant repe're
X, = (L(P.),F(P.) )
dans un
(L,F) , les regions (I) et (II) (fig 1) ne contiennent aucun
point image d'un point
P1
Zn+
de
fig. 1 Remarque 1 et
F
ou
Les fonctions
vF
(A , \i , v
XL
(en considerant
A, Aw.,...,Aw )
strictement positifs) definissent les
memes F.N.A. Remarque 2
La definition usuelle correspond au cas
F(P) = H ( p o , P / ]
p n ) = MaxC
Nous ne connaissons pas de methode de calcul pour
n > 1 .
n = 1 , il s'agit de l'algorithme des fractions continues. ment, T.W. Cusick (I) a donne une methode pour ou
1 , w. , vjy
n = 2
est une base d'un corps cubique a conjugues complexes,
nerons au §2 une methode generale pour le cas n > 2
Recen-
dans le cas
a condition de connaitre l'unite fondamentale du corps. pour
Pour
n = 2 .
Nous donLe probleme
reste a ma connaissance un probleme ouvert.
Remarque 3
Si
1,w,,,...,w
sont (IJ- li ne"airement dependants,
non tous rationnels on peut encore montrer l'existence d'une suite en supposant que la fonction L(P) > 0 } .
(L,F)
est injective sur
{P £ Z
,
Dans ce cas l'existence d'un algorithme permettant de
construire les F.M.A. devrait permettre de determiner les relations de de*pendance.
Nais H.R.P. Fergusson et R.W. Forcade (III) viennent
de definir un algorithme resolvant cette question.
D'autre part,
pour la construction de rationnels verifiant le bon degre d'approximation nous n'avions pas a considerer ce cas. Remarque 4
J.C. Lagarias (IV) a etudie une generalisation de
la notion de meilleure approximation sous forme simultanee et
249 relativement a differente norme. Si il existe des constantes et de F telles cque pour tour
L
0 < L(P) < 1
C/j et C2 ne dependant que de P dans Z n + ve"rifiant
on ait
c^HCP) < FCP) < c 2 H(P)
(2)
Dans ce cas la condition (1) est ve*rifie*e et on a: THIzOR^NE 1 .
Si
F
ste une constante il existe
est une fonction distance verifiant (2), c~ = c~(L,F) c~(L,F)
effectivement
calculable
telle que les F-M.A., P, , verifient L(Pk)F(Pk+1)n , o3
(3)
L ( P k ) H ( P k ) n < c 3 c^ n
(4)
On sait d'apres le principe des tiroirs de Dirichlet que 1' inggalite* (5) a une infinite de solution que
P
P e Z
pour
est une X-bonne approximation.
travaux de W.N. Schmidt (V), si geb.riques et si on remplace
n
X = 1 .
w/1,...,w 1 n par
On dit alors
D'autre part, d'apres les sont des nombres al-
n + e
(e > 0)
l'inegalite
(5)
ainsi obtenue n'a qu'un nombre fini de solutions. Pour demontrer le theoreme 1, il suffit de considerer dans n+
IR
le convexe defini par: |L(X)| < L(P k ) , F(X) < c
Ce convexe est un cylindre limite par deux plans ayant pour base un convexe en dimension
n .
V = c L(P, ) . K
est assez grand, il contient, d'apres le
Si
V
Son volume est proportionnel a
theoreme de Minkowski-Blichfeldt, un point entier. F
v^rifient (3) et puisque
^pk^ <
F
(pk+1^
on
a
Alors
P^+i
(4) d'apres (2).
II. ALGDRITHNE Dans le cas de
a = w.
n = 1
et en notant
H(p,q) = Iql , les H-M.A.
sont les reduites du developpement en fraction continue.
Dans (II) nous avons montre que pour toute forme quadratique positive
F
de rang
1
sur
R
, la suite des F -N.A. est la suite
250 des reduites du developpement en fraction continue. illustre alors la propriete de F
a .
si
p/q
est une reduite
L'algorithme que nous defini (II) dans le cas
Z
.
Q,
dans
Pour
n q n+1 " P n + 1 q n successives. II.1
Z
telle que
P, , Q, , P, ,
forment une base
= ±1
si
p
n/qn
et
p
n+1/qn+1
sont
deux
Existence d'une base d'approximation
Soient
1 , w. , w~
THEQRENE 2.
^P^^>Q
Soit
la
(n > 2)
On a
P
ou
vrai pour
(1,0,0) -1
et
Q, „
P. = (p ,p.,p«)
tel que
diviseur premier de •
I
e
Pour tout
formant avec
avec
p. , p^
e*gaux
Le the*oreme est done k1 < k
et con-
K
D, = 1 .
+
I K
+
Sinon soit
t e"tudions les points
P
soit
I K
I
un
de la forme
I
En exprimant
P. . , 0' . , P. K+l
P
K
On veut montrer D L L+
H.N.A.
D, = |det(P, ^Q, 1 ^ , P. D I
K+l
minimum non nul.
Q.
Supposons le vrai pour tout 1
side rons un point
P
auxiliaire
et non tous les deux nuls.
k = 1 .
Les
suite des H-M.A.
k > 1 , il existe une approximation 3 Pk et Pk_/] une base de Z
0,1
rgduites
des nombres reels Q- lineai rement inde*pen-
dants, L(P) = P Q + P ^ + P 2 W 2 ' H ^P-1 = M a x t p ^ p ^ • sont les meilleures approximations au sens habituel.
k+1 "
cP
k
°k+i = c'pi
K+l
+ bQ
+
k
+ b
les composantes de
aP
dans la base
P. , Q, , P. .
K
K
K
-,
K- I
k-i
'°k
+ a p
£P
sur la base
' k-i P, , Q, , P, . K
P
et
n = 1 , ceci correspond a la propriete
p
a
n = 2
forme quadratique repose sur l'existence d'une approximation
auxiliaire de
q|qa-p| < 1
Le theoreme 1
K
sont alors
K— I
est a composantes entieres si et seulement si ces trois entiers
sont divisibles par
I .
Mais le determinant, D. , de ce systeme
251 est multiple de modulo
£
£ .
tels que
II existe done des entiers P
soit entier.
£ ,£ ,&
Choisissons
£.
definis
dans
{-1/2 , 1/2) . Si
£2 = 0
on a
P = j U,,P k + 1
< — L _ H(P) *
+
£3?^
et done:
£_ LL((P P kk)) << L(P L(Pkk )
l ^ | * |£ 3 | £
H(Pk+1) s H(Pk+1)
L'une des deux inegalites etant stricte suivant que H(P) < HtP. +/,)
nul ou non, on a definition 1 pour
Pk+i •
fonction distance
F .
Si
Ce
&~
est
et done une contradiction avec la
t argument est aussi valable pour toute
%2 * 0 , on a
|det(P k + 1 ,P,P k )| - | ^ | D R <- 1 D, et on obtient une contradiction avec le choix de
D,
minimum non
K
nul. Corollaire P
, Q>| , P/i
la suite des F.N.A.
soit une base de
tel que
Qk+1
Pour toute fonction distance
(pk)k>0
et (2) soit
Z
PR , Q k + 1 , P R + 1
F
verifiant (1)
Si il existe
alors pour tout soit une base de
Q1
k > 0
tel que
il existe
Z3 .
Ce corollaire est encore vrai en dimension quelconque. une demonstration identique on montre qu'il existe
CL 1
Par
tels que
P
k-1 'Q k 1 ^ ' "'• ' ^k 11" 1 ^ ' P k S o i t U n e b a S e d e Z n + 1 B L ' n y P o t h e ~ s e que le resultat soit vrai au depart s.e ve*rifie, pour une fonction F
donnee en calculant
montrer que si
F
que de
(° u c'e
p. , p~
completer
P
, P.
Remarque 5
P
et
P. .
ve*rifie (2) avec
D'une maniere generale on peut 2c. ^ c~
P^j ' P7 ' • • * ' P
en
pour obtenir une base de
ou si
dimension Z
(ou
F
ne depend
n ) on peut Zn+
) .
Voronoi (VI) en gene"ralisant l'algorithme des
fractions continues, a defini des minimas relatifs qui correspondent dans le cas d'une base
(1 , a , 3)
d'un corps cubique
ment re*el aux F -meilleures approximations avec W(p
+p/.a + p 2 $ ) / ( p
sur ttj .
+p/.a + p 2 $)
ou
W
L'algorithme de Voronoi ou le
F (p
K
non totale-
, p. , p^) =
represente la norme de F
K
algorithme (Ila) donne
dans ce cas un moyen de calculer l'unite fondamentale de
K .
252 II.2
Methode de calcul (n = 2)
Dans (Ila) nous avons donne une methode de calcul des F-N.A. dans le cas Jp^
+ p^ •
n = 2 ,
F
forme quadratique.
Posons
F,(p ,p,.,p«) =
Nous avons alors:
H(P) < F 1 (P) < H(P)/2
(6)
Nous recherchons P, A sous la forme iP, . + jQ. + j?,P, . P. +/, etant determine, nous obtenons OL + I S O U S la forme enrecherchant i' , j • telsque | ij ' -i ' j | = 1 iP.^^.+jO.+^P. puis f1 par le calcul d'une partie entiere. Notons
a, = L(Q, )/L(P, ) , I
P = P(i,j)
K
= i ( P
a~ = L(P, ,)/L(P,)
K
Z
-a2Pk)
k 1
K~I
JCQk-a1Pk)
+
et ecrivons:
K
+ eP
k
= iX + jY + eP k avec
e = e(i,j)
dans
Parmi ces points H(iX +jY + £P|) . on determine calcule (i,j)
(-1,+1) . P
on recherche le minimum de
En reduisant une forme quadratique comme dans (II),
i^ , j ^
telsque
h = H(i.X + j.Y + £.Pk ) tels que
F^Ci^X + j.Y)
soit minimum.
On
puis on etudie tous les couples
H(P(i,j)) ^ h .
Tous ces points verifient d'apres
(6) F 1 (iX + jY + G P R ) < h/2 et done il suffit de consid^rer les couples
(i,j)
ve*rifiant.
F ^ i X + jY) < B f = h/2 + F1 (Pk)
(7)
On obtient comme dans (II) des majorations de
i
et
j .
II est
alors facile de decrire cette famille, de choisir le plus petit H(P(i,j))
et done d'obtenir
Notons
Pk+1 .
X = ( x r x 2 , x 3 ) , Y = (y 1 ,y 2 ,y 3 ) , A = x^ + x^ = F1 (X) 2 ,
B = x 2 y 2 + x 3 y 3 , C = y2 + y2 = F ^ Y ) 2 ,
D = AC - B 2
2 consid^rant dans
R
on obtient en X
des produits scalaires de
y
2
i(
2
) + j( X
3
des vecteurs unitaires respectivement perpendiculaires a (y 2 .y 3 ) =
) avec y
3
(x-.xj ,
253 | < B f /C7D (8) | < B f /A7D Nous n'avons pas de borne absolue pour ces majorations. peut que
D
soit petit par rapport £
nous avons en moyenne 2 a 3 couples e*tape pour obtenir
P,+ ^
et
A ,B ,C . (i,j)
II se
Dans la pratique,
h considerer h chaque
Q,+ . .
L'organigramme du calcul est le suivant:
j ,k
1)
Lecture des donne*es.
2)
Calcul de
e"gaux a
tableaux
0
PA,QA,P
ou
par 1'6tu.de de
±1 .
On range alors
jw^ + kw~
P. , Q* , P
pour
dans les
P ,Q ,R .
3)
Calcul des constantes
P ,Q ,R .
a,| , a 2 > A , B , C , D
Determination du minimum de
en fonction de
A i 2 + 2 B i j + C j 2 = F 1 (P(i,j)) 2
en conside*rant les premieres re*duites du developpement en fraction continue de 4)
-B/A .
Calcul de
Boucle sur les
(i,j)
HNIN
calcule les deux points
mum.
Determination de 5)
ij
1
- ij
1
Calcul de = ±1 .
Bp . verifiant (8). Le sous programme
P(i,j)
et choisit
H(P(i,j))
mini-
P^+i •
QL+I
en
Rangement de
determinant
i',j' tels que
P k+/] , 0 k + / ] , P k
dans
P ,Q ,R
et
retour a l'etape 3 jusqu'a un test d'arrgt.
Appli cations La methode developpee ici est valable pour toute fonction distance
G
verifiant (2). On choisit une forme quadratique
F^
telle que c 1 /F 2 (P) < G(P) < c 2 /F 2 (P] On applique les etapes 2 et 3 pour
F« .
Dans (7) et (8) et done
dans l'exemple du programme donne en II.b) il suffit de remplacer
Bf
par J- GCi^X^Ve^) • /F^T
.
Ceci nous a permis de trouver un contre exemple a la conjecture de Szekeres.
Soit
0 =
/4 .
L'algorithme de G. Szekeres 1 1 —~ + p. -g-+ p~ . La conjecture 0 est que cette suite contient toutes les meilleures approximations au donne une suite d' approximations
sens habituel.
Or en considerant
nous avons trouve que
(-10,19,-8)
p
H^(p Q ,p^,p 2 ) = Max(|p Q |, |p^| ) et
(3075,4422,-4006)
sont des
254 (1 , 6 , 9 2 )
h^.N.A. de
(1/6 2 , 1/6 , 1)
done des H ^ M . A . de
mais
ne sont pas fournies par 1' algorithrne de G. Szekeres. II.3
Cas des approximations simultanees
On dit que l'entier tanee (M.A.S.) de en notant
6
=
w. , w« Min
q
est une meilleurs approximation simul-
relativement a la norme
NCqw^-p^ , qw^-P2^
et
N
(sur
IR ) si
0 - (q , p^ > V>2^ ^e
PVP2 point correspondent on pour tout
q1
a : 0 < q' < q
on a
6 , > 6
Si ^clu-^k>n o u ^ k ^ k e s ^ ^ a s u ^ e c'es N.M.A.S. on montre comme au theoreme 2 qu'il existe R^ tel que Qj^_>| 'R k ' ^k s o ^ ^ u n e D a s e de
Z
.
Pour la methode de calcul des N.A.S. on utilise les me"mes
idees qu'au paragraphe precedent.
Lorsque
N
est la norme euclid-
ienne la methode est e*quivalente h cells utilisee par Furtwangler Hath. Annalen Bd 99 (1928).
REFERENCE (I) T.W. Cusick, Best diophantine approximation for linear ternary form (a paraitre). (II) E. Dubois, a) Approximations diophantiennes simultanees de nombres algebriques, Calcul des meilleures approximations, These, Paris, 1980. b) Calculation of F-best approximation of zero by a ternary form, Computation of units (a paraitre). (III) H.R.P. Ferguson and R.W. Forcade, Generalization of Euclidian algorithm for real numbers to all dimensions higher than two, Bull. Amer.Nath.Soc. 1_ (1979), no. 6, 912-914. (IV) J.C. Lagarias, Best simultaneous diophantine approximations II, Behaviour of consecutive best approximations (preprint). (V) W.M. Schmidt, Linear forms with algebraic coefficients, J. Number Theory 3 (1971), 253-277. (VI) Voronoi, A generalization of the algorithm of continued fractions, ThSse, Warsaw (1896) (En Russe). Delone-Fadeev, The theory of irrationalities of the third degree, Trans.Hath.Mono., 10 (1964).
Ori the group
PSL 2 (Z[i])
by J. Elstrodt, F. Grunewald, J. Mennicke
0.
The present article is an extended version of a
survey lecture given by one of the authors. It consists of two parts. In the first part, we discuss the connection of the above-mentioned group with elliptic curves defined over Q(i). This connection is almost entirely in a conjectural state. The main conjectures are analogues of the Weil jecture for elliptic curves over
con-
Q . There are links to
topology and to hyperbolic geometry. The second part is concerned with certain Poincare
series which arise natu-
rally in the study of discontinuous groups on hyperbolic 3-space. These Poincare
series are non-Euclidean analogues
of Jacobi theta functions. They are closely related to other classical functions such as Bessel functions. We study the Mellin transforms of these functions and show that they are meromorphic functions in the complex plane. One of these functions possesses a very simple functional equation. The singularities are, as usual, tied up with arithmetical questions. Invoking Siegel's main theorem on quadratic forms, we can produce information about certain sums over class numbers. Using the above mentioned generalised theta function, we can produce an explicit version of Selberg f s trace formula for
r = PSL 2 (Z[i])
Practically no proofs will be given. Parts of the details have appeared in [3], [5]. The remaining details will appear in three subsequent papers [4],[1],[7].
We thank G. Harder for useful discussions on the first part of the following results.
1.1.
We shall fix some notations.
H = H 3 = {(z,r) | z € C , r € R + } = SL ? (C) / SU ?
.
256 H
is a symmetric domain in the sense of E. Cartan.
The hyperbolic metric is given by . 2 dx ds =
/1N (1)
PSL2((E)
2
+ dy 1 r
2
+ dr
2 , z - x + ly
.
is a group of isometries, by the action
(az + e)(yz + 6) + ayr N
\ y 6 J (z'r)
r . 'N } '
6|2 + r 2 |y| 2 , ( PSL2(Z[i]) . d c 5f[i]
is an ideal, N(&) e
its norm.
r | Y = o mod a }
Q(O-) is the normal closure in
r
of the set of unipotent
elements contained in U(<X)
r (CL) . o is the subspace generated by the image of
rationalized commutator quotientgroup V(a) = r Q ( a ) a b 0 Q/u(a)
If
f
U(a) . V ± (a) , IT (a)
is the dimension over
in the
H Q .
, r(a) - dim^(V(a)).
Conjugation by the element and
r (CL)
Q(&)
Q
.j
induces involutions on V(<x)
are their of
± 1 eigenspaces.
^(a)
V~(a) .
a c fa are two ideals let Tra : r (fa)ab B Q -• V ( a ) a b H (Q
be the map induced by the transfer. We have Tra(U(6)) c Tra(U(a)) . Tra We write
V~ ,(a)
also respects the spaces
y_ € V~(flL) so that a conjugate of of some
GL2(Q(i)) 6 => a —
is of the form .Put
V~ .
for the subspace generated by all elements y_ by an appropriate element
Tra(u) with
V 1 (a) = V ± (a)/V ± n Aa.) . new old
u € V*^)
for
257 One of our first objectives of interest was the
1.2.
finite dimensional rational vectorspace its dimension
r(&) . Since one knows an explicit finite
presentation of ideal If
a
r
one may compute
r(&)
for each particular
by solving a finite linear system of equations.
CL = p
is a prime ideal of degree 1 the following table
contains some numerical information on
r(p) .
233
257
277
433
509
569
(1,0)
(1,0)
(1,0)
(0,2)
(1,0)
(0,1)
733
757
853
941
953
977
(0,1)
(1,1)
(1,0
(0,3)
(2,0)
(0,1)
137
P - N(p)
(r+(p),r~(p)) (0,1)
For all other prime ideals we have found In the
V(&) , in particular
PSL2(Z')
the analogue of
p
of degree 1 with
N(p) < 1000
r(p) = 0 . situation it is not difficult to compute r(&)
explicitely. In fact Hecke [10] and
followers have given closed formulae for these dimensions. This analogue also occurs as the dimension of an Eichler cohomology group, see [15] appendix. In our case the number r(p) , p
a prime ideal , reappears
in
many different forms. It is the first Betti number of a certain compact, closed 3-manifold
[3], [5]. For a prime ideal
p
of
degree 1 it is also the multiplicity of the irreducible representation of dimension where
p = N(p)
of
PSL ? (Z/p2)
in
N*
8
is the full congruence subgroup of modulus p [3]. + ± Note that V(Z[i]) = <0> . Hence V~(p) coincides with V~ (p) for every prime ideal p .
1.3.
N
Here we wish to discuss the construction of the action
of the Heckealgebra on prime ideal of
V (Ci) and
Z[i] , generated by
U (a) . Let q G Z[i] .
q = (q)
be a
258 Put
6 • 6(q) = f
J
. Consider the diagram
[ Tra (rQ(a) n 6 o Tra
Q
Q
is the transfer map, 6 the map induced by conjugation
with
6
; in
is the homomorphism induced by the inclusion.
We define then T(q) = in ° 1 « Tra It can be verified that Furthermore ing of
T(q)
V~(a)
and
T(q)(lT(a)) 5 IT (a) .
induces a map in the quotient v
a
-ij( ) • Hence
T(q)
V(&)
respect-
induces endomorphisms
U (a) , V (a) , V + e w (a) . All these do not depend on the
choice of the generator T(q) all
.
q
in the sequel. Put T(q)
for q . They will all be called
H(a)
for the algebra generated by
in the ring of endomorphisms of
V
also possible to define the Heckealgebra on
w (&)
V""(&)
• It: i-s and
U (a) .
In this situation one has to make an appropriate choice of the generators above for
q , see[4]. For a construction similar to the PSL 2 (Z)
see [15] appendix. For reasons explained
in the next section we can now prove:
Theorem 1.3.1.
H(&)
is a commutative, semisimple algebra.
It is diagonalizable over a totally real extension of
The letter
V
will in the following always stand for a one-
dimensional eigenspace for generated by a vector
The
a
Q .
H(&)
in
V
(a) . I f
V
is
v , then
are rational integers by construction. Our setup
makes it clear that given a computed. For
V , the
a = (8 + 13i)
existence of an eigenspace
a
can be effectively
the table in 1.2 gives the
V .
259 We give here a few examples of the corresponding eigenvalues
11
(q) -2
-4
-2
l-4i
19
-4
35
-3
-10
a. .
-3+2i -3
-3
23
More examples are discussed in [ 3] ,[ 4] . To
V
we associate
a Dirichlet series by the Euler product
(2)
L(V,s)
By elementary methods it can now be proved that Theorem 1.3.2.
L(V,s)
converges if the real part of
s
is big.
In fact what one proves here is
|a | _< cN(q)(N(q)+1) , [3l .
By analogy with the results for
PSL_(Z)
one expects that
Using more technique we can establish some further facts about the Dirichlet series
L(V,s) . To do this one uses the Weil,
Jacquet, Langlands theory for
GL.(C) . We shall now explain
the information one gets from this approach. Let x Heckecharacter of
be a
Q(i). For our purposes we interprets
as a homomorphism of the group of fractional ideals of into the multiplicative group of only x If
which are of type
A
x Q(i)
L(s) = I a N(a) is a Dirichlet series write L (s) OL —s ^ the series £ a x(&)*N(a) . Furthermore define A(V,s) = (2TT)"2s(N(0))s/2-4S-(r(s))2-L(V,s) .
for
260 Theorem 1.3.3. (i)
can be continued to a holomorphic function
L(V,s)
on the whole of
strips, (ii)
A(V,s)
satisfies the functional equation
A(V,s) - ±A(V,2-s) with a certain choice of sign, (iii)
Every
L (V,s) can also be holomorphically continued A
to the whole of
C . The continuation satisfies a
functional equation similar to the one in (ii). In the next section we shall explain how to derive this result from the theory of Weil, Jacquet, Langlands. This will also clarify the exact meaning of (iii). Consider now a one dimensional eigenspace
U
T(q)
u. and
in U (a) . Say U
is generated by
with a G Z . Define a Dirichlet series
for all Heckeoperators
L(U,s)
T(q)u. = a u
by the formula (2).
One can then prove that L(U,s) = L(x1,s)-L(x2,s) , where Xi»Xo of
are
appropriately chosen nontrivial Heckecharacters
Q(i) . The series
L(x,s)
is defined as
rV1
L(X,S) = n where
&
is the conductor of the Heckecharacter
x»
L(U,s)
satisfies then (i), (ii) of Theorem 1.3.3. but not (iii). 1.4.
Here we explain briefly
U (a) and the various
L(V,s)
the relation of
V (a),
to the theory of automorphic
forms of Weil, Jacquet, Langlands. We give a translation into the language of [18]. For this it is convenient to introduce the group
r Q (a) = {(^ fje PGL 2 (z[i]) | p * o mod
a}.
261 Note that
r Q (a) ab B Q - (r Q (a) ab s Q ) + , where the
+
denotes the
induced by conjugation with
+1 eigenspaces for the involution (_
. ) . We write
H (T (a),C)
for the first Eilenberg, Maclane cohomology group of with coefficients in
C .
^(a)
also acts as a discontinuous
group of isometries on 3-dimensional hyperbolic space H (H/F (a) , c) with
r (
H .
stands for the cohomology group computed
r (a) - invariant differential 1-forms on
the contractability of
H
H . From
and from de Rham's theorem we
deduce an identification
Put
H (H/r o (a),C)
1-forms on
for the space of harmonic, f (a)-invariant
H . A main result of [8] says that the obvious map
H1(H/ro(a),(c) - H 1 (H/r Q (a),c) is an isomorphism. In [8] there is also given a decomposition
HVH/I^W.C)
= H^usp(H/ro(a),c) $ H!nf(H/ro(a),c) .
In our special case the cuspidal part is the subspace consisting of classes of compactly supported differential forms. The
inf
component is generated by certain Eisenstein classes. The duality between homology and cohomology gives then isomorphisms
( v V ) S c)* = H^usp(H/ro(a),c) (u+(a) 0
262 Let
A(a,M ) be the space of (h,a)-automorphic functions
as defined in [18]. Here
M
stands for the unique irreducible
3-dimensional representation of U(2), the maximal compact 3 3 subgroup of GL2(C) . AQ(a,M ) c A(a,M ) is the subspace consisting of functions which are cuspidal in every cusp of fQ(a) . For definitions see [18],[12].
The discussion in [18]
and the above shows that there are isomorphisms
p :
< V a > a b * c >* "> A(a,M3)
Po:
(V+(a) B c)* - AQ(a,M3)
If A n e W (a,M )
.
stands for the space of new forms in the
sense of [12], then we have also an isomorphism
Using the results of [12] and the isomorphism
p
one gets
then Theorem 1.3.1. Weil associates to every automorphic $ € A(a,M3) a Dirichlet series Z($,s) . From [18], [12] + * it follows that if y_ € ((T (a) 8
function
for all Heckeoperators then L(
y_ £ (V (<x) El C)*
one has to use the converse
theorem in [11] to obtain (iii).
Here we
would like to add some remarks on the converse theorem
in our situation. Suppose one has a Dirichlet series
L(s) = n (1-aH N C ^ r V 1 n (1-a H
q\a
qXa
convergent in some halfplane. Assume also
a £ 2 . Suppose
one knows (i), (ii) of Theorem 1.3.3. Assume further that (iii) is satisfied for all Heckecharacters to
x
with conductor prime
a. Weils converse theorem [18],[16] implies then that
263 L(s) • L(V,s)
for an eigenspace
V <= (?QW
® Q)
• If one
knows (iii) for all Heckecharacters
x
theorem implies that
L(s) = L(V,s)
for an eigenspace
If for example
is the product of two Heckecharacters
over
Q(i)
L(s)
or a
then Langlands converse V c V (a).
Heckecharacter of a field extension then one
is in a position to employ one of the converse theorems.
1.5.
In this part we discuss the possible connection of
the eigenspaces
V c V
(a)
Q(i). An elliptic curve
E
to elliptic curves defined over
defined over
Q(i)
is represented
by a cubic equation 2 3 2 E : y + a.xy + a y = x + a«x + a.x + a. where the of
a.€ Z[i] . Let
E . By
C(E,s)
jj(E) c Z[i]
be the conductor ideal
we denote the Hasse, Weil £-function of
E
It is defined as an Eulerproduct
b qJ : ' 1 ) " V 1 ' n with appropriate
b
"S1"28"1
(l-b
€ 2 . Note that the typical Eulerfactor
is the same as in (2). It is a general belief
that
£(E,s)
has the same analytic properties as stated in Theorem 1.3.3. (i) and (ii), see e. g. [13].
One has to distinguish 3 cases: (i)
E
has complex multiplication by an order in Q(i).
Then Xt,Xo
£(E,s) = L(x,,s)*L(X2>S) °f
V
two
Heckecharacters
Q(i) • By application of the converse
theorem one finds that eigenspace
f°r
c(E,s) = L(U,s)
for an
U cr U (fl(E)) . It is not possible for a
with this property to occur in
V (tf(E)), since
L _j(U,s) cannot be holomorphically continued to X l the whole of C .
264 (ii)
E
has complex multiplication by an order in
QC/^d) Then
with
Q(/^d) $ Q(i).
C(E,s) = L(x,s) , where
character of
C(E,s) = L(V,s) (iii)
E
X
is a Hecke-
Q(/-d,i) . One finds then that for an eigenspace
V 5 V+(<J(E)).
has only trivial complex multiplications.
