THE BRAUER-MANIN OBSTRUCTION FOR CURVES. December 16, Introduction

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1 THE BRAUER-MANIN OBSTRUCTION FOR CURVES VICTOR SCHARASCHKIN December 16, Introduction Let X be a smooth projectie ariety defined oer a number field K. A fundamental problem in arithmetic geometry is to determine whether or not X has any K-rational points. In general this is a ery difficult task, and so we look instead for necessary conditions for XK) to be non-empty. Embedding K in each of its completions K we hae the local condition that XK ) must be non-empty for eery. For some classes of arieties this is sufficient in which case we say that the Hasse Principle holds), but often it is not. Manin [15] introduced subsets X B of XK ) for eery B BrX) such that 1) XK) X Br X B XK ). The sets X B are defined in 2. If X Br = then XK) must be empty. Manin s construction accounts for almost all known examples where the Hasse Principle fails. Howeer it is difficult to calculate X Br explicitly, especially if BrX) is large, as it is for cures. The aim of this paper is to gie a more tractable formulation for X Br for cures Theorem 1.2), and then use it to show that X Br = in seeral cases. In particular we show that if Γ is the cure 3x 4 + 4y 4 = 19z 4 first introduced by Bremner, Lewis and Morton in [2] then Γ Br = proided the Jacobian of Γ has finite Tate-Shafareich group. This partially answers a question of Skorobogato [21]. Along the way we deelop a test Proposition 4.3) which in practice is extremely effectie at showing that a cure has no rational points. Our notation will be as follows. Let K be a number field, and let X be a smooth projectie geometrically connected cure defined oer K. For each prime let K denote the completion of K at, let k be the residue field for each finite, and let G K = GalK/K) and G K = GalK /K ). We shall assume that XK ). Let J = Pic X) denote the Jacobian ariety of X, and let TSJ) denote the Tate-Shafareich group of J: TSJ) = ker H 1 K, J) ) H1 K, J). We assume throughout that TSJ) is finite. Finally let δ be the index of X, defined as the greatest common diisor of the degrees of finite field extensions L/K for which X has an L-rational point, or equialently, as the smallest positie degree of a diisor on X. Clearly if δ > 1 then XK) =. We proe in 2 that in this case X Br = also. Theorem 1.1. Suppose TSJ) is finite. Then there is a subset B BrX) such that X has index 1 iff X B. In particular, if X has index greater than 1 then X Br =. The remaining case occurs when X has index 1. Let be a fixed degree 1 diisor. We can use to identify Pic and Pic 1. We also hae a map ξ : XK) JK) gien by x [x ]. This is a translation of the standard embedding f P : XK) JK) which 1

2 2 VICTOR SCHARASCHKIN sends a point P to. Howeer ξ is defined oer K een if X has no K-rational point. We hae the following diagram. 2) XK) ξ JK) Thus inside JK ) we hae XK ) XK) = ξ ξ JK ) XK ) ) JK). Consider the topological closure of this set. Since ξ is a closed immersion [14, Proposition 2.3] we hae ) XK) ξ XK ) JK) where JK) denotes the closure of JK) inside the group PicX K ). Furthermore we know that X Br is always a closed set [6, Proposition 2], so that XK) X Br. Our main result is that these two closed sets containing XK) are actually equal. Theorem 1.2. Suppose that X has index 1, and that TSJ) is finite. Then XK) = ξ XK JK) ξ XK JK) = X Br Here the closure of JK) is taken inside the group PicX K ). Note that the closure of JK) is the same as its profinite completion [12, I.6.14b]. If is any other degree 1 diisor then translation by ) gies a bijection between ξ XK JK) and ξ XK JK), so that the description of X Br is not canonical. Serre showed [18, Theorem 3] that the topology induced on JK) by embedding it in JK ) is that defined by the subgroups of JK) of finite index. In particular, if JK) is a finite set then it has the discrete topology, and is unaltered on taking the completion. We thus hae the following corollary to Theorem 1.2. Corollary 1.3. Suppose JK) and TSJ) are finite. Then XK) = iff X Br =. The intersections in Theorem 1.2b) may still be difficult to calculate in general. Howeer if we project from K to the residue field k in diagram 2 we are led to consider the conditions ) 3) ξ Xk ) JK) and 4) ) ξ Xk ) JK where the closure now takes place inside PicX k ). We must check that the map to k is well defined for example, is independent of any particular choice of projectie embedding). This follows from the existence of a Néron model for J [2, Chapter IV]. Conditions 3) and 4) are amenable to testing, since JK) is finitely generated, the Xk ) are finite, and the closure is easy to understand. If condition 3) fails then XK) =. In practice this proides a powerful method of showing that a cure has no K-rational points.

