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Introduction
This paper calculates the K (1)-local homotopy of tmf ∧ tmf. The motivation behind this traces back to Mahowald's work on bo-resolutions. In his seminal papers on the subject [22,24], Mahowald was able to use the bo-based Adams spectral sequence (1) to prove the height 1 telescope conjecture at the prime p = 2, (2) and, with Wolfgang Lellmann, to exhibit the bo-based Adams spectral sequence as a viable tool for computations.
An initial difficulty with this spectral sequence is the fact that bo * bo does not satisfy Adams' flatness assumption, resulting in the E 2 -term not having a description in terms of Ext. One can still work with the spectral sequence, but one has to understand both the algebra bo * bo and the homotopy theory of bo-modules extremely well, and Mahowald's breakthrough decomposition of bo ∧ bo in terms of Brown-Gitler spectra satisfied both goals. Mahowald later initiated the study of resolutions over tmf, first known as eo 2 . Early work on this was done by Mahowald and Rezk in [25], and then developed further in the work of Behrens-Ormsby-Stapleton-Stojanoska in [4]. Again, to work with the tmf-based Adams spectral sequence, one first needs to understand of the homotopy groups π * (tmf ∧ tmf). This computation was seriously studied in [4] at the prime 2, and at the prime 3 is ongoing work of the first author and Vesna Stojanoska.
However, the E 2 -term is rather difficult to calculate since the algebra A A(2) * is very complicated. Indeed, a full computation of the Adams E 2 -term has yet to be done. The approach via the Adams spectral sequence is further complicated by the presence of differentials. Such differentials were first discovered in [25], and even more were found in [4].
Chromatic homotopy theory in principle allows the reassembly of tmf ∧tmf from its rationalization, K (1)-localizations at all primes, and K (2)-localizations at all primes. In this paper, we approach the as-yet-unstudied chromatic layer, giving a complete description of L K (1) (tmf ∧ tmf). Our main tool is a construction due to Hopkins of K (1)-local tmf as a small cell complex in K (1)-local E ∞ -rings [14].
Let us briefly mention some intuition and notation before stating the main result. First, the ring π * L K (1) tmf is essentially a graded version of the ring of functions on the p-complete moduli stack M ord ell of ordinary, generalized elliptic curves [21]. At small primes p ≤ 5, we have where j −1 is the inverse of the modular j-invariant. (Note that, at these primes, M ord ell includes the point j = ∞, corresponding to the nodal cubic, but not the point j = 0, which is supersingular for p ≤ 5.) If one writes K O for 2-complete real K -theory if p = 2, or the p-complete Adams summand for p > 2, the formula in all degrees (still for p ≤ 5) becomes This has p-torsion just at p = 2. Second, the 0th homotopy group of a K (1)-local E ∞ -ring is naturally a θ -algebra, bearing an algebraic structure studied extensively by Bousfield [7] and described briefly in our Appendix A.1. We write T(x) for the free θ -algebra on a generator x; by a theorem of Bousfield, as a ring, T(x) is polynomial on x, θ(x), θ 2 (x), and so on.
We can now state the main result.
Theorem A At primes p ≤ 5, Given this, the last remaining obstacle to a chromatic understanding of tmf * tmf is a calculation of the transchromatic map We hope to study this in future work.
Let us describe a few consequences of this result. One is a computation of the K (1)-local Adams spectral sequence based on tmf.
If tmf ∧ X is K (1)-locally pro-free over tmf, then the E 2 page of this spectral sequence is isomorphic to Ext π * L K (1) (tmf∧tmf) (π * L K (1) tmf, π * L K (1) where μ is the maximal finite subgroup of Z × p .
In particular, the spectral sequence for the sphere vanishes at E 2 above cohomological degree 1, and so collapses immediately. While the K (1)-local tmf-based Adams spectral sequence is thus uninteresting, one obtains some nontrivial information about the global tmf-based Adams spectral sequence, namely that its v 1 -periodic classes occur only on the 0 and 1 lines.
To put these results into perspective, it helps to return to bo. K (1)-locally, bo is the same as K O, and its K (1)-local co-operations algebra is simply: As bo ∧ bo is E ∞ , this ring has an alternative θ -algebraic description, namely Here b is an explicit choice of group isomorphism Z × p /μ ∼ = → Z p , and the single relation expands to a relation between b and θ(b). In the formula of Theorem A, the modular forms j −1 , j −1 also satisfy θ -algebra relations forced on them by number theory, and one obtains a relation between λ, θ(λ), and θ 2 (λ), a sort of second-order version of the bo calculation.
It is also worth noting that, for the sake of calculating Adams spectral sequences, one is interested in the coalgebra of bo * bo as much as its algebra -and the original, non-θ -algebraic calculation is actually better suited for this purpose. It is this realization, and a search for an analogue for tmf, that eventually led to the proof of Theorem B. As a final remark, our calculation also doubles as a calculation of a purely numbertheoretic object. Namely, consider the moduli problem M pair over Spf Z p that sends a p-complete ring R to the groupoid of data where E and E are ordinary generalized elliptic curves over R and φ is an isomorphism of their formal groups. Just as the structure sheaf of the moduli of generalized elliptic curves extends to a locally even periodic sheaf of E ∞ ring spectra whose global sections are (the nonconnective) Tmf [5,13], there is such a sheaf on M pair whose global sections are L K (1) (tmf ∧ tmf). Moreover, M pair is an affine scheme in the case p > 2, and has a double cover by an affine scheme in the case p = 2. In both cases, its ring of global functions R pair is exactly π 0 L K (1) (tmf ∧ tmf). We can think of this ring as a ring of "ordinary 2-variable p-adic modular functions". As examples of ordinary 2-variable p-adic modular functions, we have the functions Of course, these examples are somewhat trivial because they are really 1-variable modular functions. The results of this paper tell us that, as a θ -algebra, R pair is generated over these 1-variable functions by a single other generator. This generator is explicitly given as the generator λ described in Remark 6.2.
In fact, the θ -algebra structure on π 0 L K (1) (tmf ∧ tmf) has an equivalent definition in terms of number theory, and the generators we give can be identified in terms of modular forms. While the following is essentially a restatement of the original calculation, it is of independent enough interest to deserve explicit mention: Theorem C At the primes 2 and 3, the ring of ordinary 2-variable p-adic modular forms is generated as a θ -algebra by j −1 , j −1 , and a single other generator.

Outline of the paper
This paper is almost entirely set inside the K (1)-local category. This leads to some unusual choices about notation, for the sake of which we encourage even the expert reader to take a look at Sect. 1.2 below. In Sect. 2, we give some background information about K (1)-local homotopy theory, in particular reviewing the relevant notion of completeness and associated issues of homological algebra. Building on [1,17,19], and [3], we set up some fundamental tools, such as a relative Künneth formula, a change of rings theorem, and the theory of K (1)-local Adams spectral sequences, that we will use later on.
In Sect. 3, we study the E ∞ cone on the class ζ ∈ π −1 L K (1) S, called T ζ by Hopkins. This object was used in [14] and [21] as a partial version of tmf, and the results in this section can mostly be found in those papers. However, in the process of reading those papers, the authors found some problems with the calculation of π * T ζ (see Remark 3.29). Part of our motivation in writing down this calculation in detail is to fill these gaps.
In Sect. 5, we return to the work of Hopkins and Laures to review their construction of L K (1) tmf. Again, the material in this section can be found in [14] or [21], but we include for the reader's convenience.
In Sect. 6, we compute the K (1)-local co-operations algebra for tmf, and prove Theorems A and B.
In Sect. 7, we discuss the relationship between our results and the theory of p-adic modular forms, and prove Theorem C.
We have also included an appendix containing technical information about θalgebras and λ-rings.

