ON THE RING OF COOPERATIONS FOR 2-PRIMARY CONNECTIVE TOPOLOGICAL MODULAR FORMS. Contents. 1. Introduction Motivation: analysis of bo bo 2

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1 ON THE RING OF COOPERATIONS FOR -PRIMARY CONNECTIVE TOPOLOGICAL MODULAR FORMS M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA Contents 1. Introduction 1. Motivation: analysis of bo bo 3. Recollections on topological modular forms The Adams spectral sequence for tmf tmf and bo-brown-gitler modules The image of tmf tmf in TMF TMF Q : two variable modular forms Approximating by level structures Connective covers of TMF 0 (3) and TMF 0 (5) 47 References Introduction Note: This is a preliminary draft. In particular, parts of the last section are in the process of being written! The Adams-Novikov spectral sequence based on a connective spectrum E (E-ANSS) is perhaps the best available tool for computing stable homotopy groups. For example, HF p and BP give the classical Adams spectral sequence and the Adams Novikov spectral sequence respectively. To begin to compute with the E-ANSS, one needs to know the structure of the smash powers E k. When E is one of HF p, MU, or BP, the situation is simpler than in general, since in this case E E is an infinite wedge of suspensions of E itself, which allows for an algebraic description of the E -term. This is not the case The first author is partially supported by NSF CAREER grant DMS , second author is partially supported by NSF Postdoctoral Fellowship DMS , the third author is partially supported by NSF grant DMS , and the fourth author is partially supported by NSF Grant DMS

2 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA for bu, bo, or tmf, in which case the E page is harder to describe, and in fact, has not yet been described in the the case of tmf. Mahowald and his collaborators have studied the -primary bo-anss to a great effect: it gives the most efficient calculation of the v 1 -periodic homotopy in the sphere spectrum [LM87, Mah81]. The starting input in that calculation is a complete description of bo bo as an infinite wedge product of spectra that are a smash product of certain finite complexes with bo (as in [Mil75] and others). The finite complexes involved are the so-called integral Brown-Gitler spectra. Mahowald has worked on a similar description for tmf tmf, but concluded that no analogous result could hold. In this paper we use his insights to explore four different perspectives on -primary tmf-cooperations. While we do not arrive at a complete and closed-form description of tmf tmf, we believe our results have the potential to be very useful as a computational tool. (1) The E term of the -primary Adams spectral sequence for tmf tmf admits a splitting in terms of bo-brown-gitler modules: Ext(tmf tmf) = Ext(Σ 8i tmf bo i ). i () Modulo torsion, TMF TMF is isomorphic to a subring of the ring of integral two variable modular forms. (3) K()-locally, the ring spectrum (TMF TMF) K() is given by an equivariant function spectrum: (TMF TMF) K() Map(G /G 48, E ) hg48. (4) TMF TMF injects into a certain product of homotopy groups of topological modular forms with level structures. TMF TMF i Z, j 0 TMF 0 (3 j ) TMF 0 (5 j ). The purpose of this paper is to describe and investigate the relationship between these different perspectives Conventions. In this paper we shall always be implicitly working -locally. Homology will be taken with mod coefficients, unless specified otherwise. We will use Ext(X) to abbreviate Ext A (F, H X), the E -term of the Adams spectral sequence (ASS) for π X, and will let CA (H X) denote the corresponding cobar complex. Given an element x π X, we shall let [x] denote the coset of the ASS E -term which detects x.. Motivation: analysis of bo bo In analogy with the four perspectives described in the introduction, there are four primary perspectives on the ring of cooperations for real K-theory.

3 ON THE RING OF COOPERATIONS FOR tmf 3 (1) The spectrum bo bo admits a decomposition (at the prime ) bo bo i 0bo HZ i, where HZ i is the ith integral Brown-Gitler spectrum. () There is an isomorphism KO KO = KO KO0 KO 0 KO, and KO 0 KO is isomorphic to a subring of the ring of numerical functions. (3) K(1)-locally, the ring spectrum (KO KO) K(1) is given by the function spectrum: (KO KO) K(1) Map(Z /{±1}, KO ). (4) KO KO injects into a product of copies of KO: KO KO i Z KO..1. Integral Brown-Gitler spectra. The decomposition of bo bo above is a topological realization of a homology decomposition (see [Mah81], [Mil75]). Endow the monomials of the A -comodule H HZ = F [ 1,, 3,...] with a multiplicative weight by defining wt( i ) = i 1. The comodule H HZ admits an increasing filtration by integral Brown-Gitler comodules HZ i, where HZ i is spanned by elements of weight less than i. These A -comodules are realized by integral Brown-Gitler spectra HZ i, so that H HZ i = HZi. There is a decomposition of A(1) -comodules: H bo = (A//A(1)) =A(1) Σ 4i HZ i. This results in a decomposition on the level of Adams E -terms Ext(bo bo) = Ext(Σ 4i bo HZ i ) i 0 i 0 = Ext A(1) (Σ 4i HZ i ). This algebraic splitting is topologically realized by a splitting i 0 bo bo i 0bo HZ i. The goal of this section is to calculate the images of the maps bo HZ i bo bo in the decomposition above in order to illustrate the method used in our analysis of tmf tmf. Even in this case our perspective has some novel elements which provide a conceptual explanation for formulas obtained by Lellmann and Mahowald in [LM87].

4 4 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA.. Exact sequences relating HZ i. Just as with HZ i we define bo i to be the the submodule of (A//A(1)) = F [ 4 1,, 3,...] generated by elements of weight less than 4i. These submodules are discussed more thoroughly at the beginning of Section 4. With these in hand we have the following exact sequences: Lemma.1. There are short exact sequences of A(1) -comodules (.) (.3) 0 Σ 4j HZ j HZ j bo j 1 (A(1)//A(0)) 0, 0 Σ 4j HZ j HZ 1 HZ j+1 bo j 1 (A(1)//A(0)) 0. (Here bo i is the subspace of H bo spanned by monomials of weight 4i.) Proof. These short exact sequences are the analogs for integral Brown-Gitler modules of a pair of short exact sequences for bo-brown-gitler modules (see Propositions 7.1 and 7. of [BHHM08]). The proof is almost identical to that given in [BHHM08]. On the level of basis elements, the maps are given respectively by Σ 4j HZ j HZ j Σ 4j HZ j HZ 1 HZ j+1 i1 1 i 1 a i1 1 i {1, 1, } ( 1 a i1 i i1 where a is taken to be 4j wt( i1 i 3 ). The maps are given by 4i1+ɛ1 1 i+ɛ i3 HZ j bo j 1 (A(1)//A(0)) 3, i 3 ) {1, 1, } HZ j+1 bo j 1 (A(1)//A(0)) {4i i i3 ɛ1 3 1 ɛ 4i1, wt( 1 0, otherwise, i i3 3 ) 4j 4, where ɛ s {0, 1}. Define Ext A(1) (X) := Image ( Ext A(1) (X) v1 1 Ext A(1) (X) ). v 1 -tor The following lemma follows from a simple induction, using the fact that HZ 1 is given by 1 1. Sq 1 Sq