Nothing is known on this case, but Weil C16] ,C18] expects that
C(E,s) = L(V,s)
for a
We want to concentrate now on the case where prime ideal of such that
E
i(E) = p is a
ztil . The following table contains all
^(E)
and so that
V c= V + «$ (E)) .
is a prime ideal of degree 1 with
N(p) £ 1000
has a representing equation with "small"
coefficients
a..
0
l+i
i
i
0
233
0
257
E
l
E
2
1
-i
i
-i
E
3
i
i
1
5-i
-l-3i
257
4
i
i
1
5+4i
-12-41
257
E
5
I
-i
i
85-6i
34+274i
257
E
6
i
l-2i
0
277
E
7
l+i
-i
i
1-i
0
509
E
i+i
E
8
l+i
-i + i
1
0
-2
757
E
9
l+i
-i-i
1
-1
0
853
The curves
E
E«, E , E,, E^
are isogenous. Of course there is
an obvious coincidence of this table with the eigenspaces in V (d)
in the table of 1.2. We have also checked that the first
few hundred Eulerfactors of
£(E,s)
and of the corresponding
265 L(V,s)
coincide. For details see [4].
We have carried this computation further and also considered the fields
Q C ^ ) , Q(/^3). From our tables [4] there could
be a correspondence between the sets {Vc V (p)
is a 1-dimensional eigenspace for
p is a prime ideal in
H(p), and
Z[i]}
and {E
is an elliptic curve defined over
Q(i)
with
a prime ideal}
so that the Dirichlet series of corresponding objects are the same. Note that curves falling under (i) cannot have prime conductor!
For the eigenspaces
V c V (p)
listed in 1.2 we have also
found elliptic curves corresponding to them in the above manner. But here the conductor of the corresponding curve is not 4 but (1+i) *p . For example the curve
p
y 2 + (l+i)xy = x 3 - (l+i)x - 1 satisfies
fj(E) = (1+i) p
first few hundred
b
where
N(p) = 137. Furthermore its
coincide with the eigenvalues of the
(appropriately defined) T(Q)
on the eigenspace mentioned in
1.2.
1.6.
Here we would like to make some general comments
on the correspondence between the eigenspaces and certain abelian varieties defined over
V c V
(a)
Q(i) . All of the
following is inspired by the Eichler, Shimura theory and the Weil conjecture over
Q . See [2] for a survey of these results.
266 The first question arising is
Problem 1 Let
E
be an elliptic curve defined over Is there a
v
v
E
with
S. * e w <^( ))
(E
'
Q(i) with conductor s)=
L(V
>s)?
From the discussion in 1.5. it is clear that one has to assume that the endomorphism ring of 2[i] . In this case
E
does not contain an order of
L(E,s) = L(Xj,s)*L(x 2 ,s)
of Q ( i ) , and L(E,s) = L(U,s) for an'eigenspace
for two Heckecharacters U c U (a) .Otherwise
our tables suggest that the answer to problem 1 is 'yes'«
Problem 2 Given an eigenspace defined over
V c V (<x) is there an elliptic curve — new Q(i) so that L(V,s) = S(E,s)?
E
The answer here is 'no1. We shall explain this phenomenon in the case of an explicit example. If defined over £-function of End (A)
Q(i) we write A
A
is an abelian variety
C(A,s)
computed over
for the Hasse,Weil-
Q(i) , see [14] for a definition.
stands for the ring of endomorphisms of
over the extension
K
A
defined
of
Q(i) . We shall describe the 2 5 of the Jacobians of the curves y = x + x.
To do this we give two Heckecharacters
Xi> X 2
1
Put
0_2 = 2 + z/ ? . The group {O^/h^f
Let
G
• 0^)*
X,((x)) = x
This describes a
define
be the subgroup generated by the images of
it has index 2 . For an ideal (x) with
Let e
of
if
X2 = Xj
#
.
Heckecharacter of conductor
e . Put
Lj(s) - L(x,,s)-L(x 2 ,s)
3, 1+/-T ,
(/-2) )( (x) put
x G G
be the nontrivial character of
C-function
4 # /-T »0_ 2 .
(^.o/^'^-o^*
anc
*
267 2 then
£(Jac(y
5 = x
2 + x),s) = (Lj(s)) . This phenomenon is
described in more detail in [6]. On the other hand there is an eigenspace Note that
VcV+((l+i)12)
L,(s)
so that
L(V,s) » Lj(s) .
has the correct decomposition as an
Eulerproduct over
Q(i) . This result is obtained by appli-
cation of the converse theorem as explained in 1.4. On the other hand one can show that there is no elliptic curve
E
defined over
Q(i)
with
L,(s) =
C(E,s) .
There is also the possibility to consider a two dimensional abelian variety and
A
defined over
Q
so that
Z[i] c End.^v(
End-,(A)
that
is a quaternion algebra. Then it also happens 2 L(A,s) = (L(s)) where L(s) has the correct type of
Eulerproduct and should perhaps come from an eigenspace V c V (a) . We owe this construction to P. Deligne. These and other examples suggest that one has to modify problem 2.
Problem 2 T Given an eigenspace
V c V
(a) neW
A
defined over
A
satisfies
Q(i)
so that
Z[i] c End
(i)
is there an abelian variety ry
L(V,s)
(A)
and
= C(E,s)
and such that
rk^CEnd^A)) = 4?
Note that all counterexamples to problem 2 mentioned so far seem to be only possible if
a
is not a prime ideal.
Of course there is then
Problem 3 Given an eigenspace
V cz V
(a)
constructing a corresponding
A
is there a possibility of geometrically?
There is the suggestion in [16] to try to use the periods of a differential form
GO representing a generator of
V . One
can also bring the periods of the differential 2-form
*u
into play. But these seem to be too few data. Another approach could be to represent
A
as an emersed surface in
H/T^a) .
268 The example of elliptic curves with complex multiplication over Q(i) seems to suggest this. In section 1.8. we report on some further hints in this direction. 1.7. Here we want to report on a congruence relation for Heckeoperators stressing the correspondence discussed in 1.5. and 1.6. We discuss this phenomenon in the case of a prime ideal p with N(p) - 257 . In general there is a homomorphism cp : r Q ( p ) a b ->
defined by
••CO Let
Ker cp be its kernel and let
subgroup of
r
a o(P)
Tor(p) be the torsion
• Put
A(P) " r Q (p)
/ (Ker ip + Tor(p)) .
A computation shows that (T(q) - N(q) -l)v € Ker cp for every v_ € V (p) . Again a computation shows that A(p) is cyclic of order 4 for a prime ideal p with N(p) = 257 . This gives
4 I <•, - M(O-0 if a is the corresponding eigenvalue of T(c?) . The curves E2, E^, E^, Ec appearing in 1.6. also satisfy the same congruence with a replaced by their b . This arises from the fact that Eg is the Klein 4 group as group of Q(i)-rational points.
269 In each case we found a curve with nontrivial torsiongroup, we were able to prove the corresponding congruence for the Heckeoperator eigenvalues by the above methods.
1.8.
The quotient space
Y (p)^H
has two cusps.
Q
— —
We have
ro(P) This implies that the cusps are bounded by a sphere, not by a torus as in the general case. The quotient
r (p)^H has a
unique compactification which is a closed oriented compact 3-manifold. We have studied this phenomenon in [5l. We use the notation
We have r-p'(Mp) . By Poincare
duality, we have r = P 2 (M p )
.
Hence, in a r35! case, H_ (M ,2?) •
is a free cyclic group,
F . It is natural to look out for a
generating 2-cycle which has additional structure.
There is an abstract version a combinatorial setup. M
T
Mf
of
M
which is defined in
is a union of cells of dimension
0,1,2,3. This presentation is quite explicit, the details are given in [3]. Helling has proved that homeomorphic:
Mf
and
M
are
270 In fact, there is a translation which carries the combinatorial Mf
structure form
to
M
. There is a proof of Helling's
theorem in [19]. It is not difficult to show for the first few r = 1 cases, for Np = 137, 233, 277, that there is an abstract surface
FJJ c M f
which is of genus 2 and which generates
H ? . Using the above 3 homeomorphism, one can carry F' to H . The result is a 3 . surface F c H which has some additional structure. We have carried out the study in one particular case. Here is the result: Proposition 1.8.1.
Consider
T (p) . There is a surface
(i)
p = (11 - 4i) , Np = 137, and
Fc H
with the following properties:
F = F. U F 9 .
.
3
is the union of two geodesic surfaces
F ,F c H
(ii)
r (p)^F
is a closed surface of genus 2 .
(iii)
r (pNF
generates H 2 , i. e. = H 2 (M ,2T).
(iv)
The intersection of of
r Q (p)
F.,F?
with a fundamental domain
are Riemann surfaces of genus 0 with
5 holes each. (v)
The surface
T (p)^F
has 5 edges, corresponding to
the above holes, where the adjacent
2-cells inter-
sect under constant angles a = y (vi)
There is a group
G < V (p)
such that
(NF = rQ(pKF , i. e. certain elements of on G
F
G
identify certain edges
such that the result is a closed surface.
has the abstract presentation
It seems that in our case, H«
5
= T
6=
does not have a generator which
is a smooth closed geodesic surface. Rather, the above object seems to be natural. We do not yet know how to relate it to the elliptic curve corresponding to
p
mentioned in 1.5.
F
271 2.1
Let
P » (z,r), QI » (zff,rff) € H 33.
The metric
(1.1 (1)) defines the
hyperbolic distance d:
Cos d(P,Q) = 6(P,Q) 6(P,Q)
:
. J « ' | *
+
r 2
+
2rrf Let
T < PSL2(E)
shall take
be a discrete subgroup.
r = PSL 2 &[i]).
For
Unless mentioned otherwise, we
g €r
6) the expression
6(P,gQ)
is an Hermitian form of the matrix coefficients
+ |YZ' + 6|2 r 2 + |Y| 2 r 2 r'2} . For
P = Q = (0,1), this reduces to
It is natural to introduce the Poincare series
S*(P,Q,t) -
For
P = 0 = (0,1), and
2 .
V = PSL2(#[i]), this reduces to
. | \ a,3,Y,<S € Z [ i ] a6-$Y = 1 Notice that
(3)
3*(P,Q,t)
has as its Mellin transform
H(P,Q,s) = I —±
— .
g€r 6(P,gQ)S+1
272 It is not difficult to see that the series compact sets for
(3)
converges uniformly on
Re s > 1, and has a singularity for
s = 1, if
F \ H
has finite volume. 2.2
We shall now list a
Proposition •^(PjQjt)
S*( p ,Q,t).
few elementary properties of
2.2.1 converges uniformly on compact sets for all
t > o.
Moreover,
we have S*rP,Q,t) « 0(e~ Kt ) Proposition
2.2.2
for
t -> oo, for some
K > o.
We have d*2(t,p,Q)dv(p)
r
< oo.
rsH Here
dv(P)
denotes the hyperbolic volume element dv(P)
=
dx
<*y dr . r3
Proposition
$*(P,Q,t)
satisfies a hyperbolic differential equation:
L p %* = t 2 d*t + 3t %* - t 2 %*.
(4) Here
2.2.3
L
• L
denotes the Laplace operator with respect to the metric
V
2
ay2
dr*J
The last term on the right hand side of
(1):
8r
(4), which makes the right hand side
into a differential operator of Bessel type, results from the summation over V in the definition of
Proposition
2.2.4
finite lim sup as
%*.
The function
t2d*(P,Q,t)
has a positive lim inf and a
t •* o.
This proposition holds for arbitrary discrete groups the hyperbolic volume of
F^H
is finite.
r < PSL2((C)
such that
273 Proposition
2.2.5 j **(P,Q,t)dv(Q) - ^
Kj(t) .
K,(t)
is the usual modified Bessel function.
2.3
In order to produce more information, we have to make recourse to
spectral theory. Let
L2(r v H 3 ) denote the space of functions which are invariant under integrable over
T and square
r ^ H . As usual, this space, is made into a Hilbert space
by defining the inner product
The Laplace operator
L
j f g dv(P)
f,g
f,g €
is symmetric with respect to this inner product.
for
for f,g €
in the domain of L. There is a unique extension of L
a selfadjoint operator on
which is
L2(]>H).
The relevant Eisenstein series can be defined as follows:
(Y»6) = 0 ) The series converges for Re s > 1. Using the Hecke representation, one can easily see that continued into the. complex s-plane.
E(P,s)
on the imaginary axis and satisfies a functional equation.
(5)
can be
It is a meromorphic function without pc
j E(P,iu) ydy € L2(P4i)
One has
274 The space
L^CFNH)
(6)
has the following decomposition:
f
2
The space
L«
The space
L^ 11 ^
is the smallest closed subspace containing all funtions (5). is the closed space spanned by all eigenfunctions of
It turns out that all
e
L:
can be chosen such that they are an orthogonal
system of cusp forms. For
f € L2(I>H), the decomposition
(6) implies a
decomposition
/a
e (P) + J a(y) E (P,iy)dy .
Here the coefficients The decomposition
a and a(y) are the Fourier coefficients of n (7) converges in the Hilbert norm sense.
If f is in the domain of
L, it can be shown that
f.
(7) holds also pointwise,
and even uniformly on compact sets.
2.4.
Using the setup discussed in
2.3, one can establish the following
decomposition Theorem
(8)
2.4.1
«*
} K. (t) en(P) en(Q)
E(P,iy)E(Q,-
The theorem amounts to a computation of the Fourier coefficients. Although none of the eigenfunctions
e
is known explicitly, the computation can be
carried out using the method of Selberg transform.
The computation of the
Fourier coefficient for the Eisenstein part is a routine matter.
275 Theorem
2.4.2 lim t 2 **(P,Q,t)
(9)
Theorem
2.4.3 for
Note that the right hand side of
(9)
is independent of P
and
Q.
2.5 In 2.1, we introduced the Mellin transform H(P,Q,s) of %*. The decomposition theorem discussed in 2.3 yields the following result: Theorem
2.5.1
For
H(P,Q,s) «
/
*—
, we have
This result implies at once: Theorem 2.5.2 The function H(P,Q,s) has an analytic continuation into the whole complex s-plane. It is a meromorphic function. There is a pole of order 1 at s = 1. There are poles at s = iy n , A = - 1 - y 2 . The remaining poles are obtained by translating the imaginary axis to the left by an even integer.
276 2.6 It is natural to consider solutions of the Laplace equation which depend on the distance to a fixed point only: 5 - 6(P,Q), (10)
Q
fixed
Ltp • Acp.
The solution of (10) is a hypergeometric function, which in our case happens to be elementary:
tps(6) « 7=Lz. (6
X We can use Poincare summation over this function, obtaining F(P,Q,s) = ^ «> 8 < P ' 8Q) '
(11)
Note that F has a singularity for P = Q. The series Re s > 1. We call it a MaaB-Selberg series. The decomposition theorem yields Theorem
2.6.1
(11) converges for
The MaaB-Selberg series has the expansion
o T O I ^ T + ^ 7 ^ 2 en(P) (12)
+ I 8
—00
If s is on the imaginary axis, the path of integration must be shifted such that the two poles lie either both to the left or both to the right. The expansion
(12) converges in the Hilbert norm sense.
Using this result, and the result of
2.5, one can prove
277 Theorem
2.6.2
The Maafi-Selberg function
tinuation into the with poles at F(P,Q,s)
whole complex s-plane.
s - ±\
and
iy ,
F(P,Q,s)
has an analytic con-
It is a meromorphic function
= - 1 - y2.
*
satisfies the functional equation F(P,Q,s) = F(P,Q,-s).
We can identify the Maafi-Selberg function with another object. Theorem
2.6.3
The Maafi-Selberg function
F(P,Q,s)
is the kernel of an
integral operator which is, up to a constant factor, the inverse of the operator 2.7
- L - A . Thus
We shall now
Write
S*
F(P,Q,s)
is what is known as the resolvent kernel.
discuss an application of our results to number theory.
in the form 00
nt
d*(t) «= y c(n) e~2
c(n) -
Using elementary techniques, one can tie up the multiplicities objects from the integral theory of quadratic forms.
Lemma:
Consider the quadratic forms f * x2 + y2 + u 2 + v 2 g = (n+2) £ 2 + (n-2) n 2 ,
and the representations of X • (u,v), (13)
t
/n+2
g
by
f:
u,v € TL , o
** " V o n-2 satisfying the side conditions
c(n) with
278 u « (uu), v (14) u. = v 1
c(n)
. mod 2,
i - 1,2,3,4.
O~"l
equals the number of representations
On the other hand, the form
f
(13) satisfying (14).
has only one class in its genus.
Siegel's main theorem can be used to compute
Thus
c(n). One has to work out the
local densities and invoke Dirichlets formulae for class numbers.
The
result is Theorem
2.7.1 for
n = 0 mod 8
for
n = 1 mod 2
c(n) 32H(4n -16) For
n = 0 mod 2, put
n -4 s
m
N,
N
square-free, m
odd.
Then 48H (n -4)
for
n = 4 mod 8,
0
for
n = ±2 mod 8, N = -1 mod 8
for
n = ±2 mod 8, N = 3 mod 8
for
n = ±2 mod 8, N = 1,2 mod 4.
c(n) 2 192H n -4 2e+2
96 H
Here
H(M)
binary
counts the number of equivalence classes of positiv definite 2 2 2 quadratic forms (a,b,c) = ax + bxy + cy of discriminant -M « b - 4ac.
Notice
that non-primitive forms are included.
The form
(a,a,a)
is counted
with weight •=•. Now invoking the Tauberian theorem of Karamata, we deduce from Theorem
2.4.2:
279 Theorem
2.7.2 N
This seems to be a new result on a certain type of sums over class numbers. One can also use Theorem 2.8.
2.4.3
to produce a weak remainder term.
In this last section we shall discuss another application of our
results. On the diagonal
P = Q, the function
fundamental domain of
§ (P,Q,t)
is not integrable over the
r. However, using the theory of theta functions, it
is quite easy to isolate the terms which produce divergence of the integral.
Proposition 2.8.1: The function f S*(P, P, t) - |£ e"* r 2 for d (P, P, t) - < [ &*(P, P, t) otherwise
(15)
is integrable over the standard fundamental domain of mental domain of
T =
PSL^CZti])
standard fundamental domain for Now, on the one
r >1
T. The standard funda-
is the three-dimensional analogue of the PSL 2 (Z).
hand, we can integrate the expansion (8) of Theorem 2.4.1.
On the other hand, we can evaluate the integrals over the various conjugacy classes of
T. The result is a trace formula.
Theorem 2.8.2:
For
r = PSL (Z[i]), we have the following trace formula
(16)
IT
-t /
^- e 2t
+
E
^ y + 2 log
v
(
V
\
2—
r(|) '
2B0
•
g e"C log 2
+
£ e"C log 2n
{Y primitive}
; 8 , loB[N(Y)| - • nn, 2 2 I n=l t|N(Y)n-N(Y) n
-2n
e
|N(Y)| * e
t, 2n y
- 2((£ £
8 , log e
n=l 3t(e n - e S 2
++ EE
-2n. > y
£
C,(s),
p
~
are counted with their multiplicities. The function
£
+£
-2n. -2n > "e
0 < Re s < 1. The zeros H,
(t)
Bessel function, in the notation of [20], p. 112. The number E. (t)
(
runs through the zeros of the
k = Q(i), in the critical strip
Euler constant. The function
2
(2nr2n
n=l
Here we use the following notation, function
t, 2n 1 16ir 6ir log e
is an associated y
is the
is the exponential integral function,
see [21], p. 342. A hyperbolic or loxodromic element if it is not the power of another element of
Y € r
is called primitive,
r. The summation over
Y
runs
over a set of representatives of F-conjugacy classes of hyperbolic and loxodromic primitive elements. In jugate to
{ ^
G = PSL-(C),
such an element
° ), |c| > 1, Re £ > 0. Put
Y € r
is con-
N(Y) = £ . The number
\0 C V e = 2 + /31 is the basic unit in
Q(/3).
C =j
+
^ 3 * is a third root of
unity. Remark:
Comparing the formula (16) with Selberg^ formula in the case of
Fuchsian groups [22], p. 74 and p. 78, or with Tanigawa's formula [23], or with Venkov's formula [24], we see that all these formulae are equivalent to our explicit formula. They differ from our formula by an appropriate integral transform, which in our case happens to be a Lebedev transform, see [21], p. 398. Our method undoubtedly applies in more general situations, but we have not yet studied generalisations.
2B1
R e f e r e n c e s
[1]
Elstrodt, J.; Grunewald, F.; Mennicke, J. : Spectral theory of the Laplacian on 3-dimensional hyperbolic space and number theoretic applications. To appear
[2]
Gelbart, S. : Elliptic curves and automorphic representations. Advances in Mathematics 2\=, 235-292 (1976).
[3]
Grunewald, F,; Helling, H.; Mennicke, J.: SL (0)
over complex quadratic numberfields I.
Algebra i Logica JJ 512 - 580 (1978). (= Algebra and Logic 17, 332 - 382 (1978)J [4]
Grunewald, F.; Mennicke, J.: SL (0)
and elliptic curves.
To appear
[5]
Grunewald, F.; Mennicke, J»: Some
3-manifolds arising from
PSL 2 (Z[i]).
Archiv fur Mathematik, 35, 275-291 (1980).
[6]
Grunewald, F.; Mennicke, J.: The
c C - ffunction of the curves
y
2
= x
5
+ Dx.
To appear.
[7]
Gushoff, A.C.; Mennicke, J.; Grunewald, F. Komp lex-quadrat is che Zahlkb'rper kleiner Diskriminante und Pflasterungen des dreidimensionalen hyperbolischen Raumes. To appear.
[8]
Harder, G.: On the cohomology of In:
SL 2 (o).
Lie groups and their representations.
Edited by I.M. Gelfand. London, Hilger, 1975.
282 [9]
Harder, G. : Period integrals of cohomology classes which are represented by Eisenstein series. Preprint
1980
[10] Hecke," E.i tlber ein Fundamentalproblem aus der Theorie der elliptischen Modulfunktionen. Abh. Math. Sem. Univ. Hamburg 6, 235-257 (1928) (= Mathematische Werke, 525-547).
[11] Jacquet, H.; Langlands, R.P.: Automorphic forms on Springer
GL(2).
LNM 114 (1970).
[12] Miyake, T.: On automorphic forms on
GL«
and Hecke operators.
Annals of Math., 94, 174-189 (1971). [13]
Serre, J.P.: Le probleme des groupes de congruence pour SL^. Annals of Math. 92 (1970), 489-527.
[14]
Serre, J.P.: Facteurs locaux des fonctions zeta des varietes algebriques. Seminaire Delange - Pisot - Poitoti
[15]
(1969/1970).
Shimura, G.: Introduction to the arithmetic theory of automorphic funtions. Iwanami Shoten. and Princeton Univ. Press (1971).
[16] Weil, A.: Zeta - functions and Mellin transforms. Proc. of the Bombay Coll. on Algebraic Geometry, pp. 409-426, Bombay 1968 (= Oeuvres Scientifiques, Vol.Ill, pp. 179-196). [17] Weil, A.: On a certain type of characters of the idele-class group of an algebraic numberfield. Proc. Intern. Symb. on Algebraic Number Theory, pp. 1-7, Tokyo-Nikko 1955 (= Oeuvres Scientifiques, Vol.11, pp. 255-261).
283 [18]
Weil, A.: Dirichlet series and automorphic forms. Springer LNM 189
[19]
F. Grunewald, J. Mennicke: SL.(o)
[20]
over complex quadratic number fields II, (forthcoming).
Y. Luke: Integrals of Bessel functions, New York, 1962
[21] W. Magnus, F. Oberhettinger, R.P. Soni: Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edition, New York, 1966 [22] A. Selberg: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, Journ. Indian Math. Soc. 20 (1956), p. 47-87 [23]
Yoshio Tanigawa: Selberg trace formula for Picard groups, Proceed. Int. Symp. Algebraic number theory, Tokyo, 1977, p. 229-242
[24] A.B. Venkov: Expansion in automorphic eigenfunctions of the Laplace-Beltrami operator in classical symmetric spaces of rank one, and the Selberg trace formula. Proc. Steklov Inst. Math. 125 (1973), p. 1-48.
J. Elstrodt Mathematisches Institut der Universitat Minister Roxeler Str. 64 4400 Minister F. Grunewald Sonderforschungsbereich "Theoretische Mathematik" der Universitat Bonn Beringstr. 4 5300 Bonn J. Mennicke Fakultat fur Mathematik der Universitat Bielefeld Uhiversitatsstr. 1 4800 Bielefeld 1
SUITES A FAIBLE DISCREPANCE EN DIMENSION s H. Faure
1 INTRODUCTION. 1. 1 Le tore a s dimensionsT
est identifie au cube unite
rS
[0,l[ ; un pave P delT est le produit de s intervalles [a^9b. [ de s s [0,l[: P « II [a, , b, [; son volume est |p | - II (b, - a, ) . k-1 k-1 Etant donnee une suite X - (X ) de points d e T S , on note n n A(P, N, X) le nombre de termes de la suite X, d1indices inferieurs a N,
qui appartiennent a P ; on pose alors : E(P, N, X)= A(P, N, X)-|P|N, puis D(N, X)=
sup |E(P, N, X ) | et P€$ s
D*(N, X) -
sup |E(P, N, X ) | , s
ou ^S est l'ensemble des paves d e T la forme IT [0,b,[. k-1
et S
lTensemble des paves de
Les fonctions D et D* sont respectivement la discrepance et la discrepance a l'origine de la suite X ; elles sont reliees l V m e a 1'autre par les inegalites : D* < D < 2 s D*. Dans la suite, l?entier s est toujours suppose au moins egal a 2. 1.2 Le comportement asymptotique des fonctions D et D* quand N augmente indefiniment a ete etudie par de nombreux auteurs : K.F. Roth a d'abord etabli une minoration generale [4 ]: II existe une constante K s on ait, pour une infinite de N : D*(N)>K
s
(Log N ) s / 2 .
telle que pour toute suite infinie
285 Puis J.H. Halton a construit des suites infinies en dimension s, generalisant la suite de Van Der Corput, pour lesquelles il obtient les majorations [ 1 ] : D*(N) < A (Log N ) S avec Log A = 0 (s Log s ) . s s Ensuite I.M. Sobol a obtenu de nouvelles suites infinies, basees sur le systeme de numeration binaire, qui verifient [5 ]: D*(N) < B (Log N ) S s
avec
Log B
- 0(s Log Log s ) .
s
Tous ces resultats figurent dans un article tres detaille de H.Niederreiter [3 ] qui fait le point sur la question et qui contient une bibliographie complete jusqu'en 1977. 1.3 En travaillant avec les systemes de numeration en base quelconque, nous avons construit des suites pour lesquelles nous obtenons les majorations suivantes : D*(N) < C (Log N ) S s
avec
C
s
= °O).
Ces suites ont les plus faibles discrepances actuellement connues (voir la table en annexe). Remarquons qu f on ne sait rien pour l f instant de l!ordre exact de
D*(N) pour
s > 2.
2 DEFINITIONS ET RESULTATS. 2.1 Soit r un entier au moins egal a s (pratiquement r sera toujours un nombre premier). On appelle pave elementaire en base r un pave" de la forme
s
\
n [— k- 1 P k r Pk u, < r
de r
+
\
* [
9 p
avec u. et ^
9
k
p, entiers positifs ou nuls, et k
r pour tout k.
Soit m un entier positif ou nul et X = (X-,... ,X ) une suite r points d e T ; on dit que X est une suite de type P si tout r ,s
pave elementaire en base r de volume r seul de la suite X. Soit X - (X ) une suite de type P
>
r ,s
contient un terme et un
. une suite infinie d a n s T
; on dit que X est
si, quelquesoient m et £ entier positifs ou
286
nuls, la suite finie X = (X , ..., X ) est une m £r m + 1 (I* l)r m suite de type P r,s 2.2 Principaux risultats. Tteor&me 1. Quels que soient les entiers s > 2
et
r
premier au moins e*gal
d Sj il existe des suites de type P
r ,s De t e l l e s suites, notees S^ , seront definies au paragraphe 3 K S
ci-dessous. TMordme 2. Quels que soient les entiers s > 2, r premier au moins egal ds-\etm>0, il existe des suites de type Pm . r ,s Ces suites finies sfobtiennent aisement a partir des suites de type P
™
r,s
Remarques : Les suites de type ?^ « s °nt les LP Q ~ suites de"T
2 introduites
par I.M. Sobol1 [ 5 ] , suites egalement etudiees par S. Srinivasan [ 6 ] . Les suites de type P™ o T
et P
^ 3
sont
les
P
0~
rgseaux
de
"^"
et
consideres par Sobol 1 . ThSoreme 3. (i) Pour toute suite X de type P 2
2,
on a la majoration :
s--(Log N ) 2 + 0(Log N) pour tout N > 1.
