Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Isogeny graphs of ordinary abelian varieties   Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$
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 $\mathfrak l$-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
$(\ell, \ell)$-isogenies: those whose kernels are maximal isotropic subgroups
of the $\ell$-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.

29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio