Arithmetic of higher-dimensional algebraic varieties

DIOPHANTINE EQUATIONS: PROGRESS AND PROBLEMS 1. Introduction. A Diophantine problem over Q is concerned with the soluti...

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DIOPHANTINE EQUATIONS: PROGRESS AND PROBLEMS 1. Introduction. A Diophantine problem over Q is concerned with the solutions either in Q or in Z of a finite system of polynomial equations Fi (X1 , . . . , Xn ) = 0 (1 ≤ i ≤ m)

(1)

with coefficients in Q. Without loss of generality we can obviously require the coefficients to be in Z. A system (1) is also called a system of Diophantine equations. Often one will be interested in a family of such problems rather than a single one; in this case one requires the coefficients of the Fi to lie in some Q(c1 , . . . , cr ), and one obtains an individual problem by giving the cj values in Q. Again one can get rid of denominators. Some of the most obvious questions to ask about such a family are: (A) Is there an algorithm which will determine, for each assigned set of values of the cj , whether the corresponding Diophantine problem has solutions, either in Z or in Q? (B) For values of the cj for which the system is soluble, is there an algorithm for exhibiting a solution? For individual members of such a family, it is also natural to ask: (C) Can we describe the set of all solutions, or even its structure? (D) Is the phrase ‘density of solutions’ meaningful, and if so, what can we say about it? The attempts to answer these questions have led to the introduction of new ideas and these have generated new questions. On some of them I expect progress within the next decade, and I have restricted myself to these in the text below. Progress in mathematics usually means proven results; but there are cases where even a well justified conjecture throws new light on the structure of the subject. (For similar reasons, well motivated computations can be helpful; but computations not based on a deep feeling for the structure of the subject have generally turned out to be a waste of time.) But I have not included those problems (such as the Riemann Hypothesis and the Birch/Swinnerton-Dyer conjecture) on which I do not expect further 1

progress within so short a timescale. The reader may also find it useful to read Silverberg [36], which has the same purpose as this article but rather little overlap with it. Though the study of solutions in Z and in Q may look very similar (and indeed were believed for a long time to be so), it now appears that they are actually very different and that the theory for solutions in Q has much more structure than that for solutions in Z. The main reason for this seems to be that in the rational case the system (1) defines a variety in the sense of algebraic geometry, and many of the tools of that discipline can be used. Despite the advent of Arakelov geometry, this is much less true of integral problems. However, for varieties of degree greater than 2 it is only in low dimension that we yet know enough of the geometry for it to be useful. Uniquely, the Hardy-Littlewood method is useful both for integral and for rational problems; it was designed for integral problems but can also be applied to rational problems in projective space, because then the Fi in (1) are homogeneous and it does not matter whether we treat the variables Xν as integral or rational. There is a brief discussion of this method in §8, and a comprehensive survey in [46]. Denote by V the variety defined by the equations (1) and let V 0 be any variety birationally equivalent to V over Q. The problem of finding solutions of (1) in Q is the same as that of finding rational points on V , which is almost the same as that of finding rational points on V 0 . Hence (except possibly for Question (D) above) one expects the properties of the rational solutions of (1) to be essentially determined by the birational equivalence class of V ; and the way in which algebraic geometers classify varieties should provide at least a first rough guide to the classification of Diophantine problems — though they mainly study birational equivalence over C rather than over Q. But it does at the moment seem that even for surfaces the geometric classification needs some refinements if it is to fit the number-theoretic results and conjectures. Without loss of generality we can assume that V is absolutely irreducible. For if V has proper components defined over Q it is enough to ask the questions above for each of the proper components; and if V is the union of varieties conjugate over Q then any rational point on V lies on the intersection of these conjugates, which is a proper subvariety of V . Since we can desingularize V by a birational transformation defined over Q, it is natural to concentrate on the case when V is projective and nonsingular. The definitions and the questions above can be generalized to an arbitrary algebraic number field and the ring of integers in it; the answers are usually 2

known or conjectured to be essentially the same as over Q or Z, though the proofs can be very much harder. (But there are exceptions; for example, the modularity of elliptic curves only holds over Q.) The questions above can also be posed for other fields of number-theoretic interest — in particular for finite fields and for completions of algebraic number fields — and when one studies Diophantine problems it is essential to consider these other fields also. If V is defined over a field K, the set of points on V defined over K will always be denoted by V (K). If V (K) is not empty we say that V is soluble in K. In the special case where K = kv , the completion of an algebraic number field k at the place v, we also say that V is locally soluble at v. From now on we denote by Qv any completion of Q; thus Qv means R or some Qp . One major reason for considering solubility in complete fields and in finite fields is that a necessary condition for (1) to be soluble in Q, for example, is that it is soluble in every Qv . The condition of solubility in every Qv is computationally decidable; see §2. Moreover, at least for primes p for which the system (1) has good reduction mod p, the first step in deciding solubility in Qp is to decide whether the reduced system is soluble in the finite field GF(p) of p elements. Some geometers are already used to studying varieties over fields k which are not algebraically closed; what makes Diophantine problems special is the number-theoretic nature of the fields k. But it seems that only a few of the properties peculiar to such fields are useful in this context, so that a geometer need not learn much number-theory in order to work on Diophantine problems. On the other hand, a number-theorist would be wise to learn quite a lot of geometry. Diophantine problems were first introduced by Diophantus of Alexandria, the last of the great Greek mathematicians, who lived at some time between 300 B.C. and 300 A.D.; but he was handicapped by having only one letter available to represent variables, all the others being used in classical Greek to represent specific numbers. Individual Diophantine problems were studied by such great mathematicians as Fermat, Euler and Gauss. But it was Hilbert’s address to the International Congress in 1900 which started the development of a systematic theory. His tenth problem asked: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is soluble in rational 3

integers. Most of the early work on Diophantine equations was concerned with rational rather than integral solutions; presumably Hilbert posed this problem in terms of integral solutions because such a process for integral solutions would automatically provide the corresponding process for rational solutions also, by restricting to the special case when the equations are homogeneous. In those confident days before the First World War, it was assumed that such a process must exist; but in 1970 Matijaseviˇc showed that this was impossible. Indeed he exhibited a polynomial F (c; x1 , . . . , xn ) such that there cannot exist an algorithm which will decide for every given c whether F = 0 is soluble in integers. His proof is part of the great program on decidability initiated by G¨odel; good accounts of it can be found in [10], pp 323-378 or [9]. The corresponding question for rational solutions is still open; I am among the very few who believe that it may have a positive answer. But even if the answer to the analogue of Hilbert’s tenth problem for rational solutions is positive, one must expect that a separate algorithm will be needed for each kind of variety. Thus we shall need not one algorithm but an infinity of them. So number theorists depend on the development by geometers of an adequate classification of varieties. At the moment, such a classification is reasonably complete for curves and surfaces, but it is still fragmentary even in dimension 3; so number theorists have to concentrate on curves and surfaces, and on certain particularly simple kinds of variety in higher dimension. 2. The Hasse Principle and the Brauer-Manin obstruction. Let V be a variety defined over Q. If V is locally soluble at every place of Q, we say that it satisfies the Hasse condition. If V (Q) is not empty then V certainly satisfies the Hasse condition. What makes this remark valuable is that the Hasse condition is computable — that is, one can decide in finitely many steps whether a given V satisfies the Hasse condition. This follows from the next two lemmas. Lemma 1 Let W be an absolutely irreducible variety of dimension n defined over the finite field k = GF(q). Then N (q), the number of points on W defined over k, satisfies |N (q) − q n | < Cq n−1/2

where the constant C depends only on the degree and dimension of W and is computable. 4

This follows from the Weil conjectures, for which see §3; but weaker results which are adequate for the present application were known much earlier. Since the singular points of W lie on a proper subvariety, there are at most C1 q n−1 of them, where C1 is also computable. It follows that if q exceeds a computable bound depending only on the degree and dimension of W then W contains a nonsingular point defined over k. Now let V be an absolutely irreducible variety defined over Q. If V has good reduction at p, which happens for all but a finite computable set of primes p, denote that reduction by V˜p . If p is large enough, it follows from the remarks above that V˜p contains a nonsingular point Qp defined over GF(p). The result which follows, which is known as Hensel’s Lemma though the idea of the proof goes back to Sir Isaac Newton, now shows that V contains a point Pp defined over Qp . Lemma 2 Let V be an absolutely irreducible variety defined over Q which has a well-defined reduction mod p, denoted by Vep . If V˜p contains a nonsingular point Qp defined over GF(p) then V contains a nonsingular point Pp defined over Qp whose reduction mod p is Qp . In view of this, to decide whether V satisfies the Hasse condition one only has to check individually solubility in R and in finitely many Qp . Each of these checks can be shown to be a finite process, using ideas similar to those in the proof of Lemma 1. A family F of varieties is said to satisfy the Hasse Principle if every V contained in F and defined over Q which satisfies the Hasse condition actually contains at least one point defined over Q. Again, a family F is said to admit weak approximation if every V contained in F and defined over Q, and such that V (Q) is not empty, has the following property: given any finite set of places v and corresponding non-empty sets Nv ⊂ V (Qv ) open in the v-adic topology, there is a point P in V (Q) which lies in each of the Nv . In the special case when F consists of a single variety V , and V (Q) is not empty, we simply say that V admits weak approximation. Whether V admits weak approximation is in general not computable; for an important exception, see [43]. The most important families which are known to have either of these properties (and which actually have both) are the families of quadrics of any given dimension; this was proved by Minkowski for quadrics over Q and by Hasse for quadrics over an arbitrary algebraic number field. But many families, even of very simple varieties, do not satisfy either the Hasse Principle 5

or weak approximation. (For example, neither of them holds for nonsingular cubic surfaces.) It is therefore natural to ask Question 1 For a given family F, what are the obstructions to the Hasse Principle and to weak approximation? For weak approximation there is a variant of this question which may be both more interesting and easier to answer. For another way of stating weak approximation on V is toQsay that if V (Q) is not empty then it is dense in the adelic space V (A) = v V (Qv ). This suggests the following: Question 2 For a given V , or family F, what can be said about the closure of V (Q) in the adelic space V (A)?

However, there are families for which Question 1 does not seem to be a sensible question to ask; these probably include for example all families of varieties of general type. So one should back up Question 1 with Question 3 For what kinds of families is either part of Question 1 a sensible question to ask? The only systematic obstruction to the Hasse Principle which is known is the Brauer-Manin obstruction, though obstructions can be found in the literature which are not Brauer-Manin. Let A be a central simple algebra — that is, a simple algebra which is finite dimensional over a field K which is its centre. Each such algebra consists, for fixed D and n, of all n × n matrices with elements in a division algebra D with centre K. Two central simple algebras over K are equivalent if they have the same underlying division algebra. Formation of tensor products over K gives the set of equivalence classes the structure of a commutative group, called the Brauer group of K and written Br(K). There is a canonical isomorphism ıp : Br(Qp ) ' Q/Z for each p; and there is a canonical isomorphism ı∞ : Br(R) ' {0, 12 }, the nontrivial division algebra over R being the classical quaternions. Let B be an element of Br(Q); tensoring B with any Qv gives rise to an element of Br(Qv ), and this element is trivial for almost all v. There is an exact sequence M 0 → Br(Q) → Br(Qv ) → Q/Z → 0,

due to Hasse, in which the third map is the sum of the ıv ; it tells us when a set of elements, one in each Br(Qv ) and almost all trivial, can be generated from some element of Br(Q). 6

Now let V be a complete nonsingular variety defined over Q and A an Azumaya algebra on V — that is, a simple algebra with centre Q(V ) which has a good specialization at every point of V . The group of equivalence classes of Azumaya algebras on V is denoted by Br(V ). If P is any point of V , with field of definition Q(P ), we obtain a simple algebra A(P ) with centre Q(P ) by specializing at P . For all but finitely many p, we have ıp (A(Pp )) = 0 for all p-adic points Pp on V . Thus a necessary condition for the existence of a rational point P on V is that for every v there should be a v-adic point Pv on V such that X ıv (A(Pv )) = 0 for all A. (2)

Similarly, a necessary condition for V with V (Q) not empty to admit weak approximationQis that (2) should hold for all Azumaya algebras A and all adelic points v Pv . In each case this is the Brauer-Manin condition. It is clearly unaffected if we add to A a constant algebra — that is, an element of Br(Q). So what we are really interested in is Br(V )/Br(Q). All this can be put into highbrow language. Even without any hypotheses on V , there is an injection of Br(V ) into the ´etale cohomology group H2 (V, Gm ); and if for example V is a complete nonsingular surface, this injection is an isomorphism. If we write Br1 (V ) = ker(Br(V ) → Br(V¯ )) = ker(H2 (V, Gm ) → H2 (V¯ , Gm )), there is a filtration Br(Q) ⊂ Br1 (V ) ⊂ Br(V ). Here only the abstract structure of Br(V )/Br1 (V ) is known; and in general there is no known way of finding Azumaya algebras which represent nontrivial elements of this quotient, though in a particular case Harari [20] has exhibited a Brauer-Manin obstruction coming from such an algebra. In contrast, provided the Picard variety of V is trivial there is an isomorphism ¯ ¯ Pic(V ⊗ Q)), Br1 (V )/Br(Q) ' H1 (Gal(Q/Q), ¯ is known. and this is computable in both directions provided Pic(V ⊗ Q) (For details of this, see [8].) ¯ for arbitrary There is no known systematic way of determining Pic(V ⊗ Q) V , and there is strong reason to suppose that this is really a number-theoretic rather than a geometric problem. If V is defined over Q (rather than over an arbitrary algebraic number field) there is a tentative algorithm, depending on 7

the Birch/Swinnerton-Dyer conjecture, for determining an algebraic number ¯ =Pic(V ⊗ K), and this may field K (depending on V ) such that Pic(V ⊗ Q) ¯ but one hopes not, be the right first step towards determining Pic(V ⊗ Q); because even for so elementary a variety as a cubic surface we may need to have [K : Q] ≥ 51840. It seems to me likely that a better approach to this question will be through the Tate conjectures, for which see §3; but this is a very long-term prospect. However, it is usually possible to determine ¯ for any particular V that one is interested in. Pic(V ⊗ Q) Question 4 Is there a general algorithm (even conjectural) for determining ¯ for varieties V defined over an algebraic number field? Pic(V ⊗ Q) Lang has conjectured that if V is a variety of general type defined over an algebraic number field K then there is a finite union S of proper subvarieties of V such that every point of V (K) lies in S. (Faltings’ theorem, for which see §4, is the special case of this for curves.) This raises another question, similar to Question 4 but probably easier: Question 5 Is there an algorithm for determining Pic(V ) where V is a variety defined over an algebraic number field? The Brauer-Manin obstruction was introduced by Manin [26] in order to bring within a single framework various sporadic counterexamples to the Hasse principle for rational surfaces. The theory of this obstruction has been extensively developed by Colliot-Th´el`ene and Sansuc. In particular, for rational varieties they showed how to go back and forth between the Brauer-Manin condition and the descent condition for torsors under tori. They also showed that if there is no Brauer-Manin obstruction to the Hasse principle on a variety V then there exists a universal torsor over V which has points everywhere locally. This suggests that one should pay particular attention to Diophantine problems on universal torsors. Unfortunately, it is usually not easy to exploit what is known about the geometric structure of universal torsors. Indeed there are very few families for which the BrauerManin obstruction can be nontrivial but for which it has been shown that it is the only obstruction to the Hasse principle. (See however [12] and, subject to Schinzel’s hypothesis, [41] and [14].) It is generally believed that the Brauer-Manin obstruction is the only obstruction to the Hasse principle ¯ for rational surfaces — that is, surfaces birationally equivalent to P2 over Q. On the other hand, Skorobogatov ([37], and see also [38]) has exhibited an 8

obstruction to the Hasse principle on a bielliptic surface which is definitely not Brauer-Manin. Question 6 Is the Brauer-Manin obstruction the only obstruction to the Hasse principle for all unirational (or all Fano) varieties? For the method of universal torsors, the immediate question to address must be the following: Question 7 Does the Hasse principle hold for universal torsors over a rational surface? We can of course ask similar questions for weak approximation. Both for the Hasse principle and for weak approximation one can alternatively ask what is the most general class of varieties for which the Brauer-Manin obstruction is the only one. Colliot-Th´el`ene has suggested that this class probably includes, and may even be equal to, all rationally connected varieties. There are families F whose universal torsors appear to be too complicated to be systematically investigated, but for which it is still possible to identify the obstruction to the Hasse principle. It is sometimes possible to start from the absence of a Brauer-Manin obstruction; but there are also alternative strategies. Implementing these falls naturally into two parts: (i) Assuming that V in F satisfies the Hasse condition, one finds a necessary and sufficient condition for V to have a rational point, or to admit weak approximation. (ii) One then shows that this necessary and sufficient condition is equivalent to the Brauer-Manin condition. Both these strategies have been applied to pencils of conics, in each case assuming Schinzel’s Hypothesis; the curious reader may wish to compare the approaches in [41] and in [14]. Except for Skorobogatov’s example above, I know of no families for which it has been possible to carry out (i) but not (ii). But there are families for which it has been possible to find a sufficient condition for solubility (additional to the Hasse condition) which appears rather weak but which is definitely stronger than the Brauer-Manin condition. The obvious examples of such a condition are the various forms of what is called Condition D in [13], [42], [2] and [44]. However, in these cases it is not obvious that a condition stronger than the Brauer-Manin condition is actually necessary; and I am provisionally inclined to attribute the gap to clumsiness in the proofs. 9

Question 8 When the Brauer-Manin condition is trivial, how can one make use of this fact? 3. Zeta-functions and L-series. Let W ⊂ Pn be a nonsingular and absolutely irreducible projective variety of dimension d defined over the finite field k =GF(q), and denote by φ(q) the Frobenius automorphism of W given by φ(q) : (x0 , x1 , . . . , xn ) 7→ (xq0 , xq1 , . . . , xqn ). For any r > 0 the fixed points of (φ(q))r are precisely the points of W which are defined over GF(q r ); suppose that there are N (q r ) of them. Although the context is totally different, this is almost the formalism of the Lefschetz Fixed Point theorem, since for geometric reasons each of these fixed points has multiplicity +1. This analogy led Weil to conjecture that there should be a cohomology theory applicable in this context. This would imply that there were finitely many complex numbers αij such that r

N (q ) =

Bi 2d X X

r (−1)i αij

for all r > 0,

(3)

i=0 j=1

where Bi is the dimension of the ith cohomology group of W and the αij are the characteristic roots of the map induced by φ(q) on the ith cohomology. For each i duality implies that Bi = B2d−i and the α2d−i,j are a permutation of the q d /αij . If we define the local zeta-function Z(t, W ) by either of the equivalent relations log Z(t) =

∞ X

r

r

N (q )t /r

0

or tZ (t)/Z(t) =

r=1

∞ X

N (q r )tr ,

r=1

then (3) is equivalent to Z(t) =

P1 (t, W ) · · · P2d−1 (t, W ) P0 (t, W )P2 (t, W ) · · · P2d (t, W )

Q where Pi (t, W ) = j (1 − αij t). Each Pi (t, W ) must have coefficients in Z, and the analogue of the Riemann hypothesis is that |αij | = q i/2 . (For a fuller account of Weil’s conjectures and their motivation, see the excellent survey [24].) All this has now been proved, the main contributor being Deligne. 10

Now let V be a nonsingular and absolutely irreducible projective variety defined over an algebraic number field K. If V has good reduction at a prime p of K we can form V˜p, the reduction of V mod p, and hence form the Pi (t, V˜p). For s in C, we can now define the ith global L-series Li (s, V ) of V as a product over all places of K, the factor at a prime p of good reduction being (Pi (q −s , V˜p))−1 where q =NormK/Q p. The rules for forming the factors at the primes of bad reduction and at the infinite places can be found in [34]. These L-series of course depend on K as well as on V . In particular, L0 (s, V ) is just the zeta-function of the algebraic number field K. To call a function F (s) a (global) zeta-function or L-series carries with it certain implications: • F (s) must be the product of a Dirichlet series and possibly some Gamma-functions, and the half-plane of absolute convergence for the Dirichlet series must have the form Rs > σ0 with 2σ0 in Z. Q • The Dirichlet series must be expressible as an Euler product p fp (p−s ) where the fp are rational functions. • F (s) must have an analytic continuation to the entire s-plane as a meromorphic function all of whose poles are in Z. • There must be a functional equation relating F (s) and F (2σ0 − 1 − s). • The zeroes of F (s) in the critical strip σ0 − 1 < Rs < σ0 must lie on Rs = σ0 − 12 . In our case, the first two implications are trivial; and fortunately one is not expected to prove the last three, but only to state them as conjectures. The last one is the Riemann Hypothesis, which appears to be out of reach even in the simplest case, which is the classical Riemann zeta-function; and the third and fourth have so far only been proved in a few favourable cases. Question 9 Can one extend the list of V for which analytic continuation and the functional equation can be proved? It seems likely that any proof of analytic continuation will carry a proof of the functional equation with it. It has been said about the zeta-functions of algebraic number fields that ‘the zeta-function knows everything about the number field; we just have to 11

prevail on it to tell us’. If this is so, we have not yet unlocked the treasurehouse. Apart from the classical formula which relates hR to ζK (0) all that has so far been proved are certain results of Borel [6] which relate the behaviour of ζK (s) near s = 1−m for integers m > 1 to the K-groups of OK . I would be reluctant to claim that the L-series of a variety V contains all the information which one would like to have about the number-theoretical properties of V ; but one might hope that when a mysterious number turns up in the study of Diophantine problems on V , some L-series contains information about it. Suppose for convenience that V is defined over Q, and let its dimension be d. Even for varieties with B1 = 0 we do not expect a product like d+1 Y Y p −1 d N (p)/p or N (p) (4) p − 1 p p to be necessarily absolutely convergent. But in some contexts there is a respectable expression which is formally equivalent to one of these, with appropriate modifications of the factors at the bad primes. The idea that such an expression should have number-theoretic significance goes back to Siegel (for genera of quadratic forms) and Hardy and Littlewood (for what they called the singular series). Using the ideas above, we are led to replace the study of the products (4) by a study of the behaviour of L2d−1 (s, V ) and L2d−2 (s, V ) near s = d. By duality, this is the same as studying L1 (s, V ) near s = 1 and L2 (s, V ) near s = 2. The information derived in this way appears to relate to the Picard group of V , defined as the group of divisors defined over Q modulo linear equivalence. By considering simultaneously both V and its Picard variety (the abelian variety which parametrises divisors algebraically equivalent to zero modulo linear equivalence), one concludes that L1 (s, V ) must be associated with the Picard variety and L2 (s, V ) with the group of divisors modulo algebraic equivalence — that is, with the N´eron-Severi group of V . These remarks motivate the weak forms of the Birch/Swinnerton-Dyer conjecture (for which see §4) and the case m = 1 of the Tate conjecture below. For the strong forms (which give expressions for the leading coefficients of the relevent Laurent series expansions) heuristic arguments are less convincing; but one can formulate conjectures for these coefficients by asking what other mysterious numbers turn up in the same context and should therefore appear in the formulae for the leading coefficients. The weak form of the Tate conjecture asserts that the order of the pole of L2m (s, V ) at s = m + 1 is equal to the rank of the group of classes of 12

m-cycles on V defined over K, modulo algebraic equivalence; it is a natural generalization of the case m = 1 for which the heuristics have just been shown. For a more detailed account of both of these, including the conjectural formulae for the leading coefficients, see [45] or [40]. Question 10 What information about V is contained in its L-series? There is in the literature a beautiful edifice of conjecture, lightly supported by evidence, about the behaviour of the Li (s, V ) at integral points. The principal architects of this edifice are Beilinson, Bloch and Kato. Beilinson’s conjectures relate to the order and leading coefficients of the Laurent series expansions of the Li (s, V ) about integer values of s; in them the leading coefficients are treated as elements of C∗ /Q∗ . (For a full account see [30] or [23].) Bloch and Kato ([4] and [5]) have strengthened these conjectures by treating the leading coefficients as elements of C∗ . But I do not believe that anything like the full story has yet been revealed. 4. Curves. The most important invariant of a curve is its genus. In the language of algebraic geometry over C, curves of genus 0 are called rational, curves of genus 1 are called elliptic and curves of genus greater than 1 are of general type. But note that for a number theorist an elliptic curve is a curve of genus 1 with a distinguished point P0 on it, both being defined over the ground field K. The effect of this is that the points on an elliptic curve form an abelian group with P0 as its identity element, the sum of P1 and P2 being the other zero of the function (defined up to multiplication by a constant) with poles at P1 and P2 and a zero at P0 . A canonical divisor on a curve Γ of genus 0 has degree −2; hence by the Riemann-Roch theorem Γ is birationally equivalent over the ground field to a conic. The Hasse principle holds for conics, and therefore for all curves of genus 0; this gives a complete answer to Question (A) at the beginning of these notes. But it does not give an answer to Question (B). Over Q, a very simple answer to Question (B) is as follows: Theorem 1 Let a0 , a1 , a2 be nonzero elements of Z. If the equation a0 X02 + a1 X12 + a2 X22 = 0 is soluble in Z, then it has a solution for which each ai Xi2 is absolutely bounded by |a0 a1 a2 |. 13

Siegel [35] has given an answer to Question (B) over arbitrary algebraic number fields, and Raghavan [2] has generalized Siegel’s work to quadratic forms in more variables. The knowledge of one rational point on Γ enables us to transform Γ birationally into a line; so there is a parametric solution which gives explicitly all the points on Γ defined over the ground field. This answers Question (C). If Γ is a curve of general type defined over an algebraic number field K, Mordell conjectured and Faltings proved that Γ(K) is finite; and a number of other proofs have appeared since then. But it does not seem that any of them enable one to compute Γ(K), though some of them come tantalizingly close. For a survey of several such proofs, see [15]. Question 11 Is there an algorithm for computing Γ(K) when Γ is a curve of general type defined over an algebraic number field K? The study of rational points on elliptic curves is now a major industry, almost entirely separate from the study of other Diophantine problems. If Γ is an elliptic curve defined over an algebraic number field K, the group Γ(K) is called the Mordell-Weil group. Mordell proved that Γ(K) is finitely generated; Weil’s contribution was to extend this result to all Abelian varieties. Thanks to Mazur (see [27]) and Merel the theory of the torsion part of the Mordell-Weil group is now reasonably complete; but for the non-torsion part all that was known before 1960 is that Γ(K) can be embedded into a certain group which is finitely generated and computable. The process involved, which is known as the method of infinite descent, goes back to Fermat; for use in §6 I shall illustrate it below in a particularly simple case. By means of this process one can always compute an upper bound for the rank of the Mordell-Weil group of any particular Γ, and the upper bound thus obtained can frequently be shown to be equal to the actual rank by exhibiting enough elements of Γ(K). It was also conjectured that the difference between the upper bound thus computed and the actual rank was always an even integer, but apart from this the actual rank was mysterious. This not wholly satisfactory state of affairs has been radically changed by the Birch/Swinnerton-Dyer conjecture, the weak form of which is described at the end of this section. A survey of what is currently known or conjectured about the ranks of Mordell-Weil groups can be found in [32]. I now turn to the situation in which Γ is a curve of genus 1 defined over K but not necessarily containing a point defined over K. Let J be the Jacobian 14

of Γ, defined as a curve whose points are in one-one correspondence with the divisors of degree 0 on Γ modulo linear equivalence. Then J is also a curve of genus 1 defined over K, and J(K) contains the point which corresponds to the trivial divisor. So J is an elliptic curve in our sense. Conversely, if we fix an elliptic curve J defined over K we can consider the equivalence classes (for birational equivalence over K) of curves Γ of genus 1 defined over K which have J as Jacobian. For number theory, the only ones of interest are those which contain points defined over each completion Kv . These form a commutative torsion group, called the Tate-Shafarevich group and usually denoted by X; the identity element of this group is the class which contains J itself, and it consists of those Γ which have J as Jacobian and which contain a point defined over K. (The simplest example of a nontrivial element of a Tate-Shafarevich group is the curve 3X03 + 4X13 + 5X23 = 0 with Jacobian Y03 + Y13 + 60Y23 = 0.) Thus for curves of genus 1 the Tate-Shafarevich group is by definition the obstruction to the Hasse principle. Suppose in particular that the elliptic curve J is defined over Q and has the form Y 2 = (X − c1 )(X − c2 )(X − c3 ) where the distinguished point is taken to be the point at infinity. The three points (ci , 0) on J have order 2; they are called the 2-division points. To any rational point (x, y) on Γ there exist m1 , m2 , m3 and y1 , y2 , y3 such that mi (x − ci ) = yi2

for i = 1, 2, 3;

here the mi are really elements of Q∗ /Q∗2 but it is convenient to treat them as square-free integers. We must have m1 m2 m3 = m2 for some integer m, and my = y1 y2 y3 . Conversely the equations Yi2 = mi (X − ci ) (i = 1, 2, 3) and mY = Y1 Y2 Y3 for any m, mi with m1 m2 m3 = m2 define a curve C = C(m1 , m2 , m3 ) and a four-to-one map C → J. If C(Q) is not empty, its image under this map is a coset of 2J(Q) in J(Q), and we obtain all such cosets in this way. Thus we could find J(Q) if we could decide which C are soluble in Q. After a change of variables we can assume that the ci are in Z. Define the good primes for J as those which do not divide 2(c1 − c2 )(c2 − c3 )(c3 − c1 ); 15

then it is not hard to show that C is locally soluble at all good primes if and only if all the mi are units at all good primes. So there are only finitely many C whose solubility in Q is at all hard to decide. The curves C obtained in this way are called 2-coverings of J, and the process of obtaining them is called a 2-descent. They form a group under multiplication of the corresponding triples (m1 , m2 , m3 ). The finite subgroup consisting of those 2-coverings which are everywhere locally soluble is called the 2-Selmer group. It is easily computable; and since there is a canonical embedding of J(Q)/2J(Q) into the 2-Selmer group, this provides an upper bound for the rank of J(Q). The descent process can be continued, though with somewhat greater difficulty; for 4-descents see [11]. One can also carry out 2-descents for the more general elliptic curve Y 2 = X 3 + aX 2 + bX + c; but in order to do this one requires information about the splitting field of the right hand side. The weak form of the Birch/Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve J is equal to the order of the zero of L1 (s, J) at s = 1; the conjecture also gives an explicit formula for the leading coefficient of the power series expansion at that point. Note that this point is in the critical strip, so that the conjecture pre-supposes the analytic continuation of L1 (s, J). At present there are two well-understood cases in which analytic continuation is known: when K = Q, so that J can be parametrised by means of modular functions, and when J admits complex multiplication. In consequence, these two cases are likely to be easier than the general case; but even here I do not expect much further progress in the next decade. In each of these two cases, if one assumes the Birch/Swinnerton-Dyer conjecture one can derive an algorithm for finding the Mordell-Weil group and the order of the Tate-Shafarevich group; and in the first of the two cases this algorithm has been implemented by Gebel. (See [16].) Without using the Birch/Swinnerton-Dyer conjecture, Heegner long ago produced a way of generating a point on J whenever K = Q and J is modular; and Gross and Zagier ([18] and [19]) have shown that this point has infinite order precisely when L0 (1, J) 6= 0. Building on their work, Kolyvagin (see [17]) has shown the following. Theorem 2 Suppose that the Heegner point has infinite order; then the group J(Q) has rank 1 and X(J) is finite. 16

Kolyvagin [25] has also obtained sufficient conditions for both J(Q) and X(J) to be finite. The following result is due to Nekovar and Plater. Theorem 3 If the order of L(s, J) is odd then either J(Q) is infinite or the p-part of X(J) is infinite for every good ordinary p. If J can be parametrized by modular functions for some arithmetic subgroup of SL2 (R) then analytic continuation and the functional equation follow; but there is not even a plausible conjecture identifying the J which have this property, and there is no known analogue of Heegner’s construction. In the complex multiplication case, what is known is as follows. Theorem 4 Let K be an imaginary quadratic field and J an elliptic curve defined and admitting complex multiplication over K. If L(1, J) 6= 0, then (i) J(K) is finite; (ii) for every prime p > 7 the p-part of X(J) is finite and has the order predicted by the Birch/Swinnerton-Dyer conjecture. Here (i) is due to Coates and Wiles, and (ii) to Rubin. For an account of the proofs, see [31]. Katz has generalized (i) and part of (ii) to behaviour over an abelian extension of Q, but with the same J as before. In general we do not know how to compute X. It is conjectured that it is always finite; and indeed this assertion can be regarded as part of the Birch/Swinnerton-Dyer conjecture, for the formula for the leading coefficient of the power series for L1 (s, J) at s = 1 contains the order of X(J) as a factor. If indeed this order is finite, then it must be a square; for Cassels has proved the existence of a skew-symmetric bilinear form on X with values in Q/Z, which is nonsingular on the quotient of X by its maximal divisible subgroup. Thus finiteness implies that if X contains at most p − 1 elements of order exactly p for some prime p then it actually contains no such elements; hence an element which is killed by p is trivial, and the curves of genus 1 in that equivalence class contain points defined over K. For use later, we state the case p = 2 as a lemma. Lemma 3 Suppose that X(J) is finite and the quotient of the 2-Selmer group of J by its soluble elements has order at most 2; then that quotient is actually trivial.

17

This result will play a crucial role in §§6 and 7. 5. Generalities about surfaces. Over C a full classification of surfaces can be found in [1]. A first coarse classification is given by the Kodaira dimension κ, which for surfaces can take the values −∞, 0, 1 or 2. What also seems to be significant for the number theory (and cuts across this classification) is whether the surface is elliptic — that is, whether over C there is a map V → C for some curve C whose general fibre is a curve of genus 1. The case when the map V → C is defined over the ground field K and C has genus 0 is discussed in §6; in this case the Diophantine problems for V are only of interest when C(K) is nonzero, in which case C can be identified with P1 . When C has genus greater than 1, the map V → C is essentially unique and it and C are therefore both defined over K. By Faltings’ theorem, C(K) is then finite; thus each point of V (K) lies on one of a finite set of fibres, and it is enough to study these. In contrast, we know nothing about the case when C is elliptic. The surfaces with κ = −∞ are precisely the ruled surfaces — that is, those which are birationally equivalent over C to P1 × C for some curve C. Among these, by far the most interesting are the rational surfaces, which are birationally equivalent to P2 over C. Surfaces with κ = 0 fall into four families: • Abelian surfaces. These are the analogues in two dimensions of elliptic curves, and there is no reason to doubt that their number-theoretical properties largely generalize those of elliptic curves. • K3 surfaces, including in particular Kummer surfaces. Some but not all K3 surfaces are elliptic. • Enriques surfaces, whose number theory has been very little studied. Enriques surfaces are necessarily elliptic. • bielliptic surfaces. Surfaces with κ = 1 are necessarily elliptic. Surfaces with κ = 2 are called surfaces of general type — which in mathematics is generally a derogatory phrase. About them there is currently nothing to say beyond Lang’s conjecture stated in §2. In the next two sections I shall outline what can at present be said about rational surfaces and K3 surfaces respectively; these appear to be the two 18

most interesting families of surfaces for the number-theorist. In both cases many of the most recent results depend on one or both of two major conjectures. One of these (for the reason given near the end of §4) is the finiteness of the Tate-Shafarevich group; the other is Schinzel’s Hypothesis, which we now describe. It gives a conjectural answer to the following question: given finitely many polynomials F1 (X), . . . , Fn (X) in Z[X] with positive leading coefficients, is there an arbitrarily large integer x at which they all take prime values? There are two obvious obstructions to this: • One or more of the Fi (X) may split in Z[X]. • There may be a prime p such that for any value of x mod p at least one of the Fi (x) is divisible by p. P Clearly the second obstruction can only happen for p ≤ deg(Fi ). Schinzel’s Hypothesis is that these are the only obstructions: in other words, if neither of them happens then we can choose an arbitrarily large x so that every Fi (x) is a prime. There are various more complicated variants of this hypothesis (including ones in other algebraic number fields), but they all follow fairly easily from the hypothesis in its original form. No one in his right mind would attempt to prove Schinzel’s Hypothesis; indeed one instance is the notoriously intractable twin primes problem, which is the special case when the Fi are the two polynomials X + 1 and X − 1. But probabilistic arguments suggest that the hypothesis is in fact true. At the very least it would be perverse to look for counter-examples to results which have been proved subject to Schinzel’s Hypothesis. 6. Rational surfaces. ¿From the number-theoretic point of view, there are two kinds of rational surface: • Pencils of conics, given by an equation of the form a0 (u, v)X02 + a1 (u, v)X12 + a2 (u, v)X22 = 0

(5)

where the ai (u, v) are homogeneous polynomials of the same degree. Pencils of conics can be classified in more detail according to the number of bad fibres.

