BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Modularity of elliptic curves over totally real quartic fields not containing the square root of 5 Box, Josha


Following Wiles's 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 in which 5 is not a square, then every elliptic curve over K is modular. Thanks to strong results of Thorne and Kalyanswami, this boils down to the determination of all quartic points on a few modular curves. Some of these curves have infinitely many quartic points. In this talk I will discuss how Chabauty's method and sieving can nevertheless be used to describe such points.

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