Mathematical Research Letters

Volume 17 (2010)

Number 2

Modular units and the $q$-difference equations of Selberg

Pages: 283 – 299



Amanda Folsom (University of California at Los Angeles)


In this paper we present a subgroup of modular units that arise naturally from analytic solutions to higher order $q$-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. Further, we express these modular units in terms of Siegel functions as considered by Kubert and Lang, and show they generate the group of all units of the modular curves $X(\ell)$ with cuspidal support on $\pi^{-1}(\infty)$, where $\pi:X(\ell)\to X_0(\ell)$ is the canonical projection.

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