The elliptic gamma function I

Abstract: Recall that the exponential function is the classical example of a singly periodic entire function in one complex variable. Last talk, we learned about theta functions, which are entire functions in one complex parameter $z$ and one parameter $\tau$ in the upper half-plane, satisfying fascinating periodicity conditions (with respect to two periods) and also modular-like conditions. Special values of modifications of theta functions can produce units in abelian extensions of imaginary quadratic fields (for instance, see Rubin, 1999). These functions were generalized by (Ruijsenaars, 1994) to what are known as elliptic gamma functions, in one complex variable $z$ and two variables $\tau, \sigma$ in the upper half-plane; these functions were studied further by (Felder-Varchenko, 2000). The importance of the elliptic gamma function for us in STAGE comes from the conjectural relationship of special values of variants of this function to units in abelian extensions of complex cubic fields (Bergeron-Charollois-Garcia 2024), the subject of future talks.

In this talk, following (Felder-Varchenko, 2000), I will explain the theory of elliptic gamma functions, their periodicity conditions (with three periods, 1, $\tau$, and $\sigma$) involving a variant of the theta function, and significantly more complicated modular identities. I will also explain how these systems of identities can be understood in the context of "automorphic forms" associated to elements of certain cohomology groups of $\text{SL}_3(\mathbb{Z})\ltimes\Z^3$. Time permitting, I will also discuss the continuous extension of the elliptic gamma function to a region where $\tau$ and/or $\sigma$ lie on a subset of the real line satisfying a certain Diophantine approximation property.

Main reference: G. Felder, A. Varchenko, The Elliptic Gamma Function and $\text{SL}_3(\mathbb{Z})\ltimes\Z^3$, Adv. Math. 156 (2000), pp. 44-76

algebraic geometrynumber theory

Audience: advanced learners

---

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2026 topic: Explicit class field theory.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

---