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Biextensions alternées. (Alternating biextensions). (French) Zbl 0632.14036
Symmetric biextensions as a generalization of polarization for abelian varieties have been studied in the author’s book “Fonctions thêta et théorème du cube”, Lect. Notes Math. 980 (1983; Zbl 0558.14029). Here, antisymmetric and the subclass of alternating biextensions are introduced and compared. Alternating biextensions have, in analogy to symmetric biextensions and “cubic torsors”, as seen in the book cited above, certain additivity and descent properties. Hence, a result, which associates to an element in the Néron-Severi group of an abelian variety the corresponding Riemann form, generalizes to antisymmetric and alternating biextensions. Over \({\mathbb{C}}\) one can associate, using analytic tools, to an alternating biextension E a quadratic map \(f^ E\) and then reductions \(f^ E_ n\) \(modulo\quad n.\) The author shows that the \(f^ E_ n\) can be defined algebraically. Some results of K. Ribet are interpreted in terms of the introduced biextensions.
Reviewer: P.Cherenack

14K05 Algebraic theory of abelian varieties
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14E99 Birational geometry
Full Text: Numdam EuDML
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