Notation and conventions
The rest of this paper takes place inside the K (1)-local category, at a fixed prime p ≤ 5. To avoid notational clutter, we adopt a blanket convention that all objects are implicitly K (1)-localized and/or p-completed, unless it is explicitly stated otherwise. To be precise, this includes the following conventions for algebra: • All rings are implicitly L-completed with respect to the prime p (see Sect. 2.1, and note that the L-completion agrees with the ordinary p-completion when the ring is torsion-free). For example, by Z p [ j −1 ] we really mean the completed polynomial algebra • By ⊗ we mean the L-completed tensor product (see Sect. 2.1).
• We write Mod ∧ * for the category of L-complete graded Z p -modules, and CAlg ∧ * for the category of L-complete commutative graded Z p -algebras. • More generally, if R * is an L-complete ring, then Mod ∧ R * is the category of L-complete R * -modules and CAlg ∧ R * the category of L-complete commutative R * -algebras. If (R * , * ) is an L-complete Hopf algebroid, then Comod ∧ * is its category of L-complete comodules (see Sect. 2.3).
• Ext * is the relative Ext functor for comodules defined in Definition 2.16.
It includes the following conventions for topology: • All smash products are implicitly K (1)-localized.
• Sp is the category of K (1)-local spectra, and CAlg is the category of K (1)-local E ∞ -algebras. • P(X ) is the free K (1)-local E ∞ -algebra on a spectrum X .
We will also employ the following notation: • ω is a fixed generator of μ (so ω = −1 at p = 2). • For p > 2, g is a fixed topological generator of Z × p (for example, we can take g = ω(1+ p)). Note that g maps to a topological generator of Z × p /μ. For p = 2, g is a fixed element of Z × 2 mapping to a topological generator of Z × 2 /μ (for example, we can take g = 3). Restrictions on p). Unless otherwise stated, the results of this paper are valid only at p = 2, 3, and 5. This is primarily a matter of convenience: at these primes, there is a unique supersingular j-invariant congruent to 0 mod p, which implies that π 0 L K (1) tmf is a p-complete polynomial in the generator j −1 . At larger primes, π 0 L K (1) tmf is the p-complete ring of functions on which grows more complicated as the number of supersingular j-invariants increases, though presumably not in an essential way. Our restriction on p is also a matter of interest: it is only at p = 2 and 3 that the homotopy groups of the unlocalized spectrum tmf has torsion; at larger primes tmf * is just the ring of level 1 modular forms.
The reader will also note that the K (1)-local category behaves differently at the prime 2 than at all other primes. For example, while π * tmf has 2-and 3-torsion, π * L K (1) tmf only has torsion at the prime 2.

Complete Hopf algebroids and comodules
One often attempts to study a K (1)-local spectrum X through its completed Khomology or K O-homology, These are not just graded abelian groups, but satisfy a condition known since [19] as L-completeness. In Sect. 2.1, we review the definition of L-completeness and some basic properties of the L-complete category. Next, in Sect. 2.2, we review the important technical notion of pro-freeness, which is to be the appropriate replacement for flatness in the L-complete setting. As we have to deal with some relative tensor products of K (1)-local ring spectra, we need a relative definition of pro-freeness that is more general than that used by other authors, e.g. [17]. We use this definition to give a Künneth formula for relative tensor products in which one of the modules is profree. In Sect. 2.3, we discuss homological algebra over L-complete Hopf algebroids, a concept originally due to Baker [1], and conclude with an examination of the K (1)local Adams spectral sequence. Finally, in Sect. 2.4, we give the classical examples of the Hopf algebroids for K and K O, and describe their categories of comodules.
The results of this section should be compared with Barthel-Heard's work on the K (n)-local E n -based Adams spectral sequence [3]. While we ultimately want to write down K (1)-local Adams spectral sequences over more general bases than K itself, the work involved is substantially simplified by certain convenient features of height 1, mostly boiling down to the fact that direct sums of L-complete Z p -modules are exact-the analogue of which is not true at higher heights [17,Sect. 1.3]. The reader who wishes to do similar work at higher heights should therefore proceed with caution.

Background on L-completeness
In the category Sp of K (1)-local spectra, there is a well-known equivalence [19,Proposition 7.10] X holim i X ∧ S/ p i Replacing X by the K (1)-local smash product K ∧ X , we have an equivalence This shows that K * X is derived complete, in a sense we now make precise.
We can regard p-completion as an endofunctor of the category of abelian groups. This functor is neither left nor right exact. However, it still has left derived functors, which we write as L 0 and L 1 (the higher left derived functors vanish in this case). Since p-completion is not right exact, it is generally not the case that M ∧ p = L 0 M. There is, however, a canonical factorization of the completion map M → M ∧ p : The second map is surjective, and in fact, there is a short exact sequence [19, Theorem We also have [19,Theorem A.2(d)]

Definition 2.2 An abelian group
Being L-complete is quite close to being p-complete: for example, p-complete modules are L-complete, and if M is finitely generated, then L 0 M ∼ = M ∧ p . In particular, K * and K O * are L-complete. More generally, for any K (1)-local spectrum X , π * X is L-complete as a graded abelian group [17,Lemma 3.2].
Write Mod ∧ * for the category of L-complete graded Z p -modules. This is an abelian subcategory of the category of graded Z p -modules which is closed under extensions. It is also closed symmetric monoidal [2, Sect. A.2] under the L-completed tensor product Following our general conventions (see Sect. 1.2), we will simply write ⊗ for this tensor product, where this does not cause confusion.
Write CAlg ∧ * for the category of commutative algebra objects in Mod ∧ * . If R * ∈ CAlg ∧ * (in particular, if R * = K * or K O * ), there is an obvious abelian category of L-complete R * -modules, which we denote Mod ∧ R * .

Pro-freeness
where F * is a free graded R * -module. Say that a map R * → S * of commutative rings in Mod ∧ * is pro-free if S * is a pro-free R * -module.
Pro-free modules are projective in the category Mod ∧ R * . In this height 1 case, they are also flat in this category. As is shown below, this follows from the fact that direct sums in Mod ∧ * are exact, which is, surprisingly, not true at higher heights.

Lemma 2.4
Let R * ∈ CAlg ∧ * , and let M * be a non-zero pro-free R * -module. Then M * is faithfully flat in Mod ∧ R * , that is, the functor M * ⊗ R * · is exact and conservative.
Proof If M * is a pro-free R * -module, it is a coproduct of (possibly shifted) copies of R * in the category Mod ∧ R * . Correspondingly, M * ⊗ R * N * is a coproduct of possibly shifted copies of N * , which can be taken in Mod ∧ * . This functor is exact because coproducts in Mod ∧ * are exact [17,Proposition 1.4]. Clearly, a coproduct of copies of N * is zero iff N * is zero, which together with exactness implies conservativity.

Lemma 2.5
Pro-freeness is preserved by base change: if M * is pro-free over R * and R * → S * is a map of rings in Mod ∧ * , then M * ⊗ R * S * is pro-free over S * .
Proof Again, M * is a coproduct of copies of R * in the category Mod ∧ R * . The tensor product is a left adjoint, so distributes over this coproduct.

