Weak local-global compatibility and ordinary representations
Przemyslaw Chojecki
我们介绍了一个一般的形式主义，根据最低的要求，我们能够证明亲模块化的Fontaine-Mazur猜想。我们验证它在普通情况下使用最近的建设Breuil和Herzig。
We introduce a general formalism with minimal requirements under which we are able to prove the pro-modular Fontaine-Mazur conjecture. We verify it in the ordinary case using the recent construction of Breuil and Herzig.
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Ordinary parts and local-global compatibility at $\ell=p$
本文通过建立任意抛物子群下 $Q$-普通自对偶自守表示的相容性，证明并推广 Scholze 挠自守 Galois 表示在 $\ell=p$ 处的局部-整体相容性。
An Ordinary Rank-Two Case of Local-Global Compatibility for Automorphic Representations of Arbitrary Weight Over CM Fields
本文证明了满足普通条件的模 l 表示的秩二潜在自守性定理，并结合普通自守提升定理，确立了复数域上任意权正则代数尖点自守表示在 p≠l 情形下的局部-整体相容性。
Mod $p$ local-global compatibility for $\mathrm{GSp}_4(\mathbb{Q}_p)$ in the ordinary case
本文在偶次全实域与常情形、一般性及 Taylor–Wiles 条件下，证明模 $p$ 的 $\mathrm{GSp}_4$ 自守形式 Hecke 同型部分上的局部群作用可决定相应 Galois 表示在 $p$ 位的限制。
Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois representation such that rbar|_{Gal(F_w-bar/F_w)} is upper-triangular, maximally non-split, and generic. If rbar is automorphic, and some suitable technical conditions hold, we show that rbar|_{\Gal(F_w-bar/F_w)} can be recovered from the GL_3(F_w)-action on a space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights i…
Zero-cycles are conjectured to satisfy weak approximation with Brauer-Manin obstruction for proper smooth varieties defined over number fields. Roughly speaking, we prove that the conjecture is compatible for products of rationally connected varieties, K3 surfaces, Kummer varieties, and one curve.
The completed Kirillov model and local-global compatibility for functions on Igusa varieties
本文将GL2的普通Igusa簇上的尖点函数描述为经典模形式Kirillov模型的完备化，建立弱局部-全局相容性定理，并猜想更一般情形下的Hida理论类似物。
Weak local-global compatibility in the p-adic Langlands program for U(2)
受Emerton关于GL（2）的工作的启发，本文研究了二元有限酉群的有限集之塔的完全上同调.当p分裂（和其他技术假设得到满足）时，我们证明了GL（2）（在Q_p上）的p-adic局部Langlands对应在上同调中出现.给出了CM域上Fontaine-Mazur猜想的一个应用.
Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation $\overline{r}:\mathrm{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ is upper-triangular and satisfies certain genericity conditions at a place $w$ above $p$, and that every subquotient of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ of dimension $>2$ is Fontaine--Laffaille generic. In this paper, we show that the isomorphism class of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ …
Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\…
We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.
Local-global compatibility and the exceptional zero conjecture for GL(3)
本文研究 GL(3) 中 p 处 Steinberg 的 p-常自守表示，通过算术上同调证明自守例外零点猜想，并借助 GL_n 挠类在 p 处的局部-整体相容性推出 L-不变量相等。
Non-ordinary primes and congruences for weakly holomorphic modular forms of level 2
研究\(\Gamma_0(2)\)的权为\(k\)且仅在尖点\(\infty\)有极点的弱全纯模形式傅里叶系数算术性质，利用典范基结构等，证明模\(p\geq5\)同余式族并推导推论。
We give a characterization of toroidal (resp., semi-toric) compactifications due to Ash-Mumford-Rapoport-Tai (resp., Looijenga) as log minimal models and apply it to study weak K-moduli compactifications, giving a different proof to a theorem of Alexeev-Engel. We also discuss towards further generalization, in particular revisit Shah-Sterk compactification of moduli of polarized Enriques surfaces to show compatibility with log K-stability.
We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over Q when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a Taylor-Wiles patching argument for the completed homology in this case. As a key input, we generalize the work of Skinner-Wiles in the ordinary case. In addition, we also treat the residually irreducible case at the end of the paper. Combining with people's earlier work, we can prove the Fontaine-Mazur conjecture completely in the regular case when p is at le…
Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence for G(Q_p) we associate to an n-dimensional ordinary (i.e. Borel valued) representation rho : Gal(Q_p-bar/Q_p) to G-hat(E) a unitary Banach space representation Pi(rho)^ord of G(Q_p) over E that is built out of principal series representations. (Here, E is a finite extension of Q_p.) Our construction is inspired by the "ordinary part" of the tensor product of all fundamental algebraic representations of G. Th…
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