Endomorphism fields of abelian varieties
نویسندگان
چکیده
We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato–Tate group of the abelian variety, which had been proved for abelian surfaces by Fité–Kedlaya–Rotger–Sutherland. The proof uses Minkowski’s reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.
منابع مشابه
Cycles in the De Rham Cohomology of Abelian Varieties over Number Fields
In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of `-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper...
متن کاملGroups of Rational Points on Abelian Varieties over Finite Fields
Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of rational points on varieties from this class in terms of Newton polygons of fA(1− t).
متن کاملGroups of Points on Abelian Varieties over Finite Fields
Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of k-rational points on varieties from this class in terms of Newton polygons of fA(1− t).
متن کاملOn the Structure of Weil Restrictions of Abelian Varieties
We give a description of endomorphism rings of Weil restrictions of abelian varieties with respect to finite Galois extensions of fields. The results are applied to study the isogeny decompositions of Weil restrictions. 2000 Mathematics Subject Classification Primary: 14K15, Secondary: 11G10.
متن کاملLog Abelian Varieties over a Log Point
We study (weak) log abelian varieties with constant degeneration in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In particular, we prove the dual short exact sequence for isogenies, Poincaré complete reducibility theorem for log abelian varieties, and the semisimplicity of the endomorphism algebras of log abelian varieti...
متن کاملSplit Reductions of Simple Abelian Varieties
Consider an absolutely simple abelian variety X over a number field K. We show that if the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then Xp is absolutely simple for almost all primes p. Conversely, if the absolute endomorphism ring of X is noncommutative, then Xp is reducible for p in a set of positive density. An absolutely simple abelian variety ...
متن کامل