On étale fundamental groups of formal fibres of $p$-adic curves

Geometric stabilisation via p-adic integration

In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_κ$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of end…

$p$-进霍奇理论的叠方法

A stacky approach to $p$-adic Hodge theory

本文利用叠方法推广$p$-进霍奇理论中的比较定理，建立带系数的多种上同调比较结果，发展衍射霍奇上同调的叠方法，为波恩大学硕士论文成果。

Integral $p$-adic Hodge theory - announcement

Given a proper, smooth (formal) scheme over the ring of integers of $\mathbb C_p$, we prove that if the crystalline cohomology of its special fibre is torsion-free then the $p$-adic étale cohomology of its generic fibre is also torsion-free. In this announcement we sketch the proof, which relies on the construction of a new cohomology theory interpolating between crystalline and étale cohomology. Further details and results, including a comparison isomorphism, will be presented in the full forthcoming article.

一个具有晶体局部系系数的算术平展-晶体比较

An arithmetic étale-crystalline comparison with coefficients in crystalline local systems

本文利用栈化方法将 Colmez--Niziol 的算术平展-晶体比较定理推广到任意晶体局部系系数，并建立带系数的 Beilinson 纤维方块及 syntomic 上同调与 p 进平展上同调的比较，同时描述 Z_p 上完美 F-规范的等ogeny范畴。

On the $p$-adic local invariant cycle theorem

For a proper, flat, generically smooth scheme $X$ over a complete DVR with finite residue field of characteristic $p$, we define a specialization morphism from the rigid cohomology of the geometric special fibre to $D_{crys}$ of the $p$-adic étale cohomology of the geometric generic fibre, and we make a conjecture ("$p$-adic local invariant cycle theorem") that describes the behavior of this map for regular $X$, analogous to the situation in $l$-adic étale cohomology for $l\neq p$. Our main result is that, if $X$ has semistable reduction, this specialization map induces an isomorphism on the s…

Sections, Homotopy Rational Points and Reductions of Curves

We study unramified sections of the fundamental group sequence of smooth projective curves of genus $\geq 2$ over $p$-adic fields together with an integral model. We are particularly interested in the induced specialized sections of the special fibre and how they relate to homotopy rational points over the residue field. Under mild assumptions, such a specialized section induces a unique homotopy rational point of the special fibre that is compatible with the original section of the generic fibre in cohomological settings. We give two applications of such specialized homotopy rational points a…