Abstract. We discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and AdamsNovikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J-homomorphism.

We prove convergence of the motivic Adams spectral sequence to completions at p and η under suitable conditions. We also discuss further conditions under which η can be removed from the statement.

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We analyze the ring tmf * tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E2-term of the Adams spectral sequence for tmf ∧ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf * tmf in TMF * TMF Q admits a description in terms of 2-variable modular forms, and (3) modulo v2-torsion, tmf * tmf injects into a certain product of copies of π * TMF0(N ), for various values of N . We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf * tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ∧ tmf gives a connective cover of TMF0(3), and show that another piece gives a connective cover of TMF0(5). To help motivate our methods, we also review the existing work on bo * bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.for bo ∧ bo (respectively, tmf ∧ tmf) splits as a direct sum of Ext-groups for the integral (respectively, bo) Brown-Gitler spectra. Section 2.4 recalls some exact sequences used in [9] which allow for an inductive approach for computing Ext of bo-Brown-Gitler comodules, and introduces related sequences which allow for an inductive approach to Ext groups of integral Brown-Gitler comodules. Section 3This section is devoted to the motivating example of bo ∧ bo. Sections 3.1-3.3 are primarily expository, based upon the foundational work of Adams, Lellmann, Mahowald, and Milgram. We make an effort to consolidate their theorems and recast them in modern notation and terminology, and hope that this will prove a useful resource to those trying to learn the classical theory of bo-cooperations and v 1 -periodic stable homotopy. To the best of our knowledge, Sections 3.4 and 3.5 provide new perspectives on this subject. Section 3.1 is devoted to the homology of the HZ i and certain Ext A(1) * -computations relevant to the Adams spectral sequence computation of bo * bo.We shift perspectives in Section 3.2 and recall Adams's description of KU * KU in terms of numerical polynomials. This allows us to study the image of bu * bu in KU * KU as a warm-up for our study of the image of bo * bo in KO * KO.We undertake this latter study in Section 3.3, where we ultimately describe a basis of KO 0 bo in terms of the '9-Mahler basis' for 2-adic numerical polynomials with domain 2Z 2 . By studying the Adams filtration of this basis, we are able to use the above results to fully describe bo * bo mod v 1 -torsion elements.In Section 3.4, we link the above two perspectives, studying the image of bo * HZ i in KO * KO. Theorem 3.6 provides a complete description of this image (mod v 1 -torsion) in terms of the 9-Mahler basis.We conclude with Section 3.5 which studies a certain mapconstructed from Adams operations. We show that this map is an injection after applying π * and exhibit how it interacts with the Brown...

Abstract. For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after η-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and faithful. Along the way, we produce several results on the stable C 2 -equivariant Betti realization functor and prove convergence theorems for the p-primary C 2 -equivariant Adams spectral sequence.

Abstract. We study modular approximations Q( ), = 3, 5, of the K(2)-local sphere at the prime 2 that arise from -power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(5) and record Hill, Hopkins, and Ravenel's 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 Shimomura's 2-primary divided β-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use lowdimensional computations of the homotopy of Q(3) and Q(5) to explore the rôle of these spectra as approximations to S K(2) .

Let k be a field with cohomological dimension less than 3; we call such fields low-dimensional. Examples include algebraically closed fields, finite fields and function fields thereof, local fields, and number fields with no real embeddings. We determine the 1-column of the motivic Adams-Novikov spectral sequence over k. Combined with rational information we use this to compute π1S, the first stable motivic homotopy group of the sphere spectrum over k. Our main result affirms Morel's π1-conjecture in the case of low-dimensional fields. We also determine π1+nαS for weights n ∈ Z {−2, −3, −4}.2010 Mathematics Subject Classification. 55T15 (primary), 19E15 (secondary).

ABSTRACT. I provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-completions of the motivic spectra BPGL and kgl over p-adic fields, p > 2. The former spectrum is the algebraic Brown-Peterson spectrum at the prime 2 (and hence is part of the study of algebraic cobordism), and the latter is a certain BPGL-module that plays the role of a "connective" motivic algebraic K-theory spectrum. This is the first in a series of two papers investigating motivic invariants of p-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.