Non-vanishing of symmetric cube $L$-functions

TL;DR

This paper proves the existence of infinitely many Maass--Hecke cuspforms over [] such that their symmetric cube $L$-series do not vanish at the critical center, using spectral theory and triple product integrals.

Contribution

It establishes the non-vanishing of symmetric cube $L$-functions for infinitely many cusp forms over [], linking it to triple product integrals and cubic theta functions.

Findings

Infinitely many cusp forms with non-vanishing symmetric cube $L$-values.

Connection between triple product integrals and $L$-function non-vanishing.

Formulation of a conjecture related to the absolute value squared of the triple product.

Abstract

We prove that there are infinitely many Maass--Hecke cuspforms over the field $\mathbb{Q}[\sqrt{-3}]$ such that the corresponding symmetric cube $L$-series does not vanish at the center of the critical strip. This is done by using a result of Ginzburg, Jiang and Rallis which shows that the symmetric cube non-vanishing happens if and only if a certain triple product integral involving the cusp form and the cubic theta function on $\mathbb{Q}[\sqrt{-3}]$ does not vanish. We use spectral theory and the properties of the cubic theta function to show that the non-vanishing of this triple product occurs for infinitely many cusp forms. We also formulate a conjecture about the meaning of the absolute value squared of the triple product which is reminiscent of Watson's identity.