Annals of Combinatorics | 2019
Richaud–Degert Real Quadratic Fields and Maass Waveforms
Abstract
In this paper, we place the work of Andrews et al. (Invent Math 91(3):391–407, 1988) and Cohen (Invent Math 91(3):409–422, 1988), relating arithmetic in \\({{\\mathbb {Q}}}(\\sqrt{6})\\) to modularity of Ramanujan’s function \\(\\sigma (q)\\), in the context of the general family of Richaud–Degert real quadratic fields \\({{\\mathbb {Q}}}(\\sqrt{2p})\\). Moreover, we give the resulting generalizations of the function \\(\\sigma \\) as indefinite theta functions and invoke Zwegers’ work, (Q J Math 63(3):753–770, 2012), to prove the modular properties of the completed functions.