The supersingular locus in Siegel modular varieties with Iwahori level structure

关于亏格3超椭圆轨迹与超奇异轨迹的非横截性

On the non-Transversality of the Hyperelliptic Locus and the Supersingular Locus for $g=3$

本文研究亏格3时西格尔模叠中超椭圆轨迹与超奇异轨迹的交点性质，给出非横截交点的判别准则，并证明对无穷多个素数p存在此类交点。

关于$\mathrm{GU}(2,n-2)$ Shimura簇的超奇异轨迹

On the Supersingular Locus of the $\mathrm{GU}(2,n-2)$ Shimura Variety

本文研究$\mathrm{GU}(2,n-2)$ Shimura簇在惰性素数处约化的超奇异轨迹，通过完美化将其分解为经典Deligne-Lusztig簇上的迭代纤维积的不交并，并当$p\geq3$时证明了Oort猜想在Siegel模簇上的类比。

Level lowering on Siegel modular threefold of paramodular level

In this article we prove several level lowering results for cuspidal automorphic representations occurring in the cohomology of the Siegel modular threefold with paramodular level structure by adapting a method of Ribet in his proof of the Serre's epsilon conjecture. The proof is purely geometric and relies on the description of the supersingular locus of certain quaternionic unitary Shimura variety and an arithmetic level raising result on this Shimura variety. The heart of the proof is a comparison of the dimension of the space of vanishing cycles on the paramodular Siegel modular threefold …

Siegel modular forms (mod p) and algebraic modular forms

In a letter to Tate, Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions on an adelic double coset space constructed from the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms (mod p), proving that the systems of Hecke eigenvalues obtained from them are the same as the ones given by algebraic modular forms (mod p) on a quaternionic unitary group, as defined by Gross. The correspondence is obtained by restricting to the superspecial locus of the modu…

通过 pro-étale 基本群研究超奇异阿贝尔曲面的拟同源群

Quasi-isogeny groups of supersingular abelian surfaces via pro-étale fundamental groups

本文通过 pro-étale 基本群分类非正规概型的几何覆盖并计算曲线基本群，证明了 Siegel 三重 fold 上的 torsor 诱导出超奇异阿贝尔曲面自拟同源群与有限生成自由群的同构。

Deligne-Lusztig varieties and basic EKOR strata

Under the axiom of He and Rapoport for the stratifications of Shimura varieties, we explain a result of Görtz, He and Nie that the EKOR strata contained in the basic loci can be described as a disjoint union of Deligne-Lusztig varieties. In the special case of Siegel modular varieties, we compare their descriptions to that of Görtz and Yu for the supersingular Kottwitz-Rapoport strata and to the descriptions of Harashita, Hoeve for the supersingular Ekedahl-Oort strata.

Ekedahl-Oort strata and Kottwitz-Rapoport strata

We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a para…

Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms

In a letter to Tate (published in Israel J. Math. in 1996), J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions on an adelic double coset space constructed from the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms, proving that the systems of Hecke eigenvalues given by Siegel modular forms (mod p) are the same as the ones given by algebraic modular forms (mod p) on a quaternionic unitary group, as defined by Gross in Israel J. Math. in 1999. …

仿模西格尔三重fold的几何 Jacquet-Langlands 对应

A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds

本文研究仿模层级西格尔模三重fold的 Picard-Lefschetz 公式，证明权-单值猜想，并借助 Rapoport-Zink 一致化构造 GSp₄ 与定内形式间的几何 Jacquet-Langlands 对应，同时得到整数值结果与降层级定理。

CHL Calabi-Yau threefolds: Curve counting, Mathieu moonshine and Siegel modular forms

A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degenerat…

关于具有 $\Gamma_1(p)$-层结构的 Shimura 簇整模型的几何

On the geometry of integral models of Shimura varieties with $Γ_1(p)$-level structure

本文研究具有 $\Gamma_1(p)$-层结构的 Shimura 簇整模型，证明其几乎不正规且有时不平坦，并通过根栈构造解释局部模型图。