SHIMURA COVERINGS OF SHIMURA CURVES AND THE MANIN OBSTRUCTION. Alexei Skorobogatov. Introduction

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1 Mathematical Research Letters 1, NN (005) SHIMURA COVERINGS OF SHIMURA CURVES AND THE MANIN OBSTRUCTION Alexei Skorobogatov Abstract. We rove that the counterexamles to the Hasse rincile on Shimura curves over imaginary quadratic fields found by B. Jordan are accounted for by the Manin obstruction. Introduction Let B be an indefinite quaternion algebra over Q of reduced discriminant D. All maximal orders in B are conjugate; so let us fix one maximal order O B. Let O + O be the grou of elements of reduced norm 1. On identifying B R with M (R) we can think of O + as an arithmetic subgrou of SL (R). The comact Riemann surface O + \H, where H is the uer-half lane, is the set of comlex oints of an algebraic curve X over Q, a Shimura curve [10]. The curve X is the coarse moduli scheme over Q which classifies the isomorhism classes of airs (A, i), where A is an abelian surface and i is an embedding of O into End(A). By an abuse of language such a air is called an abelian surface with quaternionic multilication, or a QM-abelian surface. It aears that Jordan s work on global oints on Shimura curves [3] did not receive the attention it justly deserved. His aroach is based on the modular interretation and is insired by the strategy of the celebrated work of Mazur [6]. Jordan roves that certain Shimura curves have no oints defined over certain imaginary quadratic fields k; yet in some cases such Shimura curves have oints in all comletions of k. Until now it was not clear whether or not these counterexamles to the Hasse rincile can be accounted for by the Manin obstruction. One articular counterexamle on a curve of genus 3 has been treated in [11], but this exlanation was conditional on a conjectured equation of the curve. We review Jordan s aroach interreting it in terms of descent. The main idea is to consider the Shimura covering of X attached to a rime factor of D. It is the maximal étale subcovering Z X of the covering of X arameterizing triles (A, i, P ), where P is a generator of the O-module A[I ] for the unique two-sided ideal I O of reduced norm. The morhism Z X is a torsor under a constant grou scheme. A small modification of Jordan s aroach allows us to interret his results on the non-existence of global oints on X in terms of the descent erformed on a closely related torsor. (When we had nothing to add to Jordan s arguments we referred to only sketch them by referring to [3] for details.) This leads us to an exlanation of all counterexamles to the Hasse rincile stemming from his work [3] in terms of the Manin obstruction. Our main results are Theorems.4 and 3.1; a more exlicit articular case of the latter result is Corollary 3.. For the sake of comleteness let us mention that counterexamles to the Hasse rincile on Shimura curves and their Atkin Lehner quotients have been recently constructed in [], [13], [7]. All those obtained by the method of [13] are automatically accounted for by the Manin obstruction. In the end of this note we deduce from a recent result of M. Stoll [14] that the exlicit counterexamles to the Hasse rincile over Q constructed in [7] can also be exlained by the Manin obstruction. This aroach relies on well known but dee facts about modular forms

