Towards a $p$-adic Langlands programme

p-adic Jacquet-Langlands correspondence and patching

We describe two candidates for a local p-adic Jacquet-Langlands correspondence and using patching we show that they are in fact isomorphic. We then study locally algebraic vectors of the given correspondence.

GL₂(Qₚ)上的范畴p进Langlands对应

A categorical p-adic Langlands correspondence for GL_2(Q_p)

本文构建了一个从GL₂(Qₚ)的p进表示导出范畴到(phi,Gamma)-模堆上诱导凝聚sheaves导出范畴的全忠实函子，揭示了p进Langlands对应中的范畴结构。

The local Langlands conjecture for the $p$-adic inner form of Sp(4)

This paper proves the local Langlands conjecture for the non quasi-split inner form Sp(1,1) of Sp(4) over a p-adic field of characteristic 0, by studying the restriction of representations from the non quasi-split inner form GSp(1,1) of GSp(4) to Sp(1,1). The L-packets for Sp(1,1) are constructed based on the earlier work on the local Langlands correspondence for GSp(1,1) by Gan and Tantono. To parameterize them in terms of so-called S-groups, we establish and utilize the local Langlands correspondence for reductive dual groups which participate in the theta correspondence with Sp(1,1) and GSp…

$p$-adic Langlands functoriality for the definite unitary group

We formalise a notion of $p$-adic Langlands functoriality for the definite unitary group. This extends the classical notion of Langlands functoriality to the setting of eigenvarieties. We apply some results of Chenevier to obtain some cases of $p$-adic Langlands functoriality by interpolating known cases of classical Langlands functoriality.

Can we dream of a 1-adic Langlands correspondence?

After observing that some constructions and results in the $p$-adic Langlands programme are somehow independent from $p$, we formulate the hypothesis that this astonishing uniformity could be explained by a 1-adic Langlands correspondence.

Une correspondance de Jacquet-Langlands p-adique

We extend the Jacquet-Langlands'correspondence between the Hecke-modules of usual and quaternionic modular forms, to overconvergent p-adic forms of finite slope. We show that this correspondence respects p-adic families and is induced by an isomorphism between some associated eigencurves.

The Local Langlands correspondence for $\GL_n$ over $p$-adic fields

We reprove the Local Langlands Correspondence for $\GL_n$ over $p$-adic fields as well as the existence of $\ell$-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor. In contrast to the proofs of the Local Langlands Correspondence given by Henniart and Harris-Taylor our proof completely by-passes the numerical Local Langlands Correspondence of Henniart. Instead, we make use of a previous result describing the inertia-invariant nearby c…

Rigid analytic vectors of crystalline representations arising in $p$-adic Langlands

Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\math…

p进GL(2n)的中间超尖峰表示的局部朗兰兹对应

The Local Langlands Correspondence for Middle Supercuspidal Representations of $p$-adic $\text{GL}(2n)$

本文针对特征零非阿基米德局部域上的p进GL(2n)，在p不整除2n的驯顺条件下，基于极大简单型给出其中间超尖峰表示的局部朗兰兹对应的显式刻画，通过计算比较自守与伽罗瓦侧的伽马因子完成研究。

Galois表示的模栈与$\mathrm{GL}_2(\mathbb{Q}_p)$的$p$-进局部Langlands对应

Moduli stacks of Galois representations and the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$

本文为$\mathrm{GL}_2(\mathbb{Q}_p)$的$p$-进局部Langlands对应给出范畴论表述，将其嵌入模栈上的Ind-相干层范畴，并通过局部-整体相容公式与模曲线上同调联系。

On a $p$-adic extension of the Jacquet-Langlands correspondence to weight 1

We consider a novel version of the classical Jacquet-Langlands {correspondence}, explore a $p$-adic extension of the Jacquet-Langlands correspondence, and as an explicit example we find an overconvergent automorphic form of weight~1 which corresponds to a classical modular form of weight~1, using both experimental and theoretical methods.

Strong local-global compatibility in the p-adic Langlands program for U(2)

We prove that certain Galois-isotypic parts of the completed cohomology group for U(2) can be written as a completed tensor product of a representation coming from the p-adic Langlands correspondence for $GL_2(\mathbb{Q}_p)$ and a representation arising via the local Langlands correspondence in famillies of Emerton and Helm.

