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An Isaac Newton Institute Workshop

Trends in Noncommutative Geometry

Homogeneous coordinate rings for noncommutative tori and modular functions

18th December 2006

Author: Jorge Plazas (IHES)

Abstract

We analyze the homogeneous coordinate rings of real multiplication noncommutative tori defined by Polishchuk. Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving an explicit presentation of these rings in terms of their natural generators. The appearance of theta functions in these computations provides information about the rationality of the homogeneous coordinate rings. We use the modularity of these rings to obtain algebras which do no depend on the choice of a complex structure on the noncommutative torus. These algebras are obtained by an averaging process over (limiting) modular symbols.