In November 2009 I has invited as a speaker to the Intercity Number Theory Seminar.
The intercity number theory seminar is a biweekly meeting of mathematicians at varying locations in or close to Holland, most frequently in Leiden, Amsterdam, Groningen, Utrecht, Nijmegen and Eindhoven. The seminar is part of DIAMANT. The aim of the seminar is to provide a national platform for research in number theory in the broadest sense. The word Intercity refers to the type of train that participants take to get to the seminar.
My talk took place at Groningen, 13-11-2009 11:45-12:45 and it was entitled
«The discrete logarithm problem on isogenous hyperelliptic curves of genus 2″.
The absctract of the talk follows:
Abstract. In 2005, Jao, Miller, and Venkatesan proved that the DLP of elliptic curves with the same endomorhism ring is random reducible under the GRH. In this talk, we discuss a possible generalization of this result to hyperelliptic curves of genus 2 (and 3) defined over a finite field and show the difficulties involved. First, we explain the role of the endomorphism rings of the Jacobian and the polarization. Following the work of Jao, Miller and Venkatesan, we construct isogeny graphs for genus 2 curves. Specifically, we discuss the connection between isogenies and ideal classes in the Jacobian of these curves. This project is research in progress and we describe the current status of this research.