Number Theory: Paris 1992-3 (London Mathematical Society Lecture Note Series)

0). The present author improved this by replacing P4 with P3 [13] (also giving a less significant improvement on 6 to 1/300). This problem thus appears "harder" than Goldbach where a prime and a P2 suffices (6]. It has been shown by Iwaniec

[23] (see also [ 14]) that one can approximate irrationals in this way with fractions whose numerator and denominator are sums of two squares. We finish this article by giving an indication of how such results can be established. We replace a with a convergent a/q, with error 1/q2 and choose the size of our possible numerator and denominator in relation to q. To approximate with p/P3 we pick X = q8/5, and write

A={Qpa/q :pSX,Ilpalgll <X-6/2}. indicates nearest integer. We then want to show that A contains P3 numbers. To do this we need to consider how well A is distributed in arithmetic progressions. We write Here Q

Ad=#,{nEA:n-0(modd)}, Rd=Ad- al. We then wish to show that A IRdI = ° L1o X] d
(27)

gl

l

for as large a value of D as possible. Using a familiar argument (see Chapter 2 of [1]) we obtain

R-d«

X1

6-E

6 ll

d

eff [pq d] +d Y e
where L = L(d) = dXE+6 We can convert the sum over primes above into a double sum which leads us to seek estimates for mnfa e

n

ad

136

G. FIARMAN

Using the large sieve and other devices we obtain a suitable estimate when

D < X1/3. This establishes the result after employing Chen's role reversal technique [6). The reader can see from these results that there is much work to be done in this area to give a more satisfactory "contribution a 1'analyse arithmetique du continu". Additional note : since delivering the above talk (November 1992) there

has been further progress on two problems. The most important is the announcement by A. Zaharescu that the exponent 2 can be improved to 4/7 in Heilbronn's Theorem on IIan2 11, with a further improvement to 2/3 if one is only looking for infinitely many solutions. His proof uses character sums not exponential sums. As mentioned above, numerical calculation is required to obtain the best exponent p for I1apIl < p -P and Jia Chao-Hua Q. Number Theory 45 (1993), 241-253) has shown that one can take p = 4/13 (0.308...). The present author will show elsewhere how the method can be improved to yield 7/22 (0.318...).

The referee in his comments pertinently remarked that I should have mentioned in section 4 the important work of Margulis (Discrete subgroups and ergodic theory in Number Theory, trace formulas and discrete groups, Oslo 1987, pages 377-398, Academic Press, Boston 1989) whereby every irrational a has infinitely many approximations

a

712 + v2 < 162 + v2

for any e > 0.

I

Manuscrit recu le 2 septembre 1993

TOWARDS ANARI7HME77CAL ANALYSIS OF77IE CONTINUUM

137

References [11 R.C. BAKER. - Diophantine Inequalities, Clarendon Press, Oxford 1986. [2] R.C. BAKER and G. HARMAN. - On the distribution of app` modulo one,

Mathematika, 38 (1991), 170-184. [3] H. BEHNKE. - Uber die Verteilung von Irrationalitaten mod 1, Abh. Math. Sem. Hamburg, 1 (1922), 252-267. [4] E. BOREL. - Une contribution a I'analyse arithmetique du continu,

Journal de Mathematiques Pures et Appliquees, (5eme serie), 9 (1903), 329-375. [5] J.W.S. CASSELS. - Some metrical theorems in Diophantine approximation I, Proc. Cambridge Phil. Soc., 46 (1950), 209-218.

[6] J.-R. CHEN. - On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica, 16 (1973), 157-176. [7] I. DANicic. - Contributions to Number Theory, Ph. D. Thesis, London 1957. [8] R.J. DUFFIN and A.C. SCHAEFFER. - Khintchine's problem in Metric Diophantine approximation, Duke Math. J., 8 (1941), 243-255. [9] P. ERDOs. - On the distribution of the convergents of almost all real numbers, J. Number Theory, 2 (1970), 425-441. [10] P.X. GALLAGHER. - Approximation by reduced fractions, J. Math. Soc.

of Japan, 13 (1961), 342-345. [11] G.H. HARDY and J.E. LrrrLEwoOD. - The fractional part of nJB, Acta

Math., 37 (1914), 155-191. [12] G. HARMAN. - On the distribution of ap modulo one, J. London Math. Soc., (2) 27 (1983), 9-18. [13] G. HARMAN. - Diophantine approximation with a prime and an almost

prime, J. London Math. Soc., (2) 29 (1984), 13-22. [14] G. HARMAN. - Diophantine approximation with almost primes and two

squares, Mathematika, 32 (1985), 301-310. [15] G. HARMAN. - Metric diophantine approximation with two restricted variables I, Math. Proc. Cambridge Phil. Soc., 103 (1988), 197-206. [16] G. HARMAN. - Metric diophantine approximation with two restricted variables III, J. Number Theory, 29 (1988), 364-375. [17] G. HARMAN. - Some cases of the Dufn and Schaeffer conjecture, Quart. J. Math. Oxford, (2) 41 (1990), 395-404. [18] G. HARMAN. - Numbers badly approximable by fractions with prime denominator, preprint Cardiff, 1993.

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G. HARMAN

[19] D.R. HEATH-BROWN. - Diophantine approximation with square-free integers, Math. Zeit., 187 (1984), 335-344. [20] H. HEILBRONN. - On the distribution of the sequence n29 (mod 1), Quart. J. Math. Oxford, (1) 19 (1948), 249-256. [21] A. HuRwITZ. - Uber die angenaherte Darstellung der Irrationalzahlen durch rationaleBriiche, Math. Ann., 39 (1891), 279-284. [22] C. HUYGENS. - Projet de 1680-81, partiellement execute d Paris, d'un

planetaire tenant compte de la variation des vitesses des planetes dans leurs orbites supposees elliptiques ou circulaires, et consideration de diverses hypotheses sur cette variation, in CEuvres Completes de Christian Huygens, 21, 109-163, Martinus Nijhoff, La Haye (1944). [23] H. IwANIEC. - On indefinite quadratic forms in four variables, Acta Arithmetica, 33 (1977), 209-229. [24] A. KHINTCHINE. - Zdr metrischen Theorie der diophantischen Approximationen, Math. Zeit., 24 (1926), 706-714. [25] A.D. POLLINGTON and R.C. VAUGHAN. - The k-dimensional Duffin and

Schaeffer conjecture, Mathematika, 37 (1990), 190-200. [26] A.M. ROCKETF and P. SzUsz. - Continued Fractions, World Scientific, Singapore-New Jersey-London-Hong Kong 1992. [27] J.D. VAALER. - On the metric theory of Diophantine approximations,

Pacific J. Math., 76 (1978), 527-539. [28] R.C. VAUGHAN. - Diophantine approximation by prime numbers III, Proc. London Math. Soc., (3) 33 (1976), 177-192. [29] R.C. VAUGHAN. - On the distribution of ap modulo 1, Mathematika, 24 (1977), 135-141. [30] I.M. VINOGRADOV. - The method of trigonometric sums in the theory of numbers (translated from the Russian by K.F. Roth and A. Davenport), Wiley-Interscience, London 1954. Glyn HARMAN

School of Mathematics University of Wales College of Cardiff, 23 Senghenydd Road, P.O. Box 926, CARDIFF CF2 4YH

United Kingdom

Number Theory Paris 1992-93

On A-adic forms of half integral weight for SL(2)/Q Haruzo HIDA

1. - Let S be the two-fold metaplectic cover of S = SL(2)/z and fix a prime p > 5. In this short note", we want to describe a technique of lifting a family of complex automorphic representations of S(A) to a "Aadic automorphic" representation II of S(A(P°°)), where A is a one variable power series ring over an appropriate p-adically complete discrete valua-

tion ring, and A(P°°) is the adele ring A of Q the p and oo-components removed. Then we will have a A-adic version of a result of Waldspurger [Wa2]. We begin with the study of p-adic cusp forms of half integral weight

and prove in Section 3 that the classical cusp forms of weight k + . is dense in the space of p-adic cusp forms of half integral weight if k > 2 (Theorem 1). Then we study A-adic forms of half integral weight in Section 4 by combining the techniques of Wiles [Wi] (introduced for integral weights) and the representation theoretic technique of Waldspurger [Wal,2]. Taking

the limit shrinking the congruence subgroup, we get the desired A-adic representation of S(A(P°°)) (Proposition 1). Then we prove the weak multiplicity one theorem for p-ordinary A-adic automorphic representations (Theorem 2 in Section 4). Although our construction is just the combination of these two existing techniques, we get a fairly strong result on p-adic standard L-functions of G = GL(2)/Q. That is, a certain ratio of the restriction of 2-variable p-adic standard L-functions [K] to the line interpolating The author is partially supported by an NSF grant. The final touch to the paper was given while the author was visiting the Isaac Newton Institute for Mathematical

Sciences, Cambridge, England. The author acknowledges the support from the Institute for the month of April in 1993. Some part of the work presented in this note was actually done in 1988 in order to construct a p-adic standard L-functions for GL(2) restricted at the center critical line (Theorem 4). The construction of two variable p-adic standard L-functions was later done by K. Kitagawa [K] using a different method.

140

H. HIDA

the central critical values is shown to be square in the field of fractions of the Iwasawa algebra A (Theorems 3 and 4), which is the A-adic version of a result of Waldspurger ([Wa2] Corollary 2) we alluded to. A further scrutinizing of the representation we constructed might bring us a sharpening of this result giving a A-adic version of the result in M. However to make our presentation short, we will not touch this subject in the present account. Another interesting point which awaits further study is the behavior of the specialization 7rv,t=2 of irreducible factors it of 11 at weight 2. In [GS],

Greenberg and Stevens gave an interesting limit formula of the derivative

of the p-adic standard L-function at the center critical point, when the L-function has an exceptional zero at this point. This is the unique case where the specialized automorphic representation It t=2 of S(ZP°°)) (supplemented with the p-component) becomes super cuspidal at p although the integral image of 7rwt=2 under the Shimura correspondence is special and p-ordinary. Thus the study of the behavior of the other local components of Trwt=2 might cast some new insight upon the p-adic analog of the conjecture of Birch-Swinnerton Dyer formulated in [MTT]. Although I have only worked out here the result for SL(2) defined over Q, our idea works fine for SL(2) over general number fields. However, in the general case, the many variable standard p-adic L-functions defined on the spectrum of the p-adic Hecke algebra are not yet constructed.

2. - Let A be a congruence subgroup of level prime to p. When we consider modular forms of half integral weight, we assume that A is contained in ro(4). We write O1(pa) = A n r1(pa) and A(pa) = A1(p) n ro(pes). We use the same notation introduced in [H1] Sections 1 and 2 for classical modular forms. In particular, for each integer k and an algebra A, Pk+(1/2)(AI(pa); A)) stands for the space of A-integral cusp forms of half integral weight k + 2 with respect to Ai(pa), while for each integer SK,(AI (pa); A)) stands for the space of A-integral cusp forms of integral

weight ,c. Here the A-integrality is given by the q-expansion at the cusp oo. For each Dirichlet character X modulo Npa, Pk+(1/2)(ro(Npa); X; A) consists of cusp forms g in Pk+(1/2)(r1(Npa);A) with 9Ik+(1/2)a = X(d)9 for each or = I a d

)

E ro (N), where glk+(1/2)0'is the action of or defined in

[H 1J (2.2a) which is a little different from the normalization of [Sh 1] p. 447. Our normalization is :

9Ik+(1/2)0'(z) = g(a(z))j(a, z)-1J(a,

z)-k

for a = (c

b

d

J

ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)IQ

141

00

where J(o, z) = (cz + d) and j (a, z) = 0(a (z))/9(z) for 9(z) = >exp(27rin2z).

n--oo

By [HI] Theorem 2.2 or its proof, Pk+(1/2)(r1(Np°); A) is stable under this action of a E 1'o (Npa). We now give an interpretation in adelic language following [Wa2] III. We write S for the algebraic group SL(2)lz. We write S for the two fold metaplectic cover of SL(2) defined in [Wa2] II.4. Thus S(Qp), S(A) and S(R) have meaning, where A is the adele ring of Q. In other words, we have a non-splitting exact sequence of groups :

1-->{±1}--S(A)->S(A)--* 1, where A is either A, Qp or R. Now let us describe the 2-cocycle /3 giving the ), we put x (a) = d

extension S(Q,,) for a place v of Q. For each a = (a or c according as c = 0 or c # 0. We also put

sv(a) _

(c,d)v if cd

d

0, v is finite, vp(c) is odd, otherwise,

1

where (c, d)v is the Hilbert symbol at v (that is, Artin symbol (d, Qv) of d). Then we have

VC-(d'w)

= (c, d)v f for the

Qv(a> a') = (x(a) , x(aTv(-x(a)x(a'), x(aa'))vsv(a)sv(a')sv(ao') .

For a e S(Q), the product s(a) = IIvsv(av) is well defined. Similarly we may define /(a, a') = nv3(av, a',) for a, a' E S(A). Then we identify S(A) with S(A) x {±1} under the multiplication law given by (g, e)(h, e') = (gh, )0(g, h)ee'). By the product formula of the Hilbert symbol, ,Q(a, a') =

s(a)s(a')s(aa') for or, a' E S(Q). Thus a F--> (a, s(a)) gives a section : S(Q) -> S(A). We identify S(Q) with its image in S(A). We also identify the standard maximal compact subgroup S02(]R) with ][8/Z by 0 1--k cos 2x9 sin 27x9

sin 27r9 cos 27r9

Then the pull back image of S02(R) in S(R) can

be identified with ][8/2Z. We write the corresponding element r(9) in S(R) for an integer k and C, = {r(9) 10 E R/2Z}. Then r(9) - e((k + 2)9) and e(9) = exp(27ri9) is a character of C. Via (g, e) H g(i) E H, we have

S(R)/C,, = H for the upper half complex plane H. Let e : A/Q -> C be the standard additive character such that e(xo,,) = exp(27rix,,,,). We write ev for the restriction of e to Qv for each place v, and we define 'yv(t) to be the Weil's constant with respect to ev and the quadratic form tx2 on Qv [W] p. 161. We put, following [Wa2],'(t) = (t, t),,-y,, (t)-y (1)-1. Then we

142

H. HIDA

(t,t')v5'v(t)5'v(t'), 1 for arbitrary v, and ye(t) = 1 if t E Ze and s 54 2, ry52(1) = '2(5) = 1 and 5'2(3) = 5'2(7) = -i. Let Uo (N)

(a d)

E S(2) I c E Nz 5

(2 = IIC

and write Uo (N)Q for the f -component of Uo (N). Defining for or = ( a c

d)

E

S(Q2)

J 'Y2(d)(c,d)2s2(a) ry`2 (d)

if C 74 0,

if c = 0,

we can check that E extends to a character of Uo(4)2 x {±1} in S(Q2) non-

trivial on {+1}. Let x be a character of (Z/NZ)" with X(-1) = 1. For (a (u,E) E Uo(N) x {±1}, we define X(u) = X(d) if u = d). We then consider the space of functions f satisfying :

(m 1)

f(ax(u,E)r(B)) = e2(u2)EX(u)f(x)e((k + 2)6)

for a E S(Q), (u,E) E Uo(N) x {±1} and r(O) E C. We impose another condition at oo :

(m2) D f = (k(k 2- 2) ) f for k' = k+ 2 for the Casimir operator D at oo . We write Pk+(1/2) (N, X; C) for the space of functions satisfying (ml - 2)

which are cusp forms. Writing J(g, z) = cz + d for g = I a b ) E SL2 (R) and z E H, we can identify S(18) = { (g, t(g, z)) Ig E SL2 (R) , t(g, z) : holomorphic on H

with t(g, z)2 = J(g, z) }.

The product is then given by (g, t(g, z))(h, t(h, z)) = (gh, t(g, h(z))t(h, z)). We have a natural inclusion map S(l) , S(A) and S(Q) - S(A). We have

the theta series : 0(z) : E00 exp(27rin2z) defined on H. As is well known, n--oo

ON A-ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)/Q

143

putting j(ry, z) = B(y(z))/O(z), j(y, z)2 = (dl) J(ry, z) if y E ro(4). Thus ry H (y, e2(y)j(y, z)) defines an inclusion of ro(\4) into S(R). It is known that the extension splits over U1 (4) = { I a b

i

E S(7G)

I

c E 42 and

a - d - 1 mod 47L}. Thus we have by the\strong g approximation theorem that S(A) = S(Q)U1(4)S(R). We can identify these two realizations by S(A)E) (g, t(g, z)) 1-- (g, t(g., z)J(g", z)-1/2) where the square root is taken so that -ir/2 < arg(cz + d)1/2 < 7r/2. For each cusp form f E Pk+(1/2)(N, X; Q, we define F : H - C by F(z) = f((g,1))J(g,i)k+(112) for z = g(i) (g E S(R)). Then as shown in [Wa2[ Proposition 3, f H F induces an isomorphism :

(2.1)

T'k+(1/2)(ro(N), X; C)

Pk+(1/2) (N, X, (C)

When f is cuspidal, the holomorphy of F follows from (ml - 2). Let us prove the above isomorphism. We have put F(z) = f ((g, 1))J(g, i)k+(1/2) for g E S(IR). Then

F(y(z)) = f (('Y.g,1))J(yg, i)k+(1/2) Suppose y c Fo(N). Then note that

(y.g, l) _ (ygyf 1,1) _ (y, S(-Y))(g'yf 1, 1)(1, S('Y-'),3(-Y,g'yf 1))

_ (y, S(y))(g, 1)(yf 1, 1)(1, S(y-1)0(g, yf 1)Q(y,gyf 1))

Since,3 is a 2-cocycle, /3(h, k)/3(g, hk) =,3(gh, k),3(g, h). This shows

(y, s(y))(g, 1)(yf

1

1)(l, S(y-1))3(7g,yf 1)0(y,g))

Thus :

F(y(g)) =f((y,s(y))(g,l)('Yf1,1)(1,s(_ 1))3(yg,yf1)/3(y,g)))J(yg,i)k+(1/2) =f((g, l)(yf 1, 1)(1,S(7-1)0(yg,yf 1)13(y,g)))J(yg, i)k+(1/2) =S(y-1)0(yg, yf 1)Q(y, g)E2(yf 1)X(y f 1) f ((g, 1))J(yg, i)k+(1/2)

144

H. HIDA

Since J(y9, i)1/2 = Q('Y, 9)J(y, z)1/2J(9, i)1/2 and

y

_

a

c

x(") = X(d)

if

b

d) ' we see

F(y(z)) = s(y-1)/3(79,yf 1)E2(yf 1)X(d)J(7,z)1+(1/2)f(9,

1)J(9,i)k+(1/2)

= s(y-1)0(79,yf 1)E2(yf 1)X(d)F(z)J(y,z)k+(1/2)

Thus we need to prove s(ry-1)0(ryg,yf 1)E2(yf 1) = (d)%y2(d). If c = 0, the

both sides are trivial. Thus we may assume that c # 0. The case c # 0 is treated in [Wa2] p. 388.

For any open subgroup U of Uo(4), we write Tu = S(Q) n US(IR). We write Pk+(1/2) (U; C) for the space of holomorphic cusp forms on S(A) satisfying (m2) and (m' 1)

f (ax(u, e)r(O)) = E2((u2i E)) f (x)e((k + 2)9) for u E U and a E S(Q) .

Then Pk+(1/2) (Fu; C) - Pk+(1/2) (U; C). Thus we can transfer the rational

structure from the classical side to the adelic side to have the spaces Pk+(112)(U; A) for any subalgebra A of C.

3. - In this section, we first prove the density theorem of low weight classical cusp forms in the space of p-adic cusp forms of half integral weight. Using this fact, we describe another way, much closer to Weil's original definition in [W] and due to Shimura [Sh2], to define S(A). By the strong approximation theorem, we have a bijection :

{congruence subgroups of S(Z) of level prime to p} <> Z = {open subgroups of S(Z(P))}

A=

fl s(z) -,& : the closure of A in S(Z(P)),

where Z(P) = fl Z e. We put t =/P

S.(0; A) =U SK.(AI(pa); A) and Pk+(1/2) (S; A) =U Pk+(1/2) (A, (pa); A) a

a

Let 0 be the ring of Witt vectors with coefficients in an algebraic closure FP of FP and K be the field of fractions of 0. Let 1l be the completion

ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2) /Q

145

of an algebraic closure Q of Qp under its standard p-adic norm I Ip. We take an embedding : K - 1l. = QP and fix two embeddings Q -> C and Q , SlP for an algebraic closure Q of Q. Put A = 0 fl Q and S,, (A; 0) _ SK.(0; A)_®A 0, Pk+(1/2) (A; O) = Pk+(1/2) (,a;_A) ®A O. We write S(0; O)

(resp. P(0; 0)) for the p-adic completion of S, (0; 0) (resp. Pk+(1/2) (0; 0)), which is independent of ic (resp. k) if c > 2 (resp. k > 2). This fact is proven in [H2] and [H6] for integral weight and is conjectured in [H 1 ] for half integral weight. Now we can give a proof of this fact for half integral weight.

THEOREM 1. - If k > 2 and p > 3, we have an isomorphism preserving q-expansions : Pk+(1/2) (A; 0)

Pk+(3/2) (O; O) .

Proof : let U be an open subgroup of G(2) (G = GL(2) /z) and Y(U) be the corresponding open modular curve. Suppose that U D G(Zp) and we put

U(pa)=

sESIsP=(0

)modP}.

For each positive integer N, we put (N = exp( jet). Then Y., = Y(U(p-)) has a model over A = Z[1/6N, (N] for the level N of U which is the moduli space parametrizing an elliptic curve E with U-structure and a Drinfeld

style level structure at p; that is, a morphism ¢ : Z/p'Z -+ E of group schemes such that

E [O(P)] is of degree pa as a relative Cartier divisor PEZ/p'Z

(see [KM] Chapter 1 or [H71). Suppose that U C Uo(4). We can compactify Ys, adding cusps to get the proper curve X, which is regular proper over Zp [KM]. Let w/Y be the invertible sheaf corresponding to weight 1 modular

forms studied in [KM]. Let Ia be the Igusa curve containing the cusp 00 which is the irreducible component of Xq mod p'. If we consider the pordinary moduli problem 0 : pp. C E of generalized semi-stable elliptic curves, it gives an open subscheme Ua of X,, whose fiber at p is Ia-{super singular points}. Then there exists a unique invertible sheaf w1/2 on Ua such that wl/2 = Wand O E r(Uc/c, wl/2). By the q-expansion principle and p > 3, 9 is a section defined over Z P, We first suppose that U is contained in the principal congruence subgroup of level 24. Then the Dedekind ri function is a section of Ho(Uc.,w1/2). Writing w(2 + (k/2)) for w1®2 ®° IU., we

146

H. HIDA

consider the following commutative diagram : 0 1

H°(U.,w(k+

2))

H°(U.,w(k+ 2) 0 Z/p°Z)

0Z/p`xZ

177

l77

H°(U,,wl(k+ 1)) ®Z/paZ

H°(Ua,wl(k + 1) ® Z/paZ)

lp

1

H°(Ua,O(D)) 0 Z/p`Z

H°(U., O(D) 0 Z/paZ)

1 0,

E

where D is a cuspidal divisor given by div(a) =

(ords(ij))s

sEU.,orde (,j)>0

and the first horizontal maps are given by the multiplication by 77. Here we

regard D as a closed subscheme of Ua in a natural way, and O(D) is its structure sheaf. The first row is exact. When k > 2, deg (w (k + 2) ®A Q) > deg(1 x,/A ®A Q). Thus the Riemann-Roch theorem tells us the vanishing of H1(X,y, w (k+ 2)) ®AQ = H1(U.y, w (k+ 2)) ®AQ. Since w (k+ 2) is Aflat, this shows the vanishing of H1(Uy, w (k + 2 and the exactness of the second row. Since the vertical maps are injective, we have a commutative diagram whose rows are exact if k > 2 0

H°(Uy,w(k + 2)) 0 Z/ppZ

H°(U.y, w(k + 2)) 0 Z/ppZ

H°(U.y, O(D)) 0 Z/ppZ

0

E Ea

H°(U.y,w(k+2)) ®Z/ppZ

H°(Uy,w(k+ 1))O Z/p'Z

H°(Uy, O(D)) ® Z/p8Z

ONA ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2),,

147

where 0 < /3 < a < y and Ea is the modular form on U. of weight 1 with E,, - 1 modp«. Taking injective limit with respect to y, we write

H°(U,w(t)) = lim H°(Uy, w(t)) . 'Y

Then we have by the p-adic density theorem of integral weight modular forms, if k > 2 0

0

HO (U., w(k + )) ® Z/pOZ

E.

H°(U.,w(k+

®Z/p'Z 2))

2

H°(U.,w(k + 2)) 0 Z/pa7G

H°(U,,.,w(k + 1)) 0 Z/p'Z

H°(UU, O(D)) ®Z/p'7G

H°(Uc, O(D)) ®Z/p Z

.

This shows the p-adic density theorem for half integral weight if 24 1 N. If not, we just use restriction and transfer maps and recover the result in general if p > 3. Put

S,(A) = _U S,,(0; A) and P,,,(A) = _U PK,(0; A) DEZ

S(O)

= U S(0; O)

,

DEZ

and P(O) = U P(0; 0).

DEZ

AEZ

If f E S,,(A), one can find I' such that f E SK,(I;A). Then for each x E S(A(P°°)) (A(P°°) = {x E A xP = x,). = 0}), one can find I

u E f c S(A(°")) and y E S(Q) such that x = wy, where f is the closure of T in S(A(°°)) (A(°°) = {x E A I x = 0}). Some time ago, Shimura defined the action of x E S(A(P°°)) on f by f' = f I y [Sh2]. Then he showed that the action is a smooth action of S(A(P°°)) on Sk (Q( 6)), where Q(b) = Q[(N I (p, N) = 1] is the maximal abelian extension of Q unramified at p. Using Katz's theory of p-adic modular forms (see [H7] Chapter 2), it is easy to check that the action of S(A(P°°)) preserves S,, (A) and extends

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H. HIDA

to S(O) by p-adic continuity. Note that the representation of S(A(P°°)) we obtained is smooth, but not of finite type. I like to call this representation the p-adic automorphic representation of S(A(P°°)). According to Shimura [Sh2l, we can give a definition of S(A(P°°)) as follows :

S(A(PO°))={(x,v)ES(A(P°°))xGL(P(O)) I (f")2 = (f2)x for all fEP(O)}. Then we have an exact sequence : 1-* {±1 } -> S(A(P°°)) - S(A(P°°)) -1.

It is basically shown in [Sh2l that any x E S(A(P°°)) is liftable to an automorphism v of Pk+(1/2) (Q( 6)). Since x preserves A-integrality, v keeps

A-integrality and hence gives an automorphism of P(O). This shows the surjectivity of 7r. There is an alternative way of showing the surjectivity of it. One can check that the action of S(Z(P)) is liftable to half integral weight by multiplying half integral weight cusp forms by 71 (or 0), because the action

of S(Z(P)) preserves A-integral structure of integral weight cusp forms. It is easy to check the liftability of the action of upper triangular matrices. Thus by the Iwasawa decomposition, every x E S(A(P°°)) is liftable. By definition, we have a smooth p-adic "automorphic" representation of S(A(P°°)) on P(O).

Although we do not have a good action of S(Q) on P(O), we can at least define an action of the maximal split torus T(7LP) = ZP < in S(7LP). Take a subgroup A corresponding to Z E Z. Thus its level N is prime

to p. We assume that ro(4) D A. When A is a Zr algebra, we can show multiplying by 0 as done in [H 1l §3 that Pk+(1/2) (O1 (pr); A) is stable under

the action ofZN= 7LP" x (Z/NZ)" for the level N of A, which is given for (A 1 (pr); A) by f E Pk+(1/2) (3.1)

fz= f kzP

o,,, for arz E SL2 (7G) with vz

= I\

z-1 0 ) 0

Npr.

z 11 mod

This action of ZP" extends by continuity to P(O).

4. - We put W = 1 + pZP in ZP" Z. Then W = ZP as topological groups, and ZP" = W x a for the subgroup p of (p-1)-th roots of unity. Simplifying the notation, we write Pk+(1/2)(Npa; A) for Pk+(1/2)(L'1(Npa); A). We put,

for O- D E Z and a character E of W modulo pa, Pk+(1/2)(A(pa);E; A)={fEPk+(1/2)(AI(pa);A)

If I z=E(z)4f for z E W},

where A(pa) = Ai(p) n Fo(pa) and A is a ring either in SZP or in C containing all the values of e on W. We now consider the action of the

ON A-ADIC FORMS OF HALF INFEGRAL WEIGHT FOR SL(2)/Q

149

Hecke operator T(q2) for each prime q on Pk+(1/2)(A1(pa); (C). As shown in [Shl] Theorem 1.7, we know

a(n,fIT(q2))=a(p2n,f)+q-1(2)a(n,flq)+q-'a(n/g2,flg2) ifg{Np°, (4.1)

a(n,flT(g2)) = a(p2n, f) if

glNp«,

where N is the level of A and q E ZN (= ZP < x (Z/NZ)") acts on f as in (3.1). This combined with [H 11 Theorem 2.2 shows that Pk+(1/2) (O 1(pa); 0)

is stable under T(q2). In particular, we can define the idempotent e in Endo(Pk+(1/2)(A1(p"); 0)) by taking the limit :

e = n-,oo lim T(p2)n!

(4.2)

We write M°'`1 for eM for any module M with an action of e. Hereafter we allow as a base ring a finite extension of the ring of Witt

vectors with coefficients in ]FP and write the ring as 0 and its field of fractions as K. All the definitions we have given for the ring of Witt vectors carry over to this slightly general situation by extending scalar to 0 from the ring of Witt vectors. Write A = 0[[W]] for the completed group algebra of W. Then A is isomorphic to the one variable power series ring O[[X]] via u 1--f 1 + X if we fix a generator u E W. We fix an algebraic closure 1L of the quotient field L of A and consider the algebraic closure of K in QP as a subfield of E. For each normal integral domain II in 1L finite over

A, let X(II) = Homo_alg(ll, S2p) be the space of all Qp valued points of Spec(II) and A(ll) be the subset of arithmetic points, that is, those 0-algebra homomorphisms P : II -+ QP such that P(ry) = -1k(P) for an integer k(P) > 0 on a neighborhood of the identity of W. Thus sp(ry) = P(-y)ry-k(P) defines a finite order character of W, whose order will be denoted by Pr(P)-1. We write A(II; 0) = {P E A(II) 10 D P(1)}. For each congruence subgroup A (with level N) associated with ,& E Z, let ]F(O;1) be the space of II-adic cusp forms. Thus f E IF(A; II) is a formal q-expansion : (n/N, f)qn/N E ] [[q1/N]] n=o

whose specialization f(P) = E P(a(n/N, f))gn/N E P(II)[[g'/N]] at PEA(II) n=o

is a classical cusp form in Pk(p)+(1/2) (A(pr(P) ), sp; Stp) for all P E A(ll) with

sufficiently large k(P) > 0. When A = r1(N) (4 I N), we write 1F(N;11) for lP(O; II). Since A is a regular local ring of dimension 2, 1 is A-free. Fixing a

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H. HIDA

base {ij} of l over A, we can write formally that f = Ejfjij. Then it is easy to see that fj is a A-adic form. Thus P(0; II) = IP(A; A) OA II. There is another interpretation of the above space of A-adic forms. We first identify A with the measure algebra on W having values in O. Then to each f E P(0; A), we associate a p-adic measure 0 fW qdf on W having values in O[[g11N]] by (4.3)

fW Odf =

J

Oda(n/N, f)q N E O[[ql/N]].

n=1 W

Writing Xp(w) = Ep(w)wk(p) for each arithmetic point P (that is, the character of W corresponding to P), we have fW xpdf = f(P) E Pk(p)+(1/2) k(P) >> 0} (0(pr(p)), p; Q p) for sufficiently large k(P). Since {xp I

spans a dense subspace of continuous functions on W having values in K, as a measure, df has values in P(O). In particular, the new measure 0 H fW qdf s for s E S(A(P°°)) again comes from a A-adic form f s E IP(O3; A) for a suitable congruence subgroup O3 corresponding to I

Os E Z. Thus, we have a natural action of S(A(P°°)) on P(1) _ U IP(A; II). ZEZ

Similarly, we have an action of Hecke operators T(q2) and the group Z on IP(N; II). Writing c : w 1--> [w] for the tautological character of W into A,

we know that w E W acts on F(N; II) via t, that is, f I w= [w]f. Since the projector e naturally acts on Pk+(1/2) (O) and hence on P(O), e again acts on P(0; II) and P(1[). We note this fact as

PROPOSITION 1. - As long as q is prime to the level N of 0, we have Hecke operators T(q2) given by (4.1) and the ordinary projector e on IP(A; II), and the metaplectic group S(A(P°°)) naturally acts on P(II) through a smooth representation. Here the smoothness means that the stabilizer of each vector

in the representation space is open in S(A(P°°)).

We can think of the corresponding notion of II-adic cusp forms for integral weight modular forms (cf. [H5] Chapter 7). We briefly recall the definition. For o E Z, a formal q-expansion f c II[[g1/N]] is called an II-adic cusp form of integral weight if f(P) E Sk(p) (A(pr(p)), Ep; Q p) whenever P is arithmetic and k(P) is sufficiently large. We write S(A; II) for the space of II-adic cusp forms (of integral weight). Then similar to Proposition 1, we have Hecke operators T(n) (cf. [H5] Chapter 7) and the ordinary projector e on S(o; II). In this case, e is given on the space of p-adic cusp forms by

e = lim T(p)". The group S(A(P°°)) naturally acts on U S(0; II). We n-*oo DEZ

actually need to have G(A(P°°))-action (recall G = GL(2)lz). Note that

ON A-ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)/Q

151

G(A) = G(Q)G(7G)G+(R) for the identity connected component G+(R) of G(R). For each open subgroup U of G(2), we consider cusp forms f : G(A) ---> C satisfying : (M 1)

f (axu) = f (x) det(u,,,) J(u,,., i)-k for u E UC,,.R" ;

(M2)

Df= rk(k2 -2)l f; u ) xf Idu=0forallx E G(A).

(M3) J(Q\A

1

We write Sk(U; (C) for the space of functions f satisfying (M 1-3). Choosing

a complete representative set R = R(U) for G(Q)\G(A)/UG+(R) in G(2), (Ftut-1 = S(Q) f1 tUt-1S(R)) for each we can define Ft E Sk(FtUt-1; cC)

t c R by Ft(z) = f(tg)det(g)-1J(g,i)k, where g E G+(TR) such that g(i) = z. Then it is easy to see Sk(U;C) = ®tERSk(FtUt-1;(C). We then define Sk(U; A) by the image of ®tERSk(FtUt-1; A). We can take R inside

R = { (0

0

/

1 a E 7L(P) }. We always choose R in this way. Then we have

e and T(p) we/ll defined on Sk(U; Slr). Let U = {U : open subgroup of G(Z(P))}.

Write Uo = U x GL2(ZP) for U E U. Taking R(Uo) in R so that R(Uo) D R(Vo) if V D U for all U, V E U, we define S(U; II) = ®t ER(U(,)S(rtUot-1; II)

and S(l[) _ U S(U; II). Using the stability of U S(0; )<) under S(A(PO°)), UEU

pES

it is easy to check that S(I) is stable under S(A(P°°)). Since Ca a E A(P°°) basically permutes the direct summands S(II) is stable under G(A(P°°)). We thus have

011

with

S(rtuot-1; II) of S(U; II),

PROPOSITION 2. - The space S(U; II) has, as II-linear endomorphisms, the

ordinary projector e and the Hecke operators T (q) for primes q prime to the level of U. The group G(A(P°°)) acts on S(II) smoothly.

5. - Before going into a hard work, we like to give a sketch of the theory. The first main result is

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H. HIDA

THEOREM 2. - The automorphic representation of S(A(POO)) on pord (II) is

smooth and, after having extended scalar to the field of fractions of II, is a discrete direct sum of irreducible admissible representations with multiplicity at most 1.

Putting off all the details to the end of this paper for attentive readers,

we here give a sketch of the proof. It is well known that §ord(N; A) = Sord(rl(N); A) is free of finite rank over A (see [H5] Chapter 7), and if k(P) > 2 for P E A(A; 0), then (*)

Sord(A; A)/PSord(A;

A) -

Skrd(A(pr(P)), EP; 0)

.

This implies that there are only finitely many, bounded independently of weights, of complex irreducible automorphic representations of G(A) which is p-ordinary and of conductor dividing Np. On the other hand, one has the Shimura correspondence : Sh : {irreducible holomorphic automorphic representations of S(A) of weight k + 1 } --> {irreducible holomorphic automorphic

representation of G(A) of weight 2k}.

By a result of Waldspurger, there exists a bound M > 0 such that (i)

#Sh-1(7r) < M for all k, if C(7r) I Np,

where C(7r) is the conductor of 7r. If if is p-ordinary (that is, the eigenvalue of T(p2) in is a p-adic unit), Shff) is p-ordinary. Moreover, if we write V for the space of i, we have a positive bound M' independently of weights

(but depending on 0) such that (ii)

Then (i) + (ii)

dime H°(0(p), V) < M. ranko Pk+(1/2) (A(p); 0) < M"(0) independently of k

for a positive bound M"(o). Take a subset

in pord (A; A)

which is linearly independent over A. Then we can find m rational numbers nl,... , n,,,,, such that D = det(a(n1, Off)) # 0. Therefore for arithmetic P with k(P) sufficiently large and ep = id, gti(P) is and element of Pord (p); 0) and D(P) 0. In other words, {oi(P)}ti is linearly independent over 0. Therefore m < M". This implies that rankA pord (A; A) < M". As we will see later, Ford (A; A) is actually free of finite rank over A. Then all the assertion follows from the weak multiplicity one theorem of Waldspurger by reducing the A-adic reprensentation modulo P.

ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)

/Q

153

Thus we have the A-adic Shimura correspondence :

Sh : {irreducible A-adic ordinary automorphic representations of S(A)} -- {irreducible A-adic ordinary automorphic representations of G(A)}. Suppose II = Sh(II). We write 9rp = II mod P and 7rp = Sh(arp). Then 7rp for an arithmetic P is a scalar extension of classical representation if k(P) >

2. This means that one can supplement a (unique) local representation at p with irp to get a complex automorphic representation if k(P) > 2, which we again write 7rp. Similarly 9rp is associated with a complex automorphic representation of the metaplectic group if k(P) is sufficiently large, because we can only prove the metaplectic version of (*) under the assumption that k(P) is sufficiently large. Here note that 7rp # II mod P but 7rp = Sh(arp) = II mod p2, because representations of weight k + .

correspond to those of weight 2k. Here we used the group structure of Spec(A)(O) = Homgr(W,Ox) to define p2. The above fact characterizes the A-adic Shimura correspondence. By (*), the prime to p-part C(II) of the conductor of 7rp is independent of P. Moreover the central character of II can be written as L02 for a finite order even character V) modulo 4pC(II), where t is the tautological character of W into A" composed with the "norm" character : (A(P00))x 9 x --* IxI-lw-1(x) E ZPx for the Teichmuller character w. We put Op = cp2/)w k( ) for each arithmetic P. As a striking consequence of his theory, Waldspurger expressed the square of a certain ratio of two Fourier coefficients of a cusp form of half integral weight by a ratio of L-values attached to the image under the Shimura correspondence. Applying this result, we get a A-adic version of his result : I

THEOREM 3. - For each pair (m, n) of positive square free integers with m/n E HII4NPQ , we find two elements 4 and T in II such that if k(P) > 1 or 1/i2p # 1, we have : 4,(P)2 q,(P)2

L(2,7rp L(2,7rp

P1Xm)

as long as

L(2,EP ®0P'Xm) 0, where Xt is the quadratic character associated with Q(f ). 6. - We now start filling the details with the argument in Section 5. Fix a character V of (Z/NpZ) x . For each arithmetic point P E A(A), we define

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H. HIDA

a character by of ZN by Op(z) = /i(z)Xp()z-k(P) _,0ePW-k(P)(z), where z 1--> < z > is the projection to W and w is the Teichmiiller character. We now prove

PROPOSITION 3. - The dimension of P%+1/2)(A0(pr(P))>V)P;1 )) is

bounded independent of P E A(A) if k(P) > 1 (the dimension depends on ,& E Z).

