On non-abelian Lubin-Tate theory via vanishing cycles
Teruyoshi Yoshida
备注：
34 pages, some proofs updated, final version to appear in Advanced Studies in Pure Mathematics, 58 (2010), Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007)
We give a purely local proof, in the depth 0 case, of the result by Harris-Taylor which asserts that the local Langlands correspondence for GL_n over a p-adic field K realizes itself inside the vanishing cycle cohomology of the deformation space of formal O_K-modules of height n. Our proof is given by establishing the direct geometric link with the Deligne-Lusztig theory for GL_n over the residue field.
This article is the first one of a series aiming to construct an isomorphism between the p-adic Lubin-Tate and Drinfeld towers, describe this isomorphism and give applications. We construct a p-adic equivariant integral model of the Lubin-Tate tower with infinite level. This formal scheme will be later compared to another one associated to the Drinfeld tower.
Lubin-Tate spaces of dimension one are finite etale covers of the non-archimedian open unit disk. We compute certain invariants which measure the ramification of this cover over the boundary of the disk.
We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations of Gal_F x D^*, where Gal_F is the absolute Galois group of F, and D/F is the central division algebra of invariant 1/n. Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.
In this article, we investigate the alternating sum of the l-adic cohomology of the Lubin-Tate tower by the Lefschetz trace formula. Our method gives slightly stronger results than in the preceding work of Strauch.
In this article we construct the isomorphism between Lubin-Tate and Drinfeld towers at the level of points. The points we consider are the one of the theory of analytic spaces in the sens of Berkovich.
We construct a stable formal model of a Lubin-Tate curve with level three, and study the action of a Weil group and a division algebra on its stable reduction. Further, we study a structure of cohomology of the Lubin-Tate curve. Our study is purely local and includes the case where the characteristic of the residue field of a local field is two.
This is the fourth article about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the final result concerning the isomorphism that is to say the existence of an equivariant isomorphism between some blow-up of the formal scheme associated to Lubin-Tate space with infinite level constructed cellularly before and a blow-up of another formal scheme associated to Drinfeld tower. We follow the strategy of Gerd Faltings.
We analyse the mod p étale cohomology of the Lubin-Tate tower both with compact support and without support. We prove that there are no supersingular representations in the H^1_c of the Lubin-Tate tower. On the other hand, we show that in H^1 of the Lubin-Tate tower appears the mod p local Langlands correspondence and the mod p local Jacquet-Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard-Diamond-Jarvis conjecture which appears naturally in this context.
In this article, we study the De Rham cohomology of the first cover in the Lubin-Tate tower. In particular, we get a purely local proof that the supercuspidal part realizes the local Jacquet-Langlands correspondence for ${\rm GL}_n$ by comparing it to the rigid cohomology of some Deligne-Lusztig varieties. The representations obtained are analogous to the ones appearing in the $\ell$-adic cohomology if we forget the action of the Weil group. The proof relies on the generalization of an excision result of Grosse-Klönne and on the existence of a semi-stable model constructed by Yoshida for which…
This article is the last one about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the existence of an isomorphism between the compactly supported etale cohomology of the Lubin-Tate and Drinfeld towers, and more generally their equivariant cohomology complex. We also prove the existence of a geometric local Jacquet-Langlands correspondence between some equivariant rigid etale sheaves on Gross-Hopkins period space $\mathbb{P}^{n-1}$ and Drinfeld one $Ω$.
One of the goals of this article is to describe the isomorphism between Lubin-Tate and Drinfeld towers at the level of their skeletons after taking quotient by $\GL_n (Ø_F)\times Ø_D^\times$ or $I\timesØ_D^\times$ where $Ø_D$ is the maximal order in the division algebra with invariant $\frac{1}{n}$ over $F$ and $I$ a Iwahori subgroup of $\GL_n$. We give applications to the theory of canonical subgroups on Lubin-Tate spaces, the description of spherical Hecke orbits in those spaces, fundamental domains for Hecke correspondences and the Gross-Hopkins period mapping. We also study in details the …
We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of its reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers.
