1 results
Sub-Weyl subconvexity for Dirichlet
$L$-functions to prime power moduli
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 4 / April 2016
- Published online by Cambridge University Press:
- 03 November 2015, pp. 825-875
- Print publication:
- April 2016
-
- Article
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- You have access
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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.
