Global Fields

July 2 - 6, 2007, Moscow, Russia

Laboratoire J.-V. Poncelet

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Gilles Lachaud

Institut de Mathematiques de Luminy, France

An instance of Serre's conjecture on plane quartics

We describe a correspondence between from family of plane quartics over a field k to a family of abelian threefolds over k. This correspondence is the Jacobian up to isogeny. This leads to a solution of Serre's conjecture on non hyperelliptic genus 3 curves. When k is the field of complex numbers, Serre's conjecture reduces to an identity of Klein on Thetanullwerte of genus 3. We give a precise form of Klein's identity in the case considered here

Zeta functions and additive invariants

We define, following Kapranov, the zeta function over a curve defined over a field k in the framework of additive invariants, a.k.a. motivic measures. This function enjoys a functional equation. We proceed to a deconstruction of the zeta function in terms of Eisenstein series of vector bundles.