2 1000 Alexei Skorobogatov 1. A cyclic étale covering of X attached to a rime factor of D We write A Q for the ring of Q-adèles, and A Q,f for the ring of finite Q-adèles, i.e., adèles without archimedean comonents. Let H ± = C \ R. Let n : B Q be the reduced norm. We write Ô for O Z Ẑ. Let K Ô be a comact oen subgrou. Consider the toological sace of double cosets X K = B \H ± (B Q A Q,f ) /K, where B acts on the left simultaneously on both factors, and K acts on the right on the second factor. By Shimura and Deligne X K is a disjoint union of Ẑ /n(k) comact connected comlex curves, and it has a canonical model which is an irreducible algebraic curve over Q. If K = Ô, then X K is the Shimura curve X described in the introduction. This curve defined over Q is geometrically connected. We now attach a subgrou K to a rime factor of the discriminant D of B. There is a unique two-sided rime ideal I O of reduced norm (see [15],. 86). More concretely, I is the set of elements of O with reduced norm divisible by ; we have I = O. The field O/I is isomorhic to F. Define K as the set of elements of Ô which are congruent to 1 modulo I, and let X be the Shimura curve over Q which is the canonical model of X K. The natural finite flat morhism X X is a Galois covering whose Galois grou is isomorhic to F = F /{±1} if, and to F 4 Z/3 for =. The curve X is in fact the quotient of X by the action of the constant grou scheme Z/ 1 if (res. Z/3 if = ). If the curve X defined over Q is not geometrically connected. The comlex curve X Q C is a disjoint union of 1 = Z /1 + Z irreducible curves which are conjugate over Q(μ ). Each irreducible comonent is isomorhic to the Riemann surface Γ \H, where Γ = O + K. Note that Γ is normal in O + with quotient Z/( + 1). The only non-trivial element of O + that acts trivially on H is 1, and 1 Γ if is odd. Hence each irreducible comonent of X Q C is a cyclic Galois covering of X Q C of degree ( + 1)/. If = we have 1 K, so that the degree of the covering X X is 3. Following [4],. 108, for a quadratic field k we set ( ) B equal to 1 if k slits B, and 0 otherwise. Define ( ( 1 + ε() = B Q( 3) 1 + ( B Q( 1) 1 + ( B Q( 3) For we note that ε() divides ( 1)/. ) ) ))( 1 + ( k B Q( 1) for = 3; for =. )) for > 3; Lemma 1.1 (Jordan). Let. Define Z as the quotient of X by the grou subscheme Z/ε() Z/ 1. Then the natural morhism Z X is étale. Proof. Let e = hcf(ε(), ( + 1)/). Since all the irreducible comonents of X are conjugate it is enough to choose one of them, say X, and rove that the quotient of X by Z/e is an unramified covering of X. This is an immediate consequence of the fact that the image of the set of ellitic elements of O + /{±1} in O + /{±1} Γ Z/ +1 is the subgrou Z/e ([4], Pro ). QED The analogous statement for = is that the covering X X is unramified if and only if ε() = 1, that is, if and only if Q( 3) does not slit B. In the rest of the aer we assume that 5; then 6 divides ( 1)/. Define Y as the quotient of X by Z/6. Corollary 1.. Y is an X-torsor under the constant grou scheme F 1 Z/ 1 1.

3 SHIMURA COVERINGS OF SHIMURA CURVES AND THE MANIN OBSTRUCTION Proof. ε() divides 6 if 5, thus the covering Y X is a quotient of an étale covering Z X. QED Let k be a field of characteristic 0. We choose one of two ossible field isomorhisms of O/I with F. By secialization a torsor Y X under the constant grou scheme F 1 associates to a oint Q X(k) a continuous character by which the Galois grou acts on the fibre of Y X at Q. We denote it by φ Q Hom(Gal(k/k), F 1 ).. Modular interretation and local characters A natural modular interretation of X was discussed in the introduction. A similar interretation exists for X. Let (A, i) be an abelian surface with multilication by O, and let C = A[I ] be the kernel of the action of I O. Then C is an O-submodule of A[] canonically isomorhic to O/I F. The Q-curve X is the coarse moduli scheme over Q classifying triles (A, i, P ), where (A, i) is an abelian surface with multilication by O, and P is a generator of the O-module C (see [4],. 