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Fonctions thêta et théorème du cube. (French) Zbl 0558.14029
Lecture Notes in Mathematics. 980. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag. XIII, 115 p. DM 19.80; $ 8.60 (1983).
The existence of canonical theta functions associated to divisors on an arbitrary commutative group scheme over an algebraically closed field is proved, using the theorem of cube (the cube structure) and the concept of biextensions. This generalizes the existence theorem of theta functions for G an Abelian variety by I. Barsotti [Sympos. Math., Roma, Vol. 3, 247-277 (1970; Zbl 0194.522)] and D. Mumford [On the equations defining Abelian varieties. I-III, Invent. Math. 1, 287-354 (1966); 3, 75-135 and 215-244 (1967; Zbl 0219.14024)].
The proof takes the point of departure on Barsotti’s theory on theta functions, and does not use Heisenberg groups and their representations that were used by Mumford; it uses the notion of biextensions introduced by D. Mumford in his later publication in Algebr. Geom., Bombay Colloquium 1968, 307-322 (1969; Zbl 0216.331).
Reviewer: N.Yui

14K25 Theta functions and abelian varieties
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14L15 Group schemes