Anticyclotomic p-adic L-functions and Ichino's formula

 I will discuss the "anticyclotomic p-adic L-function" of Bertolini-Darmon-Prasanna, starting with an overview of what it is, some important properties it has, and how it can be used to obtain arithmetic results about rank-1 elliptic curves. I'll then talk about my work giving a new construction of this p-adic L-function, using an automorphic triple-product formula due to Ichino. I'll explain why this new construction is useful for studying the p-adic L-function (particularly in the residually reducible case) and talk about my work in progress on applications to Diophantine questions.