6,760 research outputs found
Explicit CM-theory for level 2-structures on abelian surfaces
For a complex abelian variety with endomorphism ring isomorphic to the
maximal order in a quartic CM-field , the Igusa invariants generate an abelian extension of the reflex field of . In
this paper we give an explicit description of the Galois action of the class
group of this reflex field on . We give a geometric
description which can be expressed by maps between various Siegel modular
varieties. We can explicitly compute this action for ideals of small norm, and
this allows us to improve the CRT method for computing Igusa class polynomials.
Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby
implying that the `isogeny volcano' algorithm to compute endomorphism rings of
ordinary elliptic curves over finite fields does not have a straightforward
generalization to computing endomorphism rings of abelian surfaces over finite
fields
On the Picard group of a Delsarte surface
We suggest an algorithm computing, in some cases, an explicit generating set
for the N\'eron--Severi lattice of a Delsarte surface
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
A CM construction for curves of genus 2 with p-rank 1
We construct Weil numbers corresponding to genus-2 curves with -rank 1
over the finite field \F_{p^2} of elements. The corresponding curves
can be constructed using explicit CM constructions. In one of our algorithms,
the group of \F_{p^2}-valued points of the Jacobian has prime order, while
another allows for a prescribed embedding degree with respect to a subgroup of
prescribed order. The curves are defined over \F_{p^2} out of necessity: we
show that curves of -rank 1 over \F_p for large cannot be efficiently
constructed using explicit CM constructions.Comment: 19 page
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