One of the principal analytic questions about automorphic L-functions are the so-called subconvex estimates on the size of their critical values, deeply arithmetic both in proofs and in the often spectacular consequences. In this talk, we will present our recent subconvexity bound for the central value of the L-function associated to a fixed cuspidal newform f twisted by a Dirichlet character chi of a high prime power conductor. From an adelic viewpoint, the analogy between this so-called "depth aspect" and the familiar t-aspect is particularly natural, as one is focusing on ramification at one (finite or infinite) place at a time. We prove our results by exhibiting strong cancellation between the Hecke eigenvalues of f and the values of chi, which act as twists by exponentials with a p-adically analytic phase. Among the tools, we develop p-adic counterparts to Farey dissection and van der Corput estimates for exponential sums. This is joint work with Valentin Blomer. Speaker(s): Djordje Milicevic (Bryn Mawr College)