Class invariants and cyclotomic unit groups from special values of modular units

Abstract

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order q-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which the structure of these modular unit groups and their associated cuspidal divisor class groups are also characterized, and a cuspidal divisor class number formula is given in terms of products of L-functions and compared to the classical relative class number formula within the cyclotomic fields [6, 7].