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Bi-extensions associated to divisors on abelian varieties and theta functions. (English) Zbl 0576.14043
The purpose of this paper is to construct a new purely algebraic theory of theta functions over a field of characteristic $$p\geq 0$$. Let A be an abelian variety over the field $${\mathbb{C}}$$ of complex numbers, g a (meromorphic) theta function belonging to A, and $$x_ i$$ $$(i=1,2,3)$$ the coordinate variables on 3 copies of the universal covering space $$V_ A$$ of A. If we put $$(*)\quad F(x_ 1,x_ 2,x_ 3)=g(x_ 1+x_ 2+x_ 3)g(x_ 1)g(x_ 2)g(x_ 3)/g(x_ 1+x_ 2)g(x_ 2+x_ 3\quad)g(x_ 3+x_ 1),$$ then F is a rational function on $$A\times A\times A$$, which is determined (up to a constant factor) by the divisor X of g on A. On the other hand even in the case of characteristic p$$>0$$, given an abelian variety A and a divisor X on it, a function F on $$A\times A\times A$$ is defined by X in the above sense. On the authors’ standpoint the problem is: for A find a k(A)-algebra $${\mathcal C}_ A$$, functorial with respect to A, such that for each divisor X on A (hence, for F) the equation (*) has a solution g in it. Two kinds of solutions are given to this problem: the theta functions on the Barsotti-Tate group, and those on the Tate space.
So far an abelian variety A was given first. In the last two sections 7, 8, abstract theta functions are discussed in the above two ways and using them the authors construct an (abelian) variety.
Reviewer: S.Koizumi

##### MSC:
 14K25 Theta functions and abelian varieties 14C20 Divisors, linear systems, invertible sheaves
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##### References:
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