Twisted Hilbert modular L-functions and spectral theory

TL;DR

This paper discusses spectral theory techniques to establish subconvex bounds for twisted Hilbert modular L-functions, with applications to counting representations by quadratic forms over totally real fields.

Contribution

It provides a detailed proof of a Burgess-like subconvex bound for twisted Hilbert modular L-functions using spectral theory methods.

Findings

Established a subconvex bound for twisted Hilbert modular L-functions.

Demonstrated the use of spectral theory in estimating shifted convolution sums.

Applied results to count representations by quadratic forms over totally real fields.

Abstract

These are notes for four lectures given at the 2010 CIMPA Research School "Automorphic Forms and L-functions" in Weihai, China. The lectures focused on a Burgess-like subconvex bound for twisted Hilbert modular L-functions published jointly with Valentin Blomer in the same year. They discussed the proof in some detail, especially how spectral theory can be used to estimate the relevant shifted convolution sums efficiently. They also discussed briefly an application for the number of representations by a totally positive ternary quadratic form over a totally real number field.