To prove the proposition, we prepare several lemmas. Let £ be a prime

and put

Ur=Ur,e={( a

d)esL2(z)Icomodr},

For each character x of Ze modulo £r and a Ur-module M, we write M(x)

for the x-eigenspace. That is, M(x) _ {m E M I (a Ca c

b d/

E

d

) m = x(d)m for

Ur}. When the reference to the level fr is necessary, we write

M(i'.r, x) in place of M(x).

LEMMA 1. - Let 7r be an irreducible admissible representation of the metaplectic covering group S(Qe) of SL2 (Q) and V denote its representation

space. Let x be a character of Qe modulo fr. Suppose that 7r appears as a local factor of a holomorphic automorphic representation of weight k + (1 /2) (k > 2). Then the dimension of V (.fir, x) is bounded independent of V and x (but it depends on r).

Proof : when 7r is special or principal, then we can realize it as a subquotient of the induced representation space 13,. = 13µ,eQ of a character A of the standard Borel subgroup, as in [Wal, 11.2] and [Wa2, II], for the .part ee of the standard additive character e of A/Q and a quasi character

u of the standard Borel subgroup of §(Q t). Since the left translation by the upper triangular matrices of S(Qe) is already prescribed on Bµ, any function in C3µ is determined by its restriction to SL2(Zt) x {±1}. Then for each given open compact subgroup U of SL2(Ze) x {±1}, the dimension of H°(U, Bµ) is bounded by the index 2(SL2(Z() : U). A more effective bound can be obtained using the explicit calculation of the space x)

done in [Wa2) Proposition 9 (p. 417) (see also Lemma 3 in the text). We then have dim(B,(Qr, x)) < 2(r + 1). This settles the problem in the case of non-super cuspidal representations. Let II be a holomorphic automorphic representation of S(A) of weight k + a (k > 2) having it as its factor

155

ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2),Q

at e. Let W be the space of the e-component of the automorphic representation of GL2(A) corresponding to II by the Shimura correspondence. Using the notation of [Wall V.4 (p. 99), we mean by W the f-component of V'(e, V) ®x. We know from [Cl that dim W (er, x2) < r + 1. By [Wa2) V, Proposition 5 (p. 404), V (er, x) is a subspace of the space spanned by, with the notation in (Wa2l, i,,,e o j,,,e(w)(fr,,,) for r sufficiently large (if

r > max(2ve(2) + 1, ve(C(x))) for the conductor C(x) of x), where w E W(er,x) and v E (Qell"(V)/(Qell")2). Here fr,,, is a SchwartzBruhat function on He = {x E M2(Qt) I Tr(x) = Of determined by (r, v) as specified in [Wa2) Chapter V. The choice of v E Qell" is bounded by #(Qellx/(Qell>)2) which is 4 if e > 2 and 8 if e = 2. Thus we have, for general V,

dim(V (er, x)) < 8(r + 1) for r sufficiently large . This finishes the proof. LEMMA 2. - Let it be an irreducible admissible representation of S(Q1)

with representation space V. Suppose that it is super cuspidal. Then, for sufficiently large m, T(t) annihilates V (er, x) if r > 0. Proof : note that

Uo(er)(( 0 e0 and

((e0

1)Uo(er) =

U uE7Z. /1 "Ze

uf-M ((e0 f-M ),1) _

((e0

e--M.

)

eo )'i) ((0

1

,

i)Uo(er)

u) 1). )

Thus for v E H°(Uo(er), V), we define an operator T(f-) by

vIT(em)= UEZe /1"Z

((e0

ue-'n ) e-,n

,

1) v .

The operator T(fm) coincides with the Hecke operator (TT)m acting on V (f', x) defined in [Wa2) III.3, pp. 388-389. Then we have

vit(((0

eo )'1)) I_'_Zt it( ( 10

1

,1)vdu=0

for sufficiently large m by the definition of super cuspidality. As shown in (Wa2] Lemma 4, p. 389, we know that TI = e(3-2k)/2ye(ex(e)-1T(e2) for T(e2) defined in [Shl], where ye(t) = (t, t)e-Ye(t)rye(1) 1. Thus we know the lemma from the above result. Here we should note that the definition of our space of modular forms of half integral weight is different by the character k

(=1) from that of [Wa2l, and thus we do not replace x by Xo as was done in [Wa2l for these formulas.

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H. HIDA

LEMMA 3 ([V)). - Suppose that f > 2. Let V = 13,. and let X be a continuous character of Q into Cx. We consider the Hecke operator Tyr) = Suppose that r > Sup(vt(C(X)), X(2)ry'e(2)-1f(2k-3)/2Te.

vt(C(fLX)C(ItX-1))), where C(X) is the conductor of X. Then we have the following assertions :

(i) If both µX and pX-1 are non-trivial on Zellx, then T(P) is nilpotent on V (fr; X) for r > 0; (ii) Suppose that pX-1 is unram flied but pX is ramified. Then we can decompose V(Fr; X) = N ® V(C(X); X) so that T(e2) is nilpotent on N and V (C(X); X) is one dimensional on which T(e2) acts by scalar multiplication of XV)Q

(iii) Suppose that uX is unramified but uX-1 is ramified. Then we can decompose V (Fr; X) = N ® V (C(X); X) so that T (f2) is nilpotent on N and V (C(X); X) is one dimensional on which T(2) acts by scalar multiplication

of

x()Q(2k-1)/2/l(e)

;

(iv) Suppose that both µX and

µX-1

are wuamified. Then we can de-

compose V (.fr; X) = N (D V(; X) so that (I) T(t2) is nilpotent on N, (ii) V (i?; X)

is two dimensional, and (iii) we have a base {vl, v2 } of V (2; X) such that vlIT(t2)=X(&)f(2k-l)/2µ(2-1)v1

with some constant c.

and v2 IT(j2)=x (t)e(2k-1)/2A(f)v2+cv1 l

Proof : write v(e) for the exponent of 2 in C(e) for any character of e of Z.11'. As shown in [Wa2) Proposition 9, p. 417, under the assumption v(µX-1) of r > Sup(v(X), 1), V(2'; X) 0 if and only if r > v(pX) +

As long as r > v(X) and r > 1, T(22) sends V(2r; X) to V(; X) (cf. [Wa2] Lemma 7 or [H2) (8.6)). This shows that for sufficiently large M. V (2r; X) IT(f2m) is contained in V (C(X); X) or V (f; X) if v(X) < 1.

Unless X is quadratic, v(X) = v(X2) since 2 > 2. Thus if X2

id,

then v(pX) + v(µX-1) > Max(v(µX), v(,X-1)) > v(X). If moreover both v(pX) and v(/4X-1) are positive, then v(pX) + v(pX-1) > v(X) and thus V (C(X); X) = 0. Therefore T(22) is nilpotent on V (jr; X) if X2 id and if both v(jX) and v(pX-1) are positive. Now suppose that X2 = id and both v(µX) and v(pX-1) are positive. Then if X id, then C(X) = 2

and V (C(X); X) = 0 because v(pX) + v(pX-1) > 1. If X = id, then again V(2; X) = 0 because v(µ) + v(µ) > 1. Thus T(22) is nilpotent if both v(pX) and v(µX-1) are positive. Now suppose that v(pX-1) = 0 but v(µX) > 0. Then X2 54 id because v(µX) = v(X2) > 0 (and hence v(X2) = v(X)), and V(C(X); X) is one dimensional by [Wa2] Proposition 9. Moreover

by [;a2] Proposition 10, (ii), we know that T(f2) acts on V(C(X); X) by X(f)2(k/2)-lµ(2-1) Thus we can decompose the scalar multiplication of V(fr;X) = N (3 V(C(X);X) such that on N, T(f2) is nilpotent, and on the one dimensional space V(C(X);X), T(f2) acts via the multiplication

ON A ADIc FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)/Q

of

X(f)f(k/2)-1µ(f-1). Suppose v(µX-1)

157

= v(µX) = 0 and x # id. Then

x2 = id because v(ltX) = v(X2) = 0. By [Wa2] Proposition 2, V(f; x) is 2-dimensional, and there is a base {v1, v2} of V(f; x) such that v1 V2

I T(f2) =

x(f)f(k/2)-1µ(f-1)v1

I T(f2) =

and

X(f)&/2) -11L(f)V2 + f(k/2)-21'e(f)-1x(f)(f - 1)v1

.

Thus we can decompose V (P'; x) = N ® V (t; x) such that on N, T (j2) is nilpotent and on the 2-dimensional space V (f; x), it acts by the above formula. Next suppose that v(µx-1) = v(ax) = 0 and x = id. Then V (f; x) is 2-dimensional, and we can find a base {v1, V21 by [Wa2] Proposition 10 such that (v1 + v2) E V(1; x) and

vl IT (f2) =

x(f)f(k/2)-1µ(f-1)v1

and v2 I T(f2) =

X(f)f(k/2)-1µ(f)v2

+ cv1

with some constant c. The value of c is given by [Wa2] p. 420. This shows

that V (f'; x) = N ® V (f; x) such that on N, T(f2) is nilpotent, and on the 2-dimensional space V(f;x), T(f2) is an automorphism described

as above. Finally we assume that v(µx) = 0 but v(µx-1) > 0. Then v(µX-1) = v(X-2) > 0 and hence v(x2) = v(x). Thus again by [Wa2] Propositions 9 and 10, V(C(x); x) is one dimensional and T(f2) acts on X(f)f(k/2)-1µ(f). Therefore, we can decompose it by the multiplication of V (rr; x) into V (C(x)) X) ® N, where on N, T(f2) is nilpotent and on the one-dimensional space V (C(x); x), it acts by the scalar X(f)f(k/2)-1µ(f). LEMMA 4 ([Wa l ] Proposition 18, p. 68). - Let p* be an irreducible admissible representation of PGL2 (Qe) and let p be the corresponding

irreducible admissible representation of S(Q1) via Well representation with respect to the additive character ee (1; E Qell"). Then we have

Equivalence class of p* 7r(µ> µ-1) (µ2 a) 11 2 a, jI 7 a 1/2 ) a(µ, A -1 ) (11

Equivalence class of p r{lx{

u(a1/2 a-1/2)

Supercuspidal Supercuspidal

Supercuspidal

aµxe

where et is the standard additive character of Qt and ee (x) = et(t;x) and we have used the notation of [Wa 1 ] Propositions 1 and 2.

158

H. HIDA

Here note that irf,xe (resp. &µx4) with respect to ep is isomorphic to µxev (resp. vµx{,,) with respect to e,v, and hence the right-hand side is well defined independent of the additive character. A cusp form f E S,c(A,(pr);C) is called ordinary at p if f I T(p) = Af and IAIp = 1. An automorphic representation it of GL2(A) spanned by a holomorphic primitive form f is called ordinary at p if f is ordinary at p. LEMMA 5 (e.g. [H3) § 2). - Let 7r be a unitary holomorphic automorphic representation of GL2 (A). Suppose that 7r is irreducible and ordinary at p. Then the local component 1rp of 7r is either a principal series representation it (a, )3) with unramfied a or a special representation a(a,,0) with unramified a. Let f be the primitive form of weight k on GL2 (A) belonging to it and

write µ for the central character of it and A(T(p)) for the eigenvalue of T(p) on f. Then if 7r = 7r(a, p) and a and,3 are both unramified, then a(p) + /3(p) = p(1-k)I2A(T(p)), a(p)/3(p) = µ(p) and I A(T(p))I p = 1. If7rp = 7r(a, /)) and,3 is ramified, thena(p) = p(l-k)/2A(T(p)), a(p)O(p) = µ(p) and I A(T (p)) I p = 1. If itp = a(a,,3), then 7r,,. is of weight 2 and A(T (p)) = a(p).

LEMMA 6. - Let F be a number field of finite degree. Let p be a cuspidal automorphic representation of PGL2 (FA) and let R be the set of all cuspidal automorphic representations of S(FA). Define for each integral ideal N of F,

R(p; N) = {7r c R 17rv = pv for all v outside N}

,

where 7rv denotes the corresponding representation of PGL2(Fr) via Weil representations defined in [Wa 1, V.41 (where it is written as : T see Lemma 4). Then we have

V'(e, T) ;

#R(p; N) C #{H INF,
Proof : we know from Lemma 4 (or the remark after the lemma) that

if T is principal or special, then V'(ev,Tv) =

Moreover if

x/y E (Fti )2, then V'(ev,Tv) = V'(ev,T.) for all Tv by [Wal) Theorem 2, p. 80, Proposition 28, p. 98 and [Wa2) Assertion 3, p. 394. Thus the number of isomorphism classes in {V'(ev ,Tv) I x E Fvx} for all v outside N are at most #{l lNF
Proof of Proposition 3: we only prove the assertion when A = r1(N). The general case follows from this special case because any A contains a conjugate of F1(N) for a suitable N. We shall prove the boundedness

for P E A(A) with k(P) > 1. We write x = op for a given P E A(A)

ON A-ADIc FORMS OF HALF INTEGRAL WEIGHT FOR SL(2) /Q

159

and V) : (Z/NpZ) 1 --4 QP and consider x as an idele character so that

X(w) = x(.&) for a prime element zu at any prime f outside Np. Let V be the

subspace of functions on S(A) spanned by right translations of elements in

nord

(Npr(P), OP; (C)

under the Hecke algebra of S(A). We decompose V = ®PV(p) into the sum of irreducible subspaces V(p). Then by the weak multiplicity one theorem proven by [Wal) p. 131, each irreducible representation p occurs at most once. Decompose p = ®epe into the tensor product of local representations. Then by Lemma 2, pp is either aµ or rµ for a quasi character µ : Qpx - Cx.

By the Weil representation, aµ corresponds to Q(µ,µ-1) and 7r(p, p-1), which is a representation of PGL2 (A) (Lemma 4 and [Wa 1) Proposition 27 and Lemma 70). Then the Shimura correspondence is given locally by

aµ'' o,(Fex, p-1x)

and 7rN,

i)7r(px, p-1x)

and globally by p 1-- + p* ® x, where p H p* is given via the global Weil representation. The eigenvalue for T(p2) on V(pp)(pr; x) (r = r(P)) is given as follows (Lemma 3) : if µx is unramified, then it is

(k = k(P)); if µ-1x is unramified, then

µ'1x(p)p(2k-1)/2

px(p)p(2k-1)/2

and if both px

and p-1x are ramified, it vanishes. On the other hand, these values are the eigenvalue of T (p) on V (p p O x) (pr; x2) by Lemma 5. Note that even if both

µx and p-1x are unramified, at most one eigenvalue in can be a p-adic unit in Q. Thus p corresponds to and p-1x(p)p(2k-1)/2

px(p)p(2k-1)/2

the ordinary p* of character x2 and of level at most Npr(P). Then by [H5) Theorem 7.3.3, the number of such automorphic representations occurring in S2k(P)(Npr(P),x2) is bounded independent of P if k(P) > 1. Then by Lemmas 2, 4 and 6, we know the assertion of the proposition.

We say an element f c IP(A; II) is ordinary, if for all P E .A(II) with sufficiently large k(P), fp E space of all II-adic ordinary cusp forms as Ip°rd(A; II). Then p°rd(A; II) _ Pk(P)+(1/2)(A(pr(P)),Ep;Sl1). We denote the

eP(A; II).

PROPOSITION 4. - For each 0 E Z with A C I'o (4), p rd (A; II) is free of finite rank over II.

Proof : we prove the assertion for p°rd (N; II) applying the argument of Wiles [Wi). The other cases can be treated similarly. Let A = I'1(N). Let IK be the quotient field of II, which is a finite extension of L. We put pord (N; III) = pord (N; II) 02 K. Let f1 i f2,. .., fr be a finite set of linearly independent elements in P°rd (N; II) over II. Then we can find positive integers

160

H. HIDA

nl,... , n,. so that D = det(a(n2, fj)) # 0. We now choose P E A(II) so that for all i = 1,... , r, fz(P) E

Pk(P)+(1/2)\0(pr(P))+EP QP)

and D(P)

0.

Then 0 # D(P) = det(a(nz, fj(P)) and thus ff, (P) are linearly independent. Namely, we have r < dim Pk(P)+(1/2) p, (p' (p)), eP; lP)

,

which is bounded independently of P by Proposition 3. Thus there is a maximal set {f1i f2, ... , fr} of linearly independent elements in pord(N,1[). That is, dimK Ford (N; IIK) = r < oo. For any fin prd (N;1[), we can write r

f = > cj(f)fz and Dc;,(f) E L Thus D-1(IIf1 +

+ If,) D ll ord(N;1[) and

£=1

hence prd (N; II) is of finite type over II as II-module, because II is noetherian. Now we see by definition that ]EDOrd(N; II) = npJpord(N; IIp) where P runs

over all prime ideals of height 1, lip is the localizaton at prime P and lord (N; 1p) = pord (N; II) ®Q IIp. This shows that p rd (N; II) is II-reflexive and

hence if II = A, then prd (N; A) is A-free of finite rank. Since we already know that Ford (N; 1[) = pord (N; A) ®A II, we conclude that Ford (N; 11) is II-free

of finite rank. PROPOSITION 5. - Let PEA(II). Then each f EPk(P)+(1/2) (A(pr(P)), "FP; 0)

can be lifted to an ordinary A-adic form f E p

rd (A; II)

such that f(P) = f .

Proof : it is sufficient to prove the assertion for II = A. Let E(X) E A[[q]] be the A-adic Eisenstein series (cf. 1H51 § 7.1) such that for the generator

w=1+pofW

l E(Q)=(Q(w)-1) {LP(1-k(Q),EQw-k(Q))/2+y( Q()d-1)gn1 J 00

n=1 0
in Mk(Q) (A(pr(Q)), EQ; 0P) for all Q E A(A). Then we see for the point Po of

X (A) corresponding to the trivial character of W, E(Po) = (1 -p) log(w)/p, which is a p-adic unit. We then put F = E(Po)-1E and consider the pro(A(pr(P)), duct f F inside A[[q]]. Then F f (Q) = f F(Q) E Pk(P)+k(Q)+(1/2) EPEQ;1)). We define a formal q-expansion F * f (X) by F f (EP1(w)w-kX + (Ep1(w)w-k - 1), which is a A-adic cusp form in P(N; A) ([H5]

Lemma 7.1.1). Then we see thatF*f(P) = fF(Po) = f. Then e(F*f)(P) _ (F * f (P)) I e = f by Lemma 7 and the assertion of the theorem follows.

ON A-ADIC FORMS OF HALF INTEGRAL WEIGHT F OR SL(2)/Q

161

COROLLARY 1. - For P E A(l[; 0) with sufficiently large k(P) depending

on A, we have pk(P)+(1/2)00(ord

Pr(P)), EP; 0) = pord(A; )()/Ppord(A; 1).

Proof : Choose a base fl,... , fr of POrd(A; II). We can find a> 0 so that (i) fi(P) E Pk(P)+(1/2) Ep; O) for all i and all P with

k(P) > a, and (ii) there exist integersni such that det(a(ni/N,fj))(P)#0 ifk(P)>a. Then fi (P) are linearly independent over O. Thus pord (0; II)/ppord (0 1) injects into Pk(P)+(1/2) (A(pr(p)), EP; 0). Surjectivity of the morphism follows from Proposition 5. COROLLARY 2. - Let fl,... , fr be a base of pord(A; II). Then we can find

integers nl,... , nr so that det (a(ni/N, fj)) E F. Proof : let fl, ... , fr be a base of Pk(P)+(1/2) (A(pr(p)), Ep; O). let = be a prime element of O. If det(a(ni/N, f3) - 0 mod wO for all choice of integers fl, ... , nr, then { fi mod zo} are linearly dependent and hence we can find A, E 0 not all divisible by w such that EiAi fi = 0 mod ruO. Then E Pk(P)+(1/2)(A(Pr(P)) Ep;O)

but w-1 Ai are not all in O. This contradicts to the fact that {f} forms a base. Thus we can find the ni's so that det(a(ni, fj)) E Ox. Now applying this argument to a base {fi(P)} by choosing P with sufficiently large k(P), we find that det(a(ni/N, fj))(P) E Ox which implies that det(a(ni/N, fj)) E Ix

Analogs of all the assertion so far we proved in this paragraph holds for Sord (U; II) in an obvious sense (see [H5] Chapter 7). In particular, the statement corresponding to Corollary 1 for Sord(A; II) holds if k(P) > 2.

7. - We now restate Theorem 3 in the language of p-adic Hecke algebras. Let hord(N; 0) be the p-adic ordinary Hecke algebra defined in [H5] § 7.3. Let us recall the definition. The algebra hors (N; 0) is the Asubalgebra of EndA(Sord(N; A)) generated by T(n) for all n. There is another description of the algebra. Writing h,rd(Np'; 0) for the 0-subalgebra of Endo(Skrd(Npr; 0)) generated by T(n) for all n, we have a natural isomorphism : hord(N; 0) = l4im hk''d(Npa; 0) if k > 2, which takes T(n) to a

T(n) [H2]. Under the natural pairing < h, f >= a(1, f I h), we know HomA (hors (N; 0), A) = §ord (N; A) and (7.1)

HomA (Sord (N; A), A) - hord (N; O)

.

162

H. HIDA

We have a smooth representation of G(A(P°°)) on Sord (II) = U Sord (U, II) _ UEU

eS(l<) and S(II). Thus compactly supported smooth functions on G(A(P°°))

with values in II act on §(1). We fix an algebraic closure L of L. We then consider S°rd(A) = Sord(A) ®A A as a G(A(P°°))-module for any Asubalgebra A in E. Each irreducible factor of the representation on § rd(L) of G(A(P°°)) is admissible by the control theorem ([H5] Theorem 7.3.3, which is the integral weight counterpart of Corollary 1 and is valid for all arithmetic points of weight k > 2). Pick an arithmetic point P with k(P) > 2

and consider the localization Ap at P. Then by the control theorem, Sord(Ap) ®A K(P) for K(P) = Ap/P is a semi-simple GL2(A(P°°))module. Since there are Zariski dense arithmetic points in Spec(A) at which the control theorem holds, we see that Sord (L) is semi-simple as a G(A(P°°))-module. Thus Sord(L) is a sum of irreducible subspaces. The multiplicity is one by the control theorem combined with the multiplicity one theorem in classical situation. Since the proof of the factorization theorem

in [JL] § 9 is purely algebraic, it carries over to our situation, and each irreducible factor it of §ord(L) is factored into the tensor product of local representations : it = ®e P're. Let A; hord(C; 0) -> II be a primitive Aalgebra homomorphism. Then by the control theorem, we have a unique automorphic representation ir(P) = ®e7re(P) corresponding to A mod P for

P E A(II) with k(P) > 2. Thus A corresponds a unique factor it = ir(A) of S°rd (L) and 7re(P) = ire mod P. We write V (7r) for the subspace of §ord (j[) on which S(A(P°°)) acts via it. Thus for each arithmetic point P with k(P) > 2, Ap(T(n)) = A(T(n))(P) is an algebraic number. Then for each Dirichlet character cp, we can define the complex L-function : 00

L(s, Ap

E n=1

(p(n)Ap(T(n))n_s

.

Note that L(s, ir(P)) = L(s + k (p)-', Ap) is the standard L-function of ir(P). As is well known, the L-function L(s, Ap ®cp) has a motivic interpretation. Since II is an integral domain, we see that ZC = ZP" x (Z/CZ)" E) z F--, A() E II is a character, where is the operator induced by the central action of z E (Z(P)) X C G(A(P°°)) on S(II) (see (2.1)). In particular, it restriction to µp_1 x (Z/CZ)" gives a character Oo : µP_1 x (Z/CZ)" --> QP . We regard this character as a character of ZC composing the projection : ZN -> µP_1 X (Z/CZ)" = (Z/CpZ)" and call it the character of A. We now consider the following conditions on A :

(He). Writing xe = 7r (a, 0) when Ire is principal (f a(-1) = Q(-1) = 1;

p), we have

163

ON A-ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2) /Q

(Hp). Oo = Vi2 for an even character 0 modulo N for N divisible by C and 4.

Under this condition, the automorphic representation associated to A is in the image of the Shimura correspondence (see [Wa2] Proposition 2). Now we consider the automorphism of A which takes w to IDm (w E W) for m prime to p. This ring automorphism extends to an automorphism a,,, of II if II is sufficiently large. For each P E A(ll), we denote p2 for

P 0 0'2. Then k(P2) = 2k(P) and Ep2 = Ep. As constructed in [K] and [GS], for each character cp of (Z/NpZ) 1, there is a two variable p-adic Lfunction Gp(P, Q; A ®cp) defined on X(1) x X (A) interpolating the value L(k(Q), Ap ®EQlwk(Q)cp) for (P, Q) E A(II) x A(A) with 0 < k(Q) < k(P). Here is a result slightly stronger than Theorem 3 : THEOREM 4. - Let A : h°rd (C; 0) -> II be a primitive A-algebra homomor-

phism Suppose (He) and (Hr). Then for any pair (m, n) of two square free positive integers with m/n E fl Q2, there exists an element 1 in 1K such LINp

that for all P E A(1[) with k(P) > 1, ifCp(P2, P; A ® /-IX L)

4,(P)2 _ Gp(P2, P; A

0, we have

®V)-1Xn.)Op(n/m)(n/m)k(p)-(1/2)

1Cp(P2, P; A (D V)-1X'.,,)

where Xt is the quadratic character corresponding to Q(f ). Here note that under our assumption on (m, n), (m/n) is prime to Np. Proof : we take P E A(ll) with k(P) sufficiently large. Let cp be a Dirichlet

character. Then for the least common multiple N' of C and the conductor of cp, we find A 0 cp : h(N'; 0) --* 1[ such that A 0 o(T(n)) = cp(n)A(T(n)). Then the character of A ®cp is given by 02V2 . Taking even cp with sufficiently

large 2-power conductor, we may assume that the conductor C' of A (D V is divisible by 16. If we replace A by A ®cp, the role of V) will be replaced by the L-value appearing in the Vicp. Since A ®'-1X,,, = (A ®cp) ®(p assertion of the theorem is unchanged even if we replace A by A 0 . Thus we may assume that 16 1 C (hence Tr satisfies the condition (H2) in [Wa2] p. 378). Let f be the cusp form in Pi(p)+1/2(I o(N2pr(p)) gyp; flu) which is a linear combination of the base defined in [Wa2] Theorem 1 for 7r (p2). Let

us take f E p"rd(N2;1) such that f I T(q2) = a2 0 A(T(q))f for all prime q outside Np and f(P) = c f with 0 7 c E 0. Such f exists by Corollary 1. Then by [Wa2] Corollary 2, for any Q E A(ll) such that f(Q) is classical, we have : a(m, f)2(Q)L(k(Q), AQ2 0

1GQ'X.)OQ(n/rn)(n/m)k(Q)-(1/2)

= a(n, f)2 (Q)L(k(Q), AQ2 0 OQ1X,,,,)

164

H. HIDA

To get the p-adic interpolation, we need to remove certain Euler factor at p and divide the special value by a certain period. However the Euler factor and the period are the same for n and m under the condition of the theorem. Thus using two variable p-adic L-functions, the above identity can be stated as : a(m, f)(Q)2GP(Q2, Q; A ®V)-1xn.) I

Q(n/m)(n/m)k(Q)-(1/2)

= a(nf) (Q)2GP(Q2, Q; A ® W-'X.). If £P(Q2, Q;

A®V)-1xn) = 0 for all Q

as above, the p-adieG-function GP(A®

'-1xn) vanishes. Hence there is nothing to prove. If L ,(A ®/-1xn) #0, by the assumption of the theorem, GP(A ®O-1x,,,,) # 0. Then we may assume

that GP(P2, P; ® -1xm)Gp(P2 P; A ® 0 by moving around P. Then we may assume by Theorem 1 of [Wa2] that the m-th and n-th Fourier coefficients of f are both non-zero. Therefore -IX")

0. Thus we can take 4) = a(n, f)/a(m, f). Now we have the a(m; f)a(n; f) evaluation property of 4) described in the theorem for almost all P. Note that .CP(P, Q; A ®0-1xn) for a fixed n is a p-adic analytic function of (P, Q) (see

[K] and [GS]). Thus as long as the removed Euler factor does not vanish,

we get the result. The only case where the Euler factor vanishes is the case where k(P) = 1 and the character of ir(P2) is trivial. However this case is excluded because of the vanishing of the p-adic L-function in the denominator at (P2, P). = 0 4=* L(k(P), Apt ®1/Ip1xn) = 0 if either k(P) > 1 or 02 # 1, Theorem 3 follows from Theorem 4. Since GP(P2, P;

A®z0A00-1)(n)

Manuscrit recu le 20 juin 1993

ONA-ADIC FORMS OF HALFWIEGRAL WEIGHT FOR SL(2)/Q

165

References [C] W. CASSELMAN. - On some results of Atkin and Lehner, Math. Ann.

201 (1973), 301-314. [GS] R. GREENBERG and G. STEVENS. - p-adic L-functions and p-adic periods of modular forms, Inventiones Math. 111 (1993), 407-447. [H1] H. HIDA. - p-adic L functions for base change lifts of GL2 to GL3, Perspective in Math. 11 (1990), 93-142. [H2] H. HIDA. - On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math. 128 (1988), 295-384. [H3] H. HIDA. - Nearly ordinary Hecke algebras and Galois representations of several variables, JAMI inaugural conference proceedings, 1988 May, Supplement to Amer. J. Math. (1990), 115-134. [H4] H. HIDA. - A p-adic measure attached to the zeta functions associated with two elliptic modular forms H, Ann. 1'Institut Fourier 38 No 3 (1988), 1-83. [H5] H. HIDA. - Elementary theory of L -functions and Eisenstein series, LMS Student Texts tenbfbk 26, Cambridge University Press, 1993. [H6] H. HIDA. - On nearly ordinary Hecke algebras for GL(2) over totally real fields, Adv. Studies in Pure Math. 17 (1989), 139-169. [H7] H. HIDA. - Geometric modular forms, Proc. CIMPA Summer School at Nice, 1992. [JL] H. JAcQuET and R.P. LANGLANDS. - Automorphic forms on GL(2), Lecture notes in Math. 114, 1970. [KM] N.M. KATZ and B. MAZUR. - Arithmetic moduli of elliptic curves, Ann.

of Math. Studies 108, Princeton University Press, 1985. (K] K. KITAGAWA. - On standard p-adic L functions of families of elliptic

cusp forms, preprint. [MTT] B. MAZUR, J. TATE and J. TEITELBAUM. - On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones Math. 81 (1986), 1-48. [Sh 1 ] G. SHIMURA. - On modularforms of half integral weight, Ann. of Math. 97 (1973), 440-481. [Sh2] G. SHIMURA. - On certain reciprocity laws for theta functions and modular forms, Acta Math. 141 (1978), 35-71.

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[V] M.-F. VIGNERAS. - Valeurs au centre de symetrie des fonctions L associees awcformes modulaires, Seminaire de Theorie des Nombres, Paris 1979-80, Progress in Math. 12, Birkhauser (1981), 331-356. [Wall J.-L. WALDSPURGER. - Correspondance de Shimura, J. Math. pures et appl. 59 (1980), 1-133. [Wa2] J.-L. WALDSPURGER. - Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. pures et appl. 60 (1981), 375-484. [W] A. WEIL. - Sur certain groupes d'operateurs unitaires, Acta Math. 111, 143-211. [Wi] A. WILES. - On ordinary A-adic representations associated to modular forms, Inventiones Math. 94 (1988), 529-573. Haruzo HIDA

Department of Mathematics UCLA

Los Angeles, Ca 90024 U.S.A.

Number Theory Paris 1992-93

Structures

sur les reseaux

Jacques MartinetN

PREMIERE PARTIE : rappels sur les reseaux

1. - On note E un espace euclidien de dimension n, souvent identifie par le choix d'une base orthonormee de E. La norme d'un vecteur x E E est N(x) = x.x, le carre de la norme euclidienne 11x1j. Par reseau, on entend un sous-groupe discret A de E de rang n. La norme de A est N(A) = minxEA,x#o N(x). On pose S(A) = {x E A I N(x) = N(A)} et s(A) =

Le determinant de A est le determinant de la matrice de Gram d'une base de A (matrice des produits scalaires deux a deux des vecteurs !'IS(A)I.

de la base). L'inuariant d'Hermite d e A est y (A) = N(A). det(A)et la constante d'Hermite pour la dimension it est rye,, = SUPA -y (A).

On dit qu'un reseau A est entier si le product scalaire de E est a valeurs entieres sur A, et qu'il est pair si ses vecteurs sont de norme paire. Le reseau dual de A est A* = {x E E I Vy E A, x. y E Z}. Les reseaux entiers sont les reseaux qui sont contenus dans leur dual. Ceux qui sont egaux a leur dual sont dits unimodulaires; ce sont les reseaux entiers de determinant 1.

Les reseaux que nous rencontrerons seront tous proportionnels a des reseaux entiers. Dans ce cas, it existe une plus petite norme qui les rend

entiers. On definit l'invariant de Smith d'un reseau A en considerant le reseau entier A' qui lui est ainsi associe; le couple (A'*, A') de Zmodules libres de rang n possede lui-meme un invariant de Smith (suite des "facteurs invariants" ou "diviseurs elementaires"), qui est l'invariant

L

de Smith Smith(A) de A. Si Smith(A) = (al, ... , a,,), on a a,z = 1 et Smith(A*) = (a , _an-1 , ... , a). a, 2. - Soit E' un sous-espace de E de dimension r coupant A suivant un reseau A' de E'. Alors, E'1 coupe A* suivant un reseau A'1 de E'1, et *

Recherche effectuee au sein de I'unite mixte C.N.R.S.- Enseignement Superieur U.R.M. 9936

168

J. MARTINET

l'on a entre determinants la relation

det(A') = det(A). det(A") . Considerons le cas particulier dans lequel it existe une similitude u de A sur A*, que nous prenons egale a 1'identite dans le cas unimodulaire. On associe alors a tout reseau A' comme ci-dessus le reseau relatif A' = u(E')1 n A C A. En designant par le rapport de similitude, on obtient la formule

det(A' ) = A' det(A). det(A') .

3. - Soit A un reseau. Nous appellerons defaut de perfection de A la difference entre la dimension ('2 1) de 1'espace Ends(E) des endomorphismes symetriques de E et celle du sous-espace de Ends (E) engendre par les projections sur les directions des vecteurs minimaux de A; on appelle relation d'eutaxie toute expression de 1'identite de E comme combinaison lineaire de ces projections. On dit que A est parfait s'il est de defaut nul et qu'il est eutactique s'il possede une relation d'eutaxie a coefficients positifs. On sait (theoreme de Voronoi) que A est extreme (c'est-a-dire qu'il realise un

maximum local de son invariant d'Hermite) si et seulement s'il est parfait et eutactique. (En exprimant les endomorphismes de E dans un couple de bases (13,13*) ou 13 est une base de A, on transforme ces definitions geometriques issues de [Be-M1] en les definitions classiques de la theorle des formes quadratiques.)

Une condition suffisante de perfection, due a Barnes, est 1'existence d'une section hyperplane parfaite de meme norme et de n vecteurs minimaux independant en-dehors de cette section (perfection relative). Soit Ao un reseau de dimension no. Cette condition de perfection relative est verifiee par les reseaux de dimension no+ 1 dont le determinant est minimum parmi ceux possedant A0 comme section hyperplane de meme norme. Les reseaux

faiblement lamines au-dessus de Ao sont ceux que l'on obtient en iterant le procede ci-dessus, et l'on parle de reseauxfortement lamines dans le cas de ceux qui sont de determinant minimum dans chacune des dimensions no, no + 1, no + 2.... (ce vocabulaire est emprunte a Plesken et Pohst qui ont etudie les variantes des procedes de lamination dans lesquelle on considere des reseaux entiers de norme donnee). Les reseaux lamines sans autre precision sont ceux qui ont ete obtenus par Conway et Sloane par "laminations fortes" au-dessus de Ao = {0} auquel est attribue la norme 4 ([C-S], ch. 6); pour n < 8, ces reseaux, notes A,,,, sont les renormalisations a la norme 4 des reseaux {0}, Z, A2, A3, ID4,1Th5, E6, E7, ]E8, puisque la cons-

tante d'Hermite est atteinte dans ces dimensions sur les reseaux qui leurs sont semblables ("theoreme de Blichfeldt-Vetchinkin"; Korkine et Zolotareff

pour n < 5, Barnes pour n = 6).

STRUCTURES ALGEBRIQUES SUR LES RESEAUX

169

Certains resultats de perfection et d'eutaxie que nous presentons dans cette note ont ete obtenus en utilisant deux programmes de Batut, l'un calculant le rang des projections sur les vecteurs minimaux d'un reseau defini par une matrice de Gram et indiquant s'il existe une relation a coefficients d'eutaxie egaux, et l'autre dormant une base de 1'espace des relations qui existent entre ces projections et l'identite, ainsi que divers programmes disponibles dans le systeme PART.

L'inuariant dHermite dual de A, introduit dans [Be-M1], est ^1' (A) _ (N(A)N(A*))1/2 ; sa borne superieure sur les reseaux ("constante de BergeMartinet" de [C-S31) est notee y,,,. On dit que A est dual-extreme si son invariant y,,, est un maximum local. Pour qu'il en soit ainsi, it suffit ([BeM1], 3.20) que A soit extreme et que A* soit eutactique. 4. - Rappelons les definitions de quelques reseaux classiques (cf [C-S], ch. 4). Soit (Ei), 0 < i < n (resp. 1 < i < n) la base canonique de Zn+1 (resp. de 7Ln). On pose

An = {xEZn+1I

xi=0} et IIDn={xE7LnI>,xi-Omod 2}.

Ce sont des reseaux pairs de norme 2. Le dual de D. est le reseau cubique centre, de norme 1 lorsque n est > 4. 11 est isometrique au sous-reseau de Z muni de la forme 4 E xiyi defini par les n-1 congruences x1 = X2 xn mod 2. Pour n pair > 8, soit 1D ,+ = IIDn U (El + E2 +

+ En)Dn. On obtient un reseau isometrique en considerant le2double systeme de congruences

XI -x2-...-xnmod2 et sur Zn muni de la forme 4 E Xi yi. Sous cette forme, on voit que IIDn est isometrique a son dual, et meme qu'il est unimodulaire pour n - 0 mod 4, pair pour n = 0 mod 8. On pose B8 = D8 , et l'on definit E7 (resp. E6) comme l'orthogonal dans ]E8 d'un vecteur minimal (resp. d'un sous-reseau isometrique a A2), cf. No 2 (a isometrie pres, les choix faits ci-dessus sont sans importance). Les reseaux de racines sont les sommes orthogonales de reseaux de racines irreductibles, isometriques a Z, An (n > 1), IIDn (n > 4) ou En (n = 6,7,8). Ces derniers sont extremes, ont des duals eutactiques, et sont donc aussi dual-extremes. DEUXIEME PARTIE : autour du reseau de Coxeter-Todd

5. - Soit A 1'anneau des entiers d'un corps de nombres K totalement reel ou de type C.M., de degre q. On note x H x l'involution de K (1'identite

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J. MARTINET

dans le premier cas), et l'on munit K de la forme bilineaire TrK/Q(\µ) (ou parfois d'une forme qui lui est proportionnelle), ce qui fait de A un reseau entier de I[8 0 K.

