ncts

In this talk, we consider locally harmonic Maass forms, which are non-holomorphic modular forms that are harmonic in connected components modulo certain geodesics, along which there are log-type singularities. Such forms naturally split into local polynomial parts and two other pieces that naturally appear in the theory of harmonic Maass forms. A connection between the local polynomial parts and vanishing of central L-values of quadratic twists of classical modular forms is established. In the case of weight 0 locally harmonic Maass forms and weight 2 cusp forms, a result is obtained for quadratic twists of L-functions for elliptic curves. This results in a new condition for determining whether an integer is a congruent number or not. This talk is based on joint work with Stephan Ehlen, Pavel Guerzhoy, and Larry Rolen.