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

TL;DR

This paper investigates the rational points on Galois twists of modular curves associated with elliptic curves, analyzing their L-functions, Galois representations, and implications for rational points and isogenies.

Contribution

It constructs and studies the moduli schemes $Y_E(p)$ and $X_E(p)$, analyzes their Jacobians and L-functions, and applies automorphy and Galois representation techniques to understand rational points and related conjectures.

Findings

Proved the existence of $Y_E(p)$ and $X_E(p)$ as moduli schemes.

Refined the understanding of Tate modules of Jacobians $J_E^{eta}(p)$.

Under automorphy assumptions, determined signs of functional equations of associated L-functions.

Abstract

Let $p$ be an odd prime and $E/\mathbb{Q}$ be a rational elliptic curve. There is a smooth affine curve $Y_E(p)$ whose rational points parametrize elliptic curves $F/\mathbb{Q}$ such that $F[p]$ and $E[p]$ are isomorphic Galois modules. This thesis manuscript is an attempt at applying Mazur's strategy to determine the rational points of $Y_E(p)$. One of the results that we prove, as a consequence of the approach, is the following: if $E,F/\mathbb{Q}$ are elliptic curves with isomorphic $23$-torsion Galois modules and $E$ has Weierstrass equation $y^2=x^3-23$, then $E$ and $F$ are isogenous. In Chapter 1, we prove the existence of the curve $Y_E(p)$ and its compactification $X_E(p)$ over more general bases as moduli schemes in the sense of Katz-Mazur, describe their Hecke correspondences, and construct $X_E(p)$ as a Galois twist of $X(p)$. In Chapters 2 and 3, we determine the Tate module of the Jacobians $J_E^{\alpha}(p)$ of the connected components of $X_E(p)$, thereby refining a result by Virdol. In Chapter 4, we study the factors of the $L$-function of $J_E^{\alpha}(p)$ previously determined. In general, little can be said, because proving the automorphy of the Galois representations involved seems currently out of reach. However, under suitable automorphy assumptions, we determine the signs of the functional equations satisfied by these $L$-functions. In Chapter 5, we discuss the special case where the image of the Galois action on $E[p]$ is contained in the normalizer of a nonsplit Cartan subgroup. The factors of $L(J_E^{\alpha}(p),s)$ then satisfy functional equations whose signs we determine. Using the Beilinson-Flach Euler system and results about large Galois image (from Chapter 6), we show that most factors of $J_E^{\alpha}(p)$ satisfy the Bloch-Kato in rank zero, and prove a formal immersion theorem for certain factors at all but an easily computable set of primes.