Objectives

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

1. 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 isogeny-based cryptography. Isogeny-based 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 post-quantum cryptography, i.e. cryptography resistant to attacks with quantum computers.
2. 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 path-finding problems in the isogeny graph.
3. 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.
4. Modular forms. Evidence, in terms of concrete examples, for the original Brumer-Kramer paramodularity conjecture and the Eichler-Shimura 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.
5. Group cohomology. Using Quillen’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.