5 ON THE RING OF COOPERATIONS FOR tmf 5 Lemma.4. We have Ext A(1) (HZ i 1 ) = v 1 -tor { Ext(bo i ), Ext(bsp i 1 ), i even, i odd. Here, X i denotes the ith Adams cover. We deduce the following well known result (cf. [LM87, Thm..1]). Proposition.5. Ext A(1) (HZ i ) v 1 -tor = { Ext(bo i α(i) ), Ext(bsp i α(i) 1 ), i even, i odd. Here, α(i) denotes the number of 1 s in the dyadic expansion of i. Proof. This may be established by induction on i using the short exact sequences of Lemma.1, by augmenting Lemma.4 with the following facts. (1) All v 0 -towers in Ext A(1) (HZ i ) are v 1 -periodic. This can be seen as Ext A(1) (HZ i ) is a summand of Ext(bo bo), and after inverting v 0, the latter has no v 1 - torsion. Explicitly we have () We have 0 Ext(bo bo) = F [v ±1 0, u, v ]. Ext A(1) ((A(1)//A(0)) bo j ) v 0 -tors This follows from the fact that Ext A(0) (HZ j ) = F [v 0 ], v 0 -tors = Ext A(0) (bo j ) v 0 -tors = F [v 0 ]{1, ξ 4 1,..., ξ 4j 1 }. which, for instance, can be established by induction using the short exact sequences of Lemma The cooperations of KU and bu. In order to put the ring of cooperations for bo in the proper setting, we briefly review the story for bu. We begin by recalling the Adams-Harris determination of KU KU [Ada74, Sec. II.13]. We have an arithmetic square KU KU (KU KU) (KU KU) Q ((KU KU) ) Q,

6 6 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA which results in a pullback square after applying π KU KU Map c (Z, π KU ) Q[u ±1, v ±1 ] Map c (Z, Q [u ±1 ]). Setting w = v/u, the bottom map in the above square is given by f(u, v) = u n f(1, w) (λ u n f(1, λ)). We therefore deduce that KU KU = KU KU0 KU 0 KU, and continuity implies that KU 0 KU = {f(w) Q[w ±1 ] : f(k) Z (), for all k Z () }. Note that we can perform a similar analysis for KU bu: since bu and KU are K(1)- locally equivalent, applying π to the arithmetic square yields a pullback square with the same terms on the right hand edge. KU bu Map c (Z, π KU ) Q[u ±1, v] Map c (Z, Q [u ±1 ]). We therefore deduce that KU bu = KU KU0 KU 0 bu, with KU 0 bu = {g(w) Q[w] : g(k) Z (), for all k Z () }. Consider the related space of -local numerical polynomials: NumPoly () := {h(x) Q[x] : h(k) Z (), for all k Z () }. The theory of numerical polynomials states that NumPoly () is the free Z () -module generated by the basis elements ( ) x x(x 1) (x n + 1) h n (x) := =. n n! We can relate KU 0 bu to NumPoly () by a change of coordinates. A function on Z () can be regarded as a function on Z () via the change of coordinates Z () Z () k k + 1 Observe that k(k 1) (k n + 1) k(k ) (k n + ) = n! n n! (k + 1)((k + 1) 3) ((k + 1) (n 1)) = n. n! We deduce that a Z () basis for KU 0 bu is given by (w 1)(w 3)... (w (n 1)) g n (w) = n. n! (Compare with [Ada74, Prop. 17.6(i)].)

7 ON THE RING OF COOPERATIONS FOR tmf 7 From this we deduce a basis of the image of the map bu bu KU KU. In [Ada74, p. 358] it is shown that this image is the ring bu bu v 1 -tor = (KU bu Q[u, v]) AF 0, where AF 0 means the elements of Adams filtration 0. Since the elements, u, and v have Adams filtration 1, this image is equivalently described as bu bu v 1 -tor = KU bu Z () [u/, v/]. To compute a basis for this image we need to calculate the Adams filtration of the elements of this basis {g n (w)}. Since w has Adams filtration 0 we need only compute the -divisibility of the denominators of the functions g n (w). As usual in this subject, for an integer k Z let ν (k) be the largest power of that divides k and let α(k) be the number of 1 s in the binary expansion of k. Then and so ν (n!) = n α(n) AF(g n ) = α(n) n. The following is a list of the Adams filtration of the first few basis elements: n binary AF(g n ) It follows (compare with [Ada74, Prop. 17.6(ii)]) that the image of bu bu in KU KU is the free module: bu bu v 1 -tor = Z (){ max(0,n m α(n)) u m g n (w) : n 0, m n}. The Adams chart in Figure.3 illustrates how the description of bu bu given above along with the Mahler basis can be used to identify bu bu as a bu -module inside of KU KU.

8 8 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA 15 Figure.1. bu bu g 0 g 1 g g 3 g 4 g 5 10 g 6 g 7 15 g The cooperations of KO and bo. Adams and Switzer computed KO KO along simlar lines [Ada74, Sec. II.17]. There is an arithmetic square KO KO (KO KO) (KO KO) Q ((KO KO) ) Q. This results in a pullback when applying π : KO KO Map c (Z /{±1}, π KO ) Q[u ±, v ± ] Map c (Z /{±1}, Q [u ± ]). (One can use the fact that KU is a K(1)-local C -Galois extension of KO to identify the upper right hand corner of the above pullback.) Continuing to let w = v/u, the bottom map in the above square is given by f(u, v ) = u n f(1, w ) ( [λ] u n f(1, λ ) ). We therefore deduce that KO KO = KO KO0 KO 0 KO, with KO 0 KO = {f(w ) Q[w ± ] : f(λ ) Z, for all [λ] Z /{±1}}.

9 ON THE RING OF COOPERATIONS FOR tmf 9 Again, KO bo is similarly determined: since bo and KO are K(1)-locally equivalent, applying π to the arithmetic square yields a pullback square with the same terms on the right hand edge: KO bo Map c (Z /{±1}, π KO ) Q[u ±, v ] Map c (Z /{±1}, Q [u ± ]). We therefore deduce that KO bo = KO KO0 KO 0 bo, with KO 0 bo = {f(w ) Q[w ] : f(λ ) Z, for all [λ] Z /{±1}}. To produce a basis of this space of functions we use the q-mahler bases developed in [Con00]. First note that there is an exponential isomorphism Z = Z /{±1} : k [3k ]. Taking w = 3 k, we have w = 9 k, or in other words, the functions f(w ) that we are concerned with can be regarded as functions on Z. They take the form f(9 k ) : Z = 1 + 8Z Z, where 1 + 8Z Z is the image of Z under the isomorphism given by 3 k. To apply the q-mahler basis of [Con00] with q = 9 it is important that 9 1 < 1. The q-mahler basis is a basis for numerical polynomials with domain restricted to Z. In the notation of [Con00] we have that f(9 k ) = ( ) k c n c n Z (), n n 9 where ( ) k = (9k 1)(9 k 9) (9 k 9 n 1 ) n 9 (9 n 1)(9 n 9) (9 n 9 n 1 ). Let us set Then f n (w ) = (w 1)(w 9) (w 9 n 1 ) (9 n 1)(9 n 9) (9 n 9 n 1 ). f(w ) = n c n f n (w ) c n Z (). We deduce that a basis for KO 0 bo is given by the set {f n (w )} n 0. As in the KU-case, it turns out that the image of bo bo in KO KO is given by bo bo v 1 -tor = (KO bo Q[u, v ]) AF 0. In order to compute a basis for this we once again need to know the Adams filtration of f n. One can show that ν ((9 n 1)(9 n 9) (9 n 9 n 1 )) = ν (n!) + 3n = 4n α(n).

