David Zureick-Brown: ell-adic Images of Galois for Elliptic Curves over Q (NTWS 233)

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 XH). The main result is a provisional classification of the possible images of l-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 l-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.
Link to slides: drive.google.c...
Number Theory Web Seminar: www.ntwebsemin...
Original air date:
Thursday, December 19, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 20, 2024 (12am CST, 3am AEDT, 5am NZDT)