On the $p$-adic Langlands correspondence for algebraic tori

We extend the results by R.P. Langlands on representations of (connected) abelian algebraic groups. This is done by considering characters into any divisible abelian topological group. With this we can then prove what is known as the abelian case of the $p$-adic Langlands program.

p-adic局部解析表示的Jacquet-Langlands函子

A Jacquet-Langlands functor for $p$-adic locally analytic representations

本文研究了无限层局部Shimura品种的局部解析理论，证明了不同周期sheaves在无限层的局部解析向量与p-adic李群作用无关，并展示了Jacquet-Langlands函子与局部解析向量及李代数中心字符的兼容性。

Completed cohomology of Shimura curves and a p-adic Jacquet-Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet-Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur's `level lowering' principle.

Patching and the p-adic local Langlands correspondence

We use the patching method of Taylor--Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL_n(F), F a finite extension of Q_p. We use our construction to prove many new cases of the Breuil--Schneider conjecture.

The local Langlands conjecture for GL(n) over a p-adic field, n < p

Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne group of F. This bijection is required to satisfy certain natural compatibilities, of which the most important is compatibility with local functional equations (preservation of L and epsilon factors of pairs). It is enough to construct a bijection with these properties between supercuspidal representations of GL(n,F) and n-dimensional irreducible representa…

Actions infinitésimales dans la correspondance de Langlands locale p-adique

Let V be a two-dimensional absolutely irreducible p-adic Galois representation and let Pi be the p-adic Banach space representation associated to V via Colmez's p-adic Langlands correspondence. We establish a link between the infinitesimal action of GL_2(Q_p) on the locally analytic vectors of Pi, the differential equation associated to V via the theory of Fontaine and Berger, and the Sen polynomial of V. This answers a question of Harris and gives a new proof of a theorem of Colmez: Pi has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge-Tate w…

$p$-adic sheaves on classifying stacks, and the $p$-adic Jacquet-Langlands correspondence

We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze's basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results. Our key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the …

A p-adic Labesse-Langlands transfer

We prove a p-adic Labesse-Langlands transfer from the group of units in a definite quaternion algebra to its subgroup of norm one elements. More precisely, given an eigenvariety for the first group, we show that there exists an eigenvariety for the second group and a morphism between them that extends the classical Langlands transfer. In order to find a suitable target eigenvariety for the transfer we formalise a notion of Langlands compatibility of tame levels. Proving the existence of Langlands compatible tame levels is the key to pass from the classical transfer on the level of L-packets to…

A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $ρ: W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional th…

p-进 GSpin_4、GSpin_6 及其内形式的局部朗兰兹猜想

Local Langlands Conjecture for $p$-adic $GSpin_4,$ $GSpin_6,$ and their inner forms

本文针对 p-进 GSpin_4、GSpin_6 及其内形式建立局部朗兰兹猜想，构造 L-包并验证其性质，同时在多种情形下确定了 L-包的精确大小。

Patching and the p-adic Langlands program for GL(2, Q_p)

We present a new construction of the p-adic local Langlands correspondence for GL(2, Q_p) via the patching method of Taylor--Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and global aspects of the p-adic Langlands program; in particular, it gives a new proof of many cases of the second author's local-global compatibility theorem, and relaxes a hypothesis on the local mod p representation in that theorem.

Eigenvarieties and invariant norms: Towards p-adic Langlands for U(n)

We give a proof of the Breuil-Schneider conjecture in a large number of cases, which complement the indecomposable case, which we dealt with earlier in [Sor]. In some sense, only the Steinberg representation lies at the intersection of the two approaches. In this paper, we view the conjecture from a broader global perspective. If $U_{/F}$ is any definite unitary group, which is an inner form of $\GL(n)$ over $\K$, we point out how the eigenvariety $\X(K^p)$ parametrizes a global $p$-adic Langlands correspondence between certain $n$-dimensional $p$-adic semisimple representations $ρ$ of $\Gal(\…

The local Jacquet--Langlands correspondence and congruences modulo {\ell}

Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).