19

• Del Pezzo surfaces of degree d, where 0 < d ≤ 9. Over C, such a surface is obtained by blowing up (9 − d) points of P2 in general position. It is known that Del Pezzo surfaces of degree d > 4 satisfy the Hasse principle and weak approximation; indeed those of degree 5 or 7 necessarily contain rational points. Del Pezzo surfaces of degree 2 or 1 have no aesthetic merits and have attracted little attention; it seems sensible to ignore them until the problems coming from those of degrees 4 and 3 have been solved. The Del Pezzo surfaces of degree 3 are the nonsingular cubic surfaces, which have an enormous but largely irrelevent literature, and those of degree 4 are the nonsingular intersections of two quadrics in P4 . For historical reasons, attention has been concentrated on the Del Pezzo surfaces of degree 3; but the problems presented by those of degree 4 are necessarily simpler. We consider first pencils of conics, and assume that (5) is defined over Q, the argument for an arbitrary algebraic number field not being essentially different. We can require the coefficients of the ai (u, v) to be in Z. Since the Hasse principle holds for conics, it is enough to choose u = u0 , v = v0 in such a way that (5) is locally soluble at 2, ∞ and all the odd primes which divide any of the ai (u0 , v0 ). As it stands, this appears to involve arguing in a circle; the way to make the argument respectable is as follows. Assume that (5) is everywhere locally soluble. By absorbing suitable factors into the Xi , we can ensure that the ai (u, v) are square-free and coprime. To achieve this, we have to drop the condition that the ai (u, v) are all of the same degree; but it is still true that their degrees are all even or all odd. Denote by B the set of bad places, which turns out to consist of 2, ∞, the primes which divide the discriminant of a0 (u, v)a1 (u, v)a2 (u, v) and the primes which do not exceed the degree of that product. Let S be the space of all pairs of coprime integers u0 , v0 , with the topology induced by the places of B; and let S0 be the subset of S consisting of the points at which (5) is locally soluble at every place in B. By hypothesis, S0 is not empty; and it is open in S. To obtain solubility in Q, we have to choose u0 , v0 in S0 so that (5) is locally soluble at each good prime p0 which divides one of the ai (u0 , v0 ); for solubility at the other good primes is trivial, and we have already taken care of the bad places. Let c(u, v) be the irreducible factor of that one of the ai (u, v) for which p0 |c(u0 , v0 ), and to fix ideas assume that c(u, v) divides a2 (u, v); here c(u, v) is unique because p0 does not divide the discriminant of

20

a0 a1 a2 . The condition of local solubility at p0 is (a0 (u0 , v0 )a1 (u0 , v0 ), c(u0 , v0 ))p0 = +1

(6)

where the outer bracket is the Hilbert symbol. So a necessary condition for the solubility of (5) is that all the conditions like Y (7) (a0 (u0 , v0 )a1 (u0 , v0 ), c(u0 , v0 ))p = +1 hold simultaneously for some (u0 , v0 ) in S0 , where the product is taken over all p which divide c(u0 , v0 ). What is unexpected is that this turns out to be useful, because of the following lemma. The proof of the lemma is straightforward, since the function Φ behaves like a quadratic residue symbol and can be evaluated by a Euclidean algorithm process very like that which is used for such symbols. Lemma 4 Let F (u, v), G(u, v) be homogeneous polynomials in Z[u, v], with deg(F ) even. Let B be a finite set of places of Q which contains 2, ∞ and all the primes which divide the discriminant of F G. For any coprime u0 , v0 in Z, write Y (F (u0 , v0 ), G(u0 , v0 ))p (8) Φ(B; F, G; u0 , v0 ) =

where the outer bracket on the right is the Hilbert symbol and the product is taken over all primes p not in B such that p|G(u0 , v0 ). Then Φ(u0 , v0 ) is continuous in the topology on S, and computable. In this result we take F = a0 a1 , G = c; we noted above that deg(a0 a1 ) is necessarily even. It follows that a necessary condition for the solubility of (5) is that there is a point (u0 , v0 ) in S0 such that Φ(u0 , v0 ) = +1 for all Φ which can be generated from (5) in this way. This condition is computable, and it is unsurprising (though not obvious) that it turns out to be equivalent to the Brauer-Manin condition for (5). If one assumes Schinzel’s Hypothesis, this condition is also sufficient. For suppose that u0 , v0 have been so chosen that there is only one good prime p0 which divides c(u0 , v0 ); then the product in (7) reduces to the left hand side of (6), and so (6) holds for this prime. Now choose an open set N ⊂ S0 such that (7) holds throughout N for each c(u, v); by a slightly modified version of Schinzel’s Hypothesis we can choose (u0 , v0 ) in N so that every c(u0 , v0 ) is the product of one good prime and possibly some factors in B. As c runs through all irreducible factors of a1 a2 a3 , p0 runs through all those primes for 21

which we have to verify (6). Thus (5) is everywhere locally soluble for the pair u0 , v0 , and therefore globally soluble. With minor modifications, the same argument shows that (subject to Schinzel’s Hypothesis) the Brauer-Manin obstruction is also the only obstruction to weak approximation. If there is no Brauer-Manin obstruction, this construction finds infinitely many conics in the pencil which contain rational points. But, somewhat unexpectedly, even if we know some conics of the pencil which are soluble, without Schinzel’s Hypothesis we do not know how to generate more such conics. Question 12 Given a pencil of conics and finitely many conics in the pencil each of which contains rational points, can we generate further conics of the pencil which contain rational points without using Schinzel’s Hypothesis? If we know even one rational point on a Del Pezzo surface V of degree 3 or 4, we can obtain an infinity of curves of genus 0 each of which lies on V , though they will be singular and for degree 3 it will usually not be true that each point of V (Q) lies on at least one curve of the family. But without such a point, the best we can do is to find on V a family of curves of genus 1. At first sight, it would seem that in these circumstances nothing resembling the argument above can be applied; for an essential component of that argument was that conics satisfy the Hasse principle, and this is not true for curves of genus 1. However Lemma 3 provides us with a way round this obstacle. The arguments involved are applicable to some surfaces V which are not necessarily rational, but which are elliptic with a fibration V → P1 . Consider a pencil of curves Cλ of genus 1, each of which is a 2-covering of its Jacobian Jλ . If we can choose λ in such a way that Cλ is everywhere locally soluble and at least half the elements of the 2-Selmer group of Jλ are soluble (and if we assume the finiteness of the relevent Tate-Shafarevich group), then it will follow from Lemma 3 that Cλ is soluble. For this machinery to have any chance of working, we must be able to implement the 2-descent on Jλ in a manner which is uniform in λ. This more or less requires Jλ to have its 2-division points defined over Q(λ) and therefore to have the form Y 2 = (X − c1 (λ))(X − c2 (λ))(X − c3 (λ))

(9)

where the ci (λ) are in Q(λ); but an additional trick, given in [2], enables the method to be used even if Jλ has just one 2-division point in Q(λ). 22

The details of this method are unattractive, but the strategy is as follows. (For a full account, see [13].) Without loss of generality we can assume that the ci (λ) are in Z[λ]. For any particular integral value λ0 of λ, the bad places for the equation (9) are the bad places for the system together with the primes which divide one of the ci (λ0 ) − cj (λ0 ). It was explained in §4 how to implement the 2-descent for (9). We shall eventually use Schinzel’s Hypothesis to choose λ0 so that the value of each irreducible factor of any ci (λ) − cj (λ) at λ = λ0 is the product of some bad primes for the system with one good prime. We call the latter a Schinzel prime; though it is a good prime for the system, it is a bad prime for the curve obtained by writing λ = λ0 in (9). The effect of restricting λ0 in this way is that we know those 2-coverings of (9) for λ = λ0 which are locally soluble at all good primes for the curve; they form a finite group of 2-coverings Cλ0 of Jλ which does not depend on the choice of λ0 provided it satisfies the condition above. This group certainly contains the original Cλ and the 2-coverings which correspond to the 2-division points. The next step, which involves a sophisticated analysis of the 2-descent process and also requires us to introduce additional well chosen bad primes for the system, is to find local conditions on λ0 at the bad primes of the system which ensure that for λ = λ0 • the only elements of this group which are locally soluble at all the bad places of the system lie in the subgroup generated by the original Cλ and the 2-coverings generated by the 2-division points; and • the original Cλ is locally soluble at all the bad places of the system. This is not always possible; if it is impossible, that provides an obstruction to this method of attack on the problem though not necessarily to the solubility of the system. In general this obstruction is not much stronger than the Brauer-Manin obstruction, and in some cases they are provably the same; so this is not too serious a blemish on the method. If we achieve the two properties above then solubility at the Schinzel primes turns out to be automatic. (This is an example of what seems to be a rather general phenomenon, that if one thing goes wrong, so must at least one other which is related to it.) With our enlarged set of bad places for the system, we now choose λ0 to satisfy the local conditions and the Schinzel condition in the previous paragraph. By Lemma 3, the curve (9) is soluble for this value of λ. But in contrast to what happens for pencils of conics, this kind of argument appears to provide no information about weak approximation. 23

Question 13 Can one modify the method above so that it works without any assumption about the 2-division points of Jλ ? Unfortunately it is not known (and probably is not even true) that an arbitrary Del Pezzo surface of degree 4 contains a pencil of curves of genus 1 of this particular type — and the situation for Del Pezzo surfaces of degree 3 is much worse, in that the natural curves to consider are 3-coverings rather than 2-coverings. However, for Del Pezzo surfaces of degree 4 some progress has been made. Salberger and Skorobogatov [33] have shown that the Brauer-Manin obstruction is the only obstruction to weak approximation. (Recall that weak approximation presupposes the existence of at laest one rational point.) As for the Hasse principle, the present situation is as follows. A Del Pezzo surface of degree 4 defined over the algebraic number field K has the form V : F1 (X0 , . . . , X5 ) = F2 (X0 , . . . , X5 ) = 0, where the coefficients of F1 and F2 are in K. By a linear transformation defined over an extension K1 of odd degree over K, we can separate off one of the variables; and over K1 we can find a pencil of curves Cλ of genus 1 on V for which each Jλ has one 2-division point defined over K1 (λ). Using the trick described in [2], and subject to an obstruction typical for the method, it can now be shown that V contains a point defined over K1 . A straightforward geometric argument, which does not rely on K being an algebraic number field, now shows that V contains a point defined over K. Unfortunately the overall arguments are so complicated (and so unnatural) that it is not clear whether the obstruction to the method is still simply the Brauer-Manin obstruction to the solubility of V over K; but even if it is stronger, it is not much stronger. To use and then collapse a field extension in this way is a device which probably has a number of uses. For such a collapse step to be feasible, the degree of the field extension needs to be prime to the degree of the variety; and this leads one to phrase the same property somewhat differently. Question 14 Let V be a variety defined over a field K, not necessarily of a number-theoretic kind. For what families of V is it true that if V contains a 0-cycle of degree 1 defined over K then it contains a point defined over K? As stated above, this is true for Del Pezzo surfaces of degree 4. For pencils of conics it is in general false, even for algebraic number fields K. For Del 24

Pezzo surfaces of degree 3 the question is open: I expect it to be true for algebraic number fields K but false for general fields. A variant of the method above can be applied to diagonal cubic surfaces V : a0 X03 + a1 X13 = a2 X23 + a3 X33 ,

(10)

subject to one very counterintuitive condition, which is that K, the field of definition of V , must not contain the primitive cube roots of unity. Write V in the form a0 X03 + a1 X13 = λY 3 , a2 X23 + a3 X33 = λY 3 (11) where λ is at √ our disposal. We now have two pencils of curves of genus 1, each of which is a −3-covering of its Jacobian; and we have to apply the method simultaneously to both curves. This introduces considerable additional complications, for which see [44]; and the obstruction to the method, though weak, is certainly strictly stronger than the Brauer-Manin obstruction. The reason for the condition on K is that otherwise the√curves (11) would admit complex multiplication, and the latter acts on the −3-Selmer groups; thus the order of the latter would always be an odd power of 3, whereas it has to be reduced to 9 for the method to work. (Here one factor √ 3 arises because of the 3-division points on the Jacobian defined over Q( −3).) On the other hand, in this argument we only need apply Schinzel’s Hypothesis to the single polynomial X, so that it can be replaced by Dirichlet’s theorem on primes in arithmetic progression. All this relates to Question (A). For Question (B) the only known results are for quadrics, for which see the remarks after Theorem 1. It seems reasonable to ask whether there is an analogous result for other kinds of rational surfaces; this is another problem for which the first step should probably be to use numerical search to generate a plausible conjecture. For this purpose, one needs to examine a system with not too many parameters; and this leads to the following question: Question 15 For the surface V given by (10) with the ai in Z, is there a polynomial P in the |ai | such that if V is soluble in Z then it has such a solution for which each summand is absolutely bounded by P ? The ideal answer to Question (C) would be to provide a birational map V → P2 defined over Q. However, it can be shown that such a map exists for nonsingular cubic surfaces V if and only if V (Q) is not empty and V contains a divisor defined over Q which consists of 2, 3 or 6 skew lines. (For 25

Del Pezzo surfaces of degree 4, the second condition must be replaced by the statement that V contains a divisor defined over Q which consists of one or more skew lines.) For Chˆatelet surfaces, which have the form X2 (X02 − cX12 ) = f (X2 , X3 ) where c is a non-square and f is homogeneous cubic, it is known (see [12]) that there is a finite set of parametric solutions (each in 4 inhomogeneous variables) such that each point of V (Q) is represented by one of them, though in an infinity of different ways. But in general more than one parametric solution is needed, and it was already shown in [26] that parametric solutions in only 2 variables cannot play a useful part in the process. Question 16 Is there a larger class of cubic surfaces (ideally, the class of all nonsingular cubic surfaces) for which analogous results hold? The question of parametric solutions is linked to the idea of R-equivalence. Let V be a variety defined over Q; then R-equivalence is defined as the finest equivalence relation such that two points given by the same parametric solution are equivalent. Alternatively, it is the finest equivalence relation such that for any map f : P1 → V and points P1 , P2 , all defined over Q, the points f (P1 ) and f (P2 ) are equivalent. A good deal is known about R-equivalence on cubic surfaces; in particular, it is shown in [43] that the closure of an R-equivalence class in V (A) is computable, and that the closures of two Requivalence classes are either the same or disjoint. There is an example in [12] of a Chˆatelet surface V containing two distinct R-equivalence classes, each of which has the whole of V (A) as its closure. Work of Coray and Tsfasman shows that this V is birationally equivalent to a nonsingular cubic surface. Question 17 Is the set of R-equivalence classes on a nonsingular cubic surface finite? 7. K3 surfaces and Kummer surfaces. K3 surfaces are the simplest kind of variety about whose number-theoretic properties very little is known; indeed they still present many unsolved problems even to the geometer. There are infinitely many families of K3 surfaces; the simplest of them, and the only one which will be considered in the present article, consists of all nonsingular quartic surfaces. An important special type of K3 surfaces consists of the Kummer surfaces, a phrase which can carry either of two related meanings: 26

• The quotient of an Abelian surface A by the automorphism −1; this has 16 singular points corresponding to the 16 2-division points of A. • The nonsingular surface obtained by blowing up the 16 singular points in the previous definition. One advantage of Kummer surfaces in comparison with general K3 surfaces is that for the former it is easy to determine Pic(V¯ ). Some K3 surfaces contain one or more pencils of curves of genus 1, and these pencils may even be of the kind discussed in the previous section; but one should not confine one’s attention to K3 surfaces with this additional property. For the time being, there is merit in concentrating on diagonal quartics V : a0 X04 + a1 X14 + a2 X24 + a3 X34 = 0, (12) because these contain few enough parameters to make systematic numerical experimentation possible. However, the number theory of such surfaces may be exceptional, because the geometry certainly is. Indeed Pic(V¯ ) has rank 20, which is the largest possible value for any K3 surface, and it is generated by the classes of the 48 lines on V¯ ; moreover V is a Kummer surface up to isogeny, and indeed is the Kummer surface of E × E where E is a certain elliptic curve which admits complex multiplication. One consequence of this is that V is rigid in the sense of algebraic geometry. There is an obvious map from V to the quadric surface W : a0 Y02 + a1 Y12 + a2 Y22 + a3 Y32 = 0. If a0 a1 a2 a3 is a square, and V is everywhere locally soluble, each of the two families of lines on W is defined over the ground field, and each such line lifts to a curve of genus 1 on V ; moreover the Jacobians of these curves have the form (9), so that the methods of the previous section can be applied. Martin Bright [7] has computed and tabulated Br1 (V )/Br(Q) for all V of the form (12); it is necessary to do this by computer, because there are 546 distinct cases. Assuming Schinzel’s Hypothesis and the finiteness of X, I had previously shown in [42] that over Q the Brauer-Manin obstruction is the only obstruction to the Hasse principle in the most general case in which a0 a1 a2 a3 is a square. (Most general in this context means that none of the ±ai aj is a square and a0 a1 a2 a3 is not a fourth power.) It seems reasonable to hope that the same property will still hold in all the cases for which a0 a1 a2 a3 is a square; but there are too many of them to examine individually. On 27

the other hand, there is strong numerical evidence that when a0 a1 a2 a3 is not a square the obstruction coming from Br1 (V ) is not in general the only obstruction to the Hasse principle. Question 18 What is the additional obstruction in this case? One particularly interesting example is the surface X04 + 2X14 = X24 + 4X34 ;

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this has two obvious rational points, but appears to have no others. 8. Density of rational points. So far I have ignored Question (D). It differs from the others in that it is not a birational question, but is associated with a particular embedding of the variety V in projective space. For simplicity we work over Q. A point P in Pn defined over Q has a representation (x0 , . . . , xn ) where the xi are integers with no common factor; and this representation is unique up to changing the signs of all the xi . We define the height of P to be max |xi |; a linear transformation on the ambient space multiplies heights by numbers which lie between two positive constants depending on the linear transformation. Denote by N (H, V ) the number of points of V (Q) whose height is less than H; then it is natural to ask how N (H, V ) behaves as H → ∞. This is the core question for the Hardy-Littlewood method, which when it is applicable is the best (and often the only) way of proving that V (Q) is not empty. In very general circumstances that method provides estimates of the form N (H, V ) = leading term + error term. The leading term is usually the same as one would obtain by probabilistic arguments. But such results are only valuable when it can be shown that the error term is small compared to the leading term, and to achieve this the dimension of V needs to be large compared to its degree. The extreme case of this is the following theorem, due to Birch [3]. Theorem 5 Let r1 , . . . , rm be positive odd integers, not necessarily all different. Then there exists N0 (r1 , . . . , rm ) with the following property. For any N ≥ N0 let Fi (X0 , . . . , XN ) be homogeneous polynomials with coefficients in Z and deg Fi = ri for i = 1, . . . , m. Then the Fi have a common nontrivial zero in ZN . 28

The proof falls into two parts. First, the Hardy-Littlewood method is used to prove the result in the special case when m = 1 and F1 is diagonal — that is, to show that if r is odd and N ≥ N1 (r) then c0 X0r + . . . + cN XNr = 0 has a nontrivial integral solution. Then the general case is reduced to this special case by purely elementary methods. The requirement that all the ri should be odd arises from difficulties connected with the real place; over a totally complex algebraic number field there is a similar theorem for which the ri can be any positive integers. Question 19 In Theorem 5, can the condition that all the ri are odd be replaced by the requirement on the Fi that the projective variety given by F1 = . . . = Fm = 0 has a nonsingular real point? The Hardy-Littlewood method was designed for a single equation in which the variables are separated — for example, an equation of the form f1 (X1 ) + . . . + fN (XN ) = c where the fi are polynomials, the Xi are integers, and one wishes to prove solubility for all integers c, or all large enough c, or almost all c. But it has also been applied both to several simultaneous equations and to equations in which the variables are not separated. The following theorem of Hooley [22] is the most impressive result in this direction. Theorem 6 Homogeneous nonsingular nonary cubics over Q satisfy both the Hasse principle and weak approximation. It appears that the Hardy-Littlewood method can only work for families for which N (H, V ) is asymptotically equal to its probabilistic value; in particular it seems unlikely that it can be made to work for families for which weak approximation fails. Manin has put forward a conjecture about the asymptotic density of rational solutions for certain geometrically interesting families of varieties for which weak approximation is unlikely to hold: more precisely, for Fano varieties embedded in Pn by means of their anticanonical divisors. For simplicity, we describe his conjecture only for Del Pezzo surfaces V of degrees 3 and 4. To ask about N (H, V ) is now the wrong question, for V may contain lines L defined over Q, and for any line N (H, L) ∼ AH 2 for 29

some nonzero constant A. This is much greater than the order-of-magnitude estimate for N (H, V ) given by a probabilistic argument. For the latter sugQ gests an estimate AH (N (p)/(p + 1)), where the product is taken over all primes less than a certain bound which depends on H. In view of what is said in §3, this product ought to be replaced by something which depends on the behaviour of L2 (s, V ) near s = 1. More precisely, the way in which the leading term in the Hardy-Littlewood method is obtained suggests that here we should take s − 1 to be comparable with (log H)−1 . Remembering the Tate conjecture, this gives the right hand side of (14) as a conjectural estimate for N (H, V ). But if this argument were valid, L would contain more rational points than V , even though V ⊃ L. Manin’s way to resolve this absurdity is to study not N (H, V ) but N (H, U ), where U is the open subset of V obtained by deleting the 27 or 16 lines on V . Manin conjectured that N (H, U ) ∼ AH(log H)r−1 where r is the rank of Pic(V );

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and Peyre [28] has given a conjectural formula for A. (But note that there exist Fano varieties of dimension greater than 2 for which (14) is certainly false; and it is not clear how Manin’s conjecture should be modified to cover such cases.) Various people have proved this conjecture for the cone X0 X1 X2 = X33 , and there are also results for the singular cubic surface X0 X1 X2 + X0 X1 X3 + X0 X2 X3 + X1 X2 X3 = 0, to which attention had been drawn by Birch. Heath-Brown [21] has proved that A1 H(log H)6 < N (H, U ) < A2 H(log H)6 for suitable constants A1 , A2 ; but he doubts whether his method is capable of proving an asymptotic formula. Using quite different ideas, Rudge has sketched a proof of the asymptotic formula; but the details are not yet complete. Question 20 Are there nonsingular Del Pezzo surfaces of degree 3 or 4 for which the Manin conjecture can be proved? In the first instance, it would be wise to address this problem under rather restrictive hypotheses about Pic(V ), not least because the Brauer-Manin obstruction to weak approximation occurs in the conjectural formula for A and therefore the problem is likely to be easier for families of V for which 30

weak approximation holds. A one-sided estimate for one such family is given in [39]. For Del Pezzo surfaces, the value of c for which N (H, U ) ∼ AH(log H)c is defined by the geometry rather than by the number theory, though that is not true of A. For other varieties, the corresponding statement need no longer be true. We start with curves. For a curve of genus 0 and degree d, we have N (H, V ) ∼ AH 2/d ; and for a curve of genus greater than 1 Faltings’ theorem is equivalent to the statement that N (H, V ) = O(1). But if V is an elliptic curve then N (H, V ) ∼ A(log H)r/2 where r is the rank of the Mordell-Weil group. (For elliptic curves there is a more canonical definition of height, which is invariant under bilinear transformation; this is used to prove the result above.) For pencils of conics, Manin’s question is probably not the best one to ask, and it would be better to proceed as follows. A pencil of conics is a surface V together with a map V → P1 whose fibres are conics. Let N ∗ (H, V ) be the number of points on P1 of height less than H for which the corresponding fibre contains rational points. Question 21 What is the conjectural estimate for N ∗ (H, V ) and under what conditions can one prove it? It may be worth asking the same questions for pencils of curves of genus 1. For surfaces of general type, Lang’s conjecture implies that questions about N (H, V ) are really questions about certain curves on V ; and for Abelian surfaces (and indeed Abelian varieties in any dimension) the obvious generalisation of the theorem for elliptic curves holds. But K3 surfaces pose new problems — and not ones on which any practicable amount of computation is likely to shed light. If V is a K3 surface, then we have to study not N (H, V ) but N (H, U ) where U is obtained from V by deleting the curves of genus 0 on V defined over Q, of which there may be an infinite number. One can expect that N (H, U ) ∼ A(log H)c for some constants A and c; and it seems reasonable to hope that c will be a half-integer. The surface (13) suggests that we can have c = 0, and it must be certain (though perhaps difficult to prove) that c can sometimes be strictly positive. Question 22 Can the value of c be obtained from the L-series L2 (s, V )? Question 23 If V is a Kummer surface obtained from the Abelian surface A, is c related to the rank of the Mordell-Weil group of A? 31

I am indebted to Jean-Louis Colliot-Th´el`ene for many valuable comments; but he bears no responsibility for the opinions expressed. REFERENCES [1] A.Beauville, Complex Algebraic Surfaces (2nd ed., Cambridge, 1996). [2] A.O.Bender and Sir Peter Swinnerton-Dyer, Solubility of certain pencils of curves of genus 1, and of the intersection of two quadrics in P4 , Proc. London Math. Soc (3) 83(2001), 299-329. [3] B.J.Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4(1957), 102-105. [4] S.J.Bloch, Higher regulators, algebraic K-theory and zeta-functions of elliptic curves, CRM Monograph Series 11 (AMS, 2000). [5] S.J.Bloch and K.Kato, L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, vol. I, pp. 333-400 (Birkhauser, 1990). [6] A.Borel, Cohomologie de SL2 et valeurs de fonctions zeta aux points entiers, Ann. Sc. Norm. Pisa (1976), 613-636. [7] M.Bright, Ph.D. dissertation, (Cambridge, 2002). [8] M.Bright and Sir Peter Swinnerton-Dyer, Computing the Brauer-Manin obstructions, Math. Proc. Cambridge Phil. Soc. (to appear). [9] Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry (ed. Jan Denef et al), Contemporary Mathematics, vol. 270. [10] Mathematical Developments arising from Hilbert Problems, AMS Symposia in Pure Mathematics, Vol XXVIII (ed. F.E.Browder), (Providence, 1976). [11] J.W.S.Cassels, Second descents for elliptic curves, J. reine angew. Math. 494(1998), 101-127. [12] J-L.Colliot-Th´el`ene, J-J.Sansuc and Sir Peter Swinnerton-Dyer, Intersections of two quadrics and Chˆatelet surfaces, J. reine angew. Math. 373(1987), 37-107 and 374(1987), 72-168. [13] J-L.Colliot-Th´el`ene, A.N.Skorobogatov and Sir Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134(1998), 579-650. [14] J-L.Colliot-Th´el`ene and Sir Peter Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. Reine Angew. Math., 453(1994) 49-112. [15] Rational Points, ed. G.Faltings and G. W¨ ustholz (3rd ed., Vieweg, 1992). 32

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THE COX RING OF A DEL PEZZO SURFACE VICTOR V. BATYREV AND OLEG N. POPOV Abstract. Let Xr be a smooth Del Pezzo surface obtained from P2 by blow-up of r ≤ 8 points in general position. It is well known that for r ∈ {3, 4, 5, 6, 7, 8} the Picard group Pic(Xr ) contains a canonical root system Rr ∈ {A2 × A1 , A4 , D5 , E6 , E7 , E8 }. In this paper, we prove some general properties of the Cox ring of Xr (r ≥ 4) and show its similarity to the homogeneous coordinate ring of the orbit of the highest weight vector in some irreducible representation of the algebraic group G associated with the root system Rr .

1. Introduction Let X be a projective algebraic variety over a field k. Assume that the Picard group Pic(X) is a finitely generated abelian group. Consider the vector space M Γ(X) := H 0 (X, O(D)). [D]∈Pic(X)

One wants to make it an k-algebra which is graded by the monoid of effective classes in Pic(X) such that the algebra structure will be compatible with the natural bilinear map bD1 ,D2 : H 0 (X, O(D1 )) × H 0 (X, O(D2 )) → H 0 (X, O(D1 + D2 )). However, there exist some problems in the realization of this idea. We remark that first of all there is no any natural isomorphism between H 0 (X, O(D)) and H 0 (X, O(D0 )) if [D] = [D0 ]. There exists only a canonical bijection between the linear systems |D| ∼ = |D0 | ( |D| denotes a projectivization of of the k-vector space H 0 (X, O(D))). As a consequence, the bilinear map bD1 ,D2 depends not only on the classes [D1 ], [D2 ], [D1 + D2 ] ∈ Pic(X), but also on their particular representatives. One can easily see that only the morphism s[D1 ],[D2 ] : |D1 | × |D2 | → |D1 + D2 | of the product of two projective spaces |D1 | × |D2 | to another projective space |D1 + D2 | is well-defined. For this reason, it is much more natural to consider the graded set of projective spaces G P(X) := |D| [D]∈Pic(X)

together with the all possible morphisms s[D1 ],[D2 ] ([D1 ], [D2 ] ∈ Pic(X)). Date: This version: August 3, 2003. Key words and phrases. Del Pezzo surfaces, torsors, homogeneous spaces, algebraic groups. 2000 Mathematics Subject Classification. Primary . 1

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VICTOR V. BATYREV AND OLEG N. POPOV

Inspired by the paper of Cox on the homogeneous ring of a toric variety [Cox], Hu and Keel [H-K] suggested a definition of a Cox ring M Cox(X) = R(X, L1 , . . . , Lr ) := H 0 (X, O(m1 L1 + · · · + mr Lr ) (m1 ,...,mr )∈Zr

which uses a choice of some Z-basis L1 , . . . , Lr in Pic(X) (e.g. if Pic(X) ∼ = Zr is a free abelian group). Using such a Z-basis, one obtains a particular representative for each class in Pic(X) together with a well-defined multiplication so that R(X, L1 , . . . , Lr ) becomes a well-defined k-algebra. If L01 , . . . , L0r is another Z-basis of Pic(X), then the corresponding Cox algebra R(X, L01 , . . . , L0r ) is isomorphic to R(X, L1 , . . . , Lr ). Unfortunately, we can not expect to choose a Z-basis of Pic(X) in a natural canonical way. More often one can choose in a natural way some effective divisors D1 , . . . , Dn on X such that Pic(X) is generated by [D1 ], . . . , [Dn ]. If we set U := X \ (D1 ∪ · · · ∪ Dn ) and assume that X is smooth, then P ic(U ) = 0 and we obtain the exact sequence n M 1 → k∗ → k[U ]∗ → Z[Di ] → Pic(X) → 0. i=1

Choosing a k-rational point p in U , we can split the monomorphism k∗ → k[U ]∗ , so that one has an isomorphism k[U ]∗ ∼ = k∗ ⊕ G, where G ⊂ k[U ]∗ is a free abelian group of rank n − r. The choice of a k-rational point p ∈ U allows to give another approach to the graded space Γ(X) and to the Cox algebra: Definition 1.1. Let X, U, p, D1 , . . . , Dn be as above. We consider the graded k-algebra M Γ(X, U, p) := H 0 (X, O(m1 D1 + · · · + mn Dn ) (m1 ,...,mn )∈Zn

and define Cox(X, U, p) := Γ(X, U, p)G as the k-subalgebra of all G-invariant elements in Γ(X, U, p). Since P ic(X) ∼ = Zn /G, we obtain a natural Pic(X)-grading on Cox(X, U, p). We expect that the algebra Cox(X, U, p) can de applied to some arithmetic questions about k-rational points in U ⊂ X (e.g. see [S]). Remark 1.2. If X is a smooth projective toric variety and U ⊂ X is the open dense torus orbit, then the choice of a point p ∈ U defines an isomorphism of U with the algebraic torus T , so that the subgroup G ⊂ k[U ]∗ can be identified with the character group of T . In this way, one can show that Cox(X, U, p) is isomorphic to a polynomial ring in n variables (n is the number of irreducible components of X \ U , cf. [Cox]). Let Xr be a smooth Del Pezzo surface obtained from P2 by blow-up of r ≤ 8 points in general position. It is well known that for r ∈ {3, 4, 5, 6, 7, 8} the Picard group Pic(Xr ) contains a canonical root system Rr ∈ {A2 × A1 , A4 , D5 , E6 , E7 , E8 }. Moreover, the

THE COX RING OF A DEL PEZZO SURFACE

3

natural embedding Pic(Xr−1 ) ,→ Pic(Xr ) induces induces the inclusion of root systems Rr−1 ,→ Rr . If G(Rr ) is a connected algebraic group corresponding to the root system Rr , then the embedding Rr−1 ,→ Rr defines a maximal parabolic subgroup P (Rr−1 ) ⊂ G(Rr ). We expect that for r ≥ 4 there should be some relation between a Del Pezzo surface Xr and the GIT-quotient of the homogeneous space G(Rr )/P (Rr−1 ) modulo the action of a maximal torus Tr of G(Rr ). Our starting observation is the well-known isomorphism X4 ∼ = Gr(2, 5)//T4 which follows from an isomorphism between the homogeneous coordinate ring of the Grassmaniann Gr(3, 5) = G(A4 )/P (A2 × A1 ) ⊂ P9 and the Cox ring of X4 . Another proof of this fact follows form the identification of X4 with the moduli space M0,5 of stable rational curves with 5 marked points [K]. In this paper, we start an investigation of the Cox ring of Del Pezzo surfaces Xr (r ≥ 4). It is natural to choose the classes of all exceptional curves E1 , . . . , ENr ⊂ Xr as a generating set for the Picard group Pic(Xr ). There is a natural Z≥0 -grading on Pic(Xr ) defined by the intersection with the anticanonical divisor −K. We prove some general properties of Cox rings of Del Pezzo surfaces Xr (r ≥ 4) and show their similarity to the homogeneous coordinate ring of G(Rr )/P (Rr−1 ). We remark that the homogeneous space G(Rr )/P (Rr−1 ) can be interpreted as the orbit of the highest weight vector in some natural irreducible representation of G(Rr ). Remark 1.3. Some other connections between Del Pezzo surfaces and the corresponding algebraic groups were considered also by Friedman and Morgan in [F-M]. A similar topic was considered by Leung in [Le]. In this paper, we show that the Cox ring of a Del Pezzo surface Xr is generated by elements of degree 1. This implies that the homogeneous coordinate ring of G(Rr )/P (Rr−1 ) is naturally graded by the monoid of effective divisor classes on the surface Xr (the same monoid defines the multigrading of the Cox ring of Xr ). Moreover, we obtains some results of the quadratic relations between the generators of the Cox ring of Xr . The authors would like to thank Yu. Tschinkel, A. Skorobogatov, E. S. Golod, S. M. Lvovski and E. B. Vinberg for useful discussions and encouraging remarks. 2. Del Pezzo Surfaces Let us summarize briefly some well-known classical results on Del Pezzo surfaces which can be found in [Ma, Dem, Na]. One says that r (r ≤ 8) points p1 , . . . , pr in P2 are in general position if there are no 3 points on a line, no 6 points on a conic (r ≥ 6) and a cubic having seven points and one of them double does not have the eighth one (r = 8). Denote by Xr (r ≥ 3) the Del Pezzo surfaces obtained from P2 by blowing up of r points p1 , . . . , pr in general position. If π : Xr → P2 the corresponding projective morphism, then the Picard group P ic(Xr ) ∼ = Zr+1 contains a Z-basis li , (0 ≤ i ≤ r), l0 = [π ∗ O(1)] −1 and li := [π (pi )], i = 1, . . . , r. The intersection form (∗, ∗) on P ic(Xr ) is determined in the chosen basis by the diagonal matrix: (l0 , l0 ) = 1, (li , li ) = −1 for i ≥ 1, (li , lj ) = 0 for i 6= j,. The anticanonical class of Xr equals −K = 3l0 − l1 − · · · − lr . The number

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VICTOR V. BATYREV AND OLEG N. POPOV

d := (K, K) = 9 − r is called the degree of Xr . The anticanonical system | − K| of a Del Pezzo surface Xr is very ample if r ≤ 6, it determines a two-fold covering of P2 if r = 7, and it has one base point, determining a rational map to P1 if r = 8. Smooth rational curves E ⊂ Xr such that (E, E) = −1 and (E, −K) = 1 are called exceptional curves. Theorem 2.1. [Ma] The exceptional curves on Xr are the following: (1) (2) (3) (4) (5) (6) (7)

blown-up points p1 , . . . , pr ; lines through pairs of points pi , pj ; conics through 5 points from {p1 , . . . , pr }(r ≥ 5); cubics, containing 7 points and 1 of them double (r ≥ 7); quartics, containing 8 points and 3 of them double (r = 8); quintics, containing 8 of point and 6 of them double (r = 8); sextics, containing 8 of those points, 7 of them double and 1 triple (r = 8).

The number Nr of exceptional curves on Xr is given by the following table: r 3 4 5 6 7 8 Nr 6 10 16 27 56 240 The root system Rr ⊂ Pic(Xr ) is defined as Rr := {α ∈ Pic(Xr ) : (α, α) = −2, (α, −K) = 0}. It is easy to show that Rr is exactly the set of all classes α = [Ei ] − [Ej ] where Ei and Ej are two exceptional curves on Xr such that Ei ∩ Ej = ∅. The corresponding Weyl group Wr is generated by the reflections σ : x 7→ x + (x, α)α for α ∈ Rr . There are so called simple roots α1 , . . . , αr such that the corresponding reflexions σ1 , . . . , σr form a minimal generating subset of Wr . The set of simple roots can be chosen as α1 = l1 − l2 , α2 = l2 − l3 , α3 = l0 − l1 − l2 − l3 , αi = li−1 − li , i ≥ 4. The blow up morphism Xr → Xr−1 determines an isometric embedding of the Picard lattices Pic(Xr−1 ) ,→ Pic(Xr ). This induces the embeddings for root systems, simple roots and Weyl groups Wr . For r ≥ 3, the Dynkin diagram of Rr can be considered as the subgraph on the vertices αi (i ≤ r) of the following graph: α1 e

α2 e

α4 e

α5 e

α6 e

α7 e

α8 e

α3 e

In particular, we obtain R3 = A2 × A1 , R4 = A4 , R5 = D5 , R6 = E6 , R7 = E7 , R8 = E8 . Denote by $1 , . . . , $r the dual basis to the Z-basis −α1 , . . . , −αr . Each $i is the highest weight of an irreducible representation of G(Rr ) which is called a fundamental representation. We will denote by V ($) the representation space of G(Rr ) with the highest weight $.

THE COX RING OF A DEL PEZZO SURFACE

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Definition 2.2. A dominant weight $ is called minuscule if all weights of V ($) are nonzero and the Wr -orbit of the highest weight vector is a k-basis of V ($) [G/P-I]). A dominant weight $ is called quasiminuscule [G/P-III], if all nonzero weights of V ($) have multiplicity 1 and form an Wr -orbit of $ (the zero weight of V ($) may have some positive multiplicity). One can see from the explicit description of the root systems Rr that $r is minuscule for 3 ≤ r ≤ 7, and $8 is quasiminuscule. The dimension dr of of the irreducible representation V ($r ) of G(Rr ) is given by the following table: r 4 5 6 7 8 dr 10 16 27 56 248 We will need the following statement: Proposition 2.3. Let D be a divisor on a Del Pezzo surface Xr (2 ≤ r ≤ 8) such that (D, E) ≥ 0 for every exceptional curve E ⊂ Xr . Then the following statements hold: (i) the linear system |D| has no base points on any exceptional curve E ⊂ Xr ; (ii) if r ≤ 7, then the linear system |D| has no base points on Xr at all. Proof. Induction on r. If r = 2, then there exists exactly 3 exceptional curves E0 , E1 , E2 , whose classes in the standard basis are l0 − l1 − l2 , l1 , l2 . Moreover [E0 ], [E1 ] and [E3 ] form a basis of the Picard lattice P ic(X2 ). The dual basis w.r.t. the intersection form is l0 , l0 − l1 , l0 − l2 . Therefore the conditions on D imply that [D] = n0 l0 + n1 (l0 − l1 ) + n2 (l0 − l2 ), n0 , n1 , n2 ∈ Z≥0 So it is sufficient to check that the linear systems with the classes l0 , l0 − l1 , l0 − l2 have no base points. The latter immediately follows from the fact that the first system defines the birational morphism X2 → P2 contracting E1 and E2 , the second and third linear systems define conic bundle fibrations over P1 . For r > 2, we consider a second induction on deg D = (D, −K). If there is an exceptional curve E ⊂ Xr with (D, E) = 0, then the invetible sheaf O(D) is the inverse image of an invertible sheaf O(D0 ) on the Del Pezzo surface Xr−1 obtained by the contraction of E. Since the pull back of any exceptional curve on Xr−1 under the birational morphism πE : Xr → Xr−1 is again an exceptional curve on Xr , we obtain that D0 satisfy all conditions of the proposition on Xr−1 . By the induction assumption ∗ D 0 | has no base points (r − 1 ≤ 7), |D0 | has no base points on Xr−1 . Therefore, |D| = |πE on Xr . If there is no exceptional curve E ⊂ Xr with (D, E) = 0, then we denote by m the minimal intersection number (D, E) where E runs over all exceptional curves. Since we have (E, −K) = 1 for all exceptional curves, the divisor D0 := D + mK has nonnegative intersections with all exceptional curves and there exists an exceptional curve E ⊂ Xr with (D0 , E) = 0. Since deg D0 = (D0 , −K) = (D, −K) − m(K, K) < (D, −K) = deg D, by the induction assumption, we obtain that |D0 | is base point free. If r ≤ 7, then the anticanonical linear system | − K| has no base points. Therefore, |D| = |D0 + m(−K)| is

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also base point free. In the case r = 8, | − K| does have a base point p ∈ X8 . However, p cannot lie on an exceptional curve E, because the short exact sequence 0 → H 0 (X8 , O(−K − E)) → H 0 (X8 , O(−K)) → H 0 (E, O(−K)|E ) → 0 induces an isomorphism H 0 (X8 , O(−K)) ∼ = H 0 (E, O(−K)|E ) (since deg(−K − E) = 0 0 and H (X8 , O(−K − E)) = 0). 3. Generators of Cox(Xr ) Let {E1 , . . . , ENr } be the set of all exceptional curves on a Del Pezzo surface Xr . We S r choose a k-rational point p ∈ U := Xr \ ( N i=1 Ei ) and denote the ring Cox(Xr , U, p) (see 1.1) simply by Cox(Xr ). The ring Cox(Xr ) is graded by the semigroup Meff (Xr ) ⊂ Pic(Xr ) of classes of effective divisors on Xr . There is a coarser grading on Cox(Xr ) given by M Cox(Xr )d ∼ H 0 (Xr , O(D)), = (D,−K)=d

with respect to which we shall speak about the degree d = (D, −K) of a divisor D. Proposition 3.1. The ring Cox(X3 ) is isomorphic to a polynomial ring in 6 variables k[x1 , . . . , x6 ], where xi are sections defining all 6 exceptional curves on X3 . Proof. The Del Pezzo surface X3 is a toric variety which can be descrobed as the blownup of 3 torus invariant points (1:0:0), (0:1:0) and (0:0:1) on P2 . So we can apply a general result of Cox on toric varieties [Cox] (see also 1.2). Theorem 3.2. For 3 ≤ r ≤ 8, the ring Cox(Xd ) is generated by elements of degree 1. If r ≤ 7, then the generators of Cox(Xd ) are global sections of invertible sheaves defining the exceptional curves. If r = 8, then we should add to the above set of generators two linearly independent global sections of the anticanonical sheaf on X8 . Proof. Induction on r. The case r = 3 is settled by the previous proposition. For r > 3 we choose an effective divisor D on Xr . We call a section s ∈ H 0 (Xr , O(D)) a distinguished global section if its support is contained in the union of exceptional curves of Xr (r ≤ 7), or if its support is contained in the union of exceptional curves of X8 and some anticanonical curves on X8 . Our purpose is to show that the vector space H 0 (Xr , O(D)) is spanned by all distinguished global sections. This will be proved by induction on deg D := (D, −K) > 0. We consider several cases: • If there exists an exceptional curve E such that (D, E) < 0, then H 0 (Xr , O(D)|E ) = 0 and it follows from the exact sequence H 1 (Xr , O(D)|E ) → H 0 (Xr , O(D − E)) → H 0 (Xr , O(D)) → 0 that the multiplication by a non-zero distinguished global section of O(E) induces an epimorphism H 0 (Xr , O(D − E)) → H 0 (Xr , O(D)). Since deg (D − E) = deg D −1, using the induction assumption for D0 = D −E, we obtain the required statement for D.