10 10 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA Figure.. bo bo u f 1 0 f u 4 f It follows that we have bo bo v 1 -tor = Z (){ max(0,4n m α(n)) u m f n (w) : n 0, m n, m 0 mod } Z () { max(0,4n m 1 α(n)) u m f n (w) : n 0, m n, m 0 mod } { } Z/ u m f n (w)η i n 0, m n, m 0 mod, :. i {1, }, α(n) 4n + m + i 0 Here is a list of the Adams filtration of the first several elements in the q-mahler basis: n f n in terms of g i AF(f n ) 0 g g + g g g g 7 With this information we can now give an Adams chart of bo bo..5. Calculation of the image of bo HZ i in KO KO. We now compute the image (on the level of Adams E -terms) of the composite bo HZ i bo bo KO KO. Since 1 bo Σ 4i HZ i = KO, it suffices to determine the image of the generator Because the maps e 4i bo 4i (Σ 4i HZ i ). bo Σ 4i HZ i bo bo are constructed to be bo-module maps, everything else is determined by and v 1 = u-multiplication. Consider the diagram induced by the maps bo bu,

11 ON THE RING OF COOPERATIONS FOR tmf 11 bu HF, and BP bu: bo Σ 4i HZ i bo bo bu bu BP BP HF Σ 4i HZ i HF bo HF HF. On the level of homotopy groups the bottom row of the above diagram takes the form F { 4i 1,...} F [ 4 1,, 3,...] F [ 1,, 3,...]. Since we have bo Σ 4i HZ i (HF ) Σ 4i HZ i 4i e 4i 1, it suffices to find an element b i bo 4i bo such that bo bo (HF ) bo 4i b i 1. Clearly we can take b 0 = 1 bo 0 bo. Note that we have From the equation we deduce that we have Thus we deduce that and thus BP BP (HF ) HF t 1 1. η R (v 1 ) = v 1 + t 1 BP BP bu bu t 1 v u = ug 1 (w). bu bu (HF ) HF v u 1 Since i α(i) u i f i (w) = we see that we have We therefore can take bu bu (HF ) HF ( v u ) i 4i 1. ( v u ) i modulo terms of higher AF i α(i) u i f i (w) bo bo (HF ) bo 4i 1. b i = i α(i) u i f i (w).

12 1 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA We have therefore arrived the following well-known theorem (see [LM87, Cor..5(a)]). Theorem.6. The image of the map is the submodule Ext(bo Σ 4i HZ i ) v 1 -tors Ext(bo bo) v 1 -tors F [v 0 ]{v max(0,4i m α(i)) 0 u m f i (w) : m i, m 0 mod } F [v 0 ]{v max(0,4i m 1 α(i)) 0 v 0 u m f i (w) : m i, m 0 mod } } F {u m f i (w)η j m i, m 0 mod, :. j {1, }, α(i) 4i + m + j 0 Remark.7. These are the colors in Figure The embedding into KO. Finally we consider the maps of KO-algebras given by the composite ψ 3k These result in a map of KO-algebras : KO KO 1 ψ3k KO KO µ KO. KO KO ψ3 k k Z KO. Remark.8. The map above has a modular interpretation. Let Spec(Z)//(Z/) M fg pick out Ĝm with the action of [ 1]. Then the derived global sections of Spec(Z)//(Z/) are KO. The spectrum KO KO is the global sections of the pullback (Spec(Z) Mfg Spec(Z))//(Z/ Z/). For k Z we may consider the map of stacks Spec(Z)//(Z/) (Spec(Z) Mfg Spec(Z))//(Z/ Z/) sending Ĝm to the object [3 k ] : Ĝm Ĝm. As k varies this induces the map ψ3 k. Proposition.9. The map is an injection. Proof. Consider the diagram KO KO ψ3 k k Z KO KO KO ψ3 k k Z KO (KO KO) ψ3 k k Z (KO ) Map c (Z /{±1}, (KO ) ) Map(3 Z, (KO ) ),

13 ON THE RING OF COOPERATIONS FOR tmf 13 where the bottom horizontal map is the map induced from the inclusion of groups 3 Z Z /{±1}. The vertical maps are injections, since i KO KO = 0, and i i KO = 0. The bottom horizontal map is an injection since 3 Z is dense in Z /{±1}. The result follows. We began by investigating the wedge decomposition bo Σ 4i HZ i bo bo. We end this section by explaining how the map i KO KO ψ3 k k Z KO is compatible with the Brown-Gitler decomposition. Proposition.10. The composites are equivalences after inverting v 1. bo HZ i bo bo KO KO ψ 3 i KO i Proof. This follows from the fact that f i (9 i ) = 1. Remark.11. In fact, the matrix representing the composite k ψ3 bo HZ i bo bo KO KO KO k Z is upper triangular, as we have i f i (9 k ) = { 0, k < i, 1, k = i. 3. Recollections on topological modular forms 3.1. Generalities. The remainder of this paper is concerned with determining as much information as we can about the cooperations in the homology theory tmf based on connective topological modular forms, following our guiding example of bo. Even more than in the bo case, other players will come up. First of all, we will extensively use the periodic spectrum T MF, which is the analogue of KO. In particular, we will use that this form T MF of topological modular forms arises as the global sections of the Goerss-Hopkins-Miller sheaf of ring spectra O top on the moduli stack of smooth elliptic curves M. As the associated homotopy sheaves are { π k O top ω k/, if k is even, = 0, if k is odd,

14 14 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA there is a descent spectral sequence H s (M, ω t ) π t s T MF. Morally, the connective tmf should arise as global sections of an analogous sheaf on the moduli stack of all cubic curves (i.e. allowing nodal and cuspidal singularities); however, this has not been formally carried out. Nevertheless, tmf can be constructed as an E ring spectrum from T MF as a result of the gap in the homotopy of a third, non-connective and non-periodic, version of topological modular forms associated to the compactification of M. Rationally, every smooth elliptic curve C/S is locally isomorphic to a cubic of the form y = x 3 7c 4 x 54c 6, with the discriminant = c 3 4 c 6 invertible. Here c i is a section of the line bundle ω i over the étale map S M classifying C. This translates to the fact that M Q = Proj Q[c4, c 6 ][ 1 ], which in turn implies that (T MF ) Q = Q[c 4, c 6 ][ 1 ]. The connective version has (tmf ) Q = Q[c 4, c 6 ]. Topological modular forms are, of course, not complex orientable, and just like in the case of bo, we will need the aid of a related orientable spectrum. The periodic T MF admits ring maps to several families of orientable (as well as non-orientable) spectra which come from the theory of elliptic curves. Namely, an elliptic curve C is an abelian group scheme so in particular it has a subgroup scheme C[n] of points of order n for any positive integer n. When n is invertible, C[n] is locally isomorphic to the constant group (Z/n). Rooted in this fact are the various additional structures that one can assign to an elliptic curve. In this work we will be concerned with two types, the so-called Γ 1 (n) and Γ 0 (n) level structures. A Γ 1 (n) level structure on an elliptic curve C is a specification of a point P of (exact) order n on C, whereas a Γ 0 (n) level structure is a specification of a cyclic subgroup H of C of order n. The corresponding moduli problems are denoted M 1 (n) and M 0 (n). Assigning to the pair (C, P ) the pair (C, H P ) where H P is the subgroup of C generated by P determines an étale map of moduli stacks Moreover, there are two morphisms g : M 1 (n) M 0 (n). f, q : M 0 (n) M[1/n] which are étale; f forgets the level structure whereas q quotients C by the level structure subgroup. Composing with g we obtain analogous maps from M 1 (n). We can take sections of O top over the forgetful maps and obtain ring spectra T MF 1 (n) and T MF 0 (n), ring maps T MF [1/n] T MF 0 (n) T MF 1 (n) as well as maps of descent spectral sequences H (M[1/n], ω ) π T MF [1/n] H (M? (n), ω ) π T MF? (n),

