Tate Theory Research Articles

The Lubin–Tate theory of configuration spaces: I

AbstractWe construct a spectral sequence converging to the Lubin–Tate theory, that is, Morava ‐theory, of unordered configuration spaces and identify its ‐page as the homology of a Chevalley–Eilenberg‐like complex for Hecke Lie algebras. Based on this, we compute the ‐theory of the weight summands of iterated loop spaces of spheres (parameterizing the weight operations on ‐algebras), as well as the ‐theory of the configuration spaces of points on a punctured surface. We read off the corresponding Morava ‐theory groups, which appear in a conjecture by Ravenel. Finally, we compute the ‐homology of the space of unordered configurations of particles on a punctured surface.

Examples of chromatic redshift in algebraic $K$-theory

We give a simple argument to detect chromatic redshift in the algebraic K -theory of \mathbb{E}_{\infty} -ring spectra and give two applications: we show for n\geq 1 that K(E_{n}) , the algebraic K -theory of any height n Lubin–Tate theory, has nontrivial T(n+1) -localization, and that K^{(n)}(k) , the n -fold iterated algebraic K -theory of a field k of characteristic different from the implicit prime p , has nontrivial T(n) -localization.

Console to Movie Success, Leveraging Nostalgia, and Introducing Subversion in The Super Mario Bros. Movie

This research examines the success of "The Super Mario Bros. Movie" by leveraging nostalgia and introducing subversion. The adaptation process from video game to film often fails, as seen with previous attempts like "Warcraft" (2016) and "Street Fighter: The Legend of Chun-Li" (2009). However, "The Super Mario Bros. Movie" (2023) succeeded by retaining core elements of the original game while innovatively transforming its narrative and character dynamics. The movie's success is attributed to its nostalgic appeal, faithful yet refreshed character portrayals, and the subversion of traditional gender roles. By making Princess Peach a mentor and strategist and providing Mario a hero's journey from an ordinary plumber to a savior, the film connects with both old fans and new audiences. This study utilizes Andrew Tate's theory of fidelity and transformation, along with perspectives on nostalgia, audience engagement, and gender role subversion, to analyze the movie's reception and success.

Towards a mod-p Lubin–Tate theory for GL2 over totally real fields

In this paper, we show that the conjectural mod [Formula: see text] local Langlands correspondence can be realized in the mod [Formula: see text] cohomology of the Lubin–Tate towers. The proof utilizes a well-known conjecture of Buzzard–Diamond–Jarvis [8, Conjecture 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field [Formula: see text] and of mod [Formula: see text] local Langlands correspondence as given by Emerton–Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13].

The Slice Spectral Sequence of a 𝐶₄-Equivariant Height-4 Lubin–Tate Theory

We completely compute the slice spectral sequence of the C 4 C_4 -spectrum B P ( ( C 4 ) ) ⟨ 2 ⟩ BP^{(\!(C_4)\!)}\langle 2 \rangle . This spectrum provides a model for a height-4 Lubin–Tate theory with a C 4 C_4 -action induced from the Goerss–Hopkins–Miller theorem. In particular, our computation shows that E 4 h C 12 E_4^{hC_{12}} is 384-periodic.

Suffering and the dilemmas of pediatric care: a response to Tyler Tate.

In a recent article, Tyler Tate argues that the suffering of children-especially children with severe cognitive impairments-should be regarded as the antithesis of flourishing, where flourishing is relative to one's individual characteristics and essentially involves receiving care from others. Although initially persuasive, Tate's theory is ambiguous in several ways, leading to significant conceptual problems. By identifying flourishing with receiving care, Tate raises questions about the importance of care that he does not address, giving rise to a bootstrapping problem. By making flourishing relative to an individual's circumstances, Tate is forced to confront questions about exactly how relative it can be, suggesting the possibility that, on his view, to flourish is simply to be however one is. In an attempt to surmount these problems, I offer a revision and restatement of Tate's view that defines the relationship between individualized flourishing and the more conventional, species-relative concept, and describe more clearly the role that care should play with respect to flourishing-one that is instrumental and not merely constitutive. Even this restated view, however, fails to answer difficult questions about how one should respond to the medical needs of some children, highlighting the fact that a conceptual analysis of suffering may do little, in the end, to untangle ethical dilemmas in the care of severely ill children.

