Abelian varieties

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Abelian integral

In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. Suppose given a Riemann surface S and on it a Differential form O that is everywhere on S holomorphic, and fixing a point P on S from which to integrate. We can regard >P Q>O as a multi-valued function f(Q), or (better) an honest function of the chosen path C drawn on S from P to Q. Since S will in general be multiply-connected, one should specify C, but the value will in fact only depend on the homology class, of C modulo cycles on S...

Abelian variety of CM type

In mathematics, an abelian variety A defined over a field (mathematics) K is said to have CM-type if it has a large enough commutative ring subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century. One of the major achievements in algebraic number theory and algebraic geometry of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension of an algebraic variety d > 1. The..