15 ON THE RING OF COOPERATIONS FOR tmf 15 obtained by pulling back. situation -locally. In particular, for any odd integer n we have such a We use the ring map f : T MF [1/n] T MF 0 (n) induced by the forgetful f : M 0 (n) M[1/n] to equip T MF 0 (n) with a T MF [1/n]-module structure. With this convention, the map q : T MF [1/n] T MF 0 (n) induced by the quotient map on the moduli stacks does not respect the T M F [1/n]-module structure. However, one can uniquely extend q to (3.1) T MF [1/n] T MF [1/n] T MF [1/n]. q Ψ n T MF 0 (n) Another way to define Ψ n is as the composition of f q with the multiplication on T MF 0 (n). Finally, we will be interested in the morphism φ [n] : M[1/n] M[1/n]. This is the étale map induced by the multiplication-by-n isogeny on an elliptic curve, and the induced map φ [n] : TMF[1/n] TMF[1/n] can be thought of as an Adams operation on TMF[1/n]. In Section 6 below, we will make heavy use of the maps Ψ 3 and Ψ 5. Their usefulness is due to the relative ease with which their behavior on non-torsion homotopy groups can be computed. 3.. Details on tmf 1 (3) as BP. The significance of bu in the computation of bo bo was that at the prime, bu is a truncated Brown-Peterson spectrum BP 1 with a ring map bo bu which upon K(1)-localization becomes the inclusion of homotopy fixed points (KUˆ) hc KUˆ and in particular, the image of KOˆ KUˆ in homotopy is describable as certain invariant elements. By work of Lawson- Naumann [LN1], we know that there is a -primary form of BP obtained from topological modular forms; this will be our analogue of bu in the tmf-cooperations case. Lawson-Naumann study the (-local) compactification of the moduli stack M 1 (3). Given an elliptic curve C (over a -local base), it is locally isomorphic to a Weierstrass curve of the form y + a 1 xy + a 3 y = x 3 + a 4 x + a 6. A point P = (r, s) of order 3 is an inflection point of such a curve; transforming the curve so that the given point P is moved to have coordinates (0, 0) puts C in the form (3.) y + a 1 xy + a 3 y = x 3. This is the universal equation of an elliptic curve together with a Γ 1 (3) level structure. The discriminant of this curve is = (a 3 1 7a 3 )a 3 3, and M 1 (3) Proj Z () [a 1, a 3 ][ 1 ]. Consequently, π T MF 1 (3) = Z () [a 1, a 3 ][ 1 ]. Lawson- Naumann show that the compactification M 1 (3) Proj Z () [a 1, a 3 ] also admits a

16 16 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA sheaf of E -ring spectra, giving rise to a non-connective and non-periodic spectrum T mf 1 (3) with a gap in its homotopy allowing to take a connective cover tmf 1 (3) which is an E ring spectrum with π tmf 1 (3) = Z () [a 1, a 3 ]. This spectrum is complex oriented such that the composition of graded rings Z () [v 1, v ] BP (MU () ) tmf 1 (3) is an isomorphism [LN1, Theorem 1.1], where the v i are Hazewinkel generators. Of course, the map BP tmf 1 (3) classifies the p-typicalization of the formal group associated to the curve (3.), which starts as [Sil86, IV.], [?]. F (X, Y ) = X + Y a 1 XY a 3 X 3 Y 3a 3 X Y + a 3 XY 3 a 1 a 3 X 4 Y a 1 a 3 X 3 Y a 1 a 3 X Y 3 a 1 a 3 XY 4 + O(X, Y ) 6, We used Sage to compute the logarithm of this formal group law, from which we read off the coefficients l i [Rav86, A.1.7] in front of X i as l 1 = a 1, l = a3 1 + a 3, 4 l 3 = a a 4 1a a 1 a 3 8 Now the formula [Rav86, A.1.1] pl n = 0 i<n... l i v i n i (in which l 0 is understood to be 1) allows us to recursively compute the map BP tmf 1 (3). For the first few values of n, we have that v 1 a 1 v a 3 v 3 7a 1 a 3 (a a 3 )... We can do even more with this orientation of tmf 1 (3), as is a morphism of Hopf algebroids. BP BP tmf 1 (3) tmf 1 (3) Recall that BP BP = Z () [v 1, v,... ][t 1, t,... ] with v i and t i in degree ( i 1) and right unit η R : BP BP BP determined by the fact [Rav86, A.1.7] that η R (l n ) = 0 i n l i t i n i with l 0 = t 0 = 1 by convention. On the other hand, tmf 1 (3) tmf 1 (3) Q = Q[a 1, a 3, ā 1, ā 3 ] and the right unit tmf 1 (3) tmf 1 (3) tmf 1 (3) sends a i to ā i. With computer aid from Sage and/or Magma, we can recursively compute the images of each t i in

17 ON THE RING OF COOPERATIONS FOR tmf 17 tmf 1 (3) tmf 1 (3); as an example, we include here the first three values t 1 1 (ā 1 a 1 ), t 1 8 (4ā 3 + ā 3 1 a 1 ā 1 + a 1ā 1 4a 3 3a 3 1), and t (480ā 1ā 3 16a 1 ā ā 4 1ā 3 16a 1 ā 3 1ā 3 + 8a 1ā 1ā 3 16a 3 1ā 1 ā 3 + 3a 1 a 3 ā 3 + 4a 4 1ā ā 7 1 4a 1 ā a 1ā 5 1 4a 3 ā a 3 1ā a 1 a 3 ā a 4 1ā 3 1 3a 1a 3 ā 1 a 5 1ā 1 + 3a 3 1a 3 ā 1 + 0a 6 1ā 1 496a 1 a 3 508a 4 1a 3 7a 7 1) and rather than urging the reader to analyze the terms, we simply point out the exponential increase of their number. What will allow us to simplify and make sense of these expressions is using the Adams filtration in 3.4 below The relationship between T MF 1 (3) and T MF and their connective versions. As we mentioned already, the forgetful map f : M 1 (3) M is étale; moreover, f ω = ω. As a consequence, we have a Čech descent spectral sequence E 1 = H p (M 1 (3) M(q+1), ω ) H p+q (M, ω ), giving in particular that the modular forms H 0 (M, ω ) can be computed as the equalizer of the diagram (3.3) H 0 (M 1 (3), ω ) p 1 H 0 (M 1 (3) M M 1 (3), ω ), p in which p 1 and p are the left and right projection maps. The interpretation is that the M-modular forms MF are precisely the invariant M 1 (3)-modular forms. To be more explicit, note that M 1 (3) M M 1 (3) classifies tuples ((C, P ), (C, P ), ϕ) of elliptic curves with a point of order 3 and an isomorphism ϕ : C C of elliptic curves which does not need to preserve the level structures. This data is locally given by (3.4) C : y + a 1 xy + a 3 y = x 3 C : y + a 1xy + a 3y = x 3 ϕ : x u x + r y u 3 y + u sx + t, such that the following relations hold (3.5) sa 1 3r + s = 0 sa 3 + (t + rs)a 1 3r + st = 0 r 3 ta 3 t rta 1 = 0. (Note: For more details on this presentation of M 1 (3), see the beginning of [Sto, 4]; the relations follow from the general transformation formulas in [Sil86, III.1] by observing that the coefficients a even must remain zero.) Hence, the diagram (3.3) becomes Z () [a 1, a 3 ] Z () [a 1, a 3 ][u ±1, r, s, t]/( )