Lemma 2.6
Suppose that R * ∈ CAlg ∧ * and M * ∈ Mod ∧ R * . Suppose also that R * is p-torsion-free. Then M * is pro-free over R * iff M * is p-torsion-free and M * / p is free over R * / p. Proof Suppose that M * is pro-free over R * , and write M * = L 0 α n α R * . By the exact sequence (2.1), M * is the same as the p-completion of α n α R * , and is, in particular, p-torsion-free. By [19,Proposition A.4], which is clearly free over R * / p (and flat, in particular).
For the converse, suppose that M * is L-complete and p-torsion-free and M * / p is free over R * / p. Again using (2.1), we see that the natural surjection M * → (M * ) ∧ p is an isomorphism, so that M * is honestly p-complete. Choose generators for M * / p as an R * / p-module, and lift them to a map φ : F * → M * from a free graded R * -module, which is an isomorphism mod p. Again, we observe that L 0 (F * ) = (F * ) ∧ p , that it is p-torsion-free, and that L 0 (F * )/ p = F * / p. Applying the snake lemma to the diagram of graded Z p -modules we see that multiplication by p is an isomorphism on ker(φ ∧ ) and coker(φ ∧ ). Both of these are L-complete graded Z p -modules, and this implies that they are zero, by [ (Here, as always, the coproduct is taken in the K (1)-local category).
Proof Suppose that M * is pro-free over R * . Choose generators x α ∈ M n α such that the natural map becomes an isomorphism after L-completion. Each x α corresponds to a map of spectra S n α → M, and they assemble to a map of K (1)-local R-modules This is an equivalence by a result of Hovey [17,Theorem 3.3], which states that the functor π * sends (K (1)-local) coproducts to (L-complete) direct sums. The converse also follows from Hovey's result.
Note that Hovey's proof uses the same, height-1-specific fact that direct sums are exact in Mod ∧ R * .

Proposition 2.8
Suppose that R is a K (1)-local homotopy commutative ring spectrum and M and N are R-modules, such that M * is pro-free over R * . Then the natural map of L-complete modules, is an isomorphism.
Proof By the previous lemma, we can write M as a wedge of suspensions of R, (using the fact that the K (1)-local smash product is a left adjoint, so distributes over the K (1)-local coproduct). Thus, Using Hovey's theorem again [17,Theorem 3.3], we obtain where F * is the free graded R * -module on generators in the degrees n α . By [19, A.7], It is clear that this isomorphism is induced by the natural map.

Homological algebra of L-complete Hopf algebroids
We now turn to the problem of homological algebra over an L-complete Hopf algebroid. We begin with some definitions generalizing those of [1].
Note that χ gives an isomorphism between * as a left R * -module and * as a right R * -module, so that * is also pro-free as a right R * -module.
Remark 2. 10 We should point out that in this K (1)-local setting, we impose the condition that * is pro-free over R * , as opposed to the more common condition that * is flat over R * in the unlocalized situation. This is required to produce an appropriate L-complete version of Ext (cf. [1,3]). In light of this, we often require a pro-freeness condition rather than a flatness condition (e.g. proposition 2.20).

Remark 2.11
At heights higher than 1, one has to deal with the fact that the left and right units generally do not act in the same way on the generators ( p, u 1 , . . . , u n−1 ) with respect to which L-completeness is defined. Thus, Baker's definition has the additional condition that the ideal ( p, u 1 , . . . , u n−1 ) is invariant. At height 1, this condition is trivial.
* for the category of left * -comodules.

Lemma 2.13 The category of left * -comodules is abelian, and the forgetful functor
is an exact sequence of R * -modules, and f is a map of * -comodules. A coaction map can then be defined on K * via the diagram The bottom sequence is exact because * is flat in Mod ∧ R * , by Lemma 2.4. One checks that this structure makes K * a comodule by the usual diagram chase. A similar proof works for cokernels.
Again, this is generally not true at heights higher than 1, because * may not be flat-see [3, Sect. 2.2].

Definition 2.14 An extended comodule is one of the form
When working with uncompleted Hopf algebroids, one next constructs enough injectives in the comodule category by showing that a comodule extended from an injective R * -module is injective [27, A1.2.2]. One cannot do this in this case, because Mod ∧ * does not have enough injectives [2, Section A.2]. For example, if I is an injective L-complete Z p -module containing a copy of Z/ p, then one can inductively construct extensions Z/ p n → I and thus a nonzero map Z/ p ∞ → I -but this means that I is not L-complete. Thus, one instead has to use relative homological algebra. We take the following definitions from [3, Sect. 2].

Definition 2.15
A relative injective comodule is a retract of an extended comodule. A relative monomorphism of comodules is a comodule map M * → N * which is a split injection as a map of R * -modules. A relative short exact sequence is a sequence where the image of f is the kernel of g, and f is a relative monomorphism. A relative injective resolution of a comodule M * is a sequence

(a) Every comodule has a relative injective resolution. (b) The definition of Ext above is independent of the choice of resolution. (c) We have
is a relative monomorphism into a relative injective.) Statement (d) is then trivial, as we can take N * to be its own relative injective resolution. For (e), we use (c) and the adjunction

Definition 2.18
The primitives of a comodule M * are the R * -module which are naturally identified with a sub-R * -module of M * . If M * is extended, M * = * ⊗ R * K * , then the primitives of M * are the submodule 1 ⊗ K * .
In the following lemma and proof, all tensor products are over R * .

Lemma 2.19 A tensor product of an extended comodule with an arbitrary comodule
where the source has diagonal coaction and the target is extended. The map Proof This is an L-complete version of [15, Lemma 1.1.5], and the same proof works here. The formula for the primitives follows from the following observations. Define (Note that the map is part of the structure of the Hopf algebroid (R * , * ), though the multiplication on * itself may not factor through the R * -module tensor product * ⊗ * .) For fixed C * , g is a natural transformation of functors of M * valued in Mod ∧ R * . In the case M * = R * , it is an isomorphism (and is precisely the inverse given in Hovey's proof). Thus, g is an isomorphism for all pro-free modules M * , using exactness of the direct sum, and an isomorphism for all M * using the right exactness of the tensor product.

Proposition 2.20 Let R be a K (1)-local homotopy commutative ring spectrum such that R
Proof This spectral sequence is the same as the Bousfield-Kan homotopy spectral sequence of the cosimplicial object This is of the form By Proposition 2.8, we have which is a resolution of R * X by extended comodules, so that the E 2 page is precisely We next discuss convergence of the spectral sequence. The Bousfield-Kan spectral sequence converges conditionally to the homotopy of its totalization, so this spectral sequence converges conditionally to π * X if and only if the map is an equivalence. Questions of this type were first studied by Bousfield [6], and in the local case by Devinatz-Hopkins [10]. We recall their definitions here: Definition 2.21 [10, Appendix I] Let R be a K (1)-local homotopy commutative ring spectrum. The class K (1)-local R-nilpotent spectra is the smallest class C of K (1)local spectra such that: (2) C is closed under retracts and cofibers, (3) and if X ∈ C and Y is an arbitrary K (1)-local spectrum, then X ∧ Y ∈ C.
Finally, we write down a change of rings theorem, generalizing [18,Theorem 3.3].
is an isomorphism, and such that there exists a map g :

is the identity. Then for any A -comodule M, the induced map
is an isomorphism.
This statement can probably be obtained via the method of [16], but rather than taking a further detour into L-complete stacks, we have instead followed [9] (where this theorem is proved in the very similar setting of complete Hopf algebroids). We begin with some definitions and lemmas. In

Lemma 2.24 Let
Proof This is a variant of the Yoneda lemma. One can find the ring map A → B by evaluating φ on the object of h (B, B ) (B) corresponding to id B , and likewise one can find the map A → B by evaluating φ on the morphism of h (B, B ) ( B ) corresponding to id B . That these define an actual morphism of Hopf algebroids requires checking the commutativity of various diagrams of L-complete rings, which can be done in a similar fashion.
As Grpd is really a (2,1)-category, the functor category Fun(CAlg ∧ * , Grpd) is as well. We say that a morphism of L-complete Hopf algebroids is an equivalence if it is an equivalence in this functor category. In other words, f : Moreover, τ * g * and f * induce the same map on cohomology, Proof This is the L-complete verison of [9, 1.15, 1.17], and has the same proof.
is an equivalence of categories. Moreover, the induced map is an isomorphism.
Proof This follows immediately from the previous lemma.

