Reducibility mod p of integral closed subschemes in projective

Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group

Let $E$ be an optimal elliptic curve over $\Q$ of prime conductor $N$. We show that if for an odd prime $p$, the mod $p$ representation associated to $E$ is reducible (in particular, if $p$ divides the order of the torsion subgroup of $E(\Q)$), then the $p$-primary component of the Shafarevich-Tate group of $E$ is trivial. We also state a related result for more general abelian subvarieties of $J_0(N)$ and mention what to expect if $N$ is not prime.

Bertini theorems for hypersurface sections containing a subscheme over finite fields

We show the existence of a hypersurface that contains a given closed subscheme of a projective space over a finite field and intersects a smooth quasi-projective scheme smoothly, under some condition on the dimension. This generalizes a Bertini theorem by Poonen and is the finite field analogue of a Bertini theorem by Altman and Kleiman. Furthermore, we add the possibility of modifying finitely many local conditions of the hypersurface. We show that the condition on the dimension is fulfilled for schemes with simple normal crossings and give an application to embeddings into smooth schemes.

具有Parahoric级结构的Hodge型Shimura簇整模型子概形的紧化

Compactifications of subschemes of integral models of Hodge-type Shimura Varieties with Parahoric level structures

本文证明了在具有连通parahoric级结构的Hodge型Shimura簇的Kisin-Pappas整模型特殊纤维上，中心叶、Igusa簇及各类分层结构均为良好放置，推广了Lan与Stroh的相关结果。

An explicit bound on reducibility of mod $\mathfrak{l}$ Galois image for Drinfeld modules of arbitrary rank and its application on the uniformity problem

Suppose we are given a Drinfeld Module $ϕ$ over $\mathbb{F}_q(t)$ of rank $r$ and a prime ideal $\mathfrak{l}$ of $\mathbb{F}_q[T]$. In this paper, we prove that the reducibility of mod $\mathfrak{l}$ Galois representation $${\rm{Gal}}(\mathbb{F}_q(T)^{\rm{sep}}/\mathbb{F}_q(T))\rightarrow {\rm{Aut}}(ϕ[\mathfrak{l}])\cong {\rm{GL}}_r(\mathbb{F}_\mathfrak{l})$$ gives a bound on the degree of $\mathfrak{l}$ which depends only on the rank $r$ of Drinfeld module $ϕ$ and the minimal degree of place $\mathcal{P}$ where $ϕ$ has good reduction at $\mathcal{P}$. Then, we apply this reducibility bound t…

任意秩 Drinfeld 模的 Galois 表示可约性的 Masser-Wüstholz 界

Masser-Wüstholz bound for reducibility of Galois representations for Drinfeld modules of arbitrary rank

本文为无复乘的任意秩 Drinfeld 模建立了模 $\mathfrak{l}$ Galois 表示不可约性的显式界，从而将数域椭圆曲线情形的 Masser-Wüstholz 界推广到函数域。

An invariant subbundle of the KZ connection mod $p$ and reducibility of $\hat{sl}_2$ Verma modules mod $p$

We consider the KZ differential equations over $\mathbb C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the space of polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work by V. Schechtman and the author. The module of these polynomial solutions defines an invariant subbundle of the associated KZ connection modulo $p$. We describe the algebraic equations for that subbundle and argue that the equations cor…