18 18 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA (where denotes the relations (3.5)) with p 1 being the obvious inclusion and p determined by a 1 u(a 1 + s) a 3 u 3 (a 3 + ra 1 + t). which is in fact a Hopf algebroid representing M (). Note that we do not need to localize at but only to invert 3 to obtain this presentation. As a consequence of this discussion we can explicitly compute that the modular forms MF are the subring of MF 1 (3) generated by (3.6) c 4 = a 4 1 4a 1 a 3, c 6 = a a 3 1a 3 16a 3, and = (a 3 1 7a 3 )a 3 3, which in particular determines the map T MF T MF 1 (3) on non-torsion elements Adams filtrations. The maps BP tmf 1 (3) and BP BP tmf 1 (3) tmf 1 (3) respect the Adams filtration (henceforth AF), which allows us to determine the AF in the right hand sides. Recall that AF (v i ) = 1, i 0 where as usual, v 0 =. Consequently, AF (a 1 ) = AF (a 3 ) = 1, which in turn implies via (3.6) that AF (c 4 ) = 4, AF (c 6 ) = 5, AF ( ) = 4. More precisely, modulo higher Adams filtration we have c 4 a 4, c 6 16a 3 8a 3, a 4 3. Note that the Adams filtration of each t i is zero Supersingular elliptic curves and K()-localizations. At the prime, there is a unique isomorphism class of supersingular elliptic curve; one representative is the Weierstrass curve C : y + y = x 3 over F. Recall that a supersingular elliptic curve is one whose formal completion at the identity section Ĉ is a formal group of height two.1 Under the natural map M M fg from the moduli stack of elliptic curves to the one of formal groups sending an elliptic curve to its formal completion at the identity section, the supersingular elliptic curves (in fixed characteristic) are sent to the (unique up to isomorphism, by Cartier s theorem) formal group of height two in that characteristic. Let M ss denote a formal neighborhood of the supersingular point C of M, and let Ĥ() denote a formal neighborhood of the characteristic point of height two of M fg. Formal completion yields a map M ss Ĥ() which is used to explicitly describe the K()-localization of T MF (or equivalently, tmf) in terms of Morava E-theory. 1 As opposed to an ordinary elliptic curve whose formal completion has height one. These two are the only options.

19 ON THE RING OF COOPERATIONS FOR tmf 19 The formal stack Ĥ() has a pro-galois cover by Spf W(F 4)[[u 1 ]] for the big Morava Stabilizer group G. The Goerss-Hopkins-Miller theorem implies in particular that this quotient description of Ĥ() has a derived version, namely the stack Spf E //G, where E is a Lubin-Tate spectrum of height two. As we are working with elliptic curves, we take the Lubin-Tate spectrum associated to the formal group Ĉ over F, and G = Aut F (Ĉ). Let G denote the automorphism group of C; it is a finite group of order 48 given as an extension of the binary tetrahedral group with the Galois group of F 4 /F. Then G embeds in G as a maximal finite subgroup and Spf E is a Galois cover M ss for the group G. In particular, taking sections of the structure sheaf O top over M ss gives the K()-localization of T MF which is equivalent to E hg. Moreover, we have K()-local equivalences (T MF T MF ) K() Hom c (G /G, E ) hg x G\G /G E h(g xgx 1 ). The decomposition on the right hand side is interesting though we will not pursue it further in this work. The interested reader is referred to Peter Wear s explicit calculation of the double coset in [?]. 4. The Adams spectral sequence for tmf tmf and bo-brown-gitler modules 4.1. Brown-Gitler modules. (Mod ) Brown-Gitler spectra were introduced in [BG73] to study obstructions to immersing manifolds, but immediately found use in studying the stable homotopy groups of spheres [Mah77], [Coh81] and many other places. As discussed in Section, Mahowald, Milgram, and others have used integral Brown-Gitler modules/spectra to decompose the ring of cooperations of bo [Mah81], [Mil75], and much of the work of Davis, Mahowald, and Rezk on tmfresolutions has been based on the use of bo-brown-gitler spectra [MR09],[DM10], [BHHM08]. In this section we recapitulate and extend this latter body of work. Generalizing the discussion of Section, we consider the subalgebra of of the dual Steenrod algebra We have (A//A(i)) = F [ 1 i+1 i,,..., i+1, i+,... ]. H HF = A, H HZ = (A//A(0)), H bo = (A//A(1)), H tmf = (A//A()). The algebra (A//A(i)) admits an increasing filtration by defining wt( i ) = i 1 ; then every element has filtration divisible by i+1. The Brown-Gitler submodule N i (j) is defined to be the subspace spanned by all monomials of weight less than or equal to i+1 j, which is also an A -subcomodule.

20 0 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA The modules N 1 (j) through N 1 (j) are known to be realizable by the mod- (classical), integral, and bo-brown-gitler spectra respectively, and are usually denoted by (HF ) j, HZ j, and bo j, since we have HF lim (HF ) j HZ lim HZ j bo lim bo j For clarifying notation we shall continue the convention we adopted in Section and use underline notation to refer to the corresponding sub-comodules of the dual Steenrod algebra, so that we have (HF ) j := H (HF ) j = N 1 (j) HZ j := H HZ j = N 0 (j) bo j := H bo j = N 1 (j) It is not known if tmf-brown-gitler spectra tmf j exist in general, though we will still define tmf j := N (j). The spectrum N 3 (1) is not realizible, by the Hopf-invariant one theorem. There are algebraic splittings of A(i) -comodules: (A//A(i)) = Σ i+1j N i 1 (j). This splitting is given by the sum of maps: (4.1) j Σ j+1 N i 1 (j) (A//A(i)) i1 i 1 1 a i1 i 3 where the exponent a above is chosen such that the monomial has weight i+1 j. It follows that there are algebraic splittings (4.) Ext(HZ HZ) = Ext(Σ j (HF ) j ), (4.3) Ext(bo bo) = Ext(Σ 4j HZ j ), Ext(tmf tmf) (4.4) = Ext(Σ 8j bo j ). These algebraic splittings can be realized topologically for i 1 [Mah81]: HZ HZ Σ j HZ (HF ) j, j bo bo j Σ 4j bo HZ j. However, the corresponding splitting was shown by Davis, Mahowald, and Rezk [MR09], [DM10] to fail for tmf: tmf tmf j Σ 8j tmf bo j. Indeed, they observe that in tmf tmf the homology summands Σ 8 tmf bo 1, and Σ 16 tmf bo

21 ON THE RING OF COOPERATIONS FOR tmf 1 are attached non-trivially. We shall see in Section 7 that our methods recover this fact. 4.. Rational calculations. Note that we have tmf tmf Q = Q[c4, c 6, c 4, c 6 ]. Consider the (collapsing) v 0 -inverted ASS v0 1 Ext A() (Σ 8i bo i ) tmf tmf Q. i In this section we explain the decomposition imposed on the E -term of this spectral sequence from the decomposition on the E -term. In particular, given a torsionfree element x tmf tmf, this will allow us to determine which bo-brown-gitler module supports it in the E -term of the ASS for tmf tmf. Recall from Section 3 that tmf 1 (3) BP. In particular, we have H (tmf 1 (3)) = A//E[Q 0, Q 1, Q ]. We begin by studying the map between v 0 -inverted ASS s induced by the map tmf tmf 1 (3). 0 Ext, A() (F ) π tmf Q We have 0 Ext, E[Q 0,Q 1,Q ] (F ) π tmf 1 (3) Q where the v i s have (t s, s) bidegrees: 0 Ext, E[Q 0,Q 1,Q ] (F ) = F [v ±1 0, v 1, v ] v 0 = (0, 1) v 1 = (, 1) v = (6, 1) Recall from Section 3 that π tmf 1 (3) Q = Q[a 1, a 3 ], and that v 1 = [a 1 ], v = [a 3 ]. We of course have π tmf Q = Q[c 4, c 6 ], with corresponding localized Adams E -term where the [c i ] s have (t s, s) bidegrees: 0 Ext, A() (F ) = F [v ±1 0, c 4, c 6 ] [c 4 ] = (8, 4) [c 6 ] = (1, 5)