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• If there exists an exceptional curve E such that (D, E) = 0, then O(D) is the inverse image of a sheaf O(D0 ) on the Del Pezzo surface Xr−1 obtained by the contraction of E. Therefore we have an isomorphism H 0 (Xr , O(D)) ∼ = H 0 (Xr−1 , O(D0 )) and, by the induction assumption for r − 1, we obtain the required statement for D, because distinguished global sections of O(D0 ) lift to distinguished global sections of O(D). • If D = −K, (or, equivalently, if (D, E) = 1 for every exceptional curve E), then O(D)|E ) is isomorphic to OE (1) and we have H 1 (Xr , O(D)|E ) = 0 together with the exact sequence 0 → H 0 (Xr , O(D − E)) → H 0 (Xr , O(D)) → H 0 (Xr , O(D)|E ) → 0, where H 0 (Xr , O(D)|E ) is 2-dimensional. Since one has deg (D − E) = deg D − 1. Using the induction assumption for D0 = D − E, it remains show that there exists two linearly independent distinguished global sections of O(D) such that their restriction to E are two linearly independent global sections of O(D)|E . We describe these two distinguisched sections explicitly for each value of r ∈ {4, 5, 6, 7, 8}. Without loss of generality we can assume that [E] = l1 . If r = 4, then we write the anticanonical class −K = 3l0 − l1 − · · · − l4 in the following two ways: −K = (l0 − l1 − l2 ) + (l0 − l3 − l4 ) + (l0 − l2 − l3 ) + l2 + l3 = (l0 − l1 − l3 ) + (l0 − l2 − l4 ) + (l0 − l2 − l3 ) + l2 + l3 . These two decompositions of −K determine two distinguished global sections of O(−K) with support on 5 exceptional curves. The projections of these sections under the morphism X4 → P2 are shown below in Figure 1. @re3 @ r1 E

r4 @ @

@e r2 @

r4 @re3 @ @ = @ 1 @ re2 E r @

Figure 1. Two distinguished anticanonical classes for r = 4. The restriction of the first section to E vanishes at the intersection point q1 of E with the exceptional curve with the class l0 − l1 − l2 . The restriction of the second section to E vanishes at the intersection point q2 of E with the exceptional curve with the class l0 − l1 − l3 . It is clear that q1 6= q2 . So the distinguished anticanonical sections are linearly independent. If r = 5, then we write the anticanonical class as −K = 3l0 − l1 − · · · − l5 = (l0 − l1 − l2 ) + (l0 − l3 − l4 ) + (l0 − l4 − l5 ) + l4 = (l0 − l1 − l5 ) + (l0 − l2 − l3 ) + (l0 − l3 − l4 ) + l3 .

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The corresponding distinguished anticanonical sections vanish at two different intersection points of E with the exceptional curves belonging to the classes l0 − l1 − l2 and l0 − l1 − l5 . If r = 6, then we write the anticanonical class as −K = 3l0 − l1 − · · · − l6 = (l0 − l1 − l2 ) + (l0 − l3 − l4 ) + (l0 − l5 − l6 ) = (l0 − l1 − l6 ) + (l0 − l5 − l4 ) + (l0 − l3 − l2 ). The corresponding distinguished anticanonical sections vanish at two different intersection points of E with the exceptional curves belonging to the classes l0 − l1 − l2 and l0 − l1 − l6 . If r = 7, then we write the anticanonical class as −K = 3l0 − l1 − · · · − l7 = (2l0 − l1 − l2 − l3 − l4 − l5 ) + (l0 − l6 − l7 ) = (2l0 − l7 − l6 − l5 − l4 − l3 ) + (l0 − l2 − l1 ). The corresponding distinguished anticanonical sections vanish at two different intersection points of E with the exceptional curves belonging to the classes 2l0 − l1 − l2 − l3 − l4 − l5 and l0 − l2 − l1 . If r = 8, then deg D − E = 0. Therefore, H 0 (X8 , O(D − E)) = 0 (see the proof of 2.3) and we have an isomorphism H 0 (X8 , O(D)) ∼ = H 0 (X8 , O(D)|E ). So H 0 (X8 , O(D)|E ) is generated by the restrictions of the anticanonical sections and we’re done. • If (D, E) ≥ 1 for all exceptional curves E and D 6= −K, then we denote by m the minimum of the numbers (D, E) for all exceptional curves. Let E0 be an exceptional curve such that (D, E0 ) = m ≥ 1. We define D0 = D − E0 and D00 := D + mK. By 2.3, |D0 | and |D00 | have no base points (if r ≤ 7). Moreover, D00 can be seen as zero of a distinguished global section s ∈ H 0 (Xr , O(D + mK)) whose support does not contain the exceptional curve E0 (if r ≤ 8). We have the short exact sequence 0 → H 0 (Xr , O(D0 )) → H 0 (Xr , O(D)) → H 0 (Xr , O(D)|E0 ) → 0. By the induction assumption, the space H 0 (Xr , O(D0 )) is generated by distinguished global sections. It remains to show that there exist distinguished global sections of O(D) such that their restriction to E0 generate the space H 0 (Xr , O(D)|E0 ). Since (−mK, E0 ) = (D, E0 ) = m, the space H 0 (Xr , O(D)|E0 ) is isomorphic to H 0 (Xr , O(−mK)|E0 ). Since (D00 , E0 ) = 0 the distinguished global section s ∈ H 0 (Xr , O(D + mK)) nowhere vanish on E0 . Therefore the multiplication by the distinguished global section s defines a homomorphism H 0 (Xr , O(−mK)) → H 0 (Xr , O(D))

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whose restriction to E0 is an isomorphism H 0 (Xr , O(−mK)|E0 ) ∼ = H 0 (Xr , O(D)|E0 . Therefore, it is enough to show that restrictions of the distinguished global sections of O(−mK) to E0 generate the space H 0 (Xr , O(−mK)|E0 ). Our previous considerations have shown this for m = 1. The general case m ≥ 1 follows now immediately from the fact that the homomorphism H 0 (Xr , O(−K)) → H 0 (E0 , OE0 (1)) is surjective and the space H 0 (E0 , OE0 (m)) is spanned by tensor products of m elements from H 0 (E0 , OE0 (1)). Corollary 3.3. The semigroup Meff (Xr ) ⊂ Pic(Xr ) of classes of effective divisors on a Del Pezzo surfaces Xr (2 ≤ r ≤ Xr ) is generated by elements of degree 1. These elements are exactly the classes of exceptional curves if r ≤ 7 and the classes of exceptional curves together with the anticanoncal class for r = 8. Proposition 3.4. If D is an effective divisor of degree ≥ 2 on X8 , then the vector space H 0 (X8 , O(D)) is spanned by distinguished global sections of O(D) with supports only on exceptional curves. Proof. By 3.2 and 3.3, it is sufficient to check the statement for D = −2K and for D = −K + E for any exceptional curve. The latter case immediately follows from 3.2, because D = −K + E is the pull back of the anticanonical sheaf on X7 obtained by the contraction of E. In the case D = −2K, we obtain 120 distinguished global sections of O(D) from 120 pairs of exceptional curves Ei , Ej such that (Ei , Ej ) = 3: −2K = 6l0 − 2l1 − . . . − 2l8 = l1 + (6l0 − 3l1 − 2l2 . . . − 2l8 ). Remark 3.5. Since H 0 (Xr , O(E)) is 1-dimensional for each exceptional curve E ⊂ Xr , we can choose a nonzero section xE ∈ H 0 (Xr , O(E)) which is determined up to multiplication by a nonzero scalar. Therefore the affine algebraic variety A(Xr ) := SpecCox(Xr ) is embedded into the affine space ANr on which the maximal torus Tr ⊂ G(Rr ) acts in a canonical way such that the space ANr can identified with the representation space V ($r ) of the algebraic group G(Rr ) (if r ≤ 7). In the case r = 8, all 240 exceptional curves on X8 can be similarly identified with all non-zero weights of the adjoint representation of G(E8 ) in V ($8 ). The space V ($8 ) contains the weight-0 subspace of dimension 8, but the ring Cox(Xr ) has only 2-dimensional space of anticanonical sections. Thus, there is no any identification of degree-1 homogeneous component of Cox(X8 ) with the representation space V ($8 ) of G(E8 ). 4. Quadratic relations in Cox(Xr ) Let us denote P (Xr ) := Proj Cox(Xr ). If 4 ≤ r ≤ 7, then P (Xr ) is canonically embedded into the projective space PNr −1 ( Nr is the number of exceptional curves on Xr ). For any exceptional curve E ⊂ X, we consider the open chart UE ⊂ PNr −1 defined by the condition xE 6= 0. Thus, we obtain an open covering of P (Xr ) by Nr affine subsets UE ∩ P (Xr ). We denote by A(Xr ) ⊂ ANr the affine cone over P (Xr ).

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Proposition 4.1. The ring Cox(X4 ) is isomorphic to the subring of all 3 × 3-minors of a generic 3 × 5-matrix. In particular, the projective variety P (X4 ) is isomorphic to the Grassmannian Gr(3, 5). Proof. To describe the multiplication in R(X5 ), one needs to choose a basis in Pic(X5 ). We choose the basis l0 , . . . , l4 , as in Section 2. We choose a divisor in each basis class: l0 being the preimage of the line at infinity w.r.t. a blow-down on P2 , li the exceptional fibers of this blow-down. We identify the representatives of each Picard class with the P subsheaves O( ci li ) of the constant sheaf k(X5 ). Then the multiplication in the ring is just the multiplication of the corresponding functions in the function field k(X5 ) of X5 . We choose the functions that represent xE ’s in the following way: let x : y : z be the homogeneous coordinates on P2 and let (xi : yi : zi ), i = 1, . . . , 4, be the coordinates of the blown-up points. Consider the matrix   x1 x2 x3 x4 x/z M =  y1 y2 y3 y4 y/z  . z1 z 2 z3 z4 1 Let MI for I ⊂ [1, 5], |I| = 3, denote the maximal minor of M consisting of the columns with numbers in I, taken in the natural order. Then we set xli := M[1,4]\{i} for i ∈ [1, 4], xl0 −li −lj := M{i,j,5} for 1 ≤ i < j ≤ 4. All these functions lie in the corresponding O(D)’s and are non-zero because the points are in general position. It is known that the generators of the homogeneous coordinate ring of G(3, 5) are naturally identified with the maximal minors of a generic 3 × 5-matrix. We send these generic minors into the corresponding minors of the matrix above. This determines a homomorphism of the homogeneous coordinate ring of G(3, 5) to Cox(X5 ), as the equations of the Grassmann variety, being the relations between generic minors, hold for any particular minors. As R(X5 ) is generated by xE ’s, it is surjective. This homomorphism respects the Picard grading. Therefore this homomorphism respects the Z-grading. As it is surjective, it induces a closed embedding of P (X5 ) into G(3, 5). As both varieties are irreducible of dimension 6 (because ), it is an isomorphism of varieties, therefore they coincide as subvarieties in P9 , therefore we have an isomorphism of rings. And we see that Cox(X5 ) is defined by quadratic relations, as the homogeneous coordinate ring of G(3, 5) is. The article [G/P-I] describes a k-basis for the homogeneous coordinate ring of G/P in the case, when P is a maximal parabolic subgroup containing a Borel subgroup B such that the fundamental weight $ corresponding to P is minuscule (see 2.2). It also shows that this ring is defined by quadratic relations. A way to write explicitly the quadratic relations for the orbit of the highest weight vector for any representation of a semisimple Lie group is given in [Li]. A more geometric approach to these quadratic equations is contained in the proof of Theorem 1.1 in [L-T]: Proposition 4.2. The orbit G/P$ of the highest weight vector in the projective space PV ($) is the intersection of the second Veronese embedding of PV ($) with the subrepresentation V (2$) of the symmetric square S 2 V ($). Moreover, these quadratic relations generate the ideal of G/P$ ⊂ PV ($).

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We expect that the following statement is true: Conjecture 4.3. The ideal of relations between the degree-1 generators of Cox(Xr ) is generated by quadrics for all 4 ≤ r ≤ 8. Proposition 4.4. Let Xr−1 the Del Pezzo surface obtained by the contraction of E on Xr . Then there exist an isomorphism UE ∩ P (Xr ) ∼ = A(Xr−1 ). Proof. The coordinate ring of the affine variety UE ∩ P (Xr ) consists of all fractions D f /xdeg such that f ∈ H 0 (Xr , O(D)). Let πE : Xr → Xr−1 be the contraction of E. E Then any divisor class [D] ∈ Pic(Xr ) can be uniquely represented as sum [D0 ] + k[E] ∗ (Pic(X where [D0 ] = πE r−1 )). Such a fraction is a meromorphic section over D − (deg D)E with possible poles at E, and two such fractions are equal exactly when they determine the same section (xE is evidently a nonzerodivisor from the definition of the multiplication), so the affine coordinate ring is M M H 0 (Xr \ E, O(D)) = H 0 (Xr−1 \ πE (E), π∗ O(D)), D∈Pic(Xr ) deg D=0

D∈Pic(Xr \E)

where π : Xr → Xr−1 is the blow-down of E, because the condition deg D = 0 determines a unique extension of each divisor from Xr \E to Xr and the blow-down is an isomorphism outside E. Now, in the last sum one needn’t exclude πE (E) because it is a point on a normal surface Xr−1 , so the last sum is just our ring Cox(Xr−1 ) for a Del Pezzo surface Xr−1 with r one smaller, i.e. the affine coordinate ring of A(Xr−1 ). Corollary 4.5. The singular locus of the algebraic varieties P (Xr ) and A(Xr ) has codimension 7. Proof. Since A(X3 ) ∼ = A6 , we obtain that P (X4 ) is a smooth variety covered by 10 affine charts which are isomorphic to A6 . Using the isomorphism P (X4 ) ∼ = Gr(3, 5) (see 4.1 ), we obtain that A(X4 ) has an isolated singularity at 0. Therefore, the singular locus of P (X5 ) consists of 16 isolated points. The singular locus of P (X6 ) is 1-dimensional and the singular locus of P (X7 ) is 2-dimensional. Definition 4.6. A divisor class D of a sum of two exceptional curves intersecting with multiplicity 1, or, equivalently, satisfying (D, D) = 0, (D, −K) = 2, is called a ruling, as the corresponding invertible sheaf determines a conic bundle Xr → P1 . Lemma 5.3 of [F-M] says that the Weyl group acts transitively on rulings. Let D be a sum of two exceptional curves which meet each other. Then it has several such decompositions which determine several monomials from Γ(D). One can see that for r ≥ 4 there are more such monomials than the dimension of this space, so they are linearly dependent and it determines quadratic relations between the generators of Cox(Xr ). If the curves coincide or do not intersect, then the first case in the proof of Proposition 3.2 shows that the decomposition is unique and there are no relations. Because of the Picard grading one has to look only at such homogeneous relations, living over one divisor.

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The expectations are that the ring is defined by these relations, and for r = 3, 4 this follows from Lemma 3.1 and Proposition 4.1, but in general we have proved only a weaker result: Theorem 4.7. For 4 ≤ r ≤ 7, the ring Cox(Xr ) is defined by quadratic relations up to radical. Proof. Let us look more closely at the relations for r = 4, 5, 6. A sum D of two intersecting exceptional curves can be written as l0 − l1 in a suitable basis, so the global sections of O(D) correspond to linear homogeneous polynomials on P2 that vanish at the point l1 , therefore dim Γ(D) = 2, so there is a linear relation between any three monomials over D (but not between two, as they have different divisors). For r ≤ 6 the exceptional curves have only simple intersections, so if E + F = E 0 + F 0 are two such decompositions, then 0 = (E, E + F ) = (E, E 0 + F 0 ) requires (E, E 0 ) = (F, F 0 ) = 0 as the last two intersection numbers are nonnegative. We see that D = l0 − l1 = li + (l0 − l1 − li ), i ≥ 2 admits r − 1 such decompositions, so for r = 4, 5, 6 every monomial xE xF , (E, F ) 6= 0 is equal to a polynomial in some xE 0 , (E 0 , E) = 0, i.e. the affine coordinate ring of the quadratic variety in the affine chart xE 6= 0 is generated by {xF /xE |(E, F ) = 0}, i.e. by the variables corresponding to a blow-down of E. To show the italicized feature for r = 7 one has in addition to express the xE 0 /xE with (E, E 0 ) = 2 in terms of xF /xE , (E, F ) ≤ 1, because then the variables xF /xE , (E, F ) = 1 can be reduced to those with intersection zero, as we’ve just shown. But E + E 0 = −K and a basis of sections over it was described in the proof of the proposition 3.2, namely, let E1 be a point, E2 and E3 two lines through it, then xEi xEi0 is the basis (one can see from [Ma, 26.9] that ∀E∃!E 0 (E, E 0 ) = 2, so the involution ·0 is well-defined). If E is another point (which it without loss of generality is), then writing xE xE 0 in terms of this basis one obtains the desired result. The quadratic relations determine another variety in the same projective space, which contains the torsor. We need to show that these varieties coincide, and it suffices to show it in every affine chart. Let U = {xE 6= 0} be this chart, Xr−1 the blowdown of E. Let Rq (Xr−1 ) be the ring defined by quadratic relations for Xr−1 , Rq (Xd )U the coordinate ring of the quadratic variety for r in our affine chart, R(Xd )U = R(Xr−1 ) the coordinate ring of P (Xd ) in the chart = the homogeneous coordinate ring of P (Xr−1 ). Then it follows from the italicized feature that one has surjections Rq (Xr−1 ) → Rq (Xr )U → R(Xr )U , because the relations that we had for Xr−1 hold for the lifts of the sections to Xr . Now by induction on r, the basis being r = 3 and no relations at all, we can assume that in a chart the map between the rings Rq (Xr−1 ) → R(Xr−1 ) is the factorization modulo radical, so the varieties coincide in a chart, as we need. References [Cox] [Dem]

[F-M]

D. Cox. The Homogeneous Coordinate Ring of a Toric Variety. // J. Algebr. Geom. 4, No.1, 17–50 (1995). M. Demazure. Surfaces de Del Pezzo. I. II. III. IV. V. // S´eminaire sur les singularit´es des surfaces, Cent. Math. Ec. Polytech., Palaiseau 1976–77, Springer Lect. Notes Math. 777, 21–69 (1980). R. Friedman, J. W. Morgan. Exceptional Groups and del Pezzo Surfaces. math.AG/0009155.

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[G/P-I] C. S. Seshadri. Geometry of G/P. I: Theory of standard monomials for minuscule representations. // C. P. Ramanujam. — A tribute. Collect. Publ. of C. P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 207–239 (Springer-Verlag, 1978). [K] M. M. Kapranov. Chow Quotients of Grassmannians. I, I.M. Gelfand Seminar, S. Gelfand, S. Gindikin eds., Advances in Soviet Mathematics vol. 16, part 2., A.M.S., 29-110 (1993). [G/P-III] V. Lakshmibai, C. Musili, C. S. Seshadri. Geometry of G/P. III: Standard monomial theory for a quasi-minuscule P. // Proc. Indian Acad. Sci., Sect. A, Part III 88, No.2, 93–177 (1979). [H-K] Y. Hu, S. Keel. Mori Dream Spaces and GIT. math.AG/0004017. [Hm] J. E. Humphreys. Linear algebraic groups. Graduate Texts in Mathematics, 21. Springer-Verlag, 1975. [Le] N. C. Leung. ADE-bundle over rational surfaces, configuration of lines and rulings. math.AG/0009192. [Li] W. Lichtenstein. A system of quadrics describing the orbit of the highest weight vector. // Proc. Am. Math. Soc. 84, No.4, 605–608 (1982). [L-T] G. Lancaster, J. Towber. Representation-Functors and Flag-Algebras for the Classical Groups. I. // J. Algebra 59, No.1, 16–38 (1979). [Ma] Yu. I. Manin. Cubic Forms. Algebra, Geometry, Arithmetic. North-Holland Mathematical Library. Vol. 4. North-Holland, 1974. [Na] M. Nagata. On rational surfaces. I: Irreducible curves of arithmetic genus 0 or 1. // Mem. Coll. Sci., Univ. Kyoto, Ser. A 32, 351–370 (1960). On rational surfaces. II. // Mem. Coll. Sci., Univ. Kyoto, Ser. A 33, 271–293 (1960). [S] A. N. Skorobogatov. On a theorem of Enriques-Swinnerton-Dyer. // Ann. Fac. Sci. Tiulouse Math. (6), 2 , no.3, 429-440 (1993). ¨ t Tu ¨bingen, Auf der Morgenstelle 10, Tu ¨bingen DMathematisches Institut Universita 72076, Germany E-mail address: [email protected] Department of Algebra, Faculty of Mathematics, Moscow State University, Moscow 117234, Russia E-mail address: [email protected]

Counting rational points on threefolds Niklas Broberg∗

Per Salberger

July 24, 2003 Abstract Let X ⊂ P4 be an irreducible hypersurface and ε > 0 be given. We show that there are O(B 3+ε ) resp. O(B 55/18+ε ) rational points on P4 lying on X when X is of degree d ≥ 4 resp. d = 3. The implied constants depend only on d and ε.

1

Introduction

Let Q be an algebraic closure of Q and Pn be projective n-space over Q. That n+1 is, Pn is the set of one-dimensional linear subspaces of Q . A point on Pn is n+1 said to be rational if it represents a subspace of Q generated by an element in Qn+1 . The height of a rational point p ∈ Pn (Q) is given by H(p) = max(|Z0 | , . . . , |Zn |), where Z0 , . . . , Zn are relatively prime integers such that p = [Z0 , . . . , Zn ]. For a Zariski closed subset X of Pn , let N (X, B) be the counting function N (X, B) = # {p ∈ Pn (Q) ∩ X : H(p) ≤ B} , where Pn (Q) is the set of rational points on Pn . In the case where X is defined by an irreducible form F (Z0 , . . . , Zn ) of degree d ≥ 2, Heath-Brown [4] conjectured that N (X, B) = On,d,ε (B n−1+ε ) for any ε > 0. The implied constant should thus not depend on F , only on n, d, and ε. In the same paper he verified the conjecture for curves, surfaces, and for quadrics of any dimension. In [1], Browning proved the conjecture for non-singular hypersurfaces in P4 of degree at least four. In this paper we shall prove the following result. Theorem 1. Let X ⊂ P4 be a hypersurface defined by an irreducible form F (Z0 , . . . , Zn ) with coefficients in Q. Then the following holds for any ε > 0: ( Od,ε (B 3+ε ) if d ≥ 4 N (X, B) = Oε (B 55/18+ε ) if d = 3. ∗ While working on this paper, the author was supported by the EC network Arithmetic Algebraic Geometry.

1

2

Preliminaries

In this section we recollect some known estimates of counting functions. We also state and prove some results that we use in the proof of Theorem 1. First note that: • We follow the convention that a subvariety X of Pn is a closed subset that is not necessarily irreducible. A hypersurface of Pn is a subvariety of codimension one. All varieties are defined over Q. • We shall often count rational points of bounded height on Pn which lie on X even if X is not defined over Q. The same situation occur for subvarieties of the dual space Pn∗ of Pn when we count hyperplanes Γ of Pn defined over Q for which Γ ∩ X is reducible. We shall also use coordinates over Q for Grassmannians. • If X ⊂ Pn is a hypersurface, then we define the degree of X to be the degree of the corresponding reduced scheme. That is, we let d = deg(X) be the minimal degree among all forms defining X. This implies that an intersection Λ ∩ X with a linear subspace Λ ⊂ Pn not contained in X may have lower degree than X. • Our calculations involve numerous constants. To avoid introducing the constants explicitly, we use the following notation. Suppose that f1 and f2 are functions such that fi (B) ≥ 0 for all B ≥ 1. We write f1 (B) p1 ,...,pk f2 (B), if there exists a positive constant C, depending only on the parameters p1 , . . . , pk , such that f1 (B) ≤ Cf2 (B) for all B ≥ 1. We write f1 (B) p1 ,...,pk f2 (B), if f1 (B) p1 ,...,pk f2 (B) and f2 (B) p1 ,...,pk f1 (B). • The Grassmannian G(k, n) of k-dimensional linear subspaces of Pn is assumed to be embedded into projective space by the Pl¨ ucker embedding. In particular, we identify G(n − 1, n) with the dual projective space Pn∗ . The height of a rational linear subspace Λ ⊂ Pn is by definition the height of its Pl¨ ucker coordinates. According to [6, Chapter I, Corollary 5I], we have H(Λ) n det(Λ), where

Λ = x ∈ Zn+1 : [x] ∈ Λ ∪ {0}

is the lattice associated to Λ and det(Λ) is the volume of a fundamental domain of Λ.

2

2.1

Results from the Geometry of Numbers

The following result is well-known [4, Lemma 1(iii)]. It is one of the principal results from the subject known as the Geometry of Numbers, and it is one of the key tools in the proof of Theorem 1. Lemma 2.1. Let Λ ⊂ Zn be a lattice of dimension P m. Then Λ has a basis b1 , . . . , bm such that if one writes x ∈ Λ as x = j λj bj , then λj n |x| / |bj | .

Moreover one has det(Λ) n

m Y

|bj | .

j=1

The following result is a consequence of [6, Chapter I, Corollary 5J]. Lemma 2.2. Suppose that a1 , . . . , ak are linearly independent n-dimensional vectors with integer components and let Λ = {x ∈ Zn : a1 .x = · · · = ak .x = 0} . Then Λ ⊂ Zn is a lattice of dimension n − k and det(Λ) n

k Y

|aj | .

j=1

2.2

Bad linear sections

The homogeneous ideal of a hypersurface X ⊂ Pn of degree d is generated by a single homogeneous polynomial F (Z0 , . . . , Zn ) of degree d. This means that hypersurfaces in Pn of degree d are parametrised by points in P(Q[Z0 , . . . , Zn ]d ), where Q[Z0 , . . . , Zn ]d is the vector space of homogeneous polynomials of degree d in n+1 variables. Let V (F ) denote the hypersurface in Pn given by the zero locus of F ∈ Q[Z0 , . . . , Zn ]d . The set of pairs (Λ, F ) ∈ G(k, n) × P(Q[Z0 , . . . , Zn ]d ) for which Λ ∩ V (F ) is an irreducible variety of dimension k − 1 and degree d is an open subset of G(k, n) × P(Q[Z0 , . . . , Zn ]d ). We denote the complement of this open subset by Φn,d,k . The following result is well-known but the proof is so short that we reproduce it here. Lemma 2.3. Let X ⊂ Pn be an irreducible hypersurface of degree d and dimension at least two. Then the set of hyperplanes Γ for which the linear section Γ ∩ X is reducible or of degree less than d is a proper closed subset of P n∗ . Furthermore, this closed subset is cut out by hypersurfaces of degrees bounded solely in terms of n and d. The number of required hypersurfaces is also bounded in terms of n and d.

3

Proof. By choosing a basis of Q[Z0 , . . . , Zn ]d , we may identify P(Q[Z0 , . . . , Zn ]d ) with PN for some N . Suppose that the ideal of Φn,d,n−1 ⊂ Pn∗ ×PN is generated by bihomogeneous polynomials Gi (Z0 , . . . , Zn ; W0 , . . . , WN ) for i = 1, 2, . . . , m.

(2.1)

The set of hyperplanes Γ for which Γ ∩ X is reducible or of degree less than d is then the common zero locus of the polynomials (2.1), where W0 , . . . , WN are the coefficients of any homogeneous polynomial generating the ideal of X. Since a general hyperplane section Γ ∩ X is irreducible [3, Proposition 18.10], all of the polynomials (2.1) cannot vanish identically on Pn∗ . Lemma 2.4. Let X ⊂ P4 be an irreducible hypersurface of degree d and let V ⊂ P4∗ be the set of hyperplanes Γ with the following property. There is a point p ∈ Γ ∩ X such that for every two-plane Λ ⊂ Γ either Λ is contained in X or Λ ∩ X contains an irreducible component of degree less than d. Then we have the following. (a) V is a closed subset of P4∗ , and if V 6= P4∗ , then V is cut out by hypersurfaces of degrees bounded in terms of d. The number of required hypersurfaces is also bounded in terms of d. (b) If Y = Γ ∩ X is an irreducible surface for some hyperplane Γ ∈ V , then Y is a cone over a plane curve. (c) V = P4∗ if and only if X is a cone over a plane curve with respect to a vertex line. Proof. As in the proof of Lemma 2.3, we identify P(Q[Z0 , . . . , Z4 ]d ) with PN . Let Ψ ⊂ P4∗ × P4 × PN × G(2, 4) be the set of four-tuples (Γ, p, F, Λ) such that F (p) = 0, p ∈ Λ, Λ ⊂ Γ, and (Λ, F ) ∈ Φ4,d,2 . Let π be the projection map from Ψ to P4∗ × P4 × PN . Then Ψ is a projective variety and the function λ(q) = dim(π −1 (q)) is an upper-semicontinuous function on the image π(Ψ) [3, Corollary 11.13]. In particular, Ω = {(Γ, p, F ) ∈ π(Ψ) : λ(Γ, p, F ) ≥ 2} is a subvariety of P4∗ × P4 × PN . Now the set of two-planes Λ ⊂ P4 for which p ∈ Λ and Λ ⊂ Γ for some (Γ, p) ∈ P4∗ × P4 is a two-dimensional linear subspace of G(2, 4). The fibre π −1 (Γ, p, F ) is contained in this linear subspace. Hence, Ω is the set of triples (Γ, p, F ) such that F (p) = 0 and such that Λ ⊂ V (F ) or Λ ∩ V (F ) contains an irreducible component of degree less than d for every two-plane Λ ⊂ Γ containing p. Let Σ be the projection of Ω on P4∗ × PN , and let Gi (Z0 , . . . , Zn ; W0 , . . . , WN ) for i = 1, 2, . . . , m, (2.2) be bihomogeneous polynomials generating the ideal of Σ. If W0 , . . . , WN are the coefficients of some homogeneous polynomial generating the ideal of X, then V is the common zero locus of the polynomials (2.2). This proves (a). 4

Next we consider (b). Let Y = Γ ∩ X be an irreducible hyperplane section and assume that p is a point of Y such that Λ ∩ X contains an irreducible component of degree less than d for every two-plane Λ ⊂ Γ containing p. Let π : Ye → Y be the blow-up of Y at p. There is, then, a unique map ψ : Ye → P2 which extends the projection map Y 99K P2 from p to some two-plane P2 ⊂ Γ. If ψ is surjective, then ψ −1 (L) is irreducible of degree d for a general line L ⊂ P2 [2, Theorem 1.1]. This contradicts the assumption that π(ψ −1 (L)) is reducible for every line L ⊂ P2 . Hence, the map ψ is not surjective, so Y is cone over a plane curve with vertex p. To prove (c) we assume that V = P4∗ and consider the incidence correspondence Ω ⊂ P4∗ × X consisting of all pairs (Γ, p) such that Λ ⊂ X or Λ ∩ X contains an irreducible component of degree less than d for every two-plane Λ ⊂ Γ containing p. It follows from the proof of (a) that Ω is a closed subset of P4∗ × X. Since V is the projection of Ω ⊂ P4∗ × X on the first factor, we have that the dimension of Ω is at least four. According to (b), we can then find a point p ∈ X and a family of hyperplanes {Γλ } through p such that Γλ ∩ X are all cones with the common vertex p. It follows that X is a cone with vertex p over Γ ∩ X for any hyperplane Γ ⊂ P4 which does not contain p. Let Γ ⊂ P4 be such that Γ ∩ X is a cone over a plane curve C ⊂ X with vertex q ∈ X. Then X is a cone over C with two different vertices p and q. This proves the first implication of (c). The other one is immediate.

2.3

Linear subspaces of hypersurfaces

In this section we state some elementary results about linear subspaces contained in a hypersurface X ⊂ Pn . Let Fk (X) ⊂ G(k, n) denote the Fano variety of kplanes contained in the variety X ⊂ Pn . It can be shown that the number of irreducible components and the dimensions of the irreducible components of Fk (X) can be bounded in terms of d. Lemma 2.5. Let X ⊂ Pn be an irreducible hypersurface and assume that X is not a hyperplane. Then the dimension of Fn−2 (X) is at most one and Fn−2 (X) contains no lines. Proof. Suppose that Y is anSirreducible component of Fn−2 (X) of dimension at least one. Then the variety Λ∈Y Λ has dimension at least n−1 and is therefore equal to X. In particular, every point on X belongs to an (n − 2)-plane Λ in Y . Consider the incidence correspondence Ψ = {(p, Λ) ∈ X × Y : p ∈ Λ} . The fibre of Ψ over an (n − 2)-plane Λ is irreducible of dimension n − 2. The variety Ψ is therefore irreducible of dimension dim(Y )+n−2 [3, Theorem 11.14]. Now if an (n − 2)-plane Λ ⊂ X contains the point p, then Λ must lie in the projective tangent space Tp (X) of X at p. For a non-singular point p ∈ X, the dimension of X ∩ Tp (X) is n − 2, so the fibre of Ψ over a general point of X is finite. The dimension of Ψ is thus at most n − 1. Hence, the dimension of Y is 5

S at most one. To finish the proof we note that Λ∈Y Λ is a hyperplane when Y is a line. Since X is irreducible, Fn−2 (X) cannot contain any lines. The following lemma is a modification of Example 19.11 on page 244 in [3]. Lemma 2.6. Let X ⊂ Pn be the surface swept out by the lines parametrised by an irreducible curve C ⊂ G(1, n). Then the degree of X does not exceed the degree of C. Proof. The degree of X is by definition the cardinality of the intersection Λ ∩ X for a general (n − 2)-plane Λ ⊂ Pn . Assume that Λ ∩ X contains deg(X) points. Now every point of Λ ∩ X belongs to a line L ∈ C that meets Λ. The locus of lines L ∈ G(1, n) that meet Λ is a hyperplane section Γ ∩ G(1, n). If C is contained in Γ, then every line L ∈ C meet Λ. That is, we have a regular map C → Λ given by L 7→ L ∩ Λ. But C is irreducible so the image of this map is irreducible. Hence, Λ ∩ X contains only one point so that the degree of X is one. If C is not contained in Γ, then there are at most deg(C) points in Γ ∩ C. Hence, Λ ∩ X contains at most deg(C) points. Lemma 2.7. Let X ⊂ P4 be an irreducible hypersurface of degree d, and let C(p, X) be union of lines on X passing through p ∈ X. Then the number of irreducible components and the degrees of the irreducible components of C(p, X) can be bounded in terms of d. Proof. Let Pij , for 0 ≤ i < j ≤ 4, be the Pl¨ ucker coordinates on G(1, 4), and assume that p = [1, 0, 0, 0, 0]. The lines in P4 that pass through p are then parametrised by the points in the three-dimensional plane Λ ⊂ G(1, 4) which is defined by Pij = 0 for 0 < i < j ≤ 4. The map φ : C(p, X) \ p → Λ, sending a point q to the line through p and q, is given by [Z0 , Z1 , Z2 , Z3 , Z4 ] 7→ [Z1 , Z2 , Z3 , Z4 , 0, 0, . . . ]. The map φ can thus be identified with the projection π of C(p, X) \ p to the hyperplane Z0 = 0 in Pn . Under this identification, the image of π is equal to Λ ∩ F1 (X). Since the number of irreducible components and the degrees of the irreducible components of F1 (X) are bounded in terms of d, the same is true for Λ ∩ F1 (X) and C(p, X).

2.4

Estimates for counting functions

It this section we list those known estimates for counting functions that we use in the proof of Theorem 1. (E1) Let Λ be a k-dimensional linear subspace of Pn . If Λ contains k +1 linearly independent rational points of height at most B, then Λ is defined over Q and B k+1 N (Λ, B) n . H(Λ) 6

To see this, let b1 , . . . , bk be a basis of the lattice Λ = x ∈ Zn+1 : [x] ∈ Λ ∪ {0}

with the properties stated in Lemma 2.1. Since Λ contains k + 1 linearly independent rational points of height at most B, we must have |bi | n B. Hence, B k+1 B k+1 n . N (Λ, B) n |b0 | · · · |bk | H(Λ) (E2) If X ⊂ Pn is an irreducible variety of degree d and dimension r, then N (X, B) n,d B r+1 . This is proved for hypersurfaces in [4, Theorem 1]. The general result follows by a standard projection argument (see for example the proof of Lemma 1 in [1]). (E3) If X ⊂ Pn is an irreducible variety of degree d ≥ 2 and dimension r, then N (X, B) n,d,ε B r+1/d+ε , for every ε > 0 [5]. (E4) If X ⊂ Pn is an irreducible curve of degree d, then N (X, B) n,d,ε B 2/d+ε , for every ε > 0. This estimate is proved for plane curves in [4, Theorem 3]. As in (E2), the general estimate follows by a projection argument. (E5) Let Λ ⊂ Pn be a two-plane which is defined over the rational numbers. If X ⊂ Λ is a non-singular curve of degree d ≥ 2, then N (X, B) n,d,ε 1 +

B 2/d+ε , H(Λ)2/3d

for every ε > 0. This follows from Theorem 3 and Lemma 1(iii) of [4]. (E6) If X ⊂ Pn is an irreducible surface of degree d ≥ 2, then N (X, B) n,d,ε B 2+ε , for every ε > 0 [1, Lemma 1]. (E7) If X ⊂ Pn is a quadratic hypersurface of rank at least three, then N (X, B) n,ε B n−1+ε , for every ε > 0 [4, Theorem 2] 7

3

Proof of Theorem 1

The idea of the proof is simple. We cover the set p ∈ P4 (Q) : H(p) ≤ B

by a finite collection I of linear subspaces Λ ⊂ P4 , and put [ (Λ ∩ X). Σ= Λ∈I

We then have N (X, B) = N (Σ, B)

and

dim(X) > dim(Σ).