D*(N,X)<
16(Log 2 ) Z (ii) Soient s > 2 et toute suite X dje type P r ,s
r impair (premier) au moins 6gal a s ; pour on a la maj'oration :
D*(N,X) < -i-(-Hli )S(Log N ) S + 0((Log N) S " ! ) pour tout N > 1. s! 2 Log r Remarques : 2 Les suites de type ?^ 2
sont
les
lesquelles Sobol1 obtient le majorant
LP
o"su^"tes
d e
^
Pour
*-= 1,04..., alors que
2(Log 2) Z
287 3
j-0,39....
16(Log 2) Soit q
s
le premier nombre premier au moins egal a s ; alors le
1 qs " * s majorant C = — ( ) tend vers 0 quand s tend vers lTinfini ; S s! 2Logq s les suites de type P sont dfautant meilleures que s est grand alors r ,s que les majorants associes aux suites de Halton et aux LP - suites de SobolT tendent vers lfinfini avec s. En adaptant la methode qui permet de remplacer (r-1) par dans les majorations, on peut diviser par 2
la constante obtenue
par Meijer [ 2 ] pour les suites de Halton (j) 8
ainsi :
r-1
; on obtient 1'" # *' g s
_ .
D*(N,cj>
) < (Log N ) 8
8
l'"*' s
S
)+ 0((Log N) 8 " 1 ) .
n (-^
k=l
2Log g k
Les demonstrations des theoremes ci dessus ainsi que dfautres resultats annexes seront publiees dans un article a paraitre prochainement.
3 DEFINITION DES SUITES s£ . K
S 3.1 Soit C = (( )) la matrice des coefficients binomiaux (matriP ce triangulaire dfordre infini). Pour tout entier t > 1, on a C = (tn P ( n ) ) ; en particulier C = I (mod t ) , I etant la matrice unite. 3.2 Soit r un entier au moins egal a 2, et soit A
des reels
d el a forme k r 00
s'ecrit x = I
avec n > l e t O < k < r
I1ensemble
; s i x € A
r -i-l x. r J (avec les x. tous nuls sauf un nombre fini), J
J 00
-i-l i on definit y = Cx (mod r) par y = I y. r J avec y. = I (.) x.(modr). J J J X j-0 i>j J f 3.3 Soit alors n > 1 et n - 1 = I a.(n)r l ecriture de (n-l)en j-0 J base r ; posons x = I a.(n)r n ja0 j
J
, puis x = C n
x
n
(mod r) pour
1 < k < r. Si s est un entier compris entre 2 et r, soient rj,...,r g3 s entiers compris entre 1 et r ; posons R = (r.,..-., r ) ; nous definissons
si
s
288 r alors la suite S
, -— g a termes dans IT par : s
s En particulier si s = r et r = k , on obtient la suite de terme general S (n) = (x ,. . . ,x ) , et dans le cas r = 2 , n retrouve une L1?Q~ suite etudiee par Sobol' (6.4 [5 ] ) . La demonstration du theoreme 1 se ramene a montrer qu f un determinant, generalisant le determinant de Van Der Monde, n f est pas nul modulo r. Le theoreme 2 se deduit facilement du theoreme 1. Pour le theoreme 3, on distingue les cas r = 2 et r impair ; ce dernier cas repose sur une methode de reduction par recurrence portant la longueur de la sequence X
(cf. 2.1) et sur la dimension s.
289 ANNEXE Comparaison des oonstantes A , B et C . Pour les suites de Halt on (f)
ou p, est le k-ieme nombre
s pk " 1 premier, on : A = II ( ) (voir la derniere remarque suivant le S k=l 2Logp k theoreme 3 ) . Pour les suites LP
de Sobol', on a B =
T
S
T 2S . /T
, ou T est n\ S
S
s s!(Log 2) un entier dont lTordre de grandeur est compris entre et Log Log s sLogs(4.5 [5 ]); les premieres valeurs sont les suivantes :
s
2
3
4
5
6
7
T
0
1
3
5
8
11
Pour les suites de type P
9
8
9
10
11
12
13
15
19
23
27
31
35
avec q premier nombre premier
qg,s
s
superieur ou egal a s on a : C
=
=• 16(Log 2)
et
) s pour s > 3.
C =— (— S s ! 2 Log q s
On en deduit le tableau suivant : s
2
3
4
A
.65
.81
1 .25
2.62
6 . 13
17.3
52.9
90580
B
1 .04
1 .00
1 .44
1 .66
3 .20
5.28
15.2
647
.099
.024
. 0041
.0088
s
C
s
.39
. 12
5
6
•
018
7
8
13
.000010
290 BIBLIOGRAPHIE [ 1 ] HALTON (J.H.).- On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerisohe Mathematik, 2, p. 84 - 90, 1960.
[ 2 ] MEIJER (H.G.).- The discrepancy of a g-adic sequence, Indag Math., 30, P. 54 - 66, 1968.
[ 3 ] NIEDERREITER (H.).- Quasi Monte-Carlo methods and pseudo-random numbers, Bull, of A.M.S., 84, n° 6 , p. 9571041, 1978.
[ 4 ] ROTH (K.F.).- Irregularities of distribution, Mathematika, 1, part. 2, p. 73 - 79, 1954.
[ 5 ] SOBOL1 (I.M.).- On the distribution of points in a cube and the approximate evaluation of integrals, U.S.S.R. Computational Math, and Math. Physios, 7, n°4 , p. 86 - 112, 1967. Voir la bibliographie de [ 3] pour les autres publications de I.M. SOBOL 1 .
[ 6 ] SRINIVASAN (S.).- On two dimensional Hammersleyfs sequences, J. of. Number Theory, 10, n°4 , p. 421 - 429, 1978.
CANONICAL DIVISIBILITIES OF VALUES OF p-ADlC L-FUNCTIONS By Georges GRAS
1. INTRODUCTION AND GENERALITIES The classical theory of p-adic L-functions of abelian fields may be regarded as a purely analytic one (p-adically) ; if so, one obtains results of the kind that we recall later (Res. 1 to 5), which constitute a synthesis of congruence properties of Bernoulli numbers. However, this theory contains important arithmetic aspects, which are independant of the p-adic ones, and due for instance to the fact that the values at s = 0 and 1 of these functions are connected with the order and the structure of certain groups attached to the arithmetic of abelian fields . We recall the main facts : 1. 1 Arithmetic aspects Analytic formulas. We write u^v when u v~
is a p-adic unit.
If k is an imaginary abelian field, and if &(k) denotes the pm (k) , ,v g r o u p of r e l a t i v e c l a s s e s of k, w e h a v e | # ( k ) | ^ p ° 11 - B ( oc ) ,
a
z
where oc runs over all odd primitive characters of k, p ° Q(k) |M(k)| (Q(k)
=
' «
1 or 2 is the unit index of k, M-(k) is the group of
roots of unity contained in k). Recall the relation which expresses Bernoulli numbers in terms of the L -functions : P
-iBl(a~1)(
I-CTV))
= i L p ( O , •) , H f - e o T 1
(1.1)
where 9 is the Teichmuller character mod q - 4 or p according as p - 2 or not. If k is real,
let ?T(k) be the Z -torsion module of the Galois
group of the maximal abelian p-ramified p-extension of k ; then (see [ C ] , App. I, a n d [ G 5 ] , th. 2. 1) we have I ?T(k)| «
n
p
o(k)
rr II
1 = L (1, ty), where ty runs over all non trivial primitive
P *^1 n (k) characters of k, p = [ k^O^ : Q], where Q^ is the cyclotomic
Z -extension of Q. P Conjectures on abelian fields. Beyond these relations, we have the "main conjecture11 which expresses a remarkable link between L
-
functions and the structure of certain modules of Iwasawa's theory (for a statement see [ c ] ,
5) ; it is connected with the fact that par-
tial products of Bernoulli numbers (resp. of L_ -functions at s = 1), corresponding to irreducible p-adic characters 0, express the o r der of suitable groups depending only on 0 : &(0) in the odd case, ^ ( 0 ) in the even one (for the definitions, see [ G i ] , I, 2, and [ G 5 ] , 3, 5 ; see also 2. 1 and 3. 1 in §§ 2, 3). More precisely : Conjecture 1. 1. If 0 is odd (and if, for any a | 0 , \|/ = 6 a~ m
character of Q^), we have
#(0)| ^ p
,
(0)
11 -x B A&
is not a
v
J, where
m (0) = O except in the case p = 2 and cc is of p-power order, in which case m (0) = 1. o Conjecture 1. 2. If 0 is even (and not a character of 0^), we have
\z(0)1
« n ^ L (i,\u). 1. 2 Analytic results To simplify, we consider only primitive characters. For such
a character j3, we call : vR the valuation, of the ring Z (f3) of v a lues of P, such that v« [Q (P)J=Z,
p
(resp. P ) the component of
p-power order (resp. of order prime to p) of P, Kg the f i e l d fixed by the kernel of P , and f g the conductor of P. If c i s a rational p r i me to p, we put c 9
(c) = 1 + q p
u, (u, p) = 1.
Let ty be an even character , and let s be any element of Z Result 1. 1. I f \ | f ^ 1 ,
:
then v , (^ L (s, \|i)) ^ 0 except i f \|i i s a charac-
ter of QTO, in which case v ( - L (s, \|/)) ) = -1. Result 1. 2. (see [ R ] ,
4, [ G 3 ] , IV, cor. 2). If * f 1 then a N. S. C.
to have v ^ ( I L (s, <|))^ 1 i s : v.
( I L (s, %J) ^ 0 or there exists
a prime number Z ^ p , Z \ f, , such that <; (£) = 1.
293 Result 1. 3. (see [ R ] , 5, [G33, cor, to prop. V4). If \|f is of p-power order then a N. S. C. to have v. ( 1 L (s, \|i)) = 0 is that /
,
/
\
P
£
S ^
pn(C)
=1
p
( e | f r e^p). Now, introduce the i n v a r i a n t Mty) (see £ R ] , 5, or [ G 3 ] , prop. V2) : Result 1.4. (see [ G 3 ] , prop. V 2, V 3, V 4 ) . We have M t ) = M
P n ( £ ) ( « l f ^ «^P» * o ( € ) s s i ) J M * o ) ^ 0 i f
have \(\|/)>1 i f and only i f v . ( | L (s, T|l))^1. Result 1. 5. ( s e e [ G r ] ) . If
in Z (ty), when ty is
of order divisible by p. We do not know how to obtain these divisibilities by analytic methods in full generality ; the problem of knowing if it is logically possible or not remains open (notice that the two approaches are very distinct : the "analytic" side is based on the study of explicit p-adic measures ("of Stickelberger"), with, in addition, a very deep property of "regularity" implied by F e r r e ro - Washington's result ; the "arithmetic" side is based on class field theory, on the complex analytic formula of class numbers, as well as the fact that Leopoldt's conjecture is true for abelian fields). Of course, in the area of the theory developed, at the present time, by Mazur and Wiles, for the "main conjecture" in the abelian case of composite conductor, such divisibilities will probably appear, in so far as this theory contains, in some sense, an "explicit" form of class field theory. In all the sequel, ty is an even primitive character of order divisible by p ; moreover we suppose that \|f is not a character of Q^ because in this case everything is known. 2. CANONICAL DIVISIBILITIES AT s = 0 Define a=6 \|f*~
and let 0 (resp. W) be the p-adic character
294 above oc (resp. oc ). Let L be the field K
and let K be its subfield
of index p. The norm map N,
/ K induces an exact sequence (where C£ de-
notes the class group in ordinary sense) : Next,
1
(2. 1)
we have the formula of ^-classes fixed by Gal ( L / K )
( [ G 2 ] , th. I I 1) :
where u^ = 1 for pf 2 , u^ = | ( ^ E ( K ) n N L , K L
/E(K)ZJ
|
for p = 2 ,
E(K) is the group of units of K ; D(£) is the degree over Q of the splitting field of £ in the p-subextension of L/Q,
and r =0 except in
the case i|f = 1, in which case p = 1 or p is the index
cp cp As |C€ ( L / K ) I is a divisor of | C £ ( L ) | , it follows, from ^P( 1) ( r +S D(£)) i 2. 1 and 2. 2, that p is a divisor of | #(0)| (in the case p = 2, we neglect the index u^ which does not have any canonical divisor) ; then we have the following general conjecture which is a direct consequence of conjecture 1 . 1 : Conjecture 2. 1. Let \|f be even and not a character of O^ ; let a be the primitive character 8 \|r~ .
T n e n
v a
( o B i *a~ ) ) - - ^ - r
+
^DU)
( \ ^ \^€ |f a , CCQ(6) = 1 J, where 6 = 0 except in the case p = 2 and t = 1, in which case 6 = 1 . By 1. 2, this gives a lower bound for v J - L
( 0 , \|f)J*
In some cases, conjecture 1. 1 is proved, and conjecture 2. 1 is then true ; for example : Theorem 2. 1. When the p-adic character above oc Q, then v a ( I B 1 ( a
j1
))>-6-r+E
D(£)
3. CANONICAL D I V I S I B I L I T I E S
is rational over
(t \ f a , CLQU) = i ) . ATsa1
Now we call 0 (resp. ®) the p-adic character above ty (resp.
The principle is then the same as for the odd case ; it is ba-
295 sed on the analogue, for the groups fc , of the formula of invariant ^-classes. The properties of the groups ^ have been studied in [ G 5 ] where we have shown the existence of a norm map N. an extension map J i /
K
/ ^ and
Un any extension L / K of real abelian fields)
which have, formally, the same properties as the analogous maps for class groups ; especially, as with relative classes, J i /
K
is i n ~
jective (this fact needs the exactness of Leopoldt ! s conjecture for abelian fields). If we apply this to L / K with L_=K ^ and K subfield of index p in L,
we obtain the exact sequence :
1-
r
(0)- ^ ( L )
9
-
X (K)*-1
(3. 1)
and the formula ( [ G 5 ] , th. 3. 2 and 3. 3) :
(3.2)
where D(6) is the degree over Q of the splitting field of I in the psubextension of L / Q , and,
and P = 0 or 1 is defined by : p = 1 if % ^ 1 ,
for * o = 1, P =Min(^n o (k)+ 1 , - . . , n(£) + f ( £ ) , • . . ) - n Q (k)
e|f . , £ ^ p ) , where p f ( £ ) is the residual degree of I in L / Q . Then we remark that | ^ - ( L / K ) and,
| is a divisor of | X (L) | ,
cp( 1) (p -1+E D(£)) consequently, by 3. 1 and 3. 2, p is a divisor of
1*2" (0)1 ; thus we have the following general conjecture which is a direct consequence of conjecture
1.2:
Conjecture 3. 1. Let % be even and not a character of O^,. Then ^
^
p
p
i
r
o
A particular and interesting case where conjecture 1. 2 is t r i vial is the case ty - 1 (it is also true when *P is rational over Q) : Theorem 3. 1. Let ty be even of p-power order and not a character of Q^,
then v^ ( I
L p ( 1 , i|f))>P - 1 +
S
D(£) ( e | f ^ f €
4. REMARKS ON THE VALUATION OF ~ Lp(s, It is easy to find examples where the canonical divisibilities give, or, on the contrary, do not give the exact valuation of
296 1 L (s, \|/) at s = 0 or 1: z p Example 4. 1. Consider the case p = 2, with \|/ of order 2 and s = 1 : An elementary computation, involving the 2-adic analytic formula of class number, shows that for all integers m of the form m = 4t u2 + - 1, t ^ 1 , u odd, we have, for the character i|f such that K, = Q(V"m) : , ( 1 L 2 * 1 > * v = l ~ 1+ x V h ^ ' But f o r such a c h a r a c t e r , ( i J. \
where h is the
class number of Q(\[m).
t h . 3. 1 g i v e s v , ( ^ L
( 1 , \|/)) > p - 1 +
I! ^ v ^ ' ^ i l i ' Z/2), and, in general, equality does not occur (es£^2 * pecially as soon as t i s large enough). However there exists a particular and remarkable situation for which the divisibilities of conjectures 2. 1 and 3. 1 are verified, and in some sense, the best possible ; the proof which needs F e r rero-Washington's result i s analytic, and this gives a partial answer for the problem raised in the introduction : Example 4. 2. Replace % by the character % Y , n ^ O , where Y is a character of order p of Q^ ; we know that for n large enough v, (o L ( S >^Y )) = M 1») i s independant of n and s 6 Z and that we n
have Mi|l) = X(\|/ o )+ T.
pm
io\ /
\
' (£ | f. , £ f p , \|iQ(£) = 1 ) (Res. 1.4).
We
w i l l compare this expression of X(ty) with the values p r e d i c t e d by conjectures 2. 1 and 3. 1. B e f o r e , we remark that f o r n large enough, D(£) = p n ( ^ f o r I ^ p ; f u r t h e r 9(£)= 1 f o r all Z \ f , £ f p. (i) D i v i s i b i l i t y at s = 0. Here
we have a = 9 \|f \ ~
1
,
L=K
a
and K subfield of index p i n L . Case ty r \.
Then 6 = r = 0, and conjecture 2. 1 p r e d i c t s
S D ( £ ) , thus \(\|f ) +
2
a
D ( £ ) ^ Z D(£) ( « | f a ,
o
^> = i) J
then we see the appearance of the f o l l o w i n g phenomena : i f cCQ(p) = 1 , we obtain \(\|J ) ^ D(p) = 1 ; show that this i s not a c o n t r a d i c t i o n and f o r this v e r i f y d i r e c t l y that X(\|i ) ^ 1 : f o r p f 2 , implies L
(0, \liQ) = 0, and M ^ ) -
1 (see Res.
the r e s u l t of [ G r ] r e c a l l e d i n Res.
1.5,
1 . 4 ) ; f o r p = 2, i t i s
which can also be shown
a r i t h m e t i c a l l y : by 1. 1, we have \ L_ (0, \|j ) = ^ £
2i
1.1
O
O0Q = iif^ 1 h e r e ) ; c o n s i d e r t h e e x t e n s i o n L ' / K 1 ,
B, (9f
2
1
L'=K
Q
) (because O
,
, K'=K, o
by 2. 2 a n d the f a c t that 2 s p l i t s i n K1, we s e e that
; o
297 1
I C£ (L'/K ) I |C£(K') I is an even integer, therefore the quo• cp op tient C8(L_') / j f / , C£(K!) is not trivial ; then it is sufficient to i
/
\
apply th. 1 of CG4] which implies that the valuation of ^ B A$ ty J i s not zero ; thus \(\|i ) — 1. Case \|i
= 1. If p = 2, r = 1 ; i f p ^ 2 , as ty i s not a character
of Q ^ , there exists £ r p totally ramified in L/Q(M-(K)),
and an easy
computation shows that, for n large enough, M-(K) i s not contained in the group of local norms at Z (in L / K ) , therefore, r = 1. Then the lower bound given by conjecture 2. 1 i s - 6 - 1 +E D(£) yZ I f a , oc (£) = 1J : for p r 2 , 6 = 0;asCC
=9 here , p i s excluded
from the summation, and the lower bound i s Mty), b e c a u s e \ ( f )=-1 here. If p = 2 , then 6 = 1 and oc
=1 ; therefore the term D(p) = 1 ap-
pears in the summation and the lower bound i s s t i l l X(\|i). (ii) D i v i s i b i l i t y at s = 1. Here we have L_ = K, Y
, and K sub-
f i e l d of index p in L_. Case ty ^ 1. We have P = 1, Mty ) —0,
and the expression for
X(\|i) proves immediately conjecture 3. 1 for the characters tyY (n large enough). Case ty = 1 . An elementary computation shows that P ( c o r r e s ponding to tyY ) equals 0 for n large enough ; th. 3. 1 implies X(\1O>-1+
E
p n * ' ( e | f , £ ^ p ) , but X ( \ 1 / Q ) = - 1 , and the lower bound
is stil I an equality : Theorem 4. 1. When n i s large enough, conjectures 2. 1 and 3. 1 c o r responding to the characters \|/Y
a
^e true.
In conclusion, we point out the following problem, justified by the study of the above example ; this will be studied in another paper : problem. Let \|f be even and not a character of Q^. Is it true that v
\li \5 ^" ^ S j w ^ s DOunc ' ec '> independently of s € Z , by
pi
r
£r
o
P
5. THE CASE OF P-ADIC ZETA FUNCTION We will prove a canonical divisibility for its residue at s = 1 ; it is not based on those (conjectured) for L -functions, but uses
298 a direct proof. It is interesting to note that the divisibility obtained is exactely what would give the application of conjecture 3. 1. If k is a real abelian field, C (s, k) = fl L (s, \|f) , where i|i runs over all primitive characters of k ; the residue at s= 1 is given by: Res P
C k : Q ]
(k)»(i - i ) 2 P
-
1
n ^
1
k z
(1,*)
(5.1)
p
Now introduce the groups ^ (x)> relative to the irreducible rational characters of k, for which we have the formula ([G5] , n (X) , th. 3. 1) : | Z (X) I ~ P fl 5 L n (1, * ) , where n (%) - 0 except p
• Ix
in the case X character of Q^, in which case n n ((X) - 1. We have n 1 J^ 1 I LP (1,\|l)« n |T(X)I P , thus :
f1
XT 1
P
- n (k)
°
n
T 1 Decompose k in the form k = k'k
TT(X)I
(5.2)
([k 1 : Q] p-power , [k
: Q]
prime to p) ; for all subfields L of k cyclic over k , we call K its subfield of index p, when L r k . There exists a semi-simple deX ^ (L) = © ^C (|_) ° , X
composition of the form
j. ^o
running over all
*
rational characters of k ; if L f k , and if X (L/K) denotes the kernel of N. / ^ : % (L) "* "J(K), we have the exact sequence Xo 12r*(L/K) - ^ ( L ) ° - T(K) X ° -- 1 , and it is easy to verify * )XXo - 12r
that *% (L/K) ° = T ( X
)> where X
ding to the field ( k ' n L y K
is the character correspon-
. Then we have, for each L ^ k
:
o
| T ( D | I r ( K ) ! - 1 = n I T ( X J : ) | ; therefore T] \Z (X)| xo xt 1 Tl
lT(X)|
TJ
\Zi\-)\
mula giving | T ^ L / K ) ! pP(L)"
] +t(L)
,
i T t K ) ! " 1 . Now we use the global f o r -
([G5],
Rem.3 . 5 ) : I T ^
where P(L) = O or 1 is computed in k ' n L / Q
with reference to the unit character, and where t(L_) is the number of prime ideals of K which ramify in L/K and do not divide p. Then
299 T ( D | | T(K)T1
is a multiple of pPU-)-1+t(L)^
a n db y
^
1 a n d
5. 2, we obtain (where v is the usual valuation on Z ) : Theorem 5. 1. Let k be a real abelian field, and let k
be its maxi-
mal subfield of degree prime to p. Then the residue at s = 1 of C (s, k)
verifies : v^Res (k)) ^ - 1 - nQ(k) + E (^(L) - 1 + t(L)j +
([ k : 0] - i)v(2), where L runs over all the subfields of k cyclic nQ(k) over k and distinct from k , n (k) is defined by p ° = [k^f^:®],
P(L_) = O or 1, and t(L) is the number of prime ideals of
K (subfield of index p in L) which ramify in L/K and do not divide P. Remark. Res. 1. 5 allows us to add, to the right member of the above inequality, the term (d-i)v(2), where d is the degree of the splitting field of 2 in k .
REFERENCES J. COATES, p-adic L-functions and Iwasawa's theory, Proceedings of Durham Symposium (1975), Academic Press, 1977, p. 269-353. [Gil
G. GRAS, Etude d'invariants relatifs aux groupes des classes des corps abeliens, Asterisque,4i-42, 1977, p. 35-53.
[ G 2 ] G. GRAS, Nombre de ^-classes invariantes, application aux classes des corps abeliens, Bull. Soc. Math. France, 106, 1978, p. 337-364. [ G 3 ] G. GRAS, Sur la construction des fonctions L p-adiques abeliennes, Sem. Delange-Pisot-Poitou, 20e annee, n° 22, 1978/79. [ G 4 ] G. GRAS, Sur I'annulation en 2 des classes relatives des corps abeliens, C. R. Math. Rep. Acad. Sci. Canada, 1, n° 2, 1979, p. 107-110. [ G 5 ] G. GRAS, Module de torsion de la p-extension abelienne pramifiee maximale d'un corps abelien reel et fonctions L padiques (a paraitre). [Gr]
R. GREENBERG, On 2-adic L-functions and cyclotomic i n variants, Math. Z. , t. 159, 1978, p. 37-45.
[R]
K. RIBET, p-adic L-functions attached to characters of ppower order, Sem. Delange-Pisot-Poitou, 19e annee, 1977/78, n° 9. Georges GRAS Universite de Franche-Comte - Faculte des Sciences Mathematiques, E. R. A. au C. N. R. S. n° 070654 F - 25030 Besan^on Cedex
MINIMAL ADDITIVE BASES AND RELATED PROBLEMS George P. Grekos Department of Mathematics University of Crete Iraklio, Crete, Greece
1. MINIMAL ASYMPTOTIC BASES Let
h
be a positive integer.
A subset
B
nonnegative integers is said to be an asymptotic
of the set
if all but finitely many natural numbers are sums of arily distinct, elements of is called minimal of order
h .
B .
An asymptotic basis
if no proper subset of
B
H
of
basis of order
h
h , not necessB
of order
h
is an asymptotic basis
Minimal additive bases have been studied by Erdos,
Hartter, Nathanson, and Stohr [3, 4, 6, 7, 9-12, 14, 15, 17, 2 0 ] . 2. THE ORDER OF A MINIMAL BASIS Erdos and Nathanson [12] raised the question whether it is possible for a set
A
to be simultaneously a minimal asymptotic
basis of two different orders. totic basis of order then
A
h
A
is a minimal asymp-
is certainly a minimal asymptotic basis of order
proved in [12] that if a set then
If the set
and an asymptotic basis of order
A
A
k < h , k .
It is
is an asymptotic basis of order
cannot be a minimal asymptotic basis of order
words, there is no minimal asymptotic basis of orders
2 ,
4 ; in other 2 , 3 and
4.
We prove the following result. THEOREM. and
There is no minimal asymptotic
basis of orders
2
3 . This means that a minimal asymptotic basis of order
a minimal asymptotic basis of any order
k > 2 .
2
is not
To prove the the-
orem we use the following lemma. LEMMA. (A.1) 0
Let
A
belongs to
be a set of nonnegative
(A.2) for any two elements that If
A
a*
of
A
a , a'
of
does not belong to
is an asymptotic
element order
a - a'
integers such that
A , and
basis of order
the set
A \ {a*}
A
with
a > a' > 0 , we' have
A . 2 , then for every nonzero is an asymptotic
basis of
3 . Notation .
If
A ,B
are nonempty sets of integers, let
A + B
301 denote the set consisting of all sums of the form and
b e B .
A +b
If
B
instead of
integer
has a single element, say
A + {b} .
h > 2 ,
We also define
a +b
where
a £ A
B = {b} , we write
2A = A + A
and, for any
hA = A + (h-1)A .
Proof of the lemma.
Let
a*
be a positive element of
A .
We
shall show that the set B = A \ {a*} is an asymptotic basis of order have that
2B
is a subset of
3 . 3B .
Since
0
belongs to
B , we
Hence, it suffices to show that
all but finitely many elements of the set C = N \ (2B) belong to
3B .
If
C
is finite, there is nothing to prove.
Let
C - {c r c 2 ,...,o.....} where
Since
A
large
c.
for all
is an asymptotic basis of order belongs to
2A .
But
i > i^ , we have that
c. c.
2 , every sufficiently
does not belong to
2B . Hence,
is of the form
c. = a ( i ) * a* a(l) e A ,
where
Now, if B to
with
a(i) 1
a
of
A .
lement of
c.
is large enough, we can choose an element
0 < b < c.
