Abstract. 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 Rognes's Galois correspondence extends to the profinite setting. We show that the function spectrum

Introduction ix 0.1. Background and motivation ix 0.2. Subject matter of this book xvii 0.3. Organization of this book xxiii 0.4. Acknowledgments xxv Chapter 1. p-divisible groups 1.1. Definitions 1.2. Classification Chapter 2. The Honda-Tate classification 2.1. Abelian varieties over finite fields 2.2. Abelian varieties over F p Chapter 3. Tate modules and level structures 3.1. Tate modules of abelian varieties 3.2. Virtual subgroups and quasi-isogenies 3.3. Level structures 3.4. The Tate representation 3.5. Homomorphisms of abelian schemes Chapter 4. Polarizations 4.1. Polarizations 4.2. The Rosati involution 4.3. The Weil pairing 4.4. Polarizations of B-linear abelian varieties 4.5. Induced polarizations 4.6. Classification of weak polarizations Chapter 5. Forms and involutions 5.1. Hermitian forms 5.2. Unitary and similitude groups 5.3. Classification of forms Chapter 6. Shimura varieties of type U (1, n − 1) 6.1. Motivation 6.2. Initial data 6.3. Statement of the moduli problem 6.4. Equivalence of the moduli problems 6.5. Moduli problems with level structure 6.6. Shimura stacks v vi CONTENTS Chapter 7. Deformation theory 7.1. Deformations of p-divisible groups 7.2. Serre-Tate theory 7.3. Deformation theory of points of Sh Chapter 8. Topological automorphic forms 8.1. The generalized Hopkins-Miller theorem 8.2. The descent spectral sequence 8.3. Application to Shimura stacks Chapter 9. Relationship to automorphic forms 9.1. Alternate description of Sh(K p ) 9.2. Description of Sh(K p ) F 9.3. Description of Sh(K p ) C 9.4. Automorphic forms Chapter 10. Smooth G-spectra 10.1. Smooth G-sets 10.2. The category of simplicial smooth G-sets 10.3. The category of smooth G-spectra 10.4. Smooth homotopy fixed points 10.5. Restriction, induction, and coinduction 10.6. Descent from compact open subgroups 10.7. Transfer maps and the Burnside category Chapter 11. Operations on TAF 11.1. The E ∞ -action of GU (A p,∞ ) 11.2. Hecke operators Chapter 12. Buildings 12.1. Terminology 12.2. The buildings for GL and SL 12.3. The buildings for U and GU Chapter 13. Hypercohomology of adele groups 13.1. Definition of Q GU and Q U 13.2. The semi-cosimplicial resolution Chapter 14. K(n)-local theory 14.1. Endomorphisms of mod p points 14.2. Approximation results 14.3. The height n locus of Sh(K p ) 14.4. K(n)-local TAF 14.5. K(n)-local Q U Chapter 15. Example: chromatic level 1 15.1. Unit groups and the K(1)-local sphere 15.2. Topological automorphic forms in chromatic filtration 1 Bibliography Index

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. Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the 'filtered root invariant' which takes values in the E 1 term of the E-Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2. We use the filtered root invariants to compute some low-dimensional root invariants of v 1 -periodic elements at the prime 3. We also compute the root invariants of some infinite v 1 -periodic families of elements at the prime 3.

We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological André-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable vn-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computations to the algebraic geometry of isogenies between Lubin-Tate formal groups.Date: December 21, 2017.