110, see also [1], III, 1.1). Note that a QM-abelian surface (A, i) is not necessarily defined over the residue field Q(P ) of the corresonding oint P on X, called the field of moduli of (A, i). In Thm. 1.1 of [3] Jordan roves that (A, i) is defined over a field k containing Q(P ) if and only if k slits B. Let (A, i) be a QM-abelian surface defined over a field k of characteristic 0. There is only one non-trivial O-submodule in A[], that is, C, hence it is defined over k. Our choice of a field isomorhism of C = O/I with F defines a character ρ A : Gal(k/k) F coming from the Galois action on C. Changing the isomorhism has the effect of relacing ρ A by ρ A, and the similar effect on φ Q. Lemma.1. Let k be a field of characteristic 0 which slits B, and let Q X(k). If (A, i) is a QM-abelian surface defined over k reresented by Q, then ρ 1 A = φ Q. Proof. This is a consequence of the modular interretation of the morhism X X and the fact that Y is the quotient of X by Z/6. QED Proosition. (Jordan). Let k be a number field which slits B. Let v be a rime of k which does not divide. Then for any Q v X(k v ) the character φ Qv is unramified. Perhas a most natural way to rove this roosition would be to show that the étale morhism of Q-curves Y X extends to an étale morhism of roer regular Z[1/]-schemes Y X. In other words, the torsor Y X under the constant grou scheme F 1 extends to a torsor over Sec(Z[1/]). This would imly that if v does not divide, then for any Q v X(k v ) the character φ Qv is unramified. (This argument does not require the assumtion that k slits B.) Jordan s roof, which we now sketch, is indirect but it establishes some other useful facts as by-roducts. Proof of Proosition.. (cf. [3], Sections 3 and 4) Any oint Q v X(k v ) reresents a QM-abelian surface (A, i) defined over k v. To this surface one canonically associates a character ρ A as above (well defined u to relacing ρ A by ρ A ). By Lemma.1 it suffices to show that ρ1 A is unramified away from. For this we need to show that if I(v) is the inertia subgrou of the maximal abelian extension kv ab /k v, then ρ A (I(v)) 1 = 1. Let l be a rime invertible in F v. It is known that A has otentially good reduction. By the Néron Ogg Shafarevich theorem this imlies that the kernel N of the action of I(v) on the Tate module T l (A) is a subgrou of I(v) of finite index (and N does not deend on l). Set Φ(A/k v ) = I(v)/N. Choose a Frobenius element σ Gal(k v /k v ). Let Ẑ Gal(k v/k v ) be the

4 10004 Alexei Skorobogatov closure of the infinite cyclic grou generated by σ, and let K/k v be the extension cut out by the subgrou NẐ. This is a totally ramified extension with the Galois grou Gal(K/k v) = Φ(A/k v ). By the Néron Ogg Shafarevich theorem the abelian surface A K has good reduction. Let à be the secial fibre. We have canonical isomorhisms T l (A) = T l (A K ) = T l (Ã). By the universal roerty of the Néron models the action of Φ(A/k v ) on A K = A k K via K extends uniquely to an action on the Néron model of A K. Since Gal(K/k v ) acts trivially on the residue field, the action of this grou on à is by F v-automorhisms of à (which commute with the action of O). Thus the reresentation of Φ(A/k v ) in T l (A) factors through an injective homomorhism Φ(A/k v ) Aut O (Ã/F v) (this argument goes back to Serre and Tate, see [9], Thm. ). The analysis of the automorhism grous over finite fields in Section of [3] then shows that an abelian quotient of Φ(A/k v ) is a cyclic grou of order 1,, 3, 4, or 6. We assumed that 5, and in this case these are the only ossible sizes of Φ(A/k v ). Now assume that is invertible in F v. To comlete the roof choose l =. Since N acts trivially on T (A) it acts trivially on C T (A)/, hence ρ A (I(v)) = ρ A (Φ(A/k v )). Thus ρ 1 A (I(v)) = 1. QED Proosition.3 (Jordan). Let k be an imaginary quadratic field which slits B, and such that a rime > 7, 1 mod 4, is inert in k. Denote by the unique rime of k over. Then for any Q X(k ) the restriction of the character φ Q to the inertia subgrou of the Galois grou of the maximal abelian extension of k is surjective onto F 1. Proof. As in the beginning of the revious roof any oint Q X(k ) reresents a QM-abelian surface (A, i) defined over k. To this surface one associates a character ρ A. By Lemma.1 we can write ρ 1 A for φ Q. Let U be the grou of units of the ring of integers of