Lemma 2.27
Let C be a small category. Suppose that f : F → G is a natural transformation of functors C → Grpd such that: (ii) for each c ∈ C and x ∈ G(c), there is an essential lift of x -in other words, a pair (iii) and these lifts can be chosen functorially in c ∈ C. In other words, there is a choice of essential lift for every c and x such that, given h : Then f is an equivalence in the functor 2-category Fun(C, Grpd).

Remark 2.28
If f is only assumed objectwise fully faithful and essentially surjective, then an attempt to construct an inverse will, in general, only produce a pseudonatural transformation g : G → F. Thus, some additional hypothesis like (iii) above is required.

Proof of Lemma 2.27
Let ( x, α x ) be the lifts functorial in C described in (ii) and (iii). Define g : G → F as follows: This exists by (i).
One has to check that g is a natural transformation, or in other words that, given This is an immediate consequence of (iii). It remains to show that f and g are inverse equivalences, or in other words that there are natural transformations For y ∈ F(c), let η y : y → g c f c (y) be the unique lift of the identity morphism It is easy to check that these are natural transformations for each c, and natural in c.

Proof of Proposition 2.23
It suffices to prove that f is an equivalence of Hopf algebroids. We do this by checking the conditions of Lemma 2.27, meaning that for each R, the functor of groupoids is fully faithful and essentially surjective, and the essential lifts can be chosen functorially in R. Given x, y ∈ h (B, B ) (R), we can identify Maps(x, y) with the set of maps φ : . This proves that f * is fully faithful.
An object x ∈ h (A, ) (R) is given by a ring map x : A → R. Precomposing with g : Thus, f * is essentially surjective. Moreover, as the essential lifts are given by precomposing with a morphism of rings, they are clearly functorial in R.

The Hopf algebroids for K and KO
The K -theory spectrum has a group action by Z × p via E ∞ ring maps. For k ∈ Z × p , we write ψ k for the corresponding endomorphism of K , called the kth Adams operation. On homotopy, writing u for the Bott element, we have (2.29) The group Z × p has a maximal finite subgroup μ of order p − 1 (if p is odd) or 2 (if p = 2), and we write K O = K hμ . (This agrees with the p-completion of the real K -theory spectrum at p = 2 and 3). Then K O inherits an action by the topologically cyclic group Z × p /μ, which we also refer to as an action by Adams operations. The Adams operations give us a way to analyze the completed cooperations algebras as adjoint to the map Likewise, there is a map Theorem 2.30 (cf. [16]) The map is an isomorphism. It induces an isomorphism of Hopf algebroids where the latter has the following Hopf algebroid structure: is given by sending a function f to the function

Analogous statements hold for K O.
Note that η L = η R in degree zero, so that K 0 K and K O 0 K O are Hopf algebras.

Remark 2.31
The reader should note that it follows from Mahler's theorem that K * K and K O * K O are pro-free over K * and K O * respectively.

Remark 2.32
The cooperations algebra K * K carries two actions by Adams operations, coming from the two copies of K . Given f ∈ K 0 K , we can represent f both as a map f : S 0 → K ∧ K and as an element of Maps cts (Z and Now suppose that M * is an L-complete K * K -comodule with coaction ψ M * . Then there is a map Here Hom Mod ∧ * is the ordinary space of maps between Z p -modules, which is automatically L-complete when the modules are L-complete [2, Sect. A.2]. As Mod ∧ * is closed symmetric monoidal, this map is adjoint to one of the form In the case where M * is p-complete, this defines a continuous group action by Z × p on M * . If M * is merely L-complete, then one still gets a group action by Z × p on M * , and the only reasonable definition of "continuous group action" appears to be that it extends to a map of L-complete modules of the form (2.35). In either case, we call this the action by Adams operations on M * . Of course, if M * is the completed K -theory of a spectrum X , M * = π * L K (1) (K ∧ X ), then this action is induced by the Adams operations on K .
If M * is p-complete then the standard relative injective resolution of M * , is isomorphic to the complex of continuous Z × p -cochains, Thus, we can identify the relative Ext of Definition 2.16 with continuous group cohomology: Similar remarks apply to K O: a K O * K O-comodule M * has a continuous group action by Z × p /μ, and if M * is p-complete, we have (Again, if M * is merely L-complete, then one should instead take these Ext groups as a definition of continuous group cohomology with coefficients in M * !) One recovers the familiar K (1)-local Adams spectral sequences based on K O as an immediate consequence.

Proposition 2.36 Let X be a K (1)-local spectrum. Then there is a strongly convergent Adams spectral sequence
The spectral sequence always collapses at the E 2 page.
Proof The calculation of the E 2 pages follows from the above discussion and Proposition 2.20. Since Z × p /μ has cohomological dimension 1, the spectral sequence collapses at E 2 , and in particular, converges strongly. To establish that the limit is π * X , we must show that every K (1)-local X is K (1)-local K O-nilpotent (see Proposition 2.22). But the sphere is a fiber of copies of K O (see (3.1) below), so S is K (1)-local K Onilpotent, so the same is true for arbitrary X .

Cones on
There is a class ζ in π −1 L K (1) S which vanishes in the homotopy of K (1)-local tmf (as well as K (1)-local K and K O), simply because these spectra have no nontrivial homotopy in degree −1. As a result, the cone C(ζ ) and the E ∞ -cone T ζ on ζ mediate between the sphere and tmf. We describe these spectra in this section, which is mostly an exposition of material found in [14].

The spectrum cone on
Recall from Sect. 1.2 that g is a fixed topological generator of Z × p (or, when p = 2, a fixed element of Z × 2 mapping to a topological generator of Z × 2 /μ), and that ω is a fixed generator of μ. The fiber sequence gives a long exact sequence on homotopy groups Recall that the action of ψ g on π 0 K O is trivial, so the connecting homomorphism gives an isomorphism This isomorphism does depend on the choice of topological generator g. We let ζ := ∂(1), and we define C(ζ ) to be the cone on ζ , i.e. the cofibre Since π −1 K O = 0, we get a morphism of cofibre sequences (3. 2) The morphism ι is a nullhomotopy of η • ζ .
Since ζ is nullhomotopic in K O, the top cofibre sequence in (3.2) splits after In fact, there is a canonical splitting, coming from the diagram We see that

Proposition 3.4 Under the morphism
the element a is mapped to the constant function 1 and b is mapped to the unique group homomorphism sending g to 1.
Proof We use the formulas from Theorem 2.30 and (2.34). For the sake of brevity, let a and b be the images of a and b under K O ∧ι, which we think of as continuous functions from the topologically cyclic group Z × p /μ to Z p . Since m(a) = 1, by Theorem 2.30, the function a satisfies a(1) = 1. We also have and by (2.34), together with the fact that ψ g acts trivially on K O 0 , a(g −1 n) − a(n) = 0 for any n ∈ Z × p /μ. Together with continuity of a, this implies that a is constant. Applying the same arguments to b, we obtain It follows that b(g k ) = k for any k ∈ Z, and by continuity, for any k ∈ Z p .

Corollary 3.5 The map
Proof One just has to observe that the functions a, b are linearly independent in

Corollary 3.7 We have
where the Adams operations fix a and satisfy is free on the same generators as a K -module. Since the generators of K * C(ζ ) are in the image of K O * C(ζ ), they are fixed by ψ ω .

