CADORET, ANNA

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TitleThe $ell$-primary torsion conjecture for abelian surfaces withreal multiplication (Algebraic Number Theory and RelatedTopics 2010)Author(s)CADORET, ANNACitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2012), B32: 195-204Issue Date2012-07URL http://hdl.handle.net/2433/196242RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Ko^kyu^roku BessatsuB32 (2012), 195{204The `-primary torsion conjecture for abelian surfaceswith real multiplicationByAnna CadoretAbstractWe prove that the Bombieri-Lang conjecture implies the `-primary torsion conjecture forabelian surfaces with real multiplication.x 1. IntroductionLet O be the ring of integers of a quadratic real eld extension of Q with discrimi-nant D and let X(n) denote the coarse moduli scheme for the stack of polarized abeliansurfaces with real multiplication by O and n-level structure (See Subsection 2.2 forprecise denitions). The irreducible components of the X(n) are normal, separated, ge-ometrically connected surfaces (dened over a number eld depending on the involveddata) and it is known that the smallest integer n(D) such that X(n(D)) is of generaltype is 1 except for nitely many cases, where it is 2 or 3 (Theorem 2.2).The Bombieri-Lang conjecture (Conjecture 2.1) predicts that if X is a surface ofgeneral type over a number eld k then the set of k-rational points X(k) is not Zariski-dense in X. Thus, conjecturally, polarized abelian surfaces with real multiplication byO and n(D)-level structure dened over k are parametrized by a closed subschemeS ,! X(n(D)) whose irreducible components have dimension  1.For such a scheme S and any abelian scheme A ! S, works of A. Tamagawa andthe author show that, for any integer d  1 and prime ` the `-primary rational torsionis uniformly bounded in the bres at closed points s 2 S whose residue degree [k(s) : k]is  d (Theorem 2.4). So, assuming the Bombieri-Lang conjecture, one can expectthat for any number eld k and prime ` the k-rational `-primary torsion of polarizedReceived March 30, 2011. Revised May 11, 2011.2000 Mathematics Subject Classication(s): Primary: 11G10; Secondary: 14D10, 14D22.Centre de Mathematiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau cedex,FRANCE.e-mail: anna.cadoret@math.polytechnique.frc 2012 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.196 Anna Cadoretabelian surfaces with real multiplication by O and n(D)-level structure dened over kis uniformly bounded.The only diculty stems from the fact that, for n(D)  2, the coarse moduli schemeX(n) is not ne hence there is no universal abelian surface with real multiplication byO and n(D)-level structure over it, to which one could apply directly the uniformboundedness result of A. Tamagawa and the author. However, this can be overcome byelementary cohomological and rigidication technics.The main result we obtain in this short note is the following.Theorem 1.1. Assume that the Bombieri-Lang conjecture holds. Then for anynumber eld k and prime ` there exists an integer N := N(O; k; `) such that, for anypolarized abelian surface A with real multiplication by O and n(D)-level structure denedover k:jA(k)[`1]j  `N :In Section 2 we gather the several ingredients involved in the proof of Theorem 1.1.The proof itself is carried out in Section 3.Acknowledgements: I would like to thank Pete L. Clark for suggesting the idea ofthis note as well as Marc-Hubert Nicole, Matthieu Romagny and Akio Tamagawa fortheir interest and constructive comments.x 2. PreliminariesGiven a scheme S over a eld k and an integer d  1, we will writeSd := fs 2 S j [k(s) : k]  dg;where k(s) denotes the residue eld of S at s.x 2.1. Bombieri-Lang conjectureGiven a eld k of characteristic 0, let P(k) (resp. B(k)) denote the category ofall schemes smooth, projective and geometrically connected over k (resp. of all nor-mal schemes separated, of nite type and geometrically connected over k). Also, let denote the birational equivalence on P(k) (resp. B(k)), it follows from Nagata'scompactication theorem and