Soit a un ideal de A stable par l'involution de K. On considere sur A' les congruences suivantes : moda (CO Al-A2-..._A,,,,, (C2) (C'2)

=0 mod a =0 mod a2,

qui definissent des reseaux de dimension n = qm. La congruence C'2 n'interviendra qu'en meme temps que la congruence C' I et seulement lorsque m est un multiple de la norme de a. En notant d le discriminant du corps K (i.e. le determinant du reseau A), on trouve pour les determinants des reseaux definis par les congruences C l (resp. C2, resp. Cl et C'2) les valeurs IdImNK/Q(a)2(m-I) (resp. IdImNK/Q(a)2 resp. jdImNK/Q(a)2m).

On observe que, pour m E a, le reseau defini par Cl ou par Cl et C'2 est encore entier lorsqu'on munit A de la forme -LTr(A7) : on a en effet A77

=AlDµt=mA1µ1moda.

i

Les determinants donnes ci-dessus sont alors a diviser par mq"`.

6. - Dans les numeros 6 a 9, sauf dans la remarque 8.3, A est l'anneau Z[w] (w2 + w + 1 = 0) des entiers d'Eisenstein. Les congruences C 1, C2, C'2 ont ete considerees dans les annees cinquante par Coxeter, Coxeter et Todd, et Barnes ([Cox), [Cox-T], [Bar]).

Soit n = 2r + t. Le reseau Lr de Barnes est forme des elements de Ar+t qui verifient la congruence C2 et dont les t dernieres coordonnees

sont reelles. Les reseaux Ln sont parfaits pour n > 5 et r > 2 ([Bar] ; cela se voit par reduction a la dimension 5 en utilisant des arguments de perfection relative). Pour n = 2r > 6, ces reseaux sont extremes et dual extremes ([Bar], [Be-M1]). Dans le cas n = 6, r = 3, considere initialement par Coxeter ([Cox]), on trouve un reseau semblable a E6*, et l'on obtient donc E6 par la congruence C 1 avec la forme Tr. s est defini par les congruences C 1 Le reseau de Coxeter-Todd, note K12, et C3, avec la forme Tr. Cette definition par congruences, jointe au fait que

s a son dual, montre que x H lwx est un isomorA -- A2 est semblable phisme de K12 sur son dual, un resultat note par Conway et Sloane, qui 11

1'interpretent en faisant remarquer que K12 est Z[w]-unimodulaire ([C-S], ch. 4, § 9). Une variante de cette construction, analogue a la definition de 1, 1, 1 1, 1). Dn (cf. n° 3) consiste en 1'adjonction a L62 du vecteur 1

i (l

Sous cette forme, on voit immediatement que K12 est extreme et dualextreme (on a des resultats analogues dans toutes les dimensions multiples de 6 et

STRUCTURES ALGEBRIQUES SUR LES RRSEAUX

171

> 12).

Il est facile de verifier que le reseau A6 - IE6 se plonge dans K12. Comme les reseaux A,,, realisent la constante 7,n pour n < 6, ce sont les reseaux de plus petit determinant contenus dans K12 pour les dimensions comprises entre 0 et 6; c'est la serie K,. pour 0 < n < 6. La methode du no 2, appliquee

en prenant u = (x H 11 l.x), permet de construire une suite descendante D K7 D K6 de reseaux dont les determinants sont K12 D K11 . minimaux pour les dimensions comprises entre 12 et 6. On obtient une suite K, 0 < n < 12 en raccordant les deux suites en dimension 6. Cela est bien connu depuis Leech (et egalement entre les dimensions 12 et 24 que nous examinerons plus loin), cf. IC-S], ch. 6, § 1. Toutefois, comme on va le voir, ces plongements ne sont pas compatibles avec les Z[w]-structures qui existent naturellement sur D4 et sur 1E6 (on a rencontre une telle structure dans le cas de IE6, et l'on peut identifier ID4 a l'ordre de Hurwitz 931 des quaternions usuels sur Q, puts plonger A dans 931 par

w I. -1+i+9+k 2

7. - Nous nous interessons maintenant a des reseaux A pour lesquels le produit scalaire est de la forme Tr o h ou h : A -f A est une forme hermitienne (nous dirons simplement Z[w]-reseaux), et nous considerons les plongements qui sont des isometries pour les structures hermitiennes, ce qui est plus restrictif que Metre seulement une isometrie pour la structure euclidienne qui s'en deduit. Le theoreme suivant sera demontre au no 9. 7.1. THEOREME. - Soit A un Z[w]-reseau entier de norme 4. Si n = 4

et si det(A) est < 81 (resp. si n = 6 et si det(A) est < 243), alors A est Z[w]-semblable a D4 ou a L4 (resp. a E6 ou a E6) Ces reseaux ont des A-bases (el, e,2) (resp. (el, e2, e3)) formees de vecteurs minimaux,

et sont definis par les suites de produits scalaires (el.e2i e1.we2) (resp. (el.e2, el.we2 i el.e3, el.we3, e2.e3, e2.w63)); des choixpossibles pour ces quatre reseaux sont les suites (0, 2), (1, 1) (resp. (0, 2, 0, 2, 0, 0), (1,1,1, 1, 1, 1)). [Le th. 7.1 prouve en particulier l'unlcite a Z[w]-isometrie pres des reseaux ID4 et E6. Felt a demontre un resultat analogue par une formule de masse pour le reseau K12 dans son article [Fe] consacre aux reseaux Z[w]-unimodulaires. Des resultats d'unicite concernant en particulier IID4 sur Z[(8] et sur Z[(121 et ]E6 sur Z[(9] figurent dans [Be-M2], th. 4.3 et 4.6].

En examinant les produits scalaires entre vecteurs minimaux de K12, on s'apercoit qu'iI n'est pas possible de plonger A4 - IID4 dans K12 en tant que Z[w]-reseau, et donc non plus A6 - E6. En revanche, la definition de K12 montre que 1'on peut plonger L et L3 - E*. En utilisant la methode du no 2, on construit une suite croissante de Z[w]-reseaux Kn (n pair) plonges dans

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J. MARTINET

K12, que l'on complete pour n impair en prenant le reseau de determinant minimum parmi ceux qui sont contenus dans Kn+1 et contiennent Kn_1. Ces reseaux Kn, comme les K, , sont bien definis a un automorphisme de K12 pres.

On voit tout de suite que 1'on a Ki = Al - Z, K2 = A2 - A2, K111 = K11, K12 = K12, et que, pour 4 < n < 8, Kn est isometrique a Lr avec r = L J . Le reseau K3, connu des cristallographes (cf. [C-S31) est caracterise a2similitude pres comme le reseau d'invariant ry3 minimum parmi ceux qui satisfont l'inegalite s > 5. Une verification informatique a partir d'une matrice de Gram de K12 montre :

7.2. PRoPosrrioN. - Les reseaux Kn (resp. Kn) sont parfaits sauf pour n = 7 et n = 8 (resp. n = 3 et n = 4) oft le defaut de perfection est egal a 1. Le tableau suivant decrit les principaux invariants des reseaux Kn :

K3 K4 Ke Ks K Ks

reseau det(Kn)

36

81

s(Kn)

5

9

Ks

Kio

162 243 486

729

972

972

36

54

81

135

15

Smith(K') i 12.3 i 9.32 6.33

27

35

18.33 I' 9.3 4

36.3 62.33

Signalons que les reseaux Kio et Kio* (pour lequel on a s = 120) sont extremes et donc dual-extremes. [Voici une construction explicite de ces deux reseaux. Par division par 1 - w, on transforme le vecteur minimal (0, 0, 0, 0, 1 - w, -(1 - w)) de K12 en le vecteur (0, 0, 0, 0, 1, -1) de K12, dont l'orthogonal permet de definir Kip par les congruences Al =- 1\2 =- A3 = 1\4 = A5 mod a et Al + A2 + A3 + 1\4 - A5 = 0 mod a2 sur Z(w)5 muni de la forme hermitienne AA + A2A2 + A3A3 + A4A4 + 2A5A5. On volt que les 135 couples de vecteurs minimaux de K10 sont representes par 34 = 81 vecteurs de composantes de la forme w' et 6.9 = 54 vecteurs obtenus par permutation des 4 premieres composantes de vecteurs de la forme (w'(1 w), -wj (1 - w), 0, 0, 0). On obtient le reseau dual en remplacant dans la forme hermitienne 2.A5A5 par 2A5A5 et en divisant par 1 - w, et les 120 couples de vecteurs minimaux proviennent de 81 vecteurs comme ci-dessus, de 4.9 = 36 vecteurs obtenus par permutation des 4 premieres composantes de vecteurs de la forme (w'(1 - w), 0, 0, 0, -wj (1 - w)) et des 3 vecteurs (0, 0, 0, 0, 3w')].

Grace a des programmes de Batut, on verifie qu'il existe dans les cas de K11 et de K9* une unique relation d'eutaxie. Pour une indexation convenable des directions de vecteurs minimaux, elles ont les formes respectives 41

12

Id

= d

i=2

pi

et

Id = p1 + d

- i=2

pi.

STRUCTURES ALGI;BRIQUES SUR LES RESEAUX

173

On montre que la section de K11 (resp. de KO) par 1'hyperplan orthogonal a la premiere direction minimale definit le reseau Kio (resp. K$), alors que

les autres directions minimales sont asociees a K1o (resp. a des reseaux K8 isometriques au reseau P8 de Barnes, note A(2) dans IC-S], ch. 8, § 6) [Ces proprietes d'eutaxie s'interpretent par 1'existence de deux orbites de plans hexagonaux engendres par des vecteurs minimaux dans K12 = K12 et dans Kio , signalons que K10 possede une section parfaite K9 de meme determinant (972) que

K9, mats avec s = 82 au-lieu de s = 81, les reseaux Kg et Kg ont ete trouves par Barnes ([Bar], II, p. 221)].

8. - En plongeant K12 dans le reseau de Leech A24 et en utilisant la methode du no 2, on complete la suite Kn jusqu'a la dimension 24. II est clair que Yon obtient une suite de sections de A24 qui sont de determinant minimum parmi les reseaux contenant ou contenus dans K12, et que l'on a la relation de symetrie det(Kn) = det(K24_n). On peut proceder de ]a meme facon avec la serie Kn. On commence par munir A24 d'une Z[w]-structure compatible avec le plongement K12 -f A24; on indiquera dans la quatrieme partie comment realiser un tel plongement sur un ordre maximal du corps de quatemion de centre Q ramp en {3, oo}, ce qui est un resultat plus precis. La methode du no 2 permet de prolonger

la suite K,, jusqu'a la dimension 24, les reseaux obtenus etant des Z[w]reseaux pour n pair; on a les relations de symetrie det(K,',) = det(K24_n) et les egalites K13 = K13 et K,, = An pour n = 22,23,24 (alors que la coincidence de Kn et de An a lieu des la dimension 18) ; on definit de meme un reseau K16 a partir de K$ .

Pour etudier ces reseaux K, au-dela de la dimension 12, on utilise la determination par Plesken et Pohst ([PI-PI) des reseaux faiblement lamines

pour la norme 4 au-dessus de K12. Ces auteurs ont trouve un reseau en dimension 13, qui est K13 = K13, deux en dimension 14 qui sont K14 et K14, puis, au-dessus de l'un d'eux, qui ne peut etre que K14, une suite de reseaux de determinants det(K'' ), uniques a isometrie pres, sauf en dimension 16 oit it y a deux reseaux, que l'on distingue par leurs invariants s, qui prennent les valeurs 1218 et 1224. 8.1. TrICOREME. - Le reseau K16 est le reseau d'invariant s = 1224.

Demonstration (1) ( H. NAPIAS). On repere le reseau K22 dans A24 a l'aide de matrices de Gram, on construit la suite descendante des K. jusqu'a la dimension 17, et l'on distingue les sections Kl6 et K6 par leurs orthogonaux. [Elle a egalement montre qu'un seulement parmt les 37 vecteurs minimaux de K17 a pour orthogonal Klg dans K17, resultat analogue a ceux que l'on a observes pour Kli et K9*1.

(2) Par adjonction a L2 du vecteur vi = 1 1 (1, 1, 1, 1, 1, 1, 0, 0, ...)

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J. MARTINET

pour 2m > 12 puis du vecteur v2 -1, 0, 1, -1, 0, 1, -1, 0, 0, ...) pour 2m > 16, on obtient des reseaux de determinant 3'n puis 3ri-2, obtenant K12 pour m = 6, puis un reseau A de determinant 36 pour m = 8, qui contient visiblement K12 ainsi qu'une suite de sections en dimensions n = 15,14, 13 de determinants det(K,,). On a ainsi construit celui des deux reseaux de Plesken-Pohst qui est K16, et l'on verifie facilement que 1'on a s(A) = 1224. [On construct K18 par adjonetion a L98 de vi, v2 et v3 = ll

(0, 0, 0,1,1,1,1,1,1),

et l'on en deduct des constructions explicites de K17 et de Kist.

8.2. Remarque. - Les reseaux Kn et K;,, n > 12 et K16 sont parfaits. Cela se voit en controlant la perfection relative a partir de la dimension 12. Il est probable, mais non demontre, que les constructions par lamination pour

une norme donnee donnent dans ce cas particulier les reseaux faiblement

lamines au-dessus de K12, un resultat qui entrainerait directement la perfection, comme dans le cas des reseaux An consideres par Conway et Sloane.

8.3. Remarque. - Prenons A = Z[(9] et a = ((1 - (9)2), et soit R6m le reseau defini par la congruence E Ai - 0 mod a4 sur ((a2)-,9 T). On a R6 - E6 ([Cral), d'ou R12 ^' K12 et (R18, 1 1C9 (1, 1, 1)) -- K1'8; on trouvera dans [Bay-M] une construction de K12 comme module de rang 1 sur Q((21).

9. - Nous demontrons maintenant le th.7. 1. 9.1. LEMME. - Soit A un reseau pair de dimension n et de norme m. mny,-n. Alors, A possede n Supposons uerifiee l'inegalite det(A) < 1,+2 vecteurs minimaux independants.

Demonstration. L'inegalite de Minkowski sur les minima successifs d'un reseau montre qu'il existe des vecteurs el, C 2 ,- .. , en de A verifiant

l'inegalite N(el)N(e2) ... N(en) < y7 det(A), qui entraine que l'on a N(el)N(e2) ... N(en) < + n. On peut supposer ces vecteurs ranges par normes croissantes. On montre que ce sont des vecteurs minimaux en raisonnant par recurrence sur leur indice. On a en effet

N(el) ... < '"+2m''; si l'on suppose que el, ... , ei_1 sont de norme m, on trouve pour la norme de ei les inegalites N(ei) < N(ei_l)N(ei)n-i+l

m(

)1/(n-i+l) < m+2, donc N(ei) < m puisque A est pair, d'oU 1'egalite

N(ei) = m,

U

[Pour n = 2,3,. .. , 8, la borne du lemme est egale a 18,48,96, 192, 288, 324, 3241.

9.2. TiiEOREME. - SoitA un Z [w] -reseau entier de norme4, de dimension 4 (resp.6), et de determinant < 96 (resp. < 288). Alors, A est Z [w]-semblable

STRUCTURES ALGEBRIQUES SUR LES RI;SEAUX

175

d D4 ou a L2 (resp. a E6 ou a ]EE). En particulier, les Z[w]-structures sur les reseauxD4, L2 et E6 sont uniques a Z[w]-isometrie pres.

Demonstration. Le lemme montre que A contient n = 4 (resp. n = 6) vecteurs minimaux independants et donc, compte tenu de 1'action de Z[w], qu'il contient un sous-reseau A' possedant une base de la forme (x, wx, y, wy) (resp. (x, wx, y, wy, z, wz)). Le reseau A' est determine par la

donnee des produits scalaires a1 = x. y, bi = x.wy (resp. a1 = x.y, b1 = x.wy, a2 = x.z, b2 = x.wz, a3 = y.z, b3 = y.wz) qui sont majores par 2 en valeur absolue. On a det(A') < 2n det(A2)n/2 = 12n/2 et det(A') > 24 det(D4) (resp. det(A') > 26 det(E6), d'ou la majoration [A A'] < 3 avec

egalite seulement pour n = 6, A' _- A2 I A2 I A2 et [A A'] = 3 (car 2 est inerte dans Q[w]), cas dans lequel A est de determinant 26.3, donc semblable a E6, et oii 1'existence d'autres vecteurs minimaux que ceux des orbites de x, y, z permet encore de supposer que l'on a A' = A.

On peut supposer que x.y est > 0 et minimum parmi les valeurs absolues des produits scalaires des vecteurs minimaux de A' appartenant a deux orbites distinctes, et utiliser 1'automorphisme w'-4 w2 de Z[w] pour echanger x.wy et x.w2y. On voit tout de suite que, en dimension 4, A = A' est obtenu en prenant pour (al, b1) l'un des 4 couples (0, 0), (0, 1), (0, 2), (1, 1), conduisant a des reseaux de determinants respectifs 144, 121, 64, 81, d'ou le theoreme dans cc cas. Dans le cas de la dimension 6, nous avons d'abord prouve "a la main"

1'assertion d'unicite de la Z[w]-structure de Es, qui entraine le resultat analogue pour E6. On observe pour cela que le produit scalaire de deux vecteurs minimaux de EE nest jamais nul, cc qui permet de definir A' en prenant pour (al, bl, (12, b2i a3, b3) l'une des suites (1, 1, 1, 1,1, 1) ou (1, 1, 1, 1, 1, -2), et la seconde se ramene a la premiere en remplacant y par x + w2y. On acheve la demonstration en controlant sur ordinateur qu'il n'y a pas de determinant dans l'intervalle ] 192, 243[, et que la valeur 243 du determinant, lorsqu'elle ne provient pas d'une suite sans produit scalaire nul, correspond a un reseau de norme 2. [Variante pour l'assertion d'unicite concernant 1E6 : on considere un vecteur minimal x de E6 ; on verifie que le reseau 1E6 fl (Z [w]x) L, qui est de norme 2 et de determinant

9, est isometrique a A2 I A2 ; on en deduit que E6 s'identifie a un reseau de la forme (A2 I A2 I A2, 11. (x, y, z)), et l'on observe que x, y, z doivent etre des unites de Z[w] pour que le nombre de vecteurs minimaux soft superieur a celui de A2 I A2 I A2. Par multiplications a droite par des unites, on se ramene au cas

x = y = z = 11.

10. - Nous terminons cette premiere partie par quelques remarques sur la constante ryn. Sloane, dans une lettre a 1'auteur ([S!)), a donne pour

176

J. MARTINET

certaines dimensions < 24 des exemples de reseaux sur lesquels l'invariant

yn prend des valeurs relativement grandes. Pour n < 9, ce sont ceux de [Be-M1], 4.6 et 4.7. Pour n = 10, it indique la valeur 4 pour ryn2, atteinte sur deux reseaux semblables a leur dual (dont D+), cf. [C-S3]; le couple (K'0, Kio*) fournit la meme valeur. Le resultat propose est le meme pour "ii, atteint en particulier sur les reseaux Ail et K11, qui sont tous deux dual-extremes ([Be-M 11, § 4, (a) pour le premier, n° 7 ci-dessus pour le second).

H. Napias a montre que Yon -yn2(K18) = 8 et yn2(K21) = 9. Nous avons rencontre pour la premiere fois le reseau K18 dans un travail de Souvignier ([Soul) consacre aux sous-groupes maximaux de Gln,(Z), dont nous avons extrait le premier exemple d'un reseau L de dimension 21 avec ry21(L) > y'21(A21) (L et son dual sont extremes, et l'on a y21(A21) = 8 <

y21(L) = 8,4 < y21(K21) = 9). Le reseau K20 donne le meme resultat (y202

= 8) que A20 cite dans [Si].

Tous ces reseaux sont dual-extremes. TROIsiEME PARTIE : autour du reseau de Barnes Wall

11. - Soit H2 ou simplement H le corps de quaternions "usuels" de centre Q ramifie en 2 et a l'infini, muni de sa base (1, i, j, k) verifiant

les relations i2 = j2 = -1 et ij = -ji = k, d'ou l'on deduit les relations supplementaires k2 = -1,jk = -kj = i,ki = -ik = j, et soft 9J12 ou simplement 971 l'ordre maximal des quaternions de Hurwitz,

de base (1,i, j,w = -1+2'+k) sur Z; c'est l'unique ordre de H contenant strictement l'ordre 0 de base (1, i, j, k). Notons a l'ideal bilatere engendre

(a gauche ou a droite) par 1 + i; on a a = {x = a + bi + cj + dk E £ a + b + c + d - 0 mod 2}. Munis de la forme Trd(xy), a et 971 s'identifient respectivement aux reseaux D4 et ID*, et l'on2obtient E8 en considerant sur 971 x 931 la congruence >, - JI mod a (ou par adjonction a 9J1 x 971 muni de la forme Trd(xg) de l'element 1+ti (1, 1)). Soit m > 0 un entier et soit n = 4m. On on munit 971 de la forme Tr(Aji) et l'on pose J. = { (A,, , A.) E 9311T I Al + ... Am = 0 mod a}. On verifie

que J,, est un reseau de norme 4, primitif sauf pour n = 4 ou n = 8 ou l'on trouve une renormalisation de IlD4 et de E8, dont le dual s'identifie a .L.(9J1)m. En identifiant D4 a la derniere composante de J,J, et en coupant par les orthogonaux des sections de ID* semblables a {0}, A1, A2, A3, llD4. on

obtient des reseaux Jn, Jn_1, J,t_2i J,i,_3 et un reseau qui s'identifie a J,_4, ce qui definit Jn pour tout n. On a ainsi construit les analogues pour 992 des reseaux L Lnj2J construits par Barnes sur 1'anneau des entiers d'Eisenstein. 11.1. PROPOSITION. - Pour tout n > 1, J,, est un reseau entier de norme 4, qui est une section hyperplane de Jam,+1. It possede les invariants suivants :

STRUCTURES ALGEBRIQUES SUR LES RESEAUX

n = 4h n = 4h + 1 n = 4h + 2 n = 4h + 3

det(Jn) = 22h+4 det(Jn) = 22h+5 det(Jn) = 3.2(2h+4) det(Jn) = 22h+6

177

s = 12h(4h - 3) s = 4h(12h - 7) s = 12h(4h - 1) s = 3(16h2 + 4h + 1)

It est parfait quelque soit n, extreme pour n - 0 ou 1 mod 4, et est dualextreme lorsque n est divisible par 4. En outre, pour n < 12, Jn est un reseau [amine An, et l'on a plus precisement J12 ^-, A 2 et J11 - Aii". Enfin, pour n = 4h + 2 > 9 et 2 E {1, 2, 3}, Jn est de norme e+1 et la configuration S(J,n) est semblable a S(AQ).

Demonstration. Le calcul du determinant et du nombre de vecteurs minimaux ne presente pas de difficult. On verifie aussi facilement que Jn est relativement parfait par rapport a Jn_1, d'ofi 1'assertion de perfection.

On montre que Jn est extreme pour n - 0, 1 mod 4 en montrant qu'il contient un reseau de norme 4 semblable a D, ce qui assure qu'il est dualextreme pour n = 4m vu que (M)m est eutactique. La suite des valeurs des determinants pour 0 < n < 12 montre tout de suite qu'il s'agit de reseaux lamines, que l'invariant s permet d'identifier en dimensions 11 et 12. Enfin, on determine S(J,ry) par recurrence descendante en identifiant J,, a une projection de Jn+1

12. - L'analogue pour l'ordre de Hurwitz des reseaux D8 = E8 et K12 est le reseau A16 de Barnes-Wall (la notation A16 des reseaux lamines est justifiee ci-dessous), que l'on peut definir au choix par le double systeme de congruences Al = A2 = A3 = A4 mod a et Al + A2 + A3 + A4 = 0 mod a2

sur 9314 muni de la forme 2 >

1

Trd(AJi) ou par adjonction a J16 de

[Plus generalement, on definit de facon analogue un reseau J;, pour tout n > 16 divisible par 8, ayant le meme determinant (4'') que 931''1.

On demontre comme dans les cas de DI et de K12 le resultat suivant : 12.1. PROPOSITION. - L'application q'-4 q(1 + i) A16 stir son dual; en particulier A16 est de norme 2.

est une isometric de

Il resulte de cette proposition que, pour tout vecteur minimal x de Ai6, 931x est un reseau de dimension 4 isometrique a ID4. Les orthogonaux de ses sections minimales {0}, A1, A2, A3i )
sions 16,15,14,13,12 de determinants respectifs 256, 512, 768,1024,1024 dont le dernier est visiblement isometrique a J12 f-- A12 ", ce qui prouve bien qu'il s'agit du reseau lamine de dimension 16, et que les sections considerees sont A16, A15, A14, A'13 ATh 12

178

J. MARTINET

De meme qu'il y a 2 orbites de reseaux A2 dans K12, it y a 3 orbites de reseaux D4 dans A16, et I'on constate que les reseaux de dimension 12 que 1'on obtient par orthogonalite sont Ail X, Am12 id, Ail" On construit A g" a 1'aide des deux dernieres orbites (et Ami' A l'interieur de A12" et de Amid comme iI se doit), mais on verifie qu'il nest pas possible de plonger Amid 13

dans A16. Ce resultat se voit egalement en utilisant la construction des reseaux faiblement lamines jusqu'a la dimension 24 par Plesken et Pohst ([P-L]), qui ont en outre montre que le plongement de Amid 13 est possible dans A17-

13. - Considerons toujours la suite des reseaux lamines A, en nous limitant a ceux qui sont des9J1-modules, ce qui impose que n soit divisible par 4. Conway et Sloane ([C-S I]) ont construit la "serie principale" A0 C A4 -J(D4 C A8 - IE8 C A12 X C A16 C A20 C A24 C ... C A48

dont le terme de dimension 24 est le reseau de Leech et dont les termes de dimension superieure ne sont sans doute qu'une possibilite parmi d'autres. 11 s'agit de reseaux qui sont lamines au sens fort en tant que Z-reseaux et qui possedent une 9R-structure, celle de A,,_4 etant induite par celle de A. L'unicite de ces reseaux en tant que Z-reseaux a ete etablie par Conway et Sloane jusqu'a la dimension 24 sauf bien sur en dimension 12. La question de l'unicite en tant que 931-reseaux se pose, ainsi que celle de 1'existence pour les reseaux de dimension 12 autres que Al " 13.1. PROPOSITION. - Un 931-reseau A de dimension 12, de determinant 1024 et de norme 4 dont le dual est de norme 1 est semblable a Ail X ou a Ail"

les configurations respectives des vecteurs minimaux etant respectivement celles de de IID4 I IID4 I IID4 et de D4 .

Demonstration. Pour tout vecteur x minimal de A*, 9JZx est un reseau isometrique a D* ; les sections de A par les orthogonaux des reseaux semblables a {0}, Al, A2, A3, D4 qu'il contient constituent une suite

decroissante de sections de A de normes au moins 4 et de determinants 1024,1024, 768, 512, 256. Le dernier terme de la suite est d'invariant d'Hermite 2, et est donc de norme 4 et isometrique a A8 - E8. Il s'en suit que ces sections prises pour les dimensions croissantes de 8 a 12 sont des reseaux lamines. Comme les 931-reseaux ont un invariant s divisible par 12 (le nombre de couples fu d'unites de SJJi), on dolt exclure A12d. Enfin, la valeur des invariants s (respectivement 12 et 36) determine les structures des ensembles de vecteurs minimaux des duals. 13.2. PROPOSITION. - Le reseau A12"" possede tine structure de 931-reseau.

STRUCTURES ALGgBRIQUES SUR LES RkSEAUX

179

Demonstration. Sigrist([Sil) en utilisant une generalisation de l'algorithme de Voronoi pour les reseaux quaternioniens (cf. [Be-M-SI), puis Lai'hem (ILal) au cours d'une recherche de reseaux quaternioniens entiers de norme 4 et de dimension 12, ont trouve un reseau de dimension 12, de determinant 1024, de dual de norme 1 avec s = 312. La proposition 13.1 entraine que ce reseau est isometrique a A12i", qui se trouve ainsi muni d'une structure de fit-reseau. En ce qui concerne 1'unicite a isometrie hermitienne pres des structures

sur les reseaux lamines de petite dimension, elle est connue dans les cas suivants : A4 (parce qu'il y a une seule classe dans fit), A8 et A24 (traites par Quebbemann dans IQ] a l'aide d'une formule de masse), A16 (Quebbemann,

communication privee). Comme le sous-groupe unitaire de Aut(A16) est transitif sur S(A16), les inclusions A12 c A16 compatibles avec la 931structure A16 imposent que A12 soit en fait Z-isometrique a A 2 . On a encore dans ce cas un resultat d'unicite 13.3. PROPOSITION.- A EJ 1-isometrie pres, le reseau A 12 X porte Line unique

fit-structure.

Demonstration. Son dual est de norme 1 et de determinant 1024 et ses vecteurs minimaux engendrent un reseau de determinant 43 (prop. 13.1), donc d'indice 4, i.e. de "931-indice" a = (1 + i). Par homothetie, on obtient

le reseau A - 9R 1 fit 1 931 tandis que A12 "* devient un reseau A' engendre par adjonction a A d'un vecteur 1+i v ou v = (x, y, z), x, y, z, E 911. Comme on ne change pas A' en remplacant x, y, z par des elements qui leurs sont congrus modulo a, on peut les supposer dans {0, 1, w, w2 }, et 1'egalite

S(A') = S(A) impose que x, y, z soient non nuls. Par multiplications a droite par l'une des unites 1, w, w2, on se ramene au cas ou x = y = z = 1, ce qui montre immediatement que A' est egal au dual de Am' muni de la 911-structure qui a servi a le definir au n° 11. [Un raisonnement analogue permet de traiter le cas de 1E8 : on choisit un vecteur

minimal x de E8; en considerant l'orthogonal de 931x, on plonge 911 1 931 11)4 1 I1D4 dans E8, et l'on reconstruct E8 par adjonction a 1D) 1 1D) d'un vecteur de la forme i1i (x, y). On peut de meme traiter le cas de A16 en l'identifiant au reseau defini par l'adjonction a (1 + i)(931 1 931 1 931 1 931) des 4 vecteurs ( 1 , 1,0,0), (0, 1 , 1 , 0), (0, 0, 1, 1), (1, w, w2, 1), qui engendrent un code de poids 2 sur IF4, et aussi retrouver la prop. 13.3 en identifiant Ail' a ((1 + i)(931 1 931 1 931), (1, 1, 0), (0, 1, 1))l.

Les resultats que nous venons de dormer permettent de determiner pour l'essentiel la suite des reseaux lamines munis de 931-structures jusqu'a la dimension 24 : on a la serie principale de [C-Sll decrite plus haut, une bifurcation en dimension 8 vers le reseau A1" 2 qui est un cul-de-sac vu le resultat d'unicite pour la dimension 16 (et qui pourrait ne pas etre unique en

180

J. MARTINET

tant que 931-reseau, mais c'est peu probable), et peut-etre des bifurcations en dimension 16 vers des reseaux A20 munis de fit-strucures exotiques, et qui seraient alors des culs-de-sac vu le resultat d'unicite pour la dimension 24; cette eventualite est elle aussi peu probable.

14. - On definit de facon naturelle le procede de lamination (au sens faible comme au sens fort) au-dessus d'un 931-reseau A0 dans ]'ensemble des 931-reseaux : it s'agit de suites croissantes de 931-reseaux de meme norme que A0, le plongement d'un reseau dans le suivant etant compatible a ]'action de 931, les determinants verifiant les conditions de minimalite forte ou faible. On s'interesse ici aux laminations fortes dans le cas ou A0 est le reseau de dimension 0 auquel est attribuee la norme 4. Supposons demontre pour une certaine dimension n < 44 que les laminations au sens ci-dessus conduisent a un reseau An lamine au sens usuel, et considerons le terme A' = A' '+4 suivant. Posons m = N(A'*). Pour tout x E A*, le 931-reseau 931x est semblable a 1
semblables a ID4, A3, A2 et A1, qui sont les reseaux les plus denses dans les dimensions 4,3,2, 1; leurs determinants sont 4 m4, 2 m3, 4 m2 et m. Il en resulte que les sections de determinant minimum de A dans les dimensions n - 1, n - 2, n - 3, n - 4, que nous notons An+3, A'+2, An+1, A'' , ont pour determinants les determinants des orthogonaux des reseaux contenus dans 931x, c'est-a-dire m det(A), 4m2 det(A), 2m3 det(A) et 4m4 det(A). U caractere minimal de det(A;j montre que l'on a det(An) < det(An), done en fait det(An) = det(An), ce qui entraine 1'inegalite det(An+1) > det(An+l). Or, Conway et Sloane ont calcule jusqu'a la dimension 48 les determi-

nants des An. Le determinant do de An s'obtient a partir de sa valeur en dimensions < 4 (1,4, 12, 32, 64) par les formules do = 216-nd8_n pour 0 < n < 8, do = 216-ndn_8 pour 8 < n < 16, do = d24_n pour 12 < n < 24 et do = 224-ndn_24 pour 24 < n < 48 (cf. [C-S], ch. 6). II est alors facile de verifier que la minoration de det(An+i) par det(An+1) entraine 1'egalite de ces deux determinants, et que la valeur de m qui s'en deduit entraine det(A,t+4), i.e. que les reseaux lamines sur l'ordre de 1'egalite Hurwitz en dimension n + 4 sont des reseaux lamines au sens usuel. Vu que A8 -- 1E8 est le plus dense des reseaux de dimension 8, on deduit du raisonnement ci-dessus que les 931-reseaux lamines (en tant que 931-reseaux) de dimension 12 sont A12° et A12 a". Le resultat analogue est probablement vrai dans les dimensions 16, 20, 24, pour lesquelles it est generalement conjecture que A16, A20, A24 sont les reseaux les plus denses. Il semble possible de traiter le cas de la dimension 16 en verifiant a la facon

de [La] qu'il n'y a pas d'autres 931-reseaux de norme 4 en dimension 12 que A12° et A12 X. Nous conjecturons plus, a savoir que ces deux reseaux realisent le maximum de ]'invariant `Y12 sur les 932-reseaux.

181

STRUCTURES ALGEBRIgUES SUR LES RESEAUX

QUATRIEME PARTIE : au-deli de la dimension 16

15. - Soit H un corps de quaternions totalement defini. Cela signifie que le centre F de H est un corps de nombres totalement reel et que la norme reduite NrdHIF est positive a toutes les places infinies de F. Etant donne a > > 0 de F, la forme 7L-bilineaire (A, p) - TrF/Q (aTrdH/F (Aµ)) est definie positive. Les ordres maximaux de H deviennent ainsi des reseaux de R ® H, et l'on construit d'autres reseaux par congruences selon un procede deja utilise, cf. no 5 et no 11. Nous examinons dans cette quatrieme partie

des exemples dans lesquels F est soit le corps Q, le corps H n'etant plus le corps des quaternions usuels, soit un corps quadratique reel, le corps H n'etant ramifie qu'aux deux places reelles de F, renvoyant a [Bay-M] pour d'autres exemples. Soit 931 un ordre maximal de H. Pour un ideal premier p non nul de F, it y a seulement deux possibilites : CAS RAMIFIE. On a p931 = ' 32, 9J1/T est une extension quadratique de

ZF/p, et le complete en T de H est un corps gauche. CAS DECOMPOSE. On a 931/p931 ^ M2(ZF/p) et le complete de H en p931

est isomorphe a M2 (Fp). On utilisera surtout la variante suivante des doubles congruences qui

sont intervenues aux no 5 et 11 : on choisit deux ideaux a gauche T et T' dans 9931 au-dessus d'un ideal maximal p de F non ramifie dans H,

en se limitant au cas ou p est l'ideal au-dessus de (2) suppose inerte dans F/Q, et l'on considere pour un entier m > 0 le reseau de )R ® H'' muni du produit scalaire (a,µ) 2Trp/Q(aTrdH/F(Aµ)), de dimension n = 4[F: Q]m, defini sur Al

=A2=...=A,,,,mod T et

=0 mod T'.

Al

[S'il y a de la decomposition au-dessus de 2 dans F/Q, on peut choisir un couple T T' pour chaque Ideal de F au-dessus de 2].

15.1. MiEOREME. - Sous les hypotheses ci-dessus, le reseau defini par la condition (*) est entier et pair et de norme > 4 lorsque m est > 3.

Demonstration. La demonstration de l'integralite se fait par completion en p (notee par le symbole -), ce qui permet de ramener les calculs de norme

reduite a des calculs de determinants d'ordre 2. Par une identification convenable de l'algebre locale a une algebre de matrices, on peut faire en sorte que l'on ait K

K ZK/ \ZK

\2K p/

et

, - \p 2 K

182

J. MARTINET

Identifiant alors Ai E 931 a une matrice de la forme

que les congruences de la condition (*) deviennent

(xi \ zi

Yi

ti), on constate m

yi - y1 mod p et ti - t1 mod p (1 < i < m),

et

xi i=1

i=1

zi

0 mod

Comme la norme reduite dans une algebre de matrices n'est autre que le determinant, on a m

m

A Ai = T riti - yizi = (Exi)t] - (Ezi)J1 = Omodp, i=1

i=1

i=1

i=1

ce qui prouve qu'il s'agit d'un reseau entier pair. Pour minorer la norme de A _ (A1, A2, ... , A,n) suppose non nul, on distingue trois cas :

si les Ai ne sont pas dans T, on a N(A) > m min Nrd(A) > 3minNrd(Ai), donc N(A) > 4;

Si les A, sont dans T et si deux d'entre eux sont non nuls, on a N(A) > 4 puisque les produits A Ai sont dansT n ZF = 2ZF; si un seul des Ai est non nul, c'est un element de 'a3 l', 43' = p931, d'ol

encore le resultat dans ce cas.

16. - Nous prenons maintenant pour H 1'algebre H3 de centre Q ramifiee en 3 et a l'infini, munie de sa base (1, i, j, k) verifiant les relations

i2 = -1, j2 = -3,ij = -ji = k, et donc les relations supplementaires

k2 = -3,jk = -kj = 3i,ki = -ik = j, et pour ordre maximal l'ordre 9713 = 931 de base 1, i, w, iw sur Z ou w = - 2 jest une racine de

]'unite d'ordre 3. (Le choix de 931 importe peu, les ordres maximaux de H etant conjugues, comme dans le cas des quaternions de Hurwitz.) Les unites de 93Z sont {±1, ±w, ±w2, ±i, ±iw, ±iw2 } ; elles forment un groupe isomorphe au

groupe quaternionien d'ordre 12. Le theoreme ci-dessous donne une construction explicite d'une structure de 9313-reseau sur K12, dont ]'existence a ete prouvee it y a peu par Gross (IGro]) :

16.1. THEOREME. - Le reseau construit a l'aide de la condition (*) avec m = 3 sur L'ordre 9313 est isometrique au reseau de Coxeter-Todd.

