Mark Behrens

The homotopy fixed point spectra of profinite Galois extensions

Let E be a k-local profinite G-Galois extension of an E ∞ -ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rogness Galois correspondence extends to the profinite setting. We show that the function spectrum F A ((E hH ) k , (E hK ) k ) is equivalent to the localized homotopy fixed point spectrum ((E||G/H]]) hK ) k , where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action in terms of the derived functor of fixed points.

Congruences between modular forms given by the divided β family in homotopy theory

We characterize the 2‐line of the p ‐local Adams‐Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p 5. We give a similar characterization of the 1‐line, reinterpreting some earlier work of A Baker and G Laures. These results are then used to deduce that, for ‘ a prime which generates Z p , the spectrum Q.‘/ detects the and families in the stable stems. 55Q45; 55Q51, 55N34, 11F33

On the homotopy of Q(3) and Q(5) at the prime 2

We study modular approximations Q(l), l = 3,5, of the K(2)-local sphere at the prime 2 that arise from l-power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(l) and record Hill, Hopkins, and Ravenels computation of the homotopy groups of TMF_0(5). Using these tools and formulas of Mahowald and Rezk for Q(3) we determine the image of Shimomuras 2-primary divided beta-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use low-dimensional computations of the homotopy of Q(3) and Q(5) to explore the role of these spectra as approximations to the K(2)-local sphere.

Buildings, elliptic curves, and the K(2)-local sphere

We investigate a dense subgroup Γ of the second Morava stabilizer group given by a certain group of quasi-isogenies of a supersingular elliptic curve in characteristic p. The group Γ acts on the Bruhat-Tits building for GL2(ℚ) through its action on the -adic Tate module. This action has finite stabilizers, giving a small resolution for the homotopy fixed point spectrum (EhΓ 2)hGal by spectra of topological modular forms. Here, E2 is a version of Morava E-theory and Gal = Gal().

Higher real K-theories and topological automorphic forms

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 EOn 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 ∈{ 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 give an explicit presentation of a global division algebra with involution in which the group G embeds unitarily.

The Goodwillie tower for S 1 and Kuhn's Theorem

We analyze the homological behavior of the attaching maps in the 2‐local Goodwillie tower of the identity evaluated at S 1 . We show that they exhibit the same homological behavior as the James‐Hopf maps used by N Kuhn to prove the 2‐primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie tower of S 1 at the prime 2. 55P65; 55Q40, 55S12

The Goodwillie tower for S1 and Kuhn’s Theorem

We analyze the homological behavior of the attaching maps in the 2‐local Goodwillie tower of the identity evaluated at S 1 . We show that they exhibit the same homological behavior as the James‐Hopf maps used by N Kuhn to prove the 2‐primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie tower of S 1 at the prime 2. 55P65; 55Q40, 55S12

A new proof of the Bott periodicity theorem

We give a simplification of the proof of the Bott periodicity theorem presented by Aguilar and Prieto. These methods are extended to provide a new proof of the real Bott periodicity theorem. The loop spaces of the groups O and U are identified by considering the fibers of explicit quasifibrations with contractible total spaces.  2002 Elsevier Science B.V. All rights reserved. AMS classification:Primary 55R45, Secondary 55R65