Substitution maps in the Robba ring

Lubin-Tate Robba环上秩1(φ,Γ)-模的ε-同构

$ε$-isomorphisms for rank one $(φ,Γ)$-modules over Lubin-Tate Robba rings

受Nakamura工作启发，针对ℚₚ有限扩张L，提出（相对）Robba环上L-解析Lubin-Tate(φ_L,Γ_L)-模的ε-同构猜想，构造秩1及三角模形变的ε-同构并满足插值性质。

Iwasawa theory of de Rham (ϕ,Γ)-modules over the Robba rings

The aim of this article is to study Bloch-Kato's exponential map and Perrin-Riou's "big" exponential map purely in terms of (ϕ,Γ)-modules over the Robba ring. We first generalize the definition of Bloch-Kato's exponential map for all the (ϕ,Γ)-modules without using Fontaine's rings B_{cris}, B_{dR} of p-adic periods and then we generalize the construction of Perrin-Riou's "big" exponential map for all the de Rham (ϕ,Γ)-modules and prove that this map interpolates our Bloch-Kato's exponential map and the dual exponential map. Finally, we prove a theorem concerning to the determinant of our "big…

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact $p$-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a "generalized Robba ring" for uniform pro-$p$ groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a self-dual locally convex…

A group-theoretic generalization of the $p$-adic local monodromy theorem

Let $G$ be a connected reductive group over a $p$-adic local field $F$. We propose and study the notions of $G$-$φ$-modules and $G$-$(φ,\nabla)$-modules over the Robba ring, which are exact faithful $F$-linear tensor functors from the category of $G$-representations on finite-dimensional $F$-vector spaces to the categories of $φ$-modules and $(φ,\nabla)$-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya's slope filtration theorem in this context, and show that $G$-$(φ,\nabla)$-modules over the Robba ring are "$G$-quasi-unipotent", which i…

Noetherian properties of Fargues-Fontaine curves

We establish that the extended Robba rings associated to a perfect nonarchimedean field of characteristic p, which arise in p-adic Hodge theory as certain completed localizations of the ring of Witt vectors, are strongly noetherian Banach rings; that is, the completed polynomial ring in any number of variables over such a Banach ring is noetherian. This enables Huber's theory of adic spaces to be applied to such rings. We also establish that rational localizations of these rings are principal ideal domains and that etale covers of these rings (in the sense of Huber) are Dedekind domains.

Fundamental groups and good reduction criteria for curves over positive characteristic local fields

In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(φ,\nabla)$-module over the bounded Robba ring $\mathcal{E}_K^\dagger$, whose underlying unipotent group (after base changing to the Amice ring $\mathcal{E}_K$) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, $p$-adic analogue of Oda's theorem that a semistable curve over a $p$-adic field has good reduction iff the Galois action on its $\ell$-adic unipotent fundamental group is unramified.

$\rm{GL}_2(\mathbb{Q}_p)$ 的局部解析表示的权变化

Change of weights for locally analytic representations of $\rm{GL}_2(\mathbb{Q}_p)$

本文研究 Robba 环上秩为2的$(φ, Γ)$-模之间的包含关系，并通过$p$-进局部Langlands对应描述相应$\rm{GL}_2(\mathbb{Q}_p)$局部解析表示之间的关系。

p-adic Langlands 对应中的块分解

Block decompositions in the p-adic Langlands correspondence

本文研究 p-adic Langlands 对应中的块分解，证明 Bernstein 中心与分布代数中心同构，并确定 Emerton-Gee 叠的连通分支与素等幂元的对应关系。