658 research outputs found
Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
Fix an ordinary abelian variety defined over a finite field. The ideal class
group of its endomorphism ring acts freely on the set of isogenous varieties
with same endomorphism ring, by complex multiplication. Any subgroup of the
class group, and generating set thereof, induces an isogeny graph on the orbit
of the variety for this subgroup. We compute (under the Generalized Riemann
Hypothesis) some bounds on the norms of prime ideals generating it, such that
the associated graph has good expansion properties.
We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and
Robert for computing explicit isogenies in genus 2, to prove random
self-reducibility of the discrete logarithm problem within the subclasses of
principally polarizable ordinary abelian surfaces with fixed endomorphism ring.
In addition, we remove the heuristics in the complexity analysis of an
algorithm of Galbraith for explicitly computing isogenies between two elliptic
curves in the same isogeny class, and extend it to a more general setting
including genus 2.Comment: 18 page
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
Generalised Fermat Hypermaps and Galois Orbits
We consider families of quasiplatonic Riemann surfaces characterised by the
fact that -- as in the case of Fermat curves of exponent -- their
underlying regular (Walsh) hypermap is the complete bipartite graph , where is an odd prime power. We will show that all these surfaces,
regarded as algebraic curves, are defined over abelian number fields. We will
determine the orbits under the action of the absolute Galois group, their
minimal fields of definition, and in some easier cases also their defining
equations. The paper relies on group-- and graph--theoretic results by G. A.
Jones, R. Nedela and M.\v{S}koviera about regular embeddings of the graphs
[JN\v{S}] and generalises the analogous question for maps treated in
[JStW], partly using different methods.Comment: 14 pages, new version with extended introduction, minor corrections
and updated reference
Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau categories
A decorated surface S is an oriented surface with punctures and a finite set
of marked points on the boundary, such that each boundary component has a
marked point. We introduce ideal bipartite graphs on S. Each of them is related
to a group G of type A, and gives rise to cluster coordinate systems on certain
spaces of G-local systems on S. These coordinate systems generalize the ones
assigned to ideal triangulations of S. A bipartite graph on S gives rise to a
quiver with a canonical potential. The latter determines a triangulated 3d CY
category with a cluster collection of spherical objects. Given an ideal
bipartite graph on S, we define an extension of the mapping class group of S
which acts by symmetries of the category. There is a family of open CY 3-folds
over the universal Hitchin base, whose intermediate Jacobians describe the
Hitchin system. We conjecture that the 3d CY category with cluster collection
is equivalent to a full subcategory of the Fukaya category of a generic
threefold of the family, equipped with a cluster collection of special
Lagrangian spheres. For SL(2) a substantial part of the story is already known
thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others. We
hope that ideal bipartite graphs provide special examples of the
Gaiotto-Moore-Neitzke spectral networks.Comment: 60 page
Extensions of the universal theta divisor
The Jacobian varieties of smooth curves fit together to form a family, the
universal Jacobian, over the moduli space of smooth marked curves, and the
theta divisors of these curves form a divisor in the universal Jacobian. In
this paper we describe how to extend these families over the moduli space of
stable marked curves (or rather an open subset thereof) using a stability
parameter. We then prove a wall-crossing formula describing how the theta
divisor varies with the stability parameter. We use that result to analyze a
divisor on the moduli space of smooth marked curves that has recently been
studied by Grushevsky-Zakharov, Hain and M\"uller. In particular, we compute
the pullback of the theta divisor studied in Alexeev's work on stable abelic
varieties and in Caporaso's work on theta divisors of compactified Jacobians.Comment: 42 pages, 5 figures. Final version. Added Section 4.1, which
describes how divisor classes other than the theta divisor var
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