Hironaka's desingularization theorem that the canonicalmapP(k)= ! B(k)= is bijective (e.g. [1, x2.1]). Thus any birational invariant dened on P(k) naturallyextends to B(k); this is in particular the case for Kodaira dimension and we say thatS 2 B(k) is of general type if its Kodaira dimension (S) is equal to its dimensiondim(S). With this convention, we can formulate the so-called Bombieri-Lang conjectureThe `-primary torsion conjecture for abelian surfaces with real multiplication 197Conjecture 2.1. Let k be a number eld and let S 2 B(k) be a surface of generaltype. Then the set of k-rational points S(k) is not Zariski-dense in S.Conjecture 2.1 was stated by E. Bombieri for surfaces and generalized by S. Langfor schemes S 2 B(k) of arbitrary dimension under the following more precise form.Let k be a number eld and let S 2 B(k) be of general type. Then there existe aclosed subscheme Z ,! S, Z 6= S such that for any nite eld extension k ,! k0 the setS(k0) n Z(k0) is nite.The Lang conjecture (and even the Bombieri-Lang conjecture) is widely open. Themost striking result is that it holds for subvarieties of abelian varieties [7].x 2.2. The stack of abelian surfaces with real multiplicationLet Q ,! K be a degree g totally real eld extension and let O denote the ringof integers of K. Let C and C+ denote the class group and strict class group of O,respectively. Fix a system J1; : : : ;Jh+ of fractional ideals of O, which, endowed withtheir natural notion of positivity, form a complete system of representatives of C+.For J one of the J1; : : : ;Jh+ , a g-dimensional J -polarized abelian scheme with realmultiplication by O is a triple (A; ; ), where:- A! S is an abelian scheme of relative dimension g;-  : O ,! EndS(A) is a ring homomorphism;-  : (MA;M+A) ~!(J ;J +) is a polarization that is, an O-linear isomorphism of etalesheaves between the invertibleO-moduleMA of all symmetricO-linear homomorphismsfrom A to A_ and J , identifying the positive cone of polarizationsM+A with the totallyreal elements J + in J .Let Q(J ) denote the eld of denition of J and SO;g;J ! spec(Q(J )) the etalestack of g-dimensional J -polarized abelian schemes with real multiplication by O. SetSO;g :=G1ih+SO;g;Ji :One can furthermore endow J -polarized abelian schemes with real multiplication by Owith a n level-structure, that is an injective O-linear homomorphism of etale sheaves: : n Z D 1 ,! A;whereD is the dierent of Q ,! K. Let SO;g;J (n) and SO;g(n) denote the correspondingstacks.The stacks SO;g(n), n  0 are Deligne-Mumford stacks locally of nite presenta-tion and with nite inertia; let c : SO;g;J (n) ! SO;g;J (n) denote their coarse modulischemes. For n  3, c : SO;g;J (n) ! SO;g;J (n) is a ne moduli scheme. See [12], [14,Chap. X], [6] for more details.198 Anna CadoretFor g = 2, the coarse moduli schemes SO;2;J (n) are 2-dimensional normal schemes,separated and geometrically connected over Q(J ). The analytic space associated withSO;2;J (n)Q(J ) C can be identied with PSLn2 (O J ) n H2. HereH = fz 2 C j im(z) > 0gand PSLn2 (OJ ) = SLn2 (OJ )=f1g, where SLn2 (OJ ) is the subgroup of SL2(k)of matrices:  a bc d!with a  d mod J , c 2 nJ and b 2 nJ 1. Using this analytic description, one canshow the following [9], [10], [14, Thm. VII.3.3 and VII.3.4].Theorem 2.2. For any real quadratic eld extension Q ,! K with ring of inte-gers O and discriminant D one has- SO;2;J is of general type except if D = 5; 8; 12; 13; 17; 21; 24; 28; 29; 33; 37; 40; 41; 44; 53;56; 57; 60; 61; 65; 69; 73; 77; 85; 88; 92; 93; 105; 120; 140; 165 (we will then say that Dis exceptional).