The E ∞ -cone on
The previous subsection allows us to start the analysis of the E ∞ -cone on ζ . Recall from Sect. 1.2 that we write CAlg for the category of K (1)-local E ∞ -algebras, and P(X ) for the free E ∞ -algebra on X .

Definition 3.8
The spectrum T ζ is defined by the following homotopy pushout square in the category CAlg.
Just as C(ζ ) classifies nullhomotopies of ζ in spectra equipped with a map from S 0 , T ζ classifies nullhomotopies of ζ in E ∞ -algebras. That is, there is a natural equivalence of mapping spaces where Sp S 0 / is the category of spectra equipped with a map from S 0 . In particular, there is a canonical morphism C(ζ ) → T ζ , and a canonical factorization making K O a commutative T ζ -algebra. We also have the following.

Proposition 3.11
Let R be any E ∞ -algebra such that π −1 R = 0. Then there is an equivalence in CAlg R : Proof Smashing R with the pushout diagram for T ζ produces a pushout diagram This morphism is itself adjoint to the map ζ : S −1 → S 0 → R ∧ S 0 in S-modules. As π −1 R = 0, this map is null, which implies R ∧ ζ is null in Rmodules. Thus the morphism R ∧ ζ in CAlg R is adjoint to the null morphism. So the pushout diagram is in fact the pushout of the following

Corollary 3.12 There is an equivalence of K O-algebras
More explicitly, we can choose this equivalence so that the following diagram commutes: Here, the map is the unit on the left summand, and the inclusion of the generator on the right one. This allows us to calculate the K O-homology of T ζ completely.

Corollary 3.13
As a θ -algebra over K O * , Proof This is a consequence of the E ∞ equivalence K O ∧ T ζ K O ∧ P(S 0 ), McClure's theorem A.6, and the commutativity of (3.2). Since b is in the image of K O 0 C(ζ ), its Adams operations follow from Corollary 3.6. As the Adams operations on K O * are known and ψ g commutes with θ , the calculation of ψ g (b) determines the Adams operations on all of K O * T ζ = K O * ⊗ T(b). Tensoring up to K , one also gets the formula for K * T ζ .

The homotopy groups of T
In this subsection we compute the homotopy groups of T ζ . This has been done before in [14] and [21]. As this calculation is important for the work on co-operations to follow, we review it here in detail.
We may approach the homotopy groups of T ζ using the K O-based Adams spectral sequence, which we saw in Proposition 2.36 takes the form 14) The key point of Hopkins' calculation in [14] is as follows: Theorem 3.15 [14,21]. The K O-homology of T ζ is an extended K O * K O-comodule. More specifically, there is an isomorphism of K O * K O-comodules

and T( f ) has trivial coaction.
This allows an immediate derivation of π * T ζ .

Corollary 3.16 The homotopy groups of T ζ are
Proof By Proposition 2.17, the cohomology of an extended comodule is concentrated in degree zero, and The proof of Theorem 3.15 will take up the remainder of this section. As it is somewhat involved, let us give an outline first. The map π : which we also denote by π . This is a map of θ -algebras and of K O 0 K O-comodules, and there are also natural Hopf algebra structures on both objects making it a Hopf algebra map. We also consider the leaky λ-ring structures of Definition A.10. Using all this structure, we prove that the Hopf algebra kernel is just T( f ), and construct a coalgebra splitting. This implies that K O 0 T ζ is an induced K O 0 K O-comodule by a general theorem about Hopf algebras. Finally, one can explicitly construct Z × p /μinvariant elements in nonzero degrees of K O * T ζ , multiplication by which allows us to transport the result in degree zero to nonzero degrees.
We claim that This is true for i = 0. Suppose it has been proven for i = 0, . . . , n − 1. Then and thus . . ] by the monomials b n 0 0 b n 1 1 · · · in which all n i < p and all but finitely many of the n i are zero. By Lemma 2.6, T(b) is pro-free over T( f ). In particular, the unit map is an injection.
It will be helpful to make the identification using the continuous group isomorphism By proposition 3.4, b ∈ K O 0 K O goes to the identity under this identification.
This is a θ -algebra map, determined by the fact that π(b) = id. By Proposition A.4, ψ p (π(b)) = π(b). Thus, there is an induced map where T( f ) → Z p sends all θ k ( f ) to 0. Definition 3. 20 We give T(b) and Maps cts (Z p , Z p ) the leaky λ-ring structures L(T(b)), L(Maps cts (Z p , Z p )) of Definition A. 10. In each of these λ-rings, the Adams operations ψ k associated to the λ-ring structure are the identity for k prime to p, while ψ p is equal to the operation ψ p associated to the θ -algebra structure.
By Example A.11, the λ-operations on φ ∈ Maps cts (Z p , Z p ) are given by n .

Lemma 3.21 The map
is a map of λ-rings.
Proof This map is obtained by applying the functor L to a map of ψ-θ -algebras.

Proposition 3.22 The map
is an isomorphism.
Proof First, let's show the map is surjective. Since the map T(b) → Maps cts (Z p , Z p ) is a map of λ-rings with id Z p in its image, λ k (id) is also in its image for all k ∈ N.
Observe that λ k (id) is precisely the binomial coefficient function β k : x → x k . It is a theorem of Mahler [29, 4.2.4] that Maps cts (Z p , Z p ) is generated (as a complete Z p -module) by the binomial functions β k for k ∈ N. Thus the map π : T(b) → Maps cts (Z p , Z p ) is surjective.
We now introduce an alternative description of Maps cts (Z p , Z p ). Any element of Z p has a unique description a = We claim that π(b n ) ≡ α n mod p for all n. We proceed by induction: first, π(b 0 ) = id is congruent to α 0 mod p. Suppose we have shown that Thus, applying π to (3.23) and using the fact that ψ p is the identity on Maps cts (Z p , Z p ), we get id ≡ α 0 + pα 1 + · · · + p n−1 α n−1 + p n π(b n ) mod p n+1 .
But of course id = p i α i on the nose, so solving for π(b n ) gives We can now compute the kernel of π . First note that it contains each This is just because it's a θ -algebra map whose kernel contains f , and was needed to define the map π in the first place. We want to show that the θ n ( f ) generate the kernel of π . But we know that Since Maps cts (Z p , Z p ) is a free complete Z p -module, we have that and so ker(π/ p) ∼ = ker(π ) ⊗ F p .
Since T(b) is p-adically complete and torsion free, it follows that the elements ψ p (b n ) − b n also generate ker(π ), concluding the proof. For the second statement, it suffices to show that the given map π : T(b) → Maps cts (Z p , Z p ) is a Hopf algebra map. This can be checked after tensoring with Q, in which case it suffices to check that π(ψ p n (b)) is still primitive. However, we have seen that each ψ p n (b) goes to the identity of Z p , which is primitive in Maps cts (Z p , Z p ).

Lemma 3.25
The map π : T(b) → Maps cts (Z p , Z p ) admits a coalgebra section s : Proof By Mahler's theorem cited above, Maps cts (Z p , Z p ) is a free complete Z pmodule on the binomial functions β k : x → x k , for k ∈ N. In the proof of Proposition 3.22, we saw that, in terms of the λ-ring structure on T(b), π(λ k (b)) = β k . We can define a continuous Z p -module section by It remains to see that this is also a coalgebra section. It follows from Lemma A.13 that the coproduct is a morphism of λ-algebras.
The binomial functions have comultiplication Therefore, So s is a coalgebra map.
Equipped with the above lemmas, we can finally prove Theorem 3.15. We begin by proving the degree zero part.

