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 interest. For example, it gives a precise solution to the problem of extending automorphisms of the $p$-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible.
This is an announcement of some new computational methods in stable homotopy theory, in particular, methods for using the cohomology of small-height Morava stabilizer groups to compute the cohomology of large-height Morava stabilizer groups. As an application, the cohomology of the height four Morava stabilizer group is computed at large primes (its rank turns out to be $3440$). Consequently we are able to formulate a plausible conjecture on the rank of the large-primary cohomology of the Morava stabilizer groups at all heights.
The Morava stabilizer groups play a dominating role in chromatic stable ho-motopy theory. In fact, for suitable spectra X, for example all finite spectra, thechromatic homotopy type of X at chromatic level n \textgreater{} 0 and a given prime p islargely controlled by the continuous cohomology of a certain p-adic Lie group Gn,in stable homotopy theory known under the name of Morava stabilizer group oflevel n at p, with coefficients in the corresponding Morava module (En)$\star$X.
We compute recursive approximations of the action of the height $h \geq 2$ Morava stabilizer group on the associated Lubin-Tate deformation ring. We then specialize to the case $h=3$ and $p>2$ to calculate the action explicitly. These results are new for $h>2$ and agree with computations by Lader at height $h=2$.
Oka and the second author considered the cohomology of the second Morava stabilizer algebra to study nontriviality of the products of beta elements of the stable homotopy groups of spheres. In this paper, we use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres: $\al_1\ga_t$, $\be_2\ga_t$, $\lrk{\al_1,\al_1,\be_{p/p}^p}\ga_t\be_1$ and $\lrk{\be_1,p,\ga_t}$ for $t$ with $p\nmid t(t^2-1)$ for a prime number $p>5$.
We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(\mathbb{G}_2, E_t)$, at $p=2$, for $0\leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the $d_3$-differentials in the homotopy fixed point spectral sequence for the $K(2)$-local sphere spectrum. These cohomology groups and differentials play a central role in $K(2)$-local stable homotopy theory.
In this note we show that the n-th Morava E-cohomology group of a finite spectrum with action of the n-th Morava stabilizer group can be recovered from the (n+1)-st Morava E-cohomology group with action of the (n+1)-st Morava stabilizer group.
Let MS_2 be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over \br{FF}_p, O the ring of endomorphisms of C, and \ell a topological generator of Z_p^x (respectively Z_2^x/{+-1} if p = 2). We show that for p > 2 the group Γ\subseteq O[1/\ell]^x of quasi-endomorphisms of degree a power of \ell is dense in MS_2. For p = 2, we show that Γis dense in an index 2 subgroup of MS_2.
The problem addressed is the classification up to conjugation of the finite subgroups of the (classical) Morava stabilizer group S_n and the extended Morava stabilizer group G_n(u) associated to a formal group law F of height n over the field F_p of p elements. A complete classification in S_n is provided for any n and p. Furthermore, we show that the classification in the extended group also depends on F and its associated unit u in the ring of p-adic integers. We provide a theoretical framework for the classification in G_n(u), we give necessary and sufficient conditions on n, p and u for th…
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…
The fixed point spectra of Morava E-theory $E_n$ under the action of finite subgroups of the Morava stabilizer group $\mathbb{G}_n$ and their K(n)-local Spanier--Whitehead duals can be used to approximate the K(n)-local sphere in certain cases. For any finite subgroup F of the height 2 Morava stabilizer group at p=2 we prove that the K(2)-local Spanier--Whitehead dual of the spectrum $E_2^{hF}$ is $Σ^{44}E_2^{hF}$. These results are analogous to the known results at height 2 and p=3. The main computational tool we use is the topological duality resolution spectral sequence for the spectrum $E_…
We investigate a dense subgroup Gamma of the second Morava stabilizer group given by a certain group of quasi-isogenies of a supersingular elliptic curve in characteristic p. The group Gamma acts on the Bruhat-Tits building for GL_2(Q_l) through its action on the l-adic Tate module. This action has finite stabilizers, giving a small resolution for the homotopy fixed point spectrum (E_2^hGamma)^hGal by spectra of topological modular forms. Here, E_2 is a version of Morava E-theory and Gal = Gal(barF_p/F_p).
The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology theory BP, Johnson-Wilson theory $E(n)$, or Morava $E$-theory $E_n$. For nice spaces the $E_2$-term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between $\text{Ext}_{\mathcal{U}_{Γ_B}}(B,M)$ and $\text{Ext}_{\mathcal{U}_{E_{n*}E_n}/I_{n}}(E_{n*}/I_{n},E_{n*}\otimes_{BP_*} M)$ where $(B,Γ_B)$ denotes the Hopf Algebroid $(v_n^{-1}BP_*/I_{n}, v_n^{-1}BP_*BP/I_{n} )$. We show that the latter grou…
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…
Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava stabilizer group $Γ$. We study the derived stack $\operatorname{Spf} E/Γ$. Descent-theoretic techniques allow us to deduce a theorem of Hopkins-Mahowald-Sadofsky on the $K(n)$-local Picard group, as well as a recent result of Barthel-Beaudry-Stojanoska on the Anderson duals of higher real $K$-theories.
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.
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…
We compute the Picard group of the category of $K(2)$-local module spectra over the ring spectrum $E^{hC_4}$, where $E$ is a height 2 Morava $E$-theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K(2)$-local $E$-modules in genuine $C_4$-spectra. We show that in addition to a cyclic subgroup of order 32 generated by $ E\wedge S^1$ the Picard group contains a subgroup of order 2 generated by $E\wedge S^{7+σ}$, where $σ$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO(C_4)$-g…
In this paper, we study the cohomology of the Morava stabilizer algebra $S(3)$. As an application, we show that for $p \geq 7$, if $s\not \equiv 0, \pm 1 \,\, mod \,p $, $n\not \equiv 1 \,\, mod\, 3$, $n>1$, then $ζ_nγ_s$ is a nontrivial product in $π_*(S)$ by Adams-Novikov spectral sequence, where $ζ_n$ is created by R. Cohen \cite{Co}, $γ_s$ is a third periodic homotopy elements.
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 $2$…
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.
Given a maximal finite subgroup G of the nth Morava stabilizer group at a prime p, we address the question: is the associated higher real K-theory EO_n a summand of the K(n)-localization of a TAF-spectrum associated to a unitary similitude group of type U(1,n-1)? We answer this question in the affirmative for p in {2, 3, 5, 7} and n = (p-1)p^{r-1} for a maximal finite subgroup containing an element of order p^r. We answer the question in the negative for all other odd primary cases. In all odd primary cases, we to give an explicit presentation of a global division algebra with involution in wh…
The Exotic $K(2)$-Local Picard Group at the Prime $2$
University of Colorado, Boulder（科罗拉多大学博尔德分校）；Texas A&M University（德克萨斯农工大学）；Northwestern University（西北大学）；Institut de Recherche Mathématique Avancée, C.N.R.S. et Université de Strasbourg（斯特拉斯堡大学法国国家科学研究中心高级数学研究所）；Lycée Charles Jully（夏尔·朱利中学）；University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
