Irreducibility of the Siegel moduli spaces with parahoric level structure

Irreducibility of the Hilbert-Blumenthal moduli spaces with parahoric level structure

We determine the number of irreducible components of the reduction mod p of any Hilbert-Blumenthal moduli space with a parahoric level structure, where p is unramified in the totally real field.

On Atkin-Lehner correspondences on Siegel spaces

We introduce a higher dimensional Atkin-Lehner theory for Siegel-Parahoric congruence subgroups of $GSp(2g)$. Old Siegel forms are induced by geometric correspondences on Siegel moduli spaces which commute with almost all local Hecke algebras. We also introduce an algorithm to get equations for moduli spaces of Siegel-Parahoric level structures, once we have equations for prime levels and square prime levels over the level one Siegel space. This way we give equations for an infinite tower of Siegel spaces after N. Elkies who did the genus one case.

Irreducibility and p-adic monodromies on the Siegel moduli spaces

We generalize the surjectivity result of the $p$-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any $p$-rank stratum. We discuss irreducibility and connectedness of some $p$-rank strata of the moduli spaces with parahoric level structure. Finer results are obtained on the Siegel 3-fold with Iwahori level structure.

The supersingular loci and mass formulas on Siegel modular varieties

We describe the supersingular locus of the Siegel 3-fold with a parahoric level structure. We also study its higher dimensional generalization. Using this correspondence and a deep result of Li and Oort, we evaluate the number of irreducible components of the supersingular locus of the Siegel moduli space $\calA_{g,1,N}$ for arbitrary $g$.

任意抛物子群水平下基本非分歧 $GU(1,n-1)$ Rapoport-Zink 空间的 Bruhat-Tits 分层

The Bruhat-Tits stratification for basic unramified $GU(1,n-1)$ Rapoport-Zink spaces at arbitrary parahoric level

本文对签名 $(1,n-1)$ 的基本非分歧酉 Rapoport-Zink 空间在任意抛物子群水平下的约化特殊纤维进行分层，证明了闭分层的光滑性、不可约性并计算了维数，同时利用 Bruhat-Tits 指标描述了分层间的关联关系。

Compactification minimale et mauvaise reduction

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…