$p$-adic properties of modular shifted convolution Dirichlet series

TL;DR

This paper explores the $p$-adic properties of shifted convolution Dirichlet series for modular forms, showing that certain generating functions are linear combinations of $p$-adic modular forms, revealing deep $p$-adic structure.

Contribution

It establishes that for specific cases, the generating functions are linear combinations of at most two weight 2 weakly holomorphic $p$-adic modular forms, advancing understanding of their $p$-adic properties.

Findings

Generated functions are linear combinations of mixed mock modular forms and quasimodular forms.

In special cases, these functions are linear combinations of at most two weight 2 $p$-adic modular forms.

Provides new insights into the $p$-adic structure of shifted convolution Dirichlet series.

Abstract

Hoffstein and Hulse recently introduced the notion of shifted convolution Dirichlet series for pairs of modular forms $f_1$ and $f_2$. The second two authors investigated certain special values of symmetrized sums of such functions, numbers which are generally expected to be mysterious transcendental numbers. They proved that the generating functions of these values in $h$-aspect are linear combinations of mixed mock modular forms and quasimodular forms. Here we examine the special cases when $f_1=f_2$ where, in addition, there is a prime $p$ for which $p^2$ divides the level. We prove that the mixed mock modular form is a linear combination of at most two weight 2 weakly holomorphic $p$-adic modular forms.