De Rham cohomology of modular curves via modular forms

Abstract: Computing rational points on nonsplit Cartan modular curves is a difficult open problem. The most successful method — Quadratic Chabauty — can so far only solve this problem for prime levels N = 13,17,19. In this talk I will report on joint work in progress with Isabel Rendell, in which we complete the first step of Quadratic Chabauty, namely the computation of a basis of de Rham cohomology and of the Hodge filtration, for levels N = 23,29. Crucially, we avoid using a plane model of the curve, but instead phrase everything in terms of modular forms.

number theory

Audience: researchers in the topic

Five College Number Theory Seminar