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arXiv · 2109.10434

Non-vanishing of symmetric cube $L$-functions

Jeff Hoffstein·Junehyuk Jung·Min Lee
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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.

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Jeff Hoffstein, Junehyuk Jung, Min Lee. 2021-09-21. Non-vanishing of symmetric cube $L$-functions. https://arxiv.org/abs/2109.10434

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