22 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA Recall also from Section 3 that the formulas for c 4 and c 6 in terms of a 1 and a 3 imply that the map of E -terms of spectral sequences above is injective, and is given by (4.5) Corresponding to the isomorphism [c 4 ] [a 4 1], [c 6 ] [8a 3]. π tmf Q = HQ tmf there is an isomorphism of localized Adams E -terms Since the decomposition 0 Ext A() (F ) = 0 Ext A(0) ((A//A()) ). A//A() = Σ 8j bo j is a decomposition of A() -comodules, it is in particular a decomposition of A(0) - comodules, and there is therefore a decomposition (4.6) v0 1 Ext A() (F ) = v0 1 Ext A(0) (Σ 8j bo j ) Proposition 4.7. Under the decomposition (4.6), we have j 0 Ext A(0) (Σ 8j bo j ) = F [v ±1 0 ]{[ci1 4 ci 6 ] : i 1 + i = j} j 0 Ext A() (F ). Proof. Statement () of the proof of Lemma.5 implies that we have 0 Ext A(0) (bo j ) = F [v 0 ±1 4i ]{ 1 : 0 i j}. Using the map (4.1), we deduce that we have 0 Ext A(0) (Σ 8j bo j ) = F [v 0 ±1 8i1 ]{ 1 4i : i 1 + i = j} Ext A(0) ((A//A()) ). Consider the diagram: (4.8) H tmf H tmf 1 (3) BP BP HZ tmf HZ tmf 1 (3) tmf 1 (3) tmf 1 (3) The map HQ tmf HQ tmf 1 (3) tmf 1 (3) tmf 1 (3) Q BP BP H tmf 1 (3) = F [ 1,, 3, 4,...] sends t i to i. Thus the elements8i1 1 4i H tmf t 4i1 1 ti BP BP

23 ON THE RING OF COOPERATIONS FOR tmf 3 have the same image in H tmf 1 (3). However, using the formulas of Section 3, we deduce that the images of t 1 and t in are given by Since the map t 1 (ā 1 + a 1 )/, tmf 1 (3) tmf 1 (3) Q = Q[a 1, a 3, ā 1, ā 3 ] t (4ā 3 a 1 ā 1 4a 3 a 3 1)/8 + terms of higher Adams filtration. tmf 1 (3) tmf 1 (3) Q HQ tmf 1 (3) = Q[a 1, a 3 ] of Diagram (4.8) sends ā i to a i and a i to zero, we deduce that the image of t 1 and t in HQ tmf 1 (3) is t 1 a 1 /, t a 3 / + terms of higher Adams filtration. It follows that under the map of v 0 -localized ASS s induced by the map tmf tmf 1 (3): v0 1 Ext A() (F ) v0 1 Ext E[Q0,Q 1,Q ] (F ) we have 8i1 1 4i [a 1 /] 4i1 [a 3 /] i. Therefore, by (4.5), we have (in 0 Ext A(0) ((A//A()) )) and the result follows. 8i1 1 4i = [c 4 /16] i1 [c 6 /3] i Corresponding to the Künneth isomorphism for HQ, there is an isomorphism 0 Ext A(0) (M N) = 0 Ext A(0) (M) F[v ±1 0 ] Ext A(0) (N). In particular, since the maps can be identified with the maps 0 Ext(tmf Σ 8j bo j ) 0 Ext(tmf tmf) 0 Ext A(0) ((A//A()) ) F[v ±1 0 ] v 1 0 Ext A(0) (Σ 8j bo j ) we have the following corollary. Corollary 4.9. The map 0 Ext A(0) ((A//A()) ) F[v ±1 0 ] v 1 0 Ext A(0) ((A//A()) ) 0 Ext(tmf Σ 8j bo j ) 0 Ext(tmf tmf) obtained by localizing (4.4) is the cannonical inclusion F [v ±1 0, [c 4], [c 6 ]]{[ c 4 ] i1 [ c 6 ] i : i 1 + i = j} F [v ±1 0, [c 4], [c 6 ], [ c 4 ], [ c 6 ]].

24 4 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA 4.3. Exact sequences relating the bo-brown-gitler modules. In order to proceed with integral calculations we use analogs of the short exact sequences of Section. Lemmas 7.1 and 7. from [BHHM08] state that there are short exact sequences (4.10) (4.11) 0 Σ 8j bo j bo j (A()//A(1)) tmf j 1 Σ 8j+9 bo j Σ 8j bo j bo 1 bo j+1 (A()//A(1)) tmf j 1 0 of A() -comodules. These short exact sequences provide an inductive method of computing Ext A() (bo j ) in terms of Ext A(1) computations and Ext A() (bo i 1 ). We briefly recall how the maps in the exact sequences (4.10) and (4.11) are defined. On the level of basis elements, the maps are given respectively by 4i1 1 Σ 8j bo j bo j Σ 8j bo j bo 1 bo j+1 4i1 1 i i3 3 1 a i i3 3 {1, 1, 4, 3 } ( 1 a where a is taken to be 8j wt( 4i1 (4.1) (4.13) are given by 8i1+4ɛ1 1 4i+ɛ {8i 1 1 4i1 4i1 i 3 i3 i 3 i 3 i3 4 ). The maps bo j (A()//A(1)) tmf j 1, bo j+1 (A()//A(1)) tmf j 1 i3+ɛ3 3 i4 4 4i i3 3 i4 4ɛ1 4 1 ɛ ɛ3 8i1 3, wt( 1 0, otherwise, 4, i3 4 ) {1, 1, 4, 3 } 4i i3 3 i4 4 ) 8j 8, where ɛ s {0, 1}. The only change from the integral Brown-Gitler case is that while the map (4.13) is surjective, the map (4.1) is not. The cokernel is spanned by the submodule F { } Σ 8j 8 bo j 1 (A()//A(1)) tmf j 1. We therefore have an exact sequence bo j (A()//A(1)) tmf j 1 Σ 8j+9 bo j 1 0 We give some low dimensional examples. We shall use the shorthand M M i [k i ] to denote the existence of a spectral sequence Ext s k i,t+k i A() (M i ) Ext s,t A() (M).