We may thus apply the sharp estimates (E4) and (E6) from Section 2.4. To determine a suitable set I, we apply the results from Section 2.1. Let p = [Z0 , . . . , Z4 ] be a point of P4 such that Z0 , . . . , Z4 are relatively prime integers. According to [4, Lemma 1(i)], the set Λ1 = (x0 , . . . , x4 ) ∈ Z5 : Z0 x0 + · · · + Z4 x4 = 0 is a lattice of dimension four and q det(Λ1 ) = Z02 + · · · + Z42 H(p).

Lemma 2.1 states that there exists a basis b1 , b2 , b3 , b4 of Λ1 such that |b1 | |b2 | |b3 | |b4 | det(Λ1 ). Without loss of generality we may assume that |b1 | |b2 | det(Λ1 )1/2 . Let

Λ2 = x ∈ Z5 : b1 .x = b2 .x = 0

and apply Lemma 2.1 again to find a basis x1 , x2 , x3 of the lattice Λ2 such that |x1 | |x2 | |x3 | det(Λ2 ) |b1 | |b2 | . The last inequality of (3.1) follows from Lemma 2.2. Finally let Λ3 = a ∈ Z5 : a.x1 = a.x2 = a.x3 = 0

and apply Lemma 2.1 to find a basis a1 , a2 of Λ3 such that |a1 | |a2 | det(Λ3 ) |x1 | |x2 | |x3 | . Then (Z0 , . . . , Z4 ) ∈ Λ2 , Λ2 = x ∈ Z5 : a1 .x = a2 .x = 0 , 8

(3.1)

and, |a1 | |a2 | det(Λ2 ) H(p)1/2 . This shows that there exists a rational two-plane Λ ⊂ P4 containing p such that H(Λ) H(p)1/2 . It also shows that Λ = Γ1 ∩ Γ2 for some rational hyperplanes Γ1 , Γ2 in P4 such that H(Γ1 )H(Γ2 ) H(Λ). Let A be a positive constant and let I ⊂ P4∗ (Q) × P4∗ (Q) be the set of pairs (Γ1 , Γ2 ) of hyperplanes such that (i) Γ1 6= Γ2 , (ii) H(Γ1 )H(Γ2 ) ≤ AH(Γ1 ∩ Γ2 ), where Γ1 ∩ Γ2 is considered as an element of G(2, 4), (iii) H(Γ1 ) ≤ AB 1/4 , and (iv) H(Γ2 ) ≤ AB 1/2 /H(Γ1 ). Provided that A is large enough, we have p ∈ P4 (Q) : H(p) ≤ B ⊂

[

(Γ1 ∩ Γ2 ).

(Γ1 ,Γ2 )∈I

From the discussion above it follows that we may choose A independently of B. This defines I and Σ. The next step of the proof is to use the estimates from Section 2.4 to estimate N (Σ, B). The set I can be partitioned into three subsets: I1 is the set of (Γ1 , Γ2 ) ∈ I such that Γ1 ∩ X contains an irreducible component of degree less than d. I2 is the set of (Γ1 , Γ2 ) ∈ I such that Γ1 ∩ X is irreducible of degree d but Γ1 ∩ Γ2 ∩ X contains an irreducible component of degree less than d. I3 is the set of (Γ1 , Γ2 ) ∈ I such that Γ1 ∩ Γ2 ∩ X is an irreducible curve of degree d. Let Σi =

[

(Γ1 ∩ Γ2 ∩ X) for i = 1, 2, 3.

(Γ1 ,Γ2 )∈Ii

Then N (Σ, B) ≤ N (Σ1 , B) + N (Σ2 , B) + N (Σ3 , B).

3.1

Estimate of N (Σ1 , B)

Let J be the set of rational hyperplanes Γ ⊂ P4 such that H(Γ) ≤ AB 1/4 and such that Γ ∩ X is reducible or of degree less than d = deg(X). Consider an irreducible component Y ⊂ Γ ∩ X for some Γ ∈ J. If Y is not a two-plane, then N (Y, B) d,ε B 2+ε 9

according to (E6). If Y is a two-plane such that all points of height at most B on P4 (Q) ∩ Y lie on a line, then N (Y, B) B 2 according to (E2). Hence, N (Σ1 , B) d,ε B 2+ε |J| + N 0 (X, B),

(3.2)

where N 0 (X, B) is the number of rational points on P4 of height at most B lying on the union of all two-planes in X that contain three non-collinear rational points on P4 of height at most B. By Lemma 2.3 and (E2), the cardinality of J is Od (B) so the first term of (3.2) is Od,ε (B 3+ε ). By the following lemma, the second term is also Od,ε (B 3+ε ). Lemma 3.1. Let N 0 (X, B) be the number of rational points on P4 of height at most B lying on the union of all two-planes in X that contain three non-collinear rational points on P4 of height at most B. Then, N 0 (X, B) d,ε B 3+ε . Moreover, if X is a cone with respect to two different rational vertex points of height at most B, then N (X, B) d,ε B 3+ε . Proof. If a two-plane Λ ⊂ X contains three non-collinear rational points on P4 of height at most B, then H(Λ) ≤ A0 B 3 for some constant A0 . Furthermore, Λ contains O(B 3 /H(Λ)) rational points of height at most B according to (E1). Hence, X B3 N 0 (X, B) ≤ , H(Λ) Λ∈F2 (X)(Q) H(Λ)≤A0 B 3

where F2 (X) ⊂ G(2, 4) is the Fano variety of two-planes in X. By Lemma 2.5, the dimension of F2 (X) is at most one and F2 (X) contains no lines. The number of Λ ∈ F2 (X)(Q) with H(Λ) T for some T ≥ 1 is thus Od,ε (T 1+ε ) according to (E2). Hence, X

Λ∈F2 (X)(Q) T
B 3 1+ε B3 d,ε T B 3+ε , H(Λ) T

when T B 3 . By summing over dyadic intervals, we get N 0 (X, B) d,ε B 3+ε . To prove the second statement, let p and q be two different rational points on P4 of height at most B which are vertex points of X. If every point of P4 (Q)∩X of height at most B belongs to the line L containing p and q, then N (X, B) B 2 10

according to (E2). If there is some point of P4 (Q) ∩ X of height at most B that does not belong to the line L, then every point of P4 (Q) ∩ X belongs to a two-plane in X that contains three non-collinear rational points on P4 of height at most B. In that case N (X, B) = N 0 (X, B) d,ε B 3+ε , by what we just proved.

3.2

Estimate of N (Σ2 , B)

Consider an irreducible component Y ⊂ Γ1 ∩ Γ2 ∩ X for some (Γ1 , Γ2 ) ∈ I2 . If the degree of Y is at least three, then N (Y, B) d,ε B 2/3+ε according to (E4). If the degree of Y is two, then N (Y, B) ε 1 +

B 1+ε H(Γ1 ∩ Γ2 )1/3

according to (E5). If Y is a line such that Y contains at most one rational point of height at most B, then N (Y, B) ≤ 1. Hence, N (Σ2 , B) d,ε B 2/3+ε |I2 | X B 1+ε + H(Γ1 ∩ Γ2 )1/3

(3.3)

(Γ1 ,Γ2 )∈I2

+ |I2 | + N (Z, B),

where Z ⊂ X is the union of all lines L ⊂ Σ2 that contain two different rational points of height at most B. Note that if X is a cone over a plane curve, then Γ1 ∩ Γ2 ∩ X is a union of lines for every pair (Γ1 , Γ2 ) ∈ I2 . Thus, the first two terms of (3.3) do not appear in this case. Lemma 3.2. We have 10/3+ε

# {(Γ1 , Γ2 ) ∈ I2 : H(Γi ) ≤ Ti } d,ε T15 T2

+ T15−η T24 ,

where η = 1 unless X is a cone over a plane curve in which case η = 0. Proof. Let f : Γ1 → P3 be an isomorphism for some hyperplane Γ1 ⊂ P4 such that Γ1 ∩ X is irreducible. Let g : P4∗ \ {Γ1 } → P3∗ be the projection map induced by f −1 . By Lemma 2.3, the set of two-planes Λ ⊂ P3 for which Λ ∩ f −1 (X) contains an irreducible component of degree less than d = deg(X) is a proper closed subset V ⊂ P3∗ . The set of hyperplanes 11

Γ2 ∈ P4∗ \{Γ1 } for which Γ1 ∩Γ2 ∩X is reducible is thus contained in the proper closed subset W = g −1 (V ) of P4∗ . There are two cases to consider. If V does not contain any two-planes, then W does not contain any hyperplanes. Hence, the number of (Γ1 , Γ2 ) ∈ I2 with Γ1 fixed and H(Γ2 ) ≤ T2 is 10/3+ε Od,ε (T2 ) by (E3) and (E7). Note that W is cut out by Od (1) hypersurfaces of degrees Od (1) since V is. The number of Γ1 ∈ P4∗ (Q) of height at most T1 is O(T15 ). Hence, the number of pairs (Γ1 , Γ2 ) ∈ I2 such that V does not contain 10/3+ε any two-planes and H(Γi ) ≤ Ti is Od,ε (T15 T2 ). If V contains a two-plane, then W contains a hyperplane. The best estimate for the number of (Γ1 , Γ2 ) ∈ I2 with Γ1 fixed and H(Γ2 ) ≤ T2 is therefore Od (T24 ). This is the trivial estimate (E2). Lemma 2.4 states that the set of Γ1 for which V contains a two-plane is contained in a hypersurface in P4∗ of degree Od (1), provided that X is not a cone over a plane curve. In this case there are Od (T14 ) such Γ1 ∈ P4∗ (Q) of height at most T1 , again according to (E2). In the general case we have O(T15 ) hyperplanes. Hence, the number of pairs (Γ1 , Γ2 ) ∈ I2 such that V contains a two-plane and H(Γi ) ≤ Ti is Od (T15−η T24 ), where η = 1 unless X is a cone over a plane curve in which case η = 0. We can use Lemma 3.2 to estimate the cardinality of I2 . The number of (Γ1 , Γ2 ) ∈ I2 with T < H(Γ1 ) ≤ 2T for some T B 1/4 is ( 1/2 5−η 1/2 10/3+ε B 25/12+ε if η = 1, B B 5 4 T + T d,ε T T B 5/2+ε if η = 0. By summing over dyadic intervals we get ( B 25/12+ε |I2 | d,ε B 5/2+ε

if η = 1, if η = 0.

Consequently, the first term of (3.3) is Od,ε (B 3+ε ). In order to estimate the second term of (3.3) we divide the ranges of both Γ1 and Γ2 into dyadic intervals. If T1 B 1/4 and T1 T2 B 1/2 , then X B 1+ε B 1+ε 5 10/3+ε 4 4 T T + T T d,ε 1 2 1 2 H(Γ1 ∩ Γ2 )1/3 (T1 T2 )1/3 (Γ1 ,Γ2 )∈I2 Ti
B 35/12+ε . Hence, the second term of (3.3) is Od,ε (B 3+ε ). We have already seen that |I2 | = Od,ε (B 3+ε ), so it remains to estimate the very last term in (3.3). Let Z1 (T ) be the union of all lines L ⊂ Z with H(L) > T , for some T > 0, and let Z2 (T ) be the union of all lines L ⊂ Z with H(L) ≤ T . Then, N (Z, B) ≤ N (Z1 (T ), B) + N (Z2 (T ), B), for every T > 0. 12

(3.4)

Lemma 3.3. If T ≤ B 2 , then N (Z1 (T ), B) d,ε B 13/4 T 2/d−1+ε . Proof. Let J ⊂ P4∗ (Q) be the projection of I2 ⊂ P4∗ (Q) × P4∗ (Q) on the first factor. If Γ1 ∈ J, then Γ1 ∩ X is an irreducible surface of degree d. By Lemma 2.5, the dimension of F1 (Γ1 ∩ X) is at most one, and by Lemma 2.6, every one-dimensional irreducible component of F1 (Γ1 ∩ X) has degree at least d. There are thus Od,ε (R2/d+ε ) rational lines L ⊂ Γ1 ∩ X of height at most R according to (E4). If the line L is contained in Z, then L contains two different rational points on P4 of height at most B. Hence, X

N (L, B) d,ε

L∈F1 (Γ1 ∩X)(Q) L⊂Z R
B 2 2/d+ε R B 2 R2/d−1+ε R

according to (E1). The cardinality of J is O(B 5/4 ), so X N (L, B) d,ε B 13/4 R2/d−1+ε .

(3.5)

L⊂Z R
By summing over dyadic intervals, we get N (Z1 (T ), B) d,ε B 13/4 T 2/d−1+ε . Note that we only have to sum over O(log B) dyadic intervals since H(L) B 2 for every line L that contains two rational points of height at most B. Lemma 3.4. If T ≤ B 2 , then N (Z2 (T ), B) d,ε B 3+ε + B 2+ε T (3+1/d)/2+ε . Proof. Every rational line in P4 contains a rational point p such that H(p) A00 H(L)1/2 for some positive constant A00 . This is a consequence of Lemma 2.1 and 2.2. Hence, [ Z2 (T ) ⊂ (Z ∩ C(p, X)), p∈X(Q) H(p)≤A00 T

where C(p, X) is the cone of lines in X with vertex p. Suppose that C(p, X) = X for some rational point p ∈ X and let J ⊂ P4∗ (Q) be the projection of I2 ⊂ P4∗ (Q) × P4∗ (Q) on the first factor. There are O(B) hyperplanes Γ1 ∈ J passing through p, and there are Od,ε (R2/d+ε ) rational lines of height at most R on the irreducible surface Γ1 ∩ X for any Γ1 ∈ J (see the proof of Lemma 3.3). We are only interested in lines that contain two different 13

points of height at most B, and on such lines there are O(B 2 /R) rational points on P4 of height at most B if R ≤ B 2 . Hence, X N (L, B) d,ε B 3 R2/d−1+ε . L⊂Z∩C(p,X) R
By summing over dyadic intervals, we get N (Z ∩ C(p, X), B) d,ε B 3+ε . If there are two different rational points p and q on P4 of height at most T 1/2 ≤ B such that X = C(p, X) = C(q, X), then N (Z2 (T ), B) ≤ N (X, B) d,ε B 3+ε according to Lemma 3.1. Hence, the contribution to N (Z2 (T ), B) from the points p with C(p, X) = X is Od,ε (B 3+ε ). Now assume that C(p, X) 6= X for some rational point p ∈ X and consider an irreducible component Y ⊂ C(p, X). If Y is not a two-plane, then N (Y, B) d,ε B 2+ε according to (E2) or (E6). Note that the degree of Y is bounded in terms of d according to Lemma 2.7. If Y is a two-plane such that all points of height at most B on P4 (Q) ∩ Y lie on a line, then N (Y, B) B 2 according to (E2). Hence, X N (C(p, X), B) d,ε B 2+ε N (X, T 1/2 ) + N 0 (X, B),

(3.6)

p∈X(Q) H(p)≤A00 T 1/2 C(p,X)6=X

where N 0 (X, B) is the cardinality defined in Lemma 3.1. The first term in (3.6) is Od,ε (B 2+ε T (3+1/d)/2+ε ) according to (E3), and the second term is Od,ε (B 3+ε ) according to Lemma 3.1. Hence, N (Z2 (T ), B) d,ε B 3+ε + B 2+ε T (3+1/d)/2+ε , provided that T B 2 . 5d

If we put T = B 10d−6 in (3.4) and apply Lemma 3.3 and 3.4, then we get N (Z, B) d,ε B 3+ε + B

14

7 11 4 + 10d−6 +ε

.

3.3

Estimate of N (Σ3 , B)

Consider those pairs (Γ1 , Γ2 ) ∈ I3 with T < H(Γ1 ) ≤ 2T for some T B 1/4 . For each such pair we use Lemma 2.1 to find a basis b0 , b1 , b2 of the lattice x ∈ Z5 : [x] ∈ Γ1 ∩ Γ2 ∪ {0}. With out loss of generality, we may assume that

|b0 | ≤ |b1 | ≤ |b2 | . Let φ : Γ1 ∩ Γ2 → P2 be the map [λ0 b0 + λ1 b1 + λ2 b2 ] 7→ [λ0 , λ1 , λ2 ]. Then H(φ(p)) H(p)/ |b0 | for a rational point p ∈ Γ1 ∩ Γ2 , so N (Γ1 ∩ Γ2 ∩ X, B) d,ε

B |b0 |

2/d+ε

according to (E4). Now consider all bases b0 , b1 , b2 which satisfy Ci < |bi | ≤ 2Ci for some positive numbers Ci with C0 C1 C2 . The set of Γ ∈ P4∗ which contains the point [b0 ] is a hyperplane Λ in P4∗ , and the number of Γ ∈ Λ(Q) with H(Γ) ≤ R is O(R4 / |b0 |), provided that R |b0 | [4, Lemma 1(v)]. Hence, the number of pairs (Γ1 , Γ2 ) ∈ I3 with T < H(Γ1 ) ≤ 2T , Ci < |bi | ≤ 2Ci , and b0 fixed is 4 T ((C0 C1 C2 )/T )4 = C02 C14 C24 . C0 C0 The number of b0 with |b0 | C0 is O(C05 ). Hence, X 7−2/d 4 4 2/d+ε N (Γ1 ∩ Γ2 ∩ X, B) d,ε C0 C1 C2 B (Γ1 ,Γ2 )∈I3 T
(C0 C1 C2 )5−2/3d B 2/d+ε B 5/2+5/3d+ε . By summing over dyadic intervals we get N (Σ3 , B) d,ε B 5/2+5/3d+ε .

3.4

Conclusion

We have shown that if X ⊂ P4 is an irreducible hypersurface of degree d ≥ 3, then 11 7 5 5 N (X, B) d,ε B 3+ε + B 4 + 10d−6 +ε + B 2 + 3d +ε , for every ε > 0. It is straightforward to see that this estimate implies Theorem 1. 15

References [1] T. D. Browning, A note on the distribution of rational points on threefolds, Quart. J. Math., 54 (2003), 33–39. [2] W. Fulton and R. Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic Geometry (LNM 862), Springer-Verlag, 1981, 26–92. [3] J. Harris, Algebraic Geometry, Springer-Verlag, 1992. [4] D. R. Heath-Brown, The density of rational points on curves and surfaces, Ann. of Math., 155 (2002), 553–595. [5] J. Pila, Density of integral and rational points on varieties, Ast´erisque, 228 (1995), 183–187. [6] W. M. Schmidt, Diophantine Approximations and Diophantine Equations (LNM 1467), Springer-Verlag, 1991.

16

Remarques sur l’approximation faible sur un corps de fonctions d’une variable J.-L. Colliot-Th´el`ene et P. Gille 26 mars 2003

Introduction Soit C/k une courbe projective, lisse, connexe, d´efinie sur un corps alg´ebriquement clos k de caract´eristique nulle. On note F = k(C) le corps de fonctions de C et FM le compl´et´e de F pour la valuation vM d´efinie par un point ferm´e M de C. Soit X/F une vari´et´e lisse g´eom´etriquement connexe. L’ensemble des FM -points X(FM ) est muni d’une topologie naturelle ([Kn]). On dit que l’approximationQfaible vaut pour un ensemble fini S de points ferm´es de C si l’image diagonale de X(F ) dans M ∈S X(FM ) est dense pour la topologie produit. Si c’est le cas pour tout ensemble fini S de points ferm´es, on dit que l’approximation faible vaut pour la vari´et´e X/F . Comme l’a rappel´e B. Hassett dans son expos´e `a l’A.I.M. de Palo Alto en d´ecembre 2002, on se demande si toute F -vari´et´e lisse et g´eom´etriquement rationnellement connexe satisfait `a l’approximation faible. Le seul r´esultat g´en´eral connu `a ce jour est un th´eor`eme de Koll´ar, Miyaoka et Mori (voir [Ko] IV. 6.10), qui ne concerne qu’une version faible de l’approximation, et ce seulement pour les M ∈ C o` u X/F a bonne r´eduction. Dans cette note nous ´etablissons cet ´enonc´e pour les F -vari´et´es g´eom´etriquement rationnellement connexes qui se ram`enent par fibrations `a des espaces homog`enes de groupes lin´eaires connexes. Nous montrons par ailleurs que sous la simple hypoth`ese d’annulation des H i (X, OX ) (pour i ≥ 1) il peut y avoir d´efaut d’approximation faible. Notre exemple est une surface d’Enriques. L’outil utilis´e est une loi de r´eciprocit´e et l’existence d’un revˆetement non ramifi´e non trivial sur une telle surface. On notera que des r´esultats particuliers sur l’approximation faible ont ´et´e obtenus sur des corps de fonctions plus compliqu´es que ceux consid´er´es ici : corps de fonctions d’une variable sur le corps des r´eels ([CT], [Sch], [Du]), et corps de fonctions de deux variables sur un corps alg´ebriquement clos ([CTGiPa]).

§1. Rappels et pr´ eliminaires §1.a Approximation faible Soient K un corps et Ω un ensemble de valuations de K, distinctes deux `a deux, discr`etes de rang un. Pour tout v ∈ Ω soit Kv le compl´et´e de K en v. L’approximation faible pour les K-vari´et´es lisses g´eom´etriquement connexes se d´efinit comme dans l’introduction. On note AnK l’espace affine de dimension n sur K. Le fait suivant est bien connu : Proposition 1.1. Soient K et Ω comme ci-dessus. Soient X et Y deux K-vari´et´es lisses g´eom´etriquement connexes. S’il existe n, m ∈ N tels que les K-vari´et´es X ×F AnK soient Kbirationnellement ´equivalentes, alors l’approximation faible vaut pour X si et seulement si elle vaut pour Y . En particulier, l’approximation faible vaut pour X si et seulement si elle vaut pour un ouvert non vide de X. Proposition 1.2. Soient K et Ω comme ci-dessus. Soit f : X → Y un K-morphisme lisse de K-vari´et´es lisses g´eom´etriquement connexes, a ` fibre g´en´erique g´eom´etriquement connexe. Si 1

pour tout point M ∈ Y (K) l’approximation faible vaut pour la F -vari´et´e fibre XM = f −1 (M ), et si l’approximation faible vaut pour Y , alors elle vaut pour X. D´emonstration. Soit S ⊂ Ω un ensemble fini, et supposons donn´e pour chaque v ∈ S un point Pv ∈ X(Kv ). Soit Qv = f (Pv ). Par le th´eor`eme des fonctions implicites, il existe un voisinage ouvert ωv ⊂ Y (Kv ) contenant Qv , ´equip´e d’une section analytique σv : ωv → X(Kv ) de la projection X(Kv ) → Y (Kv ). L’approximation faible valant pour Y , on peut trouver un point F -rationnel Q ∈ Y (K) tel que pour chaque v ∈ S le point Q soit tr`es proche de Qv dans Y (Kv ) et qu’il appartienne `a ωv . Soit Z = f −1 (Q) la fibre en Q. Alors Rv = σv (Q) ∈ Z(Kv ) est tr`es proche de Pv dans X(Kv ). Par hypoth`ese, l’approximation faible vaut pour la F -vari´et´e Z. Ainsi il existe un point P ∈ Z(K) ⊂ X(K) tr`es proche de Rv dans Z(Kv ) et donc dans X(Kv ) pour chaque v ∈ S. Un tel point est tr`es proche de chaque Pv pour v ∈ S. Remarque. On ne peut esp´erer appliquer cette proposition g´en´erale que lorsque l’on a d´ej`a la propri´et´e : toute fibre non vide de f au-dessus d’un point K-rationnel de Y poss`ede un point K-rationnel. Sur un corps K = k(C) du type consid´er´e dans l’introduction, d’apr`es Graber, Harris et Starr [GHS] et d’apr`es de Jong et Starr [dJS], c’est le cas si la fibre g´en´erique de f est birationnelle `a une vari´et´e projective, lisse, g´eom´etriquement connexe et rationnellement connexe. Dans ce cas, pour les vari´et´es qui se d´evissent en vari´et´es rationnellement connexes pour lesquelles l’approximation faible a d´ej`a ´et´e ´etablie, on obtient l’approximation faible.

§1.b Cohomologie galoisienne des corps C1 Le th´eor`eme suivant regroupe des r´esultats de Springer (cas des corps C1 , qui nous suffirait ici, les corps F = k(C) et FM ' k((t))) ´etant des corps C1 ) et de Steinberg (cas g´en´eral). Th´eor`eme 1.3. Soit K un corps de caract´eristique z´ero et de dimension cohomologique 1. a) Tout K-groupe r´eductif connexe est quasi-d´eploy´e. b) Tout espace homog`ene sous un K-groupe lin´eaire connexe poos`ede un point rationnel. c) Soient G un K-groupe lin´eaire connexe et H ⊂ G un sous-groupe ferm´e. La projection G(K)-´equivariante G(K) → (G/H)(K), o` u les deux ensembles sont point´es par l’´el´ement neutre de G et par son image, se prolonge en une suite exacte d’ensembles point´es G(K) → (G/H)(K) → H 1 (K, H) → 1. d) Soit H un K-groupe lin´eaire, et soit H 0 ⊂ H sa composante connexe, sous-groupe normal de H. L’homomorphisme quotient H → H/H 0 induit une bijection H 1 (K, H) → H 1 (K, H/H 0 ). D´emonstration. a) Voir [SeCG] III.2.3 Thm 1’ p. 139. b) Voir [SeCG] III.2.4 Cor. 1 p. 141. L’´enonc´e b) peut se reformuler ainsi : pour tout F espace homog`ene X sous un K-groupe lin´eaire connexe, il existe un K-morphisme G-´equivariant de G (vu comme espace principal homog`ene `a gauche sous G) vers X. c) Combiner H 1 (K, G) = 0 (qui est une reformulation de b) dans le cas des espaces principaux homog`enes) et [SeCG] I.5.4 Prop. 36 p. 47. d) Voir [SeCG] III.2.4 Cor. 3 p. 142.

§2. Approximation faible pour les espaces homog` enes de groupes lin´ eaires connexes et pour les vari´ et´ es qui s’y ram` enent Le th´eor`eme suivant devrait ˆetre consid´er´e comme bien connu.

2

Th´eor`eme 2.1 Soit F = k(C) un corps de fonctions d’une variable sur un corps alg´ebriquement clos de caract´eristique z´ero et soit G un F -groupe lin´eaire connexe. Alors G satisfait a ` l’approximation faible. D´emonstration. Le groupe G/F est le produit semi-direct de son quotient r´eductif Gred par son radical unipotent Ru G qui est F -isomorphe (en tant que vari´et´e) `a un espace affine. Par la proposition 1.1, on est ramen´e au cas o` u G est un groupe r´eductif connexe. Comme le corps F est C1 , le groupe r´eductif G est quasi-d´eploy´e. Soit B un sous-groupe de Borel de Gred et soit T un F -tore maximal de B. La F -vari´et´e sous-jacente au radical unipotent Ru B est un espace affine, il en est de mˆeme du radical unipotent Ru B − du sous-groupe de Borel B − oppos´e `a B, et le groupe G contient un ouvert, “la grosse cellule”, isomorphe au produit (Ru B) ×F T ×F (Ru B)− ([SGA3], exp. XXII, proposition 4.1.2 page 172). Par la Proposition 1.1, on est ramen´e `a ´etablir l’approximation faible pour un F -tore T . Par consid´eration du groupe des caract`eres de T , on voit ais´ement que pout tout F -tore (ici F pourrait ˆetre un corps quelconque), il existe une suite exacte de F -tores 1→R→E→T →1 Q o` u E est un F -tore quasi-trivial, i.e. un produit i RFi /F Gm de restrictions `a la Weil du groupe multiplicatif Gm pour diverses extensions de corps Fi /F , et o` u R est un F -tore. Le tore quasitrivial E est un ouvert d’un espace affine sur F , il satisfait donc `a l’approximation faible. Soit S un ensemble fini de points ferm´es de la courbe C. Comme les compl´et´es FM , M ∈ S, du corps 1 F = k(C) sont des continue Q Q corps C1 , les groupes H (FM , R) sont nuls. Ainsi l’application Q E(K T (K ) est surjective. Comme E(K) est dense dans E(K ) → M ), il M M M ∈S M ∈S M ∈S Q s’en suit imm´ediatement que T (K) est dense dans M ∈S T (KM ). Th´eor`eme 2.2. Soit F = k(C) un corps de fonctions d’une variable sur un corps alg´ebriquement clos de caract´eristique z´ero et soit G un F -groupe lin´ Qeaire. Pour tout ensemble fini u le second S de points ferm´es de C, l’application diagonale H 1 (F, G) → M ∈S H 1 (FM , G), o` produit est un ensemble fini, est surjective. D´emonstration. D’apr`es le Th´eor`eme 1.3.d, il suffit d’´etablir le th´eor`eme lorsque G est un F -sch´ema en groupes finis. Soient Fs une clˆoture s´eparable de F et G = Gal(Fs /F ). Soit M un point ferm´e de C. Soit FM,s une clˆoture s´eparable de FM ' k((t)) et Fs ⊂ FM,s un plongement. Le groupe de ˆ le compl´et´e profini de Z. On fixe un tel Galois absolu IM = Gal(FM,s /FM ) est isomorphe `a Z, isomorphisme, i.e. on fixe un g´en´erateur profini cM ∈ IM . Fixons aussi, pour chaque M ∈ C, un plongement Fs ⊂ FM,s . Ceci d´etermine un plongement IM ⊂ G. On note A = G(Fs ) = G(FM,s ). C’est un groupe fini muni d’une action de G et donc de IM pour tout M ∈ C. Quitte `a agrandir S, on peut supposer que le F -sch´ema en groupe G provient d’un U sch´ema en groupes fini ´etale sur U , o` u U d´esigne le compl´ementaire de S dans C. En d’autres termes, on peut supposer que pour tout M ∈ / S, l’action de IM sur A est triviale. Soit H un sous-groupe ferm´e de G. L’ensemble de cohomologie H 1 (H, A) est un quotient de l’ensemble Z 1 (H, A) des 1-cocycles continus de H `a valeurs dans le groupe fini A. Cet ensemble s’identifie `a l’ensemble des homomorphismes continus de H dans le produit semi-direct A.H qui sont des sections de la projection structurale A.H → H ([SeCG], chap. I, 5.1, Exercice 1 p. 43). Si zh ∈ A, h ∈ H est le 1-cocycle, la section est donn´ee par h 7→ zh .h. Si l’on se donne un syst`eme de g´en´erateurs topologiques de H, il existe au plus un 1-cocycle dans Z 1 (H, A) prenant des valeurs donn´ees sur ces g´en´erateurs. En particulier, pour tout M , l’ensemble Z 1 (IM , A) est fini (A ´etant fini). A fortiori 1 H (FM , G) = H 1 (IM , A) est fini. 3

Soit S ⊂ C un ensemble fini de points ferm´es de C. Pour ´etablir le th´eor`eme, nous allons ´etablir l’´enonc´e a priori plus fort que que l’application naturelle de 1-cocycles continus Y Z 1 (G, A) → Z 1 (IM , A) M ∈S

est surjective. Soit g le genre de la courbe C. Notons S = {M1 , · · · , Ms }. Choisissons un point ferm´e Ms+1 ∈ C \ S. Notons H = Gal(L/F ) le groupe de Galois de la sous-extension maximale L/F de Fs /F non ramifi´ee en dehors de S ∪ Ms+1 . C’est un quotient de G, et l’on a des inclusions induites IMj ⊂ H pour j = 1, · · · , s + 1. Rappelons que pour chaque j, on a choisi ˆ L’hypoth`ese initiale faite sur un g´en´erateur topologique cj du groupe d’inertie Ij = IMj ' Z. S garantit que l’inclusion naturelle G(L) ⊂ G(Fs ) = A est une ´egalit´e. On dispose alors d’une application Z 1 (H, A) → Z 1 G, A), et pour ´etablir l’´enonc´e il suffit de montrer que l’application diagonale Y Z 1 (H, A) → Z 1 (IM , A) M ∈S

est surjective. D’apr`es le th´eor`eme d’existence de Riemann ([SeRC], §1.2) le groupe H est le groupe profini engendr´e par les g´en´erateurs a1 , b1 , · · · , ag , bg , c1 , · · · , cs , cs+1 avec l’unique relation [a1 , b1 ] · · · [ag , bg ] c1 c2 · · · cs cs+1 = 1, o` u pour tout j = 1, · · · , s, l’´el´eQ ment cj est comme ci-dessus. u zj Un ´el´ement du produit M ∈S Z 1 (IM , A) d´etermine une famille zj = zMj ∈ A, o` est la valeur du 1-cocycle sur le g´en´erateur cMj = cj . Pour tout j = 1, · · · , s, on dispose donc de l’homomorphisme IMj → A.Ij envoyant cj sur zj .cj . Cet homomorphisme s’´etend naturellement en un homomorphisme ρj : IMj → A.H. Pour d´efinir un homomorphisme continu ρ : H → A.H, il suffit de le d´efinir sur les g´en´erateurs de H, et d’assurer que la relation ci-dessus est respect´ee. D´efinissons ρ(ai ) = 1.ai et ρ(bi ) = 1.bi pour tout i = 1, · · · , g, puis ρ(cj ) = zj .cj pour j = 1, · · · , s. Pour que la relation soit respect´ee, il suffit de choisir ρ(cs+1 ) ∈ A.H tel que [a1 , b1 ] · · · [ag , bg ]z1 .c1 . . . zs .cs .ρ(cs+1 ) = 1, (par abus de langage, on note ici ai .1 = ai et bi .1 = bi ), ce qui compte tenu de la relation initiale dans H se traduit encore par −1 −1 cs+1 .c−1 s . · · · .c1 .z1 .c1 . . . zs .cs .ρ(cs+1 ) = 1.

Soit u ∈ A.H tel que ρ(cs+1 ) = u.cs+1 . Un calcul imm´ediat (moins magique qu’il n’y paraˆıt) montre que u appartient `a A ⊂ A.H, on notera donc u = zs+1 . On dispose donc d’un homomorphisme continu envoyant H dans A.H, envoyant ai sur 1.ai et bi sur 1.bi pour i = 1, · · · , g, et par ailleurs cj sur zj .cj pour j = 1, · · · , s+1. Cet homomorphisme d´efinit une section de la projection A.H → H, comme on le voit sur les g´en´erateurs. Enfin sa restriction ` a chaque sous-groupe Ij pour j = 1, · · · , s a son image dans A.Ij et co¨ıncide avec l’homomorphisme initial Ij → A.Ij . Th´eor`eme 2.3. Soit F = k(C) un corps de fonctions d’une variable sur un corps alg´ebriquement clos de caract´eristique z´ero, et soit G un F -groupe lin´eaire connexe. L’approximation faible vaut pour tout espace homog`ene sous G.

4

D´emonstration. Soit X/F un tel espace homog`ene. D’apr`es le th´eor`eme 1.3.b, on peut ´ecrire X = G/H, o` u H ⊂ G est un F -sous-groupe ferm´e de G. D’apr`es le th´eor`eme 1.3.c, et le fait que tant le corps F que les corps FM sont des corps C1 , on a un diagramme commutatif de suites exactes d’ensembles point´es G(F ) − →   y Q G(FM ) − → M ∈S

(G/H)(F )   y Q (G/H)(FM )

φ

H 1 (F, H)   y

− →

φM

− →

Q

− →

1

H 1 (FM , H) − → 1.

M ∈S

M ∈S

D’apr`es le th´eor`eme 2.2, la fl`eche verticale de droite est surjective. Pour chaque M ∈ S, soit xM ∈ (G/H)(KM ). D’apr`es le diagramme ci-dessus et ses propri´et´es, il existe x ∈ (G/H)(F ) qui Q H 1 (FM , H). Il existe alors pour chaque M ∈ S a mˆeme image que la famille {PM } dans M ∈S

un ´el´ement gM ∈ G(FM ) tel que xM = gM .x. Comme l’approximation faible vaut pour G (Th´eor`eme 2.1), on peut trouver g ∈ G(F ) arbitrairement proche de chaque gM ∈ G(FM ), et alors g.X ∈ (G/H)(F ) est arbitrairement proche de chaque xM ∈ (G/H)(FM ) (chaque application G(FM ) → (G/H)(FM ) ´etant continue). Th´eor`eme 2.4. Soit F = k(C) le corps des fonctions d’une courbe projective lisse C sur un corps k alg´ebriquement clos de caract´eristique z´ero. Soit f : X → Y un F -morphisme dominant de F -vari´et´es g´eom´etriquement int`egres, dont la fibre g´en´erique est g´eom´etriquement connexe et F (Y )-birationnelle a ` un espace homog`ene d’un groupe lin´eaire connexe G sur le corps des fonctions F (Y ) de Y . Si Y satisfait a ` l’approximation faible, alors X satisfait a ` l’approximation faible. D´emonstration. Pour ´etablir le th´eor`eme, on peut d’apr`es la proposition 1.1 restreindre Y `a un ouvert et X `a l’image r´eciproque de cet ouvert. On peut donc supposer que le F (Y )groupe lin´eaire connexe est la restriction d’un Y -groupe lin´eaire fid`element plat sur Y , `a fibres connexes, soit G/Y , et que pour tout point P ∈ Y (F ) la fibre f −1 (P ) est lisse, g´eom´etriquement et F -birationnelle `a un F -espace homog`ene sous le F -groupe lin´eaire connexe fibre GP . D’apr`es le th´eor`eme 2.3 et la proposition 1.1, toute telle fibre satisfait `a l’approximation faible. Il r´esulte alors des hypoth`eses et de la proposition 1.2 que X satisfait `a l’approximation faible. Les exemples non triviaux abondent (par exemple non trivial, on entend des exemples de F -vari´et´es qui ne sont pas n´ecessairement F -birationnelles `a un espace projectif). Les plus ´evidents sont les surfaces fibr´ees en coniques (de dimension au moins un) au-dessus de la droite projective P1F , et plus g´en´eralement les fibr´es en quadriques, resp. en vari´et´es de Severi-Brauer au-dessus d’un espace projectif de dimension arbitraite. Dans les cas cit´es, la d´emonstration de l’approximation faible se fait bien sˆ ur `a moindres frais : outre l’´el´ementaire proposition 1.2, on utilise le fait que toute F -quadrique de dimension au moins 1 poss`ede un F -point (cas particulier du th´eor`eme de Tsen remontant `a Max Noether) et est donc F -birationnelle `a un espace projectif sur F , resp. le fait que toute vari´et´e de Severi-Brauer sur F est F -isomorphe `a un espace projectif sur F (th´eor`eme de Tsen et th´eorie ´el´ementaire des vari´et´es de Severi-Brauer, due `a F. Chˆatelet). D´egageons le : Corollaire 2.5. Soit F = k(C) comme ci-dessus et soit X/F une surface de Del Pezzo degr´e 4, c’est-` a-dire une intersection compl`ete lisse de deux quadriques dans P4F . L’approximation faible vaut pour X. 5

D´emonstration. Comme F est un corps C1 , on a X(F ) 6= ∅. Comme F est infini, il existe donc un point F -rationnel R non situ´e sur l’une quelconque des 16 droites (sur une clˆoture alg´ebrique de F ) contenues dans X ([Ma], Chap. IV, §30, Theorem 30.1 p. 162). En ´eclatant R, on obtient une surface cubique lisse Y ⊂ P3F qui contient une droite d´efinie sur F (la courbe exceptionnelle image inverse de R). Le pinceau des 2-plans de P3F passant par cette droite d´efinit sur Y une structure de surface fibr´ee en coniques au-dessus de P1F , et l’on a vu ci-dessus que l’approximation faible vaut pour une telle surface. Par la proposition 1.1, l’approximation faible vaut donc aussi pour X. Remarque. En utilisant la proposition 1.2, on d´eduit de ce r´esultat l’approximation faible pour toute F -vari´et´e X intersection compl`ete lisse de deux quadriques dans PnF , n ≥ 4. Pour n ≥ 5, on peut aussi d´eduire l’approximation faible de [CTSSD], Theorem 3.27 p. 80. Pour n ≥ 6, la situation est encore plus simple : dans ce cas, toute telle vari´et´e X est F -birationnelle `a un espace projectif ([CTSSD], Theorem 3.2 p. 60; Theorem 3.4 p. 62). Toute F -surface projective, lisse, g´eom´etriquement rationnellement connexe est F -birationnelle soit `a une surface fibr´ee en coniques au-dessus de P1F , soit `a une surface de Del Pezzo de degr´e d, avec 1 ≤ d ≤ 9. Toute F -surface de Del Pezzo de degr´e d ≥ 5 est F -birationnelle a` P2F , donc satisfait `a l’approximation faible. Nous venons de voir que cette derni`ere propri´et´e vaut pour d = 4. La question de savoir si l’approximation faible vaut reste ouverte pour les F -surfaces de Del Pezzo de degr´e 3 (surfaces cubiques lisses), et a fortiori pour les F -surfaces de Del Pezzo de degr´e 2 et 1.

