L-functions and rational points for Galois twists of modular curves

Overview & Synopsis

In this thesis, we discuss the existence of elliptic curves over a general base scheme for which the p-torsion is a given finite étale scheme, where p is a given prime. We first prove that this problem is representable in the sense of Katz and Mazur by a smooth affine curve, then describe another way to construct the compactification of such a curve over a base field, as a twist of X(p) by a two-dimensional Galois representation modulo p. We then study the L-function of such a twisted modular curve over the field of rationals and compute the sign of the functional equation that its factors are conjectured to satisfy. When the aforementioned Galois representation has small image, we show that this decomposition can be refined further, and we prove that the associated L-functions satisfy a functional equation whose sign we also determine. We then demonstrate the rank zero Bloch-Kato conjecture for every such factor without complex multiplication using the Euler system of Beilinson-Flach elements and extending big Galois image results due to Loeffler. We finally provide an application of such a result by giving an explicit bound on the bad reduction of elliptic curves such that the image of the Galois action on its p-torsion is contained in the normalizer of a nonsplit Cartan subgroup, under the additional hypothesis that certain Rankin-Selberg L-functions do not vanish, which is known to hold in certain cases.