Honda–Tate theory for Shimura varieties

A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are nonempty provided that the corresponding group G is quasisplit at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda–Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.

Lubin–Tate theory and overconvergent Hilbert modular forms of low weight

Let K be a finite extension of ℚp and let Γ be the Galois group of the cyclotomic extension of K. Fontaine’s theory gives a classification of p-adic representations of Ga( $$\overline K /K$$ ) in terms of (φ, Γ)-modules. A useful aspect of this classification is Berger’s dictionary which expresses invariants coming from p-adic Hodge theory in terms of these (φ, Γ)-modules. In this article, we use the theory of locally analytic vectors to generalize this dictionary to the setting where Γ is the Galois group of a Lubin—Tate extension of K. As an application, we show that if F is a totally real number field and v is a place of F lying above p, then the p-adic representation of Gal( $${\overline F _v}/{F_v}$$ ) associated to a finite slope overconvergent Hilbert eigenform which is Fv-analytic up to a twist is Lubin—Tate trianguline. Furthermore, we determine a triangulation in terms of a Hecke eigenvalue at v. This generalizes results in the case F = ℚ obtained previously by Chenevier, Colmez and Kisin.

Study on the Penetration Power of ZrO2 Toughened Al2O3 Ceramic Composite Projectile into Ceramic Composite Armor.

This work aims to improve the penetration ability of a 14.5 mm standard armor-piercing projectile into ceramic/armor steel (Al2O3/RHA) composite armor. To this end, ZrO2 toughened Al2O3(ZTA) is prepared as the material for bullet tips, utilizing in situ solidification injection molding that is realized via ceramic dispersant hydrolytic degradation. The penetration power of ZTA ceramic composite projectile, compared with standard armor, against 15 mm armor steel (RHA) and 30 mm Al2O3/RHA composite armor, is studied by ballistics testing combined with numerical simulation. The Tate theory is optimized and then employed to calculate the penetration depth and bullet core’s residual mass when ZTA ceramic composite projectile penetrates into Al2O3/RHA composite armor. The results show that when penetrating RHA of 15 mm, the penetration area of ZTA ceramic composite projectile into RHA increases by 27.59% and the exit area by 42.93%. While the standard projectile fails to penetrate the 30 mm Al2O3/RHA composite armor, the ZTA ceramic composite armor-piercing projectile succeeds, with the mass loss reduced by 66.67% over the standard one. The ZTA ceramic composite bullet has a better performance than the standard bullet in penetrating RHA and Al2O3/RHA composite armors. The test results, simulation, and theoretical analysis are consistent. This study has practical values for engineering applications to design new ceramic composite bullets.

Serre–Tate theory for Calabi–Yau varieties

Abstract Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p 2 {p^{2}} of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard’s approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

Lubin-Tate extensions and Carlitz module over a projective line: an explicit connection

In this article we consider different approaches for constructing maximal abelian extensions for local and global geometric fields. The Lubin–Tate theory plays key role in the maximal abelian extension construction for local geometric fields. In the case of global geometric fields, Drinfeld modules are of particular interest. In this paper we consider the simpliest special case of Drinfeld modules for projective line which is called the Carlitz module. In the introduction, we provide motivation and a brief historical background on the topics covered in the work. In the first and second sections we provide brief information about Lubin–Tate modules and Carlitz module. In the third section we present two main results: • an explicit connection between the local and global field theory in the geometric case for projective line over finite field: it is proved that the extension tower of Carlitz module induces the tower of the Lubin–Tate extensions. • a connection between Artin maps of extensions of a function field of an arbitrary projective smooth irreducible curve and extensions of completions of local rings at closed points of this curve. In the last section we formulate different open problems and interesting directions for further research, which include generalization first result for an arbitrary smooth projective curve over a finite field and consideration Drinfeld modules of higher rank.