25 ON THE RING OF COOPERATIONS FOR tmf 5 In the notation above, we shall abbreviate M i [0] as M i. We have: (4.14) Σ 16 bo Σ 16 (A()//A(1)) Σ 4 bo 1 Σ 3 F [1] Σ 4 bo 3 Σ 4 (A()//A(1)) Σ 3 bo 1 Σ 3 bo 4 (A()//A(1)) ( Σ 3 tmf 1 Σ 48 F ) Σ 56 bo 1 Σ 56 bo 1 [1] Σ 64 F [1] Σ 40 bo 5 (A()//A(1)) ( Σ 40 tmf 1 Σ 56 bo 1 ) Σ 64 bo 1 Σ7 bo 1 [1] Σ 48 bo 6 (A()//A(1)) ( Σ 48 tmf Σ 7 F Σ 80 F [1] ) Σ 80 bo 1 Σ88 bo 1 [1] Σ 96 F [] Σ 56 bo 7 (A()//A(1)) ( Σ 56 tmf Σ 80 bo 1 ) Σ 88 bo 3 1 Σ 64 bo 8 (A()//A(1)) ( Σ 64 tmf 3 Σ 96 tmf 1 Σ 11 F Σ 104 F [1] ) Σ 11 bo 1[1] Σ 10 bo 1 Σ 10 bo 1 [1] Σ 18 F [1] In practice, these spectral sequences seem to tend to collapse. In fact, in the range computed explicitly in this paper, there are no differentials in these spectral sequences, and the authors have not yet encountered any differentials in these spectral sequences. These spectral sequences do collapse with v 0 -inverted, for dimensional reasons. In principle the exact sequences (4.10), (4.11) allow one to inductively compute Ext A() (bo j ) given Ext A() (bo k 1 ), where bo 1 is depicted below Sq 1 Sq Sq 4 1 The problem is that, unlike the A(1)-case, we do not have a closed form computation of Ext A() (bo k 1 ). These computations for k 3 appeared in [BHHM08] (the cases of k = 0, 1 appeared elsewhere). We include in Figures?? through?? the charts for Σ 8j bo j, for 0 j 6, as well as Σ 8 bo 1 in dimensions Rational behavior of the exact sequences. We finish this section with a discussion on how to identify the generators of Ext A() (Σ8j bo j ) v 0 tors. On one hand, the inclusion Ext A() (Σ 8j bo j ) v 0 tors 0 Ext A() (Σ 8j bo j ) F [v ±1 0, [c 4], [c 6 ]]{ 8i1 4i : i 1 + i = j} 0 Ext A() ((A//A()) )

26 6 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA (a) bo 0 (b) Σ 8 bo 1 Figure 4.1

27 ON THE RING OF COOPERATIONS FOR tmf 7 (a) Σ 8 bo 1 (b) Σ 16 bo Figure 4.

28 8 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA (a) Σ 4 bo 3 (b) Σ 3 bo 4 Figure 4.3

29 ON THE RING OF COOPERATIONS FOR tmf 9 (a) Σ 40 bo 5 (b) Σ 48 bo 6 Figure 4.4

30 30 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA discussed in Section 4. informs us that the h 0 -towers of Ext A() (Σ 8j bo j ) are all generated by h k 0[c 4 ] p [c 6 ] q 8i 1 1 4i for appropriate (possibly negative) values of k depending on i 1, i, p, and q. The problem lies in that the terms (4.15) (4.16) 0 Ext A() (Σ 16j (A()//A(1)) tmf j 1 ) Ext A() (Σ 16j bo j ), 0 Ext A() (Σ 16j+8 (A()//A(1)) tmf j 1 ) Ext A() (Σ 16j+8 bo j+1 ) in the short exact sequences (4.10), (4.11) are not free over F [v 0 ±1, [c 4], [c 6 ]] (however, they are free over F [v 0 ±1, [c 4]]). We therefore instead identify the generators of v0 1 Ext A() ((A//A()) ) corresponding to the generators of (4.15) and (4.16) as modules over F [v 0 ±1, [c 4]], as well as those generators coming (inductively) from (4.17) (4.18) 0 Ext A() (Σ 4j bo j ) 0 Ext A() (Σ 16j bo j ), 0 Ext A() (Σ 4j+8 bo j bo 1 ) 0 Ext A() (Σ 16j+8 bo j+1 ). in the following two lemmas, whose proofs are immediate from the definitions of the maps in (4.10), (4.11). Lemma The summands (4.15) (respectively (4.16)) are generated, as modules over F [v 0 ±1, [c 4]], by the elements a 1 8i1 4i3 3, a 8 1 8i1+4 4i3 3 (A//A()) with i 1 + i j 1 and a = 16j 8i 1 8i (respectively a = 16j + 8 8i 1 8i ). Lemma 4.0. Suppose inductively (via the exact sequences (4.10),(4.11)) that the summand v0 1 Ext A() (Σ 8j bo j ) v0 1 Ext A() ((A//A()) ) is generated by generators of the form { i1 i 1...}. Then the summand (4.17) is generated by and the summand (4.18) is generated by The remaining term { i1 i 3 } { i1 i 3 } { 1, 8 }. 4 (4.1) 0 Ext A() (Σ 4j+8 bo j 1 [1]) 0 Ext A() (bo j ) coming from (4.10) is handled by the following lemma. Lemma 4.. Consider the summand 0 Ext A(1) (Σ 4j 8 bo j 1 ) 0 Ext A(1) (Σ 16j tmf j 1 ) 0 Ext A() (Σ 16j bo j ) generated as a module over F [v 0 ±1, [c 4]] by the generators i1 4i 3, 1 8 8i1+4 4i 3 (A//A())

31 ON THE RING OF COOPERATIONS FOR tmf 31 with i 1 + i = j 1. Let x i (0 i j 1) be the generator of the summand (4.1), as a module over F [v 0 ±1, [c 4i 4], [c 6 ]] corresponding to the generator 1 bo j 1. The we have [c 6 ] 1 8 8i1+4 4i 3 = v0x 4 i + in v0 1 Ext A() (Σ 16j bo j ), where the additional terms not listed above all come from the summand 0 Ext A() (Σ 4j bo j ) 0 Ext A() (Σ 16j bo j ). Proof. This follows from the definition of the last map in (4.10), together with the fact that with v 0 -inverted, the cell (A()//A(1)) attaches to the cell 1 4 with attaching map [c 6 ]/v0. 4 Lemmas 4.19, 4.0, and 4. give an inductive method of identifying a collection of generators for v0 1 Ext A() (bo j ) which are compatible with the exact sequences (4.10), (4.11). We tabulate these below for the decompositions arising from the spectral sequences (4.14). For those summands of the form (A()//A(1)) these are generators over F [v 0 ± 1, [c 4]], for the other summands these are generators over F [v 0, [c 4 ], [c 6 ]]: bo 0 : F : 1 Σ 8 bo 1 : Σ 8 bo 1 : 8 1, 4 Σ 16 bo : Σ 16 (A()//A(1)) : 16 1, Σ 4 bo 1 : 8, 4 3 Σ 3 F [1] : 0 [c 6] Σ 4 bo 3 : Σ 4 (A()//A(1)) : , 1 4 Σ 3 bo 1 : {, 8 3} 4 { 1, 8 } 4 Σ 3 bo 4 : Σ 3 (A()//A(1)) tmf 1 : 4 3 1, 1, , , 1 3, Σ 48 (A()//A(1)) : v 4 16, Σ 56 bo 1 : 8 3, 4 4 Σ 64 F [1] : Σ 56 bo 1 [1] : v 4 Σ 40 bo 5 : Σ 40 (A()//A(1)) tmf 1 : 40 1, 0 [c 6] v 4 0 [c 6] , 1 +, v , 16 1 Σ 56 (A()//A(1)) bo 1 : { 16, 8 4 3} { 8 1, 4 } Σ 64 bo 1 : { 8 3, 4 4} { 8 1, 4 } Σ 7 bo 1 [1] : {v 4 Σ 48 bo 6 : Σ 48 (A()//A(1)) tmf : 48 Σ 7 (A()//A(1)) : [c 6] , 1 3, [c 6] } { 1, 8 } , 40 1, , 1, 4 16, , 8 0, , , , , , Σ 80 bo 1 : { 3, 8 4} 4 {, 8 3} 4 Σ 80 bo [1] v0 4 6] , v0 4 6] ,

