Divisor problem, distribution of primes in arithmetic progressions, or, say, equidistribution of rational points on varieties - all cornerstone problems in analytic number theory - can see their complexity rise in several directions. “Level aspect”, in which the modulus is increasing, is central to arithmetic applications. The so-called “depth” and “smooth” aspects, where the modulus is highly powerful (such as a prime power) or well-factorable, respectively, have recently been understood to play a distinctive role, with tools often paralleling those available in the archimedean direction.
In this talk, we will discuss several manifestations of this phenomenon. In particular, we will present our recent 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 these 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. Natural analogues of the circle method and large sieve-type inequalities and their consequences for subconvexity and moments of $L$-functions (joint work with Blomer) will also be demonstrated.