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arXiv · 2502.18727

Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect

Tengyou Zhu
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Abstract

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $\chi$ be a primitive Dirichlet character of prime power conductor $q=p^n$. For any given $\varepsilon>0$, we establish the following subconvexity bound \begin{equation*} L(1/2,f\otimes g \otimes \chi)\ll_{f,g,\varepsilon}q^{9/10+\varepsilon}. \end{equation*} The proof employs the DFI circle method with standard manipulations, including the conductor-lowering mechanism, Voronoi summation, and Cauchy--Schwarz inequality. The key input is certain estimates on the resulting character sums, obtained using the $p$-adic version of the van der Corput method.

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Tengyou Zhu. 2025-02-26. Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect. https://arxiv.org/abs/2502.18727

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