In this article we show that the quotient of the Lubin-Tate space at infinite level by the Borel subgroup of upper triangular matrices in GL(2,Q_p) exists as a perfectoid space. As an application we show that Scholze's functor H^i_et(P^1,F(pi)) is concentrated in degree one whenever pi is a principal series representation or a twist of the Steinberg representation of GL(2,Q_p).
We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of their reductions realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for the simple supercuspidal representations. The reductions of the formal models are isomorphic to the perfections of some Artin-Schreier varieties, whose cohomology realizes primitive Galois representations. We show also the Tate conjecture for Artin-Schreier varieties associated to quadratic forms.
We prove that the p-adic local Langlands correspondence for GL_2(Q_p) appears in the etale cohomology of the Lubin-Tate tower at infinity. We use global methods using recent results of Emerton on the local-global compatibility and hence our proof applies to local Galois representations which come via a restriction from global pro-modular Galois representations. We also discuss a folklore conjecture which states that the p-adic local Langlands correspondence appears in the de Rham cohomology of the Lubin-Tate tower (Drinfeld tower). We show that a study of the de Rham cohomology for perfectoid …
Let M_m be the generic fibre of the formal deformation space of a one-dimensional formal module X of finite height together with level-m-structures. We show that it is defined over the mth Lubin-Tate extension of the ground field and that it is geometrically connected over this field. The action of the covering group of M_m over M_0 on the set of geomtrically connected components is calculated as well as the action of the action of the automorphism group of X.
We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of the reductions of the formal models realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for simple supercuspidal representations in the case where the dimension of Galois representations is prime to the residue characteristic. The reductions of the formal models are isomorphic to the perfections of Artin-Schreier varieties associated to quadratic forms.
In the Lubin-Tate setting we compare different categories of $(φ_L,Γ_L)$-modules over various perfect or imperfect coefficient rings. Moreover, we study their associated Herr-complexes. Finally, we show that a Lubin Tate extension gives rise to a weakly decompleting, but not decompleting tower in the sense of Kedlaya and Liu.
Let $F$ be a finite extension of $\mathbb{Q}_p$. We determine the Lubin-Tate $(φ,Γ)$-modules associated to the absolutely irreducible mod $p$ representations of the absolute Galois group ${\rm Gal}(\bar{F}/F)$.
We define and study stacks which parametrize Lubin--Tate $(φ,Γ)$-modules. By working at a perfectoid level, we compare these with the Emerton--Gee stacks of cyclotomic $(φ,Γ)$-modules. As a consequence, we deduce perfectness of the Herr complex in the Lubin--Tate setting.
We analyze the geometry of the tower of Lubin-Tate deformation spaces, which parametrize deformations of a one-dimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their l-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working in the equal characteristic case, we find a family of affinoids in the Lubin-Tate tower with good reduction equal to a rather curious nonsingular hypersurface, whose equation we present explicitly. Gr…
We show that, for an abelian variety defined over a $p$-adic field $K$ which has potential good reduction, its torsion subgroup with values in the composite field of $K$ and a certain Lubin-Tate extension over a $p$-adic field is finite.
Let F be a non-archimedean local field. The construction of Lubin-Tate $(ϕ_q, Γ)$-modules attached to p-adic representations of $G_F$ depends on the choice of a uniformizer of F. In this paper, we give a description of a functor which relates categories of overconvergent Lubin-Tate $(ϕ_q, Γ)$-modules for different uniformizers. Further, we study this functor more explicitly for 2-dimensional trianguline representations.
Let $F$ be a non-Archimedean local field with residue field $k$ of odd characteristic, and let $B/F$ be the division algebra of rank 4. We explicitly construct a stable curve $\mathfrak{X}$ over the algebraic closure of $k$ admitting an action of $GL_2(F)\times B^\times \times W_F$ which realizes the Jacquet-Langlands correspondence and the local Langlands correspondence in its cohomology.