- SO;2;J (2) is of general type except if D = 5; 8; 12, in which case SO;2;J (3) is ofgeneral type.We set n(D) = 1 if D is non-exceptional, n(D) = 2 if D is exceptional 6= 5; 8; 12and n(D) = 3 if D = 5; 8; 12.x 2.3. Strong uniform boundedness of `-primary torsion in 1-dimensionalfamilies of abelian varietiesThe Mordell-Weil theorem asserts that for any number eld k and abelian variety Aover k the (abelian) group A(k) of k-rational points is nitely generated. In particular itstorsion subgroup A(k)tors is nite. Given a prime `, the `-primary torsion conjecture isa uniform conjectural form of Mordell-Weil theorem for the `-Sylow subgroups A(k)[`1]of A(k)tors.Conjecture 2.3 (`-primary torsion conjecture). For any integer   1- Weak form: For any number eld k and prime ` there exists an integer N :=N(; `; k) depending only on , ` and k such that for any -dimensional abelianvariety A over k one hasjA(k)[`1]j  `N :The `-primary torsion conjecture for abelian surfaces with real multiplication 199- Strong form: For any integer d  1 and prime ` there exists an integer N :=N(; `; d) depending only on , ` and d such that for any number eld k with [k :Q]  d and -dimensional abelian variety A over k one hasjA(k)[`1]j  `N :The  = 1 case of the strong `-primary torsion conjecture essentially follows fromthe fact that for any prime ` the gonality of the modular curves Y1(`n) (classifyingelliptic curves with a torsion point of order exactly `n) goes to +1 with n. Indeed,combined with Faltings-Lang-Frey [8], this implies that for any integer d  1 the setsY1(`n)d are nite for n large enough (depending on d). But if Y1(`n)d is non-emptyfor all n  0, it follows from the usual compactness argument thatlim   Y1(`n)dis non-empty as well, which contradicts Mordell-Weil theorem.For   2, the `-primary torsion conjecture is widely open. However, the followinghigher dimensional relative variant of the strong `-primary torsion conjecture holds [2],[3], [4].Theorem 2.4. Let k be a number eld, S a normal curve, separated, of nitetype and geometrically connected over k and A! S an abelian scheme. Then, for anyinteger d  1 and prime ` there exists an integer N := N(A; `; d) depending only on A,` and d such that for any s 2 Sd on hasjAs(k(s))[`1]j  `N :When applied to the 'universal elliptic scheme'E  y2 + xy   x3 + 36j   1728x+1j   1728 ! P1j n f0; 1728;+1gtheorem 2.4 yields the 1-dimensional case of the `-primary torsion conjecture.x 3. Proof of theorem 1.1We now proceed to the proof of Theorem 1.1 by combining the ingredients gatheredin Section 2.Fix a nite extension k of Q(J ) and, for simplicity, write n for n(D) and c :X (n)! X(n) for the pullback of c : SO;2;J (n)! SO;2;J (n) via spec(k)! spec(Q(J )).In any case, the canonical morphism of stacks F : X (6)! X (n) induces a commutative200 Anna CadoretdiagramX (6) F //cX (n)cX(6)f// X(n);where the bottom arrow f : X(6)! X(n) is a nite surjective morphism of surfaces. ByTheorem 2.2 (and assuming the Bombieri-Lang conjecture), the irreducible componentsof the Zariski closure X(n) of X(n)(k) in X(n) have dimension  1. Since X(n) hasnitely many irreducible components (being of nite type over k), it is enough to bounduniformly the `-primary torsion of the abelian surfaces associated with points in X (n)(k)lying over I for each irreducible component I of X(n). So, without loss of generality,we may assume that X(n) is irreducible.x 3.1. dim(X(n)) = 0 (i.e. X(n) is a singleton)Since c : X (n)! X(n) is a coarse moduli scheme, it induces a bijectionX (n)(k)=  ~!X(n)(k):In particular, any two Ai(= (Ai; i; i; i)) 2 X (n)(k), i = 1; 2 become isomorphic inX (n)(k). The conclusion then follows from Lemma 3.1 below applied to the forgetfulfunctor X (n) ! A2 and the fact that for any abelian variety A over a number eld kand integer d  1 the setAd \A(k)torsis nite [5, Lemma 3.19].Lemma 3.1. Let S be any scheme, let A ! S denote the etale stack of -dimensional abelian varieties over S and let X ! S be an algebraic stack with quasi-nite inertia1. Then for any morphism A : X ! A of bered categories, there existsan integer  := ()  1 such that for any S-eld k and x; ; y 2 X (k) such that xjkand yjk are isomorphic in X (k), there exists a nite extension k ,! kx;y with degree[kx;y : k]   such that A(x)jkx;y and A(y)jkx;y become isomorphic in A(kx;y).Here, given a morphisme of S-schemes V ! U and an object x 2 X (U), we write xjVfor the image of x via the canonical pull-back functor X (U) ! X (V ). For instance,with this notation, A(x)jkx;y is nothing but A(x)k kx;y.1Recall that the inertia stack of X is dened