$\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$

Abstract: I will discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur’s “Program B” — the classification of the possible “images of Galois” associated to an elliptic curve (equivalently, classification of all rational points on certain modular curves $X_H$). The main result is a provisional classification of the possible images of $\ell$-adic Galois representations associated to elliptic curves over $\mathbb{Q}$ and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.

I will also discuss the framework and various applications (for example: a very fast algorithm to rigorously compute the $\ell$-adic image of Galois of an elliptic curve over $\mathbb{Q}$), and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagin’s theorem to the modular curves we study.

number theory

Audience: researchers in the topic

Harvard number theory seminar