Demonstration. Le theoreme 15.1 montre qu'il s'agit d'un reseau de norme au moins 4, dont on voit tout de suite qu'il est entier en tant que 9313-reseau et de determinant 36. 11 est donc unimodulaire en tant que 9313-reseau, et donc en particulier en taut que reseau sur 1'anneau des

STRUCTURES ALGEBRIQUES SUR LES RSEAUX

183

entiers d'Eisenstein. Le theoreme de Felt ([Fe)) cite au no 7 entraine qu'il est isometrique a K12. [Le theoreme de Felt montre qu'il s'agit meme d'une isometrie en tant que 7G[tWIreseau; Ch. Bachoc vient de demontrer que K12 est meme unique a 9723-isometrle presj

17. - Soit F un corps quadratique reel, de discriminant d impair. Nous considerons maintenant le corps de quaternions H ramifie exactement aux deux places infinies de F, et nous supposons que l'unite fondamentale a de F est de norme -1, ce qui equivaut au fait que la differente de F possede

un generateur totalement positif, en l'occurence a = ev, si bien qu'un ordre maximal 971 de H, muni de la forme TrK/Q(a-1Trd(Aµ)), definit un reseau Z-isometrique a E8. Un corps de quaternions Ho de centre Q peut etre plonge (d'une infinite de facon) dans un corps gauche H du type ci-dessus : it suffit de choisir un corps F dans lequel les nombres premiers ramifies dans Ho sont inertes ou ramifies dans F/Q et de prendre H = F ®Q Ho, les invariants locaux aux places finies de F etant alors tous nuls. Un ordre arbitraire 0 de Ho etant contenu dans l'ordre 7GF4.7 de H, lequel est a son tour contenu dans un ordre maximal 931 de H. on voit que E8 peut etre muni d'une structure de i:7-reseau sur n'importe quel ordre de quaternions totalement defini sur Z. On verra plus loin d'autres exemples du meme type; signalons simplement ici qu'un resultat analogue s'applique a A16. En appliquant a l'ordre 971 la construction par le double systeme de congruences (*), on obtient un reseau unimodulaire pair A de dimension n = 8m que nous notons simplement Un, sans mettre en evidence dans la notation sa dependance a priori des choix de H, 971,'3, T'. Il est fort possible que la classe d'isometrie de Un (en tant que Z-reseau) ne depende pas de ces choix. C'est ce qu'on constate en dimension 8 (resp. 24), puisque E8 (resp. A24) est alors l'unique reseau unimodulaire pair (resp. et de norme 4, theoreme de Conway). Le cas de la dimension 32 a ete resolu par Coulangeon ([Coul), qui a caracterise U32 comme le reseau unimodulaire pair d'invariant

de Venkov maximum qui est associe au code de Reed-Muller. Quant a la dimension 16, on trouve E8 I E8, comme on le volt en considerant 1'application (A, p) - (A +;t, A - µ). [Pour n > 40, les repartitions des normes redultes dans les suites (Al, A2,. . . , Am ) definissant des vecteurs minimaux sont des permutations de (1, 1, 0, . . . , 0) ; on en

deduct 1'egalite s(Un) = 15n(n - 7) pour n > 40. Les resultats pour n = 24 et n = 32 (et aussi pour n = 40) decoulent de la theorie des fonctions O ; on a s(U24) = 98280 et s(U32) = 73440. Pour n = 32, on dolt ajouter aux 15n(n - 7) = 12000 vecteurs ci-dessus 61440 vecteurs assoctes a la repartition

(1,1,1,1); pour n = 24, on ajoute a 15n(n - 7) = 6120 la contribution des

184

J. MARTINET

permutations de la repartition (2,1,1), soft 92160 vecteurs.

On connaitlsl en dimension 40 quelques autres reseaux unimodulaires pairs de norme 4. Pour celui de McKay (cf. [C-S], ch. 8, § 5), d'apres McKay, le groupe d'automorphismes ne serait pas transitif sur ]'ensemble de ses vecteurs minimaux, ce qui entrainerait que U40 ne lul est pas isometrique. Nous Ignorons st notre reseau U4o coincide avec 1'un des reseaux construits par Eva Bayer dans [Bay] ou par Ozeki dans [Oz]].

Revenons au double systeme de congruences (*) defini par deux ideaux

a gauche maximaux T et T' au-dessus de 2 d'un ordre maximal i3 d'un corps de quaternions Ho de centre Q. Si on choisit un corps F dans lequel 2 et les nombres premiers ramifies dans Ho sont inertes, on plonge comme cidessus Ho dans H et i7 dans un ordre maximal fit de H, et ces plongements transforment le reseau A0 associe au double systeme de congruences en un reseau defini de facon analogue sur 931 a ]'aide d'ideaux maximaux audessus de 2 contenant respectivement ¶ 3 et T'. Ceci s'applique en particulier au cas on 971o est l'ordre note'9J13 au no 16

en prenant F = Q(/5) ou p est n'importe quel nombre premier congru a 5 ou 11 modulo 24, par exemple p = 5. En prenant m = 3, on en deduit une construction du reseau de Leech A24 sur 9R3, utilisee par Tits ([Ti]) dans le

cas du corps F = Q(v), et un plongement de K12 dans A24 compatible avec la 9713-structure dont nous avons muni K12 au no 16, qui justifie la construction de la serie K i au-dela de la dimension 12 que nous avons faite au no 8.

On peut faire une remarque analogue avec ]'ordre 9312 de Hurwitz. On verifie que la construction par double congruence des reseaux J4,,n faite au debut du no 12 conduit au reseau A12 " lorsque l'on prend m = 3 (lorsque m est impair, le determinant calcule au no 12 doit etre multiplie par 24), et l'on en deduit le plongement connu de A12 " dans A24 en tant que reseaux sur l'ordre de Hurwitz.

Manuscrit recu le 8 mars 1993

Pl

Je remercie Eva Bayer pour les references concernant les reseaux de dimension 40

SMUCTTJRES ALGEBRIQUES SUR LES R$SEAUX

185

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[Fe] W. FELT. - Some Lattices over Q(/), J. Algebra 52 (1978), 248-263.

186

J. MARTINET

[Gro] B. GROSS. - Group representation and lattices, J. Amer. Math. Soc. 3 (1990), 929-960. [La] M. LAIHEM. - Communication privee. [Oz] M. OZEKI. - Examples of even unimodular extremal lattices of rank 40,

J. Number Theory 28 (1989), 119-131. [P1-P2] W. PLESKEN and M. POHSr. - Constructing Integral Lattices With Prescribed Minimum. 11, Math. Comp. 60 (1993), 817-825. [Q] H.-G. QUEBBEMANN. - An application of Siegel's formula over quaternion

orders, Mathematika 31 (1984), 12-16. [Si] F. SIGRIST. - Lettre electronique du 11 septembre 1990 a l'auteur. [S1] N.J.A. SLOANE. - Lettre a l'auteur du 11 mai 1992. [Soul B. SOUVIGNIER. - Diplomarbeit, Aachen,1991.

[Til J. TITS. - Quaternions overQ(f), Leech's lattice and the sporadic group of Hall-Janko, J. Algebra 63 (1980), 56-75.

Jacques Martinet Mathematiques, Universite Bordeaux I 351, cours de la Liberation 33405 TALENCE cedex

Number Theory Paris 1992-93

Construction of Elliptic Units in Function Fields Hassan Oukhaba

i. - Introduction Let k be a global function field and Fq be its field of constants. Fix a place o0 of k. Let Ok be the Dedekind ring of elements of k regular outside of oo, and k,, be the completion of k at oo. For each finite abelian extension F of k we let OF be the integral closure of Ok in F. We know that OF is a Dedekind ring with a finite ideal class

number h(OF). As usual we denote by OF the group of units of OF, and p(F) C OF the finite multiplicative group of non zero constants of F. We have p(k) = Ox = F9 , and in general the quotient group OF/µ(F) is a free abelian group of rank rF - 1, where rF is the exact number of places of F sitting over oo. Now suppose that F C k, which means that the place oo splits completely in F/k. Suppose in addition that one of the following two conditions holds.

1) The extension F/k is unramified. 2) One, and only one, prime divisor of k ramifies in F/k and deg (oo) = 1.

Then one knows that there exists a subgroup EF of OF called the group of elliptic units of F. It is a Galois module generated by the torsion points of certain Drinfeld Ok-modules. It's elements are also obtained as finite products of special values of elliptic functions. The group EF has finite index in OF, cf. ] 10]. Actually when only one prime does ramify in F/k we had succeed to construct subgroups of finite index in OF even if deg (00) > 1,

cf. 191. Unfortunately, the index formula obtained then contains a factor depending on deg (oo) and which is hard to control. When deg (00) = 1 this factor is equal to 1 also and the index formula is just what one can expect. But in general this factor increases proportionaly to deg (00). This means that the subgroups so constructed are not sufficiently large when deg (00) > 1. Hence, one could suppose that there is possibility to obtain larger subgroups of OF, in other words to obtain more units of OF, using

188

H. OUKHABA

new techniques of constructions. This is what we propose to do in the present paper. Our aim here is to define £F, the group of elliptic units of F. We shall expose some of its interesting properties, precise the nature of its elements and calculate its index in O. As we shall see the description of £F is rather easy and almost canonical. Moreover, the "exponential function", which we are going to redefine in the next section, is the only basic material of its construction. Finally we would like to draw the attention to the work of D. Kersey, cf. [7] chap. 12 and 13, which was one of our source of inspiration.

Some supplementary notations Let F C k,,, be a finite abelian extension of k such that the place 00 splits

completely in F/k. let b C Ok be an ideal of Ok prime to the conductor of F/k. Then we will write (b, F/k) for the automorphism of F/k associated to b by the Artin map. Moreover if q is a prime ideal of Ok then qF will denote

the product of the prime ideals of OF sitting over q. Finally, if m C Ok is an ideal of Ok then we know that there exists a maximal finite abelian extension of k whose conductor divides m and which is contained in k,,,,. It will be denoted by H,,,.

2. - Some preliminaries In this section we recall some definitions and results, necessary in the sequel. The reader is invited to consult [1], [4], [9] or [11], where are proved

all the results stated here. Let S2 be the completion at o0 of the algebraic closure of k, Then we call a lattice of l every finitely generated projective Ok-module, contained into Q. To such a lattice r C 0 one can associate its exponential function defined on 1 by :

er(z)`ifnzJJ(1- z). 7Er

ry

7#0

We know that er is defined everywhere and is entire and IFq-linear. It is also an epimorphism and we have er(z) = 0 if, and only if, z E r. Moreover the equation eyr(xz) = x er(z) holds for every x E S2" and z E Q.

When F is contained into a lattice r of SI such that r and r have the same rank as Ok-modules then er and er are related by the formula : (1)

er(z) = P(r/r; er(z))

,

where P(r/r; t) is a linear polynomial whose roots are all simples and

CONSTRUCTION OF ELLIP77C UNITS IN FUNCTION FIELDS

189

constitute the finite group er(r). Its leading coefficient is

6(r

-) arll

)-i

( pEn/r R er(p) p#0

where p describe a complete system of non zero representatives of r modulo F. Let K(oo) be the constant field of k,,.. It is a finite extension of ]Fq. We have [K(oo) 1Fq] = deg (oo). Let us choose s (once of all) a sign-function of :

k,,, i.e., a co-section of the inclusion map K(oo)" -4 k' such that s(z) = 1 if Iz - 11,, < 1. Then one can associate to each lattice r of S2 of rank 1 its s-discriminant OS(r) E SZX and ar an Fq-automorphism of K(oo) such

that X6 (r, x-1 r) = A, (r)

Nx-1 s(x)°r

for all x E Ok\{0}. In the above formula, w,,, is just the number of non zero elements of K(oo), i.e., w... = K(oo)". On the other hand Nx is by definition the exact number

of congruence classes of Ok modulo the ideal xOk. One can show that A3(zr) = z-w°°OS(r) for all z E V<. Moreover we have the equation

6(r, r)- =As (r)[r:r]/Os(r) whenever r C r are lattices of SZ which have rank 1 as Ok-modules. The invariants A (a) associated to fractional ideals a of Ok are used to express the special values of L-functions at s = 0 and hence are related to analytic class number formulas, cf. [31, [51 and [61. In other respects J. Yu has shown

that they are transcendental over k, cf. 1131. However, as in the classical theory, it is possible to construct elements of SZ which are algebraic over k using the above invariants. Indeed let 06 C k,,, be the abelian closure of k in k,,. Let H(1) be the maximal subextension of 06 such that Hlll/k is unramified. Then one knows that the quotient A,(a,)/O3(a2) E H(1) for all fractional ideals al and a2 of Ok. In fact this last quotient generates in OH(1) the ideal

(alai'OH(,))w-.

Now suppose that r is a lattice of SZ of rank 1. Then the lattice t r is well defined for all idele t of k. And if p is an element of the k-vector space

kr generated by r then one can check, using the strong approximation theorem, that there exists u E SZ such that U - tvp mod. tvrv for all the places v

oc of k, where tv is the component of t at v and IF, is the

completion of r at v. The elements u that verify the property above define

190

H. OUKHABA

the same class modulo tr. We shall write tp for any representative of this class. Let or E Gal (S2/k) and t be a idele of k such that the automorphism It, k] of kab/k, associated to t by the Artin map coincide with the restriction of a to kab, cf. [ 111, then there exists a non zero element AQ (t, r) E S2" which verifies the formula er(P)a = AQ(t,

(2)

for all p E kr. It is possible to describe the behavior of A, (t, r) as a, t or r varies, cf. [ 11 ].

In fact if we suppose that O3 (r) = 1 and if the component t of t at each place v # oo is integral, then we have

A,(t, r) = 6(r, t-lr)-1, provided that s(ty) = 1. Let us observe that the automorphism Is, k] of kab/k is equal to the identity map if, and only if, t E k" x kx. Therefore the formula (2) implies that the quotient er(pl)/er(p2) E kab, for all pl and P2 in U. Moreover we have :

= et-lrt-1P1)

(er(P1)1

J

(3)

er (P2)

'

for all idele t of k.

3. - The ramified part of the group of elliptic units Let F C koo be a finite abelian extension of k. Suppose that the conductor of F/k is qn, where q is a prime ideal of Ok and n is a positive integer. Then it is possible to construct elements of F" using the values

ea-iqn(1), where a is any ideal of Ok prime to q. These elements will constitute the ramified part of the group of elliptic units of F. PROPOSITION 1. - Let B be a finite set of ideals of Ok all prime to q. Then

the product (4)

11 eb-1gn(1)nb

bEB

belongs to Hqn provided that the rational integers nb, 6 E B verify the condition

nb=0.

(5)

bEB

This proposition is equivalent to the following one.

CONSTRUC77ON OF ELLIP77C UNrIS IN FUNCTION FIELDS

191

PROPOSITION 2. - The quotient

ea-1gn(1)/eb-1gn(1) E Hqn

for all ideals a and 6 of Ok which are prime to q.

Proof : let a E Gal (SZ/k) and t a idele of k, choosen so that or is equal to the automorphism [t, k] of kab/k. Then Corollary 3.3 of [11) (see also the previous section) implies that ea-1q^(1)° = A,(t,a-1gn)et-la-1gn(t-1 1) b-lgn)et-l6-1gn(t-1

e6-lgn(1)° =

11),

where AQ(t, a-lqn) and A0(t, b-lqn) are elements of Q': which are equal in this case, cf. [I I]. So that we get the formula ea-lgn(1)

(6) Ceb-lgn(1)

et-la

)

1gn(t-1.

1)

et-16-1gn(t-1.1)

Now suppose that a is the identity map on kab, then we know that t E k" x kx. In this case the quotient on the right of the above formula is equal to ea-1qn (1)/e6-1 qn (1). This means in particular that this last quo-

tient belongs to kab. In fact using class field theory we see also that it is in Hqn.

0

PROPOSITION 3. - Let a, 6 and -0 be integral ideals of Ok, all prime to q. Then we have ea-lgn(1) (7)

(0,Hgn/k)

e6-Ign(1))

- ea-11i-1gn(1)

ea-1a-1gn(1)

Proof : proposition 3 is easily derived from above formula (6) and the o well known properties of the Artin map. PROPOSITION 4. - Let B be a finite set of integral ideals of Ok, all prime to q. Let nb, b E B, be rational integers which verify (4). Then the product 11 eb-'gn(1)nb

bEB

generates in the integral closure OHgn of Ok in Hqn the ideal

(flbnb) bEB

192

H. OUKHABA

geneProof : all we have to prove is that the quotient rates in OHgn the ideal (a OHgn)-1. So, put a df n (a, H, -1k) and consider the element cp(a) of Q' defined as follows (8)

Bo (a)

df n OS(a-1qn) [ea-lqn (1)] --

It is obvious that cp(a) depends only on or (and not on the ideal a). The behavior of this invariant is well known, cf. [9) chap. IV. In particular cp(a) E OHgn and we have cp(a)'r = W(ar), for all a, -r c Gal (Hqn/k). Moreover it verifies the following norm formula NHgn/x(l) ( \P(a))

u'k

Os(a-1) O5

a-l ) am

Fq . which implies that p(a) generates in Oxgn the ideal qH, where Wk Indeed the quotient A,(a-1)/O3(a-1q) generates in OH(,) the ideal qH(1). On the other hand the extension Hqn /H(l) is totally ramified at each prime factor of qH(,). Now since we have the relation ov(a) C

egn(1)

Yo-

o(1)

A.(qn) O3(a-iqn)

and since OS(gn)/os(a-1qn) generates in OH(1) the ideal cf. [11) Proposition 3.7, we can deduce that the quotient ea-1 qn(1)/egn(1) generates in °Hqn the ideal (a-1 OH,,)DEFINITION 1. - Let F C k,,, be a finite abelian extension of k. Suppose that the conductor of F/k is equal to qn. Then we set SF to be the sub-group of F" generated by all the norms

NlignIF

egn(1) Cea-ign(1)

where a describes the set of integral ideals of Ok which are prime to q.

4. - The unramified part of the group of elliptic units We describe below the method of construction of non zero elements of those unramified abelian extensions of k which are included into k,,,. The connexion with the torsion points of certain Drinfeld 0k-modules is explained in [101, proof of Theorem 2.2.

193

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

Let r c r be lattices of 5l such that the index [r : r] of r in r is finite. Then one can define the function

(z

,

r r) aIn S(r, r) er(z)jr:r] er(z)

z E Il,

which is well defined on the complement of r\r in Q. It vanishes on IF and, in fact, is elliptic (i.e. periodic) with respect to F. Moreover, as a rational function of er(z) its divisor is [r

:

r](0)r

-

T, (P)r. PEI'/r

On the other hand, we have the homogeneity formula

T (Az ; Ar, Ar) = T (z; r, r), for all A E W.

Also if rl C r2 C r3 are lattices of l2 such that the index [r3

:

r1] of rl

into r3 is finite, then we have rl,r2)[r3:r21W(z;

W(z; rl,r3) = `1'(z;

(9)

r2,r3)

PROPOSITION 5. - Let M, M and r be lattices of Il such that M C M, M C r and r f1 M = M. Consider the lattice r L fn r -+- M and choose S a complete system of representatives of r modulo M. Then we have the distributivity formulas (10)

(z; r' r) _ f T(z+P; M, 79), pES

(11)

SWIM)

f W(P; M, M) PES P#o

Proof : this is a simple consequence of the formula (1).

o

Now suppose that r C r are lattices of fI of rank 1. Then the value T (p ; r, r) E kab for all p E kF. Indeed, %P (p; r, r) is a product of quotients

of the form er(Pl)/er(P2), P1, P2 E kr, which are elements of kab by Theorem 3.2 of [ 11) (see also § 2 above). Moreover, the formula (3) implies the following property (12)

for all idele t of k.

''(P; r,r)]t'k] = q(t-lp; t-1r t-ir)

194

H. OUKHABA

PROPOSITION 6. - Let m 54 0 be a proper ideal of Ok. Let a be a integral ideal of Ok prime to m. Then we have

'(1 ; m, a-1m) E Hm. Moreover if 6 is a ideal of Ok prime tom then the automorphism (6, Hm/k) of Hm/k applied to W(1; m, a-lm) gives W(1; m,

a lm)(b,Hm/k) = iy(1; b-lm a 16-1m)

Proof. this Proposition is a direct consequence of (12). See also the alternative proof given in [10].

o

The above Proposition 5 and Proposition 6 have the following remarkable consequence.

PROPOSITION 7. - Let p be a prime ideal of Ok and n > 0 be a positive integer. Let a be a integral ideal of Ok prime to p. Then we have (13)

NHpn+1 /Hpn

(14)

(,P(1;

pn+l a-lpn+1)) =

NH,/HC,) (`1'(1; p, a

1p))wk

T(1; pn, a-lpn),

= b(Ok, a-l) 6(p, a-lp) Na_1

Moreover the ideal of OHpn generated by the value W (1; pn a-lpn) is pHpn where wk = OIFy .

Proof : let X be a complete system of representatives in Ok of (Ok/p)" modulo the group F9 . Then the elements of Gal (Hp/H(l)) are precisely the automorphisms (x, Hp/k), x E X. On the other hand if we put M = p,

M = a-lp and r = Ok then we have r = I + M = a-'; moreover the set {lx, E IFq and x E X} is a complete system of non zero representatives of r modulo M. Therefore the formula (14) is just the identity (11) applied to this precise case. The formula (13) is obtained from (10) proceeding as above. Now let us observe that we have the equation '(1e ; pn a-lpn)w00 = (p(1)(Na-(a,Hvn/k))

which implies in particular that W(1; pn, a-1pn) generates in OH,n the ideal pHpn

since the ideal generated by cp(1) is p1 , cf. §3.

0

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

195

Now let L C k, be an unramified abelian extension of k. Then, for a given prime ideal p of Ok we shall write Rp,L for the subgroup of L" generated by µ(L) and by all the norms NH /L (T (1; pn,

a-lpn))

where n is any positive integer and a is any integral ideal of Ok prime to p. In fact n can be fixed according to the formula (13). On the other hand if p' is a prime ideal of Ok such that the automorphism (p', L/k) is equal to (p, L/k) then we have, cf. (10) Theorem 3.2,

Rp,L (1'L = Rp, L II

LL.

The group Rp,L n OL will be noted £L,, if a = (p, L/k). DEFINITION 2. - We define RL to be the subgroup of L" generated by all Rp,L, where p is any prime ideal of Ok Hence we have RL

df n

Rp,L p

Remark 1 : the group EL of the elements of RL which are units of OL is called the group of elliptic units of L. We have £L

dfn

RL f1 O =

fi

EL".

oEGaI (L/k)

The quotient group OL /£L is finite, cf. (101. We have the index formula (15)

[OX

:

£L] =

h(OL) [H( j)

Remark 2: the formula

T(1; b-'Pn a-lb-lpn) = W(1; pn, a-'b-lpn)/`W(1; pn, b-lpn)Na, verified for all prime ideal p of Ok and all integral ideals a and b of Ok not divisible by p, shows clearly that the group Rp,L is stable under the action of the Galois group of L/k. Thus the groups RL and £L are also stable under

196

H. OUKHABA

the action of Gal (L/k). Therefore, using formula (10), one can prove that p(L)RLk is generated by p(L) and by all the quotients 8(Ok b-1) NH( 1)/L

b(a, b-1 a)

where a and b are any integral ideals of Ok which are coprime. Hence, the group RLk"'°° is generated by all the elements of L" of the form Os(OIc)l Nb Os(b-la) Os(b-1)

NH(I)IL

\ Os(a) /

where a and b are as above.

Remark 3 : using the description of µ(L)RLk just given above and the fact that the order Wk of IFq is the g.c.d. of the integers Na - 1, a ideal of Ok, one can prove that we have NH(1)/L (11 b(Ok, b-1)1b ) E RL , bEB

where 13 is a finite set of integral ideals of Ok and nb, b E 13 are rational integers such that

Enb(Nb-1)=0. bEB

DEFINITION 3. - LetT E Gal (L/k).

i) If L = Hill, then we put h = h(Ok) and dfnWooA,(b)h aH(1)(T)

where b is any fractionnal ideal of Ok such that (6,H(l)/k) = -r-1 and bh = aOk, with a E k. ii) In general, we put : OL(T) dfn =

H

T E Gal (H(1)/k) TIL = T

OH(,)(T).

197

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

One can show, cf. [ 11 ] Lemma 3.5, that the quotient aL (T1) /aL (T2) is a unit of OL. Moreover if T E Gal (L/k) then we have the property T

,L(r1)

1L(T1T) 1L(T1T)

OL(T2)

which implies in particular that NH(1)/L

(OH(,)(T1)

49L (T1)

OH(,) (r2)

aL(T2)

where T1 and T2 are automorphisms of H(1) /k such that -Ti,, = Ti, i = 1, 2.

Finally we see that the group IZ,,, Oh is generated by all the elements of L" which have the form

8L(1) Na, OL(TT') (0L(7)) CUL(T'))x

wO°(1-Na,,)[H(1):L]

Gal (L/k) and aT, is any integral ideal of Ok such that (a,,, L/k) = T'. The element x of k" is such that there exists b a ideal

where T, T' e

of Ok which verifies (b, L/k) = T and bh = XOk.

In particular, for any A E k" take b = .Ok and x =

Ah

so that T dfn =

(b, L/k) = 1 and .hw-(Na-1)[H(l) : L]

a any ideal of Ok,

na(Na - 1) for a well suited finite set U of

belongs to RL'`" °°h. As wk = aEU

integral ideals of Ok, and convenient integers na, a E U, we get COROLLARY. - The group kxwkwooh[H(1) : L] is contained into

RLkw,oh

4. - The group of elliptic units Let F C k,, be a finite abelian extension of k such that the conductor of F/k is equal to qn, where q is a prime ideal of Ok and n is a positive integer. df° We put F(1) F fl H(1) and we define a subgroup RF of F" by setting (16)

DEFINITION 4. - Let £F be the intersection of RF with the group of units of OF, i.e., (17)

£

F

dfr`

RFfl OxF

H. OUKHABA

198

We call SF the group of elliptic units of F.

Our goal in the present section is to describe the group eFkw°°h in a manner which will allow us to calculate its index in OF. Therefore we have

first to introduce some "new" elements of OF defined as the norm from Hqn down to F of the invariants W(o ), o E Gal (Hqn /k), defined by the formula (8). DEFINITION 5. - Let T E Gal (F/k). Then we put WF(T) dfn

where z E Gal (Hq.. /k) is such that TIF = T.

Remark 5 : let M C k. be a finite abelian extension of k. Let a be a integral ideal of Ok prime to the conductor of M/k. Then we know that (a,M/k) = CNa for all £ E µ(M) .

In particular if (a', M/k) _ (a, M/k) then Na' - Na mod. wM where wM dfn

Op(M). This means that we have a well defined Dirichlet character OM : Gal (M/k) -+ (7G/wM7L)" T '--' AGM (T)

given by the condition AM (T) - Nb (mod. wM),

where b is any integral ideal of Ok such that T = (b, M/k). We have 'M(T) - 1(mod. wk) for any T E Gal (M/k) so that we can make the following construction : For IM the augmentation ideal of the group ring Z[Gal (M/k)], and each integer f > 1, we define

M : Imt -* Z/mZ, with m = wM/wk, to be the following surjective morphism XPM

nT1,T2...... e

((,l

- 1)(T2 - 1) ... (Te - 1)/

T1, 72 .....Te

dfn

E

' 'Tl ,T2 ,...,Te

T1,T2......

e

M(T1) - I" CXPM(T2) - 11... CWM(Te) - 11 wk

wk

Wk

J

These operators are well defined, as will be made clear in some further work; here we just need to have IF(') and 0M) to our disposition, and the following lemma relating them :

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

199

LEMMA 1. - For any element A of IM we have

M (wkA) _ ` (l (A)

mod.

wM 1

.

wk)

Proof : obvious. dfn

F fl H(1) and, for q" the conductor of F/k; dl [FH(1) F] = [H(1) F(1)] so that FH(1)] and d2

PROPOSITION 8. -Let F(1) dfn

: [Hqn : : put d1 Z/mZ and' F(), ) : IF(,) -> 7G/m7L, d1d2 = [Hqn : F]. Let WF(1) : IF(1) with m = wF(,) /wk, be the surjective morphisms defined as above. Also put W = wkw,,,h. Then the group SF is formed of all the products

(18)

LI

TT


oEGal (F/k)

8F(,) (T)n-

-rEGal (F(1)/k)

when the elements Emo(u) of Z[Gal (F/k)] and Emr(T) of Z[Gal (F(1)/k)] are such that 1) Emo(u) E IF; 2) En,(-r) E 1F(1), i.e., EnT(T) E IF(,) and fl T"T = 1 ;

3) consider the other element EMI(T) of IF(,) defined by Tqn dfn E mo T E Gal (F(1) /k) ; then request (q", F(1)/k) and MT

dfn

o EGal (F/k) 'n IF(,)

d1TFl1)

(Mr(r))+w(2) (>nT(T)) T

T

0 mod. _F(')

(

.

wk

Proof : let us see that any element a of F" satisfying the above conditions 1) to 3) belongs to EF : so we fix elements Emo(a) of Z[Gal (F/k)] and En, (T) of Z [Gal (F(1)/k)] satisfying 1) to 3); in particular we define MT for T E Gal (F(1) /k) by

M

T

dfn

T,

ma

EGal (F/k)

of F(,)=''qn

a) First by conditions 1) and 2) we have that Emo(a) and En, (T) are elements of the augmentation ideals IF and IF(,) respectively, so that we have :

aEOF

200

H. OUKHABA

and the only thing we have to prove is : w

aE

SFw

b) Arbitrarily, choose a finite set Z of integral ideals of Ok, all prime

to q, such that the Artin map

a'-' (a, F/k) define a bijection from Z to Gal (F/k) ; thereafter put dfn

Mb = m(b F/k).

Also, for each T E Gal (F(1) /k), choose an integral ideal aT of Ok and an element xT of k" such that (a,, F(1) 1k) = T and ah = XTOk. c) Then a can be written as the product ABC where A, B and C are as follows

A dfn = NIJq

T7 eb-lgn(1)""

IF

bEZ

B

wk

dfn

CF,1) ( T )

d1m°

xd1m°

a

iEGal (F/k)

vEGal (F/k)

)

where or and T are related by a,F(l) = TTq so that T = (bq-n, F(1)/k) and xo E k" is defined by (bq-n)h1 = x
fl

dfn

nr

TEGal (F(1)/k)

By condition 1) we have A E SF d) But we know that wk = .

na(Na - 1) where U is a finite set aEU

of integral ideals of Ok and na, a E U, are rational integers. So, just put Ta = (a, F(1) 1k), a E U, and observe that the product BC is also equal to the product B'C' with

B' dfn EGal

B

d /k)1m

and B' dfn

OF,

aE

1)

oF(1)

(T) )Na aF(1)(Ta)

'na

lx (Na-)wedz

OF,1) (TTa) /

1

201

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

where as above T = 7-gn1Q,F(1) = (bq-n, F(1)/k) and X Ok = (bq-,)h; for defining C' regroup all the m, with O]F(1) = TTgn so that C,, dfn

=

ri rEGal (F(1)/k)

na

11 ( aF (1) (1) a F(1)(T Ta ) aF(1) (T)

aEU

d 1 M.,

TT

aF (T)nr (1)

OF(,) (Ta)

rEGal (F(1)/k)

Nota Bene. We have that AB and AB' are units in OF so that C and C' are units in OF(,), as was obvious by their very definition for the quotients OF(1) (T1)/OF(1) (T2) are units.

Observe now that by the end of § 4 we have Bo E RF

1

Moreover, as

we have

Nwk 1)

(( 71)k

aEU

aEU

wF(1)/wk),TEGal (F(1) /k),

F(l) (-r

we deduce from the condition 3) the fact that (F1)

(>diMr(>fla(T - 1)(Ta - 1))+>nr(T))

0(mod.WF(1)/wk)

.

aEU

Yet, see [10), the condition F?i) (Ent(t))- 0(mod. wF(1)/wk) is a necest sary and sufficient condition for the product

JI

'F(1) (t)n,,

nt(t) E IF(1),

tECal (F(1)/k)

n

to belong to £FO = RF(1) now 1) ; whence C' E RF and a E RF(j) . SF . On the other hand, let us prove that any 1

a E I RF(1). SF )f1oF

satisfy the conditions 1) to 3) : by Definition 1 and the observation at the end of § 4, the unit a may be written as a product AB with dfn

dfn

B

H { 11 (a,a')EYxY f7 [ bEZ

Na' 9F(,) (TTTa') r hw-](1-Na')d2

( 0F(1) (1)

N"q"/F

aF(1) (T.') la

OF(1)

(

eb_1

eqQ(

Tnbl

1

))

l1 J

202

H. OUKHABA

where we have made the following conventions. Y is a finite set of integral ideals of Ok. Z is a finite set of integral ideals of Ok, all prime to q ; na,a', (a, a') E Y x Y, and mb, b E Z, are rational integers. Ta = (a, F(1) /k), Ub = (b, F/k) or its restriction to F(j). Finally [aw°°h] is the w.-power of any element x E k" such that ah = XOk. Now using formula (8) and Definition 3, we can write wk h

PF(Ub)

NHgIF(e'q(1)) eq^(1) )W= wkh

(OF(Ub)

( x

PF (1) In particular we get

1 8F(1) Qb ( Tg nl) J

aEY

fJ bEZ

O9(b-lqn)

wkdl

[b-hw°°]v'kdld2

F(1) Tqn1)

[ahw_]vad2

zs(q)

xNHa)/Fa>

1PF(1)

L

[bhwo]m(,wkdld2

=1

v

na,a' (Na' - 1) ; whence in terms of automorphisms

where Va = a' EY

11 Ta k

11 T6T2bd1 = 1.

aEY

bEZ

This equality is a necessary and sufficient condition for the sum

'a (1-Ta)+Cd1

M

aEY Wk

Tb)

bEZ

to be an element of IF(1) ; yet this sum itself (which belongs to wkIF(1)) is congruent modulo IF(1) to our n, (7) which here is

E na,a'(NQ -Ta')(1 - Ta)+wkdl a,a'EY

mbTgnl(1 -Tb) bEZ

hence condition 2) is proved. Condition 1) is trivial. On the other hand, by the Lemma 1 applied to the sum (*) we have 0(2

F(j 2 ))

Y' nT(T))l T

naa

(_) (O(T )-1/

a,a'EY

Mb )(Tgnl)

+ d1

bEZ

-d

b 4 (Tq^1)

bEZ

-dl

(1 - 4'(Tb) ) 1\

Y

(1 -

Wk

J

(Tb) )

(mod. wF(') wk

(mod. wF(1)

Wk

MT(T) I

Wk

(mod. wF(1) Wk

)

CONSTRUC77ON OF ELLIPTIC UNITS IN FTJNC77ON FIELDS

203

where as above we have put MT

E

df"

mb;

o6EGul (F/k)

o61 F.(1)=-qn

hence condition 3) is also satisfied. This concludes the proof of Proposition 8.

6. - The index formula Take F C koo to be, as in section 5, a finite abelian extension of k such that the conductor of F/k is equal to q", where q is a prime ideal of Ok. We want to calculate the index of the group £F in O. The technique we will use is well known, cf 191 or [121. Let a E Gal (F/k). Then for each rational integer a > 0 we put dfn

to F(a) =

OF(1) (a)aWF(a)

where Q E Gal(F(l)/k) is such that & = aIF(1). It is obvious that ta,F(a1)/ta,F(a2) E OF, for all al, a2 E Gal (F/k). Moreover we have the action

ta'F(a1) ° Cta,F(a2)

ta,F(aia) ta,F(a2a)

for all

Cr E

Gal (F/k).

Let us denote Ta,F the subgroup of OF generated by the quotients ta,F(a)/ta,F(a'), or, a' E Gal (F/k). We know that the group Ta,F has finite index in OF, cf. [91 or [ 121. We have (19)

[OF

:

Ta,F] = wkea(F) h(hF) (Wc)[F:k]-1

where ea(F) is a positive integer, equal to 1 if F n H(l) = k ; otherwise we have (20)

ea(F)

ST

11 (1 - X((q, F n H(1)/k)) + awkh[F : F n H(1)]) X541

where x runs through the set of all non trivial characters of Gal (FnH(l) /k).

The fact that ea(F) # 0 implies that the quotients ta,F(a)/ta,F(1), a E Gal (F/k) and a 1, constitute a maximal system of independant elements of O. In particular we have NF/F(1) (T4,F)= Ta,Fn H(1).

204

H. OUKHABA

In other words the group Ta,F n H(1) is generated by the quotients to F(T)/ta F(1),T E Gal (F(1) /k) where we have put F(1) = F n H(1) and

fj

to F(T) df"1

ta,F(T),

for all T E Gal (F(1)/k).

rE Gal (F/K) 4I F.(1) =r

This leads to the identity

(Ta,F n H(1))n= Ta F n H(1), for all n > 0.

(21)

Moreover the group Ta,FnH(1) has finite index in OF(1) given by the formula [OF(1) : Ta,F n H(1)] = wkea(F)

(22)

h(OF(u (w")[F(1):k]-1 h

Nota Bene. Let us recall also that the subgroup OF(1) of OF(1) formed of all the products 11 OF(1)(r)

j

such that

rE Gal (F(1)/k)

1

nr(T) E IF(1) has a finite index in OF(1), cf. [91 or

rE Gal (F(1)/k)

[ 121, given by the following formula (23)

[OX F(

: 0 F(1)= U)k(wkw

h)[F(1):kl-1

h(OF(1)) [H(1)

:

F(1)]

PROPOSITION 9. - Let a be a positive integer. Then the group dfn

Za,F = TaWkh F F(1) has finite index in OF, given by the formula (24)

[oF :

ZaF]=Wk(Wkw.h)[F:k]-1

h(OF) [H(1)

:

F'(1)

Proof : on one hand we have the isomorphism Za,F/T"F OF(1)/Ta F n OF(1). On the other hand one can check that Ta F n OF(1) Ta F n H(1). This leads to the following identity [OF : Za,F] [OF(1)

:

(Ta,F n H(1))wkhl = [OF : Ta,F ]

[OF(1)

:

_

OF(1)1

205

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

which allows us to conclude the proof using Formulas (19), (21) and (23).o One can notice the inclusion EFkw°°h C Za,F, for all a > 0. In fact when EFkw°°h

I a then the group group of the products w,,,,

fi

(25)

can be characterized as follows. It is the

fi

ta,F(a)ma

of Cal (F/k)

OF(,) (-r)"

TE Gal (F(1)/k)

such that of Gal (F/k)

ma E IF

E

2')

nT E IF (1)

TE Gal (F(1) /k)

3') We have the congruence

Y

d1 PF(,) (

nr(T))

TEGal (F(1)/k)

where

0(mod. ww()'\

,

TEGal (F(1) /k)

M,(7-) is the element of IF(1) such that MT =

>'

mo,

T E Gal (F(1) /k).

o EGal (F/k)

LEMMA 2. - We have a well defined morphism

Za,F -, (Z/mZ), M = WF(,) /wk,

which associate to the element (25) of Za,F the lefthand side of the above congruence 3'). This morphism is onto and its kernel isjust the group EF kwoo h

so that we have

[ZaF

:

wkwooh

EF

wF'(1) Wk

Proof : all we have to prove is that the congruence d1 ' F(j)

(

nr(T))

MI(T))+WF(i) ( TE Gal (F(1) /k)

( \

lWkh

fJ ta,F aE Gal (F/k)

Wk

TE Gal (F(,) /k)

occurs whenever the element

f, TEGal (F(1)/k)

0(mod . wF(,)

OF(1)

206

H. OUKHABA

of Za,F is equal to 1. But in this case one can see easily that the product

H uEGal (F/k)

E Ta,F n H(1),

ta,F(Q)m

which means that ma = ma', if 0 F(1) = 11'F(1); and then one can write using the definition of ta,F(a) for a E Gal (F/k) ta,F(a)'n'

OF(1)(T)'n')a[F:F(1)1

= (

aEGal (F/k)

[J

TEGal (F(1)/k)

fi

x

TEGal (F(1)/k)

(f

PF(a))"*

,

o EGal (F/k)

°I FM=

where we have put m' = ma if a E Gal (F/k) and T E Gal (F(1) /k) are such that a,F(,) = T. Now the formula OF(1)(T

VF(a)m')wkix=

)1

,TE

Gal(F(i)/k),

already proved in 191, chap. IV, leads to the equality :

11

awkh[F : F(1)]-"x-4]

8FU)

=1

rEGal (F(1)/k)

which is equivalent to the condition (26)

nT + m'T + mrawkh[F : F(1)] - m;.Tq = 0

for all T E Gal (F(1) /k). Now we have d14

M.(T)) =

[F

)/F(1) (

TEGal (F(1)/k)

F(1)J mTTgn (T))

TEGal (F(1)/k)

= [H q" : H(1)] 0(1 ( F(j) )

E

m'T(TTq^1))

TEGal (F(1) /k)

Nq

"-1(Nq-1) Wk

= -g1F(j) (

m, T

TEGal (F(1)/k)

E TEGal (F(1)/k)

,

1

)-1)

Wk

m7(1 -T)(1 - Tq1)).

