Refinements to Mumford’s theta and adelic theta groups

Abstract

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 H 2 (V(X); k × ).

Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

“…Let α ∈ End 0 λ (A) be the image of γ ∈ End 0 f * λ (B) under the natural homomorphism that is induced by f . Then, by Theorem 1.4, these above conditions on the class (6) translate into the assertion that either pNrd λ ((m − i) id A +α) = 0 (7) or the polynomial pNrd λ ((N + m − i) id A +α) (8) fails to have i positive roots (counted with multiplicities) for all i ∈ {1, . .…”

“…As some additional results, which are of an independent interest, we build on the works [6], [7] and [8], for example, which have origins in Mumford's index theorem for line bundles on Abelian varieties [17, p. 156]. In this regard, our main results are Theorem 6.2 and Corollary 1.5.…”

“…We refer to [7] for the theory of theta groups that are associated to simple semihomogeneous vector bundles and to [9], building on work of Brion [4], for the main results about homogeneous Severi-Brauer varieties.…”

On cubic torsors, biextensions and Severi-Brauer varieties over Abelian varieties

“…When S = Spec k and gcd(r, char k) = 1, Lagrangian subgroups exist, [12]. Further, in this context, the nature of higher weight representations of the nondegenerate theta groups is determined in [6].…”