12,067 research outputs found

    Isogeny graphs of ordinary abelian varieties

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    Fix a prime number â„“\ell. Graphs of isogenies of degree a power of â„“\ell are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called l\mathfrak l-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as (â„“,â„“)(\ell, \ell)-isogenies: those whose kernels are maximal isotropic subgroups of the â„“\ell-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure

    Computing square-free polarized abelian varieties over finite fields

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    We give algorithms to compute isomorphism classes of ordinary abelian varieties defined over a finite field Fq\mathbb{F}_q whose characteristic polynomial (of Frobenius) is square-free and of abelian varieties defined over the prime field Fp\mathbb{F}_p whose characteristic polynomial is square-free and does not have real roots. In the ordinary case we are also able to compute the polarizations and the group of automorphisms (of the polarized variety) and, when the polarization is principal, the period matrix.Comment: accepted by Math. Comp. major revision: added computation of the group of points; examples have been exported on the rep

    On the Iwasawa Main conjecture of abelian varieties over function fields

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    We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over ZpdZ_p^d-extensions of function fields (d≥1d\geq 1) ramified at a finite set of places, and semistable abelian varieties over the arithmetic ZpZ_p-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of AA and its dual abelian variety AtA^t. This holds as well over number fields and is a consequence of a quite general algebraic functional equation.Comment: 80 pages; many relevant changes all over the paper from v1. Among the most significant ones: new introduction; proof of the functional equation for Gamma systems in more cases and some applications to CM abelian varietie
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