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arXiv 2303.09646math.NT
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Rankin-Selberg $L$-函数的$GL(1)$扭的次凸性

Subconvexity for $GL(1)$ twists of Rankin-Selberg $L$-functions

Aritra Ghosh

备注：
18 pages. arXiv admin note: text overlap with arXiv:2111.00696
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AI总结：

本文研究$SL(2,\mathbb{Z})$的本原尖点形式对应的带Dirichlet特征扭的Rankin-Selberg $L$-函数中心值次凸界，通过约化为移位卷积和问题，将指数界从$1-1/1324$改进至$1-1/28$。

摘要：

设$f$和$g$是$SL(2,\bZ)$的两个全纯或Hecke-Maass本原尖点形式，$\b$是模为奇素数$p$的本原Dirichlet特征。本文给出Rankin-Selberg $L$-函数中心值$L(s, f \otimes g \otimes \chi)$的次凸界：对任意$\varepsilon>0$，有$$L(\frac{1}{2}, f \otimes g \otimes \chi) \ll_{f,g,\varepsilon}p^{\frac{27}{28}+\varepsilon},$$其中隐含常数仅依赖于形式$f,g$和$\varepsilon$。此前凸性界的指数为$1+\varepsilon$，后被改进至$1-\frac{1}{1324}$（见\cite{HM}），本文的界将其进一步降至$1- \frac{1}{28}$。研究的核心方法是将原问题约化为$GL(2) \times GL(2)$移位卷积和问题。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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