Power operations for Morava E-theory of height 2 at the prime 2

The cohomology of S(n,k) relavent to Morava stablizer algebra

In this paper we redefine an increasing filtration on the the Hopf algebra S(n,k), From which we get a spectral sequence called May spectral sequence. As an application we computed $H^{*,*}S(n,n)$ at prime 2, $H^{*,*}S(3,2)$ at prime 3 and $H^{*,*}S(4,2)$ at prime $p\geqslant 5$

A resolution of the K(2)-local sphere at the prime 3

We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^hF where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is…

Semistable models for modular curves and power operations for Morava E-theories of height 2

We construct an integral model for Lubin-Tate curves as moduli of finite subgroups of formal deformations over complete Noetherian local rings. They are p-adic completions of the modular curves X_0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer-Lashof algebra of Morava E-theories at height 2 as local moduli of power operations in elliptic cohomology.

The Hecke algebra action on Morava E-theory of height 2

Given a one-dimensional formal group of height 2, let E be the Morava E-theory spectrum associated to its universal deformation over the Lubin-Tate ring. By computing with moduli spaces of elliptic curves, we give an explicitation for an algebra of Hecke operators acting on E-cohomology. This leads to a vanishing result for Rezk's logarithmic cohomology operation on the units of E. It identifies a family of elements in the kernel with meromorphic modular forms whose Serre derivative is zero. Our calculation finds a connection to logarithms of modular units. In particular, we work out an action…

高度 n 处 K(n-1)-局部化 Morava E-理论上的幂运算

Power Operations on $K(n-1)$-Localized Morava $E$-theory at Height $n$

本文计算了对称群的 K(n-1)-局部化 E_n 理论，给出全幂运算 ψ^p_F 的模解释及 Dyer-Lashof 代数结构，并在 n=2 时导出显式公式，揭示其与椭圆曲线、模形式和 p-可除群的联系。

Ravenel-May谱序列在大素数处立即坍塌

Ravenel's May spectral sequence collapses immediately at large primes

本文证明在大素数处，高度$n$的Ravenel-May谱序列立即坍塌，从而严格Morava稳定化群概形的模$p$上同调可视为有限维可解Lie代数的上同调并算法可计算。

The Homology of Connective Morava $E$-theory with coefficients in $\mathbb{F}_p$

Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_*(e_n;\mathbb{F}_p)$ for any prime $p$ and $n \leq 4$ up to possible multiplicative extensions. In order to accomplish this we show that the Künneth spectral sequence based on an $E_3$-algebra $R$ is multiplicative when the $R$-modules in question are commutative $S$-algebras. We then apply this result by working over $BP$ which is known to be an $E_4$-algebra.

Morava K-理论与幂次滤过

Morava K-theory and Filtrations by Powers

本文证明了基于 Morava K-理论的 Adams 谱序列的收敛性，通过幂次滤过建立 K-局部球面同调与完备化导出函子的联系，并计算了所有素数和高度下的第零个极限。

Picard groups and duality for Real Morava $E$-theories

We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{hC_2}$ is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n = 1$.

The Tate spectrum of the higher real $K$-theories at height $n=p-1$

Let $E_n$ be Morava $E$-theory and let $G \subset G_n$ be a finite subgroup of $G_n$, the extended Morava stabilizer group. Let $E_{n}^{tG}$ be the Tate spectrum, defined as the cofiber of the norm map $N:(E_n)_{hG} \to E_n^{hG}$. We use the Tate spectral sequence to calculate $π_*E_{p-1}^{tG}$ for $G$ a maximal finite $p$-subgroup, and $p$ an odd prime. We show that $E_{p-1}^{tG} \simeq \ast$, so that the norm map gives equivalence between homotopy fixed point and homotopy orbit spectra. Our methods also give a calculation of $π_*E_{p-1}^{hG}$, which is a folklore calculation of Hopkins and M…

大素数下Morava稳定子群的模p上同调

The mod p cohomology of the Morava stabilizer group at large primes

该研究针对大素数下高度为n的扩展Morava稳定子群模p上同调问题，通过形变Ravenel李代数模型、结合导出不变循环定理与约化李代数上同调新方法，证明其结果为n个生成元的外代数。

Arithmetically defined dense subgroups of Morava stabilizer groups

For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power degree is canonically a dense subgroup of the $n$-th Morava stabilizer group at $p$. We also give a variant of this result taking into account a polarization. This is motivated by a perceivable generalization of topological modular forms to more general topological automorphic forms. For this, we prove some results about approximation of local units in maximal orders which is of independent…

Every K(n)-local spectrum is the homotopy fixed points of its Morava module

Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we…

Moduli of formal A-modules under change of A

We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an $A$-height $n$ formal $A$-module, where $A$ is the ring of integers in a degree $d$ field extension of $\mathbb{Q}_p$. We then compute this map for the quadratic extensions of $\mathbb{Q}_p$ and the height $2$ Morava stabilizer group at primes $p>3$. We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and…

On Hopkins' Picard groups for the prime 3 and chromatic level 2

We give a calculation of Picard groups of K(2)-local invertible spectra and of E(2)-local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup of invertible spectra with the same Morava module as a sphere.

