Harun Kir

Understanding the nature of curves of genus 2 via quadratic forms

We investigate certain properties of (smooth) genus 2 curves. Our approach focuses on a special associated integral quadratic form, which is intrinsically attached to a curve C of genus 2. This form, known as the refined Humbert invariant, was introduced by Ernst Kani in 1994. One of the nice features of the refined Humbert invariant is its role to translate geometric questions into arithmetic ones. This property enables to solve various intriguing geometric problems related to genus 2 curves, such as determining their automorphism groups and finding their elliptic subcovers. After illustrating the usefulness of this invariant, we discuss a generalization of the (usual) Humbert surfaces, and we use this generalization to give formulas for the number of genus 2 curves with prescribed properties.

Harun Kir will also be giving a talk in the cryptography seminar on Friday.

Title (of the talk in the cryptography seminar): Expansion Propertiers of the Superspecial Isogeny Graph with Level Structure

Abstract (of the talk in the cryptography seminar):
The l-isogeny graph is a cornerstone of isogeny-based cryptography. Analyzing its expansion properties is essential for evaluating the underlying hardness of fundamental cryptographic problems. While the one-dimensional superspecial isogeny graph is a well-known Ramanujan graph with optimal expansion, the landscape changes in higher dimensions that are now significant ingredients in this research area.

In this talk, we examine the expansion of higher-dimensional isogeny graphs. Although these graphs are not Ramanujan, we demonstrate that they still exhibit strong expansion properties. Then we discuss immediate applications, including the removal of heuristics from the Costella-Smith algorithm for solving the isogeny problem in dimension 2. This work is in progress and is joint with M. Santos, O. Taïbi, and B. Wesolowski.