12,067 research outputs found
Isogeny graphs of ordinary abelian varieties
Fix a prime number . Graphs of isogenies of degree a power of
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 -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
-isogenies: those whose kernels are maximal isotropic subgroups
of the -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
We give algorithms to compute isomorphism classes of ordinary abelian
varieties defined over a finite field whose characteristic
polynomial (of Frobenius) is square-free and of abelian varieties defined over
the prime field 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
We study a geometric analogue of the Iwasawa Main Conjecture for abelian
varieties in the two following cases: constant ordinary abelian varieties over
-extensions of function fields () ramified at a finite set of
places, and semistable abelian varieties over the arithmetic -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 and its dual abelian variety
. 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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