Canonical lifts and $$\delta $$-structures

We extend the Serre–Tate theory of canonical lifts of ordinary abelian varieties to arbitrary unpolarised families of ordinary abelian varieties parameterised by a p-adic formal scheme S. We show that the canonical lift is the unique lift to W(S) which admits a $$\delta $$ -structure in the sense of Joyal, Buium, and Bousfield. We prove analogous statements for families of ordinary p-groups and p-divisible groups.

Serre–Tate theory for Shimura varieties of Hodge type

We study the formal neighbourhood of a point in the \(\mu \)-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a “shifted cascade”. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms, analogous to the Serre–Tate canonical lift of an ordinary abelian variety.

Calculations in the Generalized Lubin–Tate Theory

In this paper, various extensions of local fields are considered. For arbitrary finite extension K of the field of p-adic numbers, the maximum Abelian extension KAb/K and the corresponding Galois group can be described using the well-known Lubin–Tate theory. It is represented as a direct product of groups obtained using the maximum unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of Lubin–Tate formal groups. We consider the so-called “generalized Lubin–Tate formal groups” and extensions obtained by adding the roots of their endomorphisms to the field under consideration. Using the fact that a correctly chosen generalized formal group coincides with the classical one over unramified finite extension Tm of degree m of field K, it was possible to obtain the Galois group of the extension (Tm)Ab/K. The main result of the work, is an explicit description of the Galois group of the extension (Kur)Ab/K, where Kur is the maximum unramified extension of K. Similar methods are also used to study ramified extensions of the field K.

Calculations in generalised Lubin—Tate theory

Calculations in generalised Lubin—Tate theory

UNLIKELY INTERSECTIONS IN FINITE CHARACTERISTIC

We present a heuristic argument based on Honda–Tate theory against many conjectures in ‘unlikely intersections’ over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian. Using methods of additive combinatorics, we answer a related question of Chai and Oort where the ambient Shimura variety is a power of the modular curve.

Dimensional analysis and extended hydrodynamic theory applied to long-rod penetration of ceramics

Abstract Principles of dimensional analysis are applied in a new interpretation of penetration of ceramic targets subjected to hypervelocity impact. The analysis results in a power series representation – in terms of inverse velocity – of normalized depth of penetration that reduces to the hydrodynamic solution at high impact velocities. Specifically considered are test data from four literature sources involving penetration of confined thick ceramic targets by tungsten long rod projectiles. The ceramics are AD-995 alumina, aluminum nitride, silicon carbide, and boron carbide. Test data can be accurately represented by the linear form of the power series, whereby the same value of a single fitting parameter applies remarkably well for all four ceramics. Comparison of the present model with others in the literature (e.g., Tate's theory) demonstrates a target resistance stress that depends on impact velocity, linearly in the limiting case. Comparison of the present analysis with recent research involving penetration of thin ceramic tiles at lower typical impact velocities confirms the importance of target properties related to fracture and shear strength at the Hugoniot Elastic Limit (HEL) only in the latter. In contrast, in the former (i.e., hypervelocity and thick target) experiments, the current analysis demonstrates dominant dependence of penetration depth only by target mass density. Such comparisons suggest transitions from microstructure-controlled to density-controlled penetration resistance with increasing impact velocity and ceramic target thickness.

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.

Torsion points on elliptic curves over function fields and a theorem of Igusa

If F is a global function field of characteristic p > 3 , we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.

On congruences of Jacobi forms

We consider congruences and filtrations of Jacobi forms. More specifically, we extend Tate's theory of theta cycles to Jacobi forms, which allows us to prove a criterion for an analog of Atkin's U-operator applied to a Jacobi form to be nonzero modulo a prime.