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29 March 2024
 
  » arxiv » 2006.7819

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Sub-convexity bound for $GL(3) imes GL(2)$ $L$-functions: $GL(3)$-spectral aspect
Sumit Kumar ; Kummari Mallesham ; Saurabh Kumar Singh ;
Date 14 Jun 2020
AbstractLet $phi$ be a Hecke-Maass cusp form for $SL(3, mathbb{Z})$ with Langlands parameters $({f t}_{i})_{i=1}^{3}$ satisfying $$|{f t}_{3} - {f t}_{2}| leq T^{1-xi -epsilon}, quad , {f t}_{i} approx T, quad , , i=1,2,3$$ with $1/2 < xi <1$ and any $epsilon>0$. Let $f$ be a holomorphic or Maass Hecke eigenform for $SL(2,mathbb{Z})$. In this article, we prove a sub-convexity bound $$L(phi imes f, frac{1}{2}) ll max { T^{frac{3}{2}-frac{xi}{4}+epsilon} , T^{frac{3}{2}-frac{1-2 xi}{4}+epsilon} } $$
for the central values $L(phi imes f, frac{1}{2})$ of the Rankin-Selberg $L$-function of $phi$ and $f$, where the implied constants may depend on $f$ and $epsilon$.
Conditionally, we also obtain a subconvexity bound for $L(phi imes f, frac{1}{2})$ when the spectral parameters of $phi$ are in generic position, that is
$${f t}_{i} - {f t}_{j} approx T, quad , ext{for} , i eq j, quad , {f t}_{i} approx T , , , i=1,2,3.$$
Source arXiv, 2006.7819
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