§3. Une surface d’Enriques qui ne satisfait pas ` a l’approximation faible Soit F = k(C) comme dans l’introduction. Nous commen¸cons par d´ecrire un m´ecanisme familier dans un cadre plus d´elicat, `a savoir celui des corps de nombres. La somme des degr´es des diviseurs d’une fonction est nulle. Le complexe ainsi obtenu F ∗ → ⊕M ∈C Z → Z, o` u la derni`ere fl`eche est la somme, induit pour tout entier n > 0 un complexe F ∗ /F ∗n → ⊕M ∈C Z/n → Z/n. Pour tout M ∈ C, la fl`eche F ∗ /F ∗n → Z/n induite par l’application diviseur en M s’identifie `a ∗ ∗n la fl`eche naturelle F ∗ /F ∗n → FM /FM . Soit X une F -vari´et´e projective, lisse, g´eom´etriquement connexe. Soit F (X) son corps des fonctions et f ∈ F (X)∗ une fonction dont le diviseur est une puissance n-i`eme dans le groupe des diviseurs de X. Soit U ⊂ X un ouvert non vide sur lequel f est inversible. L’´equation f = tn d´efinit sur l’ouvert U un µn -torseur qui, grˆace `a l’hypoth`ese sur le diviseur, s’´etend en un µn -torseur Y → X. Pour tout corps L contenant F , `a ce µn -torseur est associ´e une application d’´evaluation ϕL : X(L) → H 1 (L, µn ) = L∗ /L∗n qui sur U (L) n’est autre que l’application associant `a P ∈ U (L) la classe de f (P ) dans L∗ /L∗n . Pour M ∈ C point ferm´e et L = FM , l’application ∗n ∗ ϕFM : X(FM ) → FM = Z/n est continue, i.e. localement constante. Par ailleurs, la /FM propret´e de X/F et un argument de bonne r´eduction montre que pour presque tout M ∈ C ∗ ∗n (i.e. tout M sauf un nombre fini), l’application ϕFM : X(FM ) → FM /FM = Z/n se factorise 6

∗ ∗ n par OM /(OM ) = 1, donc a une image nulle dans Z/n. Ici OM d´esigne le compl´et´e de l’anneau local de C en M , qui est isomorphe `a k[[t]]. Soit S ⊂ C l’ensemble fini des points o` u ϕFM n’est pas constante. Le torseur Y → X d´efinit donc une application X(F ) → ⊕M ∈S Z/n qui compos´ee avec la somme ⊕M ∈S Z/n → Z/n donne z´ero. Nous pouvons alors conclure :

Proposition M ∈ S, un point PM ∈ P 3.1. Dans la situation ci-dessus, soit pour chaque Q ϕ (P ) = 6 0 ∈ Z/n, alors la famille {P } ∈ X(FM ) n’est pas dans X(FM ). Si M M ∈S M Q M l’adh´erence de l’image de X(F ) dans le produit topologique M ∈S X(FM ). Remarque. L’id´ee de cette proposition n’est pas nouvelle, on a d´ej` a utilis´e des lois de r´eciprocit´es vari´ees pour d´efinir une obstruction `a l’approximation faible dans divers contexte. L’obstruction la plus connue est l’obstruction de Brauer-Manin sur un corps de nombres, qui fait intervenir le groupe H 2 (., µn ). Toujours sur un corps de nombres, on peut aussi utiliser H 1 (., µn ) (voir [Ha]). Sur un corps de fonctions d’une variable sur les r´eels, voir [CT] et [Du]. Soit A1F = SpecF [t] la droite affine, et soit A1F ⊂ C = P1k le plongement naturel. Notons F = k(P1 ) = k(t) le corps des fonctions de P1F . Soient a, b, c, d, e ∈ C, soit c(u) = t(u − a)(u − b), d(u) = et(t − c)(t − d). Supposons que le polynˆome e.c(u).d(u).(c(u) − d(u)).u(u − 1) ∈ C[u] est s´eparable de degr´e 8. Soit U ⊂ A4K , avec coordonn´ees affines x, y, z, u) la F -surface lisse d´efinie dans A4F par le syst`eme d’´equations x2 − t(u − a)(u − b) = u(u − 1)y 2 6= 0, x2 − te(u − c)(u − d) = u(u − 1)z 2 6= 0. (∗) Soit X/F un mod`ele projectif et lisse, F -minimal, de la surface U . Il existe un ouvert non vide V ⊂ U qu’on peut identifier `a un ouvert de X. Proposition 3.2. La F -surface X est une surface d’Enriques, elle satisfait en particulier H 1 (X, OX ) = 0 et H 2 (X, OX ) = 0. La fonction rationnelle d´efinie par f = u(u − 1) sur U a son diviseur sur X qui est un double. D´emonstration. Que le diviseur de f sur tout mod`ele lisse soit un double est facile `a ´etablir par des calculs valuatifs (voir [CTSkSD]). Pour le d´etail de la d´emonstration du fait que X est une surface d’Enriques, affirm´e dans [CTSkSD], voir [La]. D’apr`es ce qui a ´et´e rappel´e ci-dessus, il existe un Z/2-torseur Y au-dessus de X dont la restriction `a V est obtenue en extrayant la racine carr´ee de la fonction f (l’espace total du torseur est une surface K3). Pour tout corps L contenant F , on a une application induite ϕL : X(L) → L∗ /L∗2 , qui sur V (L) n’est autre que l’application envoyant P ∈ V (L) sur la classe de f (P ) dans L∗ /L∗2 . Proposition 3.3. Avec les notations ci-dessus, pour M 6= 0, ∞ ∈ P1 , l’image de l’application ∗ ∗2 ϕM : X(FM ) → FM /FM = Z/2 est nulle. Par ailleurs pour M = 0 et pour M = ∞, l’image de ϕM est tout le groupe Z/2. D´emonstration. Par continuit´e et par le th´eor`eme des fonctions implicites, qui garantit que V (FM ) est dense dans X(FM ), il suffit d’´etablir ces faits pour les applications U (FM ) → ∗ ∗2 FM /FM = Z/2 d´efinie par la fonction f . Soient M ∈ P1 , soit v = vM la valuation associ´ee sur F . Soit (x, y, z, u) un point de U (FM ). Supposons v(u) < 0. Des ´equations (*) il r´esulte que v(u(u − 1)) = 2v(u) est pair. Supposons alors v(u.(u − 1)) > 0, et v(u(u − 1)) ≥ 0 impair. Supposons d’abord v(t) = 0. Alors n´ecessairement v(x2 ) = v(t(u−a)(u−b) = 0 (tous deux sont pairs), et v(x2 −t(u−a)(u−b)) > 0. De mˆeme v(x2 − et(u − c)(u − d)) > 0. On a alors v(c(u) − d(u)) > 0, ce qui est impossible. L’´enonc´e pour tout v = vM avec M 6= 0, ∞ est donc ´etabli.

7

Soit v = vM avec M = 0 ∈ A1F , et donc v(t) = 1. Il existe un point (x, y, z, u) = (1/t, y, z, 1/t) ∈ U (FM ) avec v(y) = 0, v(z) = 0, et donc v(u.(u − 1)) pair. Par ailleurs il existe un point (x, y, z, u) = (0, y, z, t) avec v(y) = 0, v(z) = 0. Pour un tel point, on a v(u.(u − 1)) = 1 impair. Soit v = vM avec M = ∞ ∈ P1F , et donc v(t) = −1. Il existe des points avec v(x) < 0, v((u.(u − 1)) = 0, v(x) = v(y) = v(z) < 0. Par ailleurs il existe des points avec x = 0, u = 1/t, v(y) = v(z) = 1 et donc v(u.(u − 1)) = 1. Th´eor`eme 3.4. La surface d’Enriques X/F poss`ede des points rationnels. L’image de l’application diagonale X(F ) → X(F0 ) × X(F∞ ) n’est pas dense dans ce produit. D´emonstration. d’´equations

Fixons u = u0 ∈ C assez g´en´eral.

On obtient alors une F -courbe

x2 − t(u0 − a)(u0 − b) = u0 (u0 − 1)y 2 6= 0, x2 − et(u0 − c)(u0 − d) = u0 (u0 − 1)z 2 6= 0, contenue dans U , et rencontrant V . Cette courbe admet une F -compactification lisse Γ donn´ee en coordonn´ee homog`enes (X, Y, Z, T ) par X 2 − t(u0 − a)(u0 − b)T 2 = u0 (u0 − 1)Y 2 , X 2 − et(u0 − c)(u0 − d)T 2 = u0 (u0 − 1)Z 2 . Sur cette courbe on trouve pour T = 0 des F -points lisses (`a coordonn´ees dans C). L’application rationnelle de la F -courbe lisse Γ vers la F -vari´et´e propre X est partout d´efinie, on a donc X(F ) 6= ∅. Le reste de l’´enonc´e r´esulte de la combinaison des Propositions 3.1, 3.2 et 3.3.

Remarque. Les ´equations concr`etes de surfaces d’Enriques utilis´ees ci-dessus furent introduites dans [CTSkSD]. Lafon [La] utilise des formes tordues des ´equations (*) pour exhiber des exemples de surfaces d’Enriques sur F = C(t) et mˆeme sur F = C((t)) sans point rationnel. Nous ne doutons pas que l’on puisse utiliser de telles ´equations (tordues) pour exhiber des contre-exemples au principe de Hasse (X(FM ) 6= ∅ pour tout M ∈ P1 , mais X(F ) = ∅) reposant sur la loi de r´eciprocit´e sur F ∗ /F ∗2 utilis´ee plus haut, mais cela demandera sans doute un peu d’acharnement.

R´ ef´ erences

[CT] J.-L. Colliot-Th´el`ene, Groupes lin´eaires sur les corps de fonctions de courbes r´eelles. J. f¨ ur die reine und angew. Mathematik 474 (1996) 139-167. [CTGiPa] J.-L. Colliot-Th´el`ene, P. Gille et R. Parimala, Arithmetic of linear algebraic groups over two-dimensional fields, preprint, to appear (cf. Arithm´etique des groupes alg´ebriques lin´eaires sur certains corps g´eom´etriques de dimension deux. C. R. Acad. Sci. Paris S´er. I Math. 333 (2001), no. 9, 827-832.) [CTSkSD] J.-L. Colliot-Th´el`ene, A. N. Skorobogatov et Sir Peter Swinnerton-Dyer, Double fibres and double covers : paucity of rational points, Acta Arithmetica LXXIX.2 (1997) 113135. [CTSSD] J.-L. Colliot-Th´el`ene, J.-J. Sansuc et Sir Peter Swinnerton-Dyer, Intersections of two quadrics and Chˆatelet surfaces, I, J. f¨ ur die reine und angew. Math. 373 (1987) 37-107.

8

[dJSt] A. J. de Jong et J. Starr, Every rationally connected variety over the function field of a curve has a rational point, `a paraˆıtre. [Du] A. Ducros, L’obstruction de r´eciprocit´e `a l’existence de points rationnels pour certaines vari´et´es sur le corps des fonctions d’une courbe r´eelle. J. f¨ ur die reine und angew. Math. 504 (1998), 73-114. [GHS] T. Graber, J. Harris et J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc., 16 (2003), 57-67. ´ [Ha] D. Harari, Weak approximation and non-abelian fundamental groups. Ann. Sci. Ecole Norm. Sup. (4) 33 (2000), 467-484. [Kn] M. Kneser, Schwache Approximation in algebraischen Gruppen, Colloque de Bruxelles (1962) 41-52. [Ko] J. Koll´ar, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 32, Springer (1996) [La] G. Lafon, Une surface d’Enriques sans point sur C((t)), note en pr´eparation. [Ma] Yu. I. Manin, Cubic forms, Algebra, Geometry, Arithmetic, North-Holland (1974; seconde ´edtion, r´evis´ee et compl´et´ee, 1986). [Sch] C. Scheiderer, Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one. Invent. Math. 125 (1996), no. 2, 307-365. [SeCG] J-P. Serre, Cohomologie Galoisienne, Cinqui`eme ´edition, r´evis´ee et compl´et´ee, Springer Lecture Notes in Mathematics 5 (1994) [SeRC] J-P. Serre, Revˆetements de courbes alg´ebriques, S´eminaire Bourbaki (1991/92), expos´e 749, Ast´erisque 206 (1992), 167-182. [SGA3] Sch´emas en groupes, S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie 1962/64 (SGA 3), dirig´e par M. Demazure et A. Grothendieck, Lecture Notes in Math. 151-153, Springer-Verlag (1970).

J.-L. Colliot-Th´ el` ene, C.N.R.S., UMR 8628, Math´ ematiques, Bˆ atiment 425, Universit´ e de Paris-Sud, F-91405 Orsay France e-mail: [email protected]

Philippe Gille C.N.R.S., UMR 8628, Math´ ematiques, Bˆ atiment 425, Universit´ e de Paris-Sud, F-91405 Orsay France e-mail: [email protected]

9

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UNIVERSAL TORSORS AND COX RINGS by

Brendan Hassett and Yuri Tschinkel

Abstract. — We study the equations of universal torsors on rational surfaces.

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Generalities on the Cox ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Generalities on toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. The E6 cubic surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4. D4 cubic surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Introduction The study of surfaces over nonclosed fields k leads naturally to certain auxiliary varieties, called universal torsors. The proofs of the Hasse principle and weak approximation for certain Del Pezzo surfaces required a very detailed knowledge of the projective geometry, in fact, explicit equations, for these torsors [6], [8], [7], [20], [22], [23]. More recently, Salberger proposed using universal torsors to count rational points of The first author was partially supported by the Sloan Foundation and by NSF Grants 0196187 and 0134259. The second author was partially supported by NSF Grant 0100277.

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BRENDAN HASSETT and YURI TSCHINKEL

bounded height, obtaining the first sharp upper bounds on split Del Pezzo surfaces of degree 5 and asymptotics on split toric varieties over Q [?]. This approach was further developed in [18], [19], [2], [13]. Colliot-Th´el`ene and Sansuc have given a general formalism for writing down equations for these torsors. We briefly sketch their method: Let X be a smooth projective variety and {Dj }j∈J a finite set of irreducible divisors on X such that U := X \ ∪j∈J Dj has trivial Picard group. In practice, one usually chooses generators of the effective cone of X, e.g., the lines on the Del Pezzo surface. Consider the resulting exact sequence: ¯ ]∗ /k¯∗ −→ ⊕j∈J ZDj −→ Pic(Xk¯ ) −→ 0. 0 −→ k[U Applying Hom(−, Gm ), one obtains an exact sequence of tori 1 −→ T (X) −→ T −→ R −→ 1, where the first term is the N´eron-Severi torus of X. Suppose we have a collection of rational functions, invertible on U , which form a basis for the relations among the {Dj }j∈J . These can be interpreted as a section U →R × U , and thus naturally induce a T (X)-torsor over U , which canonically extends to the universal torsor over X. In practice, this extension can be made explicit, yielding equations for the universal torsor. However, when the cone generated by {Dj }j∈J is simplicial, there are no relations and this method gives little information. In this paper, we outline an alternative approach to the construction of universal torsors and illustrate it in specific examples where the effective cone of X is simplicial. We will work with varieties X such that the Picard and the N´eronSeveri groups of X coincide and such that the ring M Γ(X, L), Cox(X) := L∈Pic(X)

is finitely generated. This ring admits a natural action of the N´eronSeveri torus and the corresponding affine variety is a natural embedding of the universal torsor of X. The challenge is to actually compute Cox(X)

UNIVERSAL TORSORS AND COX RINGS

3

in specific examples; Cox has shown that it is a polynomial ring precisely when X is toric [9]. Here is a roadmap of the paper: In Section 1 we introduce Cox rings and discuss their general properties. Finding generators for the Cox ring entails embedding the universal torsor into affine space, which yields embeddings of our original variety into toric quotients of this affine space. We have collected several useful facts about toric varieties in Section 2. Section 3 is devoted to a detailed analysis of the unique cubic surface S with an isolated singularity of type E6 . We compute the (simplicial) ˜ and produce 10 distineffective cone of its minimal desingularization S, ˜ guished sections in Cox(S). These satisfy a unique equation and we show the universal torsor naturally embeds in the corresponding hypersurface in A10 . More precisely, we get a homomorphism from the coordinate ring ˜ and the main point is to prove its surjectivity. Here of A10 to Cox(S) we use an embedding of S˜ into a simplicial toric threefold Y , a quotient of A10 under the action of the N´eron-Severi torus so that Cox(Y ) is the polynomial ring over the above 10 generators. The induced restriction map on the level of Picard groups is an isomorphism respecting the moving cones. We conclude surjectivity for each degree by finding an appropriate birational projective model of Y and using vanishing results on it. Finally, in Section 4 we write down equations for the universal torsors (the Cox rings) of a split and a nonsplit cubic surface with an isolated singularity of type D4 . Acknowledgments: The results of this paper have been reported at the American Institute of Mathematics conference “Rational and integral points on higher dimensional varieties”. We benefited from the comments of the other participants, in particular, V. Batyrev and J.L. ColliotTh´el`ene. We also thank S. Keel for several helpful discussions about Cox rings and M. Thaddeus for advice about the geometric invariant theory of toric varieties.

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BRENDAN HASSETT and YURI TSCHINKEL

1. Generalities on the Cox ring For any finite subset Ξ of a real vector space, let Cone(Ξ) denote the closed cone generated by Ξ. Let X be a normal projective variety of dimension n over an algebraically closed field k of characteristic zero. Let An−1 (X) and Nn−1 (X) denote Weil divisors on X up to linear and numerical equivalence, respectively. Let A1 (X) and N1 (X) denote the classes of curves up to equivalence. Let NEn−1 (X) ⊂ Nn−1 (X)R denote the cone of (pseudo)effective divisors, i.e., the smallest real closed cone containing all the effective divisors of X. Let NE1 (X) ⊂ N1 (X)R denote the cone of effective curves and NM1 (X) ⊂ Nn−1 (X)R the cone of nef Cartier divisors, which is dual to the cone of effective Cartier divisors. By Kleiman’s criterion, this is the smallest real closed cone containing all ample divisors of X. Let L1 , . . . , Lr be invertible sheaves on X. For ν = (n1 , . . . , nr ) ∈ Nr write 1 r Lν := L⊗n ⊗ . . . ⊗ L⊗n . 1 r Consider the ring R(X, L1 , . . . , Lr ) :=

M

Γ(X, Lν ),

ν∈Nr

which need not be finitely generated in general. By definition, an invertible sheaf L on X is semiample if LN is globally generated for some N > 0: Proposition 1.1. — ([14], Lemma 2.8) If L1 , . . . , Lr are semiample then R(X, L1 , . . . , Lr ) is finitely generated. Remark 1.2. — If the Li are ample then, after replacing each Li by a large multiple, R(X, L1 , . . . , Lr ) is generated by Γ(X, L1 ) ⊗ . . . ⊗ Γ(X, Lr ). However, this is not generally the case if the Li are only semiample (despite the assertion in the second part of Lemma 2.8 of [14]). Indeed, let X→P1 × P1 be a double cover and L1 and L2 be the pull-backs of the polarizations on the P1 ’s to X. For suitably large n1 and n2 , Ln1 1 ⊗ Ln2 2

UNIVERSAL TORSORS AND COX RINGS

5

is very ample and its sections embed X. However, Γ(X, Ln1 1 ) ⊗ Γ(X, Ln2 2 ) ' Γ(P1 , OP1 (n1 )) ⊗ Γ(P1 , OP1 (n2 )), and any morphism induced by these sections factors through P1 × P1 . Proposition 1.3. — Let L1 , . . . , Lr be a set of invertible sheaves on X such that Lj is generated by sections sj,0 , ..., sj,dj . Assume that the Q induced morphism X→ j Pdj is birational into its image. Then the ring generated by the sj,k ’s has the same fraction field as R(X, L1 , . . . , Lr ). Proof. — Both rings have fraction field k(X)(t1 , ..., tr ), where tj is a nonzero section of Lj .

Definition 1.4. — [14] Let X be a nonsingular projective variety so that Pic(X) is a free abelian group of rank r. The Cox ring for X is defined Cox(X) := R(X, L1 , . . . , Lr ), where L1 , . . . , Lr are lines bundles so that 1. the Li form a Z-basis of Pic(X); 2. the cone Cone({L1 , . . . , Lr }) contains NEn−1 (X). This ring is naturally graded by Pic(X): for ν ∈ Pic(X) the ν-graded piece is denoted Cox(X)ν . Proposition 1.5. — [14] The ring Cox(X) does not depend on the choice of generators for Pic(X). Proof. — Consider two sets of generators L1 , . . . , Lr and M1 , . . . , Mr . Since Cone({Li }) and Cone({Mi }) contain all the effective divisors, the nonzero graded pieces of both R(X, L1 , . . . , Lr ) and R(X, M1 , . . . , Mr ) are indexed by the effective divisor classes in Pic(X). Choose isomorphisms Mj ' L(a1j ,...,arj ) , i = 1, . . . , r, A = (aij ) which naturally induce isomorphisms Γ(M ν ) ' Γ(LAν ),

Aν = (a11 ν1 + . . . + a1r νr , . . . , ar1 ν1 + . . . + arr νr ).

Thus we find R(X, L1 , . . . , Lr ) ' R(X, M1 , . . . , Mr ).

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BRENDAN HASSETT and YURI TSCHINKEL

As Cox(X) is graded by Pic(X), a free abelian group of rank r, the torus T (X) := Hom(Pic(X), Gm ) acts on Cox(X). Indeed, each ν ∈ Pic(X) naturally yields a character χν of T (X), and the action is given by t · ξ = χν (t)ξ,

ξ ∈ Cox(X)ν , t ∈ T (X).

Thus the isomorphism constructed in Proposition 1.5 is not canonical: Two such isomorphisms differ by the action of an element of T (X). It is precisely this ambiguity that makes descending the universal torsor to nonclosed fields an interesting question. The following conjecture is a special case of 2.14 of [14]: Conjecture 1.6 (Finiteness of Cox ring). — Let X be a log Fano variety. Then Cox(X) is finitely generated. Remark 1.7. — Note that if Cox(X) is finitely generated it follows trivially that NEn−1 (X) is finitely generated. Moreover, the nef cone NM1 (X) is also finitely generated. Indeed, the nef cone corresponds to one of the chambers in the group of characters of T (X) governed by the stability conditions for points v ∈ Spec(Cox(X)). These chambers are bounded by finitely many hyperplanes (see Theorem 0.2.3 in [10] for more details). It has been conjectured by Batyrev [1] that the pseudo-effective cone of a Fano variety is finitely generated. However, the finiteness of the Cox ring is not a formal consequence of the finiteness of the pseudo-effective cone. Example 1.8. — Let p1 , . . . , p9 ∈ H ⊂ P3 be nine distinct coplanar points given as a complete intersection of two generic cubic curves in the hyperplane H, and let X be the blow-up of P3 at these points. Then NE1 (X) is finitely generated but Cox(X) is not. Indeed, X is an equivariant compactification of the additive group G3a , acting by translation on the affine space P3 − H. The group action can be used to show that NE1 (X) is generated by the boundary components (see [12]). Similarly, one can show that the cone NE1 (X) is generated by classes of curves in ˜ ⊂ X of H. It the boundary components, e.g., the proper transform H

UNIVERSAL TORSORS AND COX RINGS

7

˜ is infinite [16] §1.23(4): The pencil of cubic is well-known that NE1 (H) plane curves with base locus p1 , . . . , p9 induces an elliptic fibration, 1 ˜ , H→P

˜ for which the nine exceptional curves of H→H are sections. Addition in the group law gives an infinite number of sections, which are also (−1)˜ These are also generators of NE1 (X), curves and generators of NE1 (H). ˜ since the sections (other than the nine exceptional curves) intersect H 1 negatively. It follows that NE1 (X) and NM (X) are not finitely generated and hence Cox(X) is not finitely generated (see Remark 1.7).

Proposition 1.9. — Let X be a nonsingular projective variety whose anticanonical divisor −KX is nef and big. Suppose that D is a nef divisor on X. Then H i (X, D) = 0 for each i > 0 and D is semiample. Proof. — The first assertion is a consequence of Kawamata-Viehweg vanishing [16] §2.5. The second is a special case of the Kawamata Basepointfreeness Theorem [16] §3.2. Proposition 1.9 largely determines the Hilbert function of the Cox ring: Corollary 1.10. — Retain the assumptions of Proposition 1.9. Then for nef classes ν we have dim Cox(X)ν = χ(OX (ν)). Remark 1.11. — In practice, this will help us to find generators of Cox(X).

2. Generalities on toric varieties We recall quotient constructions of toric varieties, following BrionProcesi [3], Cox [9], and Thaddeus [24]. Let T ' Grm be a torus with character group X∗ (T ). Suppose that T acts faithfully on the polynomial ring k[x1 , . . . , xn+r ] by the formula t(xj ) = χj (t)xj ,

t ∈ T,

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BRENDAN HASSETT and YURI TSCHINKEL

where {χ1 , . . . , χn+r } ⊂ X∗ (T ). Define M as the kernel of the surjective morphism χ := (χ1 , . . . , χn+r ) : Zn+r →X∗ (T ). We interpret M as the character group of the quotient torus Gn+r m /T . Set N = Hom(M, Z) so that dualizing gives (Zn+r )∗ →N →0. Let e1 , . . . , en+r and e∗1 , . . . , e∗n+r denote the coordinate vectors in Zn+r and (Zn+r )∗ ; let e¯∗1 , . . . , e¯∗n+r ∈ N denote the images of the e∗i in N . Concretely, the e¯∗i are the columns of the n×(n+r) matrix of dependence relations among the χj . Consider a toric n-fold X associated with a fan having one-skeleton {¯ e∗1 , . . . , e¯∗n+r }. In particular, we assume that none of e¯∗i is zero or a positive multiple of any of the others. The variety X is a categorical quotient of an invariant open subset U ⊂ An+r under the action of T described above (see [9] 2.1). Elements ν ∈ X∗ (T ) classify T -linearied invertible sheaves Lν on An+r and Γ(An+r , Lν ) ' k[x1 , . . . , xn+r ]ν . We have An−1 (X) ' X∗ (T ) and we can identify Γ(OX (D)) ' k[x1 , . . . , xn ]ν(D) , where ν(D) ∈ X∗ (T ) is associated with the divisor class of D. The variables xi are associated with the irreducible torus-invariant divisors Di on X (see [11] §3.4), and the cone of effective divisors NEn−1 (X) is generated by {D1 , . . . , Dn+r }. Geometrically, the effective cone in X∗ (T ) is the image of the standard simplicial cone generated by e1 , . . . , en+r under the projection homomorphism χ : Zn+r →X(T ). Recall that the moving cone Mov(X) ⊂ NEn−1 (X) is defined as the smallest closed subcone containing the effective divisors on X without fixed components. Proposition 2.1. — Retaining the notation and assumptions above, \ Cone(χ1 , . . . , χi−1 , χi+1 , . . . , χn+r ) Mov(X) = i=1,...,n+r

UNIVERSAL TORSORS AND COX RINGS

9

and has nonempty interior. Proof. — The fixed components of Γ(X, OX (D)) are necessarily invariant under the torus action, hence are taken from {D1 , . . . , Dn+r }. Moreover, Di is fixed in each Γ(X, OX (dD)), d > 0 if and only if xi divides each element of k[x1 , . . . , xn+r ]dν(D) . This is the case exactly when ν(D) ∈ Cone(χ1 , . . . , χn+r ) − Cone(χ1 , . . . , χi−1 , χi+1 , . . . , χn+r ). Suppose that the interior of the moving cone is empty. After permuting indices there are two possibilities: Either Cone(χ2 , . . . , χn+r ) has no interior, or the cones Cone(χ2 , . . . , χn+r ) and Cone(χ1 , χ3 , . . . , χn+r ) have nonempty interiors but meet in a cone with positive codimension. As the T -action is faithful, the χi span X∗ (T ). In the first case, χ2 , . . . , χn+r span a codimension-one subspace of X∗ (T ) that does not contain χ1 , so that each dependence relation c1 χ1 + . . . + cn+r χn+r = 0 has c1 = 0. This translates into e¯∗1 = 0, a contradiction. In the second case, χ3 , . . . , χn+r span a hyperplane, and χ1 and χ2 are on opposite sides of this hyperplane. Putting the dependence relations among the χi in row echelon form, we obtain a unique relation with nonzero first and second entries, and these two entries are both positive. This translates into the proportionality of e¯∗1 and e¯∗2 . We now seek to characterize the projective toric n-folds X with oneskeleton {¯ e∗1 , . . . , e¯∗n+r }. These are realized as Geometric Invariant Theory quotients An+r //T associated with the various linearizations of our T action. We consider the graded ring X X Γ(An+r , Ldν ) = k[x1 , . . . , xn+r ]dν . R := d≥0

d≥0

Proposition 2.2 (see [24] §2,3). — Retain the notation above and set X := Proj(R). 1. X is projective over k if and only if 0 is not contained in the convex hull of {χ1 , . . . , χn+r }. 2. X is toric of dimension n if ν is in the interior of Cone(χ1 , . . . , χn+r ).

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BRENDAN HASSETT and YURI TSCHINKEL

3. In this case, the one-skeleton of X is contained in {¯ e∗1 , . . . , e¯∗n+r }. Equality holds if ν is in the interior of the moving cone \ Cone(χ1 , . . . , χi−1 , χi+1 , . . . , χn+r ). i=1,...,n+r

Remark 2.3. — Our proof will show that X may still be of dimension n even when ν is contained in a facet of Cone(χ1 , . . . , χn+r ). Similary, the one-skeleton of X may still be {¯ e∗1 , . . . , e¯∗n+r } even when ν is contained in a facet of \ Cone(χ1 , . . . , χi−1 , χi+1 , . . . , χn+r ). i=1,...,n+r

Proof. — The monomials which appear in R are in one-to-one correspondence to solutions of a1 χ1 + . . . + an+r χn+r = dν,

ai ∈ Z≥0 .

In geometric terms, the monomials appearing in R coincide with the elements of Zn+r in the cone χ−1 (Cone(ν)) ∩ Cone(e1 , . . . , en+r ). By Gordan’s Lemma in convex geometry, R is generated as a k-algebra by a finite set of monomials xm1 , . . . , xms . The monomials appearing in the dth graded piece Rd coincide with elements of Zn+r in the polytope Pdν := χ−1 R (dν) ∩ Cone(e1 , . . . , en+r ). Note that χ−1 (dν) is a translate of M . For the first part, recall that Proj(R) is projective over Spec(R0 ), where R0 is the degree-zero part. Now 0 is in the convex hull of {χ1 , . . . , χn+r } if and only if there are nonconstant elements of R of degree zero. Our hypothesis just says that R0 = k and thus is equivalent to the projectivity of X over k. As for the second part, T acts on R by homotheties and thus acts n+r trivially on Proj(R), so we have an induced action of Gm /T on Proj(R). We claim this action is faithful, so the quotient is toric of dimension n. n+r Let µ1 , . . . , µn be generators for M = X∗ (Gn+r m /T ). Choose v ∈ Z in the interior of Cone(e1 , . . . , en+r ) so that χR (Cone(v)) = Cone(ν).

UNIVERSAL TORSORS AND COX RINGS

11

Replacing v by a suitably large integral multiple, we may assume each v+µi , i = 1, . . . , n, is in Cone(e1 , . . . , en+r ). If χ(v) = dν then Rd contains a set of generators for M , so the induced representation of Gn+r m /T on Rd is faithful. For the third part, we extract the fan classifying X from Pdν , following [11] §1.5 and 3.4: For each face Q of Pdν , consider the cone σQ = {v ∈ NR : hu, vi ≤ hu0 , vi for all u ∈ Q, u0 ∈ Pdν }. This assignment is inclusion reversing, so the one-dimensional cones of the fan correspond to facets of Pdν . Moreover, each facet of Pdν is induced by one of the facets of Cone(e1 , . . . , en+r ). The corresponding one-dimensional cone in NR is spanned by e¯∗i . It remains to verify that each facet of Cone(e1 , . . . , en+r ) actually induces a facet of Pdν . The hypothesis that ν is in the moving cone means that Pdν intersects each of the Cone(e1 , . . . , ei−1 , ei+1 , . . . , en+r ). If ν is in the interior of the moving cone then the intersection of Pdν with Cone(e1 , . . . , ei−1 , ei+1 , . . . , en+r ) meets the relative interior of this cone, hence this cone induces a facet of Pdν . Proposition 2.2 yields the following nice consequence: Proposition 2.4. — Let X be a complete toric variety and ν a divisor class in the interior of Mov(X). Then there exists a projective toric variety Yν , with the same one-skeleton as X, and polarized by ν. For generic T -linearized invertible sheaves on An+r , all semistable points are actually stable; hence Yν is a simplicial toric variety for generic ν (see [3] 1.2 and [9] 2.1). For the special values ν0 , contained in the walls of the chamber decomposition of [24], this fails to be the case. However, for each special ν0 , there exists a generic ν so that Cone(ν) is very close to Cone(ν0 ) and there is a projective, torus-equivariant morphism Yν →Yν0 [24] 3.11. The polarization associated to ν0 pulls back to Yν , so we obtain the following: Proposition 2.5. — Let X be a complete toric variety and ν0 a divisor class in the moving cone of X. Then there exists a simplicial projective toric variety Y , with the same one-skeleton as X, so that ν0 is semiample on Y .

BRENDAN HASSETT and YURI TSCHINKEL

12

Of course, ν0 is big when it is in the interior of the effective cone. 3. The E6 cubic surface By definition, the E6 cubic surface is given by the homogeneous equation (3.1)

S = {(w, x, y, z) : xy 2 + yw2 + z 3 = 0} ⊂ P3 .

We recall some elementary properties (see [4] for more details on singular cubic surfaces): Proposition 3.1. — 1. The surface S has a single singularity at the point p := (0, 1, 0, 0), of type E6 . 2. S is the unique cubic surface with this property, up to projectivity. 3. S contains a unique line, satisfying the equations y = z = 0. Any smooth cubic surface may be represented as the blow-up of P2 at six points in ‘general position’. There is an analogous property of the E6 cubic surface: Proposition 3.2. — The E6 cubic surface S is the closure of the image of P2 under the linear series w = a2 c x = −(ac2 + b3 ) y = a3

z = a2 b,

where Γ(P2 , OP2 (1)) = ha, b, ci. This map is the inverse of the projection of S from the double point p. The affine open subset A2 := {a 6= 0} ⊂ P2 is mapped isomorphically onto S − `. In particular, S \ ` ' A2 , so the E6 cubic surface is a compactification of A2 . Remark 3.3. — Note that S is not an equivariant compactification of G2a , so the general theory of [5] does not apply. Indeed, if S were an equivariant compactification of G2a then the projection from p would be G2a -equivariant (see [12]). Therefore, the map P2 99K S given above has to be G2a -equivariant. The only G2a -action

UNIVERSAL TORSORS AND COX RINGS

13

2 1

3

6

5

4

l

Figure 1. Dynkin diagram of E6

on P2 under which a line is invariant is the standard translation action [12]. However, the linear series above is not invariant under the standard translation action b 7→ b + βa c 7→ c + γa.

We proceed to compute the effective cone of the minimal resolution ˜ φ` : S→S. Let ` ⊂ S˜ be the proper transform of the line mentioned in Proposition 3.1. ˜ is a free abelian group of Proposition 3.4. — The Picard group Pic(S) rank seven, generated by ` and the exceptional curves of φ` . For a suitable ordering {F1 , F2 , F3 , F4 , F5 , F6 } of the exceptional curves, the intersection pairing takes the form

(3.2)

F1 F2 F3 ` F4 F5 F6 F1 −2 0 1 0 0 0 0 F2 0 −2 0 0 0 0 1 F3 1 0 −2 0 0 0 1 . ` 0 0 0 −1 1 0 0 F4 0 0 0 1 −2 1 0 F5 0 0 0 0 1 −2 1 F6 0 1 1 0 0 1 −2

˜ is simplicial and genProposition 3.5. — The effective cone NE(S) erated by Φ := {F1 , F2 , F3 , `, F4 , F5 , F6 }. Each nef divisor is contained in

BRENDAN HASSETT and YURI TSCHINKEL

14

the monoid generated by the divisors A1 A2 A3 A` A4 A5 A6

= = = = = = =

F2 + F3 + 2` + 2F4 + 2F5 + 2F6 F1 + F2 + 2F3 + 3` + 3F4 + 3F5 + 3F6 F1 + 2F2 + 2F3 + 4` + 4F4 + 4F5 + 4F6 2F1 + 3F2 + 4F3 + 3` + 4F4 + 5F5 + 6F6 2F1 + 3F2 + 4F3 + 4` + 4F4 + 5F5 + 6F6 2F1 + 3F2 + 4F3 + 5` + 5F4 + 5F5 + 6F6 2F1 + 3F2 + 4F3 + 6` + 6F4 + 6F5 + 6F6

. Moreover A` is the anticanonical class −KS˜ and its sections induce the ˜ resolution morphism φ` : S→S. Proof. — The intersection form in terms of A := {A1 , . . . , A6 } is:

(3.3)

A1 A2 A3 A` A4 A5 A6

A1 A2 A3 A` A4 A5 A6 0 1 1 2 2 2 2 1 1 2 3 3 3 3 1 2 2 4 4 4 4 2 3 4 3 4 5 6 2 3 4 4 4 5 6 2 3 4 5 5 5 6 2 3 4 6 6 6 6.

This is the inverse of the intersection matrix (3.2) written in terms of the basis Φ, so the Ai generate the dual to Cone(Φ). Observe that all the entries of matrix (3.3) are nonnegative and Cone(A) ⊂ Cone(Φ). ˜ We write D as a sum of Suppose that D is an effective divisor on S. the fixed components contained in {F1 , . . . , F6 , `} and the parts moving relative to Φ: D = MΦ + a1 F1 + . . . + a6 F6 + a` `,

a1 , . . . , a6 , a` ≥ 0.

