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Tyler Lawson

Publications and source records attributed to Tyler Lawson.

50 records · Page 3Linked to original sources
arXiv2013

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 continuous cohomology group.

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arXiv2013

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-theory into a sheaf of E_infinity-ring spectra.

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arXiv2011

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.

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arXiv2010

Automorphic forms and cohomology theories on Shimura curves of small discriminant

We compute the homotopy groups of spectra associated by a theorem of Lurie to the Shimura curves of discriminants 6, 10, and 14, beginning with a computation of integral rings of automorphic forms on these curves. As an application, we find that a generalized truncated Brown-Peterson spectrum BP<2> is an E_\infty ring spectrum at the prime 3.

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arXiv2009

Structured ring spectra and displays

We combine Lurie's generalization of the Hopkins-Miller theorem with work of Zink-Lau on displays to give a functorial construction of even-periodic commutative ring spectra, concentrated in chromatic layers 2 and above, associated to certain n by n invertible matrices with coefficients in Witt rings. This is applied to examples related to Lubin-Tate and Johnson-Wilson spectra. We also give a Hopf algebroid presentation of the moduli of p-divisible groups of height greater than or equal to 2.

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arXiv2009

Topological automorphic forms on U(1,1)

The homotopy type and homotopy groups of some spectra TAF of topological automorphic forms associated to a unitary similitude group GU of type (1,1) are explicitly described in quasi-split cases. The spectrum TAF is shown to be closely related to the spectrum TMF in these cases, and homotopy groups of some of these spectra are explicitly computed.

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arXiv2009

Topological Hochschild homology of l and ko

We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum.

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arXiv2009

Completed representation ring spectra of nilpotent groups

In this paper, we examine the `derived completion' of the representation ring of a pro-p group G_p^ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg-MacLane spectrum HZ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor R[-] from groups to ring spectra, and show that the map R[G_p^] --> R[G] becomes an equivalence after completion when G is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the p-adic Heisenberg group.

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arXiv2009

An overview of abelian varieties in homotopy theory

This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the conference "New Topological Contexts for Galois Theory and Algebraic Geometry."

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arXiv2008

Topological automorphic forms

We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type U(1,n-1). These cohomology theories of topological automorphic forms (TAF) are related to Shimura varieties in the same way that TMF is related to the moduli space of elliptic curves. We study the cohomology operations on these theories, and relate them to certain Hecke algebras. We compute the K(n)-local homotopy types of these cohomology theories, and determine that K(n)-locally these spectra are given by finite products of homotopy fixed point spectra of the Morava E-theory E_n by finite subgroups of the Morava stabilizer group. We construct spectra Q_U(K) for compact open subgroups K of certain adele groups, generalizing the spectra Q(l) studied by the first author in the modular case. We show that the spectra Q_U(K) admit finite resolutions by the spectra TAF, arising from the theory of buildings. We prove that the K(n)-localizations of the spectra Q_U(K) are finite products of homotopy fixed point spectra of E_n with respect to certain arithmetic subgroups of the Morava stabilizer groups, which N. Naumann has shown (in certain cases) to be dense. Thus the spectra Q_U(K) approximate the K(n)-local sphere to the same degree that the spectra Q(l) approximate the K(2)-local sphere.

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arXiv2006

The product formula in unitary deformation K-theory

We prove that "unitary deformation K-theory" takes products of finitely generated groups to coproducts of algebra spectra over ku, the connective K-theory spectrum. Additionally, we give spectral sequences for computing the homotopy groups of the unitary deformation K-theory of a group G and the cofiber of the Bott map in terms of PU(n)-equivariant K-theory and homology of spaces of G-representations.

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arXiv2005

Isogenies of elliptic curves and the 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.

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