(SUNY, Stony Brook)
Uniformizations and theta-constant identities
Abstract: One variable theta functions are useful tools in complex analysis, number theory and combinatorics. They provide examples of uniformizations of Riemann surfaces that can serve as models for a more general theory.
After motivating an approach to the study of one variable theta-functions through problems arising from Kleinian groups and Ahlfors' finiteness theorem, we will take a leisurely tour of an elliptic paradise and its number theoretic and combinatorial regions, including a discussion of multiplicative functions related to the Ramanujan tau-function. The emphasis will be on function theoretic derivations of theta-constant identities and their use to uniformize (compact) Riemann surfaces. If time permits we will discuss products of theta-constants that result in constant functions, identities not connected to Riemann surface theory, and infinite product expansions of theta-constant derivatives (generalizing the Jacobi derivative formula).