C , then
or
a(l) > 0 . and such that
c. - b = c.
c. - b. > c^
for an index
.
If
c. - b
b
of
belongs
j , that is
• a* - b • a ( J ) • a* ,
=b +a
J
, where
a
But this is not true. C
in
N
is
cA - b - b n • b 2
, b , a So
J
c. - b
2B ; therefore
are nonzero elements is not in
C .
The comp-
302 where
b. , b~
are elements of
B .
It follows that
c^ = b + b^ + b 2 e 3B for all large
i , and
B
is an asymptotic basis of order
3 .
Thus
the lemma is proved. Proof of the theorem. order
2 .
of order
Let
We suppose that
3
A
A
be an asymptotic basis of
is also a minimal asymptotic basis
and we shall arrive at a contradiction.
Let us first show that, without loss of generality, we can assume that the smallest element of integer
k , an integer
n
A
is zero.
and a set
X
Consider a positve
of integers.
One can
easily see that k(X+n) = kX + kn . Clearly
X
is an asymptotic basis of order
is an asymptotic basis of order
k .
k
if and only if
Moreover, for any
X +n
x e X ,
kC(X + n) \ {x+n}) = k ( X \ { x } ) + kn . We conclude that and only if
X
X+n
is a minimal asymptotic basis of order is a minimal asymptotic basis of order
becomes clear now that if sider equivalently the set
0
does not belong to A' = A + (-min
So we assume that zero belongs to The set
A
prove this, let Suppose that
a , a'
a - a*
n. < n~ < ...
A
A .
For all
n. € 2A . l i > i~
The set
a
Consider an index
.
A
To
a > a' > 0 .
is a minimal
(i = 1,2, ...) .
we have that
e A\{a}
such that
such that
n. = a * a ( i ) where
A
3 , so there are infinitely many natural
is an asymptotic basis of order
i > i~ , 2
A , one can con-
which contains zero.
A .
be two elements of
belongs to
n i i 3(A \ {a}) , But
if It
satisfies also property (A.2) of the lemma.
asymptotic basis of order numbers
A)
k k .
It follows that
2 ; therefore for all i... > i o 3 2
such that
n,- >2a . 13
303 ia-af ) + aU) for all
i > ig .
e 3(A \ {a})
This contradiction proves that property
(A.2)
holds. Thus, the set
A
satisfies all hypotheses of the lemma; hence,
it is not a minimal asymptotic basis of order
3 .
This contradic-
tion proves the theorem. 3. MAXIMAL ASYMPTOTIC NONBASES Nathanson [17] introduced the dual concept of minimal basis, that of maximal nonbasis. of order A u {b}
If the set
A
is not an asymptotic basis
h , but, for every nonnegative integer is an asymptotic basis of order
a maximal asymptotic
nonbasis of order
b £ A , the set
h , then we say that h .
A
is
Maximal nonbases have
been studied by Erdos, Hennefeld, Nathanson, and Turjanyi [5-8, 10, 16-18, 21, 2 2 ] . Let
A
be a set of nonnegative integers and
real number. A
We denote by
not exceeding
x .
A(x)
x
a positive
the number of positive elements of
In a previous paper [1], written in common
with O.-M. Deshouillers, we constructed, for every natural number h > 2 , a class of maximal asymptotic nonbases of order the smallest possible density:
each set
A
h
having
in this class satisfies
A(x) = 0 ( x 1 / h ) . 4. COMBINATORIAL ANALOGUES Let
F
denote the collection of all finite subsets of
INI .
It has been proved (see, for instance, [2]) that many properties of the semigroup
(N , +)
hold also in
(F , u)
and in
(F , n) .
We
now state some results concerning union and intersection bases for
F . 4 .1
Union bases
A subcollection of order
h
8
of
F
is called an asymptotic
if all but finitely many sets in
not necessarily distinct sets of union nonbasis of order h
h .
8 •, otherwise,
h
8 8
of order is an
An asymptotic union nonbasis
is called maximal if every subcollection of
contains properly For each
h .
B
h
8 is an asymptotic
is called minimal if every proper subcollection of of order
union basis
are unions of
An asymptotic union basis
asymptotic union nonbasis of order 8
F
is an asymptotic union basis of order
h > 2 , Nathanson [19, p.223] constructed
examples of minimal asymptotic union bases of order
h :
F
that
h .
'trivial' let
304 T.,T2 ,...,T,
be a partition of
N
into
h
nonempty sets at least
two of which are infinite; then the collection of all finite subsets of the order
T.'s h .
(1 < i < h)
is a minimal asymptotic union basis of
He also constructed
union bases of order
'nontrivial' minimal asymptotic
2 .
We proved [13] that, for any asymptotic union nonbases of order 4.2
Intersection bases
A subcollection basis of order
h
B
of
F
is called an asymptotic
if all but finitely many sets in
resented as the intersection of B . h
h > 2 , there are no maximal h .
Otherwise,
B
h
F
intersection can be rep-
not necessarily distinct sets in
is an asymptotic
intersection
nonbasis of order
[19, p.229]. If
for every
B
is an asymptotic intersection basis of order B
belonging to
tion basis of order minimal asymptotic
h
B ,
too.
B \ {B}
h
then,
is an asymptotic intersec-
We conclude that there are neither
intersection
section nonbases of any order
bases nor maximal asymptotic
inter-
h .
5. OPEN PROBLEMS 5.1
The question whether there exist minimal asymptotic bases
of two different orders 5.2
h
and
k
remains open, when
Is there a minimal asymptotic basis
satisfying the minimal density condition: 5.3 order
min(h,k) > 3 .
of order
A(x) = 0(x
h
) ?
Are there 'nontrivial' minimal asymptotic union bases of
h > 3 ? 5.4
A
This is a question of Nathanson [19].
Can we find more general sufficient conditions on the
structure of a commutative semigroup
(M , *)
that would guarantee
the existence or the nonexistence of minimal bases or maximal nonbases? For other open problems we refer to the papers of Erdos and Nathanson, for instance [11] or [19]. REFERENCES 1. J.-M. Deshouillers et G. Grekos, Proprietes extremales de bases additives, Bull . Soc . Math . France W7_ (1979), 319-335. 2. M. Deza et P. Erdos, Extension de quelques theoremes sur les densites de series d'elements de N a des series de sous-ensembles finis de N, Discrete Math. VZ. C1 9 7 5 D , 295-308. 3. P. Erdos, Einige Bemerkungen zur Arbeit von A. Stohr 'Geloste und ungelb'ste Fragen uber Basen der naturlichen Zah lenrei he' , 3. Reine Angew.Math. 197 (1957), 216-219. 4. P. Erdos and E. Hartter, Konstruktion von nichtperiodischen Minimalbasen mit der Dichte 1/2 fur die Menge der nichtnegativen
305 ganzen Zahlen, J. Reine Angew.Nath. 221 (1966), 44-47. 5. P. Erdos and N.B. Nathanson, Maximal asymptotic nonbases, Proc. Amer.Nath.Soc. ^8 (1975), 57-60. 6. P. Erdos and N.B. Nathanson, Oscillations of bases for the natural numbers, Proc . Amer. Hath . Soc . _53_ (1975), 253-258. 7. P. Erdos and N.B. Nathanson, Partitions of the natural numbers into infinitely oscillating bases and nonbases, Comment.Nath.Helvet. 5_1_ (1976) , 171-182. 8. P. Erdos and N.B. Nathanson, Nonbases of density zero not contained in maximal nonbases, 3. London Math.Soc. (2), 15 (1977), 403-405'. 9. P. Erdos and N.B. Nathanson, Sets of natural numbers with no minimal asymptotic bases, Proc.Amer.Nath.Soc. 7J3 (1978), 100-102. 10. P. Erdos and N.B. Nathanson, Bases and nonbases of square-free integers, 3. Number Theory V\_ (1979), 197-208. 11. P. Erdos and N.B. Nathanson, Systems of distinct representatives and minimal bases in additive number theory, in: Number Theory, Carbondale 1979, Springer-Verlag Lecture Notes in Nathematics, Vol. 751, 1979, pp.89-107. 12. P. Erdos and N.B. Nathanson, Ninimal asymptotic bases for the natural numbers, J. Number Theory, to appear. 13. G. Grekos, Nonexistence of maximal asymptotic union nonbases, Discrete Nath., to appear. 14. E. Hartter, Ein Beitrag zur Theorie der Ninimalbasen, 3. Reine Angew.Nath. J_£6 (1956), 170-204. 15. E. Hartter, Eine Bemerkung uber periodische Ninimalbasen fur die Nenge der nichtnegativen ganzen Zahlen, 3. Reine Angew.Nath. 214/215 (1964), 395-398. 16. 3. Hennefeld, Asymptotic nonbases which are not subsets of maximal asymptotic nonbases, Proc . Amer. Nath . Soc . _6_2_ (1977), 23-24. 17. N.B. Nathanson, Ninimal bases and maximal nonbases in additive number theory, 3. Number Theory 6_ (1974), 324-333. 18. N.B. Nathanson, s-maximal nonbases of density zero, 3. London Nath.Soc. (2), j_5 (1977), 29-34. 19. N.B. Nathanson, Oscillations of bases in number theory and combinatorics, in: Number Theory Day, New York 1976, Springer-Verlag Lecture Notes in Nathematics, Vol.626, 1977, pp.217-231. 20. A. Stohr, Gelb'ste und ungeloste Fragen uber Basen der naturlichen Zahlenreihe, I, II, 3. Reine Angew.Nath. 194 (1955), 40-65, 111-140. 21. S. Turjanyi, On maximal asymptotic nonbases of zero density, 3. Number Theory £ (1977), 271-275. 22. S. Turjanyi, Note on maximal asymptotic nonbases of zero density, Publ.Nath. Debrecen 26i (1979), 229-236.
MEAN VALUES FOR FOURIER COEFFICIENTS OF CUSP FORMS AND SUMS OF KLOOSTERMAN SUMS Henryk Iwaniec Mathematics Institute Polish Academy of Sciences ul. Sniadeckich 8,Warszawa, Poland In this paper we prove an analogue of the large sieve inequality for Fourier coefficients both of holomorphic and of realanalytic cusp forms in respect of the modular group
SL(2,Z) .
The results will be applied for estimating trilinear forms of Kloosterman sums modul
c
SCn,m;c)
over coefficients
n,m
counted with a smooth weight function.
and over For such sums
it is shown that the Linnik-Selberg conjecture holds on average. 1. PRELIMINARIES The group
T = PSL(2,Z)
H = {zj z = x + iy, y > 0}
Y
for M.
acts on the upper half-plane
as linear fractional transformations
cz + d
y = (
^) £ r , ad - be = 1 , a,b,c,d-rational integers.
denote the space of holomorphic cusp forms for
k >> 2 , k E 0 (mod 2) .
M,
V
Let
of weight
is a Hilbert space with inner product
given by y k dz
V where
V - {z e Hj 5 < x < \, |z| > 1}
dz = y
dxdy
is r-invariant measure on
is a fundamental domain and H .
It is known that
is finite dimensional generated by Poincare series k-1 P m (z;k) = ^Yk^ ) ^ (cz + d)" k e(myz) , where
r
is the cyclic group generated by
izer of the cusp
00 .
For
k > 16
Mk
are empty.
we have Lj^l
v. = dim M, =
if
k ? 2 (mod 12)
if
k - 2 (mod 12)
, 1
m > 1
y z = z +1 , the stabil-
k = 4, 6, 8, 10 and 14 ,
identically equal to zero, thus
M, K
For
P (zjk) m k = 12
are and
307 and the first Poincare series for
P m (z;k) , 1 < m < v k
form a basis
M. . K
Every cusp form
f(z)
has Fourier expansion around
°°
00
f(z) =
I a e(nz) n=1 n
.
The problem of estimating the coefficients tention.
a
received great at-
In case of
P m (z,k) =
I n= 1
p n (m,k)e(nz)
it is known that (Petersson) t A
Pn where
( m
'
k )
=
3 , . (x)
-» k " 1
/
(4iT/mnJ
r(k-1)
°°
> r
{ 6
o
-> • k r
nm
+ 2lT1
y.
1
J
o
,
1
E
is Bessel's function of order
S
(
k
1
5
k-1 , therefore the
problem can be reduced to estimating sums of Kloosterman sums.
By
A. Wei 1's result |S(n,mjc) I < ( n , m , c ) M c ) c ^ it follows that
a
<< n
(2) while the famous conjecture of
Petersson proved recently by P. Deligne [2] says that k-1 a n << x(n)n
2
.
(3)
This shows that there is a great cancellation of terms in (1).
At
the Stockholm Congress Yu.V. Linnik [8] stated CONJECTURE.
For
T > (mn)^+e
7 - S(n,mjc) << T £ c
we have
.
(4)
Shortly afterwards A. Selberg [13] extended the conjecture to all
T > (m,n) 2
.
In a first attack on (4) one is tempted to combine
Deligne's (3) with (1) but one fails. functions
J,_/](x)
The point is that the Bessel
of integral order are not sufficient to generate
nice weights like the characteristic functions of intervals for example. Very recently N.V. Kuznietsov [7] obtained the striking result
1
1
I 1 S(n,m,c) << m T 6 (log T ) 3 n m c
.
(5)
308 This is one among many important consequences of his formula (see Lemma 2) which connects sums of Kloosterman sums weighted by Bessel's functions of imaginary order with Fourier coefficients of non-holomorphic cusp forms. The theory of real-analytic cusp forms has been originated by A. Selberg [14] and H. Haass [9]. all F-invariant functions on dz
a
y dxdy
measure.
dx
V
in
The F-invariant Laplacian
3y L (F\H) , it has a point spectrum
= 0 , 1/4 < X. < X~ < ... , X. ~ j/12
spectrum.
be the space of
H , square-integrable on
has a self-adjoint extension into X
L 2 (T\H)
Let
and it has a continuous
Constant functions have L-eigenvalue
X
= 0 .
The eigen-
functions
u.Cz) with eigenvalues X. > 1/4 are called cusp forms J J or Flaass wave forms. They have Fourier-Bessel expansions u.(z) = /y J
where
(6)
7 p .(n)KiK.(27r|n|y)e(nx) J n^O J
X. = 1/4 + K .
and
p.(n)
depend on
j
and
n
only.
The
eigenfunctions of the continuous spectrum consist of Eisenstein series
E(z,s) = on the line s = I + it
(Im Y z ) S
I
s = 5 + it .
(7)
The analytic continuation of
s s
K
where
E(z,s)
on
is given by Fourier-Bessel expansion
£(s) = TT sr(s)<;(2s)
] s _ i C 2 T T I n | y eCnx)
(n) = I
dV
and
Usually the basis { U } is chosen so that j tions of all the Hecke operators
I
Kv(x)
(8) is the
d| n
modified Bessel function.
VCz) - ±
.(n) "'
I
uj
are eigenfunc-
f(^^)
/n ad=n 0
.
This is possible because L , T
, n = ±1,2,3,...
mutually commute and are
309 hermetian. with
Letting
e. = ±1
eigenvalues
T u.=T.(n)u.
one gets T. (n)
Z.(s):=
II
J J
for
n = 1,2, . . . and
p.(n) = p.(1)x.(n)
are multiplicative.
x.(n)n
s
p.(-n)=e.p.(n).
The
More precisely we have
n (1
J J
n=1 It is conjectured that |x.(n) | < x(n)
and
T * u . - e.u .
pS
p
p
.
(9)
The above statement is not yet proved. Several results are known for mean values of a different kind. In [7] Kuznietsov proved
^ 4
K. < X
Ch
* Kj
TT
which constitutes an improvement of some results of Bruggeman [1]. The sharpest estimate of
x.(n)
which occurred in print is due to
N.V. Proskurin [12] | T . (n) | < ilnln'
2. STATEMENT OF RESULTS Since
T
are symmetric
infinitely often. of the sign of ° Z.(s)
x.(n)
are real, positive or negative
A number of questions arise here about variation
x-(n) . J
In connection with the Riemann conjecture for
(that complex zeros lie on
Re s = \ ) it is important to es-
timate sums over primes like I T.lplp^ p <X J
.
(10)
We shall show that (10) is
<< X 5 + e
for almost all
j .
Precisely,
we prove
THEOREM 1. bers.
Let
Then for any
K .
j
K > 1 , N > 1
and let
a
be complex num-
e > 0 J a p.(n)| J N
the constant implied in the symbol
<< (K2 + N 1 + £ )
<<
depending on
e
alone.
A similar inequality will be proved for Fourier coefficients of holomorphic cusp forms.
Let
f. ,,...,f I ,. K
, Vk , K
be an orthonormal
310 basis of
M,
with Fourier expansion
K
I
THEOREM 2.
*j | k (n)B(nz)
Under the assumptions - ^
l
.
^
j
(4TT) K
k=0(mod2)
of Theorem 1 we have
|
' 1*j£vk
" T
z
1
,
(n)|2
N
0(N1+e))
I
I |a | 2 n N
.
(13)
Both (11) and (13) remind one of the large sieve inequality for Dirichlet's characters. coefficients
n
Oy. (n)
A corresponding result for the Fourier of Eisenstein series is just a kind of
mean-value theorem for Dirichlet polynomials (see [10]). THEOREM 3.
Under the assumptions
fK
a n" i r a
(n)| 2 dr «
of Theorem 1 we have (K2+N1+e)
Y N
|a | 2
(14)
Next we apply Theorems 1, 2 and 3 for estimating trilinear forms of Kloosterman sums G±(N,M,C) =
I I [ a b g(n,m,c)S(n,±m,c) N
with a weight function
g(n,m,c)
having the properties
Supp g(n,m,c) c [N,2N] x [M,2M] x [C,2C] g(n,m,c) q
^ 1 q
3n
1
+q q
9m
is of 2
2
+q
e > 0
class such that for
3
q
3c
THEOREM 4. for any
C
,
-q, g(n,m,c)| £ N
-q ? M
Z
(15) 0 < q/.,q2,qo ^ 2
-q, J
C
.
(16)
3
Let
N,M,C > 1
and
g(n,m,c)
be as above.
we have
G ± ( N , M , C ) << C
1
+
I
l L
the constant implied in
<<
n
^
depending on
m
e
alone.
Then
311 Remark
G~(N,M,C)
can be treated by the large sieve in-
equality
I
I I an e(n C^ ) | 2 < (C 2 + N)
I
c
N
To make use of it one must separate the variables g(n,m,c) .
|a |2 . n
J N
n , m and c in
For, we write
g ( n , m , c ) = III g ( X. , X 2 > Xo) e ( X. n + X - m + X ^ c ) dX,. with g(n ,m, c) e (-X^ n-X2m-X2c) dndmdc << << ( N X ^ + 1 ) " 1 (n2X2+ D"1(C2X3+1)"1NlvIC by partial integration.
G±(N,N,C) which
is
Hence
<< (O/Fl) ( O / N ) ( £
sharper
than
(17) i f
|aJ2)2(£
lbm|2)5
(18)
C << /FfN" .
3. KUZNIETSOV'S FORMULAS Proofs of Theorems 1, 2 and 4 depend on several formulas of Kuznietsov [6],[7] (see also Bruggeman [1] and Proskurin [11]). LEMNA 1.
.L
Let
Then for any
ch ir(K.+t)ch ir(K.-t)
J~ '
=
m,n > 1 .
J
-2
r
0
nm
v h e r e t h e path
J
t r sh i r t
7T
—i
of
the
~
TT
2it ! r sh i r t
(n)
TT j
t e U
we have
ua«. V(n)a_ M o . v(m)
2ir
-2ir
ch 7TP k ( 1 ^ 2 i r ) r 2 ch 7 r ( r + t ) c h T r ( r - t ) r
/ U
A c=1
integration
47r/nm oi— 2 c
is
^ f J J
n f
(19) „
b in.fTij C J
a half
unit
f 47r/nm
l^'i • x. *•
-l
zit
circle
c
^ du \)
V J
|u| = 1 ,
Re u > 0 .
Kuznietsov's idea of the proof is based on computing scalar product
n m
U (z,s) =
I
of two Poincare series
(Im Yz)Se(nyz)
312 in two ways.
Apart from a simple factor the right-hand side of (19)
results from computing this product by expanding
U (z,s)
into
Fourier series and applying Rankin's method while the left-hand side of (19) is a Bessel identity for
of the orthogonal basis of cusp forms series
E(z,5+it) .
u.(z)
in respect
and the Eisenstein
Another independent proof of a similar relation
was given by Bruggeman [1]. Very clever manipulation with the continuous variable
t
led
Kuznietsov [7] to (0,«0 such that LEMMA 2. Let (j)(x) be of C class on (q) 2 £ as x + oooo for q = 0,1,2,3 . cj>(0) =
=2
I
I
{>•
with
where J0(y)4>(y)
dy
£E1(mod2) Define
f For
n,m > 1
we have
1
^
A sum with
$•
in place of
-(20)
0|_. can be expressed in terms of
Fourier coefficients of holomorphic cusp forms in much similar form. LENMA 3.
For
n,m > 1
we have
± c=1
f Jo(u)
(k-1)!
I
;
(4ir/n?ii)K
» '
Cn)», J
'
K
J
,( '
(21)
K
ksO(mod2)
Proof
If
f(z)
is a cusp form of weight
Fourier coefficient is given by (Petersson) -
.
k
then its n-th
313 Hence we first deduce that .(z) and then by Parseval's identity and (1) that J : ^1
ip. • (n)ijr! ( m ) = < Pm ( * , k ) , P J*^ J> ^
_ (4,/^)k" =
1
{re5
r(k-D
.
y
from
0
and sum over
to
y l
o
s ( n
,
(«,k)>
= p
n
( m , k )
rf"
m ; c ) 3 L
—
i 1 ES ( n ' r n i o ) j k - 1 t — 5 — ) }
'
- (k-1)! i k (4Tr/nm)1 " k J. _, (y ) Q (y ) , integrate
Multiply both sides by over
2irik
nm*2iT1
n
°°
in respect of the logarithmic measure
k = 0 (mod 2) , k > 2
getting (21 ) .
y
dy
For 6 -terms we nm
to
appealed to the identity 2
(2k-1)(-1)kJ
I k=1
ZK
'
(y) = yJ (y) °
which easily follows from the recurrence relations
Vi!zl For
k -2
+J
n+1(z) = ¥ V
z )
•
one must be careful because
P (zjk) n
are defined in a
different manner (Hecke), however the relation (22) remains valid, both sides being equal to zero. Lemmas 2 and 3 will be combined for estimating In the case of
G (N,N,C)
LENNA 4.
Let
G (N,M,C) .
we need another formula of Kuznietsov [6].
n, m > 1
and
<J> (x)
be a function as in Lemma 2.
We have p ,(n)p.(m)|(K.) j
ir 77
(nm)
cn.
(n)cjo.
(m J ch i r r q) ( r ) d r
(23)
J -o
where
4.
BILINEAR FORNS OF KLODSTERNAN SUNS
In this section we investigate sums of coefficients
n
and
m .
S(n,m;c)
over the
Our arguments do not depend on the theory
of automorphic functions and they are mostly elementary.
Define
314 B(c,N) =
7 b b S ( n , m ; c ) e ( ^ - ^ 0) M OM n m c N
LEMMA 5.
Let
0 > 0 , e > 0
and
N > 1 .
Then we have
B(c,N) << c^ +£ N l\b | 2
for all
c > 1 ,
(24)
B(c,N) << (c + N + /0d\T) I|b n | 2
for all
c > 1 ,
(25)
B(c,N) << 0"^c^N^ +£ l\b | 2
for
the constant implied in
<<
c < N
depending on
£
and
0 < 2 ,
(26)
alone.
Of the three results above the last one is most crucial. first follows trivially from A. Weil (2).
The
The second is an easy
consequence of the hybrid large sieve inequality:
f
I
I II
J-T d(mod c)
I
b n^eCn § § ) || 2 « (cT C
N
+
N)£ £ | b| J
A minor difficulty arises when separating the variables in
e(20/nm/c) .
n.(x) = 1
B(c,N) for
n
(27) and
m
N < /nrn < 2N ,
will not be affected if weights
x e (1,2]
lows we demand
.
To this end we appeal to Nellin's transform.
Notice that in the range of the summation we have therefore
2
n(x)
are attached at each term.
to be of
C°°
class with
r\[/im/N) with In what fol-
Supp n(x) = [5,3] •
Then -1 n(x)e(20c
1 r 1+ico -s xN) = -ITR ( S ) X s ds Z7T1 J 1-i*
where by Mellin's inversion formula ( |t |+1) R(s) = [
2
for all
t
n(x)e(20c" 1 xN)x lt: dx
(28) t"
Jo
2
for |t|>16TT0c"1N
Thus
B(c,N)
2^ ^T ff R )
m, n
whence (25) by (27) and (28). 1 -£ Now we proceed to prove (26) for N
£
< c < N
following from (25).
c < N
, the remaining case
By the Cauchy-Schwarz inequality
315
|B(c,N) |2 < ( I |bj 2 ) IM§) I I bmS(n,m,c)e(20/hl^/c) |2 n
n
9
m
m 1 d 1 -m 9 d 9
• tllbj2)
I b b - e (1
n
m.
,T\\J
d
1
2
1
22
c
)If(n) n
d
1' 2 d 1 -d 9 20(>/m7-v/mT) f(n) = n(fi)e( c " + g — /n) = n(^) e (An + B/n) , say.
where
For the inner sum we apply the Poisson summation formula I f(n) = I t(h) , n
h£ Z
h
where
£(h) If
= I n(£)e((A-h)t + B/t)dt
.
h # A then
| A-h | > - while B/2/t < (/?-1)9/c < 2(/?-1)/c . c Therefore by partial integration q = [2/e] times f(h)<
™
Now consider the case h = A . Since h is an integer this implies d. = d 9 . If m. * m 9 we integrate by parts getting . .
I n I
" 1 M 2
.,
CN
and if m. - m 9 we take the trivial estimate f(A) << N . The summation over d = d. = d 9 , (d,c) = 1 yields Ramanujan's sum for which we have m
i"m9 _
| Y 9 ( — d) | < (m.-mo,c) . c 1 2 d(modc) (d,c)-1 Gathering all the above estimates together we complete the proof of (26). 5. PROOF OF THEOREM 2 By (22) the left-hand side of (13) is equal to i c=1 where
I n,
ar S (n, m! c.E K (l^)
(29)
2)1-1
K
ch
1_
316
=1
Jl=1
v. a n
T
|
+
0(1)
(30)
and 251-1 E K (x) - 2
(xD
I
It remains to estimate bilinear forms
To this end we first prove LEMMA 6*
For 1
s
2 F)
«(sh
2
Proof
K
For
eiby0
f
eiby3
Jo
f1
and
K > 0
we have
Cx3Q(Cx)dC S—9
(31)
9~q
Jo [ ( c h I ) 2 - C ]
b > 1
it holds (see [3], p.106) that
(y)dy = i n (b2-1 ) "* (b
In particular for f
x > 0
n = 0
differentiating over
(y)ydy - i b ( b 2 - 1 ) " 3 / 2
b
gives
.
°
Moreover for
x,y > 0
we have (see [7])
I Hence
which for
1 becomes (31). b = ch rr
Combining Lemmas 5 and 6 we shall prove that F K (c,N) « If
c>N
or
c"eN1+2e N < c < N
integral representation
a | I |a nn N
(32)
317
1 r cos(£x sin 0)d0
(33)
'Jo
from the crude estimate (sh I,
nch^2.c2]3/2
and from (24) and (25) respectively.
For c £ N we take
e * /c/N
and split up the integral (33) as follows
the last two terms arising by partial integration.
If £ > e we
apply (26) and (34) and if £ < e 2 we apply (25) and (33) in either case getting a contribution to F K (c,N)
less than
0(N1+e£
|aj2)
as claimed. Finally Theorem 2 follows from (29), (30) and (32). 6. PROOF OF THEOREM 1 j
Multiply both sides of (19) by t(sh irt)e over
n,m e (N,2N]
and integrate from
2 f°° t sh TTt D"(t/K) J^ ch irCt + + rlch r1-1- ir(t-r) -fA- -^ e
a R a m , sum
t = 0 to t - °° .
