Talks

Elira Curri, Computer Science and Engineering, Oakland University

Machine learning for moduli space of genus two curves and an application to post-quantum  cryptography

Abstract: We use machine learning to study the locus Ln of genus two curves with (n,n)-split Jacobian. More precisely we design a transformer model which given values for the Igusa invariants determines if the corresponding genus two curve is in the locus Ln, for n = 2,3,5,7. Such curves are important in isogeny based cryptography.

During this study we discover that there are no rational points p∈Ln with weighted moduli height ≤ 2 in any of L2, L3, and L5. This extends on previous work of the authors to use machine learning methods to study the moduli space of genus 2 algebraic curves.

Eyal Goren, McGill University

Supersingular elliptic curves, quaternion algebras and some applications to cryptography

Abstract: Part of the talk is expository: I will explain how supersingular isogeny graphs can be used to construct cryptographic hash functions and survey some of the rich mathematics involved. Then, with this motivation in mind, I will discuss two recent theorems by Jonathan Love and myself. The first concerns the generation by maximal orders by elements of particular norm. The second states that maximal orders of elliptic curves are determined by their theta functions.

Borys Kadets, Hebrew  University

Groups of points on abelian varieties

Abstract: I will describe some recent results on the group structure and point counts on abelian and Jacobian varieties over a small finite field. Despite the classical nature of the subject, there has been a lot of progress in the area in the past few years, which I will attempt to survey. The newest results of the talk are based on work in progress joint with Daniel Keliher.

Tony Shaska, Oakland University

Isogenies of Jacobian surfaces

Abstract: We construct a three-parameter family of nonhyperelliptic and bielliptic plane genus-three curves whose associated Prym variety is 2-isogenous to the Jacobian variety of a general hyperelliptic genus-two curve. Our construction is based on the existence of special elliptic fibrations with the section on the associated Kummer surfaces that provide a simple geometric interpretation for the rational double cover induced by the two-isogeny between the Abelian surfaces.