Subconvexity for $GL(1)$ twists of Rankin-Selberg $L$-functions
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915108156866560 |
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| author | Ghosh, Aritra |
| author_facet | Ghosh, Aritra |
| contents | Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $χ$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg $L$-functions is $L(s, f \otimes g \otimes χ)$ is given by $$L(\frac{1}{2}, f \otimes g \otimes χ) \ll_{f,g,ε}p^{\frac{27}{28}+ε} ,$$ for any $ε> 0$, where the implied constant depends only on the forms $f,g$ and $ε$. Here the convexity bound has exponent $1+ε$, which was improved to $1-\frac{1}{1324}$ (see \cite{HM}). Our bound reduces it further to $1- \frac{1}{28}$. The main ingredients is to reduce the original problem to a $GL(2) \times GL(2)$ shifted convolution sum problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09646 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Subconvexity for $GL(1)$ twists of Rankin-Selberg $L$-functions Ghosh, Aritra Number Theory Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $χ$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg $L$-functions is $L(s, f \otimes g \otimes χ)$ is given by $$L(\frac{1}{2}, f \otimes g \otimes χ) \ll_{f,g,ε}p^{\frac{27}{28}+ε} ,$$ for any $ε> 0$, where the implied constant depends only on the forms $f,g$ and $ε$. Here the convexity bound has exponent $1+ε$, which was improved to $1-\frac{1}{1324}$ (see \cite{HM}). Our bound reduces it further to $1- \frac{1}{28}$. The main ingredients is to reduce the original problem to a $GL(2) \times GL(2)$ shifted convolution sum problem. |
| title | Subconvexity for $GL(1)$ twists of Rankin-Selberg $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2303.09646 |