Locally harmonic Maass forms and central $L$-values

In this talk, we will discuss a relatively new modular-type object known as
a locally harmonic Maass form.
We will discuss recent joint work with Ehlen, Guerzhoy, and Kane with
applications to the theory of $L$-functions. In particular, we find
finite formulas for certain twisted central $L$-values of a family of
elliptic curves in terms of finite sums over canonical binary quadratic
forms. Applications to the congruent number problem will be given.