Compactification minimale et mauvaise reduction

TL;DR

This paper constructs the minimal compactification of certain Siegel modular varieties with bad reduction at a prime p, focusing on their geometric and arithmetic properties, especially in the context of parahoric level structures.

Contribution

It provides a new construction of the minimal compactification for Siegel varieties with parahoric level structures at bad reduction primes, and sketches their associated arithmetic theory.

Findings

Construction of minimal compactification at bad reduction places

Analysis of parahoric level structures on abelian schemes

Outline of the arithmetic theory of associated Siegel modular forms

Abstract

We construct the minimal compactification of some modular Siegel varieties at their bad reduction places. These varieties parametrize principally polarized abelian schemes endowed with a parahoric level structure at a prime number $p$, and with an auxiliary level structure ; such varieties have bad reduction at $p$. We also sketch the arithmetic theory of the associated Siegel modular forms. ----- Nous construisons la compactification minimale de certaines varietes modulaires de Siegel en leurs places de mauvaise reduction. Ces varietes parametrent des schemas abeliens principalement polarises munis d'une structure de niveau parahorique en un nombre premier $p$, et d'une structure de niveau auxilliaire ; elles ont mauvaise reduction en $p$. Nous esquissons egalement une theorie arithmetique des formes modulaires de Siegel associees a ces varietes.