Refinements to Mumford’s theta and adelic theta groups

Abstract

Let \(X\) be an abelian variety defined over an algebraically closed field \(k\). We consider theta groups associated to simple, semi-homogenous vector bundles of separable type on \(X\). We determine the structure and representation theory of these groups. In doing so we relate work of Mumford, Mukai, and Umemura. We also consider adelic theta groups associated to line bundles on \(X\). After reviewing Mumford’s construction of these groups we determine functorial properties which they enjoy and then realize the Neron–Severi group of \(X\) as a subgroup of the cohomology group \(\mathrm {H}^2(\mathrm {V}(X), k^{\times })\).