Global Fields

July 2 - 6, 2007, Moscow, Russia

Laboratoire J.-V. Poncelet

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Christophe Ritzenthaler

Institut de Mathematiques de Luminy, France

On isogeny classes of Jacobians over finite fields

In this talk we will consider the following problem : which isogeny classes of abelian varieties over a finite field contain the Jacobian of a curves ? We will give the history of this question through the genus 1,2 and 3 cases. This will lead us naturally to a conjecture of Serre on non hyperelliptic genus 3 curves. Moreover we will give a complete answer for the case g=3 over F_2^(2n)