Lubin-Tate Space Research Articles

δ-Forms on Lubin–Tate spaces

δ-Forms on Lubin–Tate spaces

$p$-adic equidistribution of CM points

Let X be a modular curve and consider a sequence of Galois orbits of CM points in X , whose p -conductors tend to infinity. Its equidistribution properties in X(\mathbf{C}) and in the reductions of X modulo primes different from p are well understood. We study the equidistribution problem in the Berkovich analytification X_{p}^{\operatorname{an}} of X_{\operatorname{Q}_{p}} . We partition the set of CM points of sufficiently high conductor in X_{\operatorname{Q}_{p}} into finitely many explicit basins B_{V} , indexed by the irreducible components V of the \bmod\text{-}p reduction of the canonical model of X . We prove that a sequence z_{n} of local Galois orbits of CM points with p -conductor going to infinity has a limit in X_{p}^{\operatorname{an}} if and only if it is eventually supported in a single basin B_{V} . If so, the limit is the unique point of X_{p}^{\operatorname{an}} whose mod- p reduction is the generic point of V . The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasi-canonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin–Tate space.

An intersection formula for CM cycles on Lubin–Tate spaces

We give an explicit formula for the arithmetic intersection number of complex multiplication (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 K1,K2 over a non-Archimedean local field F. Our formula works for all cases: K1 and K2 can be either the same or different, ramified or unramified over F. This formula translates the linear arithmetic fundamental lemma (linear AFL) into a comparison of integrals. As an example, we prove the linear AFL for GL2(F).

Chromatic splitting for the K(2)–local sphere at p = 2

We calculate the homotopy type of $L_1L_{K(2)}S^0$ and $L_{K(1)}L_{K(2)}S^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$-theory and $L_1$ localization with respect to $2$-local $K$ theory. In $L_1L_{K(2)}S^0$ we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology $H^\ast_c(\mathbb{G}_2,E_0)$ where $\mathbb{G}_2$ is the Morava stabilizer group and $E_0 = \mathbb{W}[[u_1]]$ is the ring of functions on the height $2$ Lubin-Tate space. We show that the inclusion of the constants $\mathbb{W} \to E_0$ induces an isomorphism on group cohomology, a radical simplification.

A proof of the linear Arithmetic Fundamental Lemma for

AbstractLet $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

A quotient of the Lubin\u2013Tate tower II

In this article we construct the quotient mathcal {M}_mathbf {1}/P(K) of the infinite-level Lubin–Tate space mathcal {M}_mathbf {1} by the parabolic subgroup P(K) subset mathrm {GL} _n(K) of block form (n-1,1) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) to arbitrary n and K/{mathbb {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 mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of mathcal {M}_mathbf {1}/P(K) when n=2, and shows that mathcal {M}_mathbf {1}/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.

On middle cohomology of special Artin–Schreier varieties and finite Heisenberg groups

Abstract We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers ( m , n ) {(m,n)} with n ≥ 3 {n\geq 3} divisible by the characteristic of the base field, and study the middle étale cohomology group of it. If m is even, the variety admits actions of some finite Heisenberg groups. We study the middle cohomology as representations of the Heisenberg groups. If m is odd, we compute the Frobenius eigenvalues of it concretely. This affine scheme comes from the reduction of a certain affinoid in a Lubin–Tate space.

A QUOTIENT OF THE LUBIN–TATE TOWER

In this article we show that the quotient${\mathcal{M}}_{\infty }/B(\mathbb{Q}_{p})$of the Lubin–Tate space at infinite level${\mathcal{M}}_{\infty }$by the Borel subgroup of upper triangular matrices$B(\mathbb{Q}_{p})\subset \operatorname{GL}_{2}(\mathbb{Q}_{p})$exists as a perfectoid space. As an application we show that Scholze’s functor$H_{\acute{\text{e}}\text{t}}^{i}(\mathbb{P}_{\mathbb{C}_{p}}^{1},{\mathcal{F}}_{\unicode[STIX]{x1D70B}})$is concentrated in degree one whenever$\unicode[STIX]{x1D70B}$is an irreducible principal series representation or a twist of the Steinberg representation of$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

Semistable models for modular curves of arbitrary level

We produce an integral model for the modular curve $$X(Np^m)$$ over the ring of integers of a sufficiently ramified extension of $$\mathbf {Z}_p$$ whose special fiber is a semistable curve in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of $$X(Np^m)$$ , which is a union of copies of a Lubin–Tate curve. In doing so we tie together non-abelian Lubin–Tate theory to the representation-theoretic point of view afforded by Bushnell–Kutzko types. For our analysis it was essential to work with the Lubin–Tate curve not at level $$p^m$$ but rather at infinite level. We show that the infinite-level Lubin–Tate space (in arbitrary dimension, over an arbitrary nonarchimedean local field) has the structure of a perfectoid space, which is in many ways simpler than the Lubin–Tate spaces of finite level.

On the Lefschetz trace formula for Lubin–Tate spaces

Abstract We give a new proof of the Lefschetz trace formula for Lubin-Tate spaces. Our proof is based on the locally finite cell decompositions of these spaces and on Mieda’s version of the Lefschetz trace formula for certain open adic spaces. This proof is rather different from the proofs of Strauch and Mieda, and it might be generalized to other Rapoport-Zink spaces as soon as there exist suitable cell decompositions. For example, in another paper we have proved a Lefschetz trace formula for some unitary group Rapoport-Zink spaces by using similar ideas as here.

Maximal varieties and the local Langlands correspondence for 𝐺𝐿(𝑛)

The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence forGL(n)GL(n)over a nonarchimedean local field. In this article we make progress toward a purely local proof of this fact. To wit, we find a family of formal schemesV\mathcal {V}such that the generic fiber ofV\mathcal {V}is isomorphic to an open subset of Lubin-Tate space at infinite level, and such that the middle cohomology of the special fiber ofV\mathcal {V}realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degreennextension). The special fiber ofV\mathcal {V}is related to an interesting varietyXX, defined over a finite field, which is “maximal” in the sense that the number of rational points ofXXis the largest possible among varieties with the same Betti numbers asXX. The varietyXXis derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.

Action of Hecke operators on cohomology of modular curves of level two

We calculate the action of the p-th Hecke operator and the inertia group on the l-adic cohomology of modular curve of level Γ0(p2) under the assumption p ≥ 13, using only a local geometrical method. We also calculate the action of the p-th Hecke operator and the inertia group on the l-adic cohomology of the Lubin-Tate space of the same level over the maximal unramified extension of \({\mathbb{Q}_p}\).

A lemma on nearby cycles and its application to the tame Lubin-Tate space

A lemma on nearby cycles and its application to the tame Lubin-Tate space

Application de Hodge-Tate duale d'un groupe de Lubin-Tate, immeuble de Bruhat-Tits du groupe linéaire et filtrations de ramification

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 GLn(OF)×OD× or I×OD×, where OD is the maximal order in the division algebra with invariant 1/n over F and I an Iwahori subgroup of GLn. 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 detail the ramification filtrations (upper and lower) and the Hodge-Tate map of a one-dimensional formal p-divisible group

The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory

The geometry of the Lubin-Tate space of deformations of a formal group is studied via an étale, rigid analytic map from the deformation space to projective space. This leads to a simple description of the equivariant canonical bundle of the deformation space which, in turn, yields a formula for the dualizing complex in stable homotopy theory.