A priori, MΦ may have fixed components, but they are not contained in Φ (however, see Lemma 3.6). It follows that MΦ intersects each element of Φ nonnegatively, i.e., it is contained in Cone(A) and thus in Cone(Φ). ˜ over We conclude that D ∈ Cone(Φ). Since A1 , ..., A6 , A` generate N1 (S)

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Z each nef divisor can be written as a nonnegative linear combination of these divisors. To see that A` is the anticanonical divisor, we apply adjunction KS˜ Fi = 0, i = 1, . . . , 6 KS˜ ` = −1. Nondegeneracy of the intersection form implies A` = −KS˜ . Since S has rational double points, the resolution map φ` is crepant, i.e., φ∗` KS = KS˜ . Thus Γ(A` ) = Γ(−KS˜ ) = Γ(−φ∗` KS ) = Γ(φ∗` OS (+1)) so the sections of A` induce φ` . Choose nonzero sections ξ1 , . . . , ξ` generating Γ(F1 ), . . . , Γ(`): Γ(F1 ) = hξ1 i , . . . , Γ(F6 ) = hξ6 i , Γ(`) = hξ` i . These are canonical up to scalar multiplication. Each effective divisor D = b1 F1 + b2 F2 + b3 F3 + b` ` + b4 F4 + b5 F5 + b6 F6 has a distinguished nonzero section ξ (b1 ,b2 ,b3 ,b` ,b4 ,b5 ,b6 ) := ξ1b1 . . . ξ6b6 ξ`b` . The distinguished section of Aj is denoted ξ α(j) . Note that we have an injective ring homomorphism ˜ (3.4) k[ξ1 , . . . , ξ6 , ξ` ]→Cox(S). ˜ D1 ≺ There is a partial order on the monoid of effective divisors of S: D2 if D2 − D1 is effective. The restriction of this order to the generators of the nef cone is illustrated in the diagram below: A6 | A5 | A4

A`

A3 A2 | A1

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BRENDAN HASSETT and YURI TSCHINKEL

Whenever D1 ≺ D2 we have an inclusion Γ(D1 ),→Γ(D2 ) which is natural up to scalar multiplication. Indeed, express D1 − D2 = b1 F1 + b2 F2 + . . . + b6 F6 + b` `,

bj ≥ 0

so we have s1 7→ ξ (b1 ,b2 ,b3 ,b` ,b4 ,b5 ,b6 ) s1 Γ(D1 ) ,→ Γ(D2 ). The homomorphism (3.4) is not surjective, and we now look for gen˜ beyond the ξj . Consider the subring erators of Cox(S) M ˜ = ˜ν Coxa (S) Cox(S) ˜ ν∈NM(S)

˜ The obtained by restricting to degrees corresponding to nef classes on S. ˜ can following lemma implies that any homogeneous element sD ∈ Cox(S) be written in the form sD = mD ξ1b1 · · · ξ6b6 ξ`b` ˜ with nonnegative exponents and mD ∈ Coxa (S). Lemma 3.6. — Let D be an effective divisor on S˜ with fixed part FD and moving part MD . Then FD is supported in {F1 , . . . , F6 , `}, and MD is a linear combination of A1 , . . . , A6 , A` with nonnegative coefficients. Proof. — Clearly MD is nef, so the description of the nef divisors in Proposition 3.5 gives the expression in terms of the Ai . Proposition 1.9 shows MD is semiample with vanishing higher cohomology; the last part of Proposition 3.5 gives the requisite positivity of the anticanonical class. Let F be a fixed component of D not supported in {F1 , . . . , F6 , `}. To arrive at a contradiction, we need to show that h0 (MD + F ) > h0 (MD ). Since MD has vanishing higher cohomology and h2 (F + MD ) = h0 (K − F − MD ) = 0 it suffices to show that χ(F + MD ) > χ(MD ).

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By Riemann-Roch, it suffices to show that F 2 + 2MD F − KS˜ F > 0 or, equivalently, F 2 + KS˜ F + 2MD F − 2KS˜ F = 2g(F ) − 2 + 2MD F − 2KS˜ F > 0. Since F is irreducible, g(F ) ≥ 0 and MD F ≥ 0 and −KS˜ F > 1, as MD is nef and −KS˜ is nonpositive only along the exceptional curves and has degree 1 only on the line ` (see Proposition 3.1). ˜ Corollary 1.10 gives the dimensions of the graded pieces of Coxa (S). We focus first on the generators of the nef cone, introducing sections τj ∈ Γ(Aj ) as needed to achieve the prescribed dimensions:

Γ(A1 ) = ξ α(1) , τ1

Γ(A2 ) = ξ α(2) , ξ α(2)−α(1) τ1 , τ2

Γ(A` ) = ξ α(`) , ξ α(`)−α(1) τ1 , ξ α(`)−α(2) τ2 , τ`

˜ ⊂ P3 by Proposition 3.5, and can be The sections of A` induce φ` : S→S identified with the coordinates w, x, y, z of Equation (3.1). Since A1 ≺ A2 ≺ A` , we have Γ(A1 ) ,→ Γ(A2 ) ,→ Γ(A` ). We can identify Γ(A1 ) = hy, zi; these correspond to projecting S from the line ` = {y = z = 0} and induce a conic bundle structure 1 ˜ . φ1 : S→P

We have Γ(A2 ) = hx, y, zi; these correspond to projecting S from the singularity p = {w = y = z = 0} and induce the blow-up realization 2 ˜ . φ2 : S→P

Therefore, we may choose τ1 , τ2 , and τ` so that y = ξ α(`)

w = ξ α(`)−α(2) τ2

z = ξ α(`)−α(1) τ1

x = τ` .

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We obtain the following induced sections for A3 , A4 , A5 , and A6 :

Γ(A3 ) = ξ α(3) , ξ α(3)−α(1) τ1 , ξ α(3)−α(2) τ2 , ξ α(3)−2α(1) τ12

Γ(A4 ) = ξ α(4) , ξ α(4)−α(1) τ1 , ξ α(4)−α(2) τ2 , ξ α(4)−α(`) τ` , ξ α(4)−2α(1) τ12

Γ(A5 ) = ξ α(5) , ξ α(5)−α(1) τ1 , ξ α(5)−α(2) τ2 , ξ α(5)−α(`) τ` , ξ α(5)−2α(1) τ12 , ξ α(5)−α(1)−α(2) τ1 τ2

Γ(A6 ) = ξ α(6) , ξ α(6)−α(1) τ1 , ξ α(6)−α(2) τ2 , ξ α(6)−α(`) τ` , ξ α(6)−2α(1) τ12 , ξ α(6)−α(1)−α(2) τ1 τ2 , ξ α(6)−2α(2) τ22 , ξ α(6)−3α(1) τ13

Equation (3.1) gives the relation

τ` ξ 2α(`) + τ22 ξ 3α(`)−2α(2) + τ13 ξ 3α(`)−3α(1) = 0. Dividing by a suitable monomial ξ β , we obtain τ` ξ`3 ξ42 ξ5 + τ22 ξ2 + τ13 ξ12 ξ3 = 0, a dependence relation in Γ(A6 ). This is the only such relation: Any other relation, after multiplying through by ξ β , yields a cubic form vanishing on S ⊂ P3 , but equation (3.1) is the only such form. It follows that the sections given above for A1 , . . . , A5 form bases for Γ(A1 ), . . . , Γ(A5 ). Since A3 ≺ A4 ≺ A5 ≺ A6 ≺ 2A` we have Γ(A3 ) ,→ Γ(A4 ) ,→ Γ(A5 ) ,→ Γ(A6 )

,→ Γ(2A` ) = w2 , wx, wy, x2 , xy, xz, y 2 , yz, z 2

and identifications

Γ(A3 ) = Γ(A4 ) = Γ(A5 ) = Γ(A6 ) =

y 2 , yz, wy, z 2

y 2 , yz, wy, xy, z 2 , wz

y 2 , yz, wy, xy, z 2

y 2 , yz, wy, xy, z 2 , wz, w2 .

The sections of A3 induce a morphism

3 ˜ φ3 : S→P

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onto a quadric surface with a single ordinary double point. The sections of A4 induce a morphism 4 ˜ φ4 : S→P with image a quartic Del Pezzo surface with a rational double point of type D5 . The sections of A5 induce a morphism 5 ˜ φ5 : S→P with image a quintic Del Pezzo surface with a rational double point of type D4 . The sections of A6 induce a morphisms 6 ˜ φ6 : S→P with image a sextic Del Pezzo surface with two rational double points, of types A1 and A2 . We summarize this analysis in the following proposition Proposition 3.7. — Every section of Aj , j = 1, 2, 3, `, 4, 5, 6, can be expressed as a polynomial in ξ1 , . . . , ξ6 , ξ` , τ1 , τ2 , τ6 . The only dependence relation among these is τ` ξ`3 ξ42 ξ5 + τ22 ξ2 + τ13 ξ12 ξ3 = 0 in Γ(A6 ). Each Aj is globally generated and induces a morphism χ−1 ˜ φj : S→P ,

χ = χ(OS˜ (Aj )).

The remainder of this section is devoted to proving the following: Theorem 3.8. — The homomorphism ˜ % : k[ξ1 , ..., ξ6 , ξ` , τ1 , τ2 , τ` ]/hτ` ξ`3 ξ42 ξ5 + τ22 ξ2 + τ13 ξ12 ξ3 i→Cox(S) is an isomorphism. If % were not injective, its kernel would have nontrivial elements in degree ν = dA` , for some d sufficiently large. These translate into homogeneous polynomials of degree d vanishing on S ⊂ P3 . All such polynomials are multiples of the cubic form defining S, which itself is a multiple of the relation we already have. It remains to show that % is surjective. By Proposition 3.5, Lemma 3.6 and the analysis of the sections of the Ai , it suffices to prove:

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Proposition 3.9. — % is surjective in degrees corresponding to nef di˜ visor classes of S. Lemma 3.10. — For any positive integers c1 , c2 , c3 , c` , c4 , c5 , c6 , the image of Γ(A1 )⊗c1 ⊗ . . . ⊗ Γ(A` )⊗c` ⊗ . . . ⊗ Γ(A6 )⊗c6 −→ Γ(c1 A1 + . . . + c6 A6 ) ˜ is a linear series embedding S. Proof. — Proposition 3.7 says that each Aj is globally generated, so if the image of Γ(A1 ) ⊗ . . . ⊗ Γ(A6 ) −→ Γ(A1 + . . . + A6 ) embed S˜ then the general result follows. We use the standard criterion: a linear series gives an embedding iff any length-two subscheme Σ ⊂ S˜ imposes two independent conditions on the linear series. First, suppose the support of Σ is not contained in the exceptional ˜ i.e., the curves F1 , F2 , F3 , F4 , F5 , F6 . Then φ` maps Σ locus of φ` : S→S, to a subscheme of length two, which imposes independent conditions on Γ(A` ), and thus independent conditions on the linear series in question. Second, suppose that Σ ⊂ Fj for some j (resp. Σ ⊂ `). Since Aj · Fj = 1 (resp. A` · ` = 1), φj maps Fj (resp. `) isomorphically onto a line. It follows that Σ imposes independent conditions on Γ(Aj ). Third, suppose that Σ is reduced with support in Fi and Fj , but is not contained in either Fi or Fj . Consider the chain of rational curves containing Fi and Fj (see Figure 3.) There exists a curve Fk in this chain so that φk (Fi ) 6= φk (Fj ), so Σ imposes independent conditions on Γ(Ak ). Fourth, suppose that Σ is nonreduced and supported in Fj but not contained in any Fi or `. The morphism φj ramifies at points where Fj meets one of the other exceptional curves, and the kernel of the tangent morphism dφj consists of the tangent vectors to the curves contracted by φj . It follows that φj (Σ) has length two and imposes independent conditions on Γ(Aj ). The polynomial ring k[ξ1 , . . . , ξ6 , ξ` , τ1 , τ2 , τ` ] is graded by the N´eron-Severi group of S˜ deg(ξj ) = Fj , j = 1, . . . , 6 deg(ξ` ) = ` deg(τj ) = Aj , j = 1, 2, `.

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˜ on the corresponding This gives an action of the N´eron-Severi torus T (S) 10 affine space A . We consider the projective toric varieties that arise as quotients of A10 ˜ As sketched in §2, these varieties have the one-skeleton by T (S). x1 = (0, 1, 2), x2 = (1, 1, 3), x3 = (1, 2, 4), x` = (2, 3, 3), x4 = (2, 3, 4) x5 = (2, 3, 5), x6 = (2, 3, 6), t1 = (−1, 0, 0), t2 = (0, −1, 0), t` = (0, 0, −1) where the xj correspond to the ξj and the tj correspond to the τj . Lemma 3.11. — Let X be a toric threefold with one-skeleton {x1 , . . . , t` } ˜ = N1 (S). ˜ Then and divisor class-group X∗ (T (S)) Mov(X) = Cone(A1 , . . . , A6 , A` ). Proof. — Proposition 2.1 reduces this to computing the intersection of the cones generated by subsets of {F1 , . . . , F6 , `, A1 , A2 , A` } with nine elements. Since A1 , A2 , A` are effective combinations of the classes F1 , . . . , F6 , and `, it suffices to compute ! \ \ ˆ . . . , A` ) Cone(F1 , . . . , `, Cone(F1 , . . . , Fˆi , . . . , A` ) . i=1,...,6

This intersection obviously contains A1 , A2 , and A` , and it is a straightforward computation to show that it also contains A3 , A4 , A5 , A6 . For the reverse inclusion, suppose that D is contained in the intersection. Con˜ we see that sidering D as a divisor on S, D · F1 , . . . , D · F6 , D · ` are all nonnegative. Thus D is an effective sum of Aj by Proposition 3.5. Combining Lemmas 3.11 and 3.10 with Propositions 2.2 and 3.7, we obtain the following ˜ Then there Proposition 3.12. — Let ν be an ample divisor on S. exists a projective toric variety Yν with one-skeleton {x1 , . . . , t` } and polarization ν, and an embedding S˜ ,→ Yν with the following properties:

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1. the divisor class group of Yν is isomorphic to the divisor class group of S˜ so that the moving cone of Yν is identified with the nef cone of ˜ S; 2. the equation for S˜ in the Cox ring of Yν is τ` ξ`3 ξ42 ξ5 + τ22 ξ2 + τ13 ξ12 ξ3 = 0 ˜ = A6 in the divisor class group of Yν ; and [S] 3. Cox(Yν ) = k[ξ1 , . . . , τ` ] and is mapped isomorphically to the image of the homomorphism %. For each toric variety Yν , we can consider the exact sequence of sheaves 0→IS˜ →OYν →OS˜ →0, ˜ Given an element θ in the where IS˜ ' OYν (−A6 ) is the ideal sheaf of S. divisor class group of Yν , we can twist to obtain 0→IS˜ (θ)→OYν (θ)→OS˜ (θ)→0. We should make precise what we mean by the twist F(θ) of a coherent sheaf F on Yν : Realize F as the sheafification of a graded module F over Cox(Yν ) (which exists by [17] Theorem 1.1, [9] Proposition 3.1), shift F by θ, and then resheafify the shifted module to obtain F(θ). Twisting respects exact sequences [9] 3.1. The anticanonical divisor of a toric variety is the sum of the invariant divisors [11] p. 89, so −KYν = F1 + . . . + F6 + ` + A1 + A2 + A` = A` + A6 and we can rewrite our exact sequence as 0→OYν (KYν + A` + θ)→OYν (θ)→OS˜ (θ)→0. ˜ we shall prove that % Suppose that θ corresponds to a nef class on S; is surjective in degree θ, thus proving Proposition 3.9 and Theorem 3.8. Since Γ(OYν (θ)) ' k[ξ1 , . . . , ξ6 , ξ` , τ1 , τ2 , τ` ]θ it suffices to show that H 1 (OYν (KYν + A` + θ)) = 0. We apply Proposition 2.5, with ν0 = A` + θ, to get a simplicial toric variety Yν on which A` + θ is nef. As A` is in the interior of the effective

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cone of Yν , A` +θ is also big. Note that Yν has finite-quotient singularities, which are log terminal [16] §5.2. The desired vanishing follows from Theorem 2.17 of [15]. Alternately, we could apply Theorem 0.1 of [17], which applies in arbitrary characteristic and obviates the need to pass to a simplicial model.

4. D4 cubic surface The strategy of the previous section can applied to other surfaces as well. Here we illustrate it in the case of a cubic surface given by the homogeneous equation S = {(x1 , x2 , x3 , w) : w(x1 + x2 + x3 )2 = x1 x2 x3 } ⊂ P3 . We summarize its properties: 1. S has a single singularity at the point (0, 0, 0, 1) of type D4 . 2. S contains 6 lines with the equations `01 := {w = x1 = 0} m01 := {x1 = x2 + x3 = 0} `02 := {w = x2 = 0} m02 := {x2 = x1 + x3 = 0} `03 := {w = x3 = 0} m03 := {x3 = x1 + x2 = 0} 3. S is given as a blow-up of P2 by the linear series x1 = u1 (u1 + u2 + u3 )2 , x2 = u2 (u1 + u2 + u3 )2 , x3 = u3 (u1 + u2 + u3 )2 , w = u1 u 2 u3 , where hu1 , u2 , u3 i = Γ(P2 , OP2 (1)). Remark 4.1. — There are two isomorphism classes of cubic surfaces with a D4 singularity [4] Lemma 4. The other class is w(x1 + x2 + x3 )2 = x1 x2 (−x1 − x2 ); it is obtained from S by substituting (w, x1 , x2 , x3 ) 7→ (t−2 w, x1 , x2 , tx3 + (t − 1)x1 + (t − 1)x2 ) and letting t→0 in the resulting equation.

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˜ Let β : S→S denote the minimal desingularization of S and `1 , `2 , `3 , m1 , m2 , m3 the strict transforms of the lines. The rational map S 99K P2 induces a 2 ˜ and let L denote the pullback of the hyperplane class. morphism S→P Let E0 , E1 , E2 , E3 be the exceptional divisors of β, ordered so that we have the following intersection matrix:

(4.1)

L E1 E2 E3 m1 m2 m3

L E1 E2 E3 m1 m2 m3 1 0 0 0 0 0 0 0 −2 0 0 1 0 0 0 0 −2 0 0 1 0 . 0 0 0 −2 0 0 1 0 1 0 0 −1 0 0 0 0 1 0 0 −1 0 0 0 0 1 0 0 −1

This is a rank seven unimodular matrix; since the Picard group of S˜ has rank seven, it is generated by L, E1 , E2 , E3 , m1 , m2 , m3 . In particular, we have E0 = L − (E1 + E2 + E3 + m1 + m2 + m3 ) and `j = L − Ej − 2mj . The anticanonical class is given by −KS˜ = 3L − (E1 + E2 + E3 ) − 2(m1 + m2 + m3 ) = `1 + `2 + `3 . ˜ is generated by Proposition 4.2. — The effective cone NE(S) Ξ := {E0 , E1 , E2 , E3 , mj , `j }. Proof. — Each effective divisor D can be expressed as a sum D = MΞ + bE0 E0 + bE1 E1 + . . . + b`3 `3 , with nonnegative coefficients, where MΞ intersects each of the elements in Ξ nonnegatively and thus is in the dual cone to Cone(Ξ). Direct computation shows that the dual to Cone(Ξ) has generators L, L − Ei − mi , 2L − Ei − 2mi , 2L − Ei − Ej − 2mi − 2mj , 2L − Ei − Ej − mi − 2mj .

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Each of these is contained in Cone(Ξ): L 2L − Ei − 2mi 2L − Ei − Ej − mi − 2mj L − Ei − mi 2L − Ei − Ej − 2mi − 2mj

= = = = =

`i + Ei + 2mi , 2`i + Ei + 2mi , `i + `j + mi , `i + mi , `i + `j .

It follows that MΞ and D are sums of elements in Ξ with nonnegative coefficients. Each of the divisors mi , `i and Ei has a distinguished nonzero section (up to a constant), denoted µi , λi and ηi , respectively. We have {λi ηi µ2i , η0 η1 η2 η3 µ1 µ2 µ3 } ⊂ Γ(L), and we may identify ui = λi ηi µ2i and u1 + u2 + u3 = η0 η1 η2 η3 µ1 µ2 µ3 after suitably normalizing the µi , λi , and ηi . The dependence relation among the sections in Γ(L) translates into (4.2)

λ1 η1 µ21 + λ2 η2 µ22 + λ3 η3 µ23 = η0 η1 η2 η3 µ1 µ2 µ3 .

An argument similar to the one given at the end of Section 3 proves that the natural homomorphism ˜ k[η0 , ..., η3 , µi , λi ]/hλ1 η1 µ21 + λ2 η2 µ22 + λ3 η3 µ23 − η0 η1 η2 η3 µ1 µ2 µ3 i→Cox(S) is an isomorphism. The cubic surface S admits an S3 -action on the coordinates x1 , x2 , x3 . In particular, it admits nonsplit forms over nonclosed ground fields. They can be expressed as follows: Let K/k be a cubic extension with Galois closure E/k. Fix a basis {γ, γ 0 , γ 00 } for K over k so that elements Y ∈ K can be represented as Y = yγ + y 0 γ 0 + y 00 γ 00 with y, y 0 , y 00 ∈ k. Choose σ ∈ Gal(E/k) so that σ and σ 2 are coset representatives Gal(E/k) modulo Gal(E/K). Then w · TrK/k (Y )2 = NK/k (Y )

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is isomorphic, over E, to S: x1 = Y = yγ + y 0 γ 0 + y 00 γ 00 x2 = σ(Y ) = yσ(γ) + y 0 σ(γ 0 ) + y 00 σ(γ 00 ) . 2 2 0 2 0 00 2 00 x3 = σ (Y ) = yσ (γ) + y σ (γ ) + y σ (γ ) Assigning elements U, V, W ∈ K to η1 , µ1 and λ1 , respectively, the torsor equation (4.2) takes the form TrK/k (U V 2 W ) = η0 NK/k (U V ). References [1] V. V. Batyrev – “The cone of effective divisors of threefolds”, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) (Providence, RI), Contemp. Math., vol. 131, Amer. Math. Soc., 1992, p. 337–352. `che – “Nombre de points de hauteur born´ee sur [2] R. de la Brete les surfaces de del Pezzo de degr´e 5”, Duke Math. J. 113 (2002), no. 3, p. 421–464. [3] M. Brion and C. Procesi – “Action d’un tore dans une vari´et´e projective”, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Progr. Math., vol. 92, Birkh¨auser Boston, Boston, MA, 1990, p. 509–539. [4] J. W. Bruce and C. T. C. Wall – “On the classification of cubic surfaces”, J. London Math. Soc. (2) 19 (1979), no. 2, p. 245–256. [5] A. Chambert-Loir and Y. Tschinkel – “On the distribution of points of bounded height on equivariant compactifications of vector groups”, Invent. Math. 148 (2002), no. 2, p. 421–452. ´le `ne and J.-J. Sansuc – “La descente sur les [6] J.-L. Colliot-The vari´et´es rationnelles. II”, Duke Math. J. 54 (1987), no. 2, p. 375–492. ´le `ne, J.-J. Sansuc and P. Swinnerton[7] J.-L. Colliot-The Dyer – “Intersections of two quadrics and Chˆatelet surfaces. I”, J. Reine Angew. Math. 373 (1987), p. 37–107. [8] , “Intersections of two quadrics and Chˆatelet surfaces. II”, J. Reine Angew. Math. 374 (1987), p. 72–168. [9] D. A. Cox – “The homogeneous coordinate ring of a toric variety”, J. Algebraic Geom. 4 (1995), no. 1, p. 17–50.

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[10] I. V. Dolgachev and Y. Hu – “Variation of geometric invari´ ant theory quotients”, Inst. Hautes Etudes Sci. Publ. Math. (1998), no. 87, p. 5–56, With an appendix by Nicolas Ressayre. [11] W. Fulton – Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry. [12] B. Hassett and Y. Tschinkel – “Geometry of equivariant compactifications of Gna ”, Internat. Math. Res. Notices (1999), no. 22, p. 1211–1230. [13] D. R. Heath-Brown – “The density of rational points on Cayley’s cubic surface”, to appear. [14] Y. Hu and S. Keel – “Mori dream spaces and GIT”, Michigan Math. J. 48 (2000), p. 331–348, Dedicated to William Fulton on the occasion of his 60th birthday. ´r – “Singularities of pairs”, Algebraic geometry—Santa [15] J. Kolla Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, p. 221–287. ´r and S. Mori – Birational geometry of algebraic vari[16] J. Kolla eties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. ˘ – “Vanishing theorems on toric varieties”, Tohoku [17] M. Mustat ¸a Math. J. (2) 54 (2002), no. 3, p. 451–470. [18] E. Peyre – “Terme principal de la fonction zˆeta des hauteurs et torseurs universels”, Ast´erisque (1998), no. 251, p. 259–298, Nombre et r´epartition de points de hauteur born´ee (Paris, 1996). , “Torseurs universels et m´ethode du cercle”, Rational points [19] on algebraic varieties, Progr. Math., vol. 199, Birkh¨auser, Basel, 2001, p. 221–274. [20] P. Salberger and A. N. Skorobogatov – “Weak approximation for surfaces defined by two quadratic forms”, Duke Math. J. 63 (1991), no. 2, p. 517–536. [21] P. Salberger – “Tamagawa measures on universal torsors and points of bounded height on Fano varieties”, Ast´erisque (1998), no. 251, p. 91–258, Nombre et r´epartition de points de hauteur born´ee (Paris, 1996).

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[22] A. Skorobogatov – Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. [23] A. N. Skorobogatov – “On a theorem of Enriques-SwinnertonDyer”, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 3, p. 429– 440. [24] M. Thaddeus – “Toric quotients and flips”, Topology, geometry and field theory, World Sci. Publishing, River Edge, NJ, 1994, p. 193–213.

RANDOM DIOPHANTINE EQUATIONS ´ FELIPE VOLOCH BJORN POONEN AND JOSE

´ le `ne and Nicholas M. Katz) (with appendices by Jean-Louis Colliot-The 1. Introduction The main result of this paper is that, in a precise sense, a positive proportion of all hypersurfaces in Pn of degree d defined over Q are everywhere locally solvable, provided that n, d ≥ 2 and (n, d) 6= (2, 2). This result is motivated by a conjecture discussed in detail below about the proportion of hypersurfaces as above that are globally solvable, i.e., have a rational point. 2. A conjecture Fix n, d ≥ 2. Let Z[x0 , .. . , xn ]d denote the set of homogeneous polynomials in Z[x0 , . . . , xn ] of degree d. Let m = n+d denote the number of monomials in x0 , . . . , xn of degree d. Define d the height h(f ) of f ∈ Z[x0 , . . . , xn ]d as the maximum of the absolute values of the coefficients of f . Let MQ be the set of places of Q, and let Qv be the completion of Q at the place v. Define Ntot (H) = #{ f ∈ Z[x0 , . . . , xn ]d : h(f ) ≤ H } = (2bHc + 1)m , N (H) = #{ f ∈ Z[x0 , . . . , xn ]d : h(f ) ≤ H, and ∃x ∈ Zn+1 \ {0} with f (x) = 0 }, Nloc (H) = #{ f ∈ Z[x0 , . . . , xn ]d : h(f ) ≤ H, and ∀v ∈ MQ , ∃x ∈ Qn+1 \ {0} with f (x) = 0 }. v The limit of N (H)/Ntot (H) as H → ∞, if it exists, will be called the proportion of globally solvable hypersurfaces. Similarly, the limit of Nloc (H)/Ntot (H) will be called the proportion of locally solvable hypersurfaces. Remark 2.1. (1) Restricting the set of f ’s to those such that f = 0 defines a smooth geometrically integral hypersurface in Pn does not change the values of these limits, since the f ’s that violate these conditions correspond to integer points on a Zariski closed subset of positive codimension in some affine space. (2) A standard argument involving M¨obius inversion shows that the values of the limits do not change if in all our counts we restrict to f ’s whose coefficients are coprime. (3) Everything we do in this paper over Q could be generalized to number fields without difficulty. We chose to work over Q to keep statements and proofs simple. Conjecture 2.2. Date: March 1, 2003. 2000 Mathematics Subject Classification. Primary 11D72; Secondary 11G35, 14G25. Key words and phrases. Diophantine equation, Hasse principle, Brauer-Manin obstruction, hypersurface, complete intersection, Weak Lefschetz Theorem, cubic surface. The first author was partially supported by a Packard Fellowship. 1

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´ FELIPE VOLOCH BJORN POONEN AND JOSE

(i) If d > n + 1, then N (H)/Ntot (H) → 0. (ii) If d < n + 1 and Q (d, n) 6= (2, 2), then N (H)/Ntot (H) → c for some real c > 0. Moreover, c = v∈MQ cv , where cv is the proportion of polynomials in Z[x0 , . . . , xn ]d with a nontrivial zero over Qv . In the case d = n + 1, we do not know what to expect. As a special case, if you write down a plane cubic, how likely is it to have a rational point? Remark 2.3. Each local proportion cv exists, since if we define Zv = { x ∈ Qv : |x|v ≤ 1 } and normalize Haar measure (Lebesgue measure if v = ∞) on the space Zm v parametrizing homogeneous polynomials of degree d in x0 , . . . , xn with coefficients in Zv , then cv is the measure of the v-adically closed subset of Zm v corresponding to homogeneous polynomials with a nontrivial zero over Qv . 3. Motivation and evidence To motivate the first part of the conjecture, consider the set of f ∈ Z[x0 , . . . , xn ]d of height at most H having a given zero a ∈ Zn+1 \ {0} with coprime coordinates. This forms a hyperplane in the parameter space Zm , and contains c(a)H m−1 /φ(a) + O(H m−2 ) integer points of height at most H, where c(a) is the (m − 1)-dimensional volume of the part of the hyperplane inside [−1, 1]m , and φ(a) is the covolume of the lattice of integer points lying on the hyperplane. Lemma 3.1 below shows that φ(a) equals the norm of the vector b formed by the monomials of degree ignore the error term, then we get P d in the coordinates of a. If wen+1 m−1 that N (H) ≤ H \{0}. Now c(a) is bounded, and a c(a)/φ(a), where a ranges over Z P it is easy to show that 1/φ(a) converges precisely when d > n + 1, so our heuristic predicts N (H) = O(H m−1 ). Since Ntot (H) ∼ (2H)m , this leads to the first part of the conjecture. Lemma 3.1. Let b be a vector in Rm with coprime integer √ coordinates and norm |b| = M . m The covolume of the lattice Λ = { x ∈ Z : hx, bi = 0 } is M . Proof. The lattice Γ = { x ∈ Zm : hx, bi ≡ 0 (mod M ) } is the inverse image of M Z under the surjection Zm → Z mapping x to hx, bi, so Γ has covolume M in Rm . On the √ other hand, Γ is the orthogonal direct sum of Λ and Zb, so the covolume of Λ is M/|b| = M . Our next proposition will be conditional on cases of the following very general conjecture. Conjecture 3.2 (Colliot-Th´el`ene). Let X be a smooth, proper, geometrically integral variety over a number field k. Suppose that X is (geometrically) rationally connected. Then the Brauer-Manin obstruction to the Hasse principle for X is the only obstruction. Remark 3.3. Conjecture 3.2 has a long history. In the special case of rational surfaces, it appeared as Question (k1) on page 233 of [CTS80] (a paper later developed as [CTS87]), based on evidence eventually published in the papers [CTCS80] and [CTS82]. Theoretical evidence and some numerical evidence have been gathered since then, in the case of rational surfaces. The conjecture was generalized to (geometrically) unirational and Fano varieties on the first page of [CTSD94]. The full version of Conjecture 3.2 was raised as a question in lectures by Colliot-Th´el`ene at the Institut Henri Poincar´e in Spring 1999, and was repeated in print on page 3 of [CT03]. Proposition 3.4. Assume Conjecture 3.2. If 2 ≤ d ≤ n, then (Nloc − N (H))/Ntot (H) → 0 as H → ∞.

RANDOM DIOPHANTINE EQUATIONS

3

Proof. By Remark 2.1, we may restrict attention to f ’s for which f = 0 defines a smooth, geometrically integral hypersurface X in Pn . The assumption d ≤ n implies that X is Fano, hence rationally connected (see Theorem V.2.13 of [Kol96]). If n ≥ 4, then by Corollary 4.2 there is no Brauer-Manin obstruction, so Conjecture 3.2 gives the Hasse principle, as desired. If d = 2, then the Hasse principle holds unconditionally. It remains to consider the case of cubic surfaces (d = n = 3). Here the Hasse principle does not always hold. But by [SD93] there is no Brauer-Manin obstruction whenever whenever the action of Gal(Q/Q) on the 27 lines is as large as possible (namely, the Weyl group W (E6 )). The Galois action on the 27 lines on the generic cubic surface over Q(a1 , . . . , a20 ) is W (E6 ) (this follows from [Tod35] and [Seg42]), so it follows by Hilbert irreducibility (see §9.2 and §13 of [Ser97]) that the same holds for a density 1 set of cubic surfaces over Q. Such cubic surfaces, under Conjecture 3.2, satisfy the Hasse principle as desired. Remark 3.5. For n large compared to d, the conclusion of Proposition 3.4 can be proved unconditionally by using the circle method. Part (ii) of Conjecture 2.2 would follow from the conclusion of Proposition 3.4 and the following result. Theorem 3.6.QIf n, d ≥ 2 and (n, d) 6= (2, 2) then Nloc (H)/Ntot (H) → c for some c > 0. Moreover, c = cv where cv is as in Conjecture 2.2. Proof. By Hensel’s Lemma, a hypersurface f = 0 will have a point in Qp if its reduction modulo p has a smooth point in Fp . If f is absolutely irreducible modulo p and p is sufficiently large (in terms of n and d), then the existence of a smooth point in Fp is ensured by the Lang-Weil estimate [LW54]. Lemmas 20 and 21 of [PS99a] will now imply the theorem, provided that we can show that the space of reducible polynomials is of codimension at least 2 in the space of all polynomials. The lower bound on the codimension follows from the inequalities n+r n + (d − r) n+d −1 + −1 ≤ −1 −2 n n n for 0 < r < d, which hold provided that n, d ≥ 2 and (n, d) 6= (2, 2). (Here r and d − r represent degrees in a potential factorization.) See also [PS99b] for an exposition of the application of the density lemmas from [PS99a]. Remark 3.7. It follows from Theorem 3.6 and part (i) of Conjecture 2.2 that for each pair (d, n) with n ≥ 2 and d > n + 1, there are hypersurfaces of degree d in Pn for which the Hasse principle fails. There does not seem to be an unconditional proof of this statement yet, when n ≥ 3. For results conditional on various conjectures see [SW95] and [Poo01]. ´ le `ne: The Brauer-Manin 4. Appendix A by Jean-Louis Colliot-The obstruction for complete intersections of dimension ≥ 3 It seems that a full proof of the following proposition has never before appeared in print, though a sketch can be found in §2 of [SW95]. Let Hi below denote ´etale cohomology (or profinite group cohomology) unless otherwise specified, and let Br X denote the cohomological Brauer group H2 (X, Gm ) of a scheme X. Proposition 4.1. Let k be a field of characteristic 0. Let X be a smooth complete intersection in Pnk satisfying dim X ≥ 3. Then the natural map Br k → Br X is an isomorphism.

´ FELIPE VOLOCH BJORN POONEN AND JOSE

4

Proof. Let k denote an algebraic closure of k, let G = Gal(k/k), and let X = X ×k k. Let p : X → Spec k denote the structure map. In the Leray spectral sequence E2p,q := Hp (k, Rq p∗ Gm )) =⇒ E p+q := Hp+q (X, Gm ), the ´etale sheaf Rq p∗ Gm on Spec k corresponds to the G-module Hq (X, Gm ). A smooth complete intersection of positive dimension is geometrically connected [Har77, Exercise III.5.5(b)], × so H0 (X, Gm ) = k , and [BLR90, p. 203] shows that H1 (X, Gm ) = H1Zariski (X, Gm ) =: Pic X. Thus the exact sequence E21,0 → E 1 → E20,1 → E22,0 → ker E 2 → E20,2 → E21,1 from the spectral sequence becomes G 0 → Pic X → Pic X → Br k → ker Br X → Br X → H1 (k, Pic X). G It remains to prove that Pic X → Pic X is an isomorphism, that H1 (k, Pic X) = 0, and that Br X = 0. For smooth complete intersections of dimension ≥ 3 in Pn , M. Noether proved that the restriction map Pic Pn → Pic X is an isomorphism (see Corollary 3.3 on p. 180 of [Har70] for a modern proof). The commutative square / Pic X _

Pic Pn

*

Pic Pn

∼

/

Pic X

shows that the injections Pic X ,→ Pic X 1

G

,→ Pic X

1

are isomorphisms, and that H (k, Pic X) = H (G, Z) = Homconts (G, Z) = 0. Finally we need to show that if Y is a complete intersection of dimension ≥ 3 in Pn over an algebraically closed field L of characteristic 0, then Br Y = 0. For each prime `, the Kummer sequence yields the exact rows of the diagram 0 −−−→ Pic(Pn )/` −−−→ H2 (Pn , Z/`Z)     y y 0 −−−→ Pic(Y )/` −−−→ H2 (Y, Z/`Z) −−−→ (Br Y )[`] −−−→ 0, where for any abelian group A, the notation A/` denotes A/`A, and A[`] is the kernel of multiplication-by-` on A. The top horizontal injection Pic(Pn )/` → H2 (Pn , Z/`Z) is an isomorphism since both groups are of rank 1 over Z/`Z. The right vertical map H2 (Pn , Z/`Z) → H2 (Y, Z/`Z) is an isomorphism by a version of the Weak Lefschetz Theorem: see Corollary 5.6 in Appendix B of this paper. The diagram then implies that (Br Y )[`] = 0. This holds for all `, and Br Y is torsion [Gro68, Proposition 1.4], so Br Y = 0. Corollary 4.2. If in addition, k is a number field, then the Brauer-Manin obstruction for X is vacuous. Proof. This follows from Proposition 4.1, since the elements of Br X coming from Br k do not give any obstruction to rational points.

RANDOM DIOPHANTINE EQUATIONS

5

5. Appendix B by Nicholas M. Katz: Applications of the Weak Lefschetz Theorem We work over an algebraically closed field k. Take as ambient space any separated V /k of finite type which is smooth, and everywhere of dimension N (i.e., each connected component of V has the same dimension N ). In V , we are given a certain number r ≥ 1 of closed subschemes Hi , each of which has the property that its complement V − Hi is affine. Define the closed subscheme X of V to be the intersection of the Hi . Its complement V − X is covered by r affine open sets, each of dimension (at most) N , namely the V − Hi . Lemma 5.1. The scheme V − X has cohomological dimension at most N + r − 1, i.e., for any constructible torsion sheaf F on V − X, we have Hi (V − X, F) = 0 for i ≥ N + r. This is a special case of Lemma 5.2. If a separated k-scheme W/k of finite type is the union of r affine opens Ui , each of dimension at most N , then W has cohomological dimension at most N + r − 1. Proof. For r = 1, this is just the Lefschetz affine theorem [SGA4 III, Expos´e XIV, Corollaire 3.2]. For general r, one proceeds by induction on r, writing W as the union of the two S open sets A := Ur and B := i

Suppose now that ` is a prime number invertible in the field k, and that F is a Z/`Z sheaf on V whose restriction to V − X is lisse, for instance Z/`Z itself. Because V − X is smooth, everywhere of dimension N , the Poincare dual of Lemma 5.1 is the vanishing of compact cohomology up through dimension N − r: Lemma 5.3. For any integer i ≤ N − r, we have Hic (V − X, F) = 0. Now use the excision sequence (this is why we need F to be a sheaf on V ) in compact cohomology . . . → Hic (V − X, F) → Hic (V, F) → Hic (X, F) → . . . to conclude Theorem 5.4. For any integer i < N − r, the restriction map Hic (V, F) → Hic (X, F) is an isomorphism. For i = N − r, this map is injective. Corollary 5.5. If V /k is in addition assumed proper, then we have the same result for non-compact cohomology: For any integer i < N − r, the restriction map Hi (V, F) → Hi (X, F) is an isomorphism. For i = N − r, this map is injective. As a special case of Corollary 5.5, we get Corollary 5.6. Suppose X is a closed subscheme of projective space PN which is defined by the vanishing of N − d homogeneous forms. Then for i < d, the restriction map Hi (PN , Z/`Z) → Hi (X, Z/`Z) is an isomorphism. For i=d, this map is injective.

