Michael Griffin

Affiliation: Princeton University

Email: mjg4@math.princeton.edu

Title Of Talk: $p$-adic harmonic Maass forms

Abstract: Harmonic Maass forms posses many intricate $p$-adic properties. Guerzhoy, Kent, and Ono, and others have extensively studied the $p$-adic properties of the $q$-series of related mock modular forms. Special values of Harmonic Maass forms also possess similarly interesting $p$-adic properties.

In his doctoral thesis, Candelori investigated a theory of integer weight $p$-harmonic Maass forms arising from the de Rham cohomology for $p$-adic modular forms. We consider a similar theory, although our approach and definitions differ somewhat from Candelori's. We construct $p$-adic analogues of classical harmonic Maass forms of weight $0$ and $1/2$ with square free level by means of the Hecke algebra. As in the classical case these forms are connected to positive weight modular forms by certain differential operators. Moreover, the coefficients of the half integral weight forms can be given as modular traces of weight $0$ forms over Heegner divisors. The complex harmonic Maass forms and their corresponding $p$-adic analogues may also be collected into an adelic theory.

As an application, we consider elliptic curves $E/\mathbb{Q}$ with square free conductor. Building on work of Bruinier and Ono, we construct a function $H_E'$, whose vanishing at Heegner points determines the vanishing of central $L$ derivatives of quadratic twists of $E$.

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