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A precise and testable modularity conjecture for rational abelian surfaces A with trivial endomorphisms, End Q A = Z, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on ℓ-division fields of semistable abelian varieties, mainly when A[ℓ] is reducible, by considering extension problems for group schemes of small rank. A[ℓ n ] be the Tate module of A. Motivated by results of [2], [47], [62] and by the compatibility with standard conjectures [56] on the Hasse-Weil L-series L(A, s), we propose the following hypothesis.Conjecture 1.1. There is a one-to-one correspondence between isogeny classes of abelian surfaces A /Q of conductor N with End Q A = Z and weight 2 non-lifts f on K(N ) with rational eigenvalues, up to scalar multiplication. Moreover, the L-series of A and f should agree and the ℓ-adic representation of T ℓ (A) ⊗ Q ℓ should be isomorphic to those associated to f for any ℓ prime to N. See section 8, added in April 2018, for a modification of Conjecture 1.1.In contrast to Shimura's classical construction from elliptic newforms, no known method yields an abelian surface from a Siegel eigenform. Rome, Banff, Shanghai, Beijing and New York. We were inspired by René Schoof, who kindly provided us with preprints. His hospitality and support to the first author during a visit to Roma III in May 2005 helped this project along. The contributions of Brooks Roberts and Ralf Schmidt as well as those of Cris Poor and David S. Yuen were decisive to our main Conjecture. We thank them heartily for that as well as for useful conversations and correspondence. We also wish to thank the referee for many useful suggestions to improve the exposition.Proof. We know from [44] that T ℓ (A) = l T l (A) is a free o ℓ -module of rank 2g/d. From the canonical isomorphism to the Tate module of the reduction, T l (A) et is a free o l -module. As a free quotient, T l (A) et is a direct summand, and so is pure. By Cartier duality, T l ( A) m is free of the same rank. Corresponding to any opolarization, there is an isogeny A → A preserving the multiplicative component and so T l (A) m and T l ( A) m also have the same rank. To show that T l (A) m is pure, one may use the fact that it is the submodule of T l (A) orthogonal to T l ( A) 0 .Since o acts by functoriality on the connected component of the special fiber of the Néron model of A, the dimensions of T p and B p in (1.5) are multiples of d.Notation 3.2.2. Write t p = dim T p for the toroidal dimension at p and τ p = t p /d. By semistability, the reduced conductor of A is N 0 A = p p τp .

This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve E defined over a number field F as one passes to quadratic extensions K of F. Let S(K) be the Selmer group for multiplication by 2 on E(K). In analogy with genus theory, we describe S(K) in terms of various objects defined over F and the local norm indices <" = dimF2£(Ft))/Norm{£(ÄH,)} for each completion Fv of F. In particular we show that dim S(K) + dim E(K)2 has the same parity as Zi". We compute i" when E has good or multiplicative reduction modulo v. Assuming that the 2-primary component of the Tate-Shafarevitch group U1(K) is finite, as conjectured, we obtain the parity of rank E(K). For semistable elliptic curves defined over Q and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.

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