Proposition 3.26 There is an isomorphism of T( f )-modules and K O 0 K O-comodules
Proof Note: For the duration of this proof, we will make all completions explicit.
We wish to show that At this point, we have maps of complete Hopf algebras together with a coalgebra section s of π . We claim that is the desired isomorphism. This uses a variant of the arguments in [26,Sect. 1]. The situation is slightly complicated by the omnipresence of completion, as well as the fact that the objects involved are not graded in any manageable way. First, we handle the completions. Let A be the uncompleted polynomial ring and likewise let B be the uncompleted polynomial ring on the θ n (b). Let C be the sub-Z p -algebra of Maps cts (Z p , Z p ) consisting of those functions which can be written as polynomials with Q p coefficients. As an uncompleted Z p -module, C is free on the β n . The sequence (3.27) restricts to a sequence of maps of Hopf algebras (3.28) such that C = B ⊗ A Z p , together with a coalgebra section s of π . Write φ for the map Since φ is the completion of φ, it suffices to prove that φ is an isomorphism of Amodules and C-comodules. Now, φ is clearly an A-module homomorphism, and it is also a map of C-comodules since s is a map of coalgebras. We will show that φ is injective by the method of [26,Proposition 1.7]. Note that C has a coalgebra grading in which the degree of β n is n. This induces filtrations on A ⊗ C and B ⊗ C, in which and likewise for B ⊗ C. Consider the map Using the comultiplicativity of s, we see that Furthermore, since ν is a left A-module map, it preserves the filtration. Thus, there is an induced map ν on associated graded objects. However, as C is the direct sum of the C q , the associated graded objects are simply A ⊗ C and B ⊗ C. Once again, one computes that As ν is a left A-module map, we can identify it with Since C is flat over Z p , this map is injective. Thus ν is injective, so φ is injective, as desired.
For surjectivity, we use a version of [26,Proposition 1.6]. Filter A as follows: the elements of filtration ≥ s are the polynomials in f , θ( f ), θ 2 ( f ), . . . all of whose terms are of degree ≥ s. Giving B the analogous filtration, the map i : A → B is a filtered A-module map, and the counit : A → Z p kills the ideal of positively filtered elements. The A-module structure on C factors through , and we give C the trivial filtration C = C ≥0 = C ≥1 = · · · . Then π : B → C is also filtered.
The direction (⇒) holds because the tensor product is right exact. For the direction (⇐), let N = coker(g). The A-module N receives a filtration in an evident way. Again using right exactness of the tensor product, we have that If Z p ⊗ A g is surjective, then Z p ⊗ A N = 0, so N = 0 by Claim 1. (Since completion is neither left nor right exact in general, we need to work with the uncompleted tensor product here.) Finally, φ : A ⊗ C → B is a filtered A-module map whose source and target are nonnegatively filtered. We have By Claim 2, φ is surjective.
Proof of Theorem 3. 15 We have already constructed an isomorphism of To extend this to a map one has identify the image of K O * in K O * T ζ , which will consist of elements which are invariant under the Adams operations. First, suppose that p > 2. Since g ∈ Z × p maps to a topological generator of Z × p /μ, we have g p−1 ∈ 1 + pZ p . Write g p−1 = 1 + h where h ∈ pZ p . Then the series . Indeed, each term has p-adic valuation at least n − v p (n!), and these converge to ∞ with n.
As K O 0 T ζ is an extended comodule, the same follows for K O * T ζ , and we obtain The isomorphism with K O * ⊗ T( f ) is given by mapping v 1 to v 1 . Now suppose that p = 2, in which case K O * is generated by η ∈ K O 1 , v = 2u 2 ∈ K O 4 , and w = u 4 ∈ K O 8 , where u ∈ K 2 is the Bott element. We have that g 2 = 1+h where h ∈ 4Z 2 . Again, this means that the series g −2b = (1 + h) −b converges, and we can define v = g −2b v, w = g −4b w. By the same arguments, K O 4 * T ζ is an etended comodule. To deal with the rest, we note that with F 2 and noting that K O 0 T ζ is flat over Z 2 , we obtain the desired result.

Remark 3.29
As we mentioned earlier, Hopkins' argument from [14] has errors. In particular, he correctly claims that the map i⊗s mult is an isomorphism. However, he argues this by asserting that the inverse to this map is given by (

1−s•π)⊗π
But this map simply cannot be the inverse, indeed it is not even injective. To see this, let β n denote the nth binomial coefficient function. The section s is a map of coalgebras and the diagonal on the β n satisfy the Cartan formula. Thus Thus, under the above map, one computes that Since s is a section, π s = 1. Note that Note that this includes the case when j = 0, in which case β j = β 0 = 1. Thus the above map has a nontrivial kernel, and so is not injective.

Co-operations for T
We saw in Proposition 3.11 that K O * T ζ ∼ = K O * ⊗ T(b). As T(b) is a completion of a polynomial ring, K O * T ζ is pro-free over K O * . Moreover, we have an equivalence of K O-modules in Sp, So it follows from Proposition 2.8 that, (4.1) Recall that the K O * K O-comodule structure is given by an action of the group Z × p /μ. In this case, the action comes from the diagonal action on the two tensor factors, so that As we saw in the previous section, the computation of π * T ζ followed from knowing that K O * T ζ was an extended comodule. The same strategy allows us to compute the co-operations algebra π * (T ζ ∧ T ζ ).

Lemma 4.2 A tensor product of extended K O 0 K O-comodules is extended. More precisely, if M and N are Z p -modules, then
Proof This is the special case of Lemma 2.19 in which both comodules are extended. The formula for the primitives follows from the formula there, using Theorem 2.30 to describe the maps.
In the following, we will frequently use x and x to denote the image of x along respectively the left and right units of a Hopf algebroid.

Theorem 4.3 There is an isomorphism of θ -algebras
We saw in the proof of Theorem 3.15 that K O * T ζ is an extended comodule. The lemma then implies that K O * (T ζ ∧ T ζ ) is extended. Using Proposition 2.36, there is an additive isomorphism By Corollary 3.16, this is isomorphic to where f and f come from the left and right copies of π 0 T ζ respectively. Note that, as the isomorphism 15 is an isomorphism of comodules but not of comodule algebras; the above isomorphism is only additive. We can nevertheless identify the multiplicative structure on π * (T ζ ∧ T ζ ) by locating the primitive elements identified above inside the ring , namely that generated by the left unit on v 1 (or by the left unit on η, v, and w if p = 2).
We still have to identify the Maps cts (Z p , Z p ) factor. Lemma 4.2 tells us that, under the isomorphism this factor is precisely where s is as defined in Lemma 3.25. That is, By Mahler's theorem, the submodule of f ∈ Maps cts (Z p × Z p , Z p ) satisfying the condition of (4.4) is spanned by Thus, the invariant Maps cts (Z p , Z p ) factor in K O 0 (T ζ ∧ T ζ ) is spanned by In particular, the sub-λ-algebra of K O 0 (T ζ ∧ T ζ ) generated by b − b contains this Maps cts (Z p , Z p ). But this is the same as the sub-θ -algebra generated by b − b. Let The formula ψ p (b) − b = f , and the analogous one for f , show that Thus, there is an epimorphism To see that this is an isomorphism, note that Proposition 3.11 implies that as π 0 (T ζ )-modules. That is, it is a free θ -algebra on two generators. But the left-hand side of Eq. (4.5) is free on the generators f and , and any nontrivial quotient of it would not be free on two generators. Thus, we have This concludes the proof.

