Tate Spaces Research Articles

δ-Forms on Lubin–Tate spaces

δ-Forms on Lubin–Tate spaces

Topological semiinfinite tensor (super)modules

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl(V⊕ΠV) and osp(V⊕ΠV) associated with a Tate space V. Here V⊕ΠV is a Z/2Z-graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories Tˆgl(V), Tˆo(V) and Tˆsp(V), and show that these categories are anti-equivalent to respective previously studied categories Tgl(V), To(V), Tsp(V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to Tˆgl(V), Tˆo(V) and Tˆsp(V). Moreover, the categories To(V) and Tsp(V) are known to be equivalent, and this implies the equivalence of the categories Tˆo(V) and Tˆsp(V). After introducing a supersymmetric setting, we establish the equivalence of the category Tˆgl(V) with the category Tˆgl(V⊕ΠV), and the equivalence of both categories Tˆo(V) and Tˆsp(V) with Tˆosp(V⊕ΠV).

$p$-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).

A guide to state–space modeling of ecological time series

AbstractState–space models (SSMs) are an important modeling framework for analyzing ecological time series. These hierarchical models are commonly used to model population dynamics, animal movement, and capture–recapture data, and are now increasingly being used to model other ecological processes. SSMs are popular because they are flexible and they model the natural variation in ecological processes separately from observation error. Their flexibility allows ecologists to model continuous, count, binary, and categorical data with linear or nonlinear processes that evolve in discrete or continuous time. Modeling the two sources of stochasticity separately allows researchers to differentiate between biological variation and imprecision in the sampling methodology, and generally provides better estimates of the ecological quantities of interest than if only one source of stochasticity is directly modeled. Since the introduction of SSMs, a broad range of fitting procedures have been proposed. However, the variety and complexity of these procedures can limit the ability of ecologists to formulate and fit their own SSMs. We provide the knowledge for ecologists to create SSMs that are robust to common, and often hidden, estimation problems, and the model selection and validation tools that can help them assess how well their models fit their data. We present a review of SSMs that will provide a strong foundation to ecologists interested in learning about SSMs, introduce new tools to veteran SSM users, and highlight promising research directions for statisticians interested in ecological applications. The review is accompanied by an in‐depth tutorial that demonstrates how SSMs can be fitted and validated in R. Together, the review and tutorial present an introduction to SSMs that will help ecologists to formulate, fit, and validate their models.

BAYESIAN STATE SPACE MODELS IN MACROECONOMETRICS

AbstractState space models play an important role in macroeconometric analysis and the Bayesian approach has been shown to have many advantages. This paper outlines recent developments in state space modelling applied to macroeconomics using Bayesian methods. We outline the directions of recent research, specifically the problems being addressed and the solutions proposed. After presenting a general form for the linear Gaussian model, we discuss the interpretations and virtues of alternative estimation routines and their outputs. This discussion includes the Kalman filter and smoother, and precision‐based algorithms. As the advantages of using large models have become better understood, a focus has developed on dimension reduction and computational advances to cope with high‐dimensional parameter spaces. We give an overview of a number of recent advances in these directions. Many models suggested by economic theory are either non‐linear or non‐Gaussian, or both. We discuss work on the particle filtering approach to such models as well as other techniques that use various approximations – to either the time state and measurement equations or to the full posterior for the states – to obtain draws.

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.

Robust estimation for discrete‐time state space models

AbstractState space models (SSMs) are now ubiquitous in many fields and increasingly complicated with observed and unobserved variables often interacting in nonlinear fashions. The crucial task of validating model assumptions thus becomes difficult, particularly since some assumptions are formulated about unobserved states and thus cannot be checked with data. Motivated by the complex SSMs used for the assessment of fish stocks, we introduce a robust estimation method for SSMs. We prove the Fisher consistency of our estimator and propose an implementation based on automatic differentiation and the Laplace approximation of integrals which yields fast computations. Simulation studies demonstrate that our robust procedure performs well both with and without deviations from model assumptions. Applying it to the stock assessment model for pollock in the North Sea highlights the ability of our procedure to identify years with atypical observations.

