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Burgess-like subconvexity for $\text{GL}_{1}$

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  04 July 2019

Han Wu
Han Wu*
Affiliation:
MA C3 604, EPFL SB MATHGEOM TAN, CH-1015, Lausanne, Switzerland email wuhan1121@yahoo.com
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Abstract

We generalize our previous method on the subconvexity problem for $\text{GL}_{2}\times \text{GL}_{1}$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, that is, the bound $|L(1/2,\unicode[STIX]{x1D712})|\ll _{\mathbf{F},\unicode[STIX]{x1D716}}\mathbf{C}(\unicode[STIX]{x1D712})^{1/4-(1-2\unicode[STIX]{x1D703})/16+\unicode[STIX]{x1D716}}$ for varying Hecke characters $\unicode[STIX]{x1D712}$ over a number field $\mathbf{F}$ with analytic conductor $\mathbf{C}(\unicode[STIX]{x1D712})$. As a main tool, we apply the extended theory of regularized integrals due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.

Keywords

subconvexityL-function over number fieldBurgess-type hybrid bound

MSC classification

Primary: 11R42: Zeta functions and $L$-functions of number fields
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 8 , August 2019 , pp. 1457 - 1499
Copyright
© The Author 2019 

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Footnotes

Research partially supported by SNF-grant 200021-125291 and DFG-SNF-grant 00021L_153647.

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