.. ^
az > >
Since
|r| Q - ( r / K ) 2 e -1-ch--irr
and the integral in (19) contributing nonnegative amount can be discarded it yields
j
-
e
- U . / K ))
2
°
9
1
where we put
A(K.N) -
I
I a r
c-1 n,m and f o r x
a
47r/inm/c
c
we put
r
(xu)
J0
By (7.12.21) of [3] we have for e~ x u
xu
9
I I anp J (n) |2 « K 3 l | a | 2 • A(K,N) n n
ch 5
c
40 d t
Re u > 0
( 3 5 )
318 Also we have (see [4], p.214) t e ' ( t / K ) sin(2t£)dt = ^
K3E
Hence ,2 ; th i. If
c < NK
(i.e.
x >> K
) we may do better when integrating by
parts.
x"1K3 [ e" C ^ K) (1 - € th £ - 2£2K2)cos(x ch V-^j Let
A (K,N) , v * 1,2,3,4
denote partial sums of
A(K,N)
V
(37) in
_j
respect of the variable c from the intervals c 1 < NK < c9 < N < 2 ' ^~ C c^ ^ N N < 4 respectively. For A^(K,N) apply (36) and (24) giving A (K,N) « 4
For
A 3 (K,N)
A 2 (K,N)
A,j(K,N)
h
"
I c" 1 Nj; |a I 2 « N
apply (36), (26) if
A 9 (K,N) << £ c ^ NK" 2
N z J |a M
apply (36) and (25) giving
A (K,N) « For
I c * ON2
2
N2
I
£ < 1
and (25) if
J |a | 2 «
£ < 1
£ > 1
giving
K N 1 + e \ |a | 2 .
n
(40)
n
and (25) if
(c"'N^+e K 2 + c" 2 N 2 K 3 e' K «
(39)
n
C
apply (37), (26) if
A ^ K j N ) <<
N 1 + e \ |a | 2 .
£ > 1
giving
) \ \ ap | 2
(KN 1 + e + K 3 N 2 e ' K ) I | a n | 2 .
(41)
Gathering (35), (38)-(41) together we get £ _^J— | Y a p.(n)| 2 << K(K 2 + N 1 + e + K 2 N 2 e ~ K ) \ K . < K c n 7 r K j N
K
|a | 2 . n
we may replace
K. > K . On taking 2 2 -K 2 we first ignore the term K N e and then we
derive (11) by partial summation.
K
319 7.
PROOF
OF THEOREM
4
As we pointed out in Section 3 it is sufficient to prove (17) for
C: >> 8TT/MN Sir/PIN .. Let Let
support in
I
(0,1)
cj)(x)
be a function of
C
class with compact
such that
' (x) |x dx < 1 .
We first show that
H ± CN,M,4>) =
I
I Iab 1 n m
N
C
|bj 2 )* . (42) To this end we apply (20) and (21) in case of in case of and
H (N,H,<j)) .
and (23) $ , cj)
(p . LENNA 7.
For all
<J> ( r) , <J> (r) , (ch
Proof power
H (I\I,N,(!>)
We have to estimate the transforms
series
K
All
r > 0
we have
7rr)<j>(r) <<
results
expansions
are
(note
(r
+1)
trivial that
x
(43)
for
r<1.
we deal
If
with
r>1
we
is in
utilize
(0,1) )
2irtx)
and integrate termwise.
A typical term to be estimated is
An application of Stirling's formula for
T U + 1+v)
and
T(£+1+2ir)
completes the proof of Lemma 7. By (20), (21), (43), (11), (13), (14) and the Cauchy-Schwarz inequality we obtain (42) for (43), for
H + (N, N, <|)) .
Analogously, by (23),
(11), (14) and the Cauchy-Schwarz inequality we obtain (42) hf (N,M,(|>) .
It remains to derive (17) from (42).
h(x,j,X2;x) = g (x^ , x^, 4TT/XTXT/X) Just for separating the variables
we get x.,x2,x
Writing
g(n,m,c) = h (n ,mj 47r/Tim/c) we write
320 =
h ( X^ , X - J X )
with h(X/],X2;x) = h(
(x. ,, x 2 ; x) e ( X^ x,j+ X 2 x 2 ) dx^ dx 2 • h (x.
By partial integration twice in each variable 7
a
xZ - ^
A
7 7
-
hCX1 , X 2 J X ) << (NX^+1)
Therefore (17) follows from (42) for
1
9
7
-
(M Z X 2 + 1)
X^X-
we deduce that
1
NM .
<J>(x) = h(X 1 ,X 2 ;x) .
We note that Theorem 1 is applied in [5] to prove that T+Tr
2/3
J
4
T2/3+£
Other applications and generalizations of the above methods for congruence subgroups will be discussed in a forthcoming paper by J.-M. Deshouillers and the author. REFERENCES 1. R.W. Bruggeman, Fourier coefficients of cusp forms, Inventiones math. £5 (1978), 1-18. 2. P. Deligne, La conjecture de Weil I, Publ.Math. I.H.E.S., 43 (1974), 273-307. 3. A. Erdelyi, W. Magnus and F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions II, McGraw-Hill, New York, 1953. 4. I.S. Gradsztejn and I.M. Ryzyk, Tables of integrals, sums, series and products (in Polish), PWN Warszawa, 1964. 5. H. Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function, Seminaire de Theorie des Nombres, Bordeaux 1979/80. 6. N.V. Kuznietsov, Petersson hypothesis for forms of weight zero and Linnik hypothesis (in RussianJ, Preprint No.02, Khab. K.H.I.I., Khabarovsk (1977) . 7. N.V. Kuznietsov, Petersson hypothesis for parabolic forms of weight zero and Linnik hypothesis. Sums of Kloosterman sums, Math. Sbornik _1_1M_ (153), No.3 (1980), 334-383. 8. Y.V. Linnik, Additive problems and eigenvalues of the modular operators, Proc.Internat.Congr.Math. (Stockholm 1962), 270-284. 9. H. Maass, Uber eine neue Art von nichtanalitischen automorphen Funktionen, Math.Ann. 121, No.2 (1949), 141-183. 10. H.L. Montgomery, Topics in Multiplicative Number Theory, Lect. Notes in Math. 227 (1971), Springer-Verlag, Berlin-New York. 11. N.V. Proskurin, Summation formulas for generalised Kloosterman sums (in Russian), Zap.Naucn.Sem. Leningrad. Otdel.Mat.Inst.Steklov. (LOMI) (1979) Vol.82, 103-135. 12. N.V. Proskurin, Estimates for eigenvalues of Hecke operators in the space of parabolic forms of weight zero (in Russian), ibid. 136143.
321 13. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. of Symposia in Pure Math. VIII, ANS, Providence (1965), 1-15. 14. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Nath.Soc. 20 (1956), 47-87.
NONSTANDARD METHODS IN DIOPHANTINE GEOMETRY by Ernst Kani
The basic methods and results in the subject of Diophantine Geometry (as set forth in the book of Lang[1i]) were developed by Mordell, Weil, Siegel and others in the 1920fs and f 30 f s. The first such result is the celebrated Theorem of Mordell-Weil which may be stated as follows. Theorem 1 (Mordell[17],Weil[30]): Let C be a curve defined over a number field K. Then the group J Q ( K ) of K-rational points of the Jacobian variety Jp of C is a finitely generated group. Siegel subsequently (in 1929) showed how one can combine this theorem with a theorem on diophantine approximations (the Thue-Siegel Theorem) to obtain: Theorem 2 (Siegel[28],Mahler[i4]): Let C c l ^ b e a protective curve of genus g ^ 1 which is defined over a number field K, and let S be a finite set of places of K. Then there exist only finitely many S-integral points P € C(K). Recall that a point P € P31 is called S-integral if all its coordinates (for some choice of homogeneous coordinates) are p-integral for all finite p £ S. Actually, Siegel proved a bit more: he also characterized those curves of genus 0 for which the assertion fails. By using this characterization, he was able to strengthen the classical Irreducibility Theorem of Hilbert considerably. Recall that Hilbertfs Theorem is the following.
323
Theorem 3 (Hilbert[7]): Every number field K is hilbertian in the sense that the following property holds: (H) For each absolutely irreducible polynomial f(T,X) with coefficients in K there exist infinitely many t € K such that f(t,X) € K[X] is irreducible. Siegel's sharpening of this theorem is as follows. Corollary; Let f(T,X) € K[T,X] be a polynomial with S integral coefficients. Then there exists a number field K1 in which f(t,X) becomes reducible for infinitely many S-integral t € Kf if and only if, after a suitable substitution of the form T
-V*
+
^-I^"1
+
—
+ a-nY~n'
f becomes reducible as a polynomial in Y. Coming back to Siegel's theorem, for curves of genus 1 it is, in a sense, best possible, since there exist such curves with infinitely many K-rational points. For curves of genus g ^ 2, however, it falls somewhat short of what one conjectures to be true, namely: Mordell's Conjecture (Mordell[17]): Every curve of genus g £ 2 defined over a number field K has only finitely many K-rational points. While this conjecture is at present still far from being settled, there exist several partial results in this direction. Theorem 4 (Mumford[18j): Let C c P11 be a curve of genus g £ 2 defined over a number field K, and suppose that the K-rational points P^e C(K) are ordered according to increasing (logarithmic) height: h(P^) ^ h(P2) ^... Then there exist constants a and b with a > 0 such
h(Pn) * e
3
^,
V n>
324 Here, the logarithmic height of a point P = ( a ^ . . . TP^ is defined by h(P) = -Z5
Min v (a.),
where the sum runs over all archimedean and non-archimedean primes of K, and v denotes the normalized valuation associated to p (written additively): N v^(a) = -log |a| *. Theorem 5 (Dem1 janenko[1j ,Manin[i6]): Suppose that C and C f are curves defined over K such that rank Hom K (J c ,J c ,) > rank J C (K). Then C f has only finitely many K-rational points. Corollary (Manin[i6]): Let N > 1 be an integer and suppose that k » 0. Then the modular curve X Q (N ) has only finitely many K-rational points. In this lecture I would like to outline how one can use the nonstandard methods of A. Robinson (cf. Robinson[22], Stroyan-Luxemburg[29]) to prove the above results. For lack of time, however, I shall mainly concentrate on the proof of the Mordell-Weil Theorem, or, more precisely, on the proofs of the following two theorems from which (by using an "infinite descent argument") the Mordell-Weil Theorem may easily be deduced (cf. Lahg[13], Mumford[19]). Theorem 6 (weak Mordell-Weil): There exists an integer m > 1 and a finite extension Kf of K such that J c (K ! )/mJ c (K f ) is finite. Theorem 7 (N6ron[2i],Tate[12],[15]): There exists a realvalued, positive definite quadratic form h j on J Q ( K ) with the following property:
325 (*) Every hj-bounded subset of J Q ( K ) is finite. In particular, hj(P) = 0 if and only if P e J Q ( K ) is a point of finite order. The quadratic form constructed in Theorem 7 is called the Nferon-Tate height on J r . As a consequence of the explicit construction of h j by means of "intersection products", the theorems of Mumford and Dem1janenko-Manin become easy corollaries, as I will briefly indicate. §1« Nonstandard Methods a) Enlargements Roughly speaking, the nonstandard methods consist of embedding each object X which we are considering in a (much) larger object *X, called an enlargement of X, such that 1) the embedding is compatible with the basic operations of set theory 2) the embedding is "functorial": each map f:X •+ Y extends uniquely to a map *f :*X -* *Y 3) the objects *X (resp. maps *f) have similar properties as X (resp. f ) . Let me explain this in some more detail in the case that X = K is a number field, which I regard as coming equipped with its canonical set M K of places. As Roquette has suggested, one might view the enlargement *K, together with its canonical set *M K of places, as a global completion of K since it possesses the following properties: (A) *K is a valued field with respect to the set *M K of places, and there exists an embedding K "* *K such that every v e M K has a unique extension *v £ *M K to *K,
326 (B) *K is an elementary extension of K (i.e. the elementary properties of K are inherited by * K ) . (C) *K is saturated: every binary relation on K which satisfies a "Cauchy condition" (more precisely: is concurrent) has a "limit" in *K. (D) A statement about K is true if and only if the "corresponding" statement about *K is true. Note that the properties (A) - (C) are analogous to the following well-known properties of a local field ft: (a) ft. is a valued field with respect to a canonical valuation $-, and there is an embedding K -> ^ such that ^ extends v. (b) K is dense in fc^ (hence, many properties of K are inherited by ft by continuity) • (c) fL, is complete with respect to v. On the other hand, property (D) has no local analogue, since it reflects the global nature of *K. By property (D), in order to prove a statement about K, it is enough to prove the corresponding statement about *K. But this seems to have made the problem harder rather than easier. Nevertheless, by using the following principle of relativization, we have added a new dimension to the problem which allows us (in certain cases) to get a better handle on it. b) Principle of Relativization Vaguely speaking, this principle consists of considering the object *X relative to> X. Depending upon the nature of the object X, there are several variants to this principle: I) X = A is an abelian group: consider the factor group *A/A.
327 II) X = r is an ordered abelian group: consider the factor group r = *r/(r>, where
standard
1) A is finite
*A/A = 0
2) A/mA is finite
*A/A is m-divisible
3) f is bounded on S
f = 0 on *S
4) f-bounded subsets of S
£(s) = 0
»
s e S
are finite
5) f is quasi-quadratic (cf. f is quadratic on *A Lang[13] f p.86) on A Let us apply this principle to the number field K: in this case, we are interested in an arithmetic of *K relative to K. For this we shall use the following construction (basically due to Robinson[23],[24]). Let v £ * M K » ky definition, v is a map from *K* onto some subgroup r <= *Et. Set f = r y /
* : *KX -* f v by ^ = pr o v, where pr:r -» f is the projection. One easily checks that M s a non-trivial valuation in the sense of Krull (i.e. -fr satisfies the strong triangle inequality) - even if v is archimedean! Remark: If v is standard» i.e. v = *w is the canonical
328 extension of a valuation w of K, then $ coincides with ft as defined in III) above. On the other hand, if v is nonstandard, then v is already trivial on K and hence ^ = v. Since, by construction, ^ is trivial on K, the set d M = jj^^ :: vv 66 *MKJ defines an arithmetic of *K relative to K. We now come to Basic Philosophy: M K gives *K the structure of a "function field" (over K ) . §2. The Geometry of *K/K To explain the "geometry" of *K/K, let me list some properties of the arithmetic object (*K,ftK) which are analogous to geometric properties shared by (certain) varieties. The first of these may be interpreted as giving *K a "protective structure". To explain this, let me recall (cf. Lang[11j) that if V is a normal protective variety over K, and cp:V •* F31 an embedding in protective space, then the degree degi (D) of a divisor D on V is defined, and hence we have a function deg^:Div(V) -» Z on the divisor group Div(V) which is additive, monotone and vanishes on principal divisors. Furthermore, it is known that for f 6 K(V), deg^p((f)0D) = 0 « f 6 K. (Here, (f) w denotes the pole divisor of (f).) Let me sum these properties up by saying: deg^ is a non-degenerate degree function on V. Now a similar property is shared by *K: if £)(*K/K) denotes the divisor group of *K with respect to M K in the sense of Weil[32] (i.e. 2)(*K/K) is the smallest subgroup of TTf„ (the product extending over all v 6 *M~) which v j\ contains all principal divisors and is closed under the formation of minima and maxima) then v/e have: Property 1: $)(*K/K) has a non-degenerate, R-valued degree function 5.
329 Note that this immediately implies Property 1 f : The only MK-holomorphic functions of *K are the constants; i.e. 0 O. = K. To construct §, use the usual (logarithmic) size func tion on the divisor group $)(K) of K which is defined by s(D) = D
v(D). K
(Note that v is normalized!) Relativizing s yields an 1Rvalued homomorphism § on $(K) = *3)(K)/<$)(K)> . By making use of the canonical identification $)(K) ss §(*K/K), we can transport & to $)(*K/K). It is then clear that § is a degree function. The fact that 5 is non-degenerate, on the other hand, is equivalent to the assertion that a number field K (or, more precisely, the protective line IP^-) has only finitely many elements of bounded height. Now *K/K is a rather special function field, as the next property shows. Property 2: *K/K is simply connected in the sense that there is no proper finite extension L of *K # K (K = the algebraic closure of K) which is unramified at all valuations of *K«K/1. This property is (basically) the nonstandard equivalent of the Theorem of Hermitef6] which states that there exist only finitely many extension fields of a number field K with bounded discriminant and degree. Property 2 seems to suggest that *K/K is "rational", but this contention has serious drawbacks. For then, by an extended version of Lurothfs theorem, one would expect that every subfield F c*K which is a function field of one variable over K is also rational, but this is not so. To see this, we will use the next property of *K/K. Property 3: A function field F/K of one variable may be embedded in *K « F has infinitely many K-rational places.
330 Since it is easy to construct elliptic curves with infinitely many K-rational points, *K has many elliptic sub fields and hence cannot be considered to be "rational!f. On the other hand, Mordell's Conjecture suggests that *K/K is "almost rational" since we can reformulate the conjecture as: (M) Every subfield F c *K which is a function field of one variable over K has genus g ^ 1. Property 3 may be generalized to function fields of higher dimension as follows. Property 3 f : A function field F/K may be K-embedded in *K if and only if the set of K-rational places of F/K is (Zariski-) dense in the set of all places of F/K. In particular, we have: Property 3": Every finitely generated subfield of *K/K has a K-rational place. The last "geometric" property of *K/K which we will consider is analogous to the Lemma of Matsusaka which is the key point in proving the Theorem of Bertini. Property 4: There exists t 6 *K \ K such that K(t) is algebraically closed in *K. Again, this property is a reformulation of a classical theorem about number fields, namely the aforementioned Irreducibility Theorem of Hilbert (Theorem 3 ) . This equivalence was established by Gilmore and Robinson[4] in 1955, and used by Roquette[27] to give a nonstandard proof of Hilbert!s Theorem. More recently, Weissauer[33] noticed that it is possible to strengthen the Gilmore-Robinson Criterion as follows.
331 Proposition 1 (Weissauer[33j): A field K is hilbertian if and only if there exists t € *K K such that the algebraic closure Q. of K(t) in *K possesses the following property: (+) There exist only finitely many valuations v p of ft^/K such that Vp(t) < ()• Weissauer used this criterion to show that the class of hilbertian fields is very large. For example: Theorem 8 (Weissauer[33]): Let R be a Krull ring of dimension ^ 2. Then the quotient field of R is hilbertian. Since, by the Theorem of Mori-Nagata (cf. Nagata[2Oj, p. 118), the integral closure of a noetherian integral domain is a Krull ring (of the same dimension), we have: Corollary: The quotient field of a noetherian integral domain of dimension ^ 2 is hilbertian. In particular, every power series field k((t.,...ft )) in n £ 2 variables is hilbertian. (This is false for n = i!) Just to show the power of Weissauer1s Criterion, let me give a proof of Hilbertfs Theorem based on it. Proof of Theorem 3: Let t a K (t not a root of unity), and let t = t^, where uu e *IN \ IN. It is claimed that t satisfies (+) above. To see this, consider the set P t = (v e *M K : v(t) < 0 } . Since t is a power of t Q , we have P.r = P. . But P. is finite (since t f K ) , x x
o .
o .
°
and hence so is P. = \$ € M~: ^(t) < 0| . To conclude the proof of the claim, we now use Lemma (Robinson[24j): Let F D K(t) be a subfield of o t which is finite over K(t). Then every valuation v p of F/K is induced by a valuation ^ of *K/K.
332 Closely connected with this lemma (which can be deduced from Property 1f by using the Riemann-Roch Theorem) is the nonstandard formulation of the Siegel-Mahler Theorem (cf. Robinson-Roquette[25]): Theorem * 2 : Let F c *K be a function field over K of genus g ^ 1. Then every valuation v p of F/K is induced by a nonstandard valuation v = ^ of *Mr,. §3. The Mordell-Weil Theorem Using the nonstandard dictionary of §1 and the fact that *J C (K) = J C (*K), we can reformulate Theorems 6 and 7 as follows.
Theorem *6: There exists an integer m > 1 and a finite extension K1 of K such that J^KO/JQCK 1 ) is m-divisible. Theorem * 7 : There exists a real-valued function q on J Q ( K ) such that the relativized function 4:JC(*K) "*"ft is a positive definite quadratic form which vanishes precisely on J C (K). Remark: To establish the equivalence between Theorems 7 and * 7 , one also needs "Tate's trick" which enables one to replace a quasi-quadratic form by a quadratic form (cf. Lang[13], p. 8 7 ) . In keeping with the philosophy explained above, let us look at the function field analogue of these theorems; i.e. let us replace the field *K by a function field L/K, keeping in mind the "geometric" properties which we have established for *K/K. Beginning with Theorem * 6 , note that the following is true: jLf L/K :1s a simply connected function field,then J C (L) = J C (K). While this fact is no longer true if L/K is not finitely generated, one can prove the following (cf. [10]).
333 Proposition 2: Suppose that L/K is simply connected and that every finitely generated subfield of L/K has a K-rational place. Suppose further that C is a curve of genus g defined over K which has a K-rational point and a non-special divisor of degree g defined over K. Then the group J C (L)/J C (K) is divisible. Since L = *K satisfies the hypotheses of the proposition by Properties 2 and 3", Theorem *6 is indeed a consequence of Proposition 2. (Note that the additional hypotheses on C can always be realized after a finite extension Kf of K.) For the proof of Theorem * 7 , we are faced with the problem of constructing a quadratic form on J C (*K). Now in the geometric analogue, the existence of such a quadratic form is well-known, at least in the case when L/K is a function field of one variable. For then we are concerned with a surface S = C x Cf which is a product of two curves (Cf being a model of the function field L/K), and the divisor group of a surface carries a canonical symmetric bilinear form (D.E) e 2, given by counting the intersection indices of the two divisors D and E on S. It is easy to see that this pairing induces a pairing on J C (L), which is a quotient of a subgroup of the divisor class group Pic(S) of S. For historic reasons, let us write a(D)
=
- (D.D),
for D f JQ(L)j
a is called the Weil metric (or Weil trace) on J"C(L). The fundamental theorem concerning this metric is the celebrated Theorem of Castelnuovo-Severi (cf. Weil[3i], Roquette[26]) which may be stated as follows. Theorem 9 (Castelnuovo-Severi): The Weil metric is positive definite on J Q ( L ) and vanishes only on J C (K). There is a striking resemblance between Theorem *7
334 and the Theorem of Castelnuovo-Severi; what remains to be done is to make the analogy precise. To begin with, we have to generalize the notion of an intersection product in such a way that it is applicable to the general situation which we have in mind. For this, we will use Roquettefs interpretation (Roquette[26]) of the intersection product. That is, we view the surface as being fibered over the curve C f , and define an intersection divisor D.E on C 1 (called divisor residue in Roquette[26]). By taking the degree of this divisor, one arrives at the previous concept of the intersection product. This method can easily be generalized to an arbitrary base variety C f (not just a curve). Somewhat more difficult is the generalization to a "non-geometric base" since the local rings involved need no longer be noetherian. Here, the term "non-geometric base" refers to the following situation: L/K M C
a field extension (of transcendence degree ^ 1) a set of valuations of L/K a curve defined over K
Then one has (cf. [9]): Theorem 10: To each pair D, E of disjoint ( = without common components) divisors of C which are defined over L, there exists an intersection divisor D.E e 5)(M,L) with the following properties: (1)
(D1 + D 2 ).E
=
(2)
D £ 0, E £ 0
=» D.E £ 0
(3)
D.E
=
D r E + D 2 .E
E.D
(4) D, E rational over K
=> D.E = 0
(5)
=* D ^ E ~ D 2 .E
D 1 ~ D 2 , deg(E) = 0
(6) If D = min max ( f ^ ) , with f ± . $ F x (where F = K(C)), and E = P is a prime divisor rational over L, then D.P = min max (f. .(P)).
335 Since for each pair D, E of divisors of C there exists a divisor Df ~ D which is disjoint from E, and since, by (5) above, the divisor class of D f .E does not depend on the choice of D 1 when deg(E) = 0, the intersection product defines a symmetric bilinear map on the group of divisor classes of degree 0 on C which are rational over L. As this group may be identified with J C (L), we obtain: Corollary; The intersection product induces a symmetric bilinear map (•••)
: J C (L) X J C (L)
-> <£(M,L),
where £(M,L) denotes the divisor class group of L/K. If, as in the case that L = *K and M = M K , the divisor (class) group possesses a degree function 6 with values in some totally ordered group r, then we can define a Weil metric with respect to 6 by setting, for D, E 6 J Q ( L ) >
=
=
-6(D.E).
One can then generalize the Theorem of Castelnuovo-Severi as follows (cf. [9]). Theorem 11: The Weil metric < , >^ is positive definite on J C (L). If, in addition, the degree function 6 is non-degenerate, then
=
6(P.(W+2P Q )) f
where W is a canonical divisor on C which is rational over K.
336 Let us now apply the above to the case that L = *K and 6 = S is the degree function on *K/K as constructed in Property 1. Then, by Theorems 10 and 11, we have a positive definite quadratic form on *J C (K) which vanishes precisely on J C (K). Thus we are through with the proof of Theorem *7 once we have verified that this quadratic form is of the form $ for some function q:Jc(K) -> R. But this is an easy exercise, since the above quadratic form is defined "geometrically". Summing up, we therefore have a canonical quadratic form h j on J Q ( K ) such that
fij(D) =
for all D e *J C (K) = J C (*K). The form h j is called the Nferon-Tate height on J C (K). We can also use intersection products to give a nonstandard interpretation of the height functions h* on the curve. Here, h» = h <> cpA, where cp^ is a rational map associated to the divisor A (cf. Lang[1i]). (Note that cp^ and h. are not uniquely defined.) Using property (6) of Theorem 10, we have: Proposition 4: If A is a K-rational divisor on C of the form A = -min (f*)* then h A (P)
=
S(P.A),
V P € *C(K) = C(*K)
Since the above equation is valid for any height function associated to A, and the right hand side depends only on A (and P ) , we see that for any two choices h^ and h^ of height functions the difference h» - h! is bounded on C(K). In other words, h. is equivalent to h|: h. ~ h 1 .. Using the Adjunction Formula, we can apply this to compute the restriction of h j to the curve C which we view as embedded in JL P via the map ^ P : P -» P-P^. o Q
Proposition 5:
hT o $D J p o
~
h,. OTD . w+2P o
337 1
§4. The Theorems of Mumford and Dem tjanenko-Manin We shall prove the following equivalent form of Mumford's theorem. Theorem 4 f : Let C be a curve of genus g ^ 2 defined over a number field K, and assume that the points P i 6 C(K) are ordered according to increasing height h w ( P . ) f where W is a canonical divisor on C. Then there exist integers N and N such that for all n ^ N
The proof of this is based on the following elementary lemma about euclidean spaces (cf. Mumford[18]). Lemma 1: Let V be a euclidean space with norm || || and inner product < , >, and let v^, v^, ... be a sequence of vectors such that ||vJ| ^ ||vp|| ^ ... • Suppose that there exist constants c > 0 and K ^ 0 such that
< v i* v j>
(D
~ ^H v i'i 2 + Hvo!!2) + *
holds for every pair i ^ j. Then there exists an integer N such that for all n with ||v || ^ 3C«K, we have
We shall apply this lemma to V = J Q ( K ) g)lR, which is finite dimensional by the Mordell-Weil Theorem and becomes a euclidean space under the N6ron-Tate height hj (cf. Lang[13] for a discussion of this). For v n we shall take vn
=
t(P a )
-
(2g-2)Pn-¥.
Thus, the proof of Theorem 4 1 will be complete once we have: Lemma 2: a) h j . y ~ 2g(2g-2)hw.
338 b) There is a constant K ^ 0 such that (1) holds with c = 2g. The nonstandard version of this lemma is as follows. Lemma * 2 : a)
= 2g(2g-2)fl(P.W), V P € C(*K).
b)
*
rank J C (K),
where E = K(C f ) is the function field of Cf over K. To see that Theorem 5f is equivalent to Theorem 5, note that by Deuring[2] or Weil[3i] Hom K (J c ,J c ,)
^
J C (E)/J C (K),
both groups being isomorphic to the group of K-rational correspondences between C and C 1 . By Property 3 of *K/K, the nonstandard reformulation of Theorem 5 1 is: Theorem * 5 ; Suppose that E c *K is a function field of one variable over K. Then for every curve C defined over K we have rank J C (E)/J C (K)
^
*rank *J C (K).
