On the occurrence of congruence multiplicities between Ramanujan's theta functions

Abstract

Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions $φ(-q)$ and $ψ(q)$. In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for $φ(-q)$ and $ψ(q)$, which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups $Γ_0(6)$ and $Γ_0(10)$. In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.