From bounds for Fourier coefficients to bounds for Mordell-Weil ranks (and beyond)

Abstract: Motivated the by the conjecture of Birch and Swinnerton-Dyer, I will explain how the spectral theory of automorphic forms on GL_2 and its two-fold metaplectic cover can be used to derive unconditional bounds for Mordell-Weil ranks of elliptic curves in certain abelian towers of number fields. The surjectivity of the archimedean local Kirillov map (or its classical manifestation in terms of Maass weight raising operators) plays a starring role here, allowing one to realize the implicit L-values in terms as Fourier-Whittaker coefficients of distinct automorphic forms. This leads to both new progress and open questions, which I will also describe.

number theory

Audience: researchers in the topic

Japan Europe Number Theory Exchange Seminar

Series comments: The purpose of the Japan Europe Number Theory Exchange Seminar is to give a opportunity for researchers in Japan and Europe to exchange their research projects by giving short talks (30 min). The target audience are researchers of any level in the area of number theory.

Start for Fall 2021: 26th October