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Abelian variety. In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in ...
Arithmetic of abelian varieties. In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms ...
Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions.Just as elliptic curves have a natural moduli space, over characteristic 0 constructed as a quotient of the upper-half plane by the action of (), [1] there is an analogous construction for abelian varieties using the Siegel upper half-space and the symplectic group ().
Mordell–Weil group. In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety defined over a number field . It is an arithmetic invariant of the Abelian variety. It is simply the group of -points of , so is the Mordell–Weil group [1][2]pg 207. The main structure theorem about this group is the ...
On abelian varieties of higher dimension, there need not be a particular choice of smallest ample line bundle to be used in defining the Néron–Tate height, and the height used in the statement of the Birch–Swinnerton-Dyer conjecture is the Néron–Tate height associated to the Poincaré line bundle on ^, the product of with its dual.
Equations defining abelian varieties. In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss ...
Selmer group. In arithmetic geometry, the Selmer group, named in honor of the work of Ernst Sejersted Selmer ( 1951) by John William Scott Cassels ( 1962 ), is a group constructed from an isogeny of abelian varieties .
Complex multiplication of abelian varieties. In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative 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.