Sub-Weyl subconvexity and short p-adic exponential sums

One of the principal questions about L-functions is the size of their  critical values. In this talk, we will
present a new subconvexity  bound for the central value of a Dirichlet L-function of a character  to a
prime power modulus, which breaks a long-standing barrier known  as the Weyl exponent. We obtain
our results by developing a new  general method to estimate short exponential sums involving p-adically 
analytic fluctuations, which can be naturally seen as a p-adic  analogue of the method of exponent pairs.
We will present the main  results of this method and the key points in its development, and  discuss the
structural relationship between the p-adic analysis and  the depth aspect.