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

207

On the other hand the above condition (26) allows us to write ,j(F(1)

nT(T)

1

m'.)(r))

F21'

/TEGal (F(l)/k)

TEGal (F(1) /k) (2) l

mT(1 - T)(1 - Tq 1) I .

1

TEGal (Fhl /k)

The Lemma 2 is proved. PROPOSITION 10. - The quotient group OF 1,6F is finite. The exact number

of its elements is given by the index formula

[o : EF]J = L

h(OF) [H(1)

:

F(1)]

Manuscrit recu le 4 decembre 1993

208

H. OUKFIABA

References [ 1 ] V.G. DRINFELD. - Elliptic modules, Math. USSR-Sbornik, 23 (1974), 561-

592. [2] S. GALOVITCH, M. ROSEN.-Theclass number ofcyclotomicfunctionfields,

Journal of number theory, 13 (1981), 363-375. [3] S. GALOVITCH, M. RosEN. - Units and class groups in cyclotomic function

fields, Journal of number theory, 14 (1982), 156-184. [4] D.R. HAVES. - Explicit class field theory for global function fields, Studies

in algebra and number theory (Rota G.C (ed)), New-York, Academic Press, (1979), 173-217. [5] D.R. HAVES. - Elliptic units in function fields, In proc. of a conference on

modem developments related to Fermat's last theorem, D. Goldfeld ed. Birkhailser, Boston, 1982. [6] D.R. HAVES. - Stickelberger elements in function fields, Compositio. Math, 55 (1985), 209-239. [7] D. KUBERT, S. LANG. - Modular Units, Grundleh der Math. Wiss., 244 (1981), ed. Springer. [8] H. OuKHABA, G. ROBERT. - Etude d'un ideal particulier associe a un caractere de Dirichlet d'ungroupefini, Seminaire de Theorie des Nombres de Bordeaux 3 (1991), 117-127. [9] H. OUKHABA. - Fonctions discriminant, formules pour le nombre de classes et unites elliptiques; le cas des corps de fonctions (associes a des courbes sur des corps finis), These (Grenoble, Institut Fourier, Juin 1991).

[10] H. OUKHABA. - Groups of elliptic Units in Global function fields, in Proceedings of the Workshop at the Ohio State University, June 17-26, 1991. 1111 H. OUKHABA. - On discriminant functions associated to Drinfeld Modules

of rank 1, Journal of number theory, 47 (1994). [12] G. ROBERT. - Unites elliptiques, Bulletin Soc. Math. France, Supplement 36, Decembre 1973. [ 13] J. Yu. - Transcendence and Drinfeld modules, Invent. Math. 83 (1986), 507-517. Hassan OUKHABA Equipe de Mathematiques URA CNRS 741 16, Route de Gray France - 25030 Besancon Cedex

Number Theory Paris 1992-93

Arbres, ordres maximaux et formes quadratiques enti8res Isabelle Pays

On salt depuis Lagrange que tout entier naturel est une somme de quatre canes, et, d'apres Jacobi (1828), que le nombre de representations d'un entier en somme de quatre canes est

r4(m) = 8 Ed (m > 1), dim 4td

les representations obtenues en permutant 1'ordre ou en changeant le signe des composantes etant comptees separement. Les preuves connues de cette formule sont de nature analytique (analyse complexe, formes modulaires, fonctions elliptiques, ...). Parmi les nombreuses references, citons E. Landau [8, pp. 146-150] qui determine le nombre de representations d'un en-

tier en somme de quatre canes en utilisant les formules sur le nombre de decompositions d'entiers en somme de deux canes (qu'il a etablies auparavant de maniere tout a fait elementaire); G.H. Hardy et E.M. Wright [7, p. 314], J.V. Uspensky et M.A. Heaslet [15, pp. 450-458], ainsi que E. Grosswald [6, pp. 30-36] donnent des preuves basees sur des identites qui peuvent etre derivees des proprietes des fonctions elliptiques ou simplement verifiees "a la main"; dans [9, p. 3331, W. Scharlau exploite le fait que la somme de quatre canes est une forme quadratique avec un seul element dans son genre pour deduire la formule de Jacobi; A. Robert [ 111 et B. Gordon [4] etablissent la formule de Jacobi a partir de resultats sur les formes modulaires (ce qui necessite un peu d'analyse complexe). Toutes ces preuves utilisent ou bien des identites un peu "mysterieuses", ou alors du materiel assez sophistique. Pour des references concernant l'origine et les developpements historiques, nous renvoyons le lecteur au recueil de L.E. Dickson [3, Chap. VIII, p. 2851. Signalons aussi un article de G. Rousseau [ 121, oft l'auteur donne un moyen pour construire des representations d'un entier en somme de quatre canes a partir de fractions continues.

I. PAYS

210

Nous proposons ici une nouvelle preuve a caractere purement algebrique et geometrique de la formule de Jacobi. Cette preuve est tout a fait elementaire : les prerequis sont a peine un peu plus qu'un cours de premier cycle en algebre. La preuve que nous presentons decoule de resultats plus generaux sur le nombre de representations d'une puissance quelconque d'un nombre premier par certaines formes quadratiques a quatre variables, obtenus au moyen d'actions de "groupes de quaternions" sur "l'arbre de SL2 (Q p)". L'article se presente comme suit : Au § 1. on rappelle la definition d'une algebre de quaternions. Les ordres maximaux dans une algebre de quaternions rationnelle permettent de definir les formes quadratiques entieres que l'on examine plus loin. Au §2 on decrit la construction de 1'arbre a partir des ordres maximaux de M2(Qp). C'est au §3 que

l'on explique la relation entre l'action d'un certain groupe sur l'arbre et les representations d'une puissance d'un nombre premier par les formes quadratiques associees (au § 1) aux ordres maximaux. On montre au §4 que, lorsque l'ordre maximal est principal, on peut obtenir le nombre de representations d'un entier quelconque (et non plus uniquement d'une puissance d'un nombre premier). Cela conduit a une nouvelle preuve de la formule de Jacobi.

1. - Algebres de quaternions et ordres Nous renvoyons aux ouvrages 11 ], (101 et 1161 pour les preuves detaillees

des resultats mentionnes dans ce paragraphe.

Soit K un corps de caracteristique differente de 2 et soient a et b deux elements non nuls de K. L'algebre de quaternions (a, b)K est l'algebre

admettant une base de quatre elements sur K, notes 1, i, j, k, avec la multiplication definie par les relations i2 = a, j2 = b, k = i.j = -j.i. Le conjugue du quaternion q = Xi + x2i + x3j + x4k, note q, est defini par

q = xi - x2i - x3j - x4k. La norme reduite du quaternion q, notee n(q), est definie par n(q) = q.q = x1 - axe - bx3 + abx4. La trace reduite du quaternion q, notee t(q), est definie par t(q) = q + = 2x1. Il est bien connu qu'une algebre de quaternions est soit a division, soit isomorphe a 1'algebre de matrices M2(K).

Les corps consideres ici sont soit le corps (global) Q des nombres rationnels, soit un des corps (locaux) Qp des nombres p-adiques ou IR le corps des nombres reels. Sur un corps local (ici IR ou Qp), it y a une unique algebre de quaternions a division, a isomorphisme pres. Sur R, it s'agit de 1'algebre des quaternions de Hamilton, IEII = (-1, -1)a. Soit H = (a, b)q. Quitte a multiplier a et b par des carres convenables, on peut supposer que a et b sont dans Z. Pour reconnaitre si Hp = (a, b)Qp est a division, on utilise le symbole de Hilbert (a, b)p. L'algebre (a, b)Q est

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

211

a division si et settlement si (a, b) p = -1. Nous renvoyons le lecteur a 114, p.391 pour le calcul de ce symbole. Notons toutefois que (a, b)P = 1 pour presque tout p (c'est-a-dire pour tout p sauf un nombre fini d'entre eux).

Le discriminant de H est le produit des nombres premiers p pour lesquels 1'algebre de quaternions H ® Q, est a division

disc(H) =

11

p.

p premier (a,6)p=-1

Soit R un anneau principal de caracteristique differente de 2, K son corps de fractions et H une algebre de quaternions sur K (nous envisageons en particulier le cas on R = Z ou Z[P] = {ap' la, n E Z} avec K= Q ou alors R = Z P, l'anneau des entiers p-adiques, avec K = Q p). Nous designons par R" le groupe multiplicatif des elements inversibles de R.

Un ordre de H sur R est un sous-R-module de H de rang 4 qui est aussi un anneau. Les elements d'un ordre ont la propriete d'etre entiers sur R, c'est-a-dire que leur trace et leur norme appartiennent a R. Un ordre maximal est un ordre qui n'est contenu proprement dans aucun autre ordre. Voici deux exemples qui nous seront utiles.

Exemple IL. Dans H = (-1, -1)q le Z-module 0' de base (1, i, j, k) est un ordre de H sur Z. De meme, le Z-module 0 engendre par 1, i, j,

a= (1+i+j+k)/2 est un ordre de H. On note que t(a) = 1 etn(a) = 1. L'ordre 0' nest pas maximal car it est contenu dans l'ordre 0. Exemple 2. Soit R un anneau principal et K son corps de fractions. Alors M2 (R) est un ordre de M2 (K).

Les formes quadratiques que nous allons examiner sont les formes normes d'ordres sur Z d'une algebre de quaternions H sur Q. Soit 0 un tel ordre et soit (el, e2, e3, e4) une base de 0 sur Z. La forme norme de 0 par rapport d la base e est

n(x) = n(> Xei) _

XiXjt(eiej)

Xi n(ei) + i<j

ou x E 0. (Nous la notons parfois qo.) Le fait que 0 est un ordre assure que la forme quadratique obtenue est a coefficients entiers. Le choix d'une autre base de 0 conduit a une forme Z-equivalente. Par abus de langage nous appelonsforme norme de 0 un representant quelconque de la classe d'equivalence. On verifie que deux ordres 0 et 0' conjugues par un automorphisme interieur de H (c'est-a-dire 0' = hOh-1 pour un certain element inversible h de H) donnent lieu a des formes quadratiques 7L-equivalentes.

I. PAYS

212

Dans 1'exemple 1 ci-dessus, la forme quadratique associee a l'ordre 0, exprimee par rapport a la base (1, i, j, a), est go(X1, X2, X3, X4) = X1 + X2 + X3 + X4 + X1X4 + X2X4 + X3X4,

tandis que celle associee a l'ordre 0, exprimee par rapport a la base (1, i, j, k), est la somme de quatre carres : qo' (X1, X2, X3, X4) = Xi + X2 + X3 +X4 .

:etude de ces formes quadratiques necessite une connaissance plus approfondie des ordres maximaux sur Z d'une algebre de quaternions rationnelle. Pour commencer, nous allons expliquer comment on peut voir facile-

ment si un ordre est maximal. Rappelons d'abord que l'on peut munir H naturellement de la forme bilineaire bt induite par la trace en posant bt(x, y) = t(xy). On a aussi besoin des definitions suivantes. Un R-reseau d'une algebre de quaternions H sur K est un sous-R-module de H de rang 4. Le discriminant d'un R.-reseau M de H, note disc(M), est le determinant de la matrice de 1'application bilineaire bt dans une base de M. On voit, en examinant la formule de changement de base, que cet element de K"

est defini a un carre de R" pres. De plus, si L est un R-reseau contenu dans M, alors disc(L) = r2disc(M) pour un certain r E R et L = M si et seulement si r E R'. Notons aussi que le discriminant d'un ordre est un element non nul de R/R"2. Le reseau "standard" M de base (1, i, j, k) a pour discriminant -(4ab)2R"2. Des lors, a nouveau par changement de bases, on voit que le discriminant

d'un reseau de H est toujours l'oppose d'un carre de K" (modulo R"2). Notons que si R = Z, R"2 est reduit a l'unite. COROLLAIRE. - Soit R un anneau principal et K son corps de fractions. Alors : 1. M2 (R) est un ordre maximal de M2 (K),

2. M2 (R) est un anneau principal, 3. tous les ordres maximaux de M2 (K) sont conjugues a M2 (R).

Demonstration 1) 11 est clair que M2(R) est un ordre. Comme disc(M2(R)) = lmodR"2, on deduit du comportement du discriminant par rapport a l'inclusion des reseaux que M2 (R) est maximal. 2) Soit I un ideal a gauche de M2(R) et soit ((x1i x2), (yl, y2)) une base du R-reseau de R2 engendre par les lignes des matrices de I. On verifie :

aisement que la matrice A ayant pour premiere ligne (xl, x2) et pour seconde ligne (y1, y2) est dans I et que I = M2 (R) A.

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

213

Si I est un ideal a droite de M2(R), on procede de maniere semblable en considerant cette fois le reseau engendre par les colonnes des matrices de I. 3) Notons 0 = M2(R) et soit O' un autre ordre maximal de M2(R). Alors, comme 0 est maximal. O'O est un ideal a droite de 0 et est donc principal.

On peut donc ecrire O'O = xO pour un certain x dans GL2(K). Par ailleurs, comme 0' est maximal, l'ordre de stabilisateurs a gauche de O'O

est egal a 0' U'ordre de stabilisateurs a gauche, 0 (I), d'un ideal I est

09(I) = {x E HjxI C I}). Comme 0,(0'0) = O9(xO) = xOx-1, on obtient 0' = xOx-1. Voici le critere qui permet de reconnaitre les ordres maximaux :

Critere. Un ordre 0 sur Z d'une algebre de quaternions H sur Q est maximal si et seulement si son discriminant est egal a I'oppose du carre du discriminant de H, c'est-d-dire si disc(o) = -(discH)2. 2. - L'arbre

Les arbres qui nous seront utiles pour etudier les nombres de representations sont les arbres associes aux groupes SL2 sur les corps locaux, qui sont des cas particuliers des immeubles de Bruhat-Tits. Nous les realisons ici a l'aide des ordres maximaux dans une algebre de quaternions deployee sur un corps local Qp (c'est-a-dire une algebre isomorphe a M2(Qp)).

Soit H une telle algebre et 0 un ordre maximal de H sur 1'anneau Z des entiers p-adiques. D'apres le corollaire ci-dessus, on sait que tous les ordres maximaux de H sont conjugues a 0, ce qui va permettre de definir une distance entre ces ordres maximaux. On introduit pour cela la valuation p-adique v : Qp -> Zp U oc normalisee par la condition v(p) = 1, et la fonction It : H --> Z U oo definie par :

µ(x)=max{nEZIxEp"O}

pourx

0

et

µ(0) = 00. Cette fonction satisfait les proprietes suivantes : PRopweTes. - Pour x, p E H et a E Q, , on a : 1. µ(x) = oo si et seulement six = 0. 2. µ(x + y) > min{µ(x), µ(J)}.

3. µ(xy) ? i(x) + µ(J)

4. µ(xa) = it(x) + v(a). 5. ;L(x) =,u(x). De plus, en designant respectivement par OX et par H" les groupes multiplicatifs des elements inversibles de 0 et de H,

a) Pourx E Hx, on ax E 0 " sietseulementsiµ(x) = µ(x-1) =0.

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b) Pour x E H', les conditions suivantes sont equivalentes : i) x E pa0" pour un certain a E Z. ii) it(x) + µ(x-1) = 0.

iii) x0x-1 = 0. c) Pour x, y E H, si x satisfait les conditions equivatentes de la propriete precedente, on a : p(xy) = µ(x) + µ(y) = µ(yx) et µ(xyx-1) = µ(y).

d) Pourx E H", µ(x-1) = µ(x) - v(n(x)) (oCl n design la norme de H).

Demonstration : les proprietes 1 a 4 sont toutes evidentes. La propriete 5 decoule immediatement du fait que tout ordre d'une algebre de quaternions est stable par la conjugaison quaternionienne (car y = t(x).1-

x). Si x E 0", alors µ(x) = 0 car les elements de p0 ne sont pas inversibles dans 0; on a alors de meme µ(x-1) = 0. Reciproquement, si /1(x) = µ(x-1) = 0, alors x et x-1 sont tous deux dans 0, done x E 0". Cela prouve la propriete (a). Si x E pa0" pour un certain a E Z, alors p(x) = a et /1(x-1) = -a, done µ(x) +EL(x-1) = 0. Inversement, six E H"

est tel que u(x) + u(x-1) = 0, soit x = pay pour a = µ(x) et pour un certain y E 0 N p0. On a alors a(y) = 0 et µ(x-1) _ -a + µ(y-1). La relation µ(x) + li(x-1) = 0 entraine alors : µ(y-1) = 0, done y E Ox par la propriete (a). Cela demontre 1'equivalence des conditions (i) et (ii) de (b). Par ailleurs, la condition (i) entraine evidemment (iii). Reciproquement, si

x satisfait la condition (iii), on ecrit encore x = pay pour a = µ(x) et pour un certain y E 0 N pO; comme 0 - M2 (Zp), it nest pas difficile de verifier qu'alors OyO = 0. Or, de la relation xOx-1 = 0, on deduit que yO = Oy; on a done

yO=Oy=OyO=O, ce qui montre que y E O" et x E paQx et acheve la demonstration de la propriete (b). Pour etablir la propriete (c), on observe que, d'apres la propriete 3, µ(xy) ? l2(x) + t1(y)

14) = µ(x-Ixy) a(x-1) + A(xy) Lorsque µ(x-1) = -µ(x), on en deduit immediatement que µ(xy) _ µ(x) + µ(y). La relation lz(yx) = µ(x) + µ(y) se demontre de maniere analogue, et la relation a(xyx-1) = µ(y) se deduit des deux precedentes. Enfin, la propriete (d) resulte des proprietes 4 et 5, car x-1 = x.n(x)-1. Soient maintenant 01 et 02 des ordres maximaux de H, et soient x1 et x2 E H" tels que :

01 =x10xi1

et

02

=x20x21.

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On pose d(01, 02) = -µ(xi 1x2) - µ(x2 1x1) E 7L.

Pour voir que la fonction d est bien definie, it faut verifier que le second

membre ne depend pas du choix de xl et x2. Si xi E H" est tel que Oi = xi0x'i 1 pour i = 1, 2, alors x'i 1xi0x%lx' = 0 pour i = 1, 2, donc, par la propriete (c) ci-dessus, on a

µ(x,l ix2) = µ(xti

I 1x2) = µ(x'1 XI) + µ(x 1x2) + µ(x2 1x'2)

1

et, de meme, 1L(x'2 1x1) = p(x'2 1x2) + p(x21xi) + ji(xi 1x'1). Des lors, µ(x1 1x2) + p(x'2 1x'1) = p(xi 1x2) + p(x21xi), ce qui prouve que d est bien definie. On a en fait d(01, 02) > 0, car, d'apres la propriete 3, µ(x1 1x2) + µ(x2 1x1) < 11(x1 1x2.x21xi) = 0.

PROPOSITION 2.1. - La fonction d est une distance sur l'ensemble des ordres maximaux de H. Cette distance est invariante par conjugaison, c'est d-dire que pour x E H" et pour 01, 02 des ordres maximaux de H,

d(x0lx-1, x02x-1) = d(01, 02)

Demonstration

:

it

est clair par definition que la fonction d est

symetrique. De la propriete (b), on deduit que d(01, 02) = 0 si et seulement Si 01 = 02. L'inegalite triangulaire decoule de la propriete 3 et l'invariance par conjugaison est evidente puisque (xx1)-1(xx2) = x1 1x2. On obtient alors l'arbre des ordres maximaux de H. TI-IEOREME 2.2. - Le graphe X dont les sommets sont les ordres maximaux de H et dont les aretes sont les couples (01i 02) d'ordres maximaux tels que d(01, 02) = 1 est un arbre, c'est-d-dire un graphe connexe et sans circuit. De plus, cet arbre est (p+ 1)-regulier, c'est d-dire que chaque sommet est l'origine de p + 1 aretes.

Demonstration : montrons d'abord que le graphe X est connexe. II suffit de montrer que tout ordre maximal 0' est lie a 0 par un chemin du graphe. On raisonne par induction sur la distance de 0 a 0. L'enonce est evident si cette distance est 1, puisqu'alors 0 et 0' sont lies par une arete. Il suffit donc de prouver que si la distance de 0 a 0' est n > 1, alors i1 existe un ordre 0" a distance n - 1 de 0 et a distance 1 de 0'. Soit 0' = xOx-1. Quitte a multiplier x par une puissance convenable

de p, on peut supposer p(x) = 0, c'est-a-dire que x E 0 N p0. Comme O/pO ^ M2 (1FP), la trace induit une forme bilineaire non degeneree sur

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O/pO; on peut donc trouver u E 0 N p0 tel que t(xu) ¢ pp, ce qui entrain bien sur que xu E 0 , pO. Soit alors y = xu + p- in (x). Comme d(O, 0') = n > 1, on a, par la propriete (d),

-IL(x-1) = v(n(x)) = n > 1, d'ou y E 0 N pO, c'est-a-dire, µ(y) = 0. Par ailleurs, x-1 y = u + p-lx,

donc µ(x-ly) _ -1, et de la relation n(x-1 y) = n(u) + p-lt(xu) + p-2n(x)

on tire : v(n(x-ly)) = -1. D'apres la propriete (d), on en deduit

-tc(x-ly) - IL(y-lx) = 1. Par ailleurs, comme v(n(x)) = it, on dolt avoir v(n(y)) = n - 1, donc

-µ(y) - 12(y-1) = n - 1. Des lors, l'ordre 0" = yOy-1 possede les proprietes requises.

Montrons ensuite que le graphe X ne contient pas de circuit. Soit 01i ... , On un chemin sans aller-retour, c'est-a-dire, (1)

f d(Oi,Oi+1) = 1 pouri = 1,...,n- 1 d(Oi, Oi+2) > 0 pour i = 1, ... , n - 2.

Pour prouver que ce chemin n'est pas un circuit, it suffit de montrer que d(Oi, On) = n - 1.

Ecrivons Oi = xiOx-1 pour i = 1 , ... , n et, pour i = 1, ... , n - 1 xi lxi+1 = pa`yi pour un certain yi E 0 N pO et ai = µ(xi lxi+1) E Z. D'apres la propriete (d), on a, pour i = 1, ... , n - 1,

p(xi+1xi) = ai - v(n(p'' yi)) = -ai - v(n(yi)) Des lors, d(Oi,Oi+1) = -N(x lxi+1) - µ(xi+lxi) = v(n(yi)), et les conditions (1) ci-dessus s'ecrivent :

v(n(yi)) = 1 pour i = 1, ... , n - 1 v(n(yiyi+l)) - 2,a(yiyi+l) > 0 pour i = 1, ... , n - 2.

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Vu la multiplicativite de la norme, la premiere condition entraine : pour v(n(yiyi+l)) = 2 pour i = 1,... , n-2; par ailleurs, comme yti E i = 1, ... , n - 1, on a yjy2+1 E 0 pour i = 1, ... , n - 2, done µ(yiyi,+1) > 0. Ces observations conduisent a reecrire les conditions ci-dessus sous la forme :

v(n(yti)) = 1 pour i = 1, ... , n - 1

(2)

I µ(ytiyti+l)

= 0 pour i = 1,...,n - 2.

Montrons alors, par induction sur m, que µ(y1... ym) = 0. C'est clair pour m = 2. Supposons donc µ(y1... yri.-1) = 0 et µ(y1 ... ym) > 0, c'est-a-dire,

yl...ym EPO. On a alors (3)

(yl

... y.-1 + pO).(ym + pO) = 0

dans O/p0.

Comme O/pO est isomorphe a une algebre de matrices carrees d'ordre 2 sur IF7,, on peut considerer yj + pO,... , ym + pO comme des operateurs lineaires sur un espace vectoriel de dimension 2 sur IFr. Ces operateurs sont non nuls puisque yj ¢ pO, et non inversibles puisque v(n(yi)) = 1; ils sont donc tous de rang 1. De meme, yl ... ym-1 + pO est de rang 1 puisque µ(y1 ... ym-1) = 0. ;equation (3) indique alors que : Im(ym + pO) = Ker(yl ... ym-1 + pO).

Par ailleurs, on a aussi Ker(yl

... y.-1 + p0) = Ker(ym-1 + pO),

donc

(ym-1 + PO) (Y. + PO) = 0

et par consequent µ(ym-lym) > 0, contrairement a l'hypothese. On a donc bien µ(y1 ... ym) = 0 pour tout m = 1, ... , n - 1. Un calcul direct donne alors d(Ol, On) = v(n(yl ... yn-1)) - 2µ(y1 ... yn-1) = n - 1, ce qui acheve de demontrer que le graphe X ne contient pas de circuit.

Pour prouver que X est (p + 1)-regulier, comme la conjugaison par tout element de H" induit un automorphisme de X et que tout ordre maximal est conjugue a 0, it suffit de montrer qu'il y a p + 1 ordres a

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distance 1 de 0. Or, it y a une correspondance bijective entre 1'ensemble des ordres a distance 1 de 0 et l'ensemble des ideaux a droite I de 0 tels

que 0 Q I Q p0, qui associe a tout ideal I son ordre de stabilisateurs a gauche :

O9(I)={xeHIxICI}

et a tout ordre 0' a distance 1 de 0 l'ideal pO'0 (pour etablir la bijectivite de cette correspondance, it est utile de remarquer que si x E 0 '. p0 et v(n(x)) = 1, alors OTTO est un ideal bilatere de 0 contenant proprement p0, donc 0770 = 0 puisque 01p0 f-- M2(]Fp) est simple. Si a present 0' est un ordre a distance 1 de 0, on peut ecrire :

0' = xOx-1 pour un certain x comme ci-dessus, d'oCi

p0'O = xOTO = xO. Reciproquement, 09(xO) = x0x-1).

Comme par ailleurs les ideaux a droite I tels que 0 12 p0 sont en bijection avec les ideaux a droite non triviaux de 0/p0 ^ M2 (IFp), qui sont au nombre de p + 1, it y a bien p + 1 ordres a distance 1 de 0.

3. - Actions de sous-groupes et representations d'entiers Soit H une algebre de quaternions sur Q et_ 0 un ordre maximal de H sur Z. La forme quadratique quaternaire a coefficients entiers que nous allons etudier est la forme norme sur 0, c'est-a-dire (voir aussi § 1) que si e = (el, e2i e3, e4) designe une base de 0, la forme s'ecrit 4

Xiei) _ i=1

Xi n(ei) + i

i<j

XiJCjt(eie )

Etant donne un nombre premier p, on se propose dans cette section d'etudier les representations des puissances de p par cette forme, c'esta-dire les solutions (XI, x2, x3, x4) E Z4 de

1'iei =

fpn

i=1

ou, ce qui revient au meme, les elements x de 0 tels que n(x) = fpn. L'ensemble de ces elements est note R(p) : R(pn) = {x E 0 1 n(x) = fpn}.

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On note aussi Rp(pn) 1'ensemble des solutions primitives, c'est-a-dire,

Rp(pn)=Ix EONpOIn(x)=fpn}. Les resultats sont tres differents suivant que p divise le discriminant de H ou non. Lorsque p ne divise pas le discriminant, 1'algebre Hp = H 0 Qp est isomorphe a M2(Qp), et 1'ordre Op = O 0 Z,, en est un ordre maximal.

Le groupe des inversibles de O[,], que l'on note O[P]x, s'identifie a un sous-groupe de Hp x.

Le theoreme suivant met en relation 1'action de ce sous-groupe sur I'arbre Xp des ordres maximaux de Hp (par conjugaison) et 1'ensemble des representations primitives de fpn par la forme norme de O. Soit On 1'ensemble des sommets de l'arbre Xp a distance n de Op qui sont dans la meme orbite que Op par l'action de O[P]". Soit - la relation d'equivalence sur 0 N {0} definie par x " y si x et y sont associes (a droite) dans 0 c'esta-dire si x-1 y est dans Ox. II est clair que si x E Rp(pn), alors tout element

y tel que y - x est aussi contenu dans Rp(pn). On peut donc considerer 1'ensemble quotient Rp(pn)/ -. On a le : THEOREME 3.1. - L'application qui a un element x de O fait correspondre t'ordre xOpx-1 de Hp definit une byection entre les ensembles Rp(pn)/ et An.

Demonstration : si x E R.p(pn), alors a(x) = 0 et v(n(x)) = n, donc, par la propriete (d) de la section precedente, µ(x-1) = -n. Des lors, d(Op, xOpx-1) = n. Montrons que l'application definie dans 1'enonce est surjective : soit 0' un element de A, it existe alors y E 0[-!]' tel que 0' = et d(Op, yOpy-1) = n. Comme les scalaires agissent trivialement, on peut yOpy-1

supposer, quitte a multiplier y par une puissance adequate de p, que µ(y) = 0, c'est-a-dire que y E O N pO. La condition d(Op, yOpy-1) = n se traduit alors par v(n(-y)) = n. Par ailleurs, comme -y E O[P]", on doit avoir n(y) E Z[Y]" = {±pk I k e Z}.

Les conditions precedentes entrainent : n(y) = fpn, donc l'ordre 0' est l'image de y par l'application decrite dans 1'enonce.

Montrons pour terminer que l'application est aussi injective. Deux elements x, y de Rp(pn) definissent le meme sommet si et seulement si XOpx- 1

= yOpy

1

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D'apres la propriete (b) de la section precedente, cette condition est equivalente a

X-1yEpaQp

pour un certain a E Z. Comme n(x-ly) = ±1, on en deduct que x-1y E H" n OP x. Par ailleurs, x-1y E O[ff] puisque x-1 = fx/pn et que T E 0. Donc,

Voici un cas ou le nombre d'elements de An est particulierement facile a calculer. TrIEOREME 3.2. - Si t'ordre maximal 0 est principal, alors transitivement sur les sommets de l'arbre Xp, et par consequent

agit

t& l=pn-1(p+1) pour tout n > 1.

Demonstration : soit 0' un ordre maximal de Hp. On sait que tous les ordres maximaux de Hp sont conjugues, donc 0' = xOpx-1 pour un certain x E Hp >'. Quitte a multiplier x par une puissance convenable de p, on peut choisir x E Op. Considerons alors l'ideal a droite I de 0 define par

I= n(OgnH)

n(x0p n H),

qi4P

c'est-a-dire l'ideal dont les localises sont Iq = °q pour q p et Ip = xOp. D'apres l'hypothese, cet ideal est principal, donc I = yO pour un certain

y E 0. Comme Iq = Oq pour q

p, on a y E Coq pour q 54 p, donc

y E CA[P]". Par ailleurs, de la relation Ip = XOP = Y°p,

on deduit que l'ordre des stabilisateurs a gauche de Ip est

'g(Ip) = X0pX-1 = JOPJ-1+

c'est-a-dire que 0' = yOpy-1. Cela prouve que 0[-!]' agit transitivement sur les sommets de 1'arbre Xp. Il en resulte en particulier que On est 1'ensemble de tous les sommets a distance n de Op. Cet ensemble contient

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pn-1(p + 1) elements, puisque I'arbre Xp est (p + 1)-regulier, d'apres le theoreme 2.2. Remarque : plus generalement, Vigneras [ 16, p.147, Prop.3.3] a montre que le nombre d'orbites de sommets de Xp sous 1'action de O[P] est egal au nombre de classes de O. La demonstration n'est pas aussi elementaire que celle du theoreme precedent, car elle utilise un theoreme puissant d'Eichler.

On a choisi au debut de ce paragraphe un nombre premier p qui ne divisait pas le discriminant de H. Voici maintenant ce qui se produit lorsqu'au contraire p divise le discriminant de H, c'est-a-dire lorsque Hp = H 0 Qp est une algebre a division. Comme precedemment, on dit que deux elements x, y E 0 sont associes (a droite) s'il existe un element inversible u E Ox tel que x = yu; on note alors x - y. THEOREME 3.3. - Lorsquep divise le discriminant de H, alors les elements

de R(pn) sont tous associes, pour tout n > 0, de sorte que le quotient R(pn)/ - contient un seul element, si R(pn) nest pas vide. Si l'ordre 0 estprincipal, alors R(pn) est non vide, pour tout n > 1.

Demonstration : supposons que R(pn) est non vide. Soient x, y E R(pn) ; alors x et y sont inversibles dans Oq = 0 0 Z. pour q # p, et donc, xOq = YOq = Oq.

En p, l'algebre de quaternions Hp = H®Qp est isomorphe a l'unique algebre de quaternions a division sur Qp, et Op est l'anneau devaluation de Hp U10, § 12]). De plus, tout ideal a droite de Op est bilatere et principal et est donc de la forme irPOp, ou 7rp est une uniformisante de Or,. Comme la valuation p-adique de n(7rp) est 1, on a en particulier x0p = 7rnop = yOp.

Ainsi, xOp = yOp pour tout p, donc xO = yO et x - y. Cela prouve que tous les elements de R(pn) sont associes. Si 0 est principal, alors l'ideal I dont les localises sont Oq pour q 54 p et 7rpOp en p est principal; soit I = 7r0 pour un certain it E O. On a alors

0

7rO 2 pO,

donc p = 7rir' pour un certain 7r' E 0, et n(7r)n(ir') = p2.

Si n(-7r) = ±1, alors 7rO = 0; si n(7r) = ±p2, alors 7rO = pO. Comme ces deux egalites sont exclues, on doit avoir n(7r) = fp,

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donc R(p) est non vide. De plus, pour tout n > 1, on a n(7rn) = ±p', donc R(pn) est non vide pour tout n > 1. Dans le cas particulier ou la forme norme est definie positive, ce qui

revient a dire que l'algebre de quaternions H est telle que H 0 IR est isomorphe a 1'algebre (-1, -1)R des quaternions d'Hamilton, it est clair que les equations n(x) = -pn n'ont pas de solution et que les equations n(x) =

pn n'en ont qu'un nombre fini. Les resultats precedents permettent de denombrer ces solutions. L'ordre 0 de l'algebre de quaternions rationnelle H etant fixe, notons r(pn) (resp. rp(pn)) le nombre de solutions (resp. de solutions primitives) x E 0 de 1'equation n(x) = pn, c'est a dire le nombre d'elements de R(pn) (resp. Rp(pn)). COROLLAIRE 3.4. - Supposons que laforme norme soit definie positive. Si p est un nombre premier qui ne divise pas le discriminant de L'algebre H, alors pour tout n > 1, rp(a)n) = 10' I Ionl oil An est L'ensemble des sommets de I'arbre Xp qui sont dans la meme orbite que O. sous l'action de 0[-1]" et a distance n de CAP, et [n/21

r(pn) = IOX I

E k=0

.

oil [n/2] est le plus grand entier inferieur ou egal a n/2. Si p est un nombre premier qui divise le discriminant de H. alors pour tout n > 1,

r(pn) = 0 ouI0xI. Si de plus L'ordre 0 est principal, alors pour tout nombre premier p qui ne divise pas le discriminant de H,

rp(pn) =

I0XI. pn-i(p+ 1)

et

np+1

lox

p

-1

-1 r(pn) = et sip divise le discriminant de H,

pourtoutn> 1 pour tout n > 1,

I

r (pn) = I C X I

pour tout n > 1.

Demonstration : Si p ne divise pas le discriminant de H, les formules pour rp(pn) resultent directement des theoremes 3.1 et 3.2 ci-dessus, puisque rp(pn) = 0 < I ' I Rp(pn)/

I.

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Les formules pour r(p') s'en deduisent, car les solutions non primitives de n(x) = p' sont de la forme x = pkxk ou Xk est solution primitive de n(x) = p"-2k. De meme, si p divise le discriminant de H, les formules pour r(p') decoulent du theoreme 3.3. Pour completer l'information donnee dans ce corollaire, remarquons que la structure du groupe O" est connue pour les ordres maximaux des algebres de quaternions definies positives : PROPOSITION 3.5. - Si 0 est un ordre maximal d'une algebre de quater-

nions H sur Q telle que H 0 R est isomorphe a (-1, -1)R, alors O" /{f1} est cyclique d'ordre 1, 2 ou 3 sauf dans deux cas :

si H = (-1, -1)Q, alors 0'/ f ± 11 est isomorphe au groupe alterne A4; si H = (-1, -3)Q, alors O" /{±1} est isomorphe au groupe symetrique S3. Demonstration : voir [ 17, th. 5, p. 269]. Exemple : revenons a 1'exemple 1 avec l'ordre Ode base (1, i, j, a) dans

1'algebre H = (-1, -1)Q. Le discriminant de 0 vaut -4 et le discriminant de H est egal a 2. Des lors, 0 est maximal. Pour voir que c'est un anneau principal, on montre que les elements de 0 satisfont un algorithme de division euclidienne ([ 13, p. 98, lemme 3]). L'ordre du groupe O" des elements inversibles de 0 est 24, d'apres la proposition 3.5. Des lors, pour tout p 2, le nombre de representations primitives de p"` par la forme : q0 (Xl, X2, X3, X4) = X1 + X2 + X3 + X4 + X1X4 + X2X4 + X3X4.

est egal a rp(p') = 24(p +

1)p"-1

(p

2),

et le nombre de representations de 2 est donne par :

r(27z)=24

n>0.

4. - Ordres principaux Dans cette section, 0 designe un ordre maximal principal dans une algebre de quaternions rationnelle H dont la forme norme est definie positive. On se propose de montrer que, grace au fait que 0 est un anneau principal, it est possible de donner le nombre de representations d'un entier

positif quelconque (et non pas seulement des puissances d'un nombre premier) par la forme norme de O.

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LEMME 4.1. - Soient a et b des entiers positifs premiers entre eux. Si

x E 0 est tel que n(x) = ab, ators it existe y, z E 0 tels que x = yz et n(y) = a, n(z) = b.

Si y et z sont des elements de 0 tels que n(y) = a et n(z) = b, alors yz0 + a0 = yO et Oyz + 0b = Oz. Demonstration : soit x E 0 tel que n(x) = ab. On considere l'ideal x0 + a0 de 0. Comme 0 est principal, on a :

x0 + a0 = yO pour un certain y de 0. Soient x = yz et a = yy'. Alors n(x) = ab = n(y)n(z) et a2 = n(y)n(y'), donc n(y) est un commun diviseur de abet de a2. Comme

a et b sont premiers entre eux, n(y) divise a. Par ailleurs, de la relation xO + aO = yO, on tire aussi

xx'+aa' = y pour certains x', a' de 0. En prenant la norme des deux cotes, on deduit :

n(y) = n(xx') + n(aa') + t(xx'aa') = n(x) n(x') + a2n(a') + at(xx'a') = a(bn(x') + an(a') + t(xx'a')). Comme x, x' et a' sont dans 0, on a t(xx'a') E Z, donc le facteur de a dans le membre de droite est un entier. Il en resulte que a divise n(y). Comme a et n(y) sont tous les deux positifs, on a que n(y) = a et donc aussi n(z) = b. Par ailleurs, si y et z sont des elements de 0 tels que n(y) = a et n(z) = b alors, comme 0 est principal, on a :

yzO+aO=dO pour un certain d dans 0. Les arguments du debut montrent que n(d) = a. Par ailleurs, de yj = a, on deduit que aO C yO. Des lors, yzO + aO = dO C yO.

Ainsi i1 existe u E 0 tel que d = yu. Comme n(d) = n(y) = a, on a n E 0" et donc d0 = yO. Avec ce lemme, on peut donner le nombre de representations d'un entier quelconque par la forme norme de 0; comme precedemment, on note

R(m) =Ix E0I n(x)=m} et

r(m) = JR(m)I, ou m est un entier (positii) quelconque.

