Objectives
The research efforts of the MELODIA project will be articulated around the five objectives below:

Supersingular elliptic curves.
The first objective of this project focuses on isogeny graphs of
supersingular elliptic curves over a finite field of
characteristic p. This family of graphs recently became the foundation of the rapidly growing field of
isogenybased cryptography. Isogenybased cryptosystems
rely on the presumed hardness of finding paths connecting two supersingular
elliptic curves in the isogeny graph. Variants of this problem are presumably
hard even for quantum algorithms, thereby allowing postquantum
cryptography, i.e. cryptography resistant to attacks with quantum computers.

Endomorphism rings.
Our second objective focuses on studying endomorphism rings of abelian
varieties of higher dimension in positive characteristic. In particular our
main goal is to generalize to abelian varieties recent results which, for the
simpler case of ordinary elliptic curves, establish an equivalence between
computing endomorphism rings and pathfinding problems in the isogeny graph.

Class polynomials.
We intend to use reductions of Hilbert class polynomials together with
canonical lifting theorems and the study of orders in
quaternion algebras to make the Deuring correspondence for supersingular
elliptic curves mentioned in Objective 1 fully explicit. This will provide a better
description of the structure of supersingular isogeny graphs.

Modular forms.
Evidence, in terms of concrete examples,
for the original BrumerKramer paramodularity conjecture
and the EichlerShimura conjecture in dimension 2
for imaginary quadratic fields consists so far
in just one abelian surface.
We will exploit the results of Objective 2
to obtain certified instances of abelian surfaces with endomorphism ring Z
and, thus, find other examples efficiently.

Group cohomology.
Using Quillenâ€™s results, Wendt proved that the
submodule of torsion elements in the cohomology of GL3 over the function
ring of an elliptic curve over a finite field is the kernel of the Quillen
homomorphism. We seek formulas for the structure of this module over the ring
of Chern classes when the elliptic curve is supersingular. For this purpose, we
will apply the knowledge gathered in Objective 3.