k, and let ω : U I() be the Artin ma. For u U we write ũ F for the reduction of u modulo. Since U is an extension of F by a ro--grou, and the homomorhism ρ A ω : U F is trivial on the ro--art, we have that ρ A factors through a homomorhism F F. Hence we can write ρ A (ω (u)) = (ũ) c for an integer c which is uniquely defined modulo 1. The Galois grou Gal(k/k) acts on A[]/C via the conjugate character ρ A, thus ρ+1 A = N F /F (ρ A ) is the cyclotomic character χ. We have χ (u) = N F /F (ũ) 1 (cf. [8], XIV, 7, remark 1). Hence c 1 mod 1. Consider C as a grou subscheme of the abelian surface with good reduction A K, where K is a totally ramified extension of k of degree n(a) = Φ(A/k ) constructed in the roof of Proosition. (for > 3 the number n(a) can be 1,, 3, 4, or 6). Then C A K extends to a finite flat grou scheme over the ring of integers O K. Raynaud s theorem imlies that n(a)c d 0 +d 1 mod 1, where 0 d i n(a), i = 1,. We are free to change the isomorhism C F ; this relaces c by c. Elementary comutations then show that for > n(a) we have, ossibly after relacing c by c, either 1c 1 mod 1, or 1c 1 + 4( 1) mod 1. In both cases c is corime to ( 1)/1. Therefore, ρ 1 A (I()) = F 1. QED The next result is an adatation of Theorem 6.1 of [3]. Theorem.4. Let B be a rational indefinite quaternion algebra ramified at a rime 11, 3 mod 4, and X be the Shimura curve defined by B. Assume that B is slit by an imaginary quadratic field k in which is inert, and denote by the unique rime of k over. Assume also that there is no surjective homomorhism from the ray class grou of k of conductor to the roduct of the class grou Cl k and Z/ 1 1. Then X(A k) Br =.

5 SHIMURA COVERINGS OF SHIMURA CURVES AND THE MANIN OBSTRUCTION Proof. We can assume that X(A k ) as otherwise there is nothing to rove. We want to rove that no family of local characters φ Qv such that Q v X(k v ) comes from a character of Gal(k/k). This means that no twist of the covering Y of X has oints everywhere locally. Then the theorem will follow from the main theorem of the descent theory of Colliot-Thélène and Sansuc alied to the torsor Y X (see [1], Thm. 6.1.). Assume for contradiction that the family of local characters φ Qv for some Q v X(k v ) comes from a character φ of Gal(k/k). Proosition. imlies that φ is unramified away from. The inertia subgrou of the Galois grou of the maximal abelian extension of k is an extension of (O k /) by a ro--grou. Any homomorhism to the grou of order ( 1)/1 is trivial on the ro--art, hence φ factors through the generalized class grou Cl k of conductor, which is the Galois grou of the ray class field of k of conductor. The natural ma Cl k Cl k fits into the exact sequence 1 (O k /) /Ok Cl k Cl k 1. By Proosition.3 the restriction of φ to (O k /) /Ok is surjective, a contradiction. QED 3. Congruence conditions Let q be a rime number. Let P (q) be the set of rime factors of the non-zero integers in the sets {a, a ± q, a ± q, a 3q }, where a q. If q we let B(q) be the set of rational indefinite quaternion algebras such that Q( q) does not slit B. Let B() be the set of rational indefinite quaternion algebras such that neither Q( 1) nor Q( ) slits B. Define C(q) B(q) as the set of algebras in B(q) with reduced discriminant divisible by a rime P (q). It is clear that the set B(q) \ C(q) is finite. The following result is an adatation of Theorem 6.3 of [3]. Theorem 3.1. Let k be an imaginary quadratic field in which q is ramified, B a quaternion algebra in C(q) which is slit by k, and X the Shimura curve defined by B. Then X(A k ) Br =. Proof. Let P (q) be a rime factor of the reduced discriminant D of B. Since D and k slits B, the rime is either inert or ramified in k. Let be the unique rime of k over. We assume that X(A k ) as otherwise there is nothing to rove. A oint Q v X(k v ) defines a character φ Qv : Gal(k v /k v ) F 1. Our goal is to rove that there does not exist a family of oints