3 THE BRAUER-MANIN OBSTRUCTION FOR CURVES 3 The closure of JK) in XK ) is mapped into its closure in Xk ). Thus if condition 4) fails and TSJ) is finite then X Br =. In 4 we show the following. Theorem 1.4. If TSJ) and JK) are finite then the following are equialent: a) XK). b) The Brauer-Manin condition holds: X Br. c) Condition 3) holds: ξ Xk JK). d) Condition 4) holds: ξ Xk JK. In 5 we use these techniques to gie some examples, including the cure Γ mentioned aboe. 2. Reduction to Index One In this section we define the set X Br, and then proe Theorem 1.1. For any field extension L/K let XL) be the set of L-rational points of X, and let BrX L ) denote the Brauer- Grothendieck group of X oer L, H 2 X spec L, G m ) étale cohomology). For any prime spec K of K let e : BrX K ) XK ) BrK ) b, x ) b x ) be the ealuation map sending x to the fibre of b at x. Let in : BrK ) Q/Z be the local inariant map. There is a pairing, [6, Lemma 1] BrX K ) XK ) Q/Z b, x ) = in e b, x ) where b ) is the image of b under BrX K ) BrX K ). From the fundamental exact sequence [22, VII.9.6] P BrK) BrK ) in Q/Z we see that if x XK) then b, x = for eery b BrX). Definition 2.1. For any subset B BrX) let X B = { x ) XK ) b, x ) = b B } and write X Br for X BrX). Equation 1) is then immediate. Definition 2.2. a) If XK ) we say that the local condition holds for X. b) If X Br we say that the Brauer-Manin condition holds for X. The local and Brauer-Manin conditions are necessary conditions for the existence of a K-rational point on X. If the local condition is also sufficient we say that the Hasse Principle holds for X. If the Brauer-Manin condition is sufficient then in the terminology of Colliot-Thélène [6] we say that the Brauer-Manin obstruction is the only obstruction to the existence of a K-rational point on X. We now proe Theorem 1.1, which follows easily from the results in [11]. The set B is chosen so that the alues of the pairing, are independent of the choice of the point

4 4 VICTOR SCHARASCHKIN x ). Thus on B the pairing degenerates to a character ϕ: B Q/Z, which is explicitly described in [11]. Proof of Theorem 1.1. The Hochschild-Serre spectral sequence yields H r G K, H s X K, G m = H r+s X K, G m ) 5) PicX K ) PicXK ) ) G K BrK) H 3 G K, K ) H 1 G K, PicX K BrX K ) since H 1 X, G m ) = PicX) and H 2 X K, G m ) =. Sequence 5 can be obtained without using the spectral sequence; see [11, 2.3] or [1, 2].) If L is any local or global field then H 3 G L, L ) = [4, VII 11.4]. The local points proide section maps BrX K ) BrK ) for eery, so that in the corresponding sequence for K, BrK ) BrX K ) is injectie. We thus hae the following diagram [11, 2.7.1]. 6) BrK) BrX K ) H 1 G K, PicX K BrK ) P in Q/Z BrX K ) H1 G K, PicX K The injectiity of BrK) BrX K ) follows from the injectiity of BrK) BrK ) and BrK ) BrX K ). Note that we hae by 5) PicXK ) ) G K = PicX K ). 7) Let BrX) = ker BrX K ) ) BrX K ). This agrees with the notation in [11], as can be seen from Lemma 2.6 of that paper. Let TSP ) = ker H 1 G K, PicX K ) H1 G K, PicX K and define the set B to be B = ker BrX K ) ) H1 G K, PicX K. Equialently, B is the preimage of TSP ) in BrX K ). Suppose b B, and let b ) be its image in BrX K ). By definition of B, b ) is the image of a uniquely determined) element a ) BrK ), and hence e b, x ) = a for any x ) XK ). Thus, x ) is independent of the choice of x, and therefore the pairing, determines a character ϕ: B Q/Z. In particular, for any x ) we hae that b, x ) = for all b iff ϕ =. That is: 8) X B ϕ =.

