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arXiv · 1805.00209

Quantitative non-vanishing of central values of certain $L$-functions on ${\rm GL}(2)\times {\rm GL}(3)$

Shingo Sugiyama·Masao Tsuzuki
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Abstract

Let $ϕ$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $ϕ$, $f$ and $\bar f$, where $f$ runs over an orthonormal basis $H_k$ of Hecke eigen elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$ of a fixed weight $k\geq 4$. As an application, we prove a quantitative non-vanishing results on the central values for the family of degree $6$ $L$-functions $L(s,ϕ\times {\rm Ad}\,f)$ with $f$ in the union of $H_k$ $({\rm K} \leq k < 2{\rm K})$ as ${\rm K}\rightarrow \infty$.

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Shingo Sugiyama, Masao Tsuzuki. 2021-07-23. Quantitative non-vanishing of central values of certain $L$-functions on ${\rm GL}(2)\times {\rm GL}(3)$. https://doi.org/10.1007/s00209-021-02886-5

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