6

´ FELIPE VOLOCH BJORN POONEN AND JOSE

Remark 5.7. In Corollary 5.6, it is enough if X is defined set-theoretically by the vanishing of N − d homogeneous forms, since X and X red have the same ´etale cohomology.

Acknowledgements This paper developed during a workshop sponsored by the American Institute of Mathematics. We thank Jean-Louis Colliot-Th´el`ene and Nick Katz for allowing us to include their results in the appendices to this paper. We thank also Jean-Louis Colliot-Th´el`ene, Jordan Ellenberg, Brendan Hassett, Roger Heath-Brown, Bill McCallum, and Trevor Wooley for many conversations which helped shape Conjecture 2.2.

References [BLR90]

Siegfried Bosch, Werner L¨ utkebohmert, and Michel Raynaud, N´eron models, Springer-Verlag, Berlin, 1990. [CT03] Jean-Louis Colliot-Th´el`ene, Points rationnels sur les fibrations, Higher dimensional varieties and rational points, Proceedings of the 2001 Budapest conference (K. B¨or¨oczky, J. Koll´ar, and T. Szamuely, eds.), Springer-Verlag, 2003, Bolyai Society Colloquium Publications. [CTCS80] Jean-Louis Colliot-Th´el`ene, Daniel Coray, and Jean-Jacques Sansuc, Descente et principe de Hasse pour certaines vari´et´es rationnelles, J. Reine Angew. Math. 320 (1980), 150–191. [CTS80] J.-L. Colliot-Th´el`ene and J.-J. Sansuc, La descente sur les vari´et´es rationnelles, Journ´ees de G´eometrie Alg´ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 223–237. [CTS82] Jean-Louis Colliot-Th´el`ene and Jean-Jacques Sansuc, Sur le principe de Hasse et l’approximation faible, et sur une hypoth`ese de Schinzel, Acta Arith. 41 (1982), no. 1, 33–53. [CTS87] Jean-Louis Colliot-Th´el`ene and Jean-Jacques Sansuc, La descente sur les vari´et´es rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492. [CTSD94] Jean-Louis Colliot-Th´el`ene and Peter Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. Reine Angew. Math. 453 (1994), 49–112. [Gro68] Alexander Grothendieck, Le groupe de Brauer. II. Th´eorie cohomologique, Dix Expos´es sur la Cohomologie des Sch´emas, North-Holland, Amsterdam, 1968, pp. 67–87. [Har70] Robin Hartshorne, Ample subvarieties of algebraic varieties, Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin, 1970. [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. [Kol96] J´ anos Koll´ ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. [LW54] Serge Lang and Andr´e Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. [Poo01] Bjorn Poonen, The Hasse principle for complete intersections in projective space, Rational points on algebraic varieties, Progr. Math., vol. 199, Birkh¨auser, Basel, 2001, pp. 307–311. [PS99a] Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. [PS99b] Bjorn Poonen and Michael Stoll, A local-global principle for densities, Topics in number theory (University Park, PA, 1997), Math. Appl., vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 241–244. [SD93] Peter Swinnerton-Dyer, The Brauer group of cubic surfaces, Math. Proc. Cambridge Philos. Soc. 113 (1993), no. 3, 449–460. [Seg42] B. Segre, The Non-singular Cubic Surfaces, Oxford University Press, Oxford, 1942.

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[Ser97]

7

Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, third ed., Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre. [SGA4 III] Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 3, Springer-Verlag, Berlin, 1973, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1963–1964 (SGA 4), Dirig´e par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305. [SW95] Peter Sarnak and Lan Wang, Some hypersurfaces in P4 and the Hasse-principle, C. R. Acad. Sci. Paris S´er. I Math. 321 (1995), no. 3, 319–322. [Tod35] J. A. Todd, On the topology of certain threefold loci, Proc. Edinburgh Math. Soc. (2) (1935), no. 4, 175–184. Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA E-mail address: [email protected] Department of Mathematics, University of Texas, Austin, TX 78712, USA E-mail address: [email protected]

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³v³

RATIONAL POINTS ON COMPACTIFICATIONS OF SEMI-SIMPLE GROUPS OF RANK 1 by

Joseph Shalika, Ramin Takloo-Bighash and Yuri Tschinkel Abstract. — We explain our approach to the problem of counting rational points of bounded height on equivariant compactifications of semi-simple groups.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Basic definitions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Heights and height integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Eisenstein integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. P-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The cuspidal spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 7 9 13 16 22 26 31

1. Introduction Let G be a linear algebraic group, S ⊂ G a subgroup and S\G the corresponding homogeneous space, all assumed to be defined over a number

The second author was partially supported by the Clay Mathematics Institute. The third author was partially supported by NSF grant 0100277.

2

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

field F . We consider G-equivariant embeddings S\G ,→ Pn . Such embeddings arise from a choice of an F -rational projective representation % : G → PGLn+1 , together with a point p0 ∈ Pn (F ) with stabilizer S. The standard height on F -rational points of Pn is defined by H : Pn (F ) → R>0 , Q x = (x0 , ..., xn ) 7→ H(x) := v Hv (x), Hv (x) := maxj (|xj |v ),

the product over all valuations v of F . More generally, one considers heights whose local factors coincide with the above at almost all v and differ from the above by a globally bounded P function at the remaining v. For example, one could choose Hv (x) = ( j |xj |2v )1/2 at a real v. We are interested in the asymptotic distribution of the number N (%, B) of F -rational points on H\G of height ≤ B, as B → ∞. In many cases, one finds (1.1)

N (%, B) ∼ c · B a log(B)b−1 ,

with a ∈ Q, b ∈ N and c ∈ R>0 (see [7],[2],[17], [4]). There is a conceptual interpretation of these constants in terms of global geometric and arithmetic invariants of the associated algebraic variety X, the closure of the G-orbit through p0 (see [7],[1], [13] and [3]). The proofs of the above asymptotics rely on harmonic analysis on corresponding adelic groups. Note that results of type (1.1) are quite nontrivial even when unitary representations of the adeles G(AF ) are well-understood. For example, we still don’t know how to treat general equivariant compactifications of the Heisenberg group (the case of bi-equivariant compactifications is considered in [16]). For semi-simple groups we need to appeal to rather nontrivial results from the theory of automorphic forms: multiplicity one, uniform bounds of matrix coefficients, Eisenstein series, spectral theory etc. We now give an outline of our approach in a special case. Let G be a split semi-simple group of adjoint type over Q Q. Consider the Cartan decomposition G(A)Q = KA+ K, where K = v Kv is a maximal compact subgroup and A+ = v A+ v (and the product is over all valuations v of Q).

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3

+ Here A+ p (resp. A∞ ) can be identified (via the logarithmic map) with the + monoid a (resp. cone a+ ∞ ) in the Lie algebra a (resp. a∞ := a ⊗ R), dual to the monoid (resp. cone) spanned by simple roots of the corresponding maximal torus A. Fix a triangulation Σ of a+ (resp. a+ ∞ for archimedean v) into simplicial subcones σ and let λ be a continuous R-valued function on a (resp. a∞ ) which is linear on every cone σ ∈ Σ. For example, we may take

(1.2)

λ(av ) = hs, av i,

where s ∈ a∗v ⊗ R and av = log(av ) ∈ av , av ∈ A(Qv ). Put qv = p for v = p, qv = e for v = ∞ and define Y Hv (λ, gv ) = qvλ(av ) and H(λ, g) := Hv (λ, gv ), v

where g = (gv ) ∈ G(AQ ), gv =

kv av kv0

with

kv , kv0

∈ Kv , a v ∈ A + v.

Problem 1.1. — Study the analytic properties of the zeta function X H(λ, γg)−s . (1.3) Z(λ, s, g) := γ∈G(F )

For λ chosen as in (1.2), the zeta function (1.3) encodes information about the distribution of rational points of bounded height on “wonderful” compactifications of G studied by de Concini and Procesi in [5]. The study of arbitrary bi-equivariant compactifications of G can be reduced to other λ. The main goal of this paper is to explain in detail how our approach works in the simplest case: P3 considered as the wonderful compactification of P GL2 over Q. The counting problem itself is trivial, but it allows us to focus on the method, which covers (verbatim) wonderful compactifications of rank one semi-simple groups of adjoint type and highlights the technical difficulties one faces for groups of higher rank. Compactifications of anisotropic forms of semi-simple groups of adjoint type are treated in [15]. 2. Basic definitions and results Notation . —

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

4

– V = VQ := {2, 3, 5, . . . , p, . . . , ∞} - valuations of Q; – G = P GL2 , A the (diagonal) torus, N upper triangular unipotent matrices, P = N A the Borel subgroup; – a ' Z - Lie algebra Q of A, a∗ its character lattice; – Kp = G(Zp ), Kf = p Kp , K∞ = SO(2) and K = K∞ × Kf ; – for v ∈ V, let G(Qv ) = Nv Av Kv and Kv A+ v Kv be the Iwasawa, resp. Cartan decompositions; – a S = Sc := a ≥ c > 0 ⊂ G(R) ,→ G(A) 1 a Siegel Q domain, G(A) = G(Q) · S · Ω, for some compact Ω ⊂ G(A); – dg = v dgv = dn da dk normalized Haar measure, Z Z dkv = 1, dn = 1; Kv

– – – – –

N (Q)\N (A)

1 0 ; volp (`) := where = 0 p` U = U(g) universal enveloping algebra of g = Lie(G(R)); ∆ ∈ U - the standard Casimir element (Laplacian); Tδ = {s ∈ C | <(s) > 4 + δ}; L2 := L2 (G(Q)\G(A)), unless noted otherwise, k · k2 the L2 -norm. vol(Kp a`p Kp ),

a`p

Represent an element in A(Qv ) in the form av 0 , 0 1 (with av ∈ Qv ) and define two local height functions: 1/2

(2.1)

Hv : gv = kv av kv0 → 7 |av |v χv,P : gv = nv av kv → 7 |av |v

Define the global heights by Y Y (2.2) H := Hv and χP := χv,P v

v

Remark 2.1. — For γ ∈ P (Q) we have χP (γg) = χP (g) (by the product 1 formula) and χ−1 P is the usual height on P (Q) = P (Q)\G(Q).

RATIONAL POINTS

5

Remark 2.2. — The group G has a (canonical) compactification a b G= ,→ P(End(V )) = P3 = {(a : b : c : d)} c d

which is equivariant for the action of G on both sides. A standard height on P3 (Q) is Y √ a2 + b2 + c2 + d2 · max(|a|p , |b|p , |c|p , |d|p ). p

Its local factors are identical with (2.1).

The main object of interest is the height zeta function X (2.3) Z(s, g) := H(γg)−s . γ∈G(Q)

The convergence of the series in the domain <(s) 0 (for fixed g) is a special case of a general fact: let X be any projective algebraic variety over a number field F , and H any height induced from a projective embedding of X. Then the height zeta function X H(x)−s Z(H, s) = x∈X(F )

converges absolutely and uniformly on compacts in the domain <(s) 0.

Proposition 2.3. — There exists a σ > 0 such that the series X H(γg)−s γ∈G(Q)

converges absolutely and uniformly on compacts in the domain Tσ ×G(A) to a function Z(s, g) which (1) is continuous in g; (2) is bounded on G(Q)\G(A); (3) has bounded ∆-derivatives. In particular, in this domain, Z(s, g) and all its ∆-derivatives are in L2 . Proof. — By reduction theory, G(A) = G(Q) · S · Ω, where Ω ⊂ G(A) is some compact and S a Siegel domain. Now we use the following easy property of the height: there exist constants c, r > 0 such that – H(γaω) ≥ cH(γ)r for all a ∈ S, ω ∈ Ω and γ ∈ G(Q).

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

6

In particular, there exists an r > 0 such that for all g ∈ G(A) one has Z(<(s), g) ≤ Z(r<(s), e) < ∞ for <(s) 0. Since ∆ commutes with the K-action, it suffices to prove (3) on matrices in A+ ∞ . Explicit formulas for the height and for ∆ give the result. A solution of Problem 1.1 in our special case is given by Theorem 2.4. — There exists an > 0 such that Z(s, e) admits a meromorphic continuation to T− with a unique simple pole at s = 4. In the analysis of Z(s, g) we use the Eisenstein series: X (2.4) E(s, g) := χ(s, γg), γ∈P (Q)\G(Q)

where χ(s, g) := χP (g)s+1/2 . The idea of the proof of Theorem 2.4 is to first establish an identity of continuous L2 -functions on G(Q)\G(A) (2.5)

Z(s, g) = Z res (s) + Z cusp (s, g) + Z eis (s, g), for <(s) 0,

where (2.6)

res

Z (s) :=

Z

G(Q)\G(A)

Z(s, g) dg =

Z

H(g)−s dg

G(A)

is the contribution from the trivial representation, Z cusp (s, g) is the projection of Z(s, g) onto the cuspidal spectrum and Z Z 1 eis (2.7) Z (s, g) := Z(s, g)E(it, g)dg E(it, g)dt 2πi R G(Q)\G(A) is the projection onto the continuous spectrum. This is accomplished in Propositions 5.1 and 7.1. Then we use (2.5) to meromorphically continue Z(s, e) (see Propositions 3.4, 7.6 and 5.1). Finally, a Tauberian theorem implies Corollary 2.5. — There is a constant c > 0, such that # {γ ∈ G(Q) | H(γ) ≤ B} = cB 4 (1 + o(1)),

as B → ∞.

RATIONAL POINTS

7

3. Heights and height integrals Let ϕp = ϕp (χp ) be a bi-Kp -invariant function on G(Qp ) such that ` p− 2 1 − χ¯p (p)2 /p 1 − χp (p)2 /p ` ` (3.1) ϕp (a`p ) = χ (p ) + χ ¯ (p ) , p p 1 + p−1 1 − χ¯p (p)2 1 − χp (p)2 for ` ≥ 1, where χp is a nontrivial unramified quasi-character of Q∗p , (3.2)

χp (p) = χ¯p (p)−1 = pαp , with parameter αp ∈ C∗ .

We write also ϕp = ϕp (s, ·), if αp = s for all p. Define Z Y (3.3) Iv (s) := Hv (gv )−s dgv , If (s) := Ip (s) G(Qv )

p

Q

and, for ϕp = ϕp (χp ) and ϕ = p ϕp , Z Y Ip (s, ϕp (χp )) := ϕp (gp )Hp (gp )−s dgp , If (s, ϕ) := Ip (s, ϕ). G(Qv )

p

Lemma 3.1. — The functions Ip (s) are holomorphic in T−2 . Moreover, If (s) is holomorphic in T0 and admits a meromorphic continuation to T−2 with an isolated simple pole at s = 4. Proof. — We have (3.4)

( p` (1 + p−1 ) volp (`) = 1

if ` > 0, if ` = 0

so that Ip (s) is given by X `s s s s s p− 2 p` = (1 − p−( 2 −1) )−1 (1 + p− 2 ) = ζp ( − 1)(1 + p− 2 ), 1 + (1 + p−1 ) 2 `≥1 where ζp is the local factor of the Riemann zeta function ζ (the sum convergesQabsolutely and uniformly on compacts in T−2 ). The Euler s product p (1 + p− 2 ) converges (uniformly on compacts in T−2 ) to a holomorphic function. It suffices to recall the analytic properties of ζ.

8

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

Lemma 3.2. — Assume that for all p, |<(αp )| < r. Then Ip (s, ϕp ) is holomorphic for <(s) > 2r + 1. Moreover, for all > 0 there exists a constant c = c() such that If (s, ϕ) is holomorphic and |If (s, ϕ)| ≤ c,

for <(s) > 2r + 3 + .

Proof. — Combining (3.1) with (3.4) we have 1 − χ¯2p (p)/p X −` s−1 ` 1 − χ2p (p)/p X −` s−1 ` 2 χ (p)+ Ip (s, ϕp ) = 1+ p p 2 χ¯p (p). p 1 − χ¯2p (p) `>0 1 − χ2p (p) `>0 For <(s) > 2r + 1 the series converges absolutely (and uniformly) to (3.5)

(1 − p−

Ip (s, ϕp ) =

s−1 2

χp (p))−1 (1 − p− ζp (s)

s−1 2

χ¯p (p))−1

.

The corresponding Euler product is holomorphic and bounded by some constant c(), provided <(s) > 2r + 3 + . Lemma 3.3. — For all > 0 and n ∈ N there is a c = c(, n) such that Z ∆n · H∞ (g∞ )−s dg∞ I∞ (s) and In,∞ (s) := G(R)

are holomorphic for all s ∈ T−2+ with absolute value bounded by c. Proof. — Write (3.6)

g=k

1

a

k 0 , with k, k 0 ∈ SO(2), 0 < a ≤ 1.

By a standard integration formula (cf. p. 142 of [11]) Z 1 Z 1 σ σ −1 ∗ 2 a (a − a) da < I∞ (σ) a 2 −1 da∗ < ∞ 0

0

(for σ > 2). A similar explicit computation proves the second claim. Proposition 3.4. — The function Z res (s) is holomorphic in T0 and admits a meromorphic continuation to T−2 with an isolated pole at s = 4. Proof. — By definition, Z Z X Z res (s) = H(γg)−s dg = G(Q)\G(A) γ∈G(Q)

G(A)

H(g)−s dg = If (s) · I∞ (s).

RATIONAL POINTS

9

It suffices to apply Lemma 3.1 and 3.3. 4. Eisenstein series Notation . — R +∞ R – R := −∞ ;

– dµn (t) := (1 + t2 )−n dt - a measure on R;

– c(s) :=

Q

v cv (s),

with cp (s) =

ζp (2s) ζp (2s+1)

Γ(s) and c∞ (s) = π Γ(s+1/2) ;

P 2<(η) – W (t) := 1 − η <(η)2 +(t−=(η)) 2 , sum over all poles, with multiplicity, of c(s) with <(η) < 0; – E(s, g) - Eisenstein series (2.4), Es := E(s, e) and Z E(s, ng)dn; EP (s, g) := N (Q)\N (A)

– truncations: X ∧T E(s, g) := EP (s, γg) and T

∧T E(s, g) := E(s, g) − ∧T E(s, g),

the sum over γ ∈ P (Q)\G(Q), with χP (γg) ≥ T > 0.

We recall some basic facts from the theory of Eisenstein series. We follow closely the exposition in [8, 9]. Theorem 4.1. — (1) the poles of E(s, g) coincide with the poles of c(s); (2) away from the poles, E(s, g) = E(1 − s, g); (3) away from the poles, Z E(s, kg) dk = E(s, e)ϕ(s, g), K

Q where ϕ(s, ·) = v ϕv (s, ·) is the spherical function of the corresponding principal series representation; R (4) G(Q)\G(A) E(it, g)dg = 0.

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

10

Proof. — Property (4) follows easily from the fact that the p-adic principal series representations corresponding to a purely imaginary parameter do not have the trivial representation as a quotient. The following key facts about Eisenstein series will be crucial for the analysis of the height zeta function. In the special case at hand, they could be deduced from standard facts about the Riemann zeta function. However, we give proofs applicable in more general situations. Theorem 4.2. — For n 0, one has R (1) R k ∧T E(it, ·)k22 dµn (t) < ∞; (2)

R

R

kE(it, ·)k22,Ω dµn (t) < ∞, where Ω ⊂ G(A) is a compact subset;

(3) for all g ∈ G(A) Z | ∧T E(it, g)|dµn (t) < ∞ and R

(4) the function g 7→

R

R

Z

R

|E(it, g)|2 dµn (t) < ∞;

|E(it, g)|dµn (t) is continuous.

We will need the following Lemma 4.3. — We have (4.1)

k ∧T E(it, ·)k22 = 2T −

c0 (it) + Ψ(it, T ), c(it)

where Ψ(it, T ) is a bounded function. Proof. — Following [9], p. 102, we get, away from the poles (4.2) k ∧T E(s, ·)k22 = (s + s¯)−1 eT (s+¯s) − |c(s)|2 e−T (s+¯s) + Ψ(s − s¯, T ), where

1 c(s)e(s−¯s)T − c(s)e−(s−¯s)T . s − s¯ The singularities from (s + s¯)−1 and (s − s¯)−1 are removable ([8], p. 231). For s = σ + it, and σ → 0 the right side of (4.2) equals 1 2T σ c0 (it) lim − |c(σ + it)|2 e−2T σ + Ψ(it, T ) = 2T − + Ψ(it, T ). e σ→0 2σ c(it) Ψ(s, T ) :=

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11

Since |c(it)| = 1, it suffices to show that Ψ is bounded near t = 0. Write e2itT − 1 e−2itT − 1 c(it) − c(it) − c(it) + . 2it 2it 2it Since c(0) ∈ R and c(it) is differentiable in t, limt→0 Ψ(it, T ) exists. Ψ(it, T ) = c(it)

Corollary 4.4. — The function s 7→ c(s) is bounded in any region 0 ≤ <(s) ≤ , =(s) ≥ 1. Proof. — Follows from the positivity of k ∧T E(s, ·)k2 and (4.2). Lemma 4.5. — For n 0, Z 0 c (it) c(it) dµn (t) < ∞. R

Proof. — The only pole of c(s) in the right half-plane is at s = 1/2. By Theorem 6.9 in [12], there is a q > 0, such that c0 (it) 1 = log(q) + 2 c(it) t +

1 4

+ (1 − W (t)).

By Theorem 7.1 of [12], there is a polynomial Q with positive coefficients such that Z +T 0 c (it) ≤ Q(T ), dt c(it) −T

for all T ∈ R>0 . By definition, for all t, W (t) > 1, so that Z +T 0 Z +T Z +T c (it) 1 (W (t) − 1)dt − dt ≥ log(q) + 2 1 dt, t +4 −T c(it) −T −T and

Z

+T

−T

W (t)dt ≤ 2T + Q(T ) +

Z

+T

−T

1 log(q) + 2 t +

1 4

dt.

In particular, there is a polynomial R with positive coefficients such that for all T ∈ R>0 one has Z +T W (t) dt ≤ R(T ). −T

12

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

Now we use the following easy statement: if a function f : R → R>0 and a polynomial R with positive coefficients are such that Z +T Z f (t) dt ≤ Q(T ) then f (t) dµn (t) < ∞, for n 0. −T

R

If follows that, for n 0, Z 0 Z c (it) dµn (t) < ∞. W (t)dµn (t) < ∞ and R R c(it)

Combining Lemma 4.3 with Lemma 4.5 we see that for all polynomials R there exists an n0 > 0 such that for all n > n0 one has Z |R(it)| · k ∧T E(it, ·)k22 dµn (t) < ∞. (4.3) J T := R

This proves (1) of Theorem 4.2. Next, fix a compact Ω ⊂ G(A), g ∈ Ω, and let k · k∞,Ω , resp. k · k2,Ω be the L∞ (Ω), resp. L2 (Ω), norms. We bound kE(it, ·)k2,Ω by

k ∧T E(it, ·)k2,Ω + k ∧T E(it, ·)k2,Ω ≤ ck ∧T E(it, ·)k∞,Ω + k ∧T E(it, ·)k2 , for some c > 0. We proceed to estimate Z |R(it)| · k ∧T E(it, ·)k2∞,Ω dµn (t). (4.4) JT := R

Observe that

∧T E(it, g) =

X

EP (it, γg)XT (χP (γg)),

γ∈S

where S is a finite subset of P (Q)\G(Q), depending only on S and T , and XT is the characteristic function of [T, ∞). Thus X X | ∧T E(it, g)| ≤ |χ(it, γg)| + |χ(−it, g)| ≤ 2 kχ(0, γ·)k∞,Ω , γ∈S

γ∈S

and the sum on the right side is bounded. It follows that the integral JT converges, for n 0. Combined with (4.3) this proves (2) of Theorem 4.2. An argument based on Sobolev’s lemma (see the appendix or the proof of Lemma 3.4.7 [14]), shows that there exists a polynomial R such that |E(it, g)| ≤ |R(it)| · kE(it, ·)k2,Ω .

RATIONAL POINTS

13

Hence the integral in (3), Theorem 4.2, is bounded by J T + JT , from (4.3) and (4.4)). This proves (3) and (4). 5. Eisenstein integrals For g ∈ G(A), and s, w ∈ C, we put (formally) Z ˆ Z(s, g)E(−w, g) dg. (5.1) Z(s, w) := G(Q)\G(A)

The main technical result of this section is the following Proposition 5.1. — For s ∈ T−1 , the integral Z 1 eis ˆ it)E(it, g) dt Z(s, (5.2) Z (s, g) := 2πi R

(1) is absolutely and uniformly convergent to a holomorphic in s and continuous in g function; (2) Z eis (s, g) ∈ L2 . R ˆ w) equals Proof. — Since K E(s, gk)dk = E(s, e)ϕ(s, g), (formally) Z(s, Z Z H(g)−s E(−w, g) dg = H(g)−s ϕ(−w, g) dg · E(−w, e). G(A)

G(A)

The local integrals Iv (s, w) :=

Z

Hv (g)−s ϕv (−w, g)dgv

G(Qv )

have been computed for v = p in (3.5) (the character corresponding to the spherical function ϕv (w, ·) is | · |w p , see (3.1) and (3.2)). We get (5.3)

If (s, w) =

Y p

Ip (s, w) =

ζ( s−1 − w)ζ( s−1 + w) 2 2 . ζ(s)

In particular, for ∈ (0, 1) and all w = σ + it with 0 ≤ σ ≤ , the map s 7→ I(s, w) = If (s, w) · I∞ (s, w)

is holomorphic for s ∈ T−1+ . Furthermore, for s and w as above, (5.4)

|I(s, σ + it)| n, (1 + t2 )−n .

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

14

Indeed, the function ϕ∞ (it, ·) is bounded. Moreover, by Lemma 4.1, it is an eigenfunction for ∆. We now apply repeatedly integration by parts (with respect to ∆) combined with Lemma 3.3 to I∞ (s, w) and use the standard bounds for ζ. We have Z Z ˆ |Z(s, it)| · |E(it, g)| dt ≤ |I(s, it)| · |E(it, e)| · |E(it, g)| dt R

R

Let K ⊂ T−1 and Ω ⊂ G(A) be compact sets. By the estimates above, for every n ∈ N, there is a constant c = c(n, K) such that Z

(5.5)

2

sup |I(s, it)| · |E(it, e)| dt ≤ c

R s∈K

Z

R

|E(it, e)|2 dµn (t).

To show the absolute and uniform convergence of the integral (5.2) for (s, g) ∈ K × Ω we need to check that, for n 0, Z

R

|E(it, e)| · kE(it, ·)k∞,Ω dµn (t) < ∞,

or, by Cauchy-Schwartz, that Z

R

|E(it, e)|2 dµn (t) < 0 and

Z

R

kE(it, ·)k∞,Ω dµn (t) < ∞,

which follows from Theorem 4.2. To prove that Z eis (s, ·) ∈ L2 , for s ∈ T−1 , write Es := E(s, e), 2πi · Z eis (s, g) = J T (s, g) + JT (s, g), T

J (s, g) :=

Z

R

I(s, it)E−it ∧T E(it, g) dt,

RATIONAL POINTS

15

similarly JT (s, g), with ∧T replaced by ∧T . We bound kJ T (s, ·)k22 as Z Z Z T | I(s, it)E−it ∧ E(it, g)dt| · | I(s, iτ )Eiτ ∧T (iτ, g)dτ |dg G(Q)\G(A) R R Z Z | ∧T E(it, g) ∧T E(iτ, g)|dgdtdτ |I(s, it)I(s, iτ )Eit Eiτ | ≤ R2 G(Q)\G(A) Z ≤ |I(s, it)I(s, iτ )Eit Eiτ | · k ∧T E(it, ·)k2 · k ∧T E(iτ, ·)k2 dtdτ R2 Z 2 T = |I(s, it)Eit | · k ∧ E(it, ·)k2 dt R

by repeatedly applying Fubini’s theorem, and then Cauchy-Schwartz to the inner integral. Thus it suffices to bound the integrals Z Z |E(it, e)|dµn (t) and k ∧T E(it, ·)k22 dµn (t), R

R

which has been accomplished in Theorem 4.2. We turn to JT (s, ·). It suffices to consider the integral over ST (since T S is compact). For T > 1 we have, for all g ∈ G(A), ∧T E(it, g) = EP (it, g) = χ(it, g) + c(it)χ(−it, g).

R Substituting this into the definition of JT (and rewriting the integral R ) we see that we need to bound the L2 (ST )-norms of Z Z I(s, w)E−w χ(w, ·)dw and I(s, w)E−w c(w)χ(−w, ·) dw <(w)=0

<(w)=0

By Corollary 4.4, c(s) is bounded in the region <(s) ∈ [0, ], =(s) ≥ 1 and we can shift the contour of integration of the first integral slightly to the left and that of the second to the right. This gives Z Z χ(−ε, ·) I(s, w)E−w dw and χ(−ε, ·) I(s, w)E−w dw. <(w)=−ε

<(w)=ε

Now it suffices to observe that, for ε > 0, Z ∞ 1 a2(−ε+ 2 ) a−1 da∗ < ∞, kχ(−ε, ·)k2,ST = T

and to use the estimate (5.4). This completes the proof of Proposition 5.1

16

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

6. P-series Definition 6.1. — Let φ : G(A) → C satisfy

– φ(ngk) = φ(g), for n ∈ N (A), k ∈ K and g ∈ G(A); – φ(ag) = φ(g), for a ∈ A(Q), g ∈ G(A); – φ : A(Q)\A(A) → C is smooth of compact support.

For g ∈ G(A), set θφ (g) :=

X

φ(γg) and θˆφ (s) :=

Z

θφ (g)E(s, g)dg.

G(Q)\G(A)

γ∈P (Q)\G(Q)

Since φ is left P (Q)-invariant and compactly supported modulo P (Q) ˆ the sum defining θφ is finite so that θφ ∈ C∞ c (G(Q)\G(A)) and θφ is meromorphic, analytic in <(s) ≥ 0, except possibly at s = 1/2. In the following we identify A∗ with A(A): a 0 a := , 0 1 write da∗ , resp. da, for the corresponding Haar measure and regard φ as ∗ ˆ∗ ∗ ∞ ∗ being (simultaneously) in C∞ c (Q · Z \A ) = Cc (R>0 ). For <(s) 0 we may use Fubini to derive Z ˆ θφ (s) = θφ (g)χ(s, g)dg P (Q)\G(A) Z = θφ,P (g)χ(s, g)dg N (A)A(A)\G(A) Z = θφ,P (a)χ(s, a)|δ(a)|−1 A da A(Q)\A(A) Z s−1/2 da∗ , θφ,P (a)|a|A = Q∗ \A∗

where θφ,P is the constant term of θφ . By Bruhat’s lemma, Z X φ(wγg) and θφ,P (g) = φ(g) + φ(wng)dn, θφ (g) = φ(g) + γ∈N (Q)

N (A)

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17

(with w the nontrivial element of the Weyl group), so that Z Z Z s−1/2 s−1/2 ∗ ˆ da + φ(wna)dn · |a|A da∗ . φ(a)|a|A θφ (s) = Q∗ \A∗

Q∗ \A∗

N (A)

To justify the above equation note that the double integral on the right is absolutely convergent. Indeed, for g ∈ G(A) and φ as above, set Z −1/2−s fφ (s, g) = da∗ . φ(ag)|a|A Q∗ \A∗

Then

1/2+s

fφ (s, nag) = |a|A fφ (s, g) and the double integral may be written as Z (6.1) fφ (s, wn)dn = c(s) · fφ (−s, e). N (A)

The Euler product defining c(s) and hence the integral (6.1) converge for <(s) 0. We have, for general φ, and, at first for <(s) 0 and then, by analytic continuation, for all s, (6.2) θˆφ (s) = fφ (−s, e) + c(s)fφ (s, e). Notice that θˆφ (it) is rapidly decreasing (fφ (it, e) is essentially the Fourier transform of a function in C∞ c (R)) and define, for g ∈ G(A), Z 1 c θˆφ (it)E(it, g)dt. (6.3) θφ (g) := 2π R ∗ c 2 Proposition 6.2. — If φ ∈ C∞ c (R>0 ) then θφ is continuous and in L .

Proof. — From (6.3) and (6.6) we have |θφc (g)| ≤ J T (g) + JT (g), with Z Z T T J (g) := | ∧ E(it, g)|dµn (t) and JT (g) := | ∧T E(it, g)|dµn (t). R

R

Both integrals are finite by Theorem 4.2, (3). To prove continuity, let Ω be a pre-compact neighborhood of g and recall that (by the same theorem) Z sup |E(it, g)|dµn (t) < ∞ for n 0.

R g∈Ω

18

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

∗ ˆ∗ ∗ Theorem 6.3. — For all φ ∈ C∞ c (Q · Z \A ) one has

θφc = θφ − hθφ , 1i1.

Proof. — We first prove the identity for constant terms: c θφ,P = θφ,P − hθφ , 1i1.

By Theorem 4.2, g 7→ Z

R

R

|E(−it, g)|dµn (t) is continuous so that Z |θˆφ (it)E(−it, ng)|dtdn < ∞.

N (Q)\N (A)

R

It follows that, for g ∈ G(A), Z Z c ˆ θφ (it)EP (−it, g)dt = θˆφ (it) (χ(it, g) + c(−it)χ(−it, g)) dt. θφ,P (g) = R

R

Since

c(−s)θˆφ (s) = fφ (s, e) + c(−s)fφ (−s, e) = θˆφ (−s) we obtain that (6.4)

c (g) θφ,P

1 = π

Z

θˆφ (it)χ(it, g)dt.

R

c We now compare θφ,P and θφ,P . We have already seen that Z s−1/2 da∗ , for <(s) 0. (6.5) θˆφ (s) = θφ,P (a)|a|A Q∗ \A∗

For <(s) ≥ 1, the function

−1/2+s

a 7→ θφ,P (a)|a|A

ˆ ∗ and in L1 (Q∗ · Z ˆ ∗ \A∗ ) = L1 (R∗ ). On the other is left-invariant under Z >0 hand, c(s) is bounded in the region 0 ≤ <(s) ≤ σ0 (for any σ0 > 0), |=(s)| ≥ 1 (by Corollary 4.4). It follows from (6.2) that in this domain Z −1/2+σ ˆ θφ (σ + it) = |a|itA da∗ θφ,P (a)|a|A Q∗ \A∗

is rapidly decreasing in t and we may apply Fourier inversion so that Z −σ+1/2 (6.6) θφ,P (a) = |a|A θˆφ (σ + it)|a|−it A dt. R

RATIONAL POINTS

19

But we also have (6.7)

c θφ,P (g)

1 = 2πi

Z

1/2−s ds. θˆφ (s)|a|A

<(s)=0

We shift the contour to <(s) = σ, taking σ > 1/2. The shift is justified by using (6.6) and the fact that fφ (σ + it, e) is rapidly decreasing in t, uniformly in σ, for |σ| ≤ σ0 . It follows that Z 1 1/2−s 1/2−s c θφ,P (a) = ds − ress=1/2 θˆφ (s)|a|A . θˆφ (s)|a|A 2πi <(s)=σ Now it suffices to compute the residue of c(s) at s = 1/2 and to see that Z Z −1 ∗ fφ (1/2, e) = θφ (g)dg, φ(a)|a|A da = Q∗ \A∗

G(Q)\G(A)

(for appropriately normalized measures). Let us now set θφcusp := θφ − hθφ , 1i · 1 − θφc .

We have proved that θφcusp is continuous, in L2 and has a vanishing constant term. Thus θφcusp is a cusp form, i.e., orthogonal to all the P-series θφ . To complete the proof of Theorem 6.3 it suffices to prove that θφc is orthogonal to all cusp forms. Since θφc ∈ L2 , it will suffice to prove that hθφc , ψ ∗ αi = 0

for all cusp forms ψ and all α ∈ C∞ c (G(A)). Replacing ψ by ψ ∗ α we may assume that ψ is rapidly decreasing (and continuous). Recall that Z θφc (g) = θˆφ (t)E(−it, g)dt. R

Since hE(−it, ·), ψi = 0, for all t ∈ R, it suffices to prove that Z Z |E(it, g)ψ(g)|dµn (t)dg < ∞ for n 0. G(Q)\G(A)

R

As before, we consider the integral Z Z T J := | ∧T E(it, g)| · |ψ(g)|dµn (t)dg S

R

20

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

and a similar integral JT with ∧T replaced by ∧T . Using Theorem 4.2, Z T J kψk2 · k ∧T E(it, ·)k2 dµn (t) < ∞. R

To treat the integral JT we decompose S = ST ∪ ST . Since ST is compact, ψ continuous and | ∧T E(it, g)| bounded in this domain, the double integral is absolutely convergent. It remains to estimate Z Z | ∧T E(it, g)| · |ψ(g)|dµn (t)dg. ST

R

To complete the proof of Theorem 6.3 observe that for T > 1, | ∧T E(it, g)| = |EP (it, g)| ≤ 2χ(0, g)

and, since ψ is rapidly decreasing, Z |ψ(g)| · χ(0, g)dg < ∞. ST

Proposition 6.4. — For <(s) 0, the function Z eis (s, ·) is orthogonal to all cusp forms and to the constant function. Proof. — Recall that Z eis (s, g) = where

1 2πi

Z

R

ˆ it)E(−it, g)dt, Z(s,

ˆ it) = I(s, it)E(it, e) with I(s, w) = Z(s,

Z

H(g)−s ϕw (g)dg.

G(A)

As above, we show first that the double integral Z Z ˆ it)E(−it, g)ψ(g)dtdg, Z(s, G(Q)\G(A)

R

is absolutely convergent, for ψ continuous and bounded, and that it converges to zero. The last integral is majorized by Z Z |E(it, e)| · |E(−it, g)| · |ψ(g)|dµn (t)dg. R

G(Q)\G(A)

RATIONAL POINTS

21

We set J

T

:=

Z Z R

G(Q)\G(A)

|E(−it, e)| · | ∧T E(it, g)| · |ψ(g)|dµn (t)dg.

and a similar integral JT (with ∧T replaced by ∧T ). By Cauchy-Schwartz, to treat J T it suffices to consider Z |E(−it, e)| · k ∧T E(it, ·)k2 dµn (t), R

and again by Cauchy-Schwartz, the two integrals Z Z 2 |E(−it, e)| dµn (t) and k ∧T E(it, ·)k22 dµn (t), R

R

which are finite by Theorem 4.2. To bound JT we decompose S = ST ∪ ST . The contribution from (the compact) ST is estimated using the boundedness of ∧T E(it, ·) · ψ(·). To treat ST recall that in this domain (for T > 1), ∧T E(it, ·) = EP (it, ·) and |EP (it, g)ψ(g)| ≤ c · χ(0, g). Using the fact that the right side is integrable on S and, once again, Theorem 4.2, we see that J T is also finite. Since Z |E(−it, e)|dµn (t) (6.8) R

is finite for n 0, the integral J T is also finite. As we have remarked, hE(−it, ·), ψ(·)i = 0, for t ∈ R, at least if ψ is rapidly decreasing. It remains to recall Proposition 4.1 (4): Z E(it, g)dg = 0. G(Q)\G(A)

This completes the proof. We will need the following ∗ Lemma 6.5. — For all φ ∈ C∞ c (R>0 ), we have

hZ eis (s, ·), θφ i = hZ(s, ·), θφc i.