K (1)-local tmf
We continue to work K (1)-locally, and fix p = 2 or 3, so that j = 0 is the unique supersingular j-invariant. It is simple to extend this story to larger primes with a single supersingular j-invariant; slightly more complicated to extend it to other primes; but in neither case is it quite as interesting. As in section 4, the statements in this section are due to [14].
Since K O 0 T ζ is the induced K O 0 K O-comodule on π 0 T ζ , any such θ -algebra map extends uniquely to a ψ-θ -algebra map Proof This is a calculation using q-expansions. See [14, 7.1].
It follows that the map q : T ζ → tmf induces a surjective map on π 0 , Thus, θ( f ) maps to some completed polynomial in j −1 . Since f ≡ j −1 mod p, this can also be written as a completed polynomial in f , say h( f ). It follows that the kernel of q is the θ -ideal generated by θ( f ) − h( f ).

Lemma 5.4
The map of θ -algebras F : Proof This is similar to Lemma 1. Again, let us write (See Theorem A.5 for θ i .) We will prove by induction that When i = 0, mod ( p, b 0 )).
Suppose that we have proved (5.5) for i = 0, . . . , n − 1. Then for these values of i, We also have Finally, because h is a completed polynomial over Z p . Putting this all together, The left-hand side is congruent to p n F(x n ) modulo this ideal by (5.6), which proves (5.5). It follows that the map makes the target into a free module over the source, by the same argument as in . By Lemma 2.6, T(b) is pro-free over T(x). This finishes the proof of the lemma.
Theorem 5.7 [14, 7.2] There is a homotopy pushout square of K (1)-local E ∞ rings, Proof Let Y be the homotopy pushout of the above square, so Since θ( f ) = h( f ) in π 0 tmf, there is a map Y → tmf, which we will show is an isomorphism on homotopy groups. We note that K O * P(S 0 ) → K O * T ζ is precisely the map of the previous lemma, tensored by K O * . By Lemma 2.5, K O * T ζ is pro-free over K O * P(S 0 ). Then by Proposition 2.8 and the previous lemma, we have the Künneth formula, By Proposition 3.26 and the proof of Theorem 3.15, we have an isomorphism (Here the quotient is by the θ -ideal generated by θ( f ) − h( f ).) Proof One has an E ∞ map T ζ → K O, which by arguments similar to the ones above fits into a pushout square of E ∞ rings The left-hand vertical map sends the θ -algebra generator This induces a map from the E ∞ cofiber of the composite, namely tmf, to the E ∞ cofiber of the right-hand map, namely K O.
On coefficients, the map r is just Despite the obvious splitting of r at the level of coefficients, it is not clear whether or not there exists an E ∞ map from K O to tmf.

Co-operations for K (1)-local tmf
The preceding Theorem 5.7 gave a presentation of K (1)-local tmf in terms of finitely many E ∞ cells. We can now use this presentation to describe the K (1)-localization of tmf ∧ tmf.

Theorem 6.1
The homotopy groups of tmf ∧ tmf are given by Proof Write F : P(S 0 ) → T ζ for the map sending the generator We saw in the previous section that F induces a pro-free map on K O-homology, and that Therefore, is an extended K O * K O-comodule, and (F ⊗ F) factors through its fixed points, which are By the arguments of Theorem 5.7, K O * (tmf ∧ tmf) is also extended, and

Remark 6.2
For a more modular presentation of this ring, recall from Proposition 5.
we can equivalently write We now consider the Hopf algebroid for K (1)-local tmf. To obtain tmf * tmf from T ζ, * T ζ , we take the θ -algebra quotient induced by the relation θ( f ) = h( f ), and the same relation for f . We obtain where again the quotient is by a θ -ideal. This formula should be compared to the analogous one for K (1)-local K Ocooperations: as a θ -algebra, we have where the last isomorphism follows from Proposition 3.22. That is, K O * K O is generated as a θ -algebra over K O * by a single generator b, with an algebraic relation between b and pθ(b). Likewise, tmf * tmf is generated over tmf * by a single generator , with an algebraic relation over the coefficient ring Z p [ f ] that relates , θ( ), and pθ 2 ( ). One can think of this as a second-order version of the θ -algebraic structure The former allows the simple computation of the K O-based Adams spectral sequence for arbitrary X : its E 2 page is just the group cohomology and as this is concentrated on two lines, the spectral sequence collapses at E 2 and always converges. As it turns out, very similar statements are true for tmf.

Proposition 6.4
The left unit tmf * → tmf * tmf is pro-free.
Proof While one can prove pro-freeness algebraically by applying Lemmas 2.5 and 2.6 to the formula (6.3), it is easier to use Laures's [21,Corollary 3], which gives an additive equivalence of homology theories and correspondingly an additive equivalence of K (1)-local spectra Thus, to show that tmf * tmf is pro-free over tmf * , it suffices to show that K O * tmf is pro-free over K O * . From Lemma 2.7, one observes that the property of K O * X being pro-free over K O * is closed under coproducts. As K O * K O is pro-free over K O * , and tmf is a coproduct of copies of K O, K O * tmf is also pro-free.

Corollary 6.5
There is an L-complete Hopf algebroid (tmf * , tmf * tmf). For any K (1)-local spectrum X , the K (1)-local Adams spectral sequence based on tmf is conditionally convergent and takes the form Proof Since tmf * → tmf * tmf is pro-free, one has an L-complete Hopf algebroid by Definition 2.9. Then by Proposition 2.20, the E 2 page of the Adams spectral sequence has the form described.
To establish convergence, one needs to show that X is K (1)-local tmf-nilpotent. Recall from [10, Appendix 1] and [6] that this is the largest class of K (1)-local spectra containing tmf and closed under retracts, cofibers, and K (1)-local smash products with arbitrary spectra. Now, multiplication by j −1 gives a cofiber sequence then shows that the sphere is K (1)-local tmf-nilpotent. This clearly implies that an arbitrary spectrum is K (1)-local tmf-nilpotent.
We can now prove Theorem B. Theorem 6.6 Suppose that tmf * X is pro-free over tmf * . Then there is a natural isomorphism Proof The ring map tmf → K O induces a map of Hopf algebroids, The map tmf * → K O * sends j −1 to zero, and thus sends f = j −1 + O( pj −1 , j −2 ) to zero as well. We have We need to identify the image of in K O * K O. Consider the commuting square The horizontal maps are both inclusions of K O * K O-primitives, and, in particular, injective. Going from tmf * tmf to is the identity map on Z p . Using the Hopf algebroid formulas found in Theorem 2.30, together with the group isomorphism Z × p /μ ∼ = Z p , we have In the notation of Lemma 2.19, the primitives are included into Maps cts (Z p × Z p , K O * ) via precomposition with . This proves that the map tmf * tmf → K O * K O sends to b. It follows that the map is an isomorphism.
Using the fiber sequence There is a pushout square of L-complete rings The top horizontal map is pro-free by Lemma 1. By Lemma 2.5, the bottom horizontal map is also pro-free.
Since tmf * X is pro-free over tmf * , we have By the change of rings theorem, Proposition 2.23, the induced map on Ext is an equivalence.

Corollary 6.7
The K (1)-local tmf-based Adams spectral sequence for the sphere collapses at E 2 , where it is concentrated on the 0 and 1 lines.
Proof This follows immediately from the above theorem and Proposition 2.36.

