Abstract. We present an algorithm to determine if the L-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor, and Berger-Harcos, we can associate to an automorphic representation a family of compatible -adic representations. Our algorithm is based on Faltings-Serre's method to prove that -adic Galois representations are isomorphic. Using the algorithm we provide the first examples of modular elliptic curves over imaginary quadratic fields with residual 2-adic image isomorphic to S 3 and C 3 .

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this, we use archimedean results from Harris, Soudry and Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface B defined over Q, which is not a restriction of scalars of an elliptic curve and satisfies the paramodularity Conjecture of Brumer and Kramer.

The main result of this paper is a new version of Newton-Hensel lifting that relates to interpolation questions. It allows one to lift polynomials in Z[x] from information modulo a prime number p = 2 to a power p k for any k, and its originality is that it is a mixed version that not only lifts the coefficients of the polynomial but also its exponents. We show that this result corresponds exactly to a Newton-Hensel lifting of a system of 2t generalized equations in 2t unknowns in the ring of p-adic integers Z p . Finally, we apply our results to sparse polynomial interpolation in Z[x].

Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves.Date: August 10, 2018. 2010 Mathematics Subject Classification. 11F46, 11Y40.1 is made precise by a conjecture of Brumer-Kramer [8, Conjecture 1.1], restricted here for simplicity.Conjecture 1.1.1 (Brumer-Kramer). To every abelian surface A over Q of conductor N with End(A) = Z, there exists a cuspidal, nonlift Siegel paramodular newform f of degree 2, weight 2, and level N with rational Hecke eigenvalues, such thatMoreover, f is unique up to (nonzero) scaling and depends only on the isogeny class of A; and if N is squarefree, then this association is bijective.Conjecture 1.1.1 is often referred to as the paramodular conjecture. As pointed out by Frank Calegari, in general it is necessary to include abelian fourfolds with quaternionic multiplication for the converse assertion: for a precise statement for arbitrary N and further discussion, see Brumer-Kramer [7, Section 8].Extensive experimental evidence [8,47] supports Conjecture 1.1.1. There is also theoretical evidence for this conjecture when the abelian surface A is potentially of GL 2 -type, acquiring extra endomorphisms over a quadratic field: see Johnson-Leung-Roberts [34] for real quadratic fields and Berger-Dembélé-Pacetti-Şengün [5] for imaginary quadratic fields. For a complete treatment of the many possibilities for the association of modular forms to abelian surfaces with potentially extra endomorphisms, see work of Booker-Sijsling-Sutherland-Voight-Yasaki [11]. What remains is the case where End(A Q al ) = Z, which is to say that A has minimal endomorphisms defined over the algebraic closure Q al ; we say then that A is typical. (We do not say generic, since it is not a Zariski open condition on the moduli space.)Recently, there has been dramatic progress in modularity lifting theorems for nonlift Siegel modular forms (i.e., forms not of endoscopic type): see Pilloni [44] for p-adic overconvergent modularity lifting, as well as recent work by Calegari-Geraghty [12, §1.2], Berger-Klosin with Poor-Shurman-Yuen [2] establishing modularity in the reducible case when certain congruences are provided, and a paper in preparation by Boxer-Calegari-Gee-Pilloni [6] establishing potential modularity over totally real fields. 1.2. Main result. For all prime levels N < 277, the paramodular conjecture is known: there are no paramodular forms of the specified type by work of Poor-Yuen [47, Theorem 1.2], and correspondingly there are no abelian surfaces by work of Brumer-Kramer [8, Proposition 1.5]. At level N = 277, there exists a cuspidal, nonlift Siegel paramodular cusp form, unique up to scalar multiple, by work of Poor-Yuen [47, Theorem 1.3]: this form is given explicitly as a rational function in Gritsenko lifts of ten weight 2 theta blocks-see (6.2.2).Our main result is as follows. Theorem 1.2.1. ...