Computation of some Moduli Spaces of covers and explicit S_n and A_n regular Q(T)-extensions with totally real fibers

A quaternionic construction of $p$-adic singular moduli

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural $p$-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of $\mathrm{SL}_2(\mathbb{Z}[1/p])$ which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article we present a similar construction of cohomology casses in which $\mathrm{SL}_2(\mathbb{Z}[1/p])$ is replaced by an order in an indefinite quaternion algebra over a totally real number field $F$. The…

A Note On Two Fiber Bundles and The Manifestations Of "Shtuka"

In this note we intend to look at the moduli stacks for global $G$-shtukas from a new perspective. We discuss a unifying interpretation of several moduli spaces (stacks) including moduli of global $G$-shtukas and (a variant of the) moduli of Higgs bundles. We view these spaces (stacks) as different fibers of a family over a scheme (stack) locally of finite type. We discuss (a relative version of) the local model theory for this family. We also consider the Hecke stacks over the moduli stack of $G$-shtukas and discuss the corresponding (motivic) Hecke operations.

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.

关于 $p$-adic 积分模方案和 PEL 类型 D 的局部模型

On $p$-adic integral moduli schemes and local models for PEL type D

本文构造了PEL类型D的平坦积分模方案及其对应的平坦正交Raport-Zink空间，并证明了Pappas-Rapoport猜想，同时描述了Schubert变种和构造了正则半稳定模型。

On Tate conjecture for the special fibers of some unitary Shimura varieties

Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s \in \mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) \times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s) \times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.

CM values of automorphic Green functions on orthogonal groups over totally real fields

Generalizing work of Gross--Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to certain quadratic spaces over an arbitrary totally real base field. As a special case, we obtain an explicit formula for the norms of the CM values of those meromorphic modular forms arising as regularized theta lifts of holomorphic Whittaker forms.