Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions

TL;DR

This paper proves the existence of infinitely many GL(3) cusp forms for which certain L-functions with GL(2) and GL(1) forms are simultaneously non-zero, using Jacquet's Relative Trace Formula.

Contribution

It introduces a novel application of Jacquet's Relative Trace Formula to establish simultaneous nonvanishing of specific L-functions for GL(3) forms.

Findings

Infinitely many GL(3) cusp forms with non-zero L(1/2, π×χ) and L(1/2, π×φ).

Expression of the average over GL(3) spectrum as a main term plus subsidiary terms.

Subsidiary terms vanish for large level, ensuring nonvanishing results.

Abstract

Fix a Dirichlet character $\chi$ and a cuspidal GL$(2)$ eigenform $\phi$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $\pi$ on GL$(3)$ such that $L(1/2, \pi \times \chi)$ and $L(1/2, \pi \times \phi)$ are simultaneously non-zero. We achieve this by use of Jacquet's Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.