Relevement de formes modulaires de Siegel

TL;DR

This paper proves that certain Siegel modular forms in characteristic p for genus 2 or 3 can be lifted to characteristic 0, extending Katz's classical result for genus 1, using advanced cohomological techniques.

Contribution

It generalizes Katz's lifting theorem from genus 1 to genus 2 and 3 for Siegel modular forms in characteristic p.

Findings

Certain Siegel modular forms of genus 2 or 3 can be lifted to characteristic 0.

The proof relies on ampleness and vanishing theorems from algebraic geometry.

The result broadens understanding of modular forms in positive characteristic.

Abstract

Dans cette note, nous montrons que certaines formes modulaires de Siegel de caract\'eristique p et de genre 2 ou 3 se rel\`event en caract\'eristique 0. Ce r\'esultat g\'en\'eralise un th\'eor\`eme classique obtenu par Katz pour les formes de genre 1. Nous utilisons des r\'esultats de Shepherd-Barron et de Hulek et Sankaran, ainsi que des th\'eor\`emes d'annulation de la cohomologie coh\'erente d\^us \`a Deligne, Illusie et Raynaud et \`a Esnault et Viehweg. ----- In this note, we show that cuspidal Siegel modular forms of characteristic p and genus 2 or 3 can be lifted to characteristic 0. This result extends a classical theorem proved by Katz for genus 1 modular forms. We use ampleness results due to Shepherd-Barron, Hulek and Sankaran, and vanishing theorems due to Deligne, Illusie, Raynaud, Esnault and Viehweg.