Abstract. This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of E∞-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms T M F and the nonperiodic and non-connective T mf . We find that Pic(T M F ) is cyclic of order 576, generated by the suspension ΣT M F (a result originally due to Hopkins), while Pic(T mf ) = Z ⊕ Z/24. In particular, we show that there exists an invertible T mf -module which is not equivalent to a suspension of T mf .

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...

It has been observed that certain localizations of the spectrum of topological modular forms are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves M, yet is only true in the derived setting. When 2 is inverted, a choice of level 2 structure for an elliptic curve provides a geometrically well-behaved cover of M, which allows one to consider T mf as the homotopy fixed points of T mf (2), topological modular forms with level 2 structure, under a natural action by GL 2 (Z/2). As a result of Grothendieck-Serre duality, we obtain that T mf (2) is self-dual. The vanishing of the associated Tate spectrum then makes T mf itself Anderson self-dual. VESNA STOJANOSKAThe duality that naturally occurs from the geometric viewpoint is not quite that of Brown and Comenetz, but an integral version thereof, called Anderson duality and denoted I Z [And69, HS05]. After localization with respect to Morava K-theory K(n) for n > 0, however, Anderson duality and Brown-Comenetz duality only differ by a shift.Elliptic curves have come into homotopy theory because they give rise to interesting one-parameter formal groups of heights one or two. The homotopical version of these is the notion of an elliptic spectrum: an even periodic spectrum E, together with an elliptic curve C over π 0 E, and an isomorphism between the formal group of E and the completion of C at the identity section.Étale maps Spec π 0 E → M 0 give rise to such spectra; more strongly, as a consequence of the Goerss-Hopkins-Miller theorem, the assignment of an elliptic spectrum to anétale map to M 0 gives anétale sheaf of E ∞ -ring spectra on the moduli stack of elliptic curves. Better still, the compactification of M 0 , which we will hereby denote by M, admits such a sheaf, denoted O top , whose underlying ordinary stack is the usual stack of generalized elliptic curves [Beh07]. The derived global sections of O top are called T mf , the spectrum of topological modular forms. This is the non-connective, non-periodic version of T mf . 2 The main result proved in this paper is the following theorem:The proof is geometric in the sense that it uses Serre duality on a cover of M as well as descent, manifested in the vanishing of a certain Tate spectrum.

Abstract. We show that the real K-theory spectrum KO is Anderson selfdual using the method previously employed in the second author's calculation of the Anderson dual of T mf . Indeed the current work can be considered as a lower chromatic version of that calculation. Emphasis is given to an algebrogeometric interpretation of this result in spectrally derived algebraic geometry. We finish by applying the result to a calculation of 2-primary Gross-Hopkins duality at height 1, and obtain an independent calculation of the group of exotic elements of the K(1)-local Picard group.

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real K-theory spectra of Hopkins and Miller at height n = p − 1, for p an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra E hG n , where En is Lubin-Tate E-theory at the prime p and height n = p − 1, and G is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension. 1 arXiv:1511.08064v1 [math.AT]