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

225

T1ii orEME 4.2. - 1. Lafonction r(m)/ IOx I est multiplicatioe, c'est-d-dire que si a et b sont des entiers positifs premiers entre eux, alors :

- lox I

r(ab)

r(a)

lox I

r(b)

lox I

2. Soit m un entier positif. On a

r(m) = I

I

(

d) dim pgcd(d,disc H)=1

Demonstration : 1. Soient a et b des entiers positifs premiers entre eux. La multiplication

dans 0 definit une application : R(a) x R(b) -i R(ab), qui est surjective d'apres le lemme 4.1. Pour demontrer la premiere partie de 1'enonce, it suffit de prouver que tout element de R(ab) est l'image de lox I elements de R(a) x R(b), puisqu'alors

r(ab) =

IR(a) x R(b)I

r(a)r(b)

IoxI

IOxI

.

Fixons x E R(ab). Si (y, z) et (y', z') sont des elements de R(a) x R(b) tels que

yz=x=y'z',

alors d'apres la seconde partie du lemme 4. 1 on a : xO + aO = yO = y'O.

Des lors, it existe un element inversible u E Ox tel que y' = yu (et donc z' = u-1z). Cela prouve que les elements de R(a) x R(b) qui ont x pour image sont les couples (yu, u-Iz), oli u E O. Le nombre de ces couples est bien egal au nombre d'elements de Ox. 2. On a deja calcule, dans le corollaire 3.4, le nombre de representations des puissances d'un nombre premier p : si p ne divise pas disc H, alors :

et si p divise disc H, r(Pn)

IoxI

226

I. PAYS

On voit ainsi que les fonctions

(r

et (T, d) prennent la mil-me valeur dIm

pgcd(d,disc H)=1

lorsque m est une puissance d'un nombre premier. Comme ces deux fonctions sont multiplicatives, elles doivent prendre la meme valeur pour tout M. Le theoreme 4.2 s'applique aux ordres principaux dans les algebres

de quaternions rationnelles definies positives, qui sont au nombre de cinq [16, p.1551. L'ordre principal est alors unique (a conjugaison pres) [16, p.26, Cor. 4.111. Voici la liste des cinq formes quadratiques (a Zequivalence pres) auxquelles le resultat s'applique ainsi que, pour chacune, la formule pour le nombre de representations d'un entier quelconque. - Le nombre de representations d'un entier positif n par la forme

Xi +X2+x3+X4 + X1X4 + X2X4 + X3X4 est

241: d, din 2{d

- Le nombre de representations d'un entier positif n par la forme

Xi

+X2+X3 +X4 +X1X4+X2X3

est

121: d, din 3$d

- Le nombre de representations d'un entier positif n par la forme

x2 + 2X2 + 5X2 + X4 + X1X2 + X1X4 + X2X4 + 5X2X3 est

6Ed, dIn 5{d

- Le nombre de representations d'un entier positif n par la forme X1 + X2 + 2X3 + 2X4 + X1X4 + X2X3 est

41: d, din 7{d

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

227

- Le nombre de representations d'un entier positif n par la forme X1 + 4X2 + 13X3 + 2X4 + X1X2 + X1X4 + X2X4 + 13X2X3 est

2> d. dln 13{d

Nous allons maintenant deduire la formule de Jacobi pour la somme de quatre carres a partir de la premiere de ces formules. La forme quadratique Xi +X2 +X3 +X4 est la forme norme de l'ordre 0' de base (1, i, j, k) dans l'algebre de quaternions H = (-1, -1)Q. Cet ordre n'est pas maximal : it est strictement contenu dans l'ordre maximal 0 de base (1, i, j, (1+i+j+k)/2). On ne peut donc pas lui appliquer la technique developpee ci-dessus. Cependant, les relations entre 0 et 0' sont telles qu'il est quand meme possible de deduire le nombre de representations d'un entier en Somme de quatre carres a partir du nombre de representations d'un entier par la forme norme de 0. Pour eviter la confusion, la notation r(m) designe, dans la fin de ce paragraphe, le nombre de representations de m par la forme norme de 0 tandis que r4(m) designe le nombre de representations de m en somme de quatre carres (qui est la forme norme de 0'). Commencons par indiquer ]a relation entre 0 et 0'. PROPOSITION 4.3. - Soit x un element non nul de 0. Si n(x) est paire, alors x E 0'; si n(x) est impaire, alors x est associe (a droite) a 8 elements de 0' (et a 16 elements de 0 N 0').

Demonstration : par rapport a la base (1, i, j, a) de 0, la forme norme s'exprime de la maniere suivante : n(xi + 2x2 + 353 + ax4) = xi + x2 + x3 + x4 + 51x4 + 52x4 + 53x4.

Des lors, pour x = x, + i52 + jX3 + ax4, n(x) -= (xl + x2 + x3)2 + (xl + x2 + 53)x4 + x4 mod 2.

Comme la forme quadratique X2 + XY + Y2 est anisotrope sur le corps a deux elements, on a donc n(x) E 2Z si et seulement si xl + X2 + 53 - 54 0 mod 2. En particulier, si n(x) est paire, alors 54 est pair, ce qui entraine :

xe0'.

Si n(x) est impaire, alors (x + 20)(5 + 20) = 1 + 20, donc T + 20 est l'inverse de x + 20 dans l'anneau quotient 0/20. Par ailleurs, on montre

I. PAYS

228

aisement (voir par exemple [13, p.1001) que x est associe a droite a un element de 0'; pour terminer la demonstration, on peut donc supposer x E 0'. Si U E 0' est tel que xu E 0', alors dans 0/20, on a u + 20 = a; (xu) + 20,

ce qui prouve que u E 0', puisque Y(xu) E 0' et 20 C 0'. On a donc xu E 0' si et seulement si u E 0", ce qui prouve que les associes a droite de x qui sont dans 0' sont en bijection avec 0", d'ou la proposition, car

10'x1=8. La formule de Jacobi se deduit alors aisement de la formule pour le nombre de representations par la forme norme de 0 : THEOREME 4.4. (Jacobi). - Le nombre de representations d'un entier positif m en somme de quatre carres est donne par

7'4(m) =8() 'd). dim 4{d

Demonstration : soit, comme precedemment :

R(m)={xE01n(x)=m}. Si m est impair, on deduit de la proposition precedente que chacune des classes d'elements associes a droite qui constituent R(m) contient 8 elements de 0' et 16 elements de 0 . 0'; donc r4(m) = 3r(m) =

d.

d1m

Si m est pair, alors d'apres ]a proposition precedente R(m) C 0', donc

r4 (m) = r(m) = 24 Ed. djm 2{d

On peut encore exprimer ce resultat comme suit :

r4(m) = 8(E d +

2d) =

dim

dim

dim

2{d

2{d

4{d

Dans les deux cas, on a donc bien le resultat annonce. manuscrit recu le 22 fevrier 1994

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

229

Bibliographie

[1] A. BLANCHARD. - Les corps non commutatifs, Presses Universitaires de France, Paris, 1970. [2] J.W.S. CASSELS. - Rational Quadratic Forms, Academic Press, London, 1978. [3] L.E. DICKSON. - History of the Theory of Numbers, vol. II, Chelsea Publishing Co., New York, 1952. [4] B. GORDON. - An Application of Modular Forms to Quadratic Forms, BA-

thesis, 1975. [5] H. GROSS. - Darstellungsanzahlen von quaternaren quadratischen Stammformen mit quadratischer Diskriminante, Comment. Math. Helv 34, (1960) 198-221. [6] E. GROSSWALD. - Representations of Intergers as Sums of Squares, SpringerVerlag, New York Berlin Heidelberg Tokyo, 1985. [7] G.H. HARDY and E.M. WRIGHT. - An Introduction to the Theory of Numbers,

5th ed., Oxford University Press, 1979. [8] E. LANDAU. - Elementary Number Theory, Chelsea Publishing Co., New York, 1966.

[9] G. ORZECH ed. - Conference on Quadratic Forms 1976, Queen's Paper in Pure and Applied Math. 46, Kingston, Ontario, Canada 1977. [10] I. REINER. - Maximal Orders, Academic Press, London 1975. [11] A. ROBERT. - Introduction to Modular Forms, Queen's Paper in Pure and Applied Math. 45, Kingston, Ontario, Canada 1976. [12] G. ROUSSEAU. - On a construction for the representation of a positive integer

as the sum offour squares, L'Enseignement Math 33, (1987) 301-306. [13] P. SAMUEL. - Theorie algebrique des nombres, Hermann, Paris 1967. [14] J.-P. SERRE. - Cours d'Arithmetique, Coll. SUP, Presses Univ. France, Paris, 1970. [15] J.V. USPENSKY and M.A. HEASLET, Elementary Number Theory, McGraw-Hill,

New York and London, 1939.

230

1. PAYS

[16] M.-F. VIGNERAS. - Arithmetique des algebres de quaternions, Lecture Notes in Math 800, Springer-Verlag, Berlin Heidelberg New York, 1980. [17] M.-F. VIGNERAS. - Simplification pour les ordres des corps de quaternions

totalement definis, J. Reine Angew. Math. 286/287, (1976) 257-287. [18] A. WEIL. - Sur les sommes de trois et quatre carres, L'Enseignement Math 20 (1974) 215-222.

Isabelle PAYS

Universite de Mons-Hainaut Avenue Maistriau, 15 B-7000 MONS BELGIQUE

Number Theory Paris 1992-93

On a conjecture that a product of k consecutive positive integers is never equal to a product of mk consecutive positive integers except for 8.9.10 = 6! and related questions T.N. SHOREY

For an integer m > 2, we consider the equation

(1) (x+1) .

(x+k) _ (y+1) . . . (y+mk) in integers x > 0, y > 0, k > 2.

We replace x + 1 by x and y + 1 by y in (1) for observing that it is identical to considering equation

x(x+1) . . (x+k-1) = y(y+l) . . . (y+mk-1) in integers x> 0, y>0, k>2. If m = 2, equation (1) has a solution given by 8.9.10 = 6! (2)

x=7,y=0,k=3.

MacLeod and Barrodale [8) observed that this is the only solution of (1) with

m = 2 and k < 5. I give their proof for k = 2. We write equation (1) with m = 2 and k = 2 (3)

(x + 1)(x + 2) _ (y + 1)(y + 2)(y + 3)(y + 4).

By putting u = y2 + 5y, we have (x + 1) (x + 2) _ (u + 4) (u + 6).

Notice that

(x+32 _ 1)2<(x+1)(x+2)< (x+2)2 4

232

T.N. SHOREY

and

(u+5- 4)2 < (u+4)(u+6) < (u+5)2. We have [%11-(x

+ 1)(x + 2)]

=

(u + 4)(u + 6)]

[

which implies that i.e.

x=u+3.

(4)

By substituting (4) in (3), we have

(u + 4)(u + 5) = (u + 4)(u + 6) and this is a contradiction. Now, I give a few comments on the proof. We have written the right hand side of equation (1) as a product of translates of u and the translates

are independent of y. This is typical of the case m = 2. In the general case, we shall be extracting k-th roots in place of square roots. For this, it is necessary for the above argument that x and y are large as compared with k. Saradha and Shorey [ 111 proved that (2) is the only solution of equation

(1) with m = 2. Further, Saradha and Shorey [12] showed that equation (1) with m E {3, 4} has no solution. Recently, Mignotte and Shorey showed

that this is also the case when m E {5, 6}. For m > 2, it is proved in [13] that equation (1) implies that max (x, y, k) < C

where C is an effectively computable(i) number depending only on m. We have not been able to replace C by an absolute constant. It is likely that equation (1) with m > 2 has no solution. (5)

Now, we give a sketch of the proof that equation (1) with m > 2 implies that max(x, y, k) is bounded by a number depending only on m.

As pointed out earlier, we secure that x and y are large as compared with k. We re-write equation (1) as

(y + 1) ... (y + mk) (mk)! k! (1)

(mk)!

k!

All the constants appearing in this paper are effectively computable.

233

ONA CONJECTURE CONCERNING 8.9.10=6!

We count the powers of 2 on both the sides to obtain

k < ord2 ( k! ') < ord2

(x + 1

k! x + k )

< max ord2 (x + i) < - 1
log (x + k) log 2

which implies that :

x>2k-k.

(6)

By equation (1), we have

xk < (y + mk)k i.e.

x < (y + mk)'.

(7)

By (6) and (7), we observe that x and y are large as compared with k. Further, we combine (6) and (7) to write (8)

2k

- k < (y +

mk)'.

If y is bounded, we observe from (8) that k is bounded and equation (1) implies that x is bounded. Thus, we may always assume that y exceeds a sufficiently large number yo depending only on m. Further, by equation (1), we observe that x > y > yo.

For extracting k-th roots on both the sides of equation (1), we need to introduce some notation. We write ink

(9)

(z + 1) . . . (z + mk) =

Aj (m, k)zmk-j. j=0

Further, we determine rational numbers

Bj=Bj(m,k) withl<j<m such that : (10)

(zm + Biz'n-1 + ....+ Bm)k =

ink

Hj (m j=0

k)zmk-j

T.N. SHOREY

234

Hj (m, k) = Aj (m, k) for 0 < j < m.

k /Bl \= A, (m, k)

kB2 + 12 I Bi = A2 (m, k) .............../......................

Therefore B1i

,

B,,,, are determined recursively.

Let z be a sufficiently large positive number as compared with k and m. The relations (11) imply that the left hand side of (9) is close to the left hand side of (10). Therefore, the k-th root of the left hand side of (9) is close to the k-th root of the left hand side of (10). We can use this observation with z replaced by x or y, since x and y are large as compared with k and m. Therefore, the k-th root of the left hand side of (1) is close

to x +

k+ 1

and the k-th root of the right hand side of (1) is close to + B,,,.. Consequently, by equation (1), we derive that ym + + x + k1 is close to y"' + Blym-1 + + B. In fact, we show that Blym-1

(12)

Ix - (ym+Blym l +...+Bm_ k+1 )kT-1 2

where , den(B,))). T = (21cm (den(Bi), On the other hand, the left hand side of (12) is at least T-1 whenever it is not equal to zero. Hence, we conclude that

k+1

(13)

x=ym+Blym-1+...+Bm_

2

We substitute (13) in (1) to derive that (14)

HH (m, k) = Aj(m, k) for 0 < j < 2m

and (15)

H2,, (m, k) - A2m(m, k)

=

k(k + 1)(k - 1) 24

235

ONA CONJECTURE CONCERNING 8.9.10=6!

Thus, we have added m - 1 relations to the relations (11) with which we started. This is the basic idea of the proof. We calculate

Bl(2,k) = 2k+ 1 , B2(2,k) = (k+ 1)(2k+ 1)/3 and

(16)

H4(2, k) - A4(2, k) = (4k5 - 5k3 + k)/90.

By (15) and (16), we derive that m > 2. Then, we apply a result of Balasubramanian 113, Appendix] to conclude from (14) that k is bounded

by a number depending only on m. Thus, there are only finitely many possibilities for k and we fix k. We put

+1) ... (Y + mk) and

£(Y) = L(O(Y), Y)

where

k+1 O(Y) = Ym + BlYm-1 + ... + Brn 2

By equation (1) and (13), £(y) =

0,

which implies that either £(Y) is a zero polynomial or y is bounded by a number yo depending only on m and k. By taking yo > yo, we conclude that £(Y) = 0.

(17)

Now, I give two proofs to exclude the possibility (17).

We assume (17). Then (18)

L(X,Y) = (X - O(Y))

(Xk-1 + R1(Y)Xk-2 +

... + Rk-1(Y))

where Rj(Y) E Q[Y] for 1 < i < k. By equating the terms independent of X in the factorisation (18), we observe that the polynomial

(Y+1)...(Y+mk)-k!

236

T.N. SHOREY

is reducible over the field of rational numbers. Now, we apply a result of Brauer and Ehrlich 151 to conclude that k!

>k-lmk - 1)! 2(

[(mk - 2)/2]!

The right hand side is an increasing function of m and the inequality is not valid for m = 4. Therefore, we derive that m = 3 which implies that k = 2. Now, by looking at the constant term of £(Y), we observe that B3 (3, 2)

- 4 = W.

This is not possible, since B3 (3, 2) is a rational number. Next, we turn to the second proof to exclude the possibility (17). We as, i,,,,, jl, sume (17). Then, there exist pairwise distinct integers ii, , jm.

such that O(Y)+1 = and b(Y)+2-(Y+ji)...(Y+j,,,.)

Thus (19)

By putting Y = -ii in (19), we have

(ji - ii) ... U. - ii) = 1 which implies that m = 2. Then, we observe from (19) that

jl + j2 = it + i2 , 3132 = i1i2 + 1. Consequently (jl - j2)2 = (

2-4

which is not possible.

This completes our sketch of the proof of (5). Now, I mention some extensions of (5). Let f (X) be a monic polynomial of positive degree with rational coefficients. For an integer m > 2, we consider the equation

(20) f(x+1). . f(x+k) = f(y+1). . f(y+mk) in integers x, y, k > 2.

237

ONA CONJECTURE CONCERNING 8.9.10=6!

If f is a power of an irreducible polynomial, Balasubramanian and Shorey [3] proved that equation (20) with

f(x+j)#0 for 1<j
(21)

implies that max(I x 1, 1 y 1, k) is bounded by a number depending only on m and f. This is an extension of (5), since (21) is satisfied whenever f(X) = X and x is a non-negative integer. Further, it is easy to observe that the assumption (21) is necessary. The proof utilises an extension of the second proof for excluding the possibility (17). The second proof depends on the fact that f (X) = X is irreducible. For extending this proof, we need that f is a power of an irreducible polynomial. Furthermore, this is the only place in the proof where the hypothesis that f is a power of an irreducible polynomial is used.

For an integer d > 1, Saradha and Shorey [14] obtained another extension of (5) by proving that if x > 0, y > 0, k > 2 are integers satisfying (22)

(y+(mk- 1)d),

x(x + d) ... (x + (k - 1)d)

then max (x, y, k) is bounded by a number depending only on d and m. In fact, this is an immediate consequence of the following result ([ 14, Theorems 1,21): For e > 0, there exists a number Cl depending only on m and e such that equation (22) with max (x, y, k) > C1 implies that

d>

(23)

y(1-E)/(+n+1)

log d > (

M m(m + 1)

- e)K

where K and M are positive numbers given by K2 = k log k

(24)

,

M2 = m(m - 1)/2.

We observe from (23), (24), (25) and (22) that max (x, y, k) is bounded by a number depending only on d and m. Further, for positive integers d1, d2 and m > 2, Saradha and Shorey [ 151 considered a more general equation than (22) : x(x +

(x + (k - 1)d1) =y(y +

(mk - 1)d2)

(25)

in integers x > 0, y > 0, k > 2.

It was shown in [15] that equation (25) with m = 2 implies that either max(x, y, k) is bounded by a number depending only on d1, d2 or k = 2, d1 = 2d2, x = y2 + 3d2y. On the other hand, equation (25) with m = 2 is

238

T.N. SHOREY

satisfied whenever the latter possiblities hold. If m > 2, it was proved in [ 151

that there exist numbers C2 and C3 depending only on dl, d2, m such that equation (25) implies that k < C2 and moreover max (x, y) < C3 unless (*) d1 /c 2" is a product of m distinct positive integers composed of primes not exceeding m and m > a(k) where

a(k) =

14 for 2 < k < 7 50 for k = 8 I exp(klogk - (1.25475)k - logk + 1.56577) for k > 9.

This includes a result on the case dl = d2 = d mentioned above, since (*) is never satisfied. Further, we derive that equation (25) with 3 < m < 14 or 3 < m < 2568, k > 9 implies that max (x, y, k) is bounded by a number depending only on d1, d2, m. Finally, Saradha, Shorey and Tijdeman (17] showed that condition (*) is not necessary. Consequently, we conclude that equation (25) with m > 2 implies that max (x, y, k) is bounded by a number depending only on d1, d2, m. This is a consequence of a more general result which we describe now. For distinct positive integers £ and

m with gcd(f, m) = 1 and £ < m, Saradha, Shorey and Tijdeman [ 17] proved that there exists a number C4 depending only on d1, d2, m such that if x > 0, y > 0 and k > 2 are integers satisfying

x(x + di)

.

(x + (fk - 1)dl) = y(y + d2) ... (y + (mk - 1)d2),

then max(x, y, k) < C4

unless f = 1, m = 2 which corresponds to equation (25) with m = 2 and we refer to the result already stated in this case. By applying the theory of linear forms in logarithms, it is shown in [ 17] that the preceding assertion is also valid for k = 1 provided that f E {2, 4} and (f, m) = (3, 4). Now, we turn to equation (25) with m = 1. In this case, there is no loss of generality

in assuming that x > y and gcd(x, y, dl, d2) = 1. Then Saradha, Shorey and Tijdeman [ 161 proved that equation (25) with m = 1 implies that there exists a number C5 depending only on d2 such that either

x=k+l,y=2,d1=1,d2=4 or

max(x, y, k) < C5.

We observe that

(k+ 1). (2k)=2.6...(4k-1) fork= 2,3....

239

ON A CONJEC'TJRE CONCERNING 8.9.10=6!

since the right hand side is equal to

2k(2k)!/(2.4... (2k)) = (2k)!/k!.

Therefore, the above possibilities for the case d1 = 1, d2 = 4 cannot be excluded. Further Saradha, Shorey and Tijdeman [18] showed that equation (25) with m = dl = 1 implies that y < k2d2/12 and furthermore k < d2 - 2 unless y 2(mod 4) and d2 = 21 for some integer 2 > 2. In the case m = 2, dl = 1, Saradha, Shorey and Tijdeman [ 18] proved that equation (25) implies that y(y + (2k - 1)d2) < (0.44)k4d2 and furthermore

k < d2 - 2 unless k < 35 and d2 = 2e for some integer 2 _> 2. These results are applied in [181 to determine all the solutions of equation (25) with m = d1 = 1, d2 E 12,3,5,6,7,9, 10} and m = 2, dl = 1, d2 E {5, 6}. Let a and b be positive integers. We consider (26)

a(x + 1) ... (x + k) =b(y+1)...(y+k+2) in integers x > 0,y > 0,k > 2,2 > 0.

Equation (1) is a particular case of (26), namely, a = b = 1 and k + 2 is an integral multiple of k. ErdOs [71 conjectured that there are only finitely many integers x > 0, y > 0, k > 2, 2 > 0 with k + 2 > 3 and x > y + k + 2 satisfying

(26). This is a difficult problem. The assumption k + 2 > 3 is to exclude Pell's equations and the assumption x > y + k + 2 is to guarantee that the two blocks of consecutive integers in equation (26) are non-overlapping. Mordell 191 proved that equation (26) with a = b = 1, k = 2, 2 = 1 implies

that x = 1, y = 0 and x = 13, y = 4. Mordell's result initiated much of research in this direction. Avanesov [ 1] confirmed a conjecture of Sierpinski

by proving that x = 0, y = 0; x = 3, y = 2; x = 14, y = 7; x = 54, y = 19 and x = 118, y = 33 are the only solutions of equation (26) with a = 3, b = 1, k = 2, 2 = 1. Tzanakis and de Weger [20] determined all the solutions of equation (26) with a = 1, b = 2, k = 2, 2 = 1. Boyd and Kisilevsky [4] showed that x = 1, y = 0; x = 3, y = 1 and x = 54, y = 18 are the only solutions of equation (26) with a = b = 1, k = 3, 2 = 1. Cohn 161 proved that equation (26) with a = 1, b = 2, k = 4, 2 = 0 is satisfied only if x = 4, y = 3. Further, Ponnudurai [ 10] showed that x = 2, y = 1 and x = 6, y = 4 are the only solutions of equation (26) with a = 1, b = 3, k = 4, 2 = 0. Let us consider equation (26) with 2 = 0. We re-write equation (26) with

2=0 as (27) (axk-byk)+Al(axk-1-byk-1)+ +Ak-1(ax-by)+Ak(a-b) = 0 where A1,. .. , Ak are given by (28)

F(z)=(z+1)...(z+k)=zk+Alzk-1+

+Ak.

240

T.N. SHOREY

If a = b, we observe from (27) that (29)

(xk

- yk) + Al

(xk-1 _ yk-1) +

... + Ak-1(x - y) =0

which implies that x = y, since all the summmands in (29) are of the same sign. Now, we assume that a # b. Shorey [191 showed that there exists a number C6 depending only on a and b such that equation (26) with £ = 0, x > y+k and k > C6 implies that the first summand in (27) is positive and the second summand in (27) is negative (then all the summands in (27) following the second one will be negative). This is equivalent to saying that

equation (26) with e = 0 and x > y + k implies that either k < C6 or k = [a + 1] where

a =log

(a)/ log (y

We have not been able to exclude the latter possibility. This is the case if we allow C6 to depend also on P(x) and P(y).(2) In fact, Shorey [191 showed

that equation (26) with .£ = 0 and x > y + k implies that max(x, y, k) is bounded by a number depending only on a, b, P(x) and P(y). Saradha and Shorey [11) extended these results to equation (26) where $ is not necessarily equal to zero. Now, we give a sketch of the proof that equation (26) with x > y + .£ + k

implies that max(x, y, k, e) is bounded by a number depending only on a, b, P(x) and P(y). The proof depends on Gel'fond - Baker theory of linear forms in logarithms. We assume (26). By (26) and (28), we observe that

0 < aF(x) - bF(y)yl = Uk + AlUk_1 + ... + AkUO

(30)

where

Uj=ax'-by'+e for0
(31)

If Uk < 0, we observe from x > y that Uj < 0 for 0 < i < k which contradicts (30). Thus Uk > 0.

(32)

We write Cl, c2, ... , c12 for positive numbers depending only on a, b, P(x)

and P(y). We may assume that x > cl with cl sufficiently large. In view (2)

For an integer v > 1, we write P(v) for the greatest prime factor of v and we

put P(0) = P(1) = 1.

ONA CONJECTURE CONCERNING 8.9.10=6!

241

of the result of Shorey [ 191 mentioned above, we may suppose that f > 0 which implies that k + $ > 3. Then, it is proved in [ 11, Corollary 21 that

x - y< c2tx(log x)/k.

(33)

As in the proof of (5), we count the power of 2 on both the sides of (26) for deriving that

f
(34)

By (33) and (34),

x - y < c4x(log x)2/k.

(35)

On the other hand, we apply an estimate of Fel'dman (see Baker [2]) on linear forms in logarithms to obtain

x - y > x(log x)-`5

(36)

By (35) and (36), k < (log x)CB.

(37)

Now, we show that

y < (log x)".

(38)

We may assume that (39)

y > (k +

Q)4'

otherwise (38) follows from (34) and (37). Further, we derive from (26), (31), (32), (37) and (39) that (40)

0 < Uk < cs((k +

£)2yk+l-1 + k2x-1).

On the other hand, we apply again the estimate of Fel'dman on linear forms in logarithms for deriving that Uk > max(xk, yk+e) ((k + E) log

which, together with (34) and (37), implies that (41)

Uk > max(xk, yk+t) (log x)-°10

x)-C9

242

T.N. SHOREY

Finally, we combine (40), (41), (34) and (37) to conclude (38). By (37), we have

x > kP(x) > k-(x) where w(x) denotes the number of distinct prime divisors of x. Therefore, there exists a prime p dividing x such that pordp(x) >

(42)

k.

Now, we count the power of p on both the sides of (26) which we re-write as

a(x+l)...(x+k) _b(y+1)...(y+k+.2) (k +

k!

By (42), we obtain 1 < ordp (a (x +

P

1) - k! (x + k))

\

, ord,(a),

J

which sharpens (34) as : (43)

$ < Cu. l

By (26),

X < (y + k + f)1+(e/k) which, together with (37), (38) and (43), implies that x < c12. This completes the proof.

We refer to [111 for more results on equation (26). For example, it is proved in [ 11 ] that equation (26) with x > y + k + f implies that : x > C7k3(log k) -4 and

x-y>C8x2/3 where C7 and C8 are positive numbers depending only on a and b.

Manuscrit recu le 10 decembre 1993

243

ONA CONJEC7IJRE CONCERNING 8.9.10=6!

References [ 1 ] E.T. AvANEsov. - Solution of a problem on polygonal numbers (Russian),

Acta Arith. 12 (1967), 409-419. [2] A. BAKER. - The theory of linear forms in logarithms, Transcendence Theory: Advances and Applications, Academic Press (1977), 1-27. [3] R. BALASUBRAMANIAN and T.N. SHOREY. - On the equation f (x + 1)

f (x + k) = f (y + 1) . . . f (y + mk), Indag. Math. N.S. 4 (1993), 257-267. [4] D.W. BOYD and H.H. KISILEVSKY. - The diophantine equation u(u + 1)(u

+ 2) (u + 3) = v(v + 1)(v + 2), Pacific Jour. Math. 40 (1972), 23-32. [5] A. BRAUER and G. EHRLICH. - On the irreducibility of certain polynomials,

Bull. Amer. Math. Soc. 52 (1946), 844-856.

[6] J.H.E. COHN. - The diophantine equation Y(Y + 1)(Y + 2)(Y + 3) _ 2X (X + 1)(X + 2) (X + 3), Pacific Jour. Math. 37 (1971), 331-335.

[7] P. ERDOs. - Problems and results on number theoretic properties of consecutive integers and related questions, Proc. Fifth Manitoba Conf. Numerical Math. (Univ. Manitoba Winnipeg) (1975), 25-44. [8] R.A. Mac LEOD and I. BARRODALE. - On equal products of consecutive

integers, Canadian Math. Bull. 13 (1970), 255-259. [9] L.J. MORDELL. - On the integer solutions of y(y + 1) = x(x + 1) (x + 2), Pacific Jour. Math. 13 (1963), 1347-135 1. [ 101 T. PONNUDURAI. - I he diophantine equation Y (Y + 1) (Y + 2) (Y + 3) = 3X (X + 1) (X + 2) (X + 3), Jour. London Math. Soc. 10 (1975), 232-240. [11] N. SARADHA and T.N. SHOREY. - On the ratio of two blocks of consecutive

integers, Proc. Indian Acad. Sci. (Math. Sci.) 100 (1990), 107-132. [12] N. SARADHA and T.N. SHOREY. - On the equation (x + 1) . . . (x + k) _ (y+ 1) . . (y+mk) with m = 3, 4, Indag. Math., N.S. 2 (1991), 489-510. [13] N. SARADHA and T.N. SHOREY. - On the equation (x + 1) . . . (x + k) _ (y + 1) . . . (y + mk), Indag. Math., N.S. 3 (1992), 79-90. [14] N. SARADHA and T.N. SHOREY. - On the equation x(x + d)

(x + (k -

1)d) = y(y+d) . . . (y+(mk-1)d), Indag. Math., N.S. 3 (1992),237-242. [15] N. SARADHA and T.N. SHOREY. - On the equation x(x + d1)

(x + (k -

1)d1) = y(y + d2) . . . (y + (mk - 1)d2), Proc. Indian Acad. Sci. (Math. SO.), 104 (1994).

244

T.N. SHOREY

[16] N.

T.N. SHOREY and R. TIJDEMAN. - On arithmetic progressions

of equal lengths with equal products, Math. Proc. Camb. Phil. Soc. (1994), to appear. [17] N. SARADHA, T.N. SHOREY and R. TIJDEMAN. - On arithmetic progressions

with equal products, Acta Arith., to appear. [18] N. SARADHA, T.N. SHOREY and R. TIJDEMAN. - On the equation x(x +

(y + (mk - 1)d), m = 1, 2, Acta Arith., to appear. [ 19] T.N. SHOREY. - On the ratio of values of apolynomial , Proc. Indian Acad.

Sci. (Math. Sci.) 93 (1984), 109-116. [20] N. TzANAIUS and B.M.M. de WEGER. - On the practical solution of the Thue equation, Jour. Number Theory 31 (1989), 99-132. T.N. SHOREY

School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400 005, India

Number Theory Paris 1992-93

Redei-matrices and applications Peter Stevenhagen

1. - Introduction In this paper we describe an algebraic method to study the structure of (parts of) class groups of abelian number fields. The method goes back to the Hungarian mathematician L. Redei, who used it to study the 2-primary part of class groups of quadratic number fields in a series of papers [[ 18]-[24]] that appeared between 1934 and 1953. The case of the l-primary part of the class group of an arbitrary cyclic extension of prime degree l was studied by Inaba [[ 121, 19401, who realized that one should look at the class group as a module over the group ring. The matter was then taken up by FrOhlich 1[61, 19541, who generalized Inaba's results by extending Redei's quadratic method to the case of a cyclic field of prime power degree. In the seventies, generalizations in the line of Inaba were given by G. Gras 1110] ]. In all cases,

one studies 1-primary parts of the class group of an abelian extension for primes l that divide the degree. Recently, completely different methods have been developed by Kolyvagin and Rubin, showing that the structure of any l-primary part of the class group of an abelian field of degree coprime to l can be described 'algebraically'. For primes dividing the degree it is not yet clear whether the approach works. The Kolyvagin-Rubin methods can be seen as refinements of the analytic class number formula, and they are more general than the Redei-FrOhlich method as they work for most 1. On the other hand, they depend on the existence of infinite collections of auxiliary prime numbers, so effective versions of the (ebotarev density theorem are needed to yield deterministic algorithms. Because of this somewhat involved nature they can only be used in practice for abelian fields of very small degree. Moreover, the method does not give any clue as to the average behaviour that is to be expected when it is applied to a family of fields. For instance, it cannot be used to compute the class number h,+ of the maximal real subfield of the n-th cyclotomic field for any n that is not very small. Also, it does not tell us

P. STEVENHAGEN

246

whether the fact that h+ is either 1 or very small, which can be shown to be the case for all n < 200 having at least two prime divisors if one assumes the generalized Riemann hypothesis [[31]], should be seen as a common occurrence.

The Redei-Frbhlich method is based only on class field theory and therefore of a rather different nature. It can be used only to describe pprimary parts of the class group when p does divide the degree of the extension, which is exactly the case that had to be excluded before. We will see that it gives in this case rise to density statements telling us how many fields in a given infinite family will have some prescribed part in their class group. An example : the class number h3p is even for some explicit collection of primes p of Dirichlet density 1/16 and divisible by 3 for a collection of density 1/18.

An application of Redei's quadratic method that goes back to Redei himself concerns criteria for the solvability of the negative Pell equation x2 - Dy2 = -1. This is a question that is closely related to the behaviour of 2-class fields, as was made clear by Scholz [[25]]. We will discuss it in the two final sections of this paper.

2. - Redei-matrices In this section, K will denote a number field that is cyclic of prime degree

1 with Galois group G = Gal(K/Q). It is our intention to study the 1-part C of the class group of K. If l = 2, our convention will be that C is the 2part of the narrow class group of K. Correspondingly, we call an extension unramified if all finite primes are unramified. The difference between the narrow and the ordinary class group that may exist is of interest only when l = 2. It will be discussed in detail the following two sections. As C is a finite abelian 1-group with a natural G-action, it is a module over the group ring over the l-adic integers Zt [G]. The norm N = >geG 9 annihilates the class group, so we can study C as a module over ZI [G] IN. If (1 denotes a primitive l-th root of unity, we have an isomorphism Z1[G]/N -2- Zl[(l] Ors

> (l

showing that A = Z1 [G] IN is a discrete valuation ring whose maximal ideal is generated by a - 1, with or a generator of G. The residue class field of A is the finite field of l elements IF1. Every finite A-module M is isomorphic to a module of the form $

11 A/A(°-1)"i i=1

with n;, E Z>1 for i = 1, 2, ... , s. Thus, we can specify the isomorphism

REDEI-MATRICES AND APPLICATIONS

247

class of the A-module M by giving the sequence of integers dimF,(M(o-1)k-1/M(°-1)k).

rk = #{i: ni > k} =

Note that r1(M) = s and that {rk(M)}k is a decreasing sequence with rk(M) = 0 for k sufficiently large. The evaluation of r1 (C) amounts to doing genus theory for the field K. More precisely, class field theory associates to C an unramified extension H of K, called the 1-class field of K, for which the Galois group Gal(H/K) corresponds to an is canonically isomorphic to C. The quotient unramified extension H1 of K that is known as the genus field of K. It is the maximal unramified extension of K that is abelian over Q, and Gal(H1/Q) is isomorphic to the elementary abelian 1-group G x C/C°-1. If x denotes a Dirichlet character generating the character group X of G that corresponds to K, we can write x as a product x = flit= 1 xi, where t is the number of primes that ramifies in the extension K/Q and xi is a character of conductor a power of some ramifying prime pi and of order 1. The conductor of xi is equal to pi if and only if pi # 1. The field H1 corresponds to the group of Dirichlet characters C/Ca-1

Xi =

ftx.

i=1

It follows that Gal(H1/K) has order 1t-1, i.e. the (a-1)-rank r1 (C) is equal to t - 1, where t is the number of ramifying primes in K/Q. The subgroup Co = C[o, -1] of G-invariant ideal classes in C is known as the subgroup of _ambiguous ideal classes. As G is cyclic and C is finite, the order of CG = H° (G, C) equals the order of H1(G, C) = C/C°-1. It is not difficult to check that Cc is generated by the t classes [pi] of the ramified primes pi of K. The order of CG is 1t-1, so there is exactly one additional relation between these classes that is independent of the obvious relations [pti]=0. The Redei-Frohlich theorem gives a description of r2(C) by combining the two descriptions of ri (C). Note that as abelian groups, we have : C/C(o-1)2

C/C4 { C/Co-1 X C

if l = 2; 1/C(°-1)

2

if l> 2,

and that the 1-rank of C/C1 is equal to the sum EI-I rk. The theorem is based on the observation that r2 (C) can be obtained from an explicit description of the natural map

0 : C[v - 1]

C/Ca-1.

P. STEVENHAGEN

248

This is a homomorphism between elementary abelian 1-groups, so it can be viewed as a linear map between vector spaces over IF1. With this terminology, the (v - 1)2-rank of C is nothing but the Fl-dimension of the kernel of 0. This dimension can be given in terms of the rank of a certain matrix over F1, called the Redei matrix of K, as follows. 1. THEOREM (Redei-Friihlich). - Let K/Q be a cyclic extension of prime

degree l with group (v) and conductor f , and let X : (Z/ f Z) * -> IF1 be a generator of its character group, with values taken in the additive group IF1. Let pl, p2, ... , pt be the primefactors off , andX = Ei-1 Xi the corresponding decomposition of X. Then the 1-primary part C of the narrow class group of K has (Q - 1)2-rank r2 (C) = t - 1 - rankF, R, where the entries ai3 E IF1 of the Redei matrix R = (ai,j )i. j=1 are defined by : ai3 = Xi(pj)

if i

j;

t

Y. aid = 0. i=1

Proof : as C[v - 1] is generated by the classes of the ramified primes pi, we have a natural surjection p : Ft --f C[ai - 1] that maps the j-th basis vector ej to the class of p;. The group C/C°-1 is canonically isomorphic to the subgroup Gal(Hi/K) of Gal(Hi/Q) under the Artin map. We know H1 explicitly from genus theory : it is the compositum of the cyclic fields Q(Xi) of conductor a power of pi corresponding to the characters Xi. Each character Xi furnishes an isomorphism Gal(Q(Xi)/Q) -' F1, and they can be combined into an isomorphism EB 1Xi : Gal(Hi/Q) - lFi. The Redei map R : Fl -> Ff is defined as the composed map R : Fit

P . C[a - 1] - C/Ca-1 => Gal(Hi/K) C Gal(H1/Q) _

E)xi ------

Fit

of vector spaces over Fl. As the kernel of p is of dimension 1, one has (2)

r2 (C) = dime [ker 0] = dimF, [ker R] - 1 = t - 1 - rankF, R,

as desired. The image of a basis vector ej is the Artin symbol of p; in Gal(Hi/K). If i j, the restriction of this symbol to Gal(Q(Xi)/Q) is the Artin symbol of pj, and this is mapped to ai3 = Xi (pj) by Xi. For the diagonal entry aii the Artin symbol and Xi (pi) are not defined, but we can use the fact

that ®Xi maps Gal(Hi/K) to the hyperplane {(ai)i E IF' : Ei_1 ai = 0}. The desired identity F_i_1 ai3 = 0 follows immediately.

