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Register a new userMirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces
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Authors
Pascal Boyer
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In my 2009 paper at Inventiones, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle $Psi$ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of $Psi$. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.
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Explicit descriptions of local integral Galois module generators in certain extensions of $p$-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dworks power series to a set of power series with coefficients in Lubin-Tate extensions of $Q_p$ to establish a structure theorem for rank one solvable p-adic differential equations. In this paper we first generalise Pulitas power series using the theories of formal group exponentials and ramified Witt vectors. Using these results and Lubin-Tate theory, we then generalise Picketts constructions in order to give an analytic representation of integral normal basis generators for the square root of the inverse different in all abelian totally, weakly and wildly ramified extensions of a p-adic field. Other applications are also exposed.
Let $F$ be a finite extension of $mathbb{Q}_p$. We determine the Lubin-Tate $(varphi,Gamma)$-modules associated to the absolutely irreducible mod $p$ representations of the absolute Galois group ${rm Gal}(bar{F}/F)$.
We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E_n$ with its $C_2$-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C_2$-fixed points.
Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;mathbb{F}_p)$ be the commutative ring spectrum of $mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of module spectra on $C^*(X;mathbb{F}_p)$ is stratified in the sense of Benson, Iyengar, Krause, providing a classification of all its localizing subcategories. We establish stratification in this sense for classifying spaces of a large class of topological groups including Kac--Moody groups as well as whenever $X$ admits an $H$-space structure. More generally, using Lannes theory we prove that stratification for $X$ is equivalent to a condition that generalizes Chouinards theorem for finite groups. In particular, this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory.
We take a direct approach to computing the orbits for the action of the automorphism group $mathbb{G}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{mathbb{G}_2} cong mathbb{F}_p$. The result is new for $p=2$ and $p=3$. For primes $pgeq 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.
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