to be IS(X ) := X XSX X ! X . Hence, inparticular, it follows from the denition of the bre product in the 2-category of S-groupoids thatfor any geometric point x 2 X () the bre IS(X )x can be identied with Aut(x). The assumptionthat X has nite inertia is only there to ensure that the automorphism group of geometric pointsis nite.The `-primary torsion conjecture for abelian surfaces with real multiplication 201Proof. Recall rst that there exists an integer B()  1 such that for any eld kand -dimensional abelian variety A over k, any nite subgroup G of Aut(A) has order B(). Indeed, let ` be a prime dierent from the characteristic of k. Then, sinceEnd(A) acts faithfully on T`(A) [11, Lemma 12.2], G is a nite subgroup of GL2(Z`).But such subgroups have order bounded by a constant depending only on  and ` [13,LG 4.27, Thm. 5]. The conclusion thus follows from the fact that we can take ` = 2 or3.Given x 2 X (k), write Tw(x=k) for the set of isomorphism classes of twists of xover k that is pairs (x0; ), where x0 2 X (k) and  : xjk ~!x0jk is an isomorphism inX (k). One has a commutative diagram of pointed sets:Tw(A(x)=k) // H1( k;Aut(A(x)jk))H1( k; A(Aut(xjk)))iiTTTTTTTTTTTTTTTTw(x=k)a//OOH1( k;Aut(xjk));b55jjjjjjjjjjjjjjjwhere  is an isomorphism of pointed sets. By the quasi-nite inertia assumption, thegroup Aut(xjk) is nite so A(Aut(xjk)) is a nite subgroup of Aut(A(xjk)) and, by theobservation above, has order  B(). Set kx := kker('), where ' :  k ! S(A(Aut(xjk)))is the permutation representation associated with the  k-set A(Aut(xjk)). Then [kx :k]  B()! and A(Aut(xjk)) is a trivial  kx-module so H1(kx; A(Aut(xjk))) can beidentied with the set of orbits ofHomGroup( kx ; A(Aut(xjk)))under the right action by inner automorphisms ofA(Aut(xjk)). Now, given y 2 Tw(x=k),setkx;y := kker(ba(y)jkx )xthen [kx;y : k]  B()!B() and ba(y)jkx;y is trivial. Since  is an isomorphism ofpointed sets and the above diagram commutes, this implies that (y)jkx;y is trivial aswell. So () = B()!B() works.x 3.2. dim(X(n)) = 1Let X(6) denote the pullback of X(n) ,! X(n) via f : X(6) ! X(n). Up to con-sidering separately the (nitely many) irreducible components of X(6), we may assumethat X(6) is irreducible. From Subsection 3.1, we may freely remove nitely many points202 Anna Cadoretfrom X(n) hence also assume that X(6) is a curve smooth (separated and of nite type)over k. Eventually, we may freely replace k by any nite extension k ,! k0 hence assumethat X(6) is geometrically connected over k.As c : X (6)! X(6) is a ne moduli scheme, there is a sectionX (6) c // X(6)ffcorresponding to a J -polarized abelian scheme with real multiplication by O and 6-level structure (A; ; ; )! X(6). For simplicity, write againA := AX(6) X(6)! X(6)for the pullback of A ! X(6) via X(6) ,! X(6). Then, from Theorem 2.4, for anyprime ` and integer d  1 there exists an integer N := N(A; `; d) such that for anyx 2 X(6)d one hasjAx(k(x))[`1]j  `N :This is true, in particular, ford = e(2);where e denotes the degree of f : X(6)! X(n). But for any A0 = (A0; ; ; ) 2 X (n)(k)above some x 2 X(n)(k) and any x0 2 f 1(x), A0 k k and F (Ax0 k(x0) k) becomeisomorphic in X (n)(k). So the conclusion, again, follows from Lemma 3.1 applied tothe forgetful functor X (n)! A2.Remark 3.2.1. The proof given here applies as it is to other similar situations. For instance ifA; denotes the stack of -dimensional abelian varieties endowed with a degree 2polarization and c : A; ! A; its coarse moduli scheme then, for any numbereld k and 2-dimensional subscheme S ,! A; such that S 2 B(k) is of generaltype, the `-primary torsion of the abelian varieties associated with points in A;(k)lying over S is uniformly bounded.2. Even in the ne moduli situation, that is A! S is an abelian scheme with S 2 B(k)of general type, our proof does not show that the Bombieri-Lang conjecture impliesthat for integer d  2 and prime ` the `-primary k(s)-rational torsion in the bresAs, s 2 Sd is uniformly bounded. The obstruction comes from the fact that theset Sd might be Zariski dense in S even if for each nite extension