Non‐vanishing of symmetric cube L$L$‐functions

Abstract

We prove that there are infinitely many Maass–Hecke cuspforms over the field Q [ − 3 ] $\mathbb {Q}[\sqrt {-3}]$ such that the corresponding symmetric cube L $L$ -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 if a certain triple product integral involving the cusp form and the cubic theta function on Q [ − 3 ] $\mathbb {Q}[\sqrt {-3}]$ does not vanish then the symmetric cube central value 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.