339
Next, we need a criterion for bounding the ranks of J C (E)/J C (K) and J C (K). To do this, let o ( , ) be the bilinear form associated to the Weil metric on J Q ( E ) hj( , ) be the bilinear form associated to the NferonTate height on J C (K). Lemma 3: a) rank Jp(E)/Jp(K) ^ n if and only if there exist D., •.., D n 6 J L» P (E) such that det (o(D.,D.)) 4 0. i i j b) rank J C (K) ^ n if and only if there exist D^, ...t D n € J C (K) such that det (h^D^D..)) 4 0. Statement a) resp. b) of the lemma follows from the Theorem of Castelnuovo-Severi resp. the Theorem of NferonTate which asserts that o resp. h j is a non-degenerate, positive definite quadratic form on Jp(E)/Jp(K) resp. J c (K)/J c (K) T o r . Corollary: If E c *K, then rank J C (E)/J C (K) ^ *rank *J C (K) if and only if det(*h J (D± ,D .)) ^ 0 for all D ^ ..., D n e J C (E) with det(a(D±,D.)) 4 0. Thus, to conclude the proof of Theorem * 5 , we need to verify the second condition of the corollary. However, this follows from: Lemma 4: There exists p 6 *ff* such that for all D^, ..., D
n
€ J
J c(*K )K )wewehave c( E ) )c cJ c(*
det(*hJ (D ±,D -
det(a(D.,D.)).
Here, as usual, the symbol a *• b (for a, b e *R) means that a - b is infinitesimal. Proof: Since det is homogeneous of degree n (and since a(D.,D.) £ Z ) , it is enough to show that ^ ( D ^ D . )
340 *• *h T (D. ,D .)/p, for some p (independent of D.and D . ) . By definition of the Weil metric, we have (1)
a(D i ,D j )
=
-degE(D±.D..),
where D..D . denotes the intersection divisor of D. and D . in E, On the other hand, by construction of the Neron-Tate height, (2)
h J (D i ,D j )
=
-§(D i .D J ),
where now the intersection divisor D..D_. is viewed J as a divisor of *K/K. Finally, by the "generalized lemma of Artin-Whaples" (Robinson-Roquette[25], p. 152), there exists an infinitely large p £ *R such that for any divisor A of E/K (3)
deg E (A)
-
§(A)/p.
(Note that since p is infinitely large, §(A)/p may be viewed as an element of *R modulo infinitesimals,) Combining (1), (2) and (3) then yields the result.
References [1] V.A. DEMfJANENKO, Rational points of a class of algebraic curves, Izv, Akad. Nauk SSSR Ser. Mat. 30 (1966), 1373-1396 = Am. Math. Soc. Transl. (2)66 (1968), 246-272. [2] M. DEURING, Arithmetische Theorie der Korrespondenzen algebraischer Funktionenk5rper I. J. reine angew. Math. VJOL ( 1 937), 161-191. [3] M. DEURING, Arithmetische Theorie der Korrespondenzen algebraischer Punktionenkorper II. J. reine angew. Math. 18^ (1940), 25-36. [4] P.C. GILMORE, A. ROBINSON, Metamathematical Considerations on the relative irreducibility of polynomials. Can. J. Math. 2 (1955), 483-489. [5] R. HARTSHORNE, Algebraic Geometry. Springer Verlag, New York, 1977. [6] C. HERMITE, Extrait d f une lettre de M. C. Hermite a M. Bourchardt sur le nombre limits d f irrationality
341
auxquelles se r6duisent les racines des Equations £ coefficients entiers complexes d f un degrfe et d f un discriminant donnfes. J. reine angew. Math. ^3 (1857), 182192 = Oeuvres I (Gauthiers-Villars,Paris7T9O5),415-428. [7] D. HILBERT, Uber die Irreduzibilitat ganzer rationaler Funktionen mit ganzzahligen Koeffizienten. J. reine angew. Math. 212 (1892), 104-129 = Gesammelte Abhandlungen II (Springer, Berlin, 1933), 264-286. [8] E. KANI, Nonstandard diophantine geometry. In: Proc. Queen's Number Theory Conference, 1979 (P. Ribenboim, ed.), Queen's Papers in Pure and Applied Math. No. 54 (Queen's Univ. Press, Kingston, 1980), 129-172. [9] E. KANI, Eine Verallgemeinerung des Satzes von Castelnuovo-Severi. J. reine angew. Math. 318 (1980),178-220. [10] E. KANI, On the Nferon-Tate height on the Jacobian of a curve. To appear. [11] S. LANG, Diophantine Geometry. Interscience, New York, 1962. [12] S. LANG, Les formes bilinfeaires de Nferon et Tate. Sem. Bourbaki 1963/1964, no. 274. [13] S. LANG, Elliptic Curves: Diophantine Analysis. Springer Verlag, Berlin, 1978. [14] K. MAHLER, Uber die rationalen Punkte auf Kurven vom Geschlecht 1. J. reine angew. Math. T£0 (1934),168-178. [15] YU. I. MANIN, The Tate height of points on an abelian variety. Its variants and applications. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1363-1390 = Am. Math. Soc. Transl. (2) 52TT966), 82-110. [16] YU. I. MANIN, The p-torsion of elliptic curves is uniformly bounded. Izv. Akad. Nauk SSSR 33 (1969), 459465 = MAth. USSR Izv. 3 (1969), 433-438. [17] L.J. MORDELL, On the rational solutions of the indeterminate equation of the third and fourth degrees. Proc. Cambridge Phil. Soc. 2J[ (1922), 179-192. [18] D. MUMFORD, A remark on Mordell's Conjecture. Am. J. Math. 82 (1965), 1007-1016. [19] D. MUMFORD, Abelian Varieties. (App. by C.P. Ramanujam and Yu.I. Manin). Oxford Univ. Press, Oxford, 1974. [20] M. NAGATA, Local Rings. Krieger Publ. Co., Huntington N.Y., 1975. [21] A. NERON, Quasi-fonctions et hauteurs sur les varifetis abfeliennes. Ann. Math. (2) 82 (1965), 249-331. [22] A. ROBINSON, Non-standard Analysis. North-Holland Publ. Co., Amsterdam, 1966.
342
[23] A. ROBINSON, Algebraic function fields and non-standard arithmetic. In: Contributions to Non-standard Analysis (W.A.J. Luxemburg, A. Robinson, eds.) NorthHolland Publ. Co., Amsterdam, 1972 = Selected Papers II (Yale Univ. Press, New Haven, 1979), 256-269. [24] A. ROBINSON. Nonstandard points on curves. J. Number Th. 5 (1973), 301-327 = Selected Papers II, 306-332. [25] A. ROBINSON, P. ROQUETTE, On the finiteness theorem of Siegel and Mahler concerning diophantine equations, J. Number Th. £ (1975), 121-176 = Selected Papers II, 370-425. [26] P. ROQUETTE, Arithmetischer Beweis der Riemannschen Vermutung in Kongruenzfunktionenkorper beliebigen Geschlechts. J. reine angew. Math. 191 (1953), 195-252. [27] P. ROQUETTE, Nonstandard aspects of Hilbert's irreducibility theorem. In: Model Theory and Algebra - a Memorial Tribute to Abraham Robinson (D.H. Saracino, V.B. Weispfenning, eds.) Lecture Notes in Math. 498, (Springer Verlag, Berlin, 1975), 231-275. [28] C.L. SIEGEL, liber einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phy.-Math.. Kl. 1929, Nr. 1 = Gesammelte Abhandlungen I (Springer Verlag, Berlin, 1966), 206-226. [29] K.D. STROYAN, W.A.J. LUXEMBURG, Introduction to the Theory of Infinitesimals. Academic Press, New York, 1976. [30] A. WEIL, Lfarithmfetique sur les courbes alg6briques. Acta Math. 52 (1928), 281-315 = Oeuvres Scien. I (Springer Verlag, New York, 1979), 11-45. [31] A. WEIL, Sur les Courbes Alg6briques et les Varifetfes qui sfen deduisents. Hermann & Cie., Paris, 1948. [32] A. WEIL, Arithmetic on algebraic varieties. Ann. Math. (2) 53 (1951), 412-444 = Oeuvres Sc. 1,450-482. [33] R. WEISSAUER, Hilbertsche Korper. Dissertation, Heidelberg, 1980.
AN ADELIC PROOF OF THE HARDY-LITTLEWOOD THEOREM ON WARING's PROBLEM Gilles Lachaud Universite de Nice 7, Avenue Bieckert Q6000 - Nice, France
1. THE GAUSS TRANSFORM Let degree
F
be an homogeneous polynomial with
d , and with coefficients in the ring
n _Z
variables, of of rational
integers. Let
of dimension
n , where
A_
is the ring of adeles of the field Q_
of rational numbers, and let [12]).
An
<J> be a standard function on the adelic vector-space x
De
the Tate character of
We define the Gauss transform of
cf>
A
[relative to
(cf.
F )
as
A where
£ e A^ and where
dx
/\n
is the Haar measure on
such that
the induced quotient measure satisfies [ JA n /Q n
dx = 1
The set F = P u {0} |x|
P
is the set of prime numbers, and the set
is the set of places of the field
Q_ (we denote by
the usual archimedean absolute value of the number If
K
is a field and if
with coefficients in
F
x eQ).
is an homogeneous polynomial
K , we set
A K (F) = {x e K n and we say that
F
| dF(x) = 0}
is strongly non-degenerate
on
K
if
A K (F) = {0} . This definition allows us to introduce the following assumptions : (SS 1)
One has
(SS 2)
The homogeneous polynomial
for every
n > 2d
and
d > 3 *
p c P .
We note that the Fermat form F(xr...,xn) -
F
is non-degenerate over
Q
344 satisfies the condition
(SS 2 ) .
We then have the following result of Igusa [5] depending on Deligne's theorem on trigonometrical sums [2,9]. THEOREM 1. gral
G- (([>,£)
under the assumptions
(SS 1) and (SS 2) the inte-
converges and defines an integrable function on
A_ .
2. THE SINGULAR SERIES If
p e ?
and if
U(t)={xeQ Since if
F
n
t e Q
we set
|F(x)=t}
.
is homogeneous, the hypersurface
t * 0 .
We denote by
(u), )
U (t)
is non-singular
the Leray form on
U (t) ;
we
have (u)t(x)) if
A dF(x) = dx
x e U (t) . We denote now by
of the hypersurface of the measures space
IL(t)
(cf. [10]) the set of adelic points
F(x) = t , and by
(w. )
.
If
<$>
u),
the restricted product
belongs to the Schwartz-Bruhat
S(A ) , we set <j>(x) a), (x) •
We
then
have
u A (t)
the
THEOREM 2 (Wei1-Igusa).
Under the assumptions
(SS 1) and
(SS 2 ) , one has S A ($,t) = GA(cJ),-t) for every
t * 0 .
The f o r m u l a decomposable
<j)(x) =
then
for
of T h e o r e m
2 implies
function:
n _ cj) (x ) P P 6 P P
x = (x ) P
P£P
e A ~
, we
have
that
if we take for
$
a
345
sA(4>,t) =
n
s^(<^,t) su ,
p e P
P
P
with
t
u (t) *P
P
p We now make the assumption that function of
_Z
$
is the characteristic
, and we just set, once this assumption is made,
sp(*p,t) - spct) and
sjt) f P
n p
s (t) . P
The Theorem 2 then also implies that we then have S (t) = 1 + p
I A(p e ) e> 1
,
with A(q)
=
I Y (a) exp (-2i7rta/q ) (a,q) = 1 q
where Y (a) = q ^
n
I exp(2i7rF(x)/q) x mod q
;
we thus recover the usual expression of the Singular Series for S f (t)
as in [1] and [4]. We have the following result: PROPOSITION 2.
If
u
A(t)
is non-empty
when
t
is suf-
ficiently large, then
Sf(t) We now assume that the homogeneous polynomial form and that
d
is £even;
we fix
t e N
and set
in such a way that if
II x || x.o| , . . .n ,o | x | ) > R o = Hax ( I I when
F
is the Euler
R = [t
] + 1,
346 x = (x, , ... , x ) e R n , i
Let
n
\\) be a
—~
C
then
function on
|x| Q < R - (1/3) , and equal to
for
x = (x,, , . . . , x ) e R 1 n —
for any
F(x) > t
.
0
.
R_ which is equal to if
|x| Q > R + (1/3) .
With this choice of
T
1
if
We set
x < t , if we define the latter as
s (t) = tCn/d)"'1Il, J
(recall that
U (1)
f
J(1)
u.) 1
u (i:
is compact).
sA(t) = so(t)sf(t)
If we put
,
we then have
S A (CJ>,T)
for
=
SA(T)
x ^ t , when
<j>
is the decomposable function on
structed from the functions
$
and
4>
A
con-
defined above.
3. THE HARDY-LITTLEWOOD THEOREM Here, we take for is even;
we set
D = 2
F the Fermat form and we assume that
d
and
N(t) = # {x e Z n | F(x) = t} for
t e N .
We are now going to give a reading, following the
adelic language, of the proof given in [1] of the HARDY-LITTLEWOOD THEOREM.
If
d > 3
and
n > dD , then
N(t) = S A (t) (1 + 0(t" 6 )) ith wi
6 > 0 . We have to specify that this assertion is equivalent to the
following:
347 N(t) = V o S f C t ) t ( n / d ) " 1 + 0 ( t C n / d ) " 1 " 9 ) once the global Singular Series has been explicited. sufficiently large, we know that
U/.(t) * $
If
t
is
(cf. [1], lemma 11);
thus the Proposition 2, joined to the above assertion, implies that N(t) > 0
and consequently
With the function N(t) = # {x £ Qn
Uptt) * (j> .
"chosen in n° 2, we have
| <J)(x) = 1
and
F(x) = t}
,
so that in fact the Hardy-Littlewood theorem establishes a relation between
N(t) =
I (J>(x) u a ct]
and
S (t) = f U
If
<j>(x) a) Cx) . A(t)
£ e A , we introduce the (finite) trigonometrical sum
where Z_
if
on
F3
\p = Jlty , denoting by p e P
and by
ty
introduced in n
fU)n =
I
ty
the
2.
the characteristic function of C°°
function with compact support
We thus have
x £ Qn
= I xUt) N(t) and consequently,
N(t) = f We set
also
f so
that
f U ) n x^-tC) d^ .
ip(x) x^x A
E,) dx
,
(*)
348 In o r d e r to e s t a b l i s h a comparison between N ( t ) and S.(t) , we a r e going to i n t r o d u c e a subset M <= A such that f and g n are n e g l i g i b l e s out o f M and such that the d i f f e r e n c e f n - g n is n e g l i g i b l e in N . F o r £ e A , w e put Q(5) •
n Max(1 , | 5pn |pn ) > p eP
and w e define t h e major M * U
set in A
€ A | | C | 0 * F f d + <S
(depending on
6 > 0 ) as
and 0 ( 5 ) * R 6 > -
The r e s t r i c t i o n of the p r o j e c t i o n from A to A / £ is i n j e c t i v e on M ; w e denote its image by M , and w e define the minor set m as the c o m p l e m e n t a r y of M in ^/O, • Using W e y l ' s i n e q u a l i t y , w e h a v e : P R O P O S I T I O N 3. If
n > dD and
dD n< 6 <
1 , then
J mm J
i >° •
A careful estimation of the behaviour of the function
g
shows: PROPOSITION 4.
If
L with
n > 2d , one has ? n-d-6 2
62 > 0 .
With t h e a i d of t h e P o i s s o n summation formula on /\ , w e can establish the P R O P O S I T I O N 5. if
with
5 < 1 , and if
n > 3D , then
03 > 0 . Since
R
~ t , the Hardy-Littlewood theorem is then a conse-
quence of relations (*) and (**)
and of Propositions 3, 4 and 5.
349 The reader will find in the article [7] detailed proofs of the preceding propositions, together with a survey of the theory of the Gauss transform.
What I would like to add, finally, is that
I hope to have given an illustration of the fact that the adelic language can be useful in itself without any intervention of class field or algebraic group theory:
like the theory of congruences,
this topic, which appeared for rather elaborate purposes, can be used for current mathematics. REFERENCES [1] H. Davenport, Analytic methods for diophantine equations and approximations, Ann Arbor Publishers, Ann Arbor, 1962. [2] P. Deligne, La conjecture de Weil I, Publ.Math. I.H.E.S. 4_3 (1974), 273-307. [3] W.J. Ellison, Waring's problem, Amer.Math. Monthly 78 (1971), 1Q-36. [4] G.H. Hardy and J.E. Littlewood, Some problems of 'Partitio Numerorum', I, II, IV, VI, in the Collected works of G.H. Hardy, vol. 1, 405-505, Oxford University Press, Oxford, 1966. [5] J.I. Igusa, On a certain Poisson formula, Nagoya Math.J. 53 (1974) 211-233. [6] J.I. Igusa, Lectures on forms of higher degree, Tata Institute of Fundamental Research Nr 59, Springer, Berlin, 1978. [7] G. Lachaud, Une presentation adelique de la Serie Singuli^re et du problSme de Waring, to appear in 'L'Enseignement Mathematique 1 . [8] T. Ono, Gauss transforms and Zeta functions, Ann. of Math. 9J_ (1970), 332-361 . [9] J.P. Serre, Majoration de Sommes exponentielles, Journees Arithmetiques de Caen, Asterisque 41-42 (1977), 111-126. [10]
A. Weil, Adeles and algebraic groups, I. A.S. Princeton, 1961.
[11] A. Weil, Sur la formule de Siegel dans la theorie des groupes classiques, Acta Math. 113 (1965), 1-87, in Oeuvres Scientifiques, vol. Ill, 71-157, Springer, Heidelberg, 1979. [12] A. Weil, Basic number theory, Grundl.Math.Wiss.Bd. 144, Springer, Heidelberg, 1967, 3rd ed. 1974.
CLASS NUMBERS OF REAL ABELIAN NUMBER FIELDS OF SMALL CONDUCTOR
By F.J. van der Linden,
Introduction For the class number of an abelian numberfield decomposition
h = h .h
maximal real subfield
K
, where
h
K
we have a
is the class number of the
of K, and
h
is a positive integer, for
which there exists an explicit formula. In this lecture we are concerned with the determination of
h
, i.e. the determination of the
class number of a real abelian number field. For an abelian number field
K
fined as the smallest positive integer with
£f
a primitive
the conductor f(K) f
for which
is de-
K c Q(c_) ,
f-th root of unity. In this lecture we show
how to compute the class numbers of most real abelian number fields of conductor
< 200 , in some cases assuming the generalized Riemann
hypothesis. For more details see [33.
351 § 1 The results In theorems 1, 2, 3 and 4 we list our results. By GRH we denote the generalized Riemann hypothesis for the zeta-function of the Hilbert class field of
Q(c r / V x) . The Euler function is denoted by r {&)
Theorem 1 Suppose that h(K) = 1 if Theorem 2 sume GRH. Then h(K) = 4
is a prime power. Then
<\>(q) < 66.
Suppose that
if
f(K) = q
f(K) = q
is a prime power, and as-
q - 163
h(K) = 1 for all other
K
for which
Suppose next that f(K) = n is not necessarily a prime power. The genus field G(K) of K is defined as the maximal totally unramified extension of K which is abelian over Q . It is contained in Q(c ), and it is equal to G*(K) n R , where G*(K) is the smallest field containing K which is a composite of abelian extensions of flj of prime power conductors. Because g(K) is a subfield of the Hilbert class field H(K) of K , it follows that the genus factor g(K) = = [G(K) : K] divides the class number h(K) . Clearly g(K) = 1 for fields of prime power conductor. Theorem 3
Suppose that
f(K) = n
K = Q(C136)+
h(K) = 2.g(K) = 2
for
h(K) =
for all other
g(K)
is not a prime power. Then
K
for which
n < 200,
g(K)
for
n = 165.
Suppose that
f(K) = n
h(K) = 2.g(K) = 2
for
n ± 148, n + 152.
is not a prime power. As-
K = Q(C]36)+
352 h(K) = 2.g(K)
if
n = 145 and
\fi45" e K,
h(K) = 4.g(K) = 4
if
n = 183 and
12 | [K : $ ] ,
h(K) =
for a l l other
g(K)
K with
n < 200.
353 § 2 The method The results of section 1 are derived by a method developed by J.M. Masley [ 4 ] , This method consists of two parts: 1) Determine an upper bound 2) Test for each prime
B
p < B
for the class number whether
h .
p |h .
1) The classical nethods from geometry of numbers lead, for most fields, to class number bounds that are too large to be useful. For fields with small conductors, however, there is a better technique, depending on the discriminant lower bounds proved by Odlyzko [5; cf. 6 ] , If nant
K
A
over
A
A11
is a totally real field of degree
n
and discrimi-
Q , then Odlyzko proved that ~
E
A > A .e for several pairs (A, E ) , like
A = 28.668
and
E = 8.0001
A = 60.704
and
E = 200.01
A -* 60.840
for
E -> oo
If we assume the generalized Riemann hypothesis for the zeta-function of
K
we can also take
A = 30.338
and E = 8.0894
A = 213.626
and E = 5.7672 x 1 0 2 6
A •> 215.333
for E -* oo
These bounds lead to class number bounds in the following way. Apply the above inequality to the Hilbert class field Since
H(K) has discriminant
yields AA so
h
h.n -E > AA .e
A
and degree
n.h over
H(K) of K . Q , this
354 n log A - log A provided that the right hand side is positive. In this way we can get class number upper bounds if the root discriminant A
satisfies
A
< A for some
A ; that is,
A 1 / n < 60.840, or when we assume GRH: A 1 / n < 215.333. In the latter case practical upper bounds are obtained only when
A
then about 160. There are only finitly many abelian
is smaler
K
satisfying
this condition. Since the root discriminant and the conductor are roughly of the same size, the conductor is a good measure of how far we can go. 2) There are two main methods to decide whether a prime divides the class number. The first ones uses the Galois action and works mostly for primes not dividing
[K : Q ] .
Definition 5 Let
L/K
prime number not dividing
be an abelian extension, and
n = [L : K ] . Then we denote by
the p-primary part of the class group of and
where
L . If
L/K
p
a
Cl (L)
is cyclic
a
is a generator of Gal (L/K) then we put * (a) C1*(L/K) = {a £ Clp (L) I a n = 1 } p $
is the n-th cyclotomic polynomial.
Theorem 6 Let number not dividing group of
L
L/K
be an abelain extension, and
[L : K ] . For the
p-primary
p
a prime
part of the class
we have a decomposition:
Cl (L) = 9 C1*(M/K) , P P where the summation is over the M
with
K c M c L
and
M/K
is
cyclic.
This theorem and the following one are due to Frohlich [1], A proof of them is given in section 3. Using theorem 6 gives us information about the p-part of the class number from the p-parts of the class number of its
sub-
355 fields, especially if
L/K
is non-cyclic. Even if
L/K
is cyclic
we can use it in may cases because of the following theorem.
Theorem 7
For
C1*(L/K) , as defined in definition 5 we have: P f
#
# C1*(L/K)
is a power of
itive integer for which
p
p
, where
f
is the smallest pos-
s 1 mod [L : K ] •
This is a very strong restriction because often we have p f > B.
For
p
dividing [K : Q]
we can make use of the following
theorem
Theorem 8 sion with
Let
GalCK/K^)
finite) primes of unramified outside cyclic extension
Proof
If we take if
K/K~ a
K
be a
p^-group. Let
and
q
P
be a set of (finite or in-
a prime of
P u {q} . If M/K Q
p-extension, i.e. a Galois exten-
K . Suppose that
p | h(K)
of degree
p
K/K Q
is
then there exists a
that is unramified outside
P.
See Masley [4] (2.6) o
P = 0
K / K Q is a
in theorem 8 we recover a result of Iwasawa [ 2 ] :
p-extension ramifying at at most one prime, then
p | h(K) -> p I h ( K Q ) . If K/KQ
is a
p | [K : dj] we choose a subfield p-extension. Suppose that
KQ
of
K
for which
p | h(K) . The extension of
M , implied by theorem 8 , is equivalent, by class field theory, to the exsistence of a certain quotient of a ray class group of
K~, In
many cases it is possible to disprove the existence of this quotient group, by calculations with the units of M
does exist we can look if
fields of conductors 136
MK
K~ . If it turns out that
is unramified over K . For the
and 183 the Hilbert class fields and class
numbers were found in this way.
When we have used the above methods, we are sometimes left
356 with a few primes power is
p < B . Usually these are primes of which a small
1 mod [K : Q]. For these primes we can try to use reflec-
tion principles or the relation between cyclotomic units and the class number. In some cases we can use the following theorem, which is a counter part to theorem 8. Theorem 9
Let
L/K
be an extension of number fields. Then
h(K) | h(L).[L : K]. If no intermediate field then
M ^ K
of
h(K) | h(L).
Proof
See Masley. [4] (2.2), n
L/K
is unramified over
K,
357 § 3 The proofs of theorem 6 and 7 Proof of theorem 6 Let
L/K
be an abelian extension with Galois group
any intermediate field note by
C(L)
a
^
^
M
of
L/K
we write
the subgroup of
elements which are left fixed by
G . For
C(M) = Cl (M). We de-
C(L)
consisting of those
Gal(L/M) . We have maps:
i : C Q O - C ( L ) G a l ( L / M ) , N : C ( L ) G a l ( L / M ) + C(M) induced by the inclusion For elements
a
of
Nx(a) = a [ L since
: M ]
and the ideal norm map respectively.
and
and
3
C(L)
R
C(L)Gal(L/M)
of
- 3CL
lN(B>
this means that
C(M) with
For a ring rator
C(M)
p / [L : M]
identify
M c L
Gal(L/M)
i
: M]
.
and
N
by means of
and a cyclic group
U
we have
are isomorphisms. We i .
of order
m , with gene-
a . We define R(U) = R[U] / <£> (a). R[U] m
where
$ m Let
is the H
m-th cyclotomic polynomial
be the set of subgroups
clic. For each
H e H
H c G
for which
G/H
is cy-
we have an natural ring homomorphism
• H : RCG] + RCG/H] -> R(G/H) . The kernel of
$
where
H
a
and
Take
is generated by
ri
generate
G
and
R = Q . Then each
{T-1 : T € H}
and
$ (a) , m
M = #G/H.
is a field, so the ideals
ker(<(>„) are maximal, and it is easy to see that they are different, n By the Chinese remainder theorem it follows that the combined ring homomorphism <J> : Q[G] - n R
g H
Is surjective. An easy calculation shows that both sides have the same
Q-dimension so
<(> is an isomorphism. Tensoring with
restricting the coefficients to
TL
Q
and
we obtain an inclusion of
2Z [G] as a subring of finite index in T7TT u TL (G/H) . This index p h € n p divides the discriminant of TL [G] over TL , which is a unit in P P
358 7L . We conclude that P
V G ] ~ ^ e H Zp This isomorphism leads, for any
7L [G] -module
A , to a decomposi-
tion A
Where With
A^
n
H
€
H *R
is the largest submodule of
A = C(L)
A
annihalited by
ker((j> ) .
we find
A R = C1*(L H /K),
TJ
where
L
is the fixed field of
H . This proves theorem 6.
Proof of theorem 7 Let group
G
L/K
be a cyclic extension of degree
generated by
C1*(L/K) , the stabilizor of orbits we find that
n , with Galois
a . If we prove that for each a
in
G
is
a ^ 1
in
{1} , then by counting
#C1*(L/K) s 1 mod n , and the theorem is
P proved. So suppose that the fixed field of
a
a a = a, where
On the other hand, since _ / \ $ n (a) a
divides
n , and L f
. Then « di <->
and
d ^ n
- «n/d-
$ (a) n
divides
£?_., a i—l
1
in the groupring
= 1 , we have NL/L,(a) = 1
Hence quired.
a
= 1 , and from
p | n
it follows that
a = 1 as re-
359 § 4 Example Let
K
be the field of conductor 95 and degree 12 over
It is cyclic over 1 over 0) , for
Q . Denote by
i = 2, 3, 4, 6. From
class number bound gives sion get
K/K-}
we get
5 \ h ( K ) . So The extension
the prime P = {p}
K/K~
[4] we get
K
Q .
of degree
h(K.) = 1 . The
h(K) < 6 . Using theorem 7 for the extenK/Q
we
is a 2-power. ramifies only at the prime
q over 5
p over 19. Using theorem 8 for the extension we see that if
M
the subfield of
3 \ h(K) , using it for the extension h(K)
M/K~ , cyclic of degree 2 theory
K.