k ,! k0 withdegree [k0 : k]  d the set S(k0) is not.Remark 3.3 (Adding structures). One can show [1, Lemma 4.11] that the `-primarytorsion conjecture is equivalent toThe `-primary torsion conjecture for abelian surfaces with real multiplication 203Conjecture 3.4. Let k be a number eld, S 2 B(k), A! S an abelian schemeand ` a prime, then there exists an integer N := N(A; `) such that As[`1](k)  As[`N ]for all s 2 S(k).When S is a surface, combining the technics of [1], [4, Lemma 3.5] and assuming theLang conjecture for surfaces, one can show that conjecture 3.4 holds provided- for all v 2 T`(A) r `T`(A) the dimension of 1(S)v as a `-adic analytic space is 3;- A[`n` ](k()) = A[`n` ] (in other words, A ! S is endowed with a full-level-`n` -structure dened over S) withn` = 1 if `  5;2 if ` = 3;3 if ` = 2.This shows that adding structures (especially full-level structures) should force the `-primary torsion conjecture to hold; a main diculty is thus to remove the hypothesisthat small full-level structures are dened over the base eld.In Theorem 1.1, we add two kind of structures: real multiplication, so that theparameter space be a surface (recall that Ag;1 is three-dimensional) and n(D)-levelstructures so that the parameter space be of general type. In view of the above consid-erations and though it makes the proof more complicated (for n(D)  3, the dicultystemming from the fact that c : SO;2;J (n) ! SO;2;J (n) is not a ne moduli schemedisappears), it is important that n(D) be as small as possible. The fact that we canchoose n(D) = 1 for almost all D and in any case n(D)  3 (so, in particular, n(D)independent of `) relies on the highly non-trivial classication theorem 2.2.References[1] Cadoret, A., Kodaira dimension of abstract modular surfaces, preprint (2011).[2] Cadoret, A. and Tamagawa, A., A uniform boundedness of p-primary torsion on abelianschemes, Invent. Math., 188 (2012), 83{125.[3] A. Cadoret and A. Tamagawa, A uniform open image theorem for `-adic representationsI, to appear in Duke Math. J.[4] Cadoret, A. and Tamagawa, A., A uniform open image theorem for `-adic representationsII, preprint (2012).[5] Cadoret, A. and Tamagawa, A., On a weak variant of the geometric torsion conjecture, J.Algebra, 346 (2011), 227{247.[6] Deligne, P. and Pappas, G., Singularites des espaces de modules de Hilbert, en les car-acteristiques divisant le discriminant, Compositio Math., 90 (1994), 59{79.204 Anna Cadoret[7] Faltings, G., The general case of Lang's conjecture, Barsotti's symposium in algebraicgeometry, Academic Press, (1994), 175{182.[8] Frey, G., Curves with innitely many points of nite degree, Israel Journal of Math. 85,(1994), 79{83.[9] Hirzebruch, F. and Van de Ven, A., Hilbert modular surfaces and the classication ofalgebraic surfaces, Inventiones Math., 23 No 1, (1974), 1{29.[10] Hirzebruch, F. and Zagier, D., Classication of Hilbert modular surfaces, Complex analysisand algebraic geometry, Iwanami Shoten and Cambridge Univ. Press, (1977), 43{77.[11] Milne, J.S., Abelian varieties, Arithmetic geometry (Cornell/Silverman, eds.), Springer-Verlag, (1986), 103{150.[12] Rapoport, M., Compactications de l'espace de modules de Hilbert-Blumenthal, Compo-sitio Math. 36, (1978), 255{335.[13] Serre, J.-P., Lie algebras and Lie groups, W.A. Benjamin, (1965).[14] Van der Geer, G., Hilbert Modular Surfaces, E.M.G. 16, Springer, (1987).

The $ell$-primary torsion conjecture for abelian surfaces with real multiplication (Algebraic Number Theory and Related Topics 2010)

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The $ell$ -primary torsion conjecture for abelian surfaces with real multiplication (Algebraic Number Theory and Related Topics 2010)

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The ellellell-primary torsion conjecture for abelian surfaces with real multiplication (Algebraic Number Theory and Related Topics 2010)

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The $ell$ -primary torsion conjecture for abelian surfaces with real multiplication (Algebraic Number Theory and Related Topics 2010)