Number Theory Seminar

The well known Jacobi theta function has relatively unexplored generalizations to corresponding functions on the $n$-fold cover of $GL(r)$. I will describe how, in joint work with Junehyuk Jung and Min Lee, we prove that there are infinitely many Maass-Hecke cusp forms over the field $\Q(\sqrt{-3})$ 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 if a certain triple product integral involving the cusp form and the cubic theta function on $\Q(\sqrt{-3})$ 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 this triple product, which is reminiscent of Watson's identity. In addition, I will discuss some of the remarkable and little understood properties of more general theta functions on the $n$-fold cover of $GL(2)$ with $n \ge 4$.