Understanding the non-vanishing of central values of L-functions is an important and notoriously hard problem in number theory. Moment methods from analytic number theory propose a general approach which leads to deep questions in exponential sums which unfortunately is out of reach for many natural families of L-functions. In this talk I will focus on ‘horizontal families’ given by twisting a fixed automorphic form by Dirichlet characters of square-free conductor (and subfamilies thereof). I will explain how in some cases one can use various regularity properties of automorphic periods to get new non-vanishing result using both archimedean and p-adic methods. This is partly based on joint work with Daniel Kriz.