𝑝-adic properties of modular shifted convolution Dirichlet series

Bringmann, Kathrin; Mertens, Michael; Ono, Ken

doi:10.1090/proc/12809

$p$-adic properties of modular shifted convolution Dirichlet series
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by Kathrin Bringmann, Michael H. Mertens and Ken Ono PDF

Proc. Amer. Math. Soc. 144 (2016), 1439-1451 Request permission

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 the $h$-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.

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