Talks
Talks
Marco Antei, University of Applied Sciences, Lucern
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AbstractAmnon Besser, Ben Gurion
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AbstractElira 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.
Ron Donagi, Department of Mathematics, Department of Mathematics, University of Pennsylvania
Isogenies for curves of higher genus
Abstract: Computation of Gauss's arithmetic-geometric mean involves iteration of a simple step, whose algebro-geometric interpretation is the construction of an elliptic curve isogenous to a given one, specifically one whose period is double the original period. A higher genus analogue should involve the explicit construction of a curve whose jacobian is isogenous to the jacobian of a given curve. The doubling of the period matrix means that the kernel of the isogeny should be a lagrangian subgroup of the group of points of order 2 in the jacobian. In genus 2 such a construction was given classically by Humbert and was studied more recently by Bost and Mestre. In this article we give such a construction for general curves of genus 3. We also give a similar but simpler construction for hyperelliptic curves of genus 3. We show that the hyperelliptic construction is a degeneration of the general one, and we prove that the kernel of the induced isogeny on jacobians is a lagrangian subgroup of the points of order 2. We show that for g at least 4 no similar construction exists, and we also reinterpret the genus 2 case in our setup. Our construction of these correspondences uses the bigonal and the trigonal constructions, familiar in the theory of Prym varieties.
Eyal Goren, Department of Mathematics, 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.
Yasuhiro Goto, Department of Mathematics, Hokkaido University of Education
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Abstract:ÂBorys Kadets, Department of Mathematics, The 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.Danny Neftin, Israel
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AbstractMichael Schein, Bar Ilan University
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AbstractTony Shaska, Department of Mathematics, 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.Ari Schindman, Hebrew University
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AbstractAlexei Skorobogatov, Imperial College
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AbstractShaul Zemel, Department of Mathematics, Hebrew University
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