Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913707546640384 |
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| author | Zhu, Tengyou |
| author_facet | Zhu, Tengyou |
| contents | Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $χ$ 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 χ)\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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect Zhu, Tengyou Number Theory Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $χ$ 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 χ)\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. |
| title | Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.18727 |