On a hybrid bound for twisted \(L\)-values
type: Item
instance of: scholarly article
title: On a hybrid bound for twisted \(L\)-values
zbMATH Open document ID: 1270.11054
author: Ritabrata Munshi
published in: Archiv der Mathematik
publication date: 2011-05-11T00:00:00Z
review text: Let \(f \in S_k(M, \psi)\) be a holomorphic newform of weight \(k\), level \(M\) and character \(\psi\). Let \(\chi\) be a primitive character of conductor \(q\). The paper proves the hybrid subconvex bound \[ L(1/2 + it, f \otimes \chi) \ll_{f} (q(1+|t|))^{\frac{1}{2} - \frac{1}{118} + \varepsilon}. \] The result is not new. A stronger result (also valid for general \(\psi\)) has been proved by \textit{G. Harcos} and the reviewer [J. Reine Angew. Math. 621, 53--79 (2008; Zbl 1193.11044); Addendum to appear in Crelle's journal], and the currently strongest and most general result on the market is due to \textit{Han Wu} [``Burgess-like subconvex bounds for \(\mathrm{GL}_2 \times mathrm{GL}_1\)'', \url{arXiv:1209.5950}]. Unlike the two aforementioned papers the present paper does not use families and amplification, but proves directly cancellation in the approximate functional equation. To this end the author writes \(f\) as a linear combination of Poincaré series (hence the proof works only for holomorphic forms), applies Poisson summation and the bilinear estimate for Kloosterman fractions of \textit{W. Duke, J. Friedlander} and \textit{H. Iwaniec} [Invent. Math. 128, No. 1, 23--43 (1997; Zbl 0873.11050)]. The author makes the interesting remark that the best possible bound for these bilinear forms would lead to a hybrid Weyl-type exponent.
reviewed by: Valentin Blomer
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 5886763
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: subconvexity
zbMATH Keywords: hybrid bound
MaRDI profile type: Publication
full work available at URL: s00013-011-0227-4
OpenAlex ID: W2045449047
cites work: Rankin-Selberg \(L\)-functions on the critical line
cites work: Hybrid bounds for twisted L-functions
cites work: A trace formula for the scalar product of Hecke series and its applications
cites work: Bilinear forms with Kloosterman fractions
cites work: The subconvexity problem for Artin \(L\)-functions
cites work: Q4830109
cites work: Uniform bound for Hecke \(L\)-functions
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DOI: 10.1007/S00013-011-0227-4
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description: scientific article; zbMATH DE number 5886763