A higher rank shifted convolution problem with applications to L-functions

TL;DR

This paper solves a specific shifted convolution problem involving GL(3) and divisor functions, and applies the result to derive asymptotic formulas for central L-values twisted by Dirichlet characters.

Contribution

It introduces a novel approach combining delta symbol methods to solve an open shifted convolution problem involving GL(3) and divisor functions.

Findings

Established an asymptotic formula for central L-values with twisted characters.

Solved an open shifted convolution problem for GL(3) and divisor functions.

Developed new techniques using delta symbol methods.

Abstract

While several instances of shifted convolution problems for GL(3) x GL(2) have been solved, the case where one factor is the classical divisor function and one factor is a GL(3) Fourier coefficient has remained open. We solve this case in the present paper. The proof involves two intertwined applications of different types of delta symbol methods. As an application we establish an asymptotic formula for central values of L-functions for a GL(3) automorphic form twisted by Dirichlet characters to moduli q < Q.