Published online by Cambridge University Press: 03 November 2015
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.
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