5 THE BRAUER-MANIN OBSTRUCTION FOR CURVES 5 Obsere that ϕ can be iewed as a map TSP ) Q/Z, and as such is exactly the connecting map supplied by the Snake Lemma. Indeed, we obtain a diagram 9) Z/δ TSJ) χ BrX) ϕ TSP ) Q/Z The column arises as follows: From the sequence Pic X K ) PicX K ) deg Z and the similar ones for X K 1) PicX K ) deg Z and using 7) and its local ersion we obtain H 1 G K, Pic X K H 1 G K, PicX K PicX K ) deg Z H1 G K, Pic X K H1 G K, PicX K The existence of local points implies that PicX K ) Z, while the image of PicX) Z called the period) is exactly δz [11, 2.5], so that the Snake Lemma gies the desired exact sequence. The map χ can be gien explicitly, in terms of the Cassels-Tate pairing:, CT : TSJ) TSJ) Q/Z. In [11, 2.11] it is shown that there exists an element β TSJ) of order δ such that for all α χα) = α, β CT. We are assuming that TSJ) is finite. Thus, CT is non-degenerate, so that χ = iff β =, and hence 11) χ = δ = 1. Thus by 8) and diagram 9) X B iff δ = 1, which proes Theorem 1.1. The following result is stated in [16]. Corollary 2.3. If X has genus 1 then XK) = iff X Br =. Proof. Let X be a genus 1 cure. We proe the contrapositie result. Thus, we assume that X Br, and must show that XK). Applying Theorem 1.1 we see that X has index 1. Let be a diisor of degree 1. The Riemann-Roch Theorem implies that l ) = 1 and the result follows by Lemma 4.1a) below.

6 6 VICTOR SCHARASCHKIN 3. Index One In the remainder of this paper we shall assume that X has index 1, with fixed degree 1 diisor. Our goal is to proe Theorem 1.2. We must show that there is a bijection of sets X Br = ξ XK JK). The strategy is to reformulate the basic pairing, in terms of the dual of diagram 6), and then use duality results of Tate and Lichtenbaum. We first refine diagram 6. From diagram 1) we see that H 1 K, J) = H 1 G K, PicX K and similarly for H 1 K, J), while from 9) we hae TSJ) = TSP ). Thus we hae 12) BrX) TSJ) BrK) BrX K ) H 1 K, J) BrK ) P in Q/Z BrX K ) T H1 K, J) W where T and W are the indicated cokernels. Let x ) XK ). The ealuation maps e, x ): BrX K ) BrK ) induce β x ) : BrX K )/BrX) Q/Z, and by definition of the pairing, we hae x ) X Br iff β x)b) = for eery b BrX K ), that is, iff β x) is the zero map. Let βx = β x). Then X Br = ker XK ) β ) ) BrXK ) BrX) We now take duals. Write ) for the exact contraariant functor Hom, Q/Z) from profinite to torsion abelian groups. We first use Tate duality on the third column of diagram 12). begins [12, p. 12] JK) JK ) The dual sequence where JK) is the closure of JK) in JK ), which is equal to its profinite completion. The prime on the product indicates that when is a real prime we must take JK ) modulo the connected component of.