Morava E-理论中平坦环谱的幂运算

Power Operations in Morava E-Theory of Flat Ring Spectra

研究通过构造K(n)完成张量积获得的E_n代数结构，描述其上的T(n)代数结构，揭示幂运算的编码方式。

两周期第一 Morava $K$-理论的代数 $K$-理论

Algebraic $K$-theory of the two-periodic first Morava $K$-theory

本文利用根伴随形式和对数THH，简化计算了$p>3$时$T(2)_*\ ext{K}(ku)$，并新得到$T(2)_*\ ext{K}(ku/p)$，其中$ku/p$为高度1的2-周期Morava $K$-理论谱。

Picard groups of higher real $K$-theory spectra at height $p-1$

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real $K$-theory spectra of Hopkins and Miller at height $n=p-1$, for $p$ an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra $E_n^{hG}$, where $E_n$ is Lubin-Tate $E$-theory at the prime $p$ and height $n=p-1$, and $G$ is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.

A descent spectral sequence for arbitrary K(n)-local spectra with explicit $E_2$-term

Let n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment pi_*(L_{K(n)}(X)) and E_2-term equal to the continuous cohomology of G_n, the extended Morava stabilizer group, with coefficients in a certain discrete G_n-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local E_n-Adams spectral sequence for pi_*(L_{K(n)}(X)), whose E_2-term is not known to always be equal to a c…

Milnor operations and the generalized Chern character

We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with action of Morava stabilizer group G_n can be recovered from the system of some height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite subspectra Z. In this note we reformulate and extend the above result. We construct a symmetric monoidal functor F from the category of E^{vee}_*(E)-precomodules to the category of K_{*}(K)-comodules. Then we show that K^*(X) is naturally isomorphic to the inverse limit of F(E^*(Z)) as a K_{*}(K)-comodule.

The Algebraic Duality Resolution at $p=2$

The goal of this paper is to develop some of the machinery necessary for doing $K(2)$-local computations in the stable homotopy category using duality resolutions at the prime $p=2$. The Morava stabilizer group $\mathbb{S}_2$ admits a norm whose kernel we denote by $\mathbb{S}_2^1$. The algebraic duality resolution is a finite resolution of the trivial $\mathbb{Z}_2[[\mathbb{S}_2^1]]$-module $\mathbb{Z}_2$ by modules induced from representations of finite subgroups of $\mathbb{S}_2^1$. Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of t…

An Orientation Map for Height p-1 Real E Theory

Let $p$ be an odd prime and let $\mathit{EO} = E_{p-1}^{hC_p}$ be the $C_p$ fixed points of height $p-1$ Morava $E$ theory. We say that a spectrum $X$ has algebraic $\mathit{EO}$ theory if the splitting of $K_*(X)$ as an $K_*[C_p]$-module lifts to a topological splitting of $\mathit{EO} \wedge X$. We develop criteria to show that a spectrum has algebraic $\mathit{EO}$ theory, in particular showing that any connective spectrum with mod $p$ homology concentrated in degrees $2k(p - 1)$ has algebraic $\mathit{EO}$ theory. As an application, we answer a question posed by Hovey and Ravenel by produc…

On conjectures of Hovey--Strickland and Chai

We prove the height two case of a conjecture of Hovey and Strickland that provides a $K(n)$-local analogue of the Hopkins--Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross--Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava $E$-theory is coherent, and that every finitely generated Morava module can be realized by a $K(n)$-local spectrum as long as $2p-2>n^2+n$. Finally, we dedu…

卢宾-泰特变形环的威利滤波

The Witt filtration of Lubin-Tate deformation rings

本文探讨了广义威利向量空间W_E的性质，展示了其在高度≥2时的丰富性，并通过威利滤波解释了Morava E理论的cofreeness。

高度为2的delta操作

Delta operation at height 2

本文研究了高度为2的摩拉维E理论在素数3处的delta幂运算，通过显式公式和Hecke作用的联系，提供了多项式delta操作的计算方法。

Topological resolutions in K(2)-local homotopy theory at the prime 2

We provide a topological duality resolution for the spectrum $E_2^{h\mathbb{S}_2^1}$, which itself can be used to build the $K(2)$-local sphere. The resolution is built from spectra of the form $E_2^{hF}$ where $E_2$ is the Morava spectrum for the formal group of a supersingular curve at the prime $2$ and $F$ is a finite subgroup of the automorphisms of that formal group. The results are in complete analogy with the resolutions of Goerss, Henn, Mahowald, and Rezk at the prime $3$, but the methods are of necessity very different. As in the prime $3$ case, the main difficulty is in identifying t…

Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2

Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E_infinity-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E_infinity-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation fits into a diagram of E_infinity-ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix we document how to organize Morava's forms of K-theor…

Polynomial behavior of the Honda formal group law

This note provides the calculation of the formal group law $F(x,y)$ in modulo $p$ Morava $K$-theory at prime $p$ and $s>1$ as an element in $K(s)^*[x][[y]]$ and one application to relevant examples.