Q v X(k v ) for all non-archimedean comletions k v of k, such that the characters φ Qv come from a character φ : Gal(k/k) F 1. Then we can conclude as in the revious theorem by aealing to Thm of [1]. As in the revious section, Q v reresents a QM-abelian surface (A v, i v ) defined over k v, to which is attached a character ρ Av such that φ Qv = ρ 1 A v. Suose on the contrary that a global character φ exists. Then the restriction of φ to Gal(k v /k v ) is φ Qv = ρ 1 A v which is unramified for v by Proosition.. Let q be the unique rime of k above q, and let Frob q Gal(k/k) ab be a Frobenius at q. By global class field theory we have an exact sequence v U v Gal(k/k) ab Cl k 0, where v ranges over all rimes of k, U v is the grou of units of the ring of integers of k v, and U v Gal(k/k) ab is defined by the Artin ma ω v. Since q = (q) the order of the image of Frob q in the class grou divides, hence Frob q comes from an element of v U v. This element can be made exlicit. Note that Frob q comes from the idèle all of whose comonents equal 1 excet the comonent at q which equals π, a uniformizer at q. Then Frob q comes from the idèle all of whose

6 10006 Alexei Skorobogatov comonents equal 1/q excet the comonent at q which equals π /q. This last idèle is in v U v, and so it is a lifting of Frob q to v U v. Since φ is unramified away from we have φ(ω v (U v )) = 1. It follows that φ(frob q) = φ(ω (q 1 )) = ρ 1 A (ω (q 1 )). (There is an alternative argument based on the global recirocity for the norm residue symbol (φ, q).) Let us first deal with the case when is inert in k. We think of q as an element of U and follow the roof of Proosition.3. We have ρ A (ω (q)) q c mod, where c is 1 modulo 1, hence ρ A (ω (q 1 )) q mod. If is ramified in k, then the arguments in the roof of Proosition.3 should be slightly modified. Now U is an extension of F by a ro--grou, the homomorhism ρ A ω factors through the natural injection F F, and we again have ρ A (ω (q)) q c mod. However in this case we only have c mod 1 ([3], Pro. 4.8). This imlies ρ A (ω (q 1 )) ±q mod. In both cases we obtain φ(frob q) = ρ 1 A (ω (q 1 )) q 1 mod. Restricting φ to Gal(k q /k q ) we get ρ 1 A q (Frob q) q 1 mod. Therefore in F we have the equality ρ Aq (Frob q) = εq, where ε F, ε 1 = 1. In the roof of Proosition. we constructed a QM-abelian surface Ãq over F q such that the action of Frob q on the Tate modules T (A q ) and T (Ãq) is the same. The characteristic olynomial of Frob q reduced modulo is the square of (t ρ Aq (Frob q))(t ρ A q (Frob q)), and so N F /F (ρ Aq (Frob q)) = (ε+ε 1 )q is the reduction modulo of an integer a of absolute value at most q. Since ε 1 = 1 we have the following ossibilities: a 0, ±q, ±q mod or a 3q mod. Assume first that q divides a. From the Honda Tate theory and the fact that ρ Aq (Frob q ) + qρ 1 A q (Frob q ) is an integer one deduces the following list of ossibilities (see [3], Thm..1): a = 0, then q = and Q( 1) slits B; a = 3, then q = 3 and Q( 3) slits B; a = q, then Q( q) slits B. Each of these conditions contradicts the assumtion that the algebra B is in C(q). Now if q and a are corime, then divides one of the non-zero integers a, a ± q, a ± q, a 3q, where a q. This contradiction finishes the roof. QED It is worth selling out the theorem in the case q =. One easily comutes P () = {, 3, 5, 7, 11}, which leads to the following Corollary 3.. Let k = Q( d), where d < 0 is congruent to or 3 modulo 4. Let B be a rational indefinite quaternion algebra slit by k, whose discriminant is divisible by a rime greater than 11, a rime congruent to 1 modulo 4 and a rime congruent to 1 or 3 modulo 8 (these rimes do not have to be distinct). Then X(A k ) Br =, where X is the Shimura curve defined by B. 4. Examles 1. In Examle 6.4 of [3] Jordan considers the case D = 39 and k = Q( 13). The corresonding Shimura curve X has oints everywhere locally over k, as can be checked using [5]. For