Modular units from quotients of Rogers-Ramanujan type $q$-series

TL;DR

This paper introduces a new family of modular units derived from Hall-Littlewood $q$-series related to Rogers-Ramanujan identities, analyzing their algebraic properties and class field generation.

Contribution

It presents a novel family of modular units from Hall-Littlewood $q$-series and establishes their algebraic and class field properties, extending Folsom's work on modular units.

Findings

The new units form a subgroup of the cyclotomic units with the same rank.

Their specializations generate the same class fields as Folsom's units.

The rank of the subgroup generated by these units is explicitly determined.

Abstract

In [4] and [5], Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan $q$-continued fraction. These units are constructed from analytic solutions to the higher-order $q$-recurrence equations of Selberg. Here, we consider another family of modular units, which are quotients of Hall-Littlewood $q$-series that appear in the generalized Rogers-Ramanujan type identities of [6]. In analogy with the results of Folsom, we provide a formula for the rank of the subgroup these units generate and show that their specializations at the cusp $0$ generate a subgroup of the cyclotomic unit group of the same rank. In addition, we prove that their singular values generate the same class fields as those of Folsom's units.