Title: Supersingular isogeny graphs, modular curves and Galois Representations.
| Speaker: | Leonardo Colo' |
| Affiliation: | University of Waterloo |
| Location: | MC 5501 |
Abstract: In this talk we will discuss the remarkable interconnection among supersingular elliptic curves, modular curves and Galois representations with a focus on cryptographic applications.
The study of supersingular isogeny graphs has come into vogue due to their applications to cryptography. This has resulted in advances in algorithms for their construction and study. The primary focus is on explicit algorithms for local transversal of an $\ell$-isogeny graph, beginning at a given supersingular point, under the hypothesis that the characteristic $p$ is large.
On the other hand, in the prior work of Mestre and Oesterl\'e, and of Pizer in the quaternion ideal setting, the motivation for studying isogeny graphs stems from the application to construction of modular forms and their associated Galois representation.
In this setting, one needs to study the global properties of the graphs, such that the supersingular points and associated $\ell$-isogeny graphs, can be enumerated.
The computation tools for cryptographic construction apply also to the investigation of the associated Galois representations,.
The method of graphs of Mestre interprets the adjacency operators of $\ell$-isogeny graphs as correspondences defining the Hecke operator $T_\ell$ acting on the basis of supersingular points. Mestre restricts to supersingular points, over $\FF_{p^2}$, on $X(1)$ or one of the genus-$0$ modular curves $X_0(N)$. In this setting, the use of explicit correspondences is en effective tool for computing $\ell$-isogenies, but becomes cumbersome for moderate $\ell$.
One goal of the present work is to generalize the modular approach to curves of higher level. This has several advantages and applications in isogeny based cryptography.
This is joint work with David Kohel (University of Marseille, France).