this articular Shimura curve the roerty X(A k ) Br = was exlicitly checked in [11] using a conjectural equation of X found by A. Kurihara. Corollary 3. does this unconditionally.. A numerical examle of Theorem.4 over the field Q( 3) can be found in [7]. It is the Shimura curve X attached to the quaternion algebra of discriminant The genus of X is 193. Comutations based on Theorem.5 of [5] show that X(Q l ) unless l = 3 or l = 107, but Theorems 5.1 and 5.4 of the same aer imly that X has oints in the comletions of Q( 3) at the rimes over 107 and 3. Thus X has oints in all comletions of Q( 3). An alication of

7 SHIMURA COVERINGS OF SHIMURA CURVES AND THE MANIN OBSTRUCTION Theorem.4 with = 107 shows that X is a counterexamle to the Hasse rincile over Q( 3) accounted for by the Manin obstruction. Indeed, whereas Cl Q( 3) Z/3, so that Cl (107) Q( Z/1717 Z/4 Z/81 Z/53, 3) Z/ ( 107 1) ClQ( 1 3) Z/ Z/9 Z/53 Z/3, and the conditions of Theorem.4 are satisfied. 3. Let us now discuss counterexamles to the Hasse rincile over Q constructed in [7]. These counterexamles use curves which are closely related to Shimura curves. Namely, consider the twist X of the Shimura curve X from the revious examle by the quadratic field k = Q( 3) with resect to the Atkin Lehner involution ω 107. Thm. 7. and Pro. 5.7 of [14] imly that this counterexamle is exlained by the Manin obstruction. More recisely, the arguments in Section 5 of [14] show that this can be achieved using the descent attached to the restriction of the covering R k/q Y R k/q X to X with resect to the canonical morhism X R k/q X = R k/q X. 4. We exlained the revious three counterexamles to the Hasse rincile by the Manin obstruction by ointing to an abelian étale covering such that no twisted form of it has oints everywhere locally. Let us consider a counterexamle to the Hasse rincile over Q for which we do not have a ready made descent argument. Let C be the quotient of X by ω 107. It is shown in [7] that X is the only twisted form of the covering X of C which has oints everywhere locally over Q. Since X (Q) = we conclude that C is a counterexamle to the Hasse rincile over Q. Let us show how to exlain this counterexamle by the Manin obstruction. M. Stoll roved ([14], Cor. 6.10) that if a smooth and rojective curve S has a non-constant morhism to an abelian variety A such that A(Q) and the Tate Shafarevich grou of A are finite, then S(Q) is in a natural bijection with the set of connected comonents of S(A Q ) Br. In articular, if S(Q) =, then S(A Q ) Br =. An aroriate combination of CM-oints on C gives a divisor of degree 1 (see [7]). This divisor defines an embedding of C into its Jacobian J such that the image of C is not contained in a translate of a roer abelian subvariety of J. It is a dee but well known result that the Jacobian of X is isogenous to the new art of the Jacobian J 0 (D) of the modular curve X 0 (D) (this uses Jacquet Langlands, Faltings, Ribet,...). It follows that J is isogenous to the ω 107 -anti-invariant art of J 0 (D) (since the Atkin Lehner involutions acting on the modular and the Shimura sides differ by 1). From the results of Kolyvagin Logachev (comlemented by Gross Zagier, Bum Friedberg Hoffstein and Murty Murty) it follows that any quotient A of J 0 (D) of analytic rank 0 also has algebraic rank 0 and a finite Tate Shararevich grou. A glance at W. Stein s tables of modular forms (htt://modular.fas.harvard.edu) shows that J 0 (D) has an ω 107 -anti-invariant quotient of analytic rank 0 of dimension 54 (defined by a certain Hecke eigenform of level 461 and weight ). This finishes the roof of our claim. It is clear that C(A Q ) Br = imlies X (A Q ) Br =. This gives an alternative aroach to the revious examle. The author thanks Kevin Buzzard, Victor Rotger, Andrei Yafaev for many helful discussions, and Michael Stoll for sending his rerint [14].

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