# Current

Unless stated otherwise, the seminar takes place on Tuesdays,
from 10 to 11, in room 385 at IMB. To get announcements, you can subscribe to the

mailing-list.

2017-05-30

10:00

Salle 385

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, in
arbitrary dimension, their structure is similar, but not identical, to
the ``volcanoesâ€™â€™ occurring as graphs of isogenies of elliptic curves.
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. These results lead to
new, provable algorithms to navigate in isogeny graphs, with
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.

2017-06-06

10:00

Salle 385

2017-06-13

10:00

Salle 385