p-ADIC PROPERTIES OF MODULAR SHIFTED CONVOLUTION DIRICHLET SERIES

Ho stein and Hulse recently introduced the notion of shifted convolution Dirichlet series for pairs of modular forms f1 and f2. 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 f1 = f2 where, in addition, there is a prime p for which p 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.