顶点算子代数的扭曲限制共形块 II：全分歧轨道曲线上的扭曲限制共形块

Topological flatness of local models in the ramified case

Local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain Shimura varieties, with parahoric level structure. We investigate the local models for groups of the form Res_{F/\Q_p} GL_n and Res_{F/\Q_p} GSp_2g where F/Q_p is a totally ramified extension, as defined by Pappas and Rapoport, and show that they are topologically flat. In the linear case, flatness can be deduced from this.

One- and Two-Point Codes over Kummer Extensions

We compute the Weierstrass semigroup at one totally ramified place for Kummer extensions defined by $y^m=f(x)^λ$ where $f(x)$ is a separable polynomial over $\mathbb{F}_q$. In addition, we compute the Weierstrass semigroup at two certain totally ramified places. We then apply our results to construct one- and two-point algebraic geometric codes with good parameters.

Ramification and descent in homotopy theory and derived algebraic geometry

We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[ΩS^3]\otimes_RΩ^1_{R/\mathbb{Z}}\to R[ΩS^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(ΩS^3;Ω^1_{R/\ma…

交错四次覆盖的全纯微分

Holomorphic differentials of alternating four covers

本文在特征二代数闭域上，对具有忠实 $A_4$ 作用且 $H$-覆盖完全分歧的曲线，确定了其全纯微分空间的 $kA_4$-模结构，证明了无穷多不可分解模类可作为直和项并给出重数公式。

关于$K(x)$的任意Kummer扩张中的间隙集

On gap sets in arbitrary Kummer extensions of $K(x)$

本文给出任意Kummer扩张中完全分歧位的间隙集显式描述，进而确定Weierstrass半群的生成集并刻画对称性，同时推广了Towse关于Weierstrass权渐近行为的公式。

Invitation to higher local fields, Part II, section 9: Local reciprocity cycles

This is an introduction to noncommutative local reciprocity maps for totally ramified Galois extensions with arithmetically profinite group. These maps in general are not homomorphisms but Galois cycles; a description of their image and kernel is included.

On reduction of Hilbert-Blumenthal varieties

Let $O_F$ be the ring of integers of a totally real field $F$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_F$-varieties of dimension $g$ modulo $p$ for a fixed prime $p$. The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by $a$-numbers on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [GO, J. Alg. Geom. 2000] on the stratifications when $p$ is unramified in $O_F$. We also prove…

克莱因四重覆盖的全纯微分

Holomorphic differentials of Klein four covers

本文研究特征二域上克莱因四重覆盖曲线的全纯微分模结构，证明其不可分解直和项由下分歧群决定，并完全刻画了基为射影直线时的分解，同时指出先前文献中的相关定理有误。

A Gross-Zagier formula for quaternion algebras over totally real fields

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields $K$ and $K^\prime$, when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in $B_{p, \infty}$, the definite quaternion a…