2 | h(K)
and
K/K~ , with
then there exists an extension
and only ramifying at
p . By class field
belongs to a quotient group of order 2 of
(0(K 3 )/p)* / (0(K 3 )* mod p) But this is a group of odd order, because -1 e <9(K~)*
N(p) = 1 9 = 3
mod 4
and
is of order 2. So we have reached a contradiction and
2 \ h ( K ) . This means that
h(K) = 1.
Literature
[1]
A. Frohlich On the class group of relative abelian fields. Quart. J. Oxf. (2) 2 98-106 (1952)
[2]
K. Iwasawa A note on class numbers of algebraic number fields. Abh. Math. Sem. Univ. Hamburg _20 257-258 (1956)
[3]
F.J. van der Linden Class number computations of real abelian number fields. To appear.
[4]
J.M. Masley Class numbers of real cyclic number fields with small conductor. Compositio Math. 37. 297-319 (1978).
[5]
A.M. Odlyzko Discriminant bounds. Unpublished tables (nov. 1976).
[6]
G. Poitou Minorations des discriminants. Sem. Bourbaki 479 (1976).
ALGEBRAIC INDEPENDENCE PROPERTIES OF VALUES OF ELLIPTIC FUNCTIONS D.W. Nasser
G. Wustholz
Department of Mathematics
Gesamthochschule Wuppertal
University of Nottingham
Gau(3stra$e 20
University Park, Nottingham, U.K.
D-5600 Wuppertal, W. Germany
1. INTRODUCTION In 1949 A.O. Gelfond published a method of obtaining
algebraic
independence results for certain values of the exponential function. Since then a number of authors have developed this method, notably W.D. Brownawell, A.A. Smelev, R. Tijdeman, and M. Waldschmidt.
A
good exposition of the most recent theorems can be found in [ 6 ] , together with references and historical remarks.
The object of this
note is to announce the natural elliptic analogues of these theorems. For the convenience of the reader we state first the known results in the exponential case.
For an integer
n > 1
let
u^,...^
be complex numbers linearly independent over the rational field and for an integer
m > 1
let
linearly independent over
Q .
THEOREM 1.
If
vy],...,v
mn > 2m + 2n
exptu.v.) are algebraically THEOREM 2.
THEOREM 3.
independent If
then at least two of the numbers
over
Q .
mn > 2m + n
then at least two of the numbers ( 1 < i < n , 1 < i < m )
independent If
over
Q .
mn > m + n
u. ,v.,exp(u . v . ) are algebraically
then at least two of the numbers (1 < i < n , 1 < j < m)
independent
over
Q
A corollary of Theorem 2, which was first proved by Gelfond n
[ 2 ] , is the algebraic independence of a * 0,1
Q ,
numbers
(1 < i < n , 1 < j < m)
u.,exp(u.v.) are algebraically
also be complex
is algebraic and
3
a
is a cubic
o2
and
a
whenever
irrationality.
361 2. THE RESULTS For the elliptic analogues we take a Weierstrass elliptic function piz)
piz)
whose invariants
g ? ,g.
are algebraic numbers.
has complex multiplication we denote by
imaginary quadratic field; multiplication, u,.,...,u
K
otherwise, if
denotes simply
Q .
K
piz)
has no complex
For an integer
n > 1
be complex numbers linearly independent over
an integer
m > 1
pendent over
Q .
let
v/),...,v
If
the associated let
K , and for
be complex numbers linearly inde-
The possible lack of symmetry in these conditions
leads to the following four analogues of the above three theorems. THEOREM 4.
if
mn > 2m + 4n
piu^v .)
then at least two of the numbers
( 1 < i < n , 1 < j < m )
are defined and are algebraically
THEOREM 5.
If
u.,p(u.v.)
mn > 2m + 2n
over
Q .
then at least two of the numbers
(1 < i < n , 1 < j < m )
are defined and are algebraically
THEOREM 5'. if
mn > m + 4n
v . ,p(LKV .)
independent
over
Q .
then at least two of the numbers ( 1 < i < n , 1 < j < m )
are defined and are algebraically
THEOREM 6.
independent
If
mn > m + 2n
u . , v.,piu.v.)
are defined and are algebraically
independent
over
Q
then at least two of the numbers
( 1 < i < n , 1 < j < m )
independent
over
Q .
From Theorem 5 or Theorem 5' we deduce that if plex multiplication over the imaginary quadratic field is cubic over
K , then for any complex number
is defined and algebraic, the numbers and are algebraically independent over
u
piz)
such that
p(3u) , p(3 2 u)
has com-
K , and
$
p(u)
are defined
Q .
It seems likely that Theorems 4, 5, 5', and 6 are the best that can be obtained by Gelfond's method as it stands.
We note that
Theorem 5 improves upon a recent result of the second author [7].
362 3. ZERO-ESTINATES The essential tools for the proofs of the results of section 2 are suitable zero-estimates for polynomials in elliptic functions. The corresponding estimates in the exponential case were obtained in a very sharp form by Tijdeman [5] using analytic methods.
But it
seems difficult to generalize these methods in the correct way.
By
contrast, our arguments make use of techniques from commutative algebra.
This approach was initiated by 3.V. Nesterenko in [4], and
developed further by Brownawell and the first author in [1]. For simplicity we state only the zero-estimate needed for the proof of Theorem 4. In the notation of section 2, let w be a complex number such that w + u.Cs.v. + ... + s v ) does not lie in the ^ l 1 1 m m period lattice of any integers
p(z)
for any integer
i
with
1 < i < n
and
s .,..., s
PROPOSITION. D > 1 , S > 0
let
degree at most
D
Suppose
mn > 2m + 4n .
P[x*,...,x
)
For real numbers
be a non-zero polynomial
of total
such that the function
f(z) = P(p(w + u^z) , . . . , p(w + u n z) ) vanishes at the points
for all integers , n
s „,..., s 1 m
o ~ N ( n / ^m/n
D > 3
(S/n)
,
M
, where
with o
N = 3
n
0 < s.,...,s 1 m
< S .
Then
/i
- 1 .
For Theorem 5 we need a similar conclusion when
f(z)
has
zeroes of high multiplicity, and for Theorem 5' we require a result on the simple zeroes of functions of the form P(z,p(w + u^z) ,.. .,p(w + u R z))
where
P(x Q ,x^,...,x n )
is a poly-
nomial whose degree in
xn
degree in
Finally for Theorem 6 we need a combination
x.,...,x
.
need not be the same as its total
of all these estimates.
4. GENERALIZATIONS There is no essential difficulty in extending all the results of section 2 to abelian functions, provided we have the appropriate zero-estimates.
As an example, we give the estimate that yields
generalizations of Theorem 4; it puts the above Proposition in the context of more general group varieties. For integers
n > 1 , N > 1
let
G
be a quasi-projective
commutative group variety of dimension
n
in complex projective
363 space of
N
dimensions, and for an integer
finitely generated subgroup of y = y(r,G) 1 < r < n sion
put
n - r ;
group of sion
G
of rank
in the following way.
F
H
= 0
if
G
otherwise let
I > 0 Z .
let
Then
U(I\G) =
min U 1
For an integer
m > 1
r
with
has no algebraic subgroup of dimen£ r
- I )/r let
be a
We define a number
For an integer
be the maximum rank of any subJ
which lies in some algebraic subgroup of
n - r .
T
G
of dimen-
.
y,.,...,Y
be generators of
V , not
necessarily linearly independent. PROPOSITION. on let
G
There is a real number
with the following property.
P(Xp, . . . ,Xp.)
X > 0
depending only
For real numbers
be a homogeneous polynomial
D > 1 ,S > 0
of degree at most
D
which vanishes at the points
for all integers
s.,...,s 1 m vanish identically on G .
with Then
0 < s„,..., s < S 1 m D > X(S/n) .
but does not
This result is proved in [3], where it is also shown that the number
y
is the natural exponent which leads to a best possible
estimate. REFERENCES 1. W.D. Brownawell and D.W. Nasser, Multiplicity estimates for analytic functions II, Duke Nath.J. 47_ (1980), 273-295. 2. A.O. Gelfond, Transcendental and algebraic numbers, Dover, New York (1960). 3. D.W. Nasser and G. Wustholz, Zero-estimates on group varieties I, to appear in Inventiones Nath. 4. J.V. Nesterenko, Bounds on the order of zeroes of a class of functions and their application to the theory of transcendental numbers, Izv.Akad.Nauk. SSSR, Ser.Mat. £[_ (1977), 253-284. 5. R. Tijdeman, On the number of zeroes of general exponential polynomials, Indagationes Nath. 3j3 (1971), 1-7. 6. N. Waldschmidt, Nombres transcendants, Lecture Notes in Nath. 402, Springer-Verlag (1974). 7. G. Wustholz, Algebraische Unabhangigkeit von Werten von Funktionen, die gewissen Differentialgleichungen genugen, J. reine angew Nath. 317 (1980), 102-119.
ESTIMATIONS ELENENTAIRES EFFECTIVES SUR LES NONBRES ALGEBRIQUES Maurice Mignotte
RESUME Si
P
est un polynSme a coefficients entiers,
algebrique qui n'est pas une racine de
P ,
S
a
un nombre
un ensemble de
places du corps ttj(a) , nous obtenons une minoration de 1'expression 11^ | P (ot) |
.
Cette minoration est parfois meilleure qu'une esti-
mation de Liouvillej c'est par exemple le cas lorsque que le degre de
a
P
est fixe,
tend vers l'infini tandis que la mesure de
a
reste bornee. 1. INTRODUCTION Le present travail constitue un prolongement de [3], ou on considerait la quantite briques non conjugues. H~ |P(a)|
, ou
P
|a-$|
a
et
3
etant des nombres alge-
Cette fois, on cherche a minorer la quantite
est un polynome a coefficients entiers,
nombre algebrique non racine de
P
et
S
a
un
un ensemble quelconque de
places du corps ttJCa) . Comme annonce dans le titre, la methode est purement elementaire, ell utilise tout au plus le principe du maximum pour les polyn6mes (exercice: en donner une preuve algebrique).
Le debut
consiste en la construction d'un polynome auxiliaire divisible par une puissance assez grande de
P , ceci grace a une variante conven-
able du lemme de Siegel. L'estimation ainsi obtenue est raisonnable en fonction des mesures de degre de
P
et
a , et particulierement bonne en fonction du
a .
Parmi les applications possibles de ce resuitat, les deux suivantes me paraissent interessantes: - minoration du plus petit nombre premier non ramifie et totalement decompose dans le corps
Q(a)
(voir le corollaire 1 au
theoreme 4 ) , - majoration du nombre de conjugues reels d'un nombre de mesure assez petite (voir le corollaire 1 au theoreme 5 ) . J'ai tenu a ne pas multiplier les corollaires curieux comme le suivant (non demontre dans le texte): Si tout
a
e > 0
est un nombre de Pisot ou de Salem, de degre on a l'inegalite
D , pour
365 Mali > | at I (oti
II II
lorsque
M + e ) D
pour
D > DQ
,
designe la distance a l'entier le plus proche) meilleure, |a|
reste borne, que 1'estimation evidente
Mali > ( |a| + 1 + e ) " D + 1
.
I.I n'est peut-etre pas exclu que les estimations figurant ici, bien que particulieres, puissent etre utiles dans certaines
demonstrations
de la theorie des nombres transcendants. Dans la suite nous utilisons les notations suivantes: si
a
est un entier algebrique de degre
les conjugues de
a
D
et si
a,,...,a^
sont
(dans le corps des complexes), la mesure de
a
est definie par la formule D n max{1 i=1
M(a) = Si
P
I ex - I }
.
1
est un polynome a coefficients complexes D n (z-z. ) i=1
P(z) = a 0
sa mesure vaut D n
M ( P ) = |a | tandis que
L (P)
ficients de
m a x C l , I z-. I )
,
designe la somme des valeurs absolues des coef-
P , e'est la longueur de
P , la hauteur
H(P)
etant le
maximum des modules de ces coefficients. Lorsque
v
est une place d'un corps de nombres et
P
un poly-
nome a coefficients entiers, il est commode de poser L(P)
si
v
est archimedienne,
L (P) = v
1
sinon.
2. UNE VARIANTE DU LEMNE DE SIEGEL LENNE 1.
Soit
P
un polynome
a coefficients
tif (c'est-a~dire dont les coefficients leur ensemble),
de degre
entiers,
sont premiers
d , dont la decomposition
primi-
entre eux dans
en facteurs irre-
P,1 ... P. k , les P. etant K ' J distincts et de degre respectif d. . Soient N et T des entiers positifs qui verifient N > dT . Alors, il existe un polynome F ductibles sur
Z[X]
est de la forme
366 non nul, a coefficients N
entiers,
divisible par
P
, de degre au plus
et qui verifie
HCF) <- {4«(N + D d * tT+1)/2
N
n[P
}^™
,
oh on a pose
Cherchons
N . ][ a.X , les a. etant les 1 X i=0 j = 1,...,k , soit a. une racine de P. dans C . J J divise F est equivalente au systeme d'equations F
inconnues.
Pour j La condition P N . I [I)®1. i=0 Z J
sous la forme
a. = 0 ,
0 < t < r.T , 1 < j < k .
1
J
Posons
J
Soient
1
si
2
sinon .
H
'
a.
A
est r e e l ,
1.1
*i.r.R
*k,1
k.r.T
I positifs qui verifient
d
^s entiers
K
l'inegalite
Une utilisation classique du principe des tiroirs montre qu'il existe des entiers au plus
a. , 0 < i < N , non tous nuls, de valeur absolue l
H , qui verifient
| J a.t^aj^l < /^. H(N;1) m ax{1.| aj | N }/* jjtt1 , pour
t = 0,...,r.T-1 et j = 1,...,k . Si n- de*signe la norme du J J corps GJ (a. ) sur Q et si b. designe le coefficient du terme de plus haut degre du polynfime P. t ceci implique
Le membre de gauche etant un entier naturel, F
(a.) = 0
sera realisee pourvu que l'on ait
la condition
367 On choisit pour
£. . .
le plus petit petit entier entier tel tel que que cette cette condition ait lieu, dans ce cas on a }
1/n-J
et j
t +1
j
La condition (1) est done verifiee si on a (H*1)) N + 1 inegalite satisfaite pour H = [{4fi(N+1)d*(T+1)/2n(P)N}N+1-dT]
,
d'ou le resultat. Le lemme 1 conduit aisement a l'assertion suivante, qui contient des resultats anterieurs de Ch. Pisot et M. Pathiaux. Corollaire 1 Soit a un nombre algebrique non nul, il existe un polynome R a coefficients entiers, non nul, qui admet a comme racine et qui verifie H(R) < MCa)
.
De plus il existe un tel polynome de degre au plus D , ou D est une fonction effectivement calculable, qui ne depend que du degre de a et de sa mesure. 3. ETUDE DE P(cO Dans toute la suite a designe un nombre algebrique de degre D > 2 , P designe un polyn6me h coefficients entiers de degre d qui ne s'annule pas en a . GrSce au lemme 1, on obtient le resultat general suivant. Proposition 1 Soit S un ensemble de places du corps dj(a) qui contient exactement s places archimediennes. Soit P un polyn6me a' coefficients entiers qui ne s'annule pas en a . On a alors, pout tout entier positif T , la minoration
368 n |p(a) i v > 4"^riCa)" d ncp)" D x exp{- j(s Log 2 + y + LogCD+dR)) - dTv - d * ( T + 1 } ou y := Log NCa) , v := Log N(P) dans l'enonce du lemme 1 .
ft
Supposons alors on a
P
P , a entier et
n |PCa)I
=
VES
de la forme a
et
n |a| v . n |PCa)| veS veS
NCP) = |a| NCP)
d*
LogCD+dT) }
,
etant definis comme
P
primitif,
,
,
on peut done se limiter au cas ou P est primitif. Lorsque P est primitif, considerons la fonction F construite au lemme 1 pour N = dT + D - 1 ; elle ne s'annule pas en a du fait qu'elle est de la forms P Q , Q e Z[X] Ccar P est primit i f ) , avec PCa) * 0 par hypothe'se et degCQ) < D = degCa) . Pour v en dehors de S on majore |FCa)| trivialement, IFCa)l v < L v CF) max{1,|a|° + d t }
.
Pour v dans S on utilise la relation majoration evidente |FCa)l v < IPCa)I^ L V CQ) max{1,|a|°}
F = P Q , qui fournit la
,
et on sait Cvoir [2] par exemple; la demonstration qui figure en [2] est purement alge*brique) que L(Q) ve*rifie LCQ) < 2 D LCF)
.
Gra^ce ^ la formule du produit n I F C a ) |Vw . n I F C a ) |Vw = 1 veS v/S et aux majorations precedentes on obtient l'inegalite 1 < f n |P(a)|T)L(F)D2DsM(a)D+dT .
369 Grace a la majoration de
H(F)
fournie par le lemme 1, on en
deduit - Log
<
n |P(a)| V veS
*
D
Log(LCF)) '
+
QLog 4 + y d + D v + Y ( s L o g 2 + y
Ds Log 2 '
+ Log(D +dT))
+
yD ' +
d
^
+
* ^
+ 1
J
Log(D +dT)
+ dT
v
,
ce qui equivaut au resultat annonce*. En appliquant la proposition pour
T = 1 , on obtient le resul-
tat tres simple suivant. THEOREME 1.
Soit
P
un polynome quadratfrei
entiers qui ne s' annule pas en
l > 4" f i 2" D ((D+d)N(a)n(P))" ( D + d )
,
|Norme(P(a))| < 4^((D+d)M(a)M(P)) D+d
.
Pour obtenir la premiere inegalite on prend I ex I
= I a | , alors
nant pour
S
s = 1 .
a coefficients
ot , on a alors
S = {v} , OLJ
La seconde inegalite s'obtient en pre-
1'ensemble des places non archimediennes du corps
0(a) , dans ce cas
s = 0 .
Un choix convenable de THEOREME 2.
T
conduit au resultat general suivant.
Sous les hypotheses de la proposition
1, on a la
minoration n iPCct) i
> 4"^(a)"dn(p)~(D+d) x
x exp {- |/(d*D(v+Log((y + 3)dD)) (sLog2 +y+ Log((y+ 3)dD))) - d*Log((y+3)dD)}
.
On prend cette fois T1 _ f / ( D ( s L °g 2 * y + Log(D*d)))1 + . L d*(v + Log(D + dJ 3 J On conclut en reportant cette valeur dans l'inegalite de la proposition 1 et en utilisant la majoration
D + dT < (y+3)dD .
Quitte a obtenir un resultat un peu moins precis, on peut deduire du theoreme 2 une inegalite plus simple.
370 Corollaire 1
Sous les hypotheses du theoreme 2, on a
> 4"QM(a)~dnCP)~(d+D) x
n |PCa)! v,S
x exp{-(|/(d*D(v+1)(s+y+1))+d*)Log((y + 3)dD) }
.
Par le meme raisonnement que dans la seconde partie du theoreTne 1, on obtient la rnajoration ci-dessous. Corollaire 2
La norme de
P(a)
verifie
|Norme(P(a))I < 4 n M ( a ) d M ( P ) d + D x x exp{|/(d*D(v+Log((y + 3)dD))(y+Log(Cy + 3)dD) ))+d*Log((y + 3)dD)}
.
Afin d'y voir plus clair, il n'est peut-etre pas inutile de consid^rer un cas particulier simple du corollaire 2. Corollaire 3
Soit
P
racine de l'unite de degre"
un polyn6me cyclotomique et
En effet, ici
y
et
Lorsque
d
et
D
£
une
D , on a alors
INorme(PU)) I < 4 . ( 3dD) d +2 / d D
irreductible, on a ft = 1
On a par exemple.
v et
.
sont nuls; de plus, comme
P
est
d* = d .
ont le meme ordre de grandeur, on en de*duit
la majoration
|Norme(PU»| < C° tandis que 1'estimation
,
e"vidente
|Norme(P(c))I * L ( P ) D fournit seulement INorme(PU)) I < 2 D d < C°
REFERENCES [1] T. Callahan, N. Newman and N. Sheingorn, Fields with a large Kronecker constant, J. of Number Th. 9_ (1977), 182-186. [2] M. Nignotte, An inequality about factors of polynomials, Math, of Comp. 28 (1974), 1153-1157.
371 [3] M. Nignotte, Approximation des nombres algebriques par des nombres algebriques de grand degre, Annales de la Faculte des Sciences de Toulouse,
CONTINUED FRACTIONS AND RELATED ALGORITHMS G.3. Rieger Universitat D-3000 Hannover
In this short lecture, let us first speak about 3 types of infinite continued fractions. Write
E, e X := [0,1] \ Q
as an ordinary continued fraction,
Define the corresponding operator
S : X -* X
by
1
Denote by
X
the Lebesgue measure on
R .
For
a e [0,1], 0 < n e Z
let
n
*
n
n
j.ug ^
In a letter to Laplace of 1812, Gauss [3] mentions lim R (a) = 0 ; oo
n
the first proofs were published by Kusmin [7] in 1928, showing R n ta) «
C ^
.
and by Levy [8] in 1929, showing R n (a) << C~ n
(1 < C 1
2
€ R) .
Knopp [6] proved already in 1926 that S
is mixing.
S
The asymptotic expansion of
is ergodic. R (a)
Furthermore,
was given by
Babenko [1] in 1976. Write now nearest integers,
£ e Y := [-5 , 3] \ Q
as a continued fraction by
e n c {-1,1}, 2 < b n e Z, b n + e R + 1 > 2 (n > 1) . sponding operator
For
T : Y •* Y
a e [0,1], 0 £ n € Z
€
Define the corre-
by
let
Y : 0 < T n (£) £ a}
e Y : 0 < T n (^) < | v- ^ < T n ( C ^ a-1}
with a certain
C^ > 1 .
T
for
a <
for
a
is mixing by [15] and especially ergodic.
Closely related are the singular continued fractions of Hurwitz (see e.g. [9], §44).
The corresponding operator is ergodic, by [14],
and an analogue of (1) holds, by [13], We now turn to finite continued fractions. Write
a/b
with
a e Z , b £ Z , 0 < a < b ,
(a,b) = 1
as an
ordinary continued fraction, a b
let
1 =
'
1
E(a,b) := n .
of Euler.
'
Let
a(b) :=
Heilbronn [5] proved
£
d
Denote by
(j) the function
'
I E(a,b) = 12Clog 2)7r~2
a/b
with
a e Z , b e Z > 0 < 2 a < b ,
Ca,b) = 1
as a continued fraction by nearest integers (see e.g. [9]> §39),
(2)
r. 2
where
let [9],
2 <
a
•.
. e Z ( 0 < j <
e
n-1
n ) , e . e {-1,1},
a . + e . > 2
J J J N(a,b) := n . It is known N(a,b) < E(a,b) §39) . By C10], we have
(0 < j
< n ) ;
J always [see e.g.
N(a,b) = 6 (log G)7T~2c|)(b) log b + 0(b(a(b)) 3 ). h I 0 < a < j , (a,b) = 1 For
a e Z , b e Z , 0 < a < b
b = g a + r , 2 / g ,
-a
< r
the
conditions
< a
determine g e Z , r e Z uniquely. Repetition leads to an expansion like (3), where now a. e Z , 2 / a. (0 < j < n) , e. e {-1,1} , J J J a. + e. > 0 (0 < j < n) and a/b has been written as a continued fraction with odd partial quotients; U(a,b) := n
let
.
(4)
By [17], we have I U(a,b) = 18 (log G)ir~2
.
Harris [4] introduces the following algorithm. For
a eZ,b
€2,0
b = g a + r , determine
< 2a < b , 2 / a b , |r| < a , 2 | r
g e Z ,r e Z
r = 0 , we are done. divisor of stops after
then
2 J[ g , g > 3 .
r * 0 , let c ,a
c
In case
be the largest odd
instead of
a ,b .
steps, say.
By [11], for every b,
uniquely;
In case
r , and we continue with H(a,b)
the conditions
> a, > 0 , d, > c, > 0
k e Z ,k > 0 with
there exist odd integers
This
375 E(a k ,b k ) s 5 ,
H(a k ,b k ) - k ,
E(c,,d.) > N(c.,d.) > k, k k k k An analogue of (2) for For every 0 < a < b < x i
5x
H(a,b)
is not known.
x > 2 , the number of pairs
a e Z ,b e Z
with
a e Z ,b £ Z
with
which do not satisfy
l o g b s E ( a
is at most
H(c,,d,) = 2 . k k
,
1.99
For every 0 < 2a < b < x
b )
, by [2].
x > 2 , the number of pairs which do not satisfy
1 log b < N(a,b) <
lQ
g
0
(3b)
- 1
(6)
/ 7 /L
is at most
3 6 x / 7 ^ , by [16].
For every 0 < a < b < x
x > 2 , the number of pairs
a e Z ,b e Z
with
which do not satisfy
llogb
(7)
7/4 is at most
36x
For every
, by [18]. x > 2 , the number of pairs
0 < 2a < b , 2 / ab ^ l o g b is at most
18x
a £ Z ,b £ Z
with
which do not satisfy
, by [16].
In connection with (5), (6), (7), (8) one shows that there are no exceptions to the upper estimate and not too many exceptions to the lower estimate.
The treatment of the exceptions to the lower
estimate of (6) and (7) is, by the way, literally the same. we have only reported.
So far
To make this lecture more satisfactory, we
prove U(a.b) < l ° 6 t b ^ log G
(0 < a < b) .
(9)
For the infinite sequence A = (a 1 ,e 1 ,a 2 ,e 2 »a 3 ,e 3 , . . . ) with
a. e Z , 2 / a. , e. £ {-1,1} , a. + e. > 0 (j > 0) J J J J J
we define
376 the derived sequence q
(q ,q,j,q2,...)
:= a q _, + e _.q _ 2 (n > 1) . b = qn
n
f
n
Comparison with (3) gives
f : Z -> Z
kfn-k
+ f
by
f
:=1,
-F, : = 1 ,
Induction gives
nn + 1 ( ^nP-n-1 nn + ( 1) G = G " > £1 G + G" /5 = f
: = 1 , q. := a^ ,
(10)
- fn_/] - -fn_2 : = 0 (n e Z) . f
q
.
Following Fibonacci, define fn
by
(n > 0) ,
k-1 f n-k-1
(n
e
Z,k
(11)
Z) .
e
f. < f 2 j _ 2 , f 2 j > 0 (j € Z) , i. > 0 (j > -3) , and consequently f
n
< io, _f , + f9, -f . , 2k-2 n-k 2k-4 n-k-1
LENNA 1. Proof and take
The sequence
k e Z
(0 < Jj < k) , e > 1 , a
We have the
n
> qH
with
. = 1 . n-k
n
q H
satisfies
This is clear for
q. > f. (0 < j < n)
there exists a
a
A
. > 3 n-j
k-1
(k < n + 1) .
n = 0
q
and
> f
(n > 0) .
n = 1 .
Let
as induction hypothesis. 0 < k < n
such that
(12)
e
n > 1 Case 1:
. = -1
We know ( 0 < j < k ) .
(13)
J
inequalities
*-q , q ->3q . .-q . ^ ( 0 < Jj < k - 1 ) n-1 H n-2n n-j n-j-1 H n-j-2
(14)
and the inequality
q
n-k+1 "
Since
3q
n-k
3f . - f._2
%
(15)
n-k- 1 "
= f. +2
Cj e Z) , (.14) gives
* f 2j-2 q n-j " f 2j-4 q n-j-1
But (16) for
q
+ q
j = k - 1
n - f 2k-2 q n-k
+
f
(0 < j < k) .
and (15) give
2k-4 q n-k-1
'
(16)
377 From this, the induction hypothesis, and (12) we deduce Case 2:
we have
hold for
q
Using
e _. = -1 (0 < j < n) .
k = n .
But (16) for
n - f 2n-4 q 1 " f 2n-6 q o q
= a. > 3
q
> f
Then (13), (14) and (16)
k = n , j = k-1
reads
'
(instead of (15)), we obtain
q > 3f o - - in = f > f M n 2n-4 2n-6 2n-2 n (10), (4), Lemma 1, and (11) imply (9). A different proof of Lemma 1 can be found in [18]. Unfortunately, many beautiful related results had to be omitted in this short lecture. REFERENCES 1. K.I. Babenko, A problem of Gauss, (Russian) Dokl.Akad. Nauk SSSR 238 (1978), 1021-1024 (English translation: Soviet Nath.Dokl. _1_9 (1978) , 1 36-140) . 2. J.D. Dixon, A simple estimate for the number of steps in the euclidean algorithm, Amer.Nath. Monthly 1971, 374-376. 3.