7 THE BRAUER-MANIN OBSTRUCTION FOR CURVES 7 Lichtenbaum in [1, Theorems 1,4] defines a pairing ψ : BrX K ) PicX K ) Q/Z b, [x ]) in e b, x ) which induces an isomorphism PicX K ) BrX K ) for eery finite. Again, if is Archimedean then we must take PicX K ) modulo the connected component of. On taking the dual of the second and third columns of diagram 12 we thus get the following. 13) ) BrX) H 1 ) K, J) BrX) TSJ) β BrX K ) XK ) ψ ξ = PicX K ) i+ JK ) T JK) The map JK ) PicX K ) is just induced by inclusion [1, 5], and we translate it by to identify Pic and Pic 1. Thus X Br is the subset of XK ) which maps into the image of T in BrX K ). After a small diagram chase we see that inside BrX K ) the image JX K ) T comes from JK). That is, X Br = ξ XK JK). 4. Calculating X Br The intersections in Theorem 1.2 are still difficult to work with. In this section we consider the more easily tested conditions 3) and 4). In particular, the topological closure is also much more easily dealt with oer the finite fields. The simple obseration that an effectie diisor of degree 1 must actually be a K-rational point generalizes as follows. Lemma 4.1. Let D be any diisor of degree 1. Then XK) if any of the following hold. a) l K D) > b) There exists a finite prime with l K D) >. c) There exist infinitely many finite primes with l k D) >. Proof. a) Let F KX) with dif )+D, and let D = D +dif. Then degd ) = 1 and D is effectie, which is only possible if D = 1 P for some P XK). b) This follows from a), since l K D) = l K D). The dimension of the ector bundle remains unchanged on tensoring with Q p. See [9, III.9.9.3]. c) Assume X is defined oer Q. We must show that l Fp D) = l Q D) almost always. We consider X as a scheme oer spec Z). A Weil diisor n i P i will correspond to a Cartier diisor for all p with X p non-singular, hence for almost all p. Thus we may discard the finitely many primes for which X has singular fibres, and let D be a Cartier diisor,

8 8 VICTOR SCHARASCHKIN and L the associated inertible sheaf, so that l Fp D) = dim H X, L p ). The dimension of H X, L p ) is essentially an upper semicontinuous function of p [9, III, 12.8], and is hence is almost eerywhere equal to the generic dimension. Lemma 4.2. Let {A i } i=1 be a countable collection of discrete topological spaces, let A be a set, and suppose f : A i=1 A i is a map of sets. Let f i : A A i be the composite of f with the i-th projection, and endow i=1 A i with the product topology. Let B i A i. Then the following are equialent. a) fa) i=1 B i. b) There exists b = b i ) i=1 B i such that for eery n N there exists a n A with f i a n ) = b i for i = 1, 2,..., n. Proof. a) = b) Fix b = b i ) fa) i=1 B i. For eery n let U n = {b 1 } {b n } i>n B i. This is an open neighbourhood of b, and hence for eery n there exists a n A with fa n ) U n fa). That is, f i a n ) = b i for i = 1, 2,..., n. b) = a) Suppose b = b i ) i=1 B i and a n A satisfy the conditions aboe. Let U = U 1 U n i>n A i be any basic open neighbourhood of b. Then U contains V = {b 1 } {b n } i>n A i. But fa n ) V so eery open neighbourhood of b meets fa), and thus b fa) i=1 B i. The topology on XK ) induces the discrete topology on Xk ), and thus we may apply Lemma 4.2 to condition 4). Write A = JK), B = ξ Xk ), and let f be the natural projection f : K k. Then 4) holds iff there exists x i ) i Xk i ) and a sequence of points a n JK) with a n ξx i ) in k i for all i < n. Proof of Theorem 1.4 We always hae a) = b) = d) and a) = c) = d). To see that d) = a) note that any sequence in JK) must contain a constant subsequence. Thus there exists a JK) with a ξx ) in k for eery. Thus a [ ] = [x ] > in PicX k ), so l k ) >. The result now follows from Lemma 4.1c). The final difficulty in checking conditions 3) and 4) is that the Jacobian may not be easy to calculate. We introduce the following ariant. Let A be any abelian ariety for which there is a nonconstant map θ : X A defined oer K. After composing θ with a translation, we may assume that θ maps some geometric point P to. Now ξ is just a translation of the standard embedding f P : XK) JK) which maps P to, and thus the uniersal property of the Jacobian [14, 6.1] implies that there exists a map λ: J A with θ = λ ξ. Hence λ maps ξ Xk JK into θ Xk AK. Thus if this intersection is empty, X Br must be too. In practice we would like a finite test. Suppose for conienence that X is defined oer Q. If take only a finite product of finite fields then the toplogy is discrete, and taking the closure has no effect. Thus we hae the following. Proposition 4.3. Let X be defined oer Q with index 1, and assume that TSJ) is finite. Let θ : X A be as aboe, and let S be a finite set of prime numbers. If θ p S XF p AQ) = then X Br = and hence XQ) =. 5. Examples In this section we consider two examples. The more interesting example is the Bremner- Lewis-Morton cure Γ : 3x 4 + 4y 4 = 19z 4 oer Q. Skorobogato [21] asked if Γ Br = for this cure. We show it is, proided TSJ) is finite. Unfortunately Γ has an infinite

