Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions
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| Format: | Preprint |
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2024
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| _version_ | 1866929634142060544 |
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| author | Aggarwal, Keshav Kumar, Sumit Kwan, Chung-Hang Leung, Wing Hong Li, Junxian Young, Matthew P. |
| author_facet | Aggarwal, Keshav Kumar, Sumit Kwan, Chung-Hang Leung, Wing Hong Li, Junxian Young, Matthew P. |
| contents | Let $f$ be a newform of prime level $p$ with any central character $χ\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\varepsilon$, and the archimedean parameter of $f$. This improves upon the previously best-known result by Harcos and Michel. Our method ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions Aggarwal, Keshav Kumar, Sumit Kwan, Chung-Hang Leung, Wing Hong Li, Junxian Young, Matthew P. Number Theory 11M41, 11F66 Let $f$ be a newform of prime level $p$ with any central character $χ\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\varepsilon$, and the archimedean parameter of $f$. This improves upon the previously best-known result by Harcos and Michel. Our method ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee. |
| title | Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions |
| topic | Number Theory 11M41, 11F66 |
| url | https://arxiv.org/abs/2412.12410 |