Connections to number theory
In this section, we recall the relationship between K (1)-local tmf and p-adic modular forms in the sense of Katz.
has kernel a connected group scheme of rank p, which the trivialization α identifies with μ p ⊆ G m . The isogeny F 0 deforms uniquely to one of the form and again α identifies the kernel of F with μ p . Then there is an induced trivialization The mapping defines a ring endomorphism ψ p : V ∞ → V ∞ . By [5,Lemma 5.4], the Z × p -action and the operator ψ p define a ψ-θ -algebra structure on V ∞ . Now, since M triv ell = Spf V ∞ is an affine formal scheme and a Galois cover of M ord ell , one can compute the cohomology of M ord ell , and in particular, the ring (M ord ell , O M ord ell ) of ordinary p-adic modular forms, as the group cohomology of Z × p acting on Spf V ∞ .
Theorem 7. 5 We have Thus, the ring of weight 0 ordinary p-adic modular forms is There are several different ways to prove this, but below, we show how it can be recovered from known information about tmf.
First, we recall from [5] that the (uncompleted) moduli of generalized elliptic curves can be equipped with a sheaf O der of locally even periodic E ∞ ring spectra, such that there is an isomorphism of sheafs of rings where ω is the line bundle of invariant differentials on M ell . The global sections of O der over M ell are the unlocalized spectrum Tmf. By standard results on E ∞ -rings, this sheaf can be pulled back to M ord ell , and Likewise, it can be lifted along the pro-étale cover M triv ell → M ord ell .
Proposition 7. 6 The global sections of O der along M triv ell are Proof The group Z × p acts on the p-complete K -theory spectrum K by E ∞ automorphisms. Therefore, the constant sheaf K on Spf Z p descends to a locally even periodic spectral sheaf on M mult fg . (This is special to height 1: for example, the moduli of pcomplete, height ≤ 2 formal groups does not admit such a spectral enrichment. ) We have Also, the line bundle π 2 O der on M ≤1 fg is isomorphic to the line bundle that sends a formal group to its invariant differentials.
Thus, the pullback square (7.3) extends to a square of p-complete nonconnective spectral schemes. Taking global sections, which send pullbacks of nonconnective spectral schemes to pushouts of E ∞ -rings [23, 1.1.5.6], we get an equivalence of p-complete E ∞ -rings But K (1)-localization is smashing in the category of p-complete spectra (because E(1)-localization is smashing in the category of all spectra). So the left-hand side is precisely L K (1) (K ∧ tmf).

Corollary 7.7 We have
Proof The stack M triv ell = Spf V ∞ is an affine formal scheme, so the line bundles ω ⊗k have no higher cohomology, and the descent spectral sequence E s,2t 2 = H s (M triv ell , ω ⊗t ) ⇒ π 2t−s (K * tmf) collapses. Moreover, the line bundle ω is trivial. In fact, given a trivialized elliptic curve there is a canonical invariant differential on E, namely the pullback of the invariant differential dT /T ∈ ω G m along α −1 . Letting u be a basis for ω over V ∞ , the claim follows.
Proof of Theorem 7. 5 We will in fact show that V Z × p ∞ = π 0 tmf at all primes, which implies the above result at p ≤ 5.
The homotopy fixed points spectral sequence computing the homotopy groups of K (1)-local tmf thus takes the form If p > 2, then Z × p has cohomological dimension 1, so the spectral sequence collapses at E 2 . As we know, π * L K (1) tmf is concentrated in even degrees at these primes, which means that the group cohomology is concentrated in H 0 . The statement is immediate at p > 2.
At p = 2, we need to do a little more work, first by understanding the spectral sequence E * , * The convergence and integrality of the expression can be checked using the qexpansion Finally, λ is related to by a (non-explicit) composition-invertible power series. Thus, the ring of ordinary 2-variable modular forms is generated as a θ -algebra by j −1 , j −1 , and log c 4 − log c 4 16 .
At the prime 3, one likewise has the θ -algebra generator log c 6 − log c 6 9 .
Proposition A. 4 The θ -algebra structures on π 0 K = π 0 K O = Z p , on K O 0 K O, and on K 0 K are all given by ψ p = id.
Proof There is a unique θ -algebra structure on Z p satisfying the requirements of Definition A.2, and it is ψ p = id. As for K 0 K (the proof for K O 0 K O is similar), the multiplication map K ∧ K → K is an E ∞ -map. By Theorem 2.30, the map induced on π 0 is Thus, Moreover, ψ p commutes with the left action of the Adams operations, which act by It follows that for every k ∈ Z × p . Thus, ψ p acts by the identity. There is an adjunction where U is the forgetful functor, and T is the free θ -algebra functor. It is described explicitly as follows: Theorem A.5 [7, 2.6, 2.9] The free θ -algebra on a single generator x is a polynomial algebra: explicitly, where the elements θ n (x) are inductively defined so that ψ p n (x) = x p n + pθ 1 (x) p n−1 + · · · + p n θ n (x).
Theta-algebras function as an algebraic approximation to K (1)-local E ∞ -algebras, as was shown in the following form by [2,28] following work of [8]. Write P : Sp → CAlg for the free K (1)-local E ∞ -algebra functor. This is given by Since is a morphism of ψ-θ algebras, we have ψ p n • = • ψ p n .
Since ψ p n is a ring homomorphism for all n, we have Thus ψ p n (b) is a Hopf algebra primitive for all n. This uniquely determines the rest of the Hopf algebra structure.
This Hopf algebra is actually fairly classical. Recall that the additive group of p-typical Witt vectors of a p-complete ring R are classified by a Hopf algebra a 0 , a 1 , . . . ].
The map that sends θ n (b) to a n is then an isomorphism T(b) → W of θ -algebras and Hopf algebras. The element ψ p n (b) goes to the primitive element of W, w n = a p n 0 + pa p n−1 1 + · · · + p n a n , which represents the nth ghost component.
where P n and P m,n are certain universal polynomials with integral coefficients which can be recovered by taking λ n to be the nth elementary symmetric polynomial in infinitely many variables.
The category Alg λ of λ-rings is also symmetric monoidal. The tensor product is the ordinary p-complete tensor product with λ-operations defined by the Cartan formula, The notions of a λ-ring and a ψ-θ -algebra are closely related. In particular, given a λ-ring we can associate to it Adams operations. Indeed, one defines ψ n (x) = ν n (λ 1 (x), . . . , λ n (x)).
Here, ν n is the polynomial so that if σ k denotes the kth elementary symmetric polynomial in infinitely many variables x i and p k = x k i , p n (x) = ν n (σ 1 (x), . . . σ n (x)).
The operation ψ p satisfies the Frobenius congruence ψ p (x) ≡ x p mod p. Thus if R is a torsion-free p-complete λ-ring, then R is a ψ-θ -algebra. A partial converse also holds.
Theorem A.9 (Bousfield, [7, Theorem 3.6]) Let R be a p-complete ψ-θ -algebra. Then R has a unique λ-ring structure in which the Adams operations are the given ψ k and ψ p .
Definition A.10 As a result, there are not one but two functors from ψ-θ -algebras to λ-rings, both of which are the identity on underlying rings. The sealed functor, S : Alg ψ,θ → Alg λ , is the one given by Bousfield's theorem, and is an equivalence on the subcategories of torsion-free algebras. The leaky functor, L : Alg ψ,θ → Alg λ , first replaces all the ψ k by the identity for k prime to p, and then applies S to the result. We will also write L for the functor Alg θ → Alg λ which sets ψ k = 1 for k prime to p and then applies S to the result.

Example A.11
Recall that Z p has a unique θ -algebra structure, in which ψ p is the identity. Thus L(Z p ) is a λ-ring in which all Adams operations are the identity. The λ-operations are given by λ n (x) = x n [7, Example 1.3].
Lemma A.12 Both S and L are symmetric monoidal functors.
Proof As the operation of replacing the prime-top Adams operations with the identity is clearly monoidal, it suffices to prove that S is monoidal. For this, it suffices to prove that the inverse operation, from λ-rings to rings with Adams operations ψ n for n ∈ Z p , preserves the obvious tensor products, which is a simple calculation.