We give an explicit formula for the arithmetic intersection number of CM cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions $K_1,K_2/F$ of non-Archimedean local fields $F$. Our formula works for all cases, $K_1$ and $K_2$ can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover …
In the Lubin-Tate setting we study pairings for analytic $(φ_L,Γ_L)$-modules and prove an abstract reciprocity law which then implies a relation between the analogue of Perrin-Riou's Big Exponential map as developed by Berger and Fourquaux and a $p$-adic regulator map whose construction relies on the theory of Kisin-Ren modules generalising the concept of Wach modules to the Lubin-Tate situation.
Let LT be a Lubin-Tate formal group attached to a finite extension of Qp. By a theorem of Lubin-Sarkis, an invertible characteristic p power series that commutes with the elements of Aut(LT) is itself in Aut(LT). We extend this result to perfectoid power series, by lifting such a power series to characteristic zero and using the theory of locally analytic vectors in certain rings of p-adic periods. This allows us to recover the field of norms of the Lubin-Tate extension from its completed perfection.
In this article we extend work of Herr from the case of cyclotomic $(φ,Γ)$-modules to the general case of Lubin-Tate $(φ,Γ)$-modules. In particular, we define generalized $φ$- and $ψ$-Herr complexes, which calculate Galois cohomology, when applied to the etale $(φ,Γ)$-modules attached to the coefficients.
We show finiteness results on torsion points of commutative algebraic groups over a $p$-adic field $K$ with values in various algebraic extensions $L/K$ of infinite degree. We mainly study the following cases: (1) $L$ is an abelian extension which is a splitting field of a crystalline character (such as a Lubin-Tate extension). (2) $L$ is a certain iterate extension of $K$ associated with Lubin-Tate formal groups, which is familiar with Kummer theory.
Following Weinstein, Boyarchenko-Weinstein and Imai-Tsushima, we construct a family of affinoids in the Lubin-Tate perfectoid space and formal models such that the cohomology of the reduction of each formal model realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for certain representations. In the terminology of the essentially tame local Langlands correspondence, the representations treated here are characterized as being parametrized by minimal admissible pairs in which the field extensions are totally ramified.
We give a self-contained proof of local class field theory, via Lubin-Tate theory and the Hasse-Arf theorem, refining the arguments of Iwasawa's book. In the revised version, (i) positive characteristic case is included, (ii) the proof of base change is simplified, (iii) the construction of Artin map is simplified, and strengthened so that we start with an arbitrary Lubin-Tate group over complete maximal unramified extension. (to appear in Annales de la Faculte des Sciences de Toulouse, 17-2 (2008))
We define some rings of power series in several variables, that are attached to a Lubin-Tate formal module. We then give some examples of (ϕ,Γ)-modules over those rings. They are the global sections of some reflexive sheaves on the p-adic open unit polydisk, that are constructed from a filtered ϕ-module using a modification process. We prove that we obtain every crystalline (ϕ,Γ)-module over those rings in this way.
In this paper, we define a family of affinoids in the tubular neighborhoods of CM points in the Lubin-Tate curve with suitable level structures, and compute the reductions of them in the equal characteristic case. By using etale cohomology theory of adic spaces due to Huber, we show that the cohomology of the reductions contributes to the cohomology of the Lubin-Tate curve. As an application, with the help of explicit descriptions of the local Langlands correspondence and the local Jacquet-Langlands correspondence due to Bushnell-Henniart via the theory of types, we prove the non-abelian Lubin…
Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(φ,Γ)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting.
Given an elliptic curve with supersingular reduction at an odd prime p, Iovita and Pollack have generalised results of Kobayashi to define even and odd Coleman maps at p over Lubin-Tate extensions given by a formal group of height 1. We generalise this construction to modular forms of higher weights.