A lower bound for coherences on the Brown-Peterson spectrum

We provide a lower bound for the coherence of the homotopy commutativity of the Brown-Peterson spectrum, BP, at a given prime p and prove that it is at least (2p^2 + 2p - 2)-homotopy commutative. We give a proof based on Dyer-Lashof operations that BP cannot be a Thom spectrum associated to n-fold loop maps to BSF for n=4 at 2 and n=2p+4 at odd primes. Other examples where we obtain estimates for coherence are the Johnson-Wilson spectra, localized away from the maximal ideal and unlocalized. We close with a negative result on Morava-K-theory.

The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited

In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result …

层结构的谱模问题与 Morava E-理论的整 Jacquet-Langlands 对偶

Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory

本文定义谱 Deligne-Mumford 栈的相对有效 Cartier 除子并证明其可表，据此构造谱阿贝尔簇层结构模空间，得到 Lubin-Tate 塔有限层的 E-无穷环谱精细化及 Morava E-理论的整 Jacquet-Langlands 对偶。

Monochromatic homotopy theory is asymptotically algebraic

In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we prod…

Lubin-Tate 空间中的不变素理想与所有高度的 Hovey-Strickland 定理

Invariant Primes in Lubin-Tate Space and Hovey-Strickland at Every Height

本文证明对任意高度和素数，Lubin-Tate 空间中 Morava 稳定化群开子群不变素理想恰为高度理想，并由此推出 Hovey-Strickland 分类。

Chromatic (co)homology of finite general linear groups

We study the Morava $E$-theory (at a prime $p$) of $BGL_d(F)$, where $F$ is a finite field with $|F|=1\pmod{p}$. Taking all $d$ together, we obtain a structure with two products $\times$ and $\bullet$. We prove that it is a polynomial ring under $\times$, and that the module of $\times$-indecomposables inherits a $\bullet$-product, and we describe the structure of the resulting ring. In the process, we prove many auxiliary structural results.

轨道空间与无限群的特征标理论和欧拉示性数

Character theory and Euler characteristic for orbispaces and infinite groups

本文将 Hopkins–Kuhn–Ravenel 特征标理论从有限群推广到具有有限模型的离散群，用中心化子给出 Morava E-理论局部化与 K(n)-理论欧拉示性数公式，并应用于 Coxeter 群、映射类群及算术群。

The Brown-Peterson spectrum is not $\mathbb{E}_{2(p^2+2)}$ at odd primes

We show that the odd-primary Brown-Peterson spectrum $\mathrm{BP}$ does not admit the structure of an $\mathbb{E}_{2(p^2+2)}$ ring spectrum and that there can be no map $\mathrm{MU} \to \mathrm{BP}$ of $\mathbb{E}_{2p+3}$ ring spectra. We also prove the same results for truncated Brown-Peterson spectra $\mathrm{BP} \langle n \rangle$ of height $n \geq 4$. This extends results of Lawson at the prime $2$.

$\mathbb{R}P^2$ 和 $\mathbb{R}P^2 \wedge \mathbb{C}P^2$ 的 $\operatorname{E}_2^{hC_6}$-同调

The $\operatorname{E}_2^{hC_6}$-homology of $\mathbb{R}P^2$ and $\mathbb{R}P^2 \wedge \mathbb{C}P^2$

本文利用同伦不动点谱序列，计算了素数2处高度2的Morava E-theory下 $\operatorname{E}_2^{hC_6}$ 与 $\mathbb{R}P^2$ 及 $\mathbb{R}P^2 \wedge \mathbb{C}P^2$ 碎积的同伦群。

Localization of Andre-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K--theory at a prime p. This paper is a thorough study of questions like the following: to what extent does the K(n)--localization, or the K(n)--homology, of a spectrum X determine the K(n)--homology of its 0th space X_0? Our methods combine techniques from modern homotopical algebra with chromatic homotopy. In particular, we use the telescopic functors of Bousfield and the author (dependent on the Nilpotence Theorem of Devanitz, Hopkins, and Smith), as wel as Topological Andre--Quillen Homology and Goodwillie calculus in nonconnective settings.

Detecting Periodic Elements in Higher Topological Hochschild Homology

Given a commutative ring spectrum $R$ let $Λ_XR$ be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime $p\geq 5$ we calculate $π_*(Λ_{S^n}H\mathbb{F}_p)$ and $π_*(Λ_{T^n}H\mathbb{F}_p)$ for $n\leq p$, and use these results to deduce that $v_{n-1}$ in the $n-1$-th connective Morava $K$-theory of $(Λ_{T^{n}}H\mathbb{F}_p)^{hT^{n}}$ is non-zero and detected in the homotopy fixed point spectral sequence by an explicit element, which class we name the Rognes class. To facilitate these calculations we introduce Multifold Hopf algebras. Each axis circle in $T^n$ gives rise to a …

Commutativity conditions for truncated Brown-Peterson spectra of height 2

An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized trun- cated Brown-Peterson spectra of height 2 as E_\infty-ring spectra. The criterion is checked for an example at the prime 2 derived from the universal elliptic curve equipped with a level Γ_1(3)-structure.