On the conversion of module representations for higher dimensional supersingular isogenies

par Julien Soumier

jeudi 11 juin 2026, 14:00 → 19:00 Europe/Paris

The Deuring correspondence between supersingular elliptic
curves and quaternionic ideals is a major tool to design efficient
algorithms for working with one dimensional abelian varieties of known
endomorphism ring. The module correspondence can be seen as a new
framework extending this correspondence in higher dimension. It is
ruled by the same idea that, given some extra information about
morphisms from an abelian variety, we can reduce hard problems to
linear algebra over a non commutative ring.

In this talk we present our results from a joint work with Aurel Page
and Damien Robert. After recalling the basis of the module
correspondence, we will describe our solution to the Principal Ideal
Problem in matrix spaces over quaternionic orders, along with an
algorithm that computes the principally polarized abelian variety
associated with an Hermitian module. We will also highlight that these
algorithms enable the computation of unpolarized isomorphisms between
maximal abelian varieties, given their module representation.