On the Harris-Venkatesh conjecture for weight one forms

Abstract

Venkatesh recently made very general conjectures regarding the relation between derived Hecke operators and a ``hidden'' action of a motivic cohomology group for an adjoint motive. These conjectures are in the setting of the cohomology of arithmetic groups. Venkatesh and Harris made an analogous conjecture in the setting of coherent cohomology in the first non-trivial case: weight one cuspidal eigenforms. This conjecture has been proved in some dihedral cases by Darmon-Harris-Rotger-Venkatesh recently. I found another approach using triple product L-functions. After some introduction on the conjecture, I will try to explain some ideas behind my method.

Speaker

Research Interests:

I am a number theorist, especially interested in the theory of modular forms. One of the central tools I have been using is the theory of modular symbols.

Here is a list of specific subjects I have been working on (roughly sorted from least to most recent).

• Eisenstein ideals, following the work of Mazur.

• Classical algebraic number theory, in particular K-groups of rings of integers.

• Galois representations and their deformations.

• p-adic uniformization of Shimura curves.

• The theory of 1-motives, developed by Deligne.

• Bianchi 3-manifolds, and their relation with arithmetic.

• The BSD conjecture and its variants, in particular the "refined" conjectures of Mazur and Tate.

Personal Homepage:

https://ymsc.tsinghua.edu.cn/en/info/1032/2464.htm