Exponential sums in the depth aspect

Many of the principal analytic questions about $L$-functions, such as the subconvexity estimates, moment evaluations, and nonvanishing of their critical values, at their core rely on estimates of associated exponential sums. In this talk, we will present new estimates for short exponential sums with phases involving $p$-adically analytic fluctuations. As applications, we obtain subconvexity bounds for Dirichlet and twisted modular $L$-functions with characters to a high prime power modulus, which are as strong as those available in the $t$-aspect. 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. Among the tools, we develop $p$-adic counterparts to Farey dissection, the circle method, and van der Corput estimates. Some of the results are joint work with V. Blomer.