Restriction of Sections for Families of Abelian Varieties
نویسنده
چکیده
Given a family of Abelian varieties over a positive-dimensional base, we prove that for a sufficiently general curve in the base, every rational section of the family over the curve is contained in a unique rational section over the entire base. 1. Main results sec-intro The starting point for this article is the following theorem. Theorem 1.1. [GHMS05, Theorem 6.2] Let B be a smooth, quasi-projective, thm-GHMS complex variety, let A→ B be an Abelian scheme over B (i.e. a family of Abelian varieties over B), and let π : T → B be a torsor for A → B. Then π is a trivial torsor if and only if for every curve C ⊂ B, the restriction TC → C is a trivial torsor for C ×B A→ C. Stated more succinctly, the following map of étale cohomology groups is injective
منابع مشابه
The Spectral Construction for a (1,8)-Polarized Fam- ily of Abelian Varieties
We extend the Spectral Construction, a technique used with great success to study and construct vector bundles on elliptically fibered varieties, to a special family of abelian surface fibered varieties. The results are motivated by requirements from Heterotic String Phenomenology where vector bundles with specified chern class are required to produce a realistic particle spectrum. Although onl...
متن کاملCycles 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 کاملPolarizations of Prym Varieties of Pairs of Coverings
To any pair of coverings fi : X → Xi, i = 1, 2 of smooth projective curves one can associate an abelian subvariety of the Jacobian JX , the Prym variety P (f1, f2) of the pair (f1, f2). In some cases we can compute the type of the restriction of the canonical principal polarization of JX . We obtain 2 families of Prym-Tyurin varieties of exponent 6.
متن کاملDefinability of Restricted Theta Functions and Families of Abelian Varieties
We consider some classical maps from the theory of abelian varieties and their moduli spaces, and prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety Sp(2g,Z)\Hg is definable in Ran,exp when restricted to Siegel’s fundamental set Fg . We also prov...
متن کاملFour-Dimensional GLV via the Weil Restriction
The Gallant-Lambert-Vanstone (GLV) algorithm uses efficiently computable endomorphisms to accelerate the computation of scalar multiplication of points on an abelian variety. Freeman and Satoh proposed for cryptographic use two families of genus 2 curves defined over Fp which have the property that the corresponding Jacobians are (2, 2)isogenous over an extension field to a product of elliptic ...
متن کامل