Number Theory Seminar: ℓ-adic images of Galois for elliptic curves over ℚ

SEMINARS, NUMBER THEORY

Speaker:

David Zureick-Brown - Amherst College

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 $Xʜ$. The main result is a provisional classification of the possible images of $\ell$-adic Galois representations associated to elliptic curves over 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 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.