REDEI-MATRICES AND APPLICATIONS

249

The Redei matrix R is by definition a singular (t x t) -matrix since the sum of its rows is zero. It is said to have maximal rank if the rank equals t - 1.

Obviously, the rank is maximal if and only if r2 (C) = 0. We will meet this condition in the next section when investigating the solvability of the negative Pell equation.

The field H2 corresponding to the quotient C/C(°-1)2 is the central 1-class field of K, i.e. the largest unramified extension E of H1 that is normal over Q and for which the group extension

0 -p Gal(E/Hl) ---+ Gal(E/Q) -) Gal(Hi/Q) -40 is a central extension. In Frdhlich's terminology [[711, the central class field

H2 is a field of class two : its Galois group 11 = Gal(H2/Q) is not in general abelian but its lower central series has length at most two. This is equivalent to saying that the commutator subgroup [S2, S2] is contained in the center of S2, or that [ci, [SZ, Ii]] = 0. The Redei-FrOhlich method enables us to obtain Gal(H2/K) from very simple rational data. More precisely, we

can determine ri = ri(C) for i = 1, 2 in terms of the prime factors of the discriminant, and this leads to (A/A°-i)rl-r2 X (A/A(U-1)2).2 Gal(H2/K) =A f (Z/2Z)rl-r2 x (Z/4Z)r2 if l = 2; if 1 > 2. l (Z/JZ)rl+r2

The first isomorphism is an isomorphism of modules over the ring A = Z1 [G] IN, the second is an isomorphism of abelian groups.

3. - Applications As a first application, we will obtain divisibility results for the real cyclotomic class numbers hn of the type discussed in the introduction. Recall that h,+ is the class number of the maximal real subfield Fn = Q(Sn + (n-1) of the cyclotomic field of conductor n.

3. LEMMA. - Suppose that l > 2 and that the l -class group C of the field K in theorem 1 has (v - 1)2-rank r2(C) = r. Then jr divides h+, with n the conductor of K.

Proof : as K is real of conductor n, it is contained in Fn. The genus field H1 of K is equal to H n F, so H2Fn/Fn is an unramified abelian extension of Fn of degree [H2 : H1] = jr. This degree divides hn by class field theory, so we are done. This lemma provides us with an easy method of constructing infinitely many n for which hn is divisible by an arbitrarily high power of a prime number

250

P. STEVENHAGEN

1. One simply takes those n for which Fn contains a subfield K of degree l over Q for which the Redei-matrix is of rank much smaller than t - 1. By taking it equal to the zero matrix, the following result is obtained. 4. THEOREM. - Let n be divisible by t distinct primes congruent to 1 mod l

such that each of these primes is an l-th power modulo all others. Then divides h, .

It-1

For fixed t, there are infinitely many pairwise coprime n satisfying the hypothesis of the theorem. This follows easily from Dirichlet's theorem on primes in arithmetic progressions. If t -1 primes congruent to 1 mod l have

been chosen such that each is an l-th power modulo the others, the t-th prime that makes the hypothesis of the theorem hold true can be chosen 1)12(t-1)]-1 This follows from an infinite collection of Dirichlet density [(1from the Gebotarev density theorem, as the condition on the t-th prime is that for each of the t -1 previous primes p, it splits completely in the field Ep that is obtained by adjoining an l-th root of unity (l and to the subfield of degree l in the p-th cyclotomic field. The fields Ep are all of degree (l -1)12 over Q, and they are linearly disjoint over Q((l). As a very special case, we obtain the claim made in the introduction that h132 is divisible by 3 for a set of primes p of Dirichlet density 1/18. For odd 1, results similar to those in the preceding theorem have been proved by Cornell and Rosen [[3], 19841 using cohomological methods that

go back to Furuta [[9]]. For t = 2 and t = 3 their results are identical to those following from lemma 3, for large t their method is better. Neither method gives any result for the prime conductor case t = 1. For l = 2, the lemma and the arguments given above have to be adapted for several reasons. First of all one has to take care of the ramification of real primes. This leads one to consider only those quadratic fields K that have real genus fields, i.e. real quadratic fields for which all odd prime divisors of the discriminant are congruent to 1 modulo 4. One only obtains a divisibility result 2r-1 h ,which is weaker than lemma 3. In order to find 2T hn one needs to show that H2 is real. This can sometimes be done using a method of Scholz [[25]]. Secondly, one has to adapt the density computation given above as the fields Ey have smaller degree, a statement equivalent to the quadratic reciprocity law. Rather than working out all details here, we give a characteristic example. It is stronger than the cohomological result in [[3]].

5. THEOREM. - Let p and q be primes congruent to 1 modulo 4 that are mutual quadratic residues, and suppose that the fourth power residue symbols (q) 4 and (P) are equal Then hr9 is even. 4 It is not difficult to see that we obtain the claim in the introduction for q = 13. For n = pq = 5 29 = 145 it follows that hi45 is even. This is one

REDEI-MATRICES AND APPLICATIONS

251

of the two values n < 200 with n not a prime power for which h+ > 1. In fact, one can use Odlyzko's discriminant minorations to show [13111 that h145 = 2.

There are no results of a similar algebraic nature in the prime conductor case t = 1, and we cannot produce infinite families of primes p for which hP is even. See [12811 for a more complete discussion.

A second application of the technique of Redei matrices arises in the study of the solvability in integers of the negative Pell equation x2 - Dy2 = -1, where D > 1 is a squarefree integer. With ED a fundamental unit in Q(om) and N the norm to Q one has x2

- Dye = -1 is solvable in integers e= NED = -1.

Indeed, if the equation is solvable there are units of norm -1, so the fundamental unit cannot have norm + 1. Conversely, if NED = -1 it may be that ED is not in Z[/], but as its cube ED always is we still get an integral solution to the equation. As it is more natural to work with discriminants

than radicands, we will further take D to be a quadratic discriminant and say that the negative Pell equation is solvable for D if the equation x2 - Dy2 = -4 has integral solutions. If the equation is solvable for D, then D is positive and -1 is a quadratic residue modulo every prime divisor

of D, so D must be in the set D of real quadratic discriminants that are not divisible by any prime congruent to 3 mod 4. A question that has been studied by many people but that is still completely open is the following. 6. PROBLEM. - Let D(-1) C D be the set of real quadratic discriminants for which the negative Pell equation is solvable. Decide whether the limit lim X-

#{D E D(-1) : D < X} 00

#{DED:D<X}

exists and if so, determine it.

This is a very hard problem, and to my knowledge is is not even known whether the liminf and the limsup of this expression are in the open interval (0,1). The relation between the solvability of the negative Pell equation and the previous section is given by the following immediate consequence of class field theory.

7. LEMMA. - The negative Pell equation is solvable for a quadratic discriminant D if and only if the narrow 2-Hilbert class field of Q(%) is real.

P. STEVENHAGEN

252

Proof : both statements are equivalent to the fact that Q(v) is a real field for which the narrow Hilbert class field coincides with the ordinary Hilbert class field.

Let H be the narrow 2-Hilbert class field of K = Q(/D). This is the situation of the preceding section, with l equal to 2. From the lemma, we see that the negative Pell equation is solvable for D if and only Hk is real for all k > 1. The condition that the genus field H1 is real is equivalent to the requirement that D is in D, since H1 is obtained from K by adjoining a square root of (-1)(P-X)/2 for each odd prime divisor p of D. If H = H1 this condition is also sufficient for solvability of the negative Pell equation. 8. LEMMA. - The negative Pell equation is solvable for D E D if the Redei matrix of Q (v/D) has maximal rank

Proof : the condition implies an equality H1 = H2, so H = H1 is real and we are done by the previous lemma. If D E D has t distinct prime divisors, the corresponding Redei matrix R is by the quadratic reciprocity law a symmetric (t x t)-matrix over F2 whose

rows and columns add up to zero. Let R' be the (t - 1) x (t - 1)-minor obtained by leaving out the last row and column from R. If D ranges over the subset Dt of D consisting of those discriminants that have exactly t distinct prime divisors, it is intuitively clear that the corresponding Redei minor R'D behaves like a random symmetric (t - 1) x (t - 1)-matrix over ]F2, i.e. that 1imX-00

#{DEDt:D<X and R'D=S} #{DEDt:D<X}

exists and does not depend on the choice of the symmetric matrix S. The statement is a reformulation of the fact that the vector consisting of the (2) Legendre symbols (P) of an element D = p1p2 ... pt is randomly distributed as a function on Dt. The details of a correct proof are not trivial. Redei's original proof [[22]] proceeds by induction on t, and so does the proof of the rediscovery of the result in [[5]]. There is also an easy way out by adapting the notion of density [[17]].

Once one knows that a discriminant D E Dt gives rise to a random symmetric matrix, one can determine how likely it is that such a matrix is non-singular. We give a slightly more general result for future reference. 9. PROPOSITION. - Let n > 1 be an integer and q a prime power. Then there are An(q) = q(n21)

fl

1
(1 - q-k)

R$DEI-MATRICES AND APPLICATIONS

253

symmetric (n x n) -matrices over the field of q elements IFq that are nonsingular. The number of matrices of arbitrary rank r c {O, 1,... n} equals [T] gAr(q), where 1'r1 q denotes the number of r-dimensional subspaces of a vector space of dimension n over IFq.

Proof : the result for q = 2 occurs in rather cumbersome terminology and with a lengthy proof in 112211. A completely elementary proof by induc-

tion on n can be found in [[131]. In order to see that the statement given there is identical to ours one needs the explicit value 1(q` - 1)

fn

flr 1(qt - 1) flti 1 (qi

Lr] q

The first half of the proposition immediately implies the second half, as symmetric matrices correspond bijectively to symmetric bilinear forms and giving a symmetric bilinear form of rank r on V = IFq is equivalent to giving

a subspace W C V of dimension n - r and a non-degenerate symmetric bilinear form of the factor space V/W. This remark also shows that the numbers An (q) can be computed inductively from the relation n

rnl

r-o r q

AT(q) = q(n21),

so it suffices to check that the given expression satisfies this relation. An elegant way of doing this is given in [[511.

We will only use the preceding proposition for q = 2 and n = t - 1, so we write At_1(2) = 2(2)at with [t/2] at = fJ(1 -

21-2j)

j=1

Set Dt(-1) = Dt n D(-1). We now know that for fixed t, we have a lower density for Dt(-1) in Dt, since the two preceding lemmas imply liminf

#{D E Dt(-1) : D< X}

#{DEDt:D<X} >at>am

00

-2j). j=1

The numerical value a, = .4194224... is already in 1[2211. The density result has been reproved in [[ 1711, [[1 Ifl and [[5]]. The formulation given by

these authors is different from Redei's, as they interpret the Redei matrix as an incidence matrix of a graph on t points.

P. STEVENHAGEN

254

The equations D = Ut>1Dt and D(-1) = Ut>1Dt(-1) make it very plausible that the lower density of D(-1) in V is not smaller than a.. However, I do not know how to prove this. The problem is that each Vt is a subset of zero density in V, and it seems non-trivial to prove a density result for D from a density result for each of the subsets Dt. The preceding argument can be further refined in order to obtain a still higher value of the lower density of Dt (-1) in Dt. In particular, we will push the limit value for t -* oo over the value 1/2 that has been suggested as a possible value [[ 17]]. However, we need to pass to the next higher level, i.e. the field H3, in order to do this.

4. - Higher levels In principle, the Redei-Frohlich method for determining r2 (C) can be extended to determine inductively all values rk (C). Having defined the first Redei map

R = R1 : Ft --> C[v - 1] -- C/Co-1 -4 Fit, one can repeat the procedure and consider the higher Redei maps Rk :

kerRk_1 -> C[Q -1] n

C(o_1)k

Just as in the case k = 1, we obtain the (o -

1

can

C(0-1)k-1

/C(o-1)k.

1)k+1-rank from this map by

rk+1(C) = dimF, [ker Rk] - 1 = rk (C) - rankF, Rk,

which is the analogue of (2). Despite the close analogy, a serious complication arises for these higher levels. For k = 1, we were able to embed c(a-1)k-1 /C(a-1)k in a canonical way in a vector space of dimension t over ]Fl. This was due to the fact that we could describe the genus field H1 very explicitly in terms of Dirichlet characters. The fields Hk for k > 2 are no longer abelian over Q, and no general method is known to describe them explicitly. This is a serious drawback that accounts for the fact that there is no generalisation of theorem 1 to higher levels that is of a comparable simplicity. For the same reason, we do not have general density results for these levels that resemble those in the preceding section.

Only in the special case where k = 1 = 2, there is a more explicit version of the theory that goes back to Redei [[22]] and was further developed by Frohlich [[7]]. We can formulate it in modern terms as follows.

Let D be a quadratic discriminant, and D = jlt=i dg its factorization into prime power discriminants. The set V of discriminantal divisors of D is defined as the set of divisors d of D of the form d = llt=1 d7 with ei E 10, 1}. This is in a natural way a vector space of dimension t over F2 with a canonical basis consisting of the divisors d1. The natural surjection

REDEI-MATRICES AND APPLICATIONS

255

V -> C[2] maps dti to the class of the ramified prime ai of K that divides dti. As the genus field H1 of K = Q(v/D) is generated over Q by the square roots dti, Kummer theory tells us that the Galois group Gal(Hi/Q) can be seen as the dual space V* = Hom(V, IF2) of V. The kernel of the Redei map R1 : V -- V * consists of divisors d E V for which the associated Artin symbol oa E Gal(H/K) is the identity on H1. The kernel of the dual map Ri : V = V** -> V* consists of those d E V for which Vd- is left invariant by the Artin symbols of all ideals that have order 2 in the class group. Note that D itself is always in this kernel. A decomposition D = Dl D2 with D1 E ker Ri is called a decomposition of the second kind. These decompositions are characterized by the fact D1 is a square modulo all prime divisors of D2 and vice versa. The prime 2

needs special attention here. Given a decomposition D = Dl D2 that is of the second kind, Redei explicitly constructs a quadratic extension of Q( Dl D2) that is cyclic of degree 4 and unramified over K. This is possible since the equation x2 - D1y2 - D2 Z2 = 0 has non-zero rational solutions by Legendre's theorem and the assumption on the decomposition. For a primitive integral solution (x, y, z) with well-chosen 2-adic behavior, the extension E that is generated over K by a square root yD, of x + y D has the desired properties. The extension E/K depends on the choice of the solution, but the quadratic extension EH1 = H1(ryD1) of H1 does not. Every element v E Gal(H2/Hi) is determined by its action on the elements 'D, for D1 E kerRi, so we can view this Galois group as a subspace of Hom(kerRi,IF2) _ (kerRi)*. With these identifications, we can describe the second Redei map : R2 :

kerRi -> C[2] fl C2 -> C2/C4

Gal(H2/Hi) C (kerRi)*

explicitly as an F2-linear map between vector spaces of dimension 1+r2(C)The 8-rank r3(C) of the narrow class group of K is given by the formula r3 (C) = r2 (C) - rankF, R2,

which is non-negative as R2 is always singular. As soon as one chooses a basis for ker R1 and for the space ker R*1

of decompositions of the second kind, R2 is given by a matrix whose entries describe the action of the Artin symbols va coming from d E ker R1

on explicit elements 7d' with d' E ker Ri . Note the equality R1 = Ri in case D is in the set D of discriminants that are of interest for the negative Pell equation. The entries of R2 are quadratic symbols of quadratic

irrationals and can be computed rather easily. Redet's paper [[2211 has numerous identities that express these `new number theoretic symbols', as he calls them, in rational terms, and FrOhlich 11811 does the same in a

P. STEVENHAGEN

256

more systematic way. However, these expressions are usually given in terms of the chosen solution of x2 - D1y2 - D2 Z2 = 0, and this makes it difficult to obtain density results in terms of the prime divisors of D. Special cases have been dealt with by Morton [[141-[1611, and density statements for the

behavior of C/C8 have been proved by the author [[26]] in the case that t - 1 prime divisors of the discriminant are fixed and the last one varies. For t = 2 this yields results that had been known for some time. In the previous section, we showed that the negative Pell equation is solvable

for all D in a subset of density at of Dt since these D have r2 (C) = 0. Following an idea that goes back to Redei [[21]] and Scholz [[25]], we can use the 8-rank theory to enlarge this set even further by looking at those D that have r2 (C) = 1. We will indicate briefly how this is done.

The density Qt of the set of D E Dt having r2 (C) = 1 follows easily from proposition 9. One has ,Qt = at if t is even and pt = (1 - 21-t)at if t is odd. Note that limt_,00)3t = limt_,00 at = a... For D as above, there is exactly one non-trivial decomposition D = D1D2, and one can show that the higher Redei matrix R2 equals R2

-

D2 4 \ (D2) 4

(D I /4

which has to be interpreted in the obvious way as a matrix over ]F2. As the biquadratic residue symbols have value f1, there are 4 possible values for this matrix, and they each occur for a set of D that has density .114,Qt in Dt. If (D )4 and (D )4 are both equal to 1, the matrix R2 is the zero matrix and H2 is strictly smaller than the 2-Hilbert class field H. In all other cases,

its rank is one and H = H2. We can determine whether H2 is real by a generalization of the argument used in proving theorem 5. One has :

H2isreal

\Da)4=1. (:;-)

It follows that H = H2 is totally complex and the Pell equation is not solvable if the biquadratic residue symbols are not equal, which happens for a collection of discriminants of density,Qt/2 in Dt. If both symbols equal

-1, the Pell equation is solvable, and this happens for a set of density Qt/4. Taking together the two collections of D for which the Pell equation is solvable, we conclude that for each fixed t, the set Dt (-1) has lower density at + 1)3t and upper density 1 - 2 Qt inside V. For increasing t these values rapidly converge to : s 4

a,, = .52428.. .

1 - 2 a = .79029.. .

REDEI-MATRICES AND APPLICATIONS

257

It remains a challenging problem to deduce any non-trivial density result for D(-1) in D. Hardly anything is known about the distribution of the ranks rk(C) when

k > 4. Some numerical data are available in the quadratic case 1121, [27]], mainly for cyclic C. The best known example is probably that of the quadratic field Q(v/=p), with p a prime congruent to 1 mod 4. In this case

the discriminant D = -4p has two prime divisors and C is a non-trivial cyclic 2-group. It follows from theorem 1 that the order of C is divisible by 4 exactly when p - mod8, and the 8-rank results quoted above imply that the order is divisible by 8 if and only if p splits completely in Q((s, V/-1-+-i),

which is a non-abelian field of degree 8 over Q. All numerical evidence suggests strongly that the order of C is divisible by 16 for a set of primes of density 1/16, but the existing techniques do not even suffice to show that this happens infinitely often. The question is closely related to the 2-adic behavior of the fundamental unit e, in the field Q(J), see [[27]].

Note added in proof It is now conjectured that the limit value in problem 6 exists and equals 1 - a,, = .5805775582..., with a,,, as in section 3, see [[29]]. The heuristics

can be extended to the case of quadratic orders [[30]]. They have been confirmed by extensive computer calculations [[1]].

Manuscrit recu le 7 septembre 1994

P. STEVENHAGEN

258

References [1] W. BosMA, P. STEVENHAGEN. - Density computations for real quadratic

units, preprint (1994). [2] H. COHN, J.C. LAGARIAS. - On the existence of fields governing the 2-

invariants of the class group of Q(Vl p) as p varies, Math. Comp. 41, 711-730 (1983). [3] G. CORNELL, M.I. RosEN. - The $-rank of the real class group of cyclotomic

fields, Compositio Math. 53, 133-141 (1984). [5] J. E. CREMONA, R.W.K. ODONI. - Some density results for negative Pell

equations; an application of graph theory, J. London Math. Soc. (2) 39, 16-28 (1989). [6] A. FROHLICH. - The generalization of a theorem of L. Redei's, Quart. J. Math. Oxford (2) 5, 130-140 (1954). [7] A. FROHLICH. - On fields of class two, Proc. Lond. Math. Soc. (3) 4, 235-256 (1954). [8] A. FROHLICH. - A prime decomposition symbol for certain non Abelian numberfields, Acta Sci. Math. 21, 229-246 (1960). [9] Y. FURUTA. - On class field towers and the rank of ideal class groups, Nagoya Math. J. 48, 147-157 (1972).

[10] G. GRAs. - Sur les 1-classes d'ideaux dans les extensions cycliques relatives de degre premier 1, Ann. Inst. Fourier, Grenoble 23,3, 1-48 (1973).

[111 J. HURRELBRINK. - On the norm of the fundamental unit, preprint, Louisiana State University (1990). [12] E. INABA. - Uber die Struktur der i-Klassengruppe zyklischer Zahlkorper vom Primzahigrad 1, J. Fac. Sci. Imp. Univ. Tokyo, section I, vol. W 2, 61-115 (1940). [13] J. MACWILLIAMS. - Orthogonal matrices over finite fields, Amer. Math. Monthly 76, 152-164 (1969). [14] P. MORTON. - Density results for the 2-classgroups of imaginary quadratic fields, J. reine angew. Math. 332, 156-187 (1982).

[15] P. MORTON. - Density results for the 2-classgroups and fundamental units of real quadratic fields, Studia Scientiarum Math. Hungarica 17, 21-43 (1982). [16] P. MORTON. - The quadratic number fields with cyclic 2-class groups, Pac. J. Math. 108, 165-175 (1983).

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259

[17] R. V. PERLis. - On the density of fields with N(e) _ -1, preprint, Louisiana State University (1990). [18] L. REDEI, H. REICHARDT. - Die Anzahl der durch 4 teilbaren Invarianten

der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. reine angew. Math. 170, 69-74 (1934). 1191 L. REDE!. - Arithmetischer Beweis des Satzes fiber die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengn.ippe im quadratischen Zahlkorper, J. refine angew. Math. 171, 55-60 (1935).

[20] L. REDEI. - Uber die Grundeinheit and die durch 8 teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkorper, J. reine angew. Math. 171, 131-148 (1935). [211 L. REDEI. - Uber einige Mittelwertfragen im quadratischen Z iilkorper, J. reine angew. Math. 174, 131-148 (1936). [22] L. REDE!. - Ein neues zahlentheoretisches Symbol mitAnwendungen auf die Theorie der quadratischen Zahikorper, J. reine angew. Math. 180, 143 (1939).

[231 L. REDEI. - Bedingtes Artinsches Symbol mit Anwendungen in der Klassenkorpertheorie, Acta Math. Acad. Sci. Hung. 4, 1-29 (1953). [241 L. REDEI. - Die 2-Ringklassengruppe des quadratischen Zahlkorpers and die Theorie derPellschen Gleichung, Acta Math. Acad. Sci. Hung. 4, 3187 (1953). [251 A. SCHOLZ. - Uber die Losbarkeit der Gleichung t2 - Due = -4, Math. Zeitschrift 39 [261 P. STEVENHAGEN. - Class groups and governing fields, Publ. Math. Fac.

Sci. Besancon, annee 1989/90, 1-94 (1990). [27] P. STEVENHAGEN. - On the 2-power divisibility of certain quadratic class

numbers, J. Number Theory 43 (1), 1-19 (1993). 1281 P. STEVENHAGEN. - Class number parity for the p-th cyclotomic field, Math. Comp. 63 no. 208 (to appear, 1994). [291 P. STEVENHAGEN. - The number of real quadratic fields having units of negative norm, Exp. Math. 2 (2), 121-136 (1993). [301 P. STEVENHAGEN. - Frobenius distributions for real quadratic orders, J. Theorie des Nombres Bordeaux (to appear, 1995). [31] F.VAN DER LINDEN. - Class number computations of real abelian number fields, Math. Comp. 39, 693-707 (1982). Peter Stevenhagen Faculteit Wiskunde en Informatica Plantage Muidergracht 24 1018 TV Amsterdam, Netherlands e-mail : psh@fwi . uva. n1

Number Theory Paris 1992-93

Decomposition of the integers as a direct sum of two subsets R. Tijdeman

1. - Introduction Two subsets A and B of a set C induce a decomposition of C if every

element of C has a unique representation a + b with a E A, b E B. Notation : C = A T B. We call A and B complementing C-pairs. A first study of such pairs arose in the forties from Hajos' proof of Minkowski's conjecture on systems of linear inequalities. Hajos reduced this conjecture to an equivalent statement on decompositions of finite abelian groups, which he was able to prove. A survey of the work on decompositions of finite abelian groups is given in Section 2. The question of characterising all complementing Z-pairs seems first to have been stated by de Bruijn in 1950. De Bruijn came to the problem while he studied bases for the integers. Let A be a finite set of integers including 0. A set of integers {bl, b2,. ..I is called an A-base whenever any integer x can be expressed uniquely in the form 00

x=

00

Eibi

i=1

(Ei E A,

IEiI < oo). i=1

if it can be rearranged in the form h2d3.... } where h denotes the cardinality of A and dl, d2, d3, .. .

An A-base is called simple {dl, hd2,

are integers. De Bruijn [21 considered the special case where the elements of A have no common factor and where h is a prime. He conjectured that under these assumptions A ® B = Z implies that B is the set of multiples of h. He remarked that a proof of his conjecture would imply that every A-base is simple. For later work on A-bases we refer to de Bruijn [61, Long and Woo [161, Swenson and Long [281. In 1974, Swenson 1271 showed that there is no effective characterisation of all complementing i-pairs. More precisely, he showed that any two finite

sets of integers A, B with the property that all sums a + b (a E A, b E B)

R. TIJDEMAN

262

are distinct, can be extended to two infinite complementing Z-pairs. For a similar construction, see Post [20]. In contrast, there is a particularly nice characterisation of all complementing Z>o-pairs. The result, which was implicit in the work of de Bruijn [5], was rediscovered by Vaidya [30]. It is obvious that A fl B = {0} and 1 E A U B. Suppose 1 E A. Then A and B are infinite complementing Z>opairs if and only if there exists an infinite sequence of integers {mi}i>1 with mi > 2 for all i, such that A and B are the sets of all finite sums of the form 00

CK)

a = E x2iM2i, b = E x2i+111'I2i+1 i-o

i-o

i

where 0<xi<mi+1fori>0andMO=1andMi= flmj fori> 1.If j=1

A or B is a finite set, a similar characterisation holds with the change that the sequence {mi} will be of finite length r and the only restriction on xr is that it be nonnegative. C.T. Long [ 151 gave a corresponding characterisation of all complementing C-pairs in case C = {0, 1, ... , n - 1}. He also showed

that in this case the number C(n) of complementing C-pairs is the same as the number of ordered nontrivial factorisations of n. The number C(n) is determined by

2C(n) = E C(d)

(n E Z>1).

din

Hansen [ 14] and Niven [ 181 generalised these results to a characterisation of the complementing pairs of the set Z>o x Z>o. Long [ 151 made the interesting

observation that it follows from the above characterisation that if A and B are infinite sets such that A ®B = Z>o then A ®(-B) = Z (see also Brown [1]). In particular, we can take for A the set of finite sums of odd powers of 2 and for B the set of finite sums of even powers of 2. Eigen and Hajian [31] showed that if A and B are infinite sets such that A ®B = Z>o, then there exists a continuum number of sets b such that A ® B = Z.

In Section 3 we formulate a conjecture which, if true, provides an inductive characterisation of all complementing sets A, B for which the cardinality of A is fixed integer n. It was already observed by Hajos [12] and de Bruijn [5] p. 240 that B is periodic if A is finite. In this way the problem is reduced to a finite problem which can be stated in terms of finite cyclic groups. If n is a prime number, then our conjecture coincides with de Bruijn's one stated above. This conjecture was proved by Sands [24] in 1957. We shall show how a proof of our conjecture can be derived from Sands' results if n is a prime power. The general case remains open.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

263

We shall further show by a combinatorial argument that if m is coprime to

n and A ®B = Z, then mA ®B = Z (we define mA = {mala E A}). On using this result we give an alternative proof of de Bruijn's conjecture.

The problem of characterising all sets B such that A $ B = Z in the special case where A consists of the finite sums of odd powers of 2 was posed to me by Yu. Ito. He was interested in the problem because of his joint research with S. Eigen and A. Hajian on exhaustive weakly wandering sequences for ergodic measure preserving transformations [71, [91, [8]. Such a characterisation will be presented in Section 4.

2. - Decomposition of finite abelian groups About one century ago Minkowski [ 171 proved the following fundamental

result on the geometry of numbers : Let h1i ... , n be homogeneous linear forms in the variables x1, ... , xn with real coefficients and determinant 1. Then there exist integers x1,. .. , xn, not all zero, such that Ill

(1)

1,...,IU

1.

Since L;1i ... , n may have integer coefficients, the equality signs cannot be deleted. Minkowski conjectured that (1) can be replaced by IS1I < 1,...,ISnI < 1

(2)

unless at least one of the linear forms has integer coefficients. Minkowski proved the statement for n < 3. Several mathematicians worked on it and in 1940 it was known to be true for n < 9. In 1941 Hajos [11] established Minkowski's conjecture in the affirmative. His proof consists of three parts : (1) reduction to some equivalent geometric statement on k-multiple lattice tiling of the unit cube, (ii) further reduction to the equivalent group theoretic statement given below.

(iii) proof of this group theoretic statement.

Hajos' theorem is very fundamental and has various aspects. Fary [101 reformulated it as a result on the structure of commutative compact topological groups. Now we state Hajos' result in terms of group theory. Let G be a finite abelian group with unit element 1. A subset of G is called a simplex if it is of the form {1, a, a2, ... , ae-1 } where a E G has order > e. Notation [a]e, or briefly [a]. It is clear that [a] is a subgroup of G if and only if a has order e. We say that G is the free product of the sets Al, ... , A. if every element

of G has a unique representation a1 Hajos' theorem reads as follows.

an with aj E A; for j = 1, .

.

.

, n.

R. TIJDEMAN

264

If G is the free product of n simplices, then at least one of the simplices is

a subgroup of G. Hajos' proof has been simplified by Redei [22] and Szele [29]. Szele [29] p. 57 conjectured that Hajos' theorem would hold true for any decomposition of the finite abelian group G. A simple example (cf. [ 131 p. 185) shows

that this is false. Let G be the cyclic group defined by a8 = 1 and let A = {1, a2}, B = {1, a, a4, a5}. Then none of A and B are subgroups of G whereas G is the free product of A and B. Note, however, that a4B = B. A subset A of G is said to be periodic whenever there exists an element g E G, g 1, such that gA = A. De Bruijn [2] conjectured that if G is a finite abelian group of order > 1 and G is the free product of the sets A and B, then A or B is periodic. He observed that the assertion is not true if G is the infinite cyclic group generated by g. Szele (cf. [ 131 p. 185) made

the same observation. He took for A the product of the subsets {1,g2}, { 1, g32}.... and for B the product of the subsets 11, g-1 }, 11, g-4 }, {1, g-16}, .... Here G is the free product of A and B and none of A and B is periodic. { 1, g8 } ,

Some years earlier, however, Redei [21] had published two examples of Hajos which show that de Bruijn's conjecture is false. The simplest example refers to the abelian group generated by the elements a, b, c of orders 4, 4, 2 respectively. This group is the free product of the nonperiodic sets {1, a}, {1, b} and {1, a2, ab2, a3b2, c, a2bc, a2b3c, b2c}.

Later, Hajos [ 131 showed that any finite cyclic group the order of which is the product of three pairwise relatively prime numbers > 1, two of which are composite, can be represented as the free product of two nonperiodic subsets. He gave an explicit example of order 180 = 9 x 4 x 5, which is the smallest number satisfying the conditions. Let us follow de Bruijn in calling a group good if any factorisation of G as a free product of A and B implies that A or B is periodic and otherwise bad. De Bruijn [3] extended Hajos' result by showing that if n = dld2d3 with (d1, d2) = 1, d3 > 1 and both d1 and d2 are composite numbers, then the cyclic group of order n is bad. The smallest order of this type is 72. De Bruijn [4] gave the explicit example (g72 = 1) g18 926 g34

90 98, 916 B : g12, 917 918 921 924 941 945 948 954 960 965, 969 A :

,

On the other hand, cyclic groups of the following orders have been proved to be good (p, q, r, s are distinct primes) : p' ' (A > 1) (Hajos [12]), pq, pqr (Redei [231), p"q (A > 1) (De Bruijn [41), p2g2, p2qr and pqrs (Sands [241).

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

265

This covers all cyclic groups. Already in 1947 Redei [211 had shown that the non-cyclic group of order p2 is good. The problem was completely solved by Sands [25, 261 who determined all good finite abelian groups. Sands [241

further proved that if the finite cyclic group G is the free product of the subsets A and B and the cardinality of A is a prime power, then either A or B is periodic. This had been conjectured by de Bruijn ([3] p. 371) for the case that the number of elements of A is prime. Hajos 1 121 proposed the question whether every decomposition of a finite abelian group G is quasiperiodic. A factorisation of G as free product of A and B is called quasiperiodic if either A or B, B say, can be split into a number of parts B1, B2, ... , B,,,, (m > 1) such that ABi = giAB1 (i = 1,. .. , m) where the elements gl,... , gn form a subgroup of G. De Bruijn's example is quasiperiodic as we can take B1 = {g12 917 918 924 941 965} and gi = 1, 92 = g36. Hajos' example is quasiperiodic as we can take A = {1, a, b, ab}, B1 = {1, a2, ab2, a3b2}, B2 = {c, a2bc, a2b3c, b2c} and gi = 1, g2 = c.

De Bruijn [4] obtained some partial result on Hajos' question.

3. - A is finite Suppose A E B = Z where A is finite. Let A = {ao, a1,. .. , an_i }. Since,

for any integer x, we have (A - x) E B = Z, we may assume without loss of generality that 0 = ao < al < ... < an_i. If x = a + b with a E A, b E B, put (x)A = a, (x)B = b. The following result of Hajos [121 and de Bruijn 121 p. 240 reduces the problem of characterising all complementing Z-pairs to a problem on finite sets which can be stated in terms of finite cyclic groups. LEMMA 1. - The sequence {(x)A}XEZ is periodic. If the period length is L

then n divides L and B is periodic mod L.

Proof : put M = an_i. Consider the nM + 1 vectors ((i)A, (i + 1)A, ... , (i + M - 1)A) for i = 0,1, ... nM.

By the box principle at least two vectors are equal, for i = s and i = t with

s < t, say. Hence (x)A = (x+t-S)A forx = s,s+1,...,s+M- 1. Suppose k is the smallest integer with k > s + M and (k)A

(k + t - $)A

-

If (k)A # 0, then put k = a + b with a E A, b E B. We infer that (b)A = 0 and s < k - M < b < k. Hence (b + t - s)A = (b)A = 0 which implies b + t - s E B. Since k + t - s = a+ (b + t - s), we obtain (k + t - S)A = a = (k)A, a contradiction. If (k + t - s)A 0, a similar argument yields a contradiction. Thus (x)A = (x + t - s)A for all x > s. By

R. TIJDEMAN

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symmetry we also have (x)A = (x+t-s)A for all x < s. Let the period length of {(x)A}xEZ be L. Since Z = U o 'jai + B}, all ai have the same density in the sequence {(x)A}xEZ. Therefore, they occur with the same frequency in one period. This implies n1 L. Since the elements of B are precisely the integers x with (x)A = 0, we have that B is periodic mod L. 0 By simple transformations each complementing Z-pair A, B with A

finite can be reduced to the standard situation that A is represented by < an_1 < L with gcd(ao, al, ... , an-1) = 1 and B is ao = 0 < a1 < a2 < < bn,._ 1 < L. Here L = nm. represented by U'- 1(bi +ZL) with 0 < b1 < Namely, 0 = a + b for some a E A, b E B. By taking A - a in place of A and B + a in place of B we have 0 E A n B. If gcd(ao, a1, ... , an_1) = d > 1, then 7G=

(3)

A

®Bjd

j forj=0,1,...,d-1

where Bj = {b E Bib - j(modd)}. The elements of A/d are coprime and we have obtained d complementing Z-pairs with coprime a's. It is obvious that B can be represented as indicated. Without any trouble we can add or subtract multiples of L from the elements of A to obtain the required structure. The problem is now reduced to the decomposition problem for the cyclic group of residue classes mod L (where we only know that L is a multiple of the cardinality of A).

It will be clear from the previous sections that a characterisation of all complementing i-pairs is not a simple matter, even if we assume one of the subsets to be finite. However, in the latter case, a kind of inductive characterisation would be possible, if we could prove the following

statement. CONJECTURE. - If A®B = Z, 0 E An B, gcdaEA a = 1 and A has exactly

n elements, then there exists a prime factor p of n such that all elements of B are divisible by p.

Suppose the statement is true. Then the elements of A are equally distributed among the residue classes mod p. We can make a splitting in

p complementing i-pairs as indicated in (3), with d = p and A and B interchanged. So the problem is reduced to the decomposition problem for

the cyclic group of residue classes mod(L/p) and the procedure can be repeated. The following examples show that p is not determined by L and n.

L=12, n=6, A=10,1,4,5,8,91, B = {0, 2(mod 12)}, p=2; L=12, n=6, A=10,1,2,6,7,81, B = {0, 3(mod l2)}, p=3.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

267

If the conjecture is true, then every such decomposition is quasiperiodic

in accordance with Hajbs' conjecture stated at the end of the previous section. For we can split A into residue classes mod p, Ao, &... , A,-1, say, and Ai +B = pZ+i for i = 0, 1, ... , p-1. If the number n of elements of A is a prime, then the conjecture implies that A represents a complete residue system mod p and every element of B is divisible by p. This is precisely the conjecture of de Bruijn stated in the introduction. A proof of this conjecture can be obtained by combining results of de Bruijn and Sands. De Bruijn ([2)), p. 241) provided an argument which implies that his conjecture is true if the following statement is true : if the finite cyclic group G is the

free product of the subsets A and B and the cardinality of A is prime, then either A or B is periodic. As remarked in the previous section, the latter statement was proved by Sands. I shall present a completely different proof of de Bruijn's conjecture (Theorem 2). Subsequently I shall extend

de Bruijn's argument to the case where the cardinality of A is a prime power. By combining this with Sands' general result we obtain a proof of my conjecture, stated above, in case n is a prime power (Theorem 3). We start with a result without any restriction on n. THEOREM 1. - Let A ®B = Z with 0 E A fl B and cardinality n of A finite.

Then, for any integer h with gcd(h, n) = 1, we have hA ® B = Z.

We need some lemmas. Let A= {ao, a1, ...,an-, } with ao = 0. LEMMA 2. - For any integer x

{(x+ao)A,(x+al)A,...,(x+an_1)A}=A, {(x-ao)A,(x-a1)A,...,(x-an_1)A} =A. Proof : suppose (x + ai) A = (x + aj) A. Then x + ai - Q1 = x + aj - /32 for some 01, 32 E B. Hence ai + 32 = aj + ,31. Since such a representation is unique, we have i = j. The proof of the second statement is similar. 0 LEMMA 3. - Let q be a prime power with gcd(n, q) = 1. Then, for any integer x, {(x + gao)A, (x + ga1)A, ... , (x + qan-1)A} = A.

