An Explicit Cohen-Style Threefield Identity

TL;DR

This paper constructs explicit formulas for a special $q$-series related to three quadratic fields, extending Cohen's identity and connecting it to partition generating functions.

Contribution

It provides a new explicit example of a Cohen-style threefield identity involving $ ext{Q}( oot{-6}{}), ext{Q}(i),$ and $ ext{Q}( oot{6}{}),$ using recent quadratic form theta series work.

Findings

Explicit formulas for the $q$-series in terms of quadratic form theta series.

Identification of an analogue of $\sigma$ with coefficients given by a partition generating function.

Extension of Cohen's identity to new quadratic fields.

Abstract

In 1988, Andrews, Dyson, and Hickerson showed that a $q$-series $\sigma$ found in Ramanujan's lost notebook and related to partitions could be interpreted as counting ideals in $\mathbb{Q}(\sqrt{6})$, and found similar formulas for $\sigma$ in terms of ideals of $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$. Cohen followed this by showing more generally that for certain triples of quadratic fields, there is abelian extension and conductor so that the ray class character theta series for all three fields coincide. In the intervening years, several $q$-series counting ideals in quadratic fields have been explored, nearly all relating to the fields explored by Andrews et al. In this paper we give an example of a ray class character theta series which stems from $\mathbb{Q}(\sqrt{-6})$, $\mathbb{Q}(i)$, and $\mathbb{Q}(\sqrt{6})$. We use recent work of Okano to give explicit formulas via quadratic form theta series with congruence conditions stemming from each field. We isolate an analogue of $\sigma$ for this series and show that it has coefficients also given by a partition generating function.