  We prove a subconvexity bound for the central value L(1/2, chi) of a
Dirichlet L-function of a character chi to a prime power modulus q=p^n of the
form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx
0.1645 < 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.