C.F. Gauss, Werke X/1, 371-374.
4. V.C. Harris, An algorithm for finding the greatest common divisor, Fibonacci Quarterly _8 (1970), 102-103. 5. H. Heilbronn, On the average length of a class of finite continued fractions, Abhandlungen aus Zahlentheorie und Analysis (zur Erinnerung an Edmund Landau), Berlin 1968. 6. K. Knopp, Mengentheoretische Behandlung einiger Probleme der diophantischen Approximationen und der transfiniten Wahrscheinlichkeiten, Math.Ann. 9_5 (1926), 409-426. 7. R.0. Kusmin, Sur une probleme de Gauss, Atti Congr.Intern. Bologne _6 (1928) , 83-89. 8. P. Levy, Sur le loi de probability dont dependent des quotients complets et incomplets d'une fraction continue, Bu11.Soc.Hath. France 5_7 (1929), 178-194. 9. 0. Perron, Die Lehre von den Kettenbruchen, 3. Auflage, Stuttgart 1954. 10. G.J. Rieger, Uber die mittiere Schrittanzah1 bei Divisionsalgorithmen, Nath.Nachr. 8_2 (1978), 157-180. 11. G.J. Rieger, On the Harris modification of the Euclidean algorithm, Fibonacci Quarterly U\_ (1976), 196. 12. G.J. Rieger, Ein Gauss-Kusmin-Levy-Satz fur Kettenbruche nach nachsten Ganzen, Hanuscripta Math. Z4 (1978), 437-448. 13. G.J. Rieger, Ein Gauss-Kusmin-Levy-Satz fur die singularen Kettenbruche im Sinn von Hurwitz, Abh. Braunschweigische Wiss.Ges. ,2_8 (1977) , 81-88. 14. G.J. Rieger, Die metrische Theorie der Kettenbruche seit Gauss, Abh. Braunschweigische Wiss.Ges. 2]_ (1977), 103-117.
378 15. G.J. Rieger, Mischung und Ergodizitat bei Kettenbruchen nach nachsten Ganzen, J. reine angew.Nath. 310 (1979), 171-181. 16. G.J. Rieger, Uber die Schrittanzahl beim Algorithmic von Harris und dem nach nachsten Ganzen, Arch.d.Nath. (in print). 17. G.J. Rieger, Ein Heilbronn-Satz fur Kettenbruche nach ungeraden Teilnennern, Math.Nachr. (in print). 18. G.J. Rieger, Uber die Lange von Kettenbruchen mit ungeraden Teilnennern, Abh. Braunschweigische Wiss.Ges. 32 (1981), 51-53.
IWASAWA THEORY AND ELLIPTIC CURVES: SUPERSINGULAR PRIMES Karl Rubin Department of Mathematics Harvard University
1 . PRELIMINARIES We are interested in the following situation. E is an elliptic curve defined over the number field F , and K c F is an imaginary quadratic field. Let 0 denote the ring of integers of K , and suppose that E has complex multiplication by an order 0' in 0 . Fix a prime p of 0 such that E has good reduction at all primes of F above p and p is prime to 6 . Write [ T ] for the endomorphism of E corresponding to .T £ 0' . If 91 is any ideal of 0 , let E_r denote the points of VI E annihilated by all endomorphisms {[x] | x e 0' n 91} . Consider the tower of number fields F n = F(E *»n + 1,) p
,
Foo = U F n n
Definition Let K : GaKF^/F) -»• 0* be the map obtained by considering the action on E «, = U E . V n V The character K is an isomorphism between Gal(Fw /F) and an open subgroup of 0* , so GaKF^/F) breaks up into GaHF^/F ) xA where A = G a K F /F) <—» (0/p) * . Further, G a K F /F ) is isomorphic oo
O
O
to an open subgroup of 0 , so GaHF^/F ) = Z e g " , where p the rational prime below p . In [1,2] Coates and Wiles studied the tower F /F in the
is
00
special case F = K , deg(p) = 1 and used their results to prove an important theorem in the direction of the Birch and Swinnerton-Dyer conjecture. The remainder of this paper is concerned with an investigation of the case deg(p) = 2 . So from now on we assume p = (p) , p a rational prime which remains inert in 0 . We will study the Z -extension F^/F by introducing the We will study the Z -extension F /F p
Iwasawa algebra
A
°o
and two modules over
A
A = U m Z p [Gal(F n /F o )] = Z [ [Gal ( F O O /F Q ) ] ] = Z n C T ^ T ^ ] Definition where the last isomorphism depends on the choice of two topological generators of Gal(F /F ) . °
oo
o
Definition
X
— — — — _ _ — _
= GaltM /F ) where
OO
00
380 is the maximal abelian
N
OO
OO
p-extension of F^ which is unramified outside the primes above Definition
Y
•-•iM.^Mdi.—• • ••„ !»!!• i i
= U /C
00
OO
where
U
00
= lim U
00
^
and U p]
p .
is the
pj
group of local units congruent to 1 in the completion F ® K ; and C00 = lim C , C being the closure of the elliptic units of to f— n n F (see [1], section 5) in U n n As we will show in §2, arithmetic of the curve
X^ is closely connected with the
E . Also,
X
and Y
OO
J
class field theory.
In [23, with
Wiles connected
with the p-adic L-function attached to E .
Y
p
are related by
OO
of degree
1 , Coates and
~
oc
Combining these facts gives interesting results relating the arithmetic of the curve with its complex L-function.
This is the motiv-
ation for the present investigation. 2. X m AND GALOIS COHOMOLOGY We first introduce the Tate-Shafarevich group variant
ill , and a slight
UJ' .
Definition UI
= kerCH 1 (E/F ) •> © H 1 (E/ (F )q ) ]
III1
= ker[H1(E/F
n
) -> © H 1 ( E / ( F n
UI^ = lim ui
)q)]
ui' = limui'
n
°°
where the first direct sum is over all primes second is over all primes not above
— » n
q of F
and the
p.
Galois cohomology gives a Kummer theory exact sequence 0 * E(F n )/p n + 1 E(F n ) - Hom(Gal(M n /f : n )'E pn + 1 ) - ( m ' ) ^ -> 0 where
N is the maximal abelian p-extension of F unramified n ^ n outside primes above p . Taking the direct limit over n yields 0 + E(F
) ® K /O oo
where
IH^tp)
p
,E oo
p
denotes the p-primary
Our main result
THEOREM.
-> Hom(X
p
here
) + W ' (p) oo
part
-*- 0
(2.1)
oo
of
lll^ .
is
UJ^(p) ^Ul^Cp) .
Sketch of proof
The theorem follows from the following local
381 is a finite extension of K and L1 = U L n 1 P n is a Z -extension of L. Then lim H (E/L ) = 0 . This implies that P > 1 1 n any element of H (E/F ) vanishes in H (E/(F ) ) for every Cl n m Q
statement.
above
Suppose
p , if
m
L
is taken large enough.
This local statement is proved by Tate duality: is dual to
lim E(L )
lim H (E/L )
where the inverse limit is taken with respect
to the norm map on E . We are therefore reduced to showing that there are no sequences of points P e E(L ) such that P = N / P ^ ^ n n m n/m n for n > m . This is just a statement about formal groups, and follows because the formal group here has height
2 .
Combining the theorem with (2.1) yields 0
+
E(F
)
9
K /0
oo
p
+
Hom(X
p
,E
p
X
(p) oo
-*• 0
.
(2.2)
~
as a A-module has free rank
It follows from this and (2.2) that either
E(F ) x K /O 00
+ Ul
oo
Greenberg [4] showed that F2(FQ) > 0 .
)
oo
or
HI (p)
p
torsion A-moduIes.
is 'very large' -- both cannot be co-
^
OO
J
<=>
Unfortunately it seems very difficult to decide
which one is large, even assuming the standard conjectures about the Tate-Shafarevich group. 3. THE STRUCTURE OF Y^ For this section we make the additional assumption that defined over
K , so
F = K .
Then the character
K
E
is
defined in §1
maps onto
0* , so A = (0/p)* = F n . Since the characters of A P pz are 0 -valued, it is useful to introduce the following notation. Definition
A-module,
A = A ®z 0
Definition
If
A-module on which which
A
= 0 [ [ T ^ T ^ ] ; and if
acts via
x
A
:
A -»• 0 *
acts, then
THEOREM.
Let
modulo
is any character and Z(x)
is any
Z
any
is the submodule of
Z
on
x •
With this notation we have
i t 0,p+1
Z
Z = Z ®. A . A
x
he
the
Z = © Z(x) • X
restriction
p -1 , we have
Further, we can choose a basis
a,B
C (x )
c
A#3 •
C (x )
=
A # £ , say, then there is an
The significance
of
K
U^Cx ) = A for
to *
U (x )
A .
and
Then if
C^tx ) = A .
so that
of this last statement is that if F
then Y (x1) = ((T7tf M X 1 ) = A © A/FA .
in
A
so
e - F»$ , and
382 Remarks isomorphisms
If
i = 0
U^tx1)*^—»A
or
p + 1 modulo
, Cootx1)0—»A
p -1 , then we get pseudoXP
•
is exceptional because
it is the eigenspace containing the p-power roots of unity. Unfortunately, the splitting of although the ideal
FA
F
A .
up to a unit in
Y (x )
is not canonical, and
is uniquely determined, this only specifies
So what can we say about the power series of the
F ?
Here we make use
'logarithmic differentiation' homomorphisms of Coates-Wiles.
Briefly, Coleman [3] has defined an injection p : U ^ — » ( 0 [[X]])* , k so for each k > 0 we have the homomorphism 4>,(u) = D (log(p(u)) on
U
, where
D
is a derivation associated to the formal group of
the kernel of reduction Proposition
mod p
Suppose
on
E .
i £ 0 , p+1
modulo
p -1 .
has a generator e such that <J>. (e) = 12 ft f 2 k = i (mod p -1) , where ft is a period of E f to
is the conductor of
E ,•
and
Then
Kip ,k)
C (x )
for every
such that
E = C/ftO1 ;
ij; is the Grossencharacter attached
E . Then the properties of the Corollary
we have every
<J>^
give us the following corollary:
With notation as in the theorem and the proposition,
F ( K( 1 +T\ ) k - 1 , K ( 1 + T 7 ) k - 1 ) = 12ft"k f " k L( i>k , k) /
for
k = i (mod p -1) . Unfortunately, the values
<J>, (3)
are still a mystery.
ever, I should remark that the power series
F
How-
is uniquely deter-
mined by the special values above. n
Proposition
For any
u
in
U^ ,
p
k
divides
(f>k(u) , where
(k-1)p - 1' This follows because the formal group that the has height two.
come from
This result is a slight improvement on a similar
result of Katz [5],- Katz also obtained congruences among the numbers n. (f>,(u)/p . Clearly, to understand fully the significance of F , -k and the interpolation properties of the numbers KIJJ ,k) , we must understand the numbers degree
4>, (3) .
In the case where
1 , this is the role of the
p
is a prime of
F-transform.
BIBLIOGRAPHY 1. Coates, 3., Wiles, A.: On the Conjecture of Birch and Swinnerton-Dyer. Inventiones math. 39 (1977), 223-251. 2. Coates, J . , Wiles, A.: On p-adic L-functions and Elliptic Units. J.Austral.Math.Soc. (A) 26 (1978), 1-25.
383 3. Coleman, R.: Division Values in Local Fields. Inventiones math. 53 (1979), 91-116. 4. Greenberg, R.: On the Structure of Certain Galois Groups. Inventiones math. 47 (1978), 85-99. 5. Katz, N.: Formal Groups and p-adic Interpolation. Societe Nath. de France Asterisque 41-42 (1977), 55-65.
ON RELATIONS BETWEEN GAUSS SUNS AND CYCLOTONIC UNITS C.-G. Schmidt Fachbereich 9 Nathematik Universitat des Saarlandes D-6600 Saarbrucken
1. INTRODUCTION For a natural number
m £ 2(4)
let
K:~ U(£ )
be the cyclo-
tomic field generated by a primitive m-th root of unity arithmetic of
K
cyclotomic units as is well known. rational prime
T X
where
p •[ m
in
(fe) : = _
K
.
The
For a prime divisor
fe
of the
we define the Gauss sums
(a/fe) fe
a mod
(a/fe)
£
is closely connected to certain Gauss sums and
x m
•c
(x mod m) ,
t r
p
denotes the m-th power residue symbol and
solute trace of the residue class field of
K mod fe .
taken over a complete set of representatives of
tr the ab-
The sum is
K mod fe .
In gen-
eral the numbers n(x) := 1 - t m are not units in
(x mod m)
K
but only suitable quotients are units.
Never-
theless we call them cyclotomic units. We consider the multiplicative relations of Gauss sums and cyclotomic units or in more detail the relation modules _,
' a m -^
R id (fe) := (a = (a 1
e Z
m-1
'
n
T [fe) X x
a
"
1}
x= 1 ('specific ideal relations'), R
n el
:= {a e Z m ~
' ,
m-1 a n T (fe) X = 1 X x=1
for all fe + m}
('simultaneous element relations'), m-1
A
Rf
:= {a e Z m "
;
a
n |n(x)| x=1
X
= 1}
('relations of cyclotomic units mod torsion'). 2. THE GAP GROUPS OF THE IDEAL RELATIONS AND THE UNIT RELATIONS For
y y mod m
set
6y
:= (0,...,0,1,0,...,0) e Z m ~ 1
with
1
385 in the y-th component. a) Davenport-Hasse
There are well known explicit relations:
(or distribution) relations [21: d-1
:= 6
V x for
d-1
dx " .lQ 6 x+ jm/d
+
6
^
jm/d
d | m , x = 1 , . . . , (m/d) - 1 , where
As an analog we have for the same
K.^x e R i d ( k )
for all
d,x
d-1 : =
Vx'
6
dx •
b) Norm relations:
\
:= 5
y
+
lQ
6
x + jm/d
For
€
y mod m
«m-y " 61 - V i
R1
* there is
£ R
idtfe)
for a11
fe
+
m
and n' := 6 - 6 £ R' . y y m-y Hasse [ 4 ] , p.465 respectively Nilnor [1] asked: generate R. ,(fe) R'
R. ,(fe)
respectively
generated by the
generated by the
Nilnor's question.
h.\
Ji, and
R'?
Let
and
w
n' .
R.
Do these relations
be the submodule of
, and
R'
the submodule of
Bass has partly answered
He proved in [ 1 ] :
R^ ® (U = R' ® U . Yamamoto [8] showed the following result by combinatorica1 methods: Let
r
be the number of rational primes dividing
p = 1(m) .
m
and take
Then we have an isomorphism of 2-elementary
abelian
groups: r 1 Rid(fe)/R1 = (Z/2Zr2 " -1' .
But there is a totally different, more structural approach which offers in addition an analogous result for A := Z m ~ 1 / <*' and for on
A
-,
J := {±1 mod m}
by: (-1)
° 6
d|m,x
We set
= 1,...,(m/d) - 1 >
we consider
:= 6 _ ) .
R' .
A
as a J-module
cal description of the quotient groups in question. see C5]. )
(J operates
This provides us with a cohomologi(For details
Mm.
386 THEOREM 1.
Ri(j(fe)/R1 « H°(J\A)
for
p = Km) .
R'/Rjj = H 1 (J,A) . Now using a formula of Sinnott [7] we can compute these cohomology groups (see [5 ] ) . 7 r-1
THEOREM 2.
H (J,A) * (Z/2Z)
,
H 1 (J,A) ~ (Z/2Z) 2
"r .
In this way we obtain again Yamamoto's result and also give a complete answer to Milnor's question. 3. THE GAP GROUP OF ELEMENT RELATIONS We now ask for the simultaneous element relations of the Gauss sums.
For
a e Z
and fe | m
m-1
set
a
X (fe) := n TX (fe) X *
x=1
and further set
R := n R. .(fe)
where the intersection is taken over
ld
all fe \ m .
All
x 9
class group
for
a e R
c h a r a c t e r s are roots of unity of R x S
2
extend to characters of the ray
— ~~ which we denote by
mod m
-*• T o r « X
m
, (a,fe) ~
h-
K X
.
S o . The values of these m Thus we get a p a i r i n g
(fe) • £
By class field theory there is a Kummer extension L := Fix( n over
K
Ker x )
and we have
R/R e l = GalCL/K)* where the star denotes the corresponding character group.
So for
all
aeR the Kummer extension L := Fix(Ker x ) measures the — a a order of the class of a_ considered as an element of R/R •, . By a theorem of Deligne [3] the extension
L /K a
is given explicitly by
adjunction of a certain product of values of the classical T-function .
387 THEOREM (Deligne). m- 1
n
For
£ e R
set
a
r(x/m)
X
x=1 Then
La = K(fta )
and the Frobenius automorphism
Deligne's proof is not yet published.
operates like
The result is reported in [3]
where it is also stated that the proof involves Hodge-theory of Fermat hypersurf aces .
In the case
a_ e R,
the theorem was indepen-
dently proved by Gross, Koblitz [3] (using a result of Katz on some p-adic cohomology of the Fermat curve) and the author [6] (using only arguments from class field theory and Kummer theory). the theorem for all
a_ e R
To prove
it would be sufficient to show
m-1 . a n (r(x/m)pr"'/r(x(p-1)/m+ 1)) X = 1 mod fe x=1 Tor
p = 1(m)
and
a_ e R
but it is not clear how to attack this
directly. REFERENCES 1. H. Bass, Generators and relations for cyclotomic units, Nagoya Math.J. 27_ (1966), 401-407. 2. H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktion in gewissen zyklischen Fallen, 3. reine angew.Math. 172 (1934), 151-182. 3. B.H. Gross and N. Koblitz, Gauss sums and the p-adic T-function, Ann.Math. W9_ (1979), 569-581. 4.
H. Hasse, Vorlesungen uber Zahlentheorie (Berlin 1964).
5. C.-G. Schmidt, Die Relationenfaktorgruppen von StickelbergerElementen und Kreiszahlen, J. reine angew.Math. 315 (1980), 60-72. 6. C.-G. Schmidt, Gauss sums and the classical F-function, Bull. London Nath.Soc. J_2 (1980), 344-346. 7. W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann.Math. 108 (1978), 107-134. 8. K, Yamamoto, The gap group of multiplicative relationship of Gaussian sums, Symp.Math. Vol.XV (1975), 427-440.
SUR LA PROXIHITE DES DIVISEURS Gerald Tenenbaum Residence Saint Sebastien, tour A, F-500 Nancy
Soit
n
un nombre entier, et,
1=d
y
1 la suite croissante de ses diviseurs.
,
Nous nous proposons ici de
presenter certains resultats lies ^ 1'etude de la fonction arithmetique E(n) :=
T(n)-1 min i-1
d X d
' i
"
Erdos a conjecture il y a plus de quarante ans que l'on a pour presque tout entier
n
1 < E(n) < 2
;
(1)
il a montre en 1948 [1] que la suite des entiers
n
satisfaisant (1)
possede effectivement une densite asymptotique mais sa methode ne permet pas d'etablir que cette densite a pour valeur
1 .
Une justification heuristique de (1) peut etre formulee ainsi: comme le nombre des valeurs distinctes des quantites df
et
parcourant les diviseurs de
U(n) := card{d|n,d'|n:Cd,d') = 1} = on a pour tout
log d'/d ,
d
n , est egal a n (2v+1) P V | |n p premier
,
n
UCn) < ou n
u)(n)
(resp. fi(n) ) designe le nombre des facteurs premiers de
compte*s sans (resp. avec) leur ordre de multiplicite; le theor^me
de Hardy et Ramanujan implique done U(n) = (log n ) l 0 g pour presque tout I
inclus dans
ment
+
n , et, l'on peut s'attendre h ce qu'un intervalle
[-log
X . (log n )
3
log
n, log n] 3
"
1
+
o(1)
et de longueur points
X
log d'/d
contienne usuelledistincts - la
conjecture d'Erdbs exprime que le nombre de ces points est au moins e*gal a
1
dans le cas
I = ]o , log 2[ .
Cet argument a conduit
389 Erdos a conjecturer qu'en fait la formule asymptotique r- r
-\
A
r T
•» 1
~
3
log
+
o(1)
fnN
B
E(n) = 1 + (log n) a lieu pour presque tout
n .
(2) II a meme annonce pouvoir etablir (2)
en 1964 [2] mais son argument s'est malheureusement avere inexact. Erdos et Hall ont recomment prouve un theoreme qui implique 1'inegalite ECn) >- 1 • (log n ) 1 " pour presque tout normal de
ECn)
lo
§
3
+
ot1)
n , mais le probleme de la majoration de l'ordre reste entier.
L'etude de
ECn)
a suscite la definition de plusieurs fonc-
tions arithmetiques: T(n,a) := card{d|n,d' |n: |log d'/d| < (log n) a } Ca < 1) UCn,a) :=card{d|n,d'|n: Cd,d•) = 1, |log d*/d| < Clog n) a } Ca < 1) T + Cn) := card{k e N : 3 d | n , 2 k < d < 2 k + 1 } g(n):=card{iCi
d
±I
d
i +1 >
Les deux premieres definitions sont dues a Hall, les deux autres a Erdos.
L'inegalite
(1) est impliquee par
T + Cn) < xCn)
C3)
et, si l'on y remplace
2
par
e
Cce qui represente une modifi-
cation sans importance), par TCn,O) > xCn)
C4)
ou encore par UCn,O) > 1
.
C5)
Enfin, notons que, comme chaque intervalle au plus un diviseur di tel que gCn) < x + (n)
.
[2 k ,2 k+1 [
contient
d.|d.+ . , on a pour tout entier
n C6)
390 Hall a etudie [6] la valeur moyenne de
T(n,a)
pour
a
dans
[0,1] : THEQREME 1 (Hall). constante positive
Pour tout reel
C(a)
a
de
[0,1]
telle gue l'on ait pour
K
il existe une infini
Un I 'l\> ~ C(a)K(loglog K ^ ( a ) , n
ou l'on a pose Pour
y(a)
=
1 *
si
a = 0 ,
et
y(a) = 0 ,
si
a * 0 .
a = 0 , la formule n'est pas explicitement prouvee dans
[6] mais on peut constater facilement que la methode de transformation de Fourier utilisee pour traiter le cas
a > 0
est encore
applicable, au prix de certaines precautions supplementaires. valeur explicite de discontinuity en
C(a)
La
est donnee dans [6]j elle presente une
a = 1 .
On deduit sans peine du the*oreme 1 que presque tout entier satisfait a"
log T(n,a) < {log(2a + 2) + o(1)} loglog n .
n
On trouvera
dans [6] ou dans [7] la demonstration, due a Erdb's, de l'inegalite log T(n,a) £ {(a+1)log 2 + o(1)} loglog n
pour presque tout
n .
Dans [7], Hall et l'auteur etablissent l'existence d'un ordre normal pour
log T(n,a) , 0 < a < 1 . THEORENE 2 (Hal1-Tenenbaum).
uniformement Ainsi, l'inegalite
par rapport a T(n,0)
a
dans
Pour presque tout entier
n
on a
[0,1] .
est usuellement de l'ordre de
x(n)
(4), si elle est verifiee pour presque tout
et n , est
pratiquement optimale et risque d'etre difficile ^ etablir.
En
revanche, l'argument heuristique presente* plus haut semble indiquer que
U(n,0)
est une fonction mieux adaptee au probl^me.
la conjecture naturelle concernant l'ordre normal de log U(n,a) = {log 3 - 1 pour presque tout U(n,0) log n
En effet,
log U(n,a)
+ a + o(1)} loglog n
est (7)
n ; en particulier, on s'attend done a ce que
soit usuellement de l'ordre d'une puissance positive de et, partant, que l'inegalite
de la re*alite*. normal de
(5) soit une forme tres affaiblie
Dans [7], nous avons etudie l'ordre moyen et l'ordre
U(n,a) .
Nous avons obtenu les resultats suivants:
391 THEORENE 3 (Hall-Tenenbaum). existe une constante positive
Pour tout reel
D(a)
a
de
[0,1]
telle que l'on ait pour
il
K
infini K)6(a)
U(n,a) ~ D(a)K(loglog n
si
1 si a
0 < a <5
y(ct) =
, et, 6(a) = l/(2a+1)
La valeur de de
si
D(a)
0 si a * 5
i ^a < 1
est explicitee dans [7]; cornme dans le cas
T(n,a) , on observe une discontinuite pour 4 (Hall-Tenenbaum). a log 3 * o(1)
S
^ S
^ ' ^
g
a = 1 .
Pour presque
tout entier
n on a
s log 3 - 1 • a • o(1)
La majoration du the"oreme 4 est en accord avec (7).
II suf-
firait done pour e*tablir (7), et par consequent (5), de montrer que l'on a asymptotiquement
I log U(n,0) > {log 3 - 1 + o(1)}K loglog K n
Erdos a souvent conjecture Cvoir [3] par exemple) que le rapport
T (n)/t(n)
tend vers
0
sur une suite de densite
1 .
A l'oppose*, Montgomery a conjecture, lors du Symposium de Theorie Analytique des Nombres de Durham, en 1979, que non seulement T
(n)/T(.n)
mais me~me
g(n)/x(n)
peut §tre minor^ par une constante
positive sur une suite d'entiers
n
de densite positive.
ment heuristique peut §tre resume ainsi: son plus petit
facteur premier;
p
quitte a ne*gliger une suite de
n
log p < Clog n)°
la moitie* au moins des diviseurs de cL|(n/p)
mais
y
d^ / ^•+ \ >
\j ]log d., log pd.[ 1 x d.|(n/p) done usuellement
0
comme
a
(log n)
log d.+/. og
dt
+i
n <
divisent P^*
ne depasse pas
< (log n) °^
que la proportion des vers
on
+ 0
n
Son argu-
un entier et
de densite* nulle, on peut supposer
soit
'
or
; de plus,
n/p , et, si amesure
•'•
de
x(n) log p , alls est
et l'on peut s'attendre I ce
contenus dans cette reunion tende °
j la conjecture necessite seule-
ment qu'elle soit majoree par une constante
< \ .
392 Dans [ 5 ] , Erdos et l'auteur etablissent la conjecture de Montgomery.
Plus precisement, on a
THEOREHE 5 (Erdb's-Tenenbaum) . designons
par
A,, (a)
suite des entiers g(n) < ai(n)j.
Pour tout
a
de
[0,1]
(resp. A-ta),) la densite superieure
n
satisfaisant
a
T (n) < axCn)
de la
(resp.
Alors
(i) pour tout reel positif
e , il existe une constante
c(e)
telle que 1'on ait A 1 (a) < c(e)a 1 ~ e pour tout
a
de
(ii) on a
[0,1] . lim A~(a) = 0 . a->0
Cela implique que non seulement ne tend pas vers A
0
T (n)/x(n)
pour presque tout
n
(resp. g(n)/x(n))
mais meme que toute suite
lim T [n] = 0 (resp. lim ^ l = 0) est de densite neA T [ n J neA T [ n J II est egalement a noter que le resultat (ii) est optimal en
telle que
nulle.
ce sens que
A~(a)
est positif lorsque
a
est positif [5].
L'ensemble de ces resultats suggere que, meme s'il est usuellement non nul, le nombre des rapports est relativement faible.
d.+/)/d.
qui sont proches de 1
Cela explique peut etre, au moins partielle-
ment, pourquoi la conjecture initiale d'Erdos reste encore a prouver. BIBLIOGRAPHIE [1] P. Erdos, On the density of some sequences of integers, Bull. Amer.Math.Soc. 5j4 (1948), 685-692. [2] P. Erdos, On some applications of probability to analysis and number theory, J. London Hath.Soc. 3j3_ (1964), 692-696. [3] P. Erdos, Some unconventional problems in number theory, Aste*risque 6_1_ (1979), 73-82. [4] P. Erdos and R.R. Hall, The propinquity of divisors, Bull. London Hath.Soc. V\_ (1979), 304-307. [5] P. Erdos et G. Tenenbaum, Sur la structure de la suite des diviseurs d'un entier, Ann.Inst. Fourier, (1) 31 (1981), 17-37. [6] R.R. Hall, The propinquity of divisors, J. London Nath.Soc. (2) j_9 (1979), 35-40. [7] R.R. Hall et G. Tenenbaum, Sur la proximite des diviseurs, Recent Progress in Analytic Number Theory, vol. 1 (1981), 103-113 CH. Halberstam and C. Hooley e d . ] , Academic Press.
