Compactification de varietes de Siegel aux places de mauvaise reduction

TL;DR

This paper develops methods to construct arithmetic toroidal compactifications of moduli stacks of principally polarized abelian varieties with parahoric level structure, extending existing techniques to cases of bad reduction.

Contribution

It extends Faltings and Chai's methods to build compactifications in cases of bad reduction for moduli of abelian varieties.

Findings

Successful construction of arithmetic toroidal compactifications for bad reduction cases

Extension of existing methods to more general reduction scenarios

Provides new tools for studying moduli spaces with bad reduction

Abstract

We construct arithmetic toroidal compactifications of the moduli stack of principally polarized abelian varieties with parahoric level structure. To this end, we extend the methods of Faltings and Chai to a case of bad reduction. ----- Nous construisons des compactifications toroidales arithm\'etiques du champ de modules des vari\'et\'es ab\'eliennes principalement polaris\'ees munies d'une structure de niveau parahorique. Pour ce faire, nous \'etendons la m\'ethode de Faltings et Chai \`a un cas de mauvaise r\'eduction.