First steps towards $p$-adic Langlands functoriality

By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the …

$p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系

Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks

本文研究了 $p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系，通过构建增强的 $L$-参数集并应用到各种群上，证明了某些猜想并简化了深度零情况。

L'espace de Drinfeld et correspondance de Langlands locale II

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld's $p$-adic symmetric space. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $\ell$-adic cohomologies, as well as the torsion freeness of the integral cohomology. During the study, we obtain some results of independent interest on the coefficient systems over the Bruhat-Tits building associated to $GL_d(K).$

Gelfand-Kirillov 维数与 $p$-进 Jacquet-Langlands 对应

Gelfand-Kirillov dimension and the $p$-adic Jacquet-Langlands correspondence

本文通过界定具有无穷小特征的酉 Banach 表示的 Gelfand-Kirillov 维数，研究 Shimura 曲线上同调与 $p$-进 Jacquet-Langlands 对应，并得到相关有限性结论。

Local Root Numbers and Spectrum of the Local Descents for Orthogonal Groups: p-adic case

We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain an explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan-Gross-Prasad conjecture.

The spectral $p$-adic Jacquet-Langlands correspondence and a question of Serre

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge-Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proo…

Programme de Langlands en bref

This is a credit mini-course in French prepared for a Summer School at the University of Sherbrooke. The course consists of three one-and-half hour lectures and three credit exercises for a class of advanced graduate students.

族中的局部 Langlands 对应：banal 情形

Local Langlands in families: The banal case

提出了族中的局部 Langlands 对应猜想，关联了 p-进群表示范畴中心、Langlands 参数叠函数环与 Gelfand-Graev 表示自同态，并在反转特定整数后对经典 p-进群证明了该猜想。

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

Using the overconvergent cohomology modules introduced by Ash and Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash-Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of \emph{p-}adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequen…

A Real Groups Construction of the Tame Local Langlands Correspondence for PGSp(4,F)

In this paper, we continue the work of the first author and give a new construction of the tame local Langlands correspondence for PGSp(4,F), where F is a p-adic field, that is analogous to the construction of the local Langlands correspondence for real groups.

On multiplicity in restriction of tempered representations of $p$-adic groups

We establish an equality between two multiplicities: one in the restriction of tempered representations of a $p$-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $\mathcal{S}$-groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered $L$-packets. This provides a formula of the multiplicity for $p$-adic groups by means of dimensions of irreducible representations of their $\mathcal{S}$-groups.

Automorphic $L$\snug-functions, intertwining operators, and the irreducible tempered representations of $p$\snug-adic groups

We give an expository account of the theory of intertwining operators for connected reductive $p$--adic groups, and their connection with automorphic $L$--functions. Our purpose is to illustrate the relation between harmonic analysis and arithmetic. In particular, we describe how the theory of Plancherel measures allows us to compute certain local Langlands $L$--functions. In order to be more self contained, we give a brief introduction to the Langlands program. The theory of $R$--groups and elliptic representations is treated here as well. Finally, we give an example which illustrates how the…

p-进群的深度零特征

On depth-zero characters of p-adic groups

本文研究了p-进群的深度零特征，证明了不同定义的等价性，并为局部朗兰兹对应的新情况提供了理论支持。

半稳定局部朗兰兹对应

The semi-stable Local Langlands Correspondence

本文梳理半稳定局部朗兰兹对应相关研究，结合岩堀理论与p进泛函分析结果，给出对应p进巴拿赫空间格约化的计算思路，可确定特定参数范围下二维半稳定表示的模p约化。

Paramètres de Langlands et Algèbres d'entrelacement

Let G be a classical p-adic group and $(ψ,ε)$ the Langlands parameter of an irreducible supercuspidal representation of a Levi subgroup of G. Using data from $(ψ,ε)$, we determine explicitly the intertwining algebra of the representation which is induced from the orbit of the supercuspidal representation associated to $(ψ,ε)$.

Geometric approach to the local Jacquet-Langlands correspondence

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that n is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic techn…

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。

Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique

We describe the de Rham complex of the étale coverings of Drinfeld's p-adic upper half-plane for GL_2(Q_p). Conjectured by Breuil and Strauch, this description gives a geometric realization of the p-adic local Langlands correspondence for certain two-dimensional de Rham representations of the absolute Galois group of Q_p.