In this article we construct the quotient M_1/P(K) of the infinite-level Lubin-Tate space M_1 by the parabolic subgroup P(K) of GL(n,K) of block form (n-1,1) as a perfectoid space, generalizing results of one of the authors (JL) to arbitrary n and K/Q_p finite. For this we prove some perfectoidness results for certain Harris-Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze's candidate for the mod p Jacquet-Langlands and the mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of p…
We show that the conjectural mod $p$ local Langlands correspondence can be realised in the mod $p$ cohomology of the Lubin-Tate towers. The proof utilizes a well known conjecture of Buzzard-Diamond-Jarvis \cite[Conj. 4.9]{BDJ10}, a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field $F$ and of mod $l(\neq p)$ local Langlands correspondence as given by Emerton-Helm \cite{EmertonHelm14}. %And then we connect the completed cohomlgy with the cohomology of Lubin-Tate towers. In the case of modular curves a similar theorem was obtained…
An etale cohomology group $W$ of some irreducible components, which is the smooth compactification of an affine curve $(X^{q^2}-X)^{q-1}=(Y^{q(q+1)}-Y^{q+1})^{q-1},$ in the stable reduction the Lubin-Tate curve of level two is related to the Lusztig theory over finite rings. In this paper, we investigate a relationship between the cohomology group $W$ and the Lusztig theory.
In this paper we formulate a conjecture on the relationship between the equivariant ε-constants (associated to a local p-adic representation V and a finite extension of local fields L/K) and local Galois cohomology groups of a Galois stable \mathbb{Z}_{p}-lattice T of V. We prove the conjecture for L/K being an unramified extension of degree prime to p and T being a p-adic Tate module of a one-dimensional Lubin-Tate group defined over \mathbb{Z}_{p} by extending the ideas of \cite{Breu} from the case of the multiplicative group \mathbb{G}_{m} to arbitrary one-dimensional Lubin-Tate groups. For…
In this paper, we compute irreducible components which appear in the stable reduction of the Lubin-Tate curve of level two, in the mixed characteristic case. We also compute the action of the central division algebra of invariant 1/2, the action of ${\rm GL}_2$, and the inertia action explicitly. As a result, in a sense, we observe that, in the cohomology group of the stable reduction of the Lubin-Tate curve for ${\rm GL}_2$, the local Langlands correspondence and the local Jacquet-Langlands correspondence for ${\rm GL}_2$ are realized for the cuspidal representations of level 1 or 1/2.
In this paper we study the analyticity of the group action of the automorphism group $G$ of a formal module $\bar{F}$ of height 2 (defined over $\overline{\mathbb{F}}_q$) on the Lubin-Tate deformation space $X$ of $\bar{F}$. It is shown that a wide open congruence group of level zero attached to a non-split torus acts analytically on a particular disc in $X$ on which the period morphism is not injective. For certain other discs with larger radii (defined in terms of quasi-canonical liftings) we find wide open rigid analytic groups which act analytically on these discs.
For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a way compatible with extension of scalars. When R is a field of characteristic l, this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals, and a new proof of its existence. When R runs over complete local algebras, this provides bijections between deformat…
The classification of the local Galois representations using $(φ,Γ)$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $(φ_q,Γ_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $(ψ_q,Γ_{LT})$-modules. We also comput…
Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork's power series to a set of power series with coefficients in Lubin-Tate extensions of $\Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulita's power series using the theories of formal group exponent…
In my 2009 paper at Inventiones, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle $Ψ$ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of $Ψ$. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.
The construction of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbf{Q}_p)$ uses in an essential way Fontaine's theory of cyclotomic $(φ,Γ)$-modules. Here \emph{cyclotomic} means that $Γ= \mathrm{Gal}(\mathbf{Q}_p(μ_{p^\infty})/\mathbf{Q}_p)$ is the Galois group of the cyclotomic extension of $\mathbf{Q}_p$. In order to generalize the $p$-adic local Langlands correspondence to $\mathrm{GL}_2(L)$, where $L$ is a finite extension of $\mathbf{Q}_p$, it seems necessary to have at our disposal a theory of Lubin-Tate $(φ,Γ)$-modules. Such a generalization has been carried out t…