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

22

Proof. — First observe that (6.9)

ˆ θˆφ iL2 (R) = hZ eis , θφ i = hZ,

1 2πi

Z

R

ˆ it)θˆφ (t)dt. Z(s,

For this it suffices to prove that Z Z ˆ it)E(it, g)θφ (g)|dtdg < ∞ |Z(s, G(Q)\G(A)

R

ˆ it) = I(s, it)E(it, e), where, by (5.4), I(s, it) is rapidly We have Z(s, decreasing in t, and Z Z |E(−it, e)E(it, g)θφ (g)|dµn (t)dg < ∞, G(A)\G(A)

R

for n 0, by Theorem 4.2. To show that ˆ θˆφ iL2 (R) = hZ, θc i hZ, φ

it suffices to check that, for <(s) 0, Z Z |Z(s, g)E(it, g)θˆφ (it)|dtdg < ∞. G(A)\G(A)

R

Since Z(s, g) is bounded in g, for <(s) 0, and θˆφ (it) is rapidly decreasing, it suffices to recall that Z |E(it, ·)|dµn (t) ∈ L2 , for n 0. R

7. The cuspidal spectrum Write G(A) = G(Q) · G(R) · Kf and let Γ = pr∞ (G(Z)). The map j : L2 (G(Q)\G(A)) → L2 (Γ\G(R)) φ 7→ φ|G(R)

is an isometry. We consider the right regular representation of G(R) on H := L2 (G(Q)\G(A))Kf = L2 (Γ\G(R)) and denote by H∞ ⊂ H the subspace of smooth vectors. If M = Γ\G(R) and ω is the gauge form on M whose accociated measure is a fixed rightinvariant measure on Γ\G(R), then H∞ = H∞ (M ) as defined in the

RATIONAL POINTS

23

Appendix. Write L20 := L20 (G(Q)\G(A)) for the space of cusp forms on G(A) and H0 = L20 (G(Q)\G(A))Kf for the closed G(R)-stable subspace of H. Let P : H → H0 be the orthogonal projection, it maps H∞ to H∞ 0 , the subspace of smooth ∞ vectors in H0 . Moreover, H∞ ⊂ H . We have decompositions 0 (7.1)

K

L20 = ⊕π Hπ ⊂ L2 (G (Q)\G(A)) , and H0 = ⊕Hπ f ,

as a G(A)-modules, into a countable direct sum of closed irreducible subspaces (each occuring with finite multiplicity). Let A0 be the set of all K π occuring in H0 . Note that each Hπ f contains an essentially unique K∞ fixed vector φπ normalized so that kφπ k2 = 1. Let H0 (M ) ⊂ L2 (M, ω) be the Hilbert subspace spanned by the φπ ’s. We remark that the φπ are necessarily in H∞ and are eigenfunctions of ∆, with eigenvalue, say, λπ . By Proposition 8.8, for φ ∈ H0 ∩ H∞ , the Fourier series X hφ, φπ iφπ (7.2) π∈A0

converges to φ, uniformly of compact sets.

We apply the above arguments to the height zeta function as follows. For <(s) 0, put Z res = hZ, 1i1 and define Z cusp so that Z(s, ·) = Z res (s) + Z cusp (s, ·) + Z eis (s, ·).

Proposition 7.1. — For <(s) 0,

Z cusp (s, ·) = P(Z(s, ·)) ∈ L20 .

Proof. — By Proposition 5.1, Z eis (s, ·) ∈ L2

for <(s) 0 and is continuous. Same holds for all its ∆-derivatives. It follows that Z cups (s, ·) ∈ L2 and is also continuous. ∗ For φ ∈ C∞ c (R>0 ) we have hZ cusp (s, ·), θφ i = hZ(s, ·), θφ i − hZ eis (s, ·), θφ i − hZ(s, ·), 1i1.

By Lemma 6.5, hZ eis (s, ·), θφ i = hZ(s, ·), θφc i and, by Theorem 6.3, θφc = θφ − hθφ , 1i1.

24

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

We get at once hZ cusp (s, ·), θφ i = 0

and Z cusp ∈ L20 . (Here we used the right K-invariance of Z cusp (s, ·).) To prove that Z cusp = P(Z), we need to show that hZ cusp (s, ·), ψi = hZ(s, ·), ψi,

for all ψ ∈ L20 . This is an immediate consequence of Proposition 6.4. We proceed to investigate the meromorphic properties of the Fourier expansion of Z cusp as in (7.2). For v = p, let ϕp be the corresponding local spherical function with parameter αp ∈ C∗ . They are given by (3.1) and (3.2). The crucial facts in our further analysis are Theorem 7.2. — [10] For π = ⊗ν πν ∈ A0 let ϕp be the normalized spherical function corresponding to πp and αp its parameter. Then |<(αp )| ≤ 1/6. Remark 7.3. — Any nontrivial uniform bound towards the Ramanujan conjecture suffices for our purposes. Theorem 7.4. — For all r > 0 there is a c > 0 such that X kφkr ∞ (7.3) Z∆ (s) := < ∞, for all n > c. n |λ π| π∈A ,λ 6=0 0

π

Proof. — Estimate the k·k∞ -norm of eigenfunctions in terms of the corresponding eigenvalues as in the Appendix and use the following fact: there exist constants c, r > 0 such that the number of linearly independent ∆eigenfunctions with eigenvalue less than B is bounded by c(1 + B r ), for all B ≥ 0 (see [12], for example). Lemma 7.5. — For all > 0 and n ∈ N there is a constant c = c(, n) such that for all s ∈ T−2/3+ and all π the function Z hZ(s, ·), φπ i = Z(s, g)φπ (g)dg G(Q)\G(A)

is holomorphic, with absolute value bounded by ckφk∞ |λπ |−n (for λπ 6= 0).

RATIONAL POINTS

25

Proof. — For φπ (with λπ 6= 0) and s such that Z(s, ·) ∈ L2 Z Z −s −n hZ(s, ·), φπ i = H(g) φπ (g) dg = λπ H(g)−s ∆n φπ (g) dg G(A) G(A) Z = λ−n ∆n H(g)−s φπ (g) dg. π G(A)

Since the (right) actions of ∆ and K commute, ∆n H(g)−s is invariant under K (which has volume 1), so that the above expression equals (7.4)Z Z Z Z −n n −s −n n −s λπ ∆ H(kg) φπ (g)dgdk = λπ ∆ H(g) φπ (kg)dkdg. K G(A)

G(A)

K

As is well-known, Z

K

φπ (kg) = ϕπ (g) · φπ (e),

where ϕπ is the spherical function attached to π, i.e., ϕπ (g) = hπ(g)φπ , φπ i, g ∈ G(A). Indeed, the functional φ 7→

Z

φ(k·)dk

K

is a bounded, left K-invariant functional on H∞ π (and Hπ ). Thus it is proportional to the functional φ 7→ hφ, φπ i. Taking φ = π(g)φπ , we find that the proportionality constant is φπ (e). Thus the integral in (7.4) is computed as Z (7.5) If (s, ϕ) · ∆n H∞ (g∞ )−s ϕ∞ (g∞ ) dg∞ · φπ (e) G(R)

Combining Lemma 3.2 with Theorem 7.2 (giving r = 1/6), we find that If (s, ϕ) is holomorphic for s ∈ T−2/3 and uniformly bounded by a constant c() for s ∈ T−2/3+ (and > 0). Since φ is bounded, Lemma 3.3 shows that Z ∆n H∞ (g∞ )−s ϕ∞ (g∞ ) dg∞ G(R)

26

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

is absolutely convergent, in T−2 , to a holomorphic function which, in T−2+ , is bounded by ckφk∞ for some constant c = c(, n). Proposition 7.6. — For all > 0 the function X hZ(s, ·), φπ iφπ (g) (7.6) Z cusp (s, g) := π∈A0

is holomorphic in s and continuous in g ∈ G(A) for all s ∈ T−2/3+ . Proof. — Combine Theorem 7.4 and Lemma 7.5. The uniform convergence on compacts Ω ⊂ G(A) follows from the estimate X X kφπ k2 ∞ sup |hZ(s, ·), φπ iφπ (g)| n,,Ω n |λπ | π g∈Ω π∈A 0

and the convergence of the spectral zeta function (7.3) for n 0. 8. Appendix Notation . — – dx = dx1 · · · dxn - Lebesgue measure on Rn , |x|2 = x21 + · · · + x2n ; – α = (α1 , . . . , αn ) ∈ Nn - a multi-index, |α| = α1 + · · · + αn and ∂ α = ( ∂x∂ 1 )α1 · · · ( ∂x∂n )αn the corresponding differential operator; ¯ 0 ⊂ B; – B, B0 ⊂ Rn - open balls such that the closure B ∞ ∞ α 2 – H (B) := {u ∈ C (B) | ∂ u ∈ L (B), ∀α}; – ∆ - a fixed second order elliptic operator on B; – u 7→ uˆ the usual Fourier transform. ∞ n For u ∈ C∞ c (B) ⊂ Cc (R ), put Z Z 2 2 2 kukB := kuk(2,0),B = |u(x)| dx =

Rn

B

and, more generally, X k∂ α ukB , kuk(2,r),B := |α|≤r

kuk2(2,−1) :=

Z

Rn

|u(x)|2 dx

u(x)|2 dx. (1 + |x|2 )−1 |ˆ

Since ∂ α preserves H∞ (B), we can extend the norm k · k(2,r),B to H∞ (B).

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27

Proposition 8.1. — For every r ≥ 2, there is a c = cr,B > 0 such that (8.1)

kuk(2,r),B0 ≤ c kukB + k∆uk(2,r−2),B ,

for all u ∈ H∞ (B).

Proof. — Induction on r. Fix ψ ∈ C∞ c (B) with 0 ≤ ψ ≤ 1 and ψ = 1 on ¯ 0 . Let v = u · ψ ∈ C∞ (B). By Corollary 6.27, p. 267 in [6], B Next we have (8.2)

kvk(2,2),B ≤ c (k∆vkB + kvkB ) .

∆v = ∆(ψ · u) = ψ · ∆u +

n X

ψj

j=1

∂u + ψ0 · u, ∂xj

with ψj (0 ≤ j ≤ n) fixed in C∞ c (B). To prove the assertion for r = 2 it suffices to show that with φ fixed in C∞ (B), ∂u (8.3) kφ · kB ≤ c1 (kukB + k∆ukB ) . ∂xj For this we write

∂u ∂(uφ) ∂φ = −u ∂xi ∂xi ∂xi and note that w := φ · u ∈ C∞ c (B). Apply (6.25), p. 262 of [6] to obtain kwk(2,1),B ≤ c k∆wk(2,−1) + kwkB . φ·

The following inequality will imply (8.3): (8.4)

k∆wk(2,−1) ≤ c1 (k∆ukB + kukB ) .

Again, for φ as above, we have

∆w = ∆(φ · u) = φ · ∆u +

n X

φj

j=1

∂u + φ0 · u, ∂xj

∂u with φj ’s fixed in To prove (8.4) it suffices to bound kφ· ∂x k(2,−1) j ∂w ∞ (for φ ∈ Cc (B)), or equivalently, k ∂xj k(2,−1) . We have in fact Z ∂w 2 2 k k = ˆ dx (1 + |x|2 )−1 x2j |w(x)| ∂xj (2,−1) n ZR Z 2 |w(x)| ˆ |w(x)|2 dx ≤ kuk2B ≤ dx ≤

C∞ c (B).

Rn

Rn

28

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

This completes the proof of (8.1) for r = 2. Suppose that r ≥ 2 and assume that the claim holds for all s with 2 ≤ s ≤ r. Choose a ball B1 ⊂ Rn with ¯ 0 ⊂ B1 ⊂ B ¯ 1 ⊂ B. B0 ⊂ B

For ψ ∈ C∞ (B1 ) and v as before, we have (8.5)

kuk(2,r),B0 ≤ kvk(2,r),B1 ≤ c(k∆vk(2,r−2),B1 + kvkB1 ),

again, by Corollary 6.27 in [6]. We need only bound k∆vk(2,r−2),B1 . For this we have first from (8.2) (8.6)

∂ α (∆v) = ∂ α (ψ · ∆u) + ∂ α (ψ0 · u) +

n X

∂ α (ψj

j=1

∂u ), ∂xj

2 where now ψj ∈ C∞ c (B1 ) (for 0 ≤ j ≤ n). It suffices to bound the L norms of the terms on the right in (8.6) (for |α| ≤ r − 2) in terms of kukB and k∆uk(2,r−2),B . By Leibniz’ rule,

k∂ α (ψ0 · u)kB1 ≤ c2 kuk(2,r−2),B1 ,

and we may use induction on r, provided r ≥ 4. Suppose then that r = 3 and set w = ψ0 · u ∈ C∞ c (B1 ). Since here |α| ≤ 1, we have trivially, k∂ α wkB1 ≤ kwk(2,1),B1 .

Applying [6] once more, this time to B1 , kwk(2,1),B1 ≤ c1 (k∆wk(2,−1) + kwkB1 ) ≤ c2 (k∆ukB1 + kukB1 ) ≤ c2 (k∆ukB + kukB ),

(the second inequality follows from (8.4)). Next we have, with |α| ≤ r−2, k∂ α (ψ∆u)kB ≤ c3 k∆uk(2,r−2),B ,

and finally k∂ α (ψj

∂u )kB1 ≤ c4 kuk(2,r−1),B ∂xj

and we may apply the induction hypothesis to arrive at (8.1).

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29

Corollary 8.2. — Suppose that r > 0 is even. Then there is a constant c = cr > 0 such that for all u ∈ H∞ (B) kuk(2,r),B0 ≤ c kukB + k∆ukB + · · · + k∆r/2 ukB .

Proof. — We use induction on r. The case r = 2 follows from Proposition 8.1. Suppose that r ≥ 4 and choose a ball B1 ⊂ Rn so that ¯ 0 ⊂ B1 ⊂ B ¯ 1 ⊂ B. B0 ⊂ B

By Proposition 8.1,

kuk(2,r),B0 ≤ c0r kukB1 + k∆uk(2,r−2),B1 .

Further, we have by induction



k∆uk(2,r−2),B1 ≤ cr/2−1 

r/2−1

X j=0



k∆j+1 ukB  .

The claim follows combining these two inequalities.

The following form of Sobolev’s lemma will be useful for our purposes. Proposition 8.3. — Let r > n/2 be an integer, B ⊂ Rn an open ball and u ∈ H∞ (B). Then u ∈ L∞ (B) and there exists a c = cr,B such that sup |u(x)| ≤ ckuk(2,r),B . x∈B

Corollary 8.4. — Assume in addition that r is even. Then there exists a c = cr,B , such that r/2 X k∆j ukB ) sup |u(x)| ≤ c(

x∈B0

j=0

Let M be an n-dimensional manifold, ω a gauge form on M (a nowhere vanishing exterior n-form) and dµ the associated volume form. We define Z 2 kf kB := |f (x)|2 dµ(x), and similarly kf k2M , B

and the corresponding space L2 (M ) = L2 (M, ω). Here B is a coordinate ¯ 0 ⊂ B. Let G be the ball in M . We also fix a ball B0 such that B0 ⊂ B Lie algebra of vector fields on M (for each x ∈ M , Gx = Tx , the tangent

30

JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

space to M at x). Let U = U(G) be its universal enveloping algebra, regarded as the algebra of differential operators on M . Define H∞ (M ) to be the space of all C∞ -functions f : M → C such that ∂f ∈ L2 (M ) for all ∂ ∈ U. We fix a second order elliptic operator ∆ ∈ U. Proposition 8.5. — Let r > n/2 be an even integer. Then there exists a c = cB > 0 such that for all f ∈ H∞ (M ) one has   r/2 X k∆j f kM  . sup |f (x)| ≤ c  x∈B0

j=0

Proof. — Easy consequence of Corollary 8.4.

Corollary 8.6. — Let K be a compact and r > n/2 an even integer. Then there exists a constant c = cK > 0 such that for all f ∈ H∞ (M ) one has   r/2 X k∆j f kM  . sup |f (x)| ≤ c  x∈K

j=0

Corollary 8.7. — For any even integer r > n/2 there exists a constant c = cr,B > 0 such that for any ∆-eigenfunction f = fλ ∈ C∞ (M ) with eigenvalue λ 6= 0 and any B0 ⊂ B as above one has sup |f (u)| ≤ c|λ|r/2 kf kB

x∈B0

Let M, ω, ∆ be as above. Assume that h∆f, f 0 i = hf, ∆f 0 i, for all f, f 0 ∈ H∞ (M ),

and let {fj }j≥1 be an orthonormal sequence of ∆-eigenfuctions in H∞ (M ). Let H0 (M ) ⊂ L2 (M ) be the Hilbert subspace spanned by the fj ’s and H0,∞ := {f ∈ H∞ (M ) | ∆j f ∈ H0 (M ) for all j}.

Proposition 8.8. — Suppose that f ∈ H0,∞ (M ). Then the series X hf, fj ifj j≥1

converges uniformly on compacts, and in particular, pointwise, to f .

RATIONAL POINTS

31

Proof. — Write ∆fj = λj fj (and note that λj ∈ R). For f ∈ H0,∞ (M ) and n ∈ N we set n X [n] hf, fj ifj . f := j=1

We have k [n]

∆f

=

n X j=1

k

h∆ f, fj ifj =

n X j=1

k

hf, ∆ fj ifj =

n X j=1

hf, fj iλkj fj = ∆k f [n] .

Consequently, for given > 0, we have k∆k (f − f [n] )kM ≤ for 1 ≤ k ≤ r/2 (with r as above), provided n 0. Since f − f [n] belongs to H∞ (M ), the proof follows immediately from Corollary 8.6.

References [1] V. V. Batyrev and Yu. I. Manin – “Sur le nombre de points rationnels de hauteur born´ee des vari´et´es alg´ebriques”, matha 286 (1990), p. 27–43. [2] V. V. Batyrev and Yu. Tschinkel – “Manin’s conjecture for toric varieties”, J. Algebraic Geom. 7 (1998), no. 1, p. 15–53. [3] V. V. Batyrev and Yu. Tschinkel – “Tamagawa numbers of polarized algebraic varieties”, in Nombre et r´epartition des points de hauteur born´ee, Ast´erisque, no. 251, 1998, p. 299–340. [4] A. Chambert-Loir and Yu. Tschinkel – “On the distribution of points of bounded height on equivariant compactifications of vector groups”, Invent. Math. 148 (2002), no. 2, p. 421–452. [5] C. De Concini and C. Procesi – “Complete symmetric varieties”, in Invariant theory (Montecatini, 1982), Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, p. 1–44. [6] G. B. Folland – Introduction to partial differential equations, Princeton University Press, Princeton, N.J., 1976, Preliminary informal notes of university courses and seminars in mathematics, Mathematical Notes.

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JOSEPH SHALIKA, RAMIN TAKLOO-BIGHASH and YURI TSCHINKEL

[7] J. Franke, Yu. I. Manin and Yu. Tschinkel – “Rational points of bounded height on Fano varieties”, Invent. Math. 95 (1989), no. 2, p. 421–435. [8] S. Gelbart and H. Jacquet – “Forms of GL(2) from the analytic point of view”, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, p. 213–251. [9] S. Gelbart and F. Shahidi – “Boundedness of automorphic Lfunctions in vertical strips”, J. Amer. Math. Soc. 14 (2001), no. 1, p. 79–107 (electronic). [10] H. H. Kim and F. Shahidi – “Functorial products for GLb2×GLb3 and the symmetric cube for GLb2”, Ann. of Math. (2) 155 (2002), no. 3, p. 837–893, With an appendix by Colin J. Bushnell and Guy Henniart. [11] A. W. Knapp – Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001, An overview based on examples, Reprint of the 1986 original. ¨ller – “The trace class conjecture in the theory of auto[12] W. Mu morphic forms”, Ann. of Math. (2) 130 (1989), no. 3, p. 473–529. [13] E. Peyre – “Hauteurs et mesures de Tamagawa sur les vari´et´es de Fano”, Duke Math. J. 79 (1995), p. 101–218. [14] D. Ramakrishnan – “Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)”, Ann. of Math. (2) 152 (2000), no. 1, p. 45–111. [15] J. Shalika, R. Takloo-Bighash and Y. Tschinkel – “Rational points on compactifications of anisotropic forms of semi-simple groups”, to appear, 2003. [16] J. Shalika and Y. Tschinkel – “Height zeta functions of equivariant compactifications of the Heisenberg group”, ArXiV:math.NT/0203093, to appear, 2001. [17] M. Strauch and Yu. Tschinkel – “Height zeta functions of toric bundles over flag varieties”, Selecta Math. (N.S.) 5 (1999), no. 3, p. 325–396.

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TRANSCENDENTAL BRAUER-MANIN OBSTRUCTION ON A PENCIL OF ELL IPTIC CURVES

OLIVIER WITTENBERG

1. Introduction Let

Br(X)

2 Hét (X, Gm )

denote the cohomological Brauer group

k be an algebraically closed extension of k . over k is said to be algebraic if it belongs to the

and

C

X.

Let

k

be a number eld

X Br(X) → Br(Xk ), transcendental number ℓ, the ℓ-primary part of the

kernel of the restriction map

otherwise; this property does not depend on the choice of Brauer group over

of a scheme

A class in the Brauer group of a projective smooth variety

k.

For any prime

ts into an exact sequence

/ (Qℓ /Zℓ )b2 −ρ

0

/ H 3 (X(C), Z){ℓ}

/ Br(XC ){ℓ}

/ 0,

b2 and ρ respectively denote the second Betti number and the Picard number of XC , and M {ℓ} denotes ℓ-primary part of M . Although this sequence does prove the non-triviality of Br(XC ) in many cases, e.g. when X is a K3 surface, transcendental classes are in general dicult to e xhibit. where

the

Almost all known instances of Brauer-Manin obstruction are thus explained by algebraic classes, the only exceptions being Harari's examples [4] with conic bundles over of curves of genus

1,

P2Q .

Besides, in the particular case of pencils

results on the Hasse principle have been obtained only unde r the assumption that the

2-

primary part of the Brauer group be vertical, and therefor e algebraic (see [3], 4.7). The rôle of transcendental elements in the Brauer-Manin obstruction thus seems worthy of investigation.

In this note we present an

example of transcendental Brauer-Manin obstruction to wea k approximation for an elliptic where elliptic means that it possesses a bration in curve s of genus noted that the class of order

H 3 (X(C), Z) = 0

for a

K3

2

i Hét

K3 surface over Q, P1Q . It should be

with a section, over

enjoys the property of being divisible (because

surface), which was not the case in Harari's examples.

2. Preliminaries: The subscript in

Br(XC )

which we will exhibit in

1,

2-descent

and the Brauer group of an elliptic curve

will be dropped, as we will only use étale cohomology. If

group scheme), n G will denote the

n-torsion subgroup ⋆ elements f, g ∈ k will

of

G.

The Hilbert symbol of a pair of

be denoted

k

G

is an abelian group (resp.

2. (f, g); it is the class of a quaternion algebra in 2 Br(k). When X is a geometrically integral variety over k and L is an extension of k , L(X) will denote the function eld of XL . The canonical morphism Br(X) → Br(k(X)) is injective if in addition X is regular; this fact will be used without further mention. Let E be an elliptic curve over k whose 2-torsion is rational. 2 ∼ → 2 E . The kernel of the evaluation map at the zero section Fix an isomorphism of k -group schemes (Z/2Z) − 0 Br(E) → Br(k) will be denoted Br (E). Br0 (E)

Lemma 2.1. The group

Let

is canonically isomorphic to

be a eld of characteristic dierent from

H 1 (k, E).

f : E → Spec(k) and the étale q > 1 by Tsen's theorem, we get an

Proof. Let us write the Leray spectral sequence for the structure mo rphism

sheaf

Gm .

Since

exact sequence

f⋆ Gm = Gm , R1 f⋆ Gm = E ⊕ Z

and

/ Br(E)

/ H 1 (k, E)

The zero section induces retractions of

Br(k) → Br(E)

Br(k)

Rq f⋆ Gm = 0

for

/ H 3 (k, Gm ) and of

/ H 3 (E, Gm ).

H 3 (k, Gm ) → H 3 (E, Gm ),

hence the lemma.

The Kummer sequence

0

/ 2E

/E

z7→2z

together with the previous lemma and the chosen isomorphism

0

(1)

Date

/ E(k)/2E(k)

δ

/ (k ⋆ /k ⋆2 )2

: April 8, 2003.

1

/E

/ 0, 2 ∼

→ 2E, (Z/2Z) − γ

/ 2 Br0 (E)

yields the exact sequence

/ 0.

2

OLIVIER WITTENBERG

We shall need explicit descriptions of the maps δ and γ . First choose distinct p, q ∈ k⋆ such that the Weierstrass equation (2) y 2 = x(x − p)(x − q)

→ (Z/2Z)2 . denes E and the points P = (p, 0) and Q = (q, 0) are respectively sent to (1, 0) and (0, 1) via 2 E −∼ It is well-known (see e.g. [9], p. 281) that δ(M ) = (x(M ) − q, x(M ) − p) for M ∈ E(k) if M 6∈ 2 E(k), that δ(P ) = (p − q, p(p − q)) and that δ(Q) = (q(q − p), q − p). f, g ∈ k ⋆ . The classes of the quaternion Br (E), and γ(f, g) = (x − p, f ) + (x − q, g).

Proposition 2.2. Let actually belong to

algebras

0

(x − p, f )

and

(x − q, g) ∈ Br(k(E))

By symmetry, it is enough to prove that γ(f, 1) = (x − p, f ) in Br(k(E)). Choose a separable closure k of k and let Gk be its Galois group over k . Likewise, choose a separable closure k(E) of k(E) and let Gk(E) be its Galois group over k(E). It follows from the Hochschild-Serre spectral sequence, Tsen's theorem and Hilbert's theorem 90 that the ination map H 2 (k, k(E)⋆ ) → Br(k(E)) is an isomorphism. Let ρ : H 1 (k, E) → ⋆ ∼ H 2 (k, k(E)⋆ /k ) denote the composition of the canonical isomorphism H 1 (k, E) − → H 1 (k, Pic(Ek )) and the

Proof.

boundary of the exact sequence 0

/ k(E)⋆ /k ⋆

/ Div(E ) k

/ Pic(E ) k

/ 0.

As shown in the annexe of [2], the diagram / Br(E)

Br(k)

θ

/ H 1 (k, E)

T

Br(k(E))

−ρ

≀

Br(k)

/ H 2 (k, k(E)⋆ )

/ H 2 (k, k(E)⋆ /k ⋆ )

commutes, where θ denotes the map which stems from the Leray spectral sequence (see lemma 2.1). This ⋆ enables us to carry out cocycle calculations for determining the image of γ(f, 1) in H 2 (k, k(E)⋆ /k ). We shall use the standard cochain complexes. Let χf : Gk → Z be the map with image in {0, 1} whose composition with the projection Z → Z/2Z is the quadratic character associated with f ∈ k⋆ /k⋆2 = H 1 (Gk , Z/2Z). The image of (f, 1) in H 1 (k, E) is represented by the 1-cocycle a : σ 7→ χf (σ)P . If M ∈ E(k), let [M ] denote the corresponding divisor on Ek . The 1-cochain with values in Div(Ek ) dened by σ 7→ χf (σ)([P ] − [0]) is a lifting of a. Its dierential (σ, τ ) 7→ (χf (σ) + χf (τ ) − χf (στ ))([P ] − [0]) is, as expected, a 2-cocycle with ⋆ values in k(E)⋆ /k , which we may rewrite as (σ, τ ) 7→ (x − p)χf (σ)χf (τ ) ; it represents the image of γ(f, 1) in ⋆ H 2 (k, k(E)⋆ /k ). Since x − p is invariant under Gk , the same formula denes a 2-cocycle on Gk with values in k(E)⋆ . We thus end up with a 2-cocycle ⋆

b : Gk(E) × Gk(E) −→ k(E) (σ, τ ) 7−→ (x − p)χf (σ)χf (τ )

which represents the image of γ(f, 1) in Br(k(E)), at least modulo Br(k), where χm now denotes the lifting with values in {0, 1} of the quadratic character on k(E) associated with m ∈ k(E)⋆ . (Note that k is separably closed in k(E), so that Gk identies with a quotient of Gk(E) .) Choose a square root s of x−p in k(E). Dividing b by the dierential of the 1-cochain σ 7→ sχf (σ) gives the 2-cocycle (σ, τ ) 7→ (−1)χx−p (σ)χf (τ ) , which does represent the image of the cup-product (x − p) ∪ f by the composite map H 1 (k(E), Z/2Z)⊗2 → H 2 (k(E), Z/2Z) → Br(k(E)). We have now proved that γ(f, 1) = (x − p, f ) in Br(k(E))/ Br(k), but the equality holds in Br(k(E)) since (x − p, f ) = (y 2 /(x − p)3 , f ) evaluates to 0 at the zero section. 3.

An actual example

The reader is referred to [4] for the denitions of weak approximation, Brauer-Manin obstruction, residue maps and unramied Brauer group. Let Ω denote the set of places of Q. Dene the polynomials p, q ∈ Q[t] by p(t) = 3(t − 1)3 (t + 3) and q(t) = p(−t). It will be useful to notice that p(t) − q(t) = 48t. Let E be the elliptic curve over Q(t) dened by (2). Denote by E its minimal proper regular model over P1Q (see [8]); it is a smooth surface over Q endowed with a proper at morphism f : E → P1Q whose generic bre is isomorphic to E . A geometric bre of f is either smooth or is a union of rational curves whose intersection numbers may be computed with Tate's algorithm [10].

TRANSCENDENTAL BRAUER-MANIN OBSTRUCTION ON A PENCIL OF ELLIPTIC CURVES

3

One nds the following reduction types, in Kodaira's notation [5]: I2 above t = 0, t = 3 and t = −3; I6 above t = 1, t = −1 and t = ∞; the other bres are smooth. Recall that a bre of type In has n irreducible components (Ci )16i6n , with (Ci .Ci+1 ) = 1, (C1 .Cn ) = 1 and (Ci .Cj ) = 0 if |j − i| > 1. Put A = γ(6t(t + 1), 6t(t − 1)) = (x − p, 6t(t + 1)) + (x − q, 6t(t − 1)) ∈ Br(E).

Proposition 3.1.

The class A ∈ Br(E) belongs to the subgroup Br(E ).

Proof. Let v be a discrete rank 1 valuation on Q(E ) whose restriction to Q is trivial, and κ be its residue eld. We shall prove that A has trivial residue at v . Let us choose a uniformiser π of v and put z˜ = zπ −v(z) for z ∈ Q(E )⋆ . It will be convenient to denote by V : Q(E )⋆ → Z × κ⋆ the group homomorphism z 7→ (v(z), [˜ z ]), where [u] denotes the class in κ of u ∈ Q(E ) if v(u) = 0. For f, g ∈ Q(E )⋆ , the residue of the quaternion algebra (f, g) at v is given by the tame symbol formula ∂v (f, g) = (−1)v(f )v(g)

h iv(g) h iv(f ) f v(g) = (−1)v(f )v(g) f˜ g˜ ∈ κ⋆ /κ⋆2 . v(f ) g

Note that it only depends on V (f ) and V (g). Furthermore, if V (f ) is a double, i.e. if v(f ) is even and f˜ is a square modulo π , then ∂v (f, g) = 1. These remarks will be used implicitly throughout the proof. Lemma 3.2.

The class (−p, 6t(t + 1)) + (−q, 6t(t − 1)) ∈ Br(Q(t)) is unramied over P1Q .

Proof. The residue at a closed point of P1Q other than t = α for α ∈ {−3, −1, 0, 1, 3, ∞} is obviously trivial. It is straightforward to check that the remaining residues are also trivial. Let us now turn to showing that ∂v (A) = 1. As A is invariant under t 7→ −t, we may assume v(p) 6 v(q). If v(x) < v(p), then V (x − p) = V (x − q) = V (x), from which we deduce thanks to (2) that V (x − p) and V (x − q) are doubles. If v(x) > v(q), then V (x − p) = V (−p) and V (x − q) = V (−q), hence the result by lemma 3.2. From now on, we may and will therefore assume v(p) 6 v(x) 6 v(q). To begin with, suppose v(p) < v(q). In this case, either v(t − 3) > 0 or v(t + 1) > 0. If v(x) = v(q), then V (x − p) = V (−p), hence ∂v (A) = ∂v (−q(x − q), 6t(t − 1)) by lemma 3.2; but with a look at (2), one nds that both v(−q(x − q)) and v(6t(t − 1)) are even. Suppose now v(x) < v(q). It follows from (2) that V (x − p) is a double, hence ∂v (A) = ∂v (x − q, 6t(t − 1)) = ∂v (x, 6t(t − 1)). If v(x) is even or if [6t(t − 1)] is a square in κ, which happens if v(t − 3) > 0, we get ∂v (A) = 1. If on the other hand v(t + 1) > 0 and v(x) is odd, then [6t(t − 1)] = 12, which (2) shows to be a square in κ. We are now left with the case v(p) = v(q) = v(x). If v(t) = 0, then v(t−3) = v(t−1) = v(t+1) = v(t+3) = 0, so v(6t(t + 1)) = v(6t(t − 1)) = 0 and it suces to prove that v(x − p) and v(x − q) are even, which follows from (2) and the equality v(p) = v(x) = v(q) = v(p − q) = 0. If v(t) < 0, then V (6t(t + 1)) = V (6t(t − 1)), so that ∂v (A) = ∂v (x, 6t(t + 1)), which is trivial since both v(x) = v(p) = 4v(t) and v(6t(t + 1)) are even. Suppose nally that v(t) > 0. If v(x − p) < v(t), then V (x − p) = V (x − q) since v(p − q) = v(t), and ∂v (A) = ∂v (x − p, (t + 1)(t − 1)) = ∂v (x − p, −1); if v(x − p) = 0, the residue is obviously trivial, and if v(x − p) > 0, which means that [˜ x] = [˜ p] = −9, (2) shows that −1 is a square in κ. We therefore assume v(x − p) > v(t), which still leads to [˜ x] = [˜ p] = −9. As v(p − q) = v(t), at least one of v(x − p) and v(x − q) is equal to v(t). In either case, (2) implies that v(x − p) + v(t) is even, so (−9)v(t) (−1)v(x−p) is a square, hence ∂v (A) = ∂v (x, 6t(t − 1)) + ∂v (x − p, (t + 1)(t − 1)) is trivial. We shall now prove the following.

The class A ∈ Br(E ) is transcendental and yields a Brauer-Manin obstruction to weak approximation on the projective smooth surface E over Q.

Theorem 3.3.

Proof. Let us rst deal with the second part of the assertion. A glance at equation (2) shows that E has a Q2 -point M2 with coordinates x = 1 and t = 2. (Indeed, this equation denes an ane surface over Q endowed with a morphism to P1Q whose smooth locus identies with an open subset of E .) Using the formula given in [7], Ch. XIV, 4, one easily checks that A(M2 ) is non-trivial. Now choose N ∈ E (Q) in the image of the zero section and let Mv ∈ E (Qv ) be equal to N for any v ∈ Ω \ {2}. This denes an adelic point (Mv )v∈Ω . The class A(M ) ∈ Br(Q) is trivial since A ∈ Br0 (E); consequently, the evaluation of A at (Mv )v∈Ω is non-trivial, which is an obstruction to weak approximation. It remains to be shown that A is transcendental. The exact sequence (1) reduces this to the computation of E(C(t))/2E(C(t)). Lemma 3.4.

The surface E is a K3 surface.

4

OLIVIER WITTENBERG

The topological Euler-Poincaré characteristic e(EC ) of EC can be expressed in terms of that of the bres and that of the base ([1], p. 97, prop. 11.4), which leads to e(EC ) = 24. Let χ(OE ) denote the Euler-Poincaré characteristic of the coherent sheaf OE . The canonical bundle KE of E is simply f ⋆ O(χ(OE )−2) (see [1], p. 162, cor. 12.3); in particular it has self-intersection 0, hence χ(OE ) = 2 by Noether's formula. We have now proved the triviality of KE . That H 1 (E , OE ) = 0 follows from χ(OE ) = 2 and Serre duality. Proof.

Lemma 3.5. The elliptic curve

E

has Mordell-Weil rank

0

over

C(t).

Proof. Let ρ(EC ) be the Picard number of EC and R be the subgroup of the Néron-Severi group NS(EC ) spanned by the zero section and the irreducible components of the bres. As follows from the output of Tate's algorithm, R has rank 20. On the other hand, ρ(EC ) 6 20 since E is a K3 surface. The Shioda-Tate formula

ρ(EC ) = rank(E(C(t))) + rank(R)

thus yields the result.

This lemma shows that the F2 -vector space E(C(t))/2E(C(t)) has dimension 2. Now the classes δ(P ) = (t, t(t − 1)(t + 3)) and δ(Q) = (t(t + 1)(t − 3), t) are independent over F2 , hence span the whole kernel of γ . On the other hand (t(t + 1), t(t − 1)) is evidently not a combination of δ(P ) and δ(Q), so that A has non-zero image in Br(C(E )) and is therefore transcendental. Remark 3.6. It is actually true that A(M ) = 0 for all M ∈ E (Q). This is a consequence of the global reciprocity law and the fact that A vanishes on E (Qv ) for all v ∈ Ω \ {2}, which can be checked by a tedious computation.

It is possible to determine 2 Br(E ) completely if one is willing to compute explicit equations for E . This involves blowing up the singular surface given by equation (2) a sucient number of times. Alternately, one may observe that all bres have type In (in other words, E → P1Q is semi-stable), and then use the equations given by Néron in this case in [6], III. Anyhow one nds that 2 Br(E ) is spanned by A modulo 2 Br(Q) after writing out all possible residues of a general class γ(f, g). On the other hand, the 2-torsion of the Brauer group of a complex K3 surface with Picard number 20 has rank 2 over F2 , so 2 Br(EC ) is strictly larger than (x, t), which 2 Br(E )/2 Br(Q). It turns out that 2 Br(EC ) is spanned by A and the class of the quaternion algebra √ √ unexpectedly belongs to Br(Q(E )) and only gets unramied after extension of scalars to Q( −1, 3). Remark 3.7.

In the semi-stable case, a computer program was written to carry out the calculations alluded to in the previous paragraph, as they often get quite lengthy. Its source code is available on request.

Remark 3.8.

Acknowledgements The author is most grateful to J-L. Colliot-Thélène for sharing unpublished notes on the topic (which contain in particular the statement of proposition 2.2), and would also like to thank him for his encouragements and many helpful conversations during the course of this research.

References 1. W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 4, Springer-Verlag, Berlin, 1984. 2. J-L. Colliot-Thélène and J-J. Sansuc, La R-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 175229. 3. J-L. Colliot-Thélène, A. N. Skorobogatov, and Sir Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. math. 134 (1998), no. 3, 579650. 4. D. Harari, Obstructions de Manin transcendantes, Number theory (Paris, 19931994), London Math. Soc. Lecture Note Ser., vol. 235, Cambridge Univ. Press, Cambridge, 1996, pp. 7587. 5. K. Kodaira, On compact analytic surfaces II, Ann. of Math. (2) 77 (1963), 563626. 6. A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. No. 21 (1964). 7. J-P. Serre, Corps locaux, Hermann, Paris, 1968. 8. I. R. Shafarevich, Lectures on minimal models and birational transformations of two dimensional schemes, Notes by C. P. Ramanujam, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, No. 37, Tata Institute of Fundamental Research, Bombay, 1966. 9. J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. 10. J. Tate, Algorithm for determining the type of a singular ber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1975, pp. 3352, Lecture Notes in Math., Vol. 476. UMR 8628, Mathématiques, Bâtiment 425, Université de Paris-Sud, F-91405 Orsay, France

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