THE LOGIC OF INFORMATION IN STATE SPACES

AbstractState spaces are, in the most general sense, sets of entities that contain information. Examples include states of dynamical systems, processes of observations, or possible worlds. We use domain theory to describe the structure of positive and negative information in state spaces. We present examples ranging from the space of trajectories of a dynamical system, over Dunn’s aboutness interpretation of fde, to the space of open sets of a spectral space. We show that these information structures induce so-called hype models which were recently developed by Leitgeb (2019). Conversely, we prove a representation theorem: roughly, hype models can be represented as induced by an information structure. Thus, the well-behaved logic hype is a sound and complete logic for reasoning about information in state spaces.As application of this framework, we investigate information fusion. We motivate two kinds of fusion. We define a groundedness and a separation property that allow a hype model to be closed under the two kinds of fusion. This involves a Dedekind–MacNeille completion and a fiber-space like construction. The proof-techniques come from pointless topology and universal algebra.

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 Rejectability of the Subjective Expected Utility Theory

AbstractState space, a key element of the Subjective Expected Utility (SEU) theory, is not observable. This implies that, in order to test the SEU theory, it is necessary to assume some state space. Consequently, if the SEU theory is rejected, then it is appropriate to conduct a robustness check; that is, to search for a different state space and a probability over that state space which together do not lead to the rejection of the SEU theory. To find such state space and probability means to SEU-rationalize the agent’s behavior. I show how to conduct the process of SEU-rationalization and determine when an SEU-rationalization is possible.

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.

Bayesian nonparametric dynamic state space modeling with circular latent states

ABSTRACTState space models are well-known for their versatility in modeling dynamic systems that arise in various scientific disciplines. Although parametric state space models are well studied, nonparametric approaches are much less explored in comparison. In this article we propose a novel Bayesian nonparametric approach to state space modeling, assuming that both the observational and evolutionary functions are unknown and are varying with time; crucially, we assume that the unknown evolutionary equation describes dynamic evolution of some latent circular random variable. Based on appropriate kernel convolution of the standard Weiner process, we model the time-varying observational and evolutionary functions as suitable Gaussian processes that take both linear and circular variables as arguments. Additionally, for the time-varying evolutionary function, we wrap the Gaussian process thus constructed around the unit circle to form an appropriate circular Gaussian process. We show that our process thus cr...

Analytic Tate Spaces and Reciprocity Laws

We consider a functional analytic variant of the notion of Tate space, namely the category of those topological vector spaces which have a direct sum decomposition where one summand is nuclear Frechet space and the other is the dual of a nuclear Frechet. We show that, both in the complex and in the p-adic setting, one can use this formalism to define symbols for analytic functions which satisfy Weil-type reciprocity laws.

Sato Grassmannians for generalized Tate spaces

We generalize the concept of Sato Grassmannians of locally linearly compact topological vector spaces (Tate spaces) to the Beilinson category of the “locally compact objects”, or Generalized Tate Spaces, of an exact category. This allows us to extend the Kapranov dimensional torsor Dim and determinantal gerbe Det to generalized Tate spaces and unify their treatment in the determinantal torsor. We then introduce a class of exact categories, that we call partially abelian exact, and prove that if the base category is so, then Dim and Det are multiplicative in admissible short exact sequences of generalized Tate spaces. We then give a cohomological interpretation of these results in terms of the Waldhausen K-theoretical space of the Beilinson category. Our approach can be iterated and we define analogous concepts for the successive categories of $n$-dimensional (generalized) Tate spaces. In particular we show that the category of Tate spaces is partially abelian exact, so we can extend the results for Dim and Det obtained for 1-Tate spaces to 2-Tate spaces, and provide a new interpretation in the context of algebraic $K$-theory of results of Kapranov, Arkhipov-Kremnizer and Frenkel-Zhu.

LOCALLY COMPACT OBJECTS IN EXACT CATEGORIES

We identify two categories of locally compact objects on an exact category [Formula: see text]. They correspond to the well-known constructions of the Beilinson category [Formula: see text] and the Kato category [Formula: see text]. We study their mutual relations and compare the two constructions. We prove that [Formula: see text] is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants, and study the exact structure of the category [Formula: see text] of Tate spaces. It is natural therefore to consider the Beilinson category [Formula: see text] as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories [Formula: see text], [Formula: see text] of countably indexed ind/pro-objects over any category [Formula: see text] can be described as localizations of categories of diagrams over [Formula: see text].

H∞ control for a singularly perturbed aircraft model

AbstractState space solutions for H∞ control are formulated for composite state feedback control of singularly perturbed systems. Scaled frequency domain realizations of dynamic systems are developed and used to generate H∞ solutions that approach the H2 solution as an H∞‐norm design parameter is allowed to approach infinity. The realizations also account for non‐orthogonal cost function terms that occur naturally in the slow subsystem design for singularly perturbed systems. Traditional H2 results are also provided. The formulation is applied to compute reduced, composite and full‐order controller gains for the longitudinal dynamics of an F‐8 aircraft.