# A Spectral reciprocity formula and non-vanishing for L-functions on GL(4)xGL(2)

## TL;DR

This paper introduces a spectral reciprocity formula for L-functions on GL(4)xGL(2) and demonstrates the existence of non-vanishing central values for these L-functions, advancing understanding of their behavior.

## Contribution

It develops a new reciprocity formula involving a balanced Voronoi summation and proves non-vanishing results for L-functions on GL(4)xGL(2).

## Key findings

- Established a spectral reciprocity formula for GL(4)xGL(2) L-functions.
- Proved the existence of non-vanishing central values for these L-functions.
- Introduced a balanced Voronoi summation involving Kloosterman sums.

## Abstract

We develop a reciprocity formula for a spectral sum over central values of L-functions on GL(4)xGL(2). As an application we show that for any self-dual cusp form Pi for SL(4,Z), there exists a Maass form pi for SL(2,Z) such that L(1/2, Pi x pi) is nonvanishing. An important ingredient is a "balanced" Voronoi summation formula involving Kloosterman sums on both sides, which can also be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL(4) Hecke eigenvalues.