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Sub-Weyl subconvexity for Dirichlet $L$-functions to prime power moduli

Part of: Zeta and $L$-functions: analytic theory Real functions: Miscellaneous topics Exponential sums and character sums Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  03 November 2015

Djordje Milićević
Djordje Milićević*
Affiliation:
Bryn Mawr College, Department of Mathematics, 101 North Merion Avenue, Bryn Mawr, PA 19010, USA email [email protected]
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Abstract

We prove a subconvexity bound for the central value $L(\frac{1}{2},{\it\chi})$ of a Dirichlet $L$-function of a character ${\it\chi}$ to a prime power modulus $q=p^{n}$ of the form $L(\frac{1}{2},{\it\chi})\ll p^{r}q^{{\it\theta}+{\it\epsilon}}$ with a fixed $r$ and ${\it\theta}\approx 0.1645<\frac{1}{6}$, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving $p$-adically analytic phases, which can be naturally seen as a $p$-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.

Keywords

L-functionssubconvexityvan der Corput’s methodp-adic analysisexponential sumscharacter sumsdepth aspectmethod of stationary phaseexponent pairs

MSC classification

Primary: 11L07: Estimates on exponential sums 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11L03: Trigonometric and exponential sums, general 11L26: Sums over arbitrary intervals 11L40: Estimates on character sums 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 26E30: Non-Archimedean analysis
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 4 , April 2016 , pp. 825 - 875
Copyright
© The Author 2015 

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