从 $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ 的 $p$-模到 $p$-进 Langlands 对应：非超尖点情形下的形变

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

本文综述了酉群 $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ 非超尖点情形下 $p$-进与 $p$-模 Langlands 对应的已知结果，并探讨了形变理论在关联这些对应中的潜力。

Deligne--Lusztig constructions for division algebras and the local Langlands correspondence, II

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups, analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig's program. Precisely, let $X$ be the $p$-adic Deligne-Lusztig ind-scheme associated to a division algebra $D$ of invariant k/n over a non-Archimedean local field $K$. We study its homology groups by establishing a Deligne-Lusztig theory for families of finite unipotent groups that arise as subquotients of $D^\times$. The homology of $X$ induces a natural corresp…

On the local Langlands correspondence

The local Langlands correspondence for GL(n) of a non-Archimedean local field $F$ parametrizes irreducible admissible representations of $GL(n,F)$ in terms of representations of the Weil-Deligne group $WD_F$ of $F$. The correspondence, whose existence for $p$-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of salient properties of the two classes of representations. The article reviews the strategies of the two proofs. Both the author's proof with Taylor and Henniart's proof are global and rely ultimat…

Hiraga-Ichino-Ikeda 猜想对于分裂 p-adic 群主级数的情形

The Hiraga-Ichino-Ikeda Conjecture for Principal Series of Split p-adic Groups

本文利用局部 Langlands 对应证明了分裂 p-adic 群主级数中不可约离散序列表示的 HII 猜想，验证了形式次数与伴随 γ-因子之间的预测公式，并先从连通中心情形推广到一般情形。

$p$-adic 域上 $\mathrm{GL}_n$ 内形式的共轭自对偶表示的奇偶性

Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields

本文证明了共轭自对偶离散序列表示的 Langlands 参数奇偶性与 Jacquet-Langlands 像奇偶性之间的一般公式，推广了 Mieda 的部分结果并变体化了 Prasad-Ramakrishnan 的自对偶结果。

Reducible principal series representations, and Langlands parameters for real groups

The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of irreducible subquotients of a reducible principal series representation of $GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities which is done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal of a similar kind for principal series representations of $GL_n({\mathbb R})$. Our investigation on principal series representations naturally led us to consider the Steinberg representation for real groups, which has curiuosly not been paid much attention to in the subject (unlike the $…

$p$-进局部 Langlands 对应与 Kisin 环

Correspondance de Langlands locale $p$-adique et anneaux de Kisin

本文借助 $\mathcal B$-进完备化与 $\mathrm{GL}_2(\mathbf Q_p)$ 的 $p$-进局部 Langlands 对应，从经典对应直接构造 Kisin 环及二维通用 Galois 表示，并统一证明超尖点情形的几何 Breuil–Mézard 猜想。

Representations of $p$-adic groups over coefficient rings

Motivated by the Langlands program in representation theory, number theory and geometry, the theory of representations of a reductive $p$-adic group over a coefficient ring different from the field of complex numbers has been widely developped during the last two decades. This article provides a survey of basic results obtained in the 21st century.

Toward the endoscopic classification of unipotent representations of $p$-adic $G_2$

We begin this paper by reviewing the Langlands correspondence for unipotent representations of the exceptional group of type $G_2$ over a $p$-adic field $F$ and present it in an explicit form. Then we compute all ABV-packets, as defined in [CFM+21] following ideas from Vogan's 1993 paper The local Langlands Conjecture, and prove that these packets satisfy properties derived from the expectation that they are generalized A-packets. We attach distributions to ABV-packets for $G_2$ and its endoscopic groups and study a geometric endoscopic transfer of these distributions. This paper builds on ear…

通过p进霍奇理论与朗兰兹对应构造4个穿孔的复射影直线上的$\text{GL}_2$型阿贝尔簇族及其在Painleve VI方程代数解中的应用

Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation

本研究借助p进霍奇理论与朗兰兹对应，构造了4穿孔射影直线上无穷多族带特定坏约化的$\text{GL}_2$型阿贝尔簇，结合相关猜想证明其可由椭圆曲线泛族挠截面参数化，且可导出Painleve VI方程的代数解。

Deligne--Langlands gamma factors in families

Let F be a p-adic field, W_F its absolute Weil group, and let k be an algebraically closed field of prime characteristic l different from p. Attached to any l-adic representation of W_F are local epsilon- and L-factors. There are natural notions of families of l-adic representations of W_F, such as the theory of Galois deformations or, more generally, families over arbitrary Noetherian W(k)-algebras. However, the epsilon and L-factors do not interpolate well in such families. In this paper it is shown that the gamma factor, which is the product of the epsilon factor with a ratio of L-factors, …