Josha Box

Modularity of elliptic curves over certain totally real quartic fields

Following Wiles' breakthrough work, it has been shown in recent years that elliptic curves over each totally real field of degree 2 (Freitas-Le Hung-Siksek) or 3 (Derickx-Najman-Siksek) are modular. We study the degree 4 case and show that if K is a totally real quartic field without any quadratic subfields, then every elliptic curve over K is modular. Thanks to strong results of Thorne and Kalyanswami, this boils down to the determination of all K-points on three modular curves. In this talk I will focus on the genus 6 curve X(b5,ns7) and explain how a relative Chabauty method can be used to study its *infinite* set of quartic points. Time permitting I will briefly discuss a possible strategy for extending this to all quartic fields, and which problems arise.