Abstract. We conjecture that if C is a curve of genus> 1 over a number field k such that C(k) = ∅, then a method of Scharaschkin (equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that C(k) = ∅. As evidence, we prove a corresponding statement in which C(Fv) is replaced by a random subset of the same size in J(Fv) for each residue field Fv at a place v of good reduction for C, and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjecture holds, and if Shafarevich-Tate groups are finite, then there exists an algorithm to decide whether a curve over k has a k-point, and the Brauer-Manin obstruction to the Hasse principle for curves over the number fields is the only one. 1. Setup Let k be a number field. Fix an algebraic closure k of k, and let G = Gal(k/k). Let C be a curve of genus g over k. (In this paper, curves are assumed to be smooth, projective, and geometrically integral.) Let C = C ×k k. Let J be the Jacobian of C, which is an abelian variety of dimension g over k. Assume that C has a G-invariant line bundle of degree 1: this

Abstract. Let k be an algebraically closed field of characteristic p. Let X(p e; N) be the curve parameterizing elliptic curves with full level N structure (where p ∤ N) and full level p e Igusa structure. By modular curve, we mean a quotient of any X(p e; N) by any subgroup of ((Z/p e Z) × × SL2(Z/NZ)) /{±1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This extends earlier work, in which the result was proved for particular sequences of modular curves, such as X0(N) for p ∤ N. We give an application to the function field analogue of a uniform boundedness statement for the image of Galois on torsion of elliptic curves. 1.

Abstract. A curve C defined over Q is modular of level N if there exists a nonconstant morphism X1(N) − → C defined over Q for some positive integer N. We present an algorithm to compute explicitly equations for modular non-hyperelliptic curves defined over Q of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between J1(N) and Jac(C) factorizes through the new part of J1(N). We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves. 1.

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. There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n. 1.

Abstract. We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2. Contents

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Abstract. We prove a conjecture of Yamauchi which states that the level N for which the new part of J0(N) is Q-isogenous to a product of elliptic curves is bounded. We also state and partially prove a higher-dimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we derive a formula for the trace of Hecke operators acting on spaces S new (N, k) of newforms of weight k and level N. We use this trace formula to study the equidistribution of eigenvalues of Hecke operators on these spaces. For any d ≥ 1, we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to d. 1.

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Abstract. In this work, we estimate the genus of the intermediate