### Modularity of elliptic curves over certain totally real quartic fields

Linfoot Number Theory Seminar

6th May 2020, 4:00 pm – 5:00 pm

Virtual Seminar, https://bluejeans.com/629681924

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.

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