Proof : let q = pk, p prime. Put D = {(a, a, ... , a) E Agla E A}. Define ByLemma2 we have

f : A9 ->Aby(a1ia2i...,aq)'--1 n-1

U f (a1, a2, ... , aq-1, aj) = A. j=0

R. TIJDEMAN

268

Hence f (a) (a E A9) assumes each element of A exactly nq-1 times. Note that f (al, a2, ... , aq) does not change value if we permute al, a2, ... , aq. If (al, a2.... , aq) contains entry aj exactly h times (j = 0, 1, ... , n - 1), then it has precisely q!

lo! ii! .. In-I! permutations in Aq. This multinomial coefficient is divisible by p, unless all but one lo, equal zero, that is (al, a 2 ,--. , aq) E D. It follows that

f assumes on Aq\D each value of A a number of times which is divisible by p. Since p-n9-1, we infer that f assumes each value of A on D. Thus f (D) = A. 0

LEMMA 4. - Let q = -1 or a prime power with gcd(n, q) = 1. Then

qA®B=Z. Proof : we first show that all numbers {qa + b}aEA, bEB are distinct.

Suppose al, a2 E A, 31, /32 E B are such that qal +,31 = qa2 + 02. If q = -1, then the assertion follows from a2 + 01 = al + /j2. Otherwise qai - /j2 = qa2 - /3i. This number has a unique representation a +,3 with a c A,,3 E B. It follows that -,0 + qa1 = a + /32, -0 + qa2 = a +,31. Hence (-,3+gal)A = (-/3+ga2)A. By Lemma 3 we obtain al = a2 and therefore 13i =/32 By Lemma 1, B is periodic mod mn for some positive integer m. Hence B consists of m residue classes mod mn. Let {bo, b1, ... , bm_1 } be a set of representatives. Since by the first paragraph all numbers qal + bj (i =

0, 1,...,n- 1, j = 0,1,...,M- 1) are in distinct residue classes modmn, these mn numbers represent a complete residue system mod mn. Hence qAED B=Z. 0

Proof of Theorem i : since h can be written as the product of prime powers and factors -1, each coprime to n, we reach the conclusion by repeated application of Lemma 4.

0

COROLLARY. - Let h be an integer with gcd(h, n) = 1. Then h(aI - a2) _ 01 -,32 for some al, a2 E A, ,Ol, /j2 E B implies al = a2, 01 = Q2.

Proof : we have hal = (hal + /32)hA = (hat + /31)hA = hat.

0

Subsequently we show how a proof of de Bruijn's conjecture can be derived from Theorem 1.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

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ThEOREM 2. - Let A®B = Z with 0 E AnB, the elements of A are coprime

and the cardinality n of A is prime. Then every element of B is divisible by n. Proof : since B is periodic mod mn for some m, we assume without loss of generality that A consists of the nonnegative integers ao = 0, a1, ... , an_1 and that bo, . . . , b,,,,_1 are integers with 0 < bo < . . . < b.,,,,_1 < mn such that B = U oi(b; + mnZ). Put Bo(z) = 1 + zb1 + zb2 + + zb^-1 and (4)

B(z) _ E zb = BO(z) (1 + zmn + z2mn + ...) =

Bo(Z) nen

bEB b>O

Note that every pole of B is an mn-th root of unity. We shall show that it is an n-th root of unity. Set Ah(z) = 1 + zhal + Zha2 + + zhan 1 . Then for every h > 0 with gcd(h, n) = 1 we have, by Theorem 1, 00

Ah(z)B(z) = > zk - Ph(z) =

(5)

1 1

k=0

z

- Ph(z)

where Ph(z) E Z[z]. Hence every pole # 1 of B is a zero of Ah. Let ( be a pole of B with (# 1. Put Sk = E o (k-; for k E Z. Since Ah(() = 0 whenever gcd(h, n) = 1, we have sh = 0 whenever gcd(h, n) = 1. By the theorem on elementary symmetric functions (formulae of Newton-Girard) we obtain, since n is prime,

IIn-1 j-o (z-(a') =zn+cn where cn is some constant. Since ao = 0, we have cn = -1. Therefore, is the complete set of n-th roots of unity. Since (ao = 1 (a1, ... , the a's are coprime, there exist integers to, ti, ... , tn_1 such that 1 = toao + tiai + + tn_lan_1. Hence, putting (i = e2nti/n ( = ((ao)to ((a1)tl ... ((an-1)tn-1 = (t for some t E Z. (an-1

Thus (is an nth root of unity. From (4) we see that every pole of B is simple and that (zmn -1) /(zn -1) divides Bo(z). This implies, by the choice of the b's, Bo(z) = (1 + zn + z2n

+... + z(,,,.-i)n)

(1 + fiZ+... + fn_izn-1)

for some coefficients fi, ... , fn-1. Since Bo has only m nonzero coefficients, = fn-1 = 0. Thus Bo(z) = (1 - zmn)/(1 Zn ) and we see that fi =

-

B(z) = (1 - zn)-1 = 1 + zn + Z2n + multiples of n.

, in other words, B consists of the 0

We use Sands' result on finite cyclic groups [241 to obtain the following generalisation of Theorem 2.

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270

THEOREM 3. - Let A ® B = Z With 0 E A fl B, gcdaEA a = 1 and A has exactly pt elements with p prime, t E Z>1. Then all elements of B are divisible by p.

Proof : let n = pt. By Lemma 1, B is periodic. Let L be its minimal period. If G denotes the group of residue classes mod L, then Z = A ® B furnishes a decomposition G = A* ® B* where A* and B* consist of the residue classes mod L determined by the elements of A and B. respectively. Note that L is divisible by pt.

For t = 1 the statement is true by Theorem 2. So suppose t > 1. We apply induction on t. It follows from Theorem 2 of Sands [24] that A* or B* is periodic. B* cannot be periodic because of the minimality of L, so it has to be A*. Note that the elements g with g + A* = A* form a subgroup Go of G. We shall show that Go contains the residue class Ll p (mod L). If not, Go contains L/q for some prime q # p. Then a E A* if and only if a+ vL/q E A* for all v E Z. Hence A* splits into subsets of size q. Since A* has pt elements, this is impossible. Thus A* is periodic mod L/p. Let A** and B** consist of the residue classes mod L/p determined by the elements of A and B, respectively. By the previous paragraph A** has pt-i elements and A** ® B** = Z/(L/p)Z. Put r = pt-1. Let A = {ao = 0, al, ... , ar_1} be a set of integers representing A**. Since gcdaEA a = 1

and every element of A is of the formal + wL/p, there exists integers + Vr_lar_1 + vrL/p = 1. vo, Vl, ... , Vr_1i yr such that voao + v1a1 + W e infer that the greatest common divisor d of do, a1, ... , aris coprime to L/p. Let h be the inverse of d mod(L/p). Then 1_

1_

1_

L

hao, hat, ... , har_1 = dao, dal, ... dar_1 (mod -) . P

Therefore, by Theorem 1 , d A = G do ao = 0, dal , ... , ar_ 1 } is a set of pt-I relatively prime numbers with d dA ® B** = hA ® B** = Z. Hence, by the induction hypothesis, all elements of B** are divisible by p. Since B** = B + (L/p)Z and p is a divisor of L/p, all elements of B are divisible

by p

0

4. - A is the set of finite sums of distinct odd powers of 2 We use the following notation. If n = f Ek b32 with b; E {0, 1} is the binary notation of n, then we write n = ±bkbk_1 ... bo. We say that bk is the first bit and bo the last. The bit b; is said to be at place j, for j = 0, 1, ... , k. If j is even, then b; is at an even place, otherwise at an odd place. If b; is the last bit 1 in the binary expansion of n, then ord2(n) = j.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

271

Let A be the set of finite sums of distinct odd powers of 2 and A the set of the finite sums of distinct even powers of 2. Yu. Ito asked me to characterise all sets B such that A ® B = Z. Obviously A ®A = Z>o, whence A ® (-A) = Z. THEOREM 4. - The above A satisfies A ® B = Z if and only if B is such

that (i) if b, b' E B with b b', then ord2 (b - b') is even, (ii) the set B is maximal with respect to (i),

(iii) -A c A+ B. The first condition says that there is an even number k such that 2k I b - b', but 2k+1 { b - b'. The second means that B cannot be enlarged without affecting (i). The third condition is equivalent to saying that for

every element a E A there is an a E A such that a+ a E -B. Still another interpretation is that any finite collection of bits at even places can be completed to some nonpositive number in B by inserting suitable bits at odd places, zeros at even places and putting a minus sign in front of the number. Recently it was proved by Eigen, Hajian and Kakutani (32) that if F is a finite set of integers, then F can be extended to a complementary set B of A if and only if (i) holds for F.

Proof : (this proof was shown to me by Yu. Ito. A simpler proof of (i) can be found in S. Eigen, A. Hajian and S. Kakutani (32), Lemma 1). (ii) Suppose b V B and ord2(b-b) is even for every b E B. Since b = a+ b for some a E A, b E B, we have b - b = a (z- A, whence ord2 (b - b) is odd. NO Obvious.

(i) Put An = 22nA and Bn = An ®B. Then the 2n sets n-1 Bn +

ej2 2i+1,

eo, E1,

,Cn_1 E {0,1},

j=0

are disjoint and their union is Z. We claim that Bn + k 22n = Bn for k E Z. For n = 0 it is clear. Suppose the claim is valid for n = m. Then Bm+1 + k 22m C B,n + k

22m

= Bm = Bm.+1 U (Bm+1 +

22m+1).

Since B,n+1 and Bm,+1 + 22m+1 are disjoint sets of the same cardinality, we conclude that adding or subtracting 22m+1 to an element from one set

yields an element from the other set. It follows by induction on III that Bm+i + 1. 22m+1 = Bm+i + 22m+1 for odd 1 and B,,,,+1 + 1. 22m+1 = Bm+1 for even 1. This proves the claim.

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272

Suppose B - B contains an element z with ord2 (z) is odd. Then

b - b' = z = (2k+1)2 21+1 = k . 221+2 +2 21+1 for some b, b' E B and k, 1 E Z. Since B C Bn for all n, we have

b=b'+k 221+2+221+1 E B1+1+221+1 Thus b E B1+, fl (B1+1 + 221+1) but these sets are disjoint.

The proof of the sufficiency part of Theorem 4 requires some lemmas. LEMMA 5. - If(i) holds, then every integer is represented at most once as

a+bwithaEA, bEB. Proof : suppose al + bi = a2 + b2 for some al, a2 E A and bl, b2 E B. Then al - a2 = b2 - bi. However, ord2(al - a2) is odd and ord2(b2 - bl) is even, unless al = a2, bl = b2. 0

LEMMA 6. - Let (i) hold. If b and b' are elements of B with bb' > 0 such that b = ±b2k-lb2k-2 ... bo, b' = ±b2k-lb2k-2 ... bo and b2j = b2i

forj=0,1,...,m-1.Then b3=b.forj=0,1,...,2m-1. Proof : clear.

LEMMA 7. - If (i) and (iii) hold, then every non-positive integer has a representation a + b with a E A and b E B.

Proof : we have 0 E A, whence 0 E A ® B. Consider n E Z 0. Put bo = no. Since bo E A, there exists some nonpositive element of B ending with bo by (iii). Let bi be the bit at place 1 of this element. Define a E {0, 1} such that, in binary notation, bibo - n1no + aiO(mod 4). Next consider n2nino + a10. Let b2 be the bit at place 2 of this sum. Then, by (iii), there is a nonpositive element in B such that the last two bits at even places are b2 and bo. Let b3 be the bit at place 3 of this element. Define a3 E {0,1 } such that b3b2bibo - n3n2nlno+a3OalO(mod 24). By considering n4n3n2n1no+a3OaiO and continuing the procedure, we eventually construct bits b2k+1, b2k,... , bl, bo and a2k+1, a2k-1, ... , a3i al such that (6)

b2k+lb2k ... bibo - -n + a2k+lOa2k-10 ... Oa1O (mod 22k+2)

and b2k+1 is the bit at place 2k + 1 of some nonpositive element b of B with b2k, b2k-2, ... , bo at the last k + 1 even places and zeros at all other even places.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

273

Note that, by Lemma 6, the bit at place 1 of b is bi (apply it for m = 1), the bit at place 3 of 6 is b3 (apply Lemma 6 for m = 2), and so on. Thus b ends with the 2k + 2 bits b2k+lb2k bibo, whence -b can be written as bibo with t E A. Further, observe that on both sides of (6) the numbers are nonnegative and less than 22k+2, by -n < 22k, so that actually in (6) both sides are equal. Put a = n - b. Then, for some t E A, t 22k+2 +b2k+lb2k

a

b

with a E A and b E B.

0

LEMMA 8. - Let B E Z be a set satisfying (i) and (iii) and such that A ® B represents every nonpositive integer, but not 1. Then every element of B - 1 has its last nonzero bit at an even place.

Proof : suppose b is a negative element of B such that the last nonzero bit of b-1 is at an odd place, at place 2m- 1, say. By (iii) there is a nonpositive bo in B with bo = b2 = = b2,,,_2 = 1 and b2k = 0 element b* _ -bt b1_1

for k > m. Since the last bit 1 of b - 1 is at place 2m - 1 we have 2m-1

Hence, by Lemma 6, b* - b(mod

22m+1). Thus

2m-1

It follows that b* -1 has zeros at all even places. Hence a :_ - (b* -1) E A and 1 = a + b* E A ® B. a contradiction. Suppose b is a positive element of B such that the last nonzero bit of b - 1 is at an odd place, at place 2m - 1 say. By (iii) there exists a negative element b' = -b11 b11_1 ... b o in B such that bo = b2 = =b2 m_2 = 1 and b2k = 0 for k > m. Since we have proved in the previous paragraph that the last nonzero bit of b' - 1 is at an even place, we find that bo = bi = _ U b2,,,,_1 = 1. Thus ord2(b - b') = 2m - 1 which contradicts (i).

Proof of the sufficiency part of Theorem 4 : by Lemmas 5 and 7 it remains to prove that every positive integer has a representation a + b with

a E A and b E B. Let n be the smallest positive integer without such a representation. Then n V B, since 0 E A. Put B = B - (n - 1). Then A ® B represents every nonpositive integer, but not 1. By Lemma 8 every element of b - 1 = B - n has its last nonzero bit at an even place. Put B* = B U {n}. Then B* is larger than B and ord2(b - b') is even for every b, b' E B* with b # b'. Thus B is not maximal with respect to (I), in contradiction to (ii). Hence every positive integer is contained in A ® B. 0

274

R. TIJDEMAN

Theorem 4 induces a similar characterisation for A. COROLLARY. - A ® B = Z if and only if B satisfies (i') if b, b' E B with b # b', then ord2 (b - b') is odd, (ii') the set B is maximal with respect to (i'),

(iii') -ACA + B. Proof: note that A=2A andA=2A U(2A+1).

'='. By 2A®2B=2Z, we have A3(2BU(2B+1))=Z. Hence 2B U (2B + 1) satisfies the conditions (i), (ii), (iii) of Theorem 3. It follows immediately, that B satisfies (i'), (ii'), (iii'). / .'. By (i') all elements of B are even or all are odd. In the latter case

we replace B by B + 1. This involves no loss of generality. Let t be the set of numbers of B divided by 2. Then t satisfies conditions (i), (ii), (iii) of Theorem 1. Thus A + B = Z. Hence 2A + 2B = 2Z and

A®B = (2A U(2A+1))®2B = 2A®2B U (2A+1)®2B = 2ZU(2Z+1) = Z. 0

It is obvious that conditions (i) and (ii) of Theorem 3 are not enough to

guarantee A ® B = Z. The set q satisfies (i) and (ii), but A ®A = Z o. Yu. Ito asked for some set of type A for which the complementing sets are characterised by (i) and (ii) only. He wondered whether A' = { (- 1) a/2 ala E

Al is such a set. P. ten Pas [19) showed that there exist sets B' and B" which both satisfy (i) and (ii) such that A' ® B' = Z and A' (D B" =,/: Z. Acknowledgement. I am indebted to Yu. Ito and J. Urbanowicz for useful discussions and to S. Eigen and Yu. Ito for remarks on an earlier version of this paper.

Manuscrit recu le 3 decembre 1993

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References

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of finite type, Tokyo J. Math. 11 (1988), 459-470. [10] I. FARY. - Die Aquivalente des Minkowski-Hajosschen Satzes in der Theorie der topologischen Gruppen, Comm. Math. Hely. 23 (1949), 283-287. [111 G. HAJ6s. - Uber einfache and mehrfache Bedeckung des n-dimensionalen Raumes mit einem Wurfelgitter, Math. Z. 47 (1941), 427-467.

[12] G. HAJ6s. - Sur la factorisation des groupes abelien, Casopis Pest. Mat. Fys. 74 (1950), 157-162. [13] G. HAJ6s. - Sur la probleme de factorisation des groupes cycliques, Acta. Math. Acad. Sci. Hungar. 1 (1950), 189-195. [14] R.T. HANSEN. - Complementing pairs of subsets in the plane, Duke Math. J. 36 (1969), 441-449.

[15] C.T. LONG. - Addition theorems for sets of integers, Pacific J. Math. 23 (1967), 107-112.

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[16] C.T. LONG and N. Woo. - On bases for the set of integers, Duke Math. J. 38 (1971), 583-590. [17] H. MINKOwsKI. - Geometrie der Zahlen, Leipzig, 1896. [18] I. NivEN. - A characterization of complementing sets of pairs of integers, Duke Math. J. 38 (1971), 193-203. [19] P. ten PAS. - Complementing sets for Z (in Dutch), Leiden, 1990. [20] K. Posr. - Problem 71, Nieuw Arch. Wisk. (3) 14 (1966), 274-275. [21] L. REDEI. - Zwei Liickensdtze caber Polynome in endlichen Primkorpern mit

Anwendung auf die endlichen Abelschen Gruppen and die Gaussischen Surnmen, Acta Math. 79 (1947), 273-290. [22] L. REDEI. - Kurzer Beweis des gruppentheoretischen Satzes von Hajbs, Comm. Math. Helv. 23 (1949), 272-282. [23] L. REDEI. - Ein Beitrag zum Problem der Faktorisation von endlichen Abelschen Gruppen, Acta Math. Acad. Sci. Hungar. 1 (1950), 197-207. [24] A.D. SANDS. - On thefactorisation offinite abelian groups, Acta Math. Acad. Sci. Hungar. 8 (1957), 65-86. [25] A.D. SANDS. - The factorisation of abelian groups, Quart. J. Math. Oxford (2) 10 (1959), 81-91. [26] A.D. SANDS. - On the factorisation of finite abelian groups II, Acta Math. Acad. Sci. Hungar. 13 (1962), 153-159. [27] C. SWENSON. - Direct sum subset decompositions of 7G, Pacific J. Math. 53 (1974), 629-633. [28] C. SWENSON and C. LONG. - Necessary and sufficient conditions for simple A-bases, Pacific J. Math. 126 (1987), 379-384.

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39 (1966), 43-44. [31] S. EIGEN and A. HAJIAN. - Sequences of integers and ergodic transformations,

Advances in Mathematics 73 (1989), 256-262. [32] S. EIGEN, A. HAIIAN and S. KAKUTANI. - Complementing sets of integers - A

result from ergodic theory, Japan J. Math. 18 (1992), 205-2 10.

R. TIJDEMAN

Mathematisch Instituut R.U. Postbus 9512 2300 RA Leiden

The Netherlands

Number Theory Paris 1992-93

CM Abelian varieties with almost ordinary reduction Yuri G. ZARHIN

In this note we discuss the Hodge group Hdg(X) of a simple Abelian variety X of CM-type. It is well-known that dimQ Hdg(X) _< dim(X). Assuming that X has somewhere good almost ordinary reduction, we prove

that dimQ Hdg(X) = dim(X) and give an explicit description of Hdg(X).

1. - Almost ordinary Abelian varieties Let A be an Abelian variety defined over a finite field k of characteristic p. We call A almost ordinary if dim(A) > 1 and it has the same Newton polygon as the product of (dim(A) - 1)-dimensional ordinary Abelian variety and a supersingular elliptic curve. This means that its set of slopes is {0, 1/2, 1}

and slope 1/2 has length 2. For example, an Abelian surface is almost ordinary if and only if it is neither ordinary nor supersingular. One may easily check that if g = dim(A) > 1 then A is almost ordinary if and only if its p-rank equals g - 1, i.e., the group of "physical" points of order p is isomorphic to (Z/pZ)9-1. Almost ordinary varieties were studied by Oort 113] in connection with the lifting problem of CM Abelian varieties to characteristic zero. In particular, he proved that each almost ordinary Abelian variety can be lifted to characteristic zero as CM Abelian variety (recall 126] that each Abelian variety over a finite field can be lifted to characteristic zero as CM Abelian variety up to an isogeny). Of course, if we start with an (absolutely) simple Abelian variety over a finite field, then its lifting will be also (absolutely) simple. It follows from ([5], Th.7; 112], Th.4. 1) that polarized almost ordinary Abelian varieties of given dimension constitute subvarieties of codimension 1 in the moduli spaces of Abelian varieties. See also [ 141.

A special case of a theorem of Lenstra and Oort [61 asserts that, for each positive integer g > 1 and for each prime number p there exists an absolutely simple almost ordinary g-dimensional Abelian variety defined Supported by C.N.R.S.

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over a certain finite field field of characteristic p. It was proven by Oort [ 13] that the endomorphism algebra of simple almost ordinary Abelian variety

(over finite field) is a number field of degree 2 dim(A). Notice (see Sect. 6.6 below), that each simple almost ordinary Abelian variety is absolutely simple.

One may easily check that each non-simple almost ordinary Abelian variety is isogenous either to the product of ordinary Abelian variety and simple almost ordinary Abelian variety or to the product of ordinary Abelian variety and a supersingular elliptic curve. Let A be an Abelian variety over a finite field k of characteristic p. We write FA for the multiplicative subgroup of C* generated by the eigenvalues of the Frobenius endomorphism of A [29, 30, 31]. It is known (1301, Sect. 2.1; [33], Sect. 4.1), that the rank rk(I'A) of FA is a positive number which does not exceed dim(A) + 1. The non-negative integer rk(I'A) - 1 is called the rank of A and denoted by rk(A) [31]. One may easily check ([31], Sect.

2.0), that 0 < rk(A) < dim(A) and rk(A) = 0 if and only if A is supersingular. Now, assume that A is simple and almost ordinary. In that case it is known ([71, Th. 5.7) that either rk(A) = dim(A) or rk(A) = dim(A) - 1. In addition, if dim(A) is even then rk(A) = dim(A), i.e., rk(FA) = dim(A) + 1. If rk(TA) = dim(A) then the endomorphism algebra of A must contain an imaginary quadratic field; see [7], Th. 3.6. H.W. Lenstra (see [31], pp. 286288) has constructed an example of 3-dimensional simple almost ordinary Abelian variety A with rk(FA) = dim(A). His construction also gives an example of a 3-dimensional absolutely simple CM Abelian variety having an almost ordinary reduction. 2. - Q-adic Lie Algebras

Let X be an Abelian variety defined over a number field K. We assume that K is sufficiently large, i.e., all endomorphisms of X are defined over K. We will also fix an embedding of K into the field C of complex numbers and consider K as a certain subfield of C. We write K(s) for the algebraic closure of K in C. We write G(K) for the Galois group of K. We write g for the dimension of X. Let E be the endomorphism algebra of X ; it is a finite-dimensional semisimple Q-algebra. For a positive integer in, we denote by X,,,, the group

{x E X(K(s)) I mx = 0}. It is well known that X. is a free Z/mZ-module of rank 2g. Let us fix a prime number £. Then one may define the ZI Tate module T1 (X) as the projective

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limit of the groups X,,,, where m runs through the set of all powers .£i and the transition maps are multiplication by 2. It is well known that T1(X) is a free Zi-module of rank 2g . Clearly, all X,,, are finite Galois submodules of X(K(s)), and the Galois actions for m = £ glue together to give rise to a continuous homomorphism

pi = pi,x : G(K)

Autz, Ti(X).

The image

Ge = Gi,x = Im(pi,x) C Autz Ti(X) is a compact £-adic Lie subgroup in Autze Ti (X ). Let us put VV (X) := TT(X) ®Ze Q

Clearly, Vi(X) is a Qi vector space of dimension 2g and one may

with a certain Zi-lattice of rank 2dim(X) in naturally identify Vi (X). In particular, AutZe Ti (X) becomes an open compact subgroup in AutQ, Vi(X). This allows us to regard pi as an .£-adic representation ([ 191):

pi = pi,x : G(K) - Autz, Te(X) C AutQ, Ve(X). We have

Gi C Autz, Ti(X) C AutQ, Vi(X). Clearly, Gi is a compact (and therefore) closed subgroup of AutQ, VV(X) and therefore is a closed 2--adic Lie subgroup. Let gi = gi,x C EndQ, Vi(X) be the Lie algebra of Gi [ 19]. A theorem of Faltings [4] asserts that of is a reductive Qt-Lie algebra, its natural representation in Vi (X) is completely reducible and the centralizer of this representation is E ®Q Qi. A theorem

of Bogomolov [1] asserts that gi is an algebraic Lie algebra containing homotheties Qtid. It follows that

gi,x = Qiid (Dg°,x. Here

9°,x := sl(Vi(X)) n gi,x is an algebraic reductive Qi-Lie algebra. Its natural representation in VV(X) is completely reducible and the centralizer of this representation is E ®Q Qi.

It is known that the rank of g° is a non-negative number which does not

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exceed g. If the equality holds then the Lie algebra is "as large as possible" and one may give an "explicit" description of go in terms of E; see (32], Th. 3.2; [331.

Let v be a non-Archimedean place of K such that X has a good reduction X (v) at v. Then GQ contains a Frobenius element Frv E Ge C AutQe V1(X)

canonically defined up to conjugation in G1 1191. If we view Frv as a linear operator in Ve (X), then its eigenvalues are just eigenvalues of the Frobenius endomorphism of X(v). In particular, if r(Frv) is the multiplicative group generated by the eigenvalues of Frv, then

I (Frv) = rX(v)

Notice, that the rank of ge is greater or equal than rk(r(Frv)) (see 1331, Corollary 2.4.1). This implies that the rank of go is greater or equal than

rk(r(Frv)) - 1 = rk(FX(v)) - 1 = rk(X(v)). We have

0 < rk(X(v)) < rkge < g = dim(X(v)). In particular, if rk(X(v)) = dim(X(v)) then rk(ge) = 9.

For example, if X(v) is a simple almost ordinary Abelian variety and g is even then (see the end of Sect. 1) rk(g°) = g

(recall that g = dim(X) = dim(X(v))).

The aim of the present paper is to prove that if X (v) is an almost ordinary Abelian variety then rk(ge°)

= 9

under an additional assumption that X is an absolutely simple Abelian variety of CM-type. (Compare with the corresponding results for Abelian varieties having a reduction of K3 type ([32], Th. 3.0 and Sect. 7.1).

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3. - Abelian varieties of CM-type Let X be an absolutely simple Abelian variety of CM-type. Then its endomorphism algebra E is a CM-field of degree 2g. We write a -4 a' for the complex conjugation on E. We write TE for the Well restriction RE/QGm of the multiplicative group Gm. Clearly, TE is a 2g-dimensional algebraic torus. Let UE be the g-dimensional algebraic subtorus of TE defined by the condition

UE(Q) = {a E TE(Q) = E* I aa' = 1}.

3.1. - The Hodge group We write V (X) for the first rational homology group H1 (X (C), Q) of X (C)

:

it is a 2g-dimensional Q--vector space. It also carries a natural

structure of 1-dimensional E-vector space. The choice of a polarization on X gives rise to a certain non-degenerate skew-symmetric bilinear form cp : V (X) X V (X) -+ Q

such that cp(ax, y) = cp(x, a'y)

for all x, y E V(X) and a E E. Let us choose a non-zero e E E with

E =-e. Then there exists a non-degenerate E-Hermitian sesquilinear form

0, : V(X) x V(X) -* E such that co(x, y) =

n'E/Q(e-i0E(x y))

where TrE/Q : E - Q is the trace map (see 121], [2], Sect. 4; [ 171, p. 531). If we change e by el then the form is multiplied by a non-zero totally real element el /e of E. The unitary group U(V (X ), 0) viewed as a Q-algebraic group does not depend on the choice of a and can be naturally identified with UE. In particular,

U(V(X), 0,) (Q) = {a E E* I aa' = 1}. Here we identify E with its image in EndQ V (X ). Its Lie algebra

uE := Lie(U(V(X), VE)) = Lie(UE) _ {a E E I a + a' = 0}.

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Let Hdg(X) be the corresponding Hodge or as it sometimes called the special Mumford-Tate group of X (see (10, 15, 17, 18, 11]). It is a connected commutative reductive algebraic Q-group. It is well-known ([ 17], p. 531)

that

Hdg(X) C U. Let f1Dg = f1DgX be its Lie algebra. Clearly,

13DgCuE={aEEIa+a'=0}. It is known that it is a commutative Q-Lie algebra, i.e., its rank and dimension coincide, and rk(f1Dg) = dirngp 11Dg < dimQUE = 9;

the equality holds true if and only if Hdg(X) = UE.

For example, it is known that this equality holds true when g is a prime (a theorem of Tankeev-Ribet [ 17, 23]). For arbitrary dimensions there is a Ribet's inequality (118], p.87) loge (2g) < dimQ Hdg(X)

(see also [81). For further properties and examples of the Hodge groups of CM-Abelian varieties see 118, 3, 8, 281. There is a well-known natural isomorphism of Qe-vector spaces

Vt(X) =V(X) ®QQ . It is known that for Abelian varieties of CM-type the Qt-Lie algebra g° is a commutative Qt-Lie algebra, i.e., its rank and dimension coincide. A theorem of Pohlman [16] asserts that the isomorphism of the Qt-vector spaces mentioned above gives us an identification 17Dg®QQt =g°

of commutative Qe-Lie algebras.

Clearly, if rk(g°) = g, then it follows easily that dimQ 13Dg = g and, therefore,

Hdg(X) = UE.

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3.2. - Remark. One may define the Hodge group Hdg(X) for any (complex) Abelian variety X not necessarily of CM-type [10]. It is a connected reductive algebraic (12-group which is commutative if and only if X is of CM-type. The Mumford-Tate conjecture [201 asserts that the 2--adic Lie algebra ge,x can be obtained from the Lie algebra of Hdg(X) by extensions of scalars from Q to Qt. The theorem of Pohlman cited above proves the MumfordTate conjecture for Abelian varieties of CM-type.

4. - Main result. The main result of the present paper is the following statement. MAIN THEOREM. - Let X be an absolutely simple g-dimensional Abelian variety of CM-type defined over a number field K and all endomorphisms of X are also defined over K. Let E be the endomorphism algebra of X. Assume that there exists a non-Archimedean place v of K such that X has a good reduction X (v) at v and X (v) is an almost ordinary Abelian variety X (v). Then Hdg(X) = UE.

In other words,

dime Hdg(X) = dim(X) = g.

4.2. - Remark. Assume that g is even and X (v) is a simple almost ordinary Abelian variety. Then dim(X(v)) = dim(X) = g is also even and, as we have already seen, g = rk(X(v)) < rk(g°) = dimQ Clag < dimQ UE = g and, therefore, dimQ ljag = dimQ UE = g.

This proves the Theorem under our additional assumptions.

4.3. - Remark. Assume that g is odd and X (v) is a simple almost ordinary Abelian variety. Then dim(X(v)) = dim(X) = g and, as we have already seen, g - 1 < rk(X(v)) < rk(ge) = dimQ fjag < dimQUE = 9 and, therefore, g - 1 < dimQ [jag = dimQ Hdg(X) < dimQ UE = 9-

4.4. - Combining the last two Remarks, we obtain that the Theorem follows from the next two lemmas.

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4.5. LEMMA. - Let X be an absolutely simple g-dimensional Abelian variety of CM-type defined over a number field K and all endomorphisms of X are also defined over K. Let E be the endomorphism algebra of X. Assume that there exists a non-Archimedean place v of K such that X has a good reduction at v and this reduction is an almost ordinary Abelian variety X (v). Then X (v) is a simple Abelian variety.

4.6. LEMMA. - Let Y be an absolutely simple g-dimensional Abelian variety of CM-type. Let E be the endomorphism algebra of Y. Assume that

dimQ Hdg(Y) = g - 1 = dimQ UE - 1. Then g is even.

4.7. - Remark. It is well-known ([ 17], Th. 0, p. 524) that the equality

Hdg(X) = UE implies that all Hodge classes on all powers of X are linear combinations of the products of divisors classes. In particular, all these Hodge classes are algebraic, i.e., the Hodge conjecture holds true for all powers of X. Since the Mumford-Tate conjecture holds true for Abelian varieties of CM-type [ 161, we obtain that all Tate classes on all powers of X are linear combinations

of the products of divisors classes. Indeed, by a theorem of Faltings [4], each 2-dimensional Tate class on an Abelian variety over a number field is a linear combination of divisor classes. In particular, all these Tate classes are algebraic, i.e., the Tate conjecture [24, 25] holds true for all powers of X.

5. - Proof of the Lemma 4.6. We start this section with the explicit description of Q-algebraic subtori in UE of codimension 1. This description had tacitly appeared in [6] and, later, was explicitly formulated and proved in [9]. In our exposition we follow 191.

Suppose E contains an imaginary quadratic subfield k. Let us define the algebraic subtorus SUE/k of UE by the condition SUE/k(Q) = {a E UE(Q) I NormE/k(a) = 1}. One may easily check that SUE/k has codimension 1 in UE. Clearly, its Lie algebra SUE/k := Lie(SUE/k) = {a E UE I TrE/k(a) = 0}.

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Here TrE/k : E --> k is the trace map. Notice, that this trace map commutes with the complex conjugation (if someone is unhappy with the definition of SUE/k by its Q-rational points then there is another description of SUE/k.

Namely, it is a Q-algebraic (connected) subtorus of UE such that its Lie algebra coincides with SUE/k). The following statement was proven in [9[, Sect. 7.3.

5.1. KEY LEMMA. - Let H be an algebraic subtorus of codimension I in UE. Then there exists an imaginary quadratic subfield k of E such that :

H = SUE/k.

5.2. - Since H := Hdg(X) is an algebraic subtorus of codimension 1 in UE, we obtain, applying the Key Lemma, that there exists an imaginary quadratic subfield k of E such that Hdg(Y) = SUE/k.

This means that CJ-0g = SUE/k.

Now, let us choose a non-zero e c k c E such that

Now, if we consider V (X) as a g-dimensional k -vector space, then the EHermitian form 0E gives rise to the k-Hermitian form '

E/k4'e : V(X) X V(X) --4k,

0(x,y) =TrE/k(0,(x,y))It follows easily that P(x, Y) = TrE/k(E-14(x, y))

for all x, y E V (X) and 0 is non-degenerate. Clearly, UE C u(V(X), ) :_

{aEEndkV(X) I 0(ax,y)+b(x,a'y)=0bx,yEV(X)} and

SUE/k C SUk(V(X),0) := {a E U(V(X),

)

1 TrV(X)/k(a) = 0}.

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286

Here

TV(X)/k : Endk V(X) - k is the usual trace map on the algebra of k-linear operators of the k-vector space V(X) (notice, that the maps TrE/k and Trv(X)/k coincide on E). So, we obtained that 49 = SUE/k C SUk(V(X),V)) It turns out that the inclusion hag C SUk(V(X),')

can be rewritten in terms of the action of k on the tangent space of X (see Well [27]). Namely, if Lie(X(C)) is the tangent space of the complexAbelian variety X then the inclusion means that Lie(X(C)) is a free k ®Q C-module ([91, Lemma 2.8; see also [ 17], p. 525). Since Lie(X(C)) is a g-dimensional complex vector space and k ®Q C = C ® C, the dimension g must be even. This ends the proof.

6. - Proof of the Lemma 4.5. By functoriality of Neron models, there is a natural embedding

E = End(X) ® Q -> End(X(v)) ® Q

and 1 E E acts on X(v) as the identity map. Notice, that E is a number field and

[E : Q] = 2 dim(X) = 2 dim(X(v)). The following proposition will be proved at the end of this Section. 6.1. PROPOSmoN. - Let Y be an Abelian variety over an arbitrary field

K and assume that the semisimple Q-algebra End°(Y) = End(Y) ® Q contains a numberfield F of degree 2 dim(Y) such that 1 E E is the identity automorphism of Y. Then there exists a K-simple Abelian variety Z over k such that Z is )C-isogenous to the power Zr of Z with r = dim(Y)/ dim(Z).

6.2. - Applying the Proposition 6.1, we obtain that there exists a k(v)simple Abelian variety Z over k(v) such that X (v) is isogenous to Zr for a certain positive integer r. In order to prove the lemma 4.5, we have only to check that

r=1.

First, notice, that each slope of the Newton polygon of X (v) has length divisible by r. Since the slope 1/2 has length 2, either r = 1 or r = 2. If r = 2 then X(v) is isogenous to Z2 and, therefore, 1/2 is the slope of the Newton polygon of Z with length 1. But it cannot be true, since the length

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of the slope 1/2 must always be even [30, 7], due to the fact that all the break-points of the Newton polygon are integral. This rules out the case r = 2. So, r = 1 and we are done.

6.3. - Proof of the Proposition 6.1. Assume that Y is not )C-isogenous to a power of a )C-simple Abelian variety. Then, using the Poincare reducibility theorem, one may easily check that there exist Abelian 1C-subvarieties Y1, Y2 C Y of positive dimensions, enjoying the following properties : a) the natural homomorphism Y1 X Y2 -> Y, (yl, y2) --' yl + y2

is an isogeny; b) Hom(Yi, Y2) = {0}, Hom(Y2, Yi) = {0}. This implies that

0 < dim(Yi) < dim(Y); 0 < dim(Y2) < dim(Y); End°(Y) = End°(Yi) ® End°(Y2). Let pri : End°(Y) --4End°(Y) be the corresponding projection homo-

morphisms. Clearly, if idy E End°(Y) is the identity automorphism of Y then pri (idy) E End° (Y) is the identity automorphism idy, of Y . This implies that Fi := pri(F) c End°(Yi) is a number field isomorphic to F; in particular, its degree equals 2 dim(Y) > 2 dim(Y) (i = 1, 2.) Now, in order to get a contradiction let us recall the following well-known fact (see [22], Sect. 5.1, Proposition 2). 6.4. SUBLEMMA. - If the endomorphism algebra of an m-dimensional Abelian variety contains a number field which, in turn, contains the identity automorphism, then the degree of this field divides 2m. In particular, it does not exceed 2m.

6.5. - Now, in order to finish the proof by coming to the contradiction, one has only to apply the Sublemma to the Abelian variety Yi of dimension m = dim(Y) and the number field Fi of degree 2 dim(Y) > dim(Y).

6.6. - Remark. Similar arguments prove that if k is a finite field and A is a gdimensional k-simple almost ordinary Abelian variety over k then A is absolutely simple. Indeed, for each extension k' of k the Abelian variety A' := A x k' is an almost ordinary Abelian variety and End° A' contains a number field End° A of degree 2g = 2 dim(A'), which, in turn, contains the

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identity automorphism. By the Sublemma, A' must be k'-isogenous to the power Z' of k'-simple Abelian variety Z. Now, the same arguments with the Newton polygons as in Sect. 6.2, prove that r = 1, i.e., A' = Z is k/-simple.

7. - Acknowledgements I am deeply grateful to H.W. Lenstra and B. Moonen for helpful discussions. This paper is a result of my stay in Paris in June-July of 1993 and I am deeply grateful to the Groupe d'Etudes sur les Problemes Diophantiens (Universite de Paris VI) for the hospitality. The support of the Universite Paris Nord is also gratefully acknowledged. I am grateful to Frans Oort who had read the manuscript and made many valuable remarks. My special thanks go to Daniel Bertrand and Larry Breen, whose efforts made my trip to France possible.

Manuscrit recu le 21 janvier 1994

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Yuri G. ZARHIN

The Pennsylvania State University, Department of Mathematics 325 McAllister Building, University Park, PA 16802, USA

e-mail address : [email protected] and Institute for Mathematical Problems in Biology Russian Academy of Sciences Pushchino, Moscow Region, 142292 Russia