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We prove a subconvexity bound for the central value L(1/2, chi) of aDirichlet L-function of a character chi to a prime power modulus q=p^n of theform L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx0.1645 < 1/6, breaking the long-standing Weyl exponent barrier. In fact, wedevelop a general new theory of estimation of short exponential sums involvingp-adically analytic phases, which can be naturally seen as a p-adic analogue ofthe method of exponent pairs. This new method is presented in a ready-to-useform and applies to a wide class of well-behaved phases including many thatarise from a stationary phase analysis of hyper-Kloosterman and other completeexponential sums.

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