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Number Theory Seminar

Thursday, October 20, 2022
4:00pm to 5:00pm
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Linde Hall 387
Theta functions on the $n$-fold metaplectic cover of $GL(2)$ and the non-vanishing at the center of the critical strip of symmetric cube $L$-series
Jeffrey Hoffstein, Department of Mathematics, Brown University,

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 $n \ge 4$.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].