9 THE BRAUER-MANIN OBSTRUCTION FOR CURVES 9 Jacobian, so Corollary 1.3 does not apply. Howeer Proposition 4.3 can be applied after some work. This example also shows that in Corollary 1.3 finiteness of the Jacobian is sufficient but not necessary to produce a triial Brauer-Manin obstruction. Theorem 5.1. The projectie cure has the following properties: Γ : 3x 4 + 4y 4 = 19z 4 a) There are points on Γ oer eery p-adic field Q p, and oer R. b) The index of Γ is 1. c) The Jacobian of Γ oer Q is infinite. d) Nonetheless, if TSJ) is finite, then Γ Br is empty and hence Γ has no rational points). Proof. Parts a) and b) are proed by Bremner, Lewis and Morton [2, IIa)], who also show that Γ has no rational points. Cassels proides an alternatie proof of this last statement is gien in [5, Appendix], and also notes that c) holds. To proe d) we show that the condition 4) fails for Γ. The Jacobian is isogenous to a product of 3 elliptic cures [5]. We work with just one of these. The map θ : x : y : z) 12y 2 z : 36x 2 y : z 3 ) sends a point on cure Γ to a point on the elliptic cure E : y 2 z = x 3 684xz 2. The map is well defined oer Q or F p with p 2, 3, 19, since at least one image co-ordinate is always non-zero. We need a description of the Mordell-Weil group of E. Lemma 5.2. The elliptic cure E has good reduction for p 2, 3, 19 and has Mordell-Weil group EQ) = Z Z/2. generated by the points P = 3 : 45 : 1) of infinite order, and T = : : 1) of order 2. Proof. The conductor of E is By Silerman [19, X.6.1] or explicit calculation, EQ) = Z r i Z/2, for some r 1. The rank can be calculated by 2-descent using the algorithm described in [8, III.6, page 84] there are no difficulties), or read from the table in [1]. Finally we must show that P is not a proper multiple of a point of smaller height. Cremona [8, III.5.1] gies the following explicit bound due to Silerman for the cure y 2 = x 3 + dx: 14) hq) ĥq) + 1 log d ) for any point Q = a/c 2, b/c 3 ) on the cure, where hq) is the naie height log max{ a, c 2 } and ĥ is the canonical height. 1 Suppose P = n Q + T where T is a torsion element and n 2, so that ĥq) 4ĥP ). This gies a < 35 and c < 56, but a computer search for points up to this height bound yields only points of the form n P and n P + T. After some experimentation, we find that can apply Proposition 4.3 with the finite set of primes {17, 31}. To complete the proof of Theorem 5.1 it is thus enough to show that no R = n P + ɛ T EQ) with n Z and ɛ {, 1} can be in 15) θγf 17 ) θγf 31 ) For conenience we use affine co-ordinates, and denote the point at infinity by.