32 3 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA Σ 56 bo 7 : Σ 56 (A()//A(1)) tmf : 56 Σ 80 (A()//A(1)) bo 1 : { 4, Σ 64 bo 8 : Σ 64 (A()//A(1)) tmf 3 : 64 Σ 96 (A()//A(1)) tmf 1 : 3, Σ 11 (A()//A(1)) : v , 1, , 1 0 [c 6] , 8 0, 16 3} 4 { 1, 8 } 4 1, , 40 Σ 88 bo 3 1 : { 8 3, 4 4} { 8, 4 3} { 8 1, 4 } 56 1, 1, , , 1, , , , , 16 3, Σ 10 bo 1 : 8 4, 4 5 Σ 18 F [1] : Σ 10 bo 1 [1] : Σ 104 bo 3 [1] : v , 8 0 [c 6] v 4 0 [c 6] 8 v0 4 [c 6] 1 8 v0 4 [c 6] , 1 4 3, , , , , v , 4 1 3, , 4 0 3, , , , 4, , , [c 6] , v0 4 [c 6] , Identification of the integral lattice. Having constructed useful bases of the summands 0 Ext A() (Σ 8j bo j ) 0 Ext A() (A//A() ) it remains to understand the lattices Ext A() (Σ 8j bo j ) v0 1 Ext A() (Σ 8j bo j ) v 0 tors This can accomplished inductively; the rational generators we identified in the last section are compatible with the exact sequences (4.10), (4.11), and Ext A() v 0 tors of the terms in these exact sequences are determined by the Ext A(1) v 0 tors computations of Section, and knowledge of Ext A() (bo k 1 ). v 0 tors Unfortunately the latter requires explicit computation for each k, and hence does not yield a general answer. Nevertheless, in this section we will give some lemmas which provide convenient criteria for identifying the i so that given a rational generator x (A//A()) (as in the previous section) we have v i 0x Ext A() ((A//A()) ) v 0 tors 0 Ext A() ((A//A()) ). We first must clarify what we actually mean by rational generator. The generators identified in the last section originate from the exact sequences (4.10), (4.15) , 1 8 3,

33 ON THE RING OF COOPERATIONS FOR tmf 33 from the generators of 0 Ext A() (M) where M is given by Case 1: M = bo k 1 Case : M = (A()//A(1)) tmf j In Case 1, the generators x of v0 1 Ext A() (M) are generators as a module over F [v 0 ±1, [c 4]] using the isomorphisms (4.3) 0 Ext A() ((A()//A(1)) tmf j ) = 0 Ext A(1) (tmf j ) = 0 Ext A ((A//A(1)) tmf j ) α = 0 Ext A(0) ((A//A(1)) tmf j ) = v0 1 Ext A(0) ((A//A(1)) ) F[v ±1 0 ] v 1 0 Ext A(0) (tmf j ) = F [v 0 ±1, [c 4]]{1, 1} 4 F F { 8i1 1 4i : i 1 + i j}. The rational generators in this case correspond to the generators x = 4ɛ 1 8i1 1 4i. In Case, the generators x of v0 1 Ext A() (M) are generators as a module over F [v 0 ±1, [c 4], [c 6 ]], using the isomorphisms (4.4) 0 Ext A() (bo k 1) = 0 Ext A ((A//A()) bo k 1) α = 0 Ext A(0) ((A//A()) bo k 1 ) = 0 Ext A(0) ((A//A()) ) F[v ±1 0 ] v 1 0 Ext A(0) (bo k 1 ) = F [v ±1 0, [c 4], [c 6 ]] F F {1, 4 1} k. The rational generators in this case correspond to the generators x {1, 4 1} k. In either case, since the maps α in both (4.3) and (4.4) arise from surjections of cobar complexes C A (N) C A(0) (N) induced from the surjection A A(0). Thus a term v i 0x C A(0) (N) representing an element in 0 Ext A(0) (N) corresponds (for i sufficiently large) to a term [ 1 ] i x + C A (N). Then we have determined an element of the integral lattice [ [1 ] i x + ] Ext A (N) v 0 tors v 1 0 Ext A (N). Lemma 4.5. Suppose that the A() -coaction on x (A//A()) satisfies ψ(x) = 4 1 y + terms in lower dimension

34 34 M. BEHRENS, K. ORMSBY, N. STAPLETON, AND V. STOJANOSKA with y primitive, as in the following cell diagram : Then v0x 3 Ext A() ((A//A()) ) v 0 tors and is represented by x y Sq 4 0 Ext A() ((A//A()) ) [ ]x + ( [ 1 ] + [ ] + [ 1 1 1] + [ 1 1] ) y in the cobar complex C A() ((A//A()) ). Proof. Since the cell complex depicted agrees with A()//A(1) through dimension 4, Ext A() of this comodule agrees with Ext A(1) (F ) through dimension 4. In particular, v0x 3 + generates Ext A() ( ) v 0 tors in this dimension. To determine the exact representing cocycle, we note that kills h 3 0h in Ext A() (F ). [ 1 ] + [ ] + [ 1 1 1] + [ 1 1] Example 4.6. A typical instance of a set of generators of (A//A()) satisfying the hypotheses of Lemma 4.5 is 4 i α Sq 4 8 i 1 α where α = 8j1 i 1 8j i is a monomial with exponents all divisible by 8. The following corollary will be essential to relating the integral generators of Lemma 4.5 to -variable modular forms in Section 5. Corollary 4.7. Suppose that x satisfies the hypotheses of Lemma 4.5. Then image of the corresponding integral generator v 3 0x + Ext A() ((A//A() )) in Ext E[Q0,Q 1,Q ] ((A//E[Q 0, Q 1, Q ]) ) is given by Proof. Note that v 3 0x + v 0 [a 1 ] y. E[Q 0, Q 1, Q ] = F [ 1,, 3 ]/( 1,, 3). Therefore the image of the integral generator of Lemma 4.5 under the map is C A() ((A//A()) ) C E[Q 0,Q 1,Q ] ((A//E[Q 0, Q 1, Q ]) ) [ ]x + [ 1 ]y

35 ON THE RING OF COOPERATIONS FOR tmf 35 and this represents v 3 0x + v 0 [a 1 ] y. Similar arguments provide the following slight refinement. Lemma 4.8. Suppose that the A() -coaction on x (A//A()) satisfies ψ(x) = 1 4 y + terms in lower dimension with y primitive, and that there exists w and z satisfying ψ(z) = 1y + terms in lower dimension and ψ(w) = 1 z + y + terms in lower dimension as in the following cell diagram : x w Sq 1 z Sq 4 Sq y Then and is represented by v 0 x Ext A() ((A//A()) ) v 0 tors 0 Ext A() ((A//A()) ) [ 1 ]x + [ 1]w + ( [ 3 1] + [ ] ) z + [ 1 ]y in the cobar complex C A() ((A//A()) ). Example 4.9. A typical instance of a set of generators of (A//A()) satisfying the hypotheses of Lemma 4.8 is 4 i 4 i α ( 8 i 1 i + + i+ 8 i 1 )α Sq 1 Sq ( 8 4 i 1 i +1 + i+1 i 8 1 )α Sq ( 8 i 1 4 i + 4 i 8 i 1 )α where α = 8j1 i 1 8j i is a monomial with exponents all divisible by 8.