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Refinements to Mumford’s theta and adelic theta groups

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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 \(\mathrm {H}^2(\mathrm {V}(X), k^{\times })\).

Résumé

Soit \(X\) une variété abélienne définie sur un corps algébriquement clos \(k\). Nous étudions des groupes thêta associés aux faisceaux localement libres sur \(X\) qui sont simples, semi-homogènes de type séparable. Nous déterminons la structure et la théorie des représentations de ces groupes. Il s’avère que nous relions des travaux de Mumford, Mukai, et Umemura. En plus, nous étudions des groupes thêta adéliques associés à des fibrés vectoriels de rang \(1\) sur \(X\). Après avoir examiné la construction de Mumford de ces groupes, nous déterminons certaines de leurs propriétés fonctorielles. Enfin nous montrons que le groupe de Néron-Severi de \(X\) peut s’identifier avec un sous-groupe du groupe de cohomologie \(\mathrm {H}^2(\mathrm {V}(X), k^\times )\).

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Notes

  1. Throughout we employ the (somewhat non-standard) terminology of [8]: if \(A\) and \(C\) are groups and \(1\rightarrow A \rightarrow B \rightarrow C \rightarrow 1\) is a short exact sequence of groups, then we say that \(B\) is an extension of \(A\) by \(C\).

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Acknowledgments

The author thanks his Ph.D. adviser Mike Roth for useful discussions. The author also acknowledges that the final writing of this work benefited from conversations with Eyal Goren, Jacques Hurtubise, and criticisms and suggestions given by the referee.

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Grieve, N. Refinements to Mumford’s theta and adelic theta groups. Ann. Math. Québec 38, 145–167 (2014). https://doi.org/10.1007/s40316-014-0024-0

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  • DOI: https://doi.org/10.1007/s40316-014-0024-0

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