$p$-adic properties of Maass forms arising from theta series

We investigate arithmetic properties of the Fourier coefficients of certain harmonic weak Maass forms of weight $1/2$ and $3/2$. Each of the forms in question is the sum of a holomorphic function and a period integral of a theta series. In particular, for any positive integer $M$ coprime to $6$ we prove that the coefficients of the holomorphic function satisfy Ramanujan-type congruences modulo $M$, and establish sufficient conditions under which they are well-distributed modulo $\ell^j$ for primes $\ell \geq 5$. As an example we show that our results apply to Ramanujan’s mock theta function $\omega(q)$.

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Published 1 January 2008