10 1 VICTOR SCHARASCHKIN p = 17 Oer F 17 P 3, 6) of order 4. n P + ɛ T. We hae the following table of alues of n = n = 1 n = 2 n = 3 ɛ = -3,-6) 2,) -3,6) ɛ = 1,) 7,3) -2,) 7,-3) A quick search shows that θγf 17 ) = {7, ±3), 1, ±5)}, so that n must be odd and ɛ = 1. p = 31 Oer F 31 P 3, 14) has order 8. We thus hae only to consider the points P + T = 11, 1), 3 P + T = 1, 1), 5 P + T = 1, 1) and 7 P + T = 11, 1). We find that θγf 31 ) = {, 2, ±2), 4, ±5), 7, ±9), 8, ), 1, ±9), 14, ±9), 19, ±1), 28, ±4)}. Hence the intersection 15) is empty. We gie a genus 2 example. The following cure was considered in [7]. Example 5.3. Let C be the smooth projectie model of the affine cure C : s 2 = 2t 3 + 7)t 3 7). Then C has points locally eerywhere, has index 1, but has no global points. Moreoer C Br =, so that the Brauer-Manin obstruction is the only obstruction for C. Proof. Coray and Manoil [7, Proposition 4.6, p 183] showed that C has index 1 but fails the Hasse Principle. They also obsered that C maps into the elliptic cure E : y 2 = x ia s, t) 2t 2, 2s), and that EQ) consists only of the point at infinity. After replacing the Jacobian with E we can apply Theorem 1.4. Proposition 4.3 is ery effectie at showing that a cure has no rational points; I hae been unable to produce an example of a cure X without a global point for which condition 4) still holds. If none exist then we would be close to a proof that the Brauer-Manin obstruction is the only one for cures. Further interesting cures can be found in [3]. References [1] B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on Elliptic Cures II, J. Reine Angew. Math. 218, 1965, [2] A. Bremner, D. J. Lewis, P. Morton, Some Varieties with Points Only in a Field Extension, Arch. Math. Basel) 43, 1984, [3] A. Bremner, Some interesting cures of genus 2 to 7, J. Number Theory 67, 1997, [4] J. W. S. Cassels and A. Fröhlich editors, Algebraic Number Theory, Proc. Instructional Conf., Brighton, 1965, Thompson, Washington, D.C., [5] J. W. S. Cassels, The Arithmetic of Certain Quartic Cures, Proc. Roy. Soc. Edinburgh Sect. A 1A, 1985, [6] J. L. Colliot-Thélène, The Hasse Principle in a Pencil of Algebraic Varieties, Contemporary Mathematics 21, 1996, [7] D. Coray, C. Manoil, On Large Picard Groups and the Hasse Principle for Cures and K3 Surfaces, Acta Arith. 76, 1996, [8] J. E. Cremona, Algorithms for Modular Elliptic Cures, Second Edition, Cambridge Uniersity Press [9] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Spring-Verlag [1] S. Lichtenbaum, Duality Theorems for Cures oer p-adic Fields, Inent. Math. 7, 1969, [11] J. S. Milne, Comparison of the Brauer Group with the Tate-Šafareič Group, J. Fac. Sci. Uni. Tokyo Sect. IA, Vol. 28:3, 1982, [12] J. S. Milne, Arithmetic Duality Theorems, Academic Press 1986.

11 THE BRAUER-MANIN OBSTRUCTION FOR CURVES 11 [13] J. S. Milne, Abelian Varieties, Arithmetic Geometry, G. Cornell and J. H. Silerman Eds, Springer- Verlag 1986, [14] J. S. Milne, Jacobian Varieties, Arithmetic Geometry, G. Cornell and J. H. Silerman Eds, Springer- Verlag 1986, [15] Yu. I. Manin, Le Groupe De Brauer-Grothendieck En Géométrie Diophantienne, Actes du Congrès International des Mathématiciens Nice, 197), 1, 197, [16] Yu. I. Manin, Cubic Forms, Second Edition, North-Holland [17] J. P. Serre, Sur les groupes de congruence des ariétés abéliennes, Iz. Akad. Nauk. SSSR 28, 1964, [18] J. P. Serre, Sur les groupes de congruence des ariétés abéliennes II, Iz. Akad. Nauk. SSSR 35, 1971, [19] J. H. Silerman, The Arithmetic of Elliptic Cures, Graduate Texts in Mathematics 16, Springer-Verlag [2] J. H. Silerman, Adanced Topics in the Arithmetic of Elliptic Cures, Graduate Texts in Mathematics 151, Springer-Verlag [21] A. N. Skorobogato, Beyond the Manin Obstruction, Preprint Noember Website: [22] J. Tate, Global Class Field Theory, Algebraic Number Theory, J. W. S. Cassels and A. Fröhlich Eds, Thompson 1967, [23] L. Wang, Brauer-Manin Obstruction to Weak Approximation on Abelian Varieties, Israel J. Math. 94, 1996,