Abstract. In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.

Abstract. This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute. 1.

We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the Fontaine-Mazur conjectures, specifically, the modularity of certain potentially Barsotti-Tate Galois representations. The proof follows the template of Wiles, Taylor-Wiles, and Breuil-Conrad-Diamond-Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered φ1-modules of C. Breuil. 1. Notation, terminology, and results Throughout this article, we let l be an odd prime, and we fix an algebraic closure Ql of Ql with residue field Fl. The fields K, L, and E will always be finite extensions of Ql inside Ql. We denote by GK the Galois group Gal(Ql/K), by WK the Weil group of K, and by IK the inertia group of K. The group IQl will be abbreviated Il. The character ωn: GQl → Fln ⊂ Fl is defined via u ωn: u ↦→

Our aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by Q ( √ 5), and to examine the conjecture that any abelian surface with RM by Q ( √ 5) is isogenous to a simple factor of the Jacobian of a modular curve X0(N) for some N. To this end, we review previous work in this area, and are able to use a criterion due to Humbert in the last century to produce a family of curves of genus 2 with RM by Q ( √ 5) which parametrizes such curves which have a rational Weierstrass point. We proceed to give a calculation of the ℓ-adic representations arising from abelian surfaces with RM, and use a special case of this to determine a criterion for the field of definition of RM by Q ( √ 5). We examine when a given polarized abelian surface A defined over a number field k with an action of an order R in a real field F, also defined over k, can be made principally polarized after k-isogeny, and prove, in particular, that this is possible when the conductor of R is odd and coprime to the degree of the given polarization.

In [3], a reduction theory for binary forms of degree 3 and 4 with integer coefficients was developed in detail, the motivation in the case of quartics being to improve 2-descent algorithms for elliptic curves over Q. In this paper we extend some of these results to forms of higher degree. One application of this is to the

Abstract. We present a method to solve in an efficient way the problem of constructing the curves given by Torelli’s theorem in dimension 3 over the complex numbers: For an absolutely simple principally polarized abelian threefold A over C given by its period matrix Ω, compute a model of the curve of genus three (unique up to isomorphism) whose Jacobian, equipped with its canonical polarization, is isomorphic to A as a principally polarized abelian variety. We use this method to describe the non-hyperelliptic modular Jacobians of dimension 3. We investigate all the non-hyperelliptic new modular Jacobians Jac(Cf) of dimension 3 which are isomorphic to Af,wheref∈Snew 2 (X0(N)), N ≤ 4000.

Abstract. We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties Af attached by Shimura to normalized newforms f ∈ S2(Γ0(N)). We present all the curves corresponding to principally polarized surfaces Af for N ≤ 500. 1

Let F be a real quadratic field, and let R be an order in F. Suppose given a polarized abelian surface (A,λ) defined over a number field k with a symmetric action of R defined over k. This paper considers varying A within the k-isogeny class of A to reduce the degree of λ and the conductor of R. It is proved, in particular, that there is a k-isogenous principally polarized abelian surface with an action of the full ring of integers of F when F has class number 1 and the degree of λ and the conductor of R are odd and coprime. 1

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