New York Journal of Mathematics

New York J. Math. 1 (1995) 39–74.

Operations and Cooperations in Elliptic Cohomology, Part I: Generalized modular forms and the cooperation algebra

Andrew Baker

Abstract. This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, Eℓℓ_{∗}Eℓℓ, while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable operations in elliptic cohomology.

We investigate the structure of the stable operation algebra Eℓℓ^{∗}Eℓℓ by first determining the dual cooperation algebra Eℓℓ_{∗}Eℓℓ. A major ingredient is our identification of the cooperation algebra Eℓℓ_{∗}Eℓℓ with a ring of generalized modular forms whoses exact determination involves understanding certain integrality conditions; this is closely related to a calculation by N. Katz of the ring of all ‘divided congruences’ amongst modular forms. We relate our present work to previous constructions of Hecke operators in elliptic cohomology. We also show that a well known operator on modular forms used by Ramanujan, Swinnerton-Dyer, Serre and Katz cannot extend to a stable operation.

Introduction

This paper is in two interelated parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra Eℓℓ_{∗}Eℓℓ, while Part II will be devoted to the more algebraic theory associated with Hecke algebras and operations in elliptic cohomology.

In our earlier paper [6], we defined operations in the ‘level 1’ version of elliptic cohomology Eℓℓ^{∗}( ) which restricted to the classical Hecke operators on the coefficient ring Eℓℓ_{∗} (defined to be a ring of modular forms for the full modular group SL_{2}(ℤ)). In the present paper we investigate the structure of the operation algebra Eℓℓ^{∗}Eℓℓ by determining the dual cooperation algebra Eℓℓ_{∗}Eℓℓ, thus following the pattern established in the case of K-theory; we also describe a category

Received August 3, 1994 Mathematics Subject Classification. 55N20, 55N22, 55S25. Key words and phrases. Elliptic cohomology, modular forms, operations and cooperations. The author acknowledges the support of the Science and Engineering Research Council, the

Max-Planck-Institut für Mathematik, Glasgow University, Johns Hopkins University, Manchester University and Osaka Prefecture whilst parts of this work were undertaken.

\begin{matrix}c \\ ◯ \end{matrix}1995 State University of New York ISSN 1076-9803/95

of modules (dually comodules) over these which are closely related to modules over Hecke algebras associated to the group SL_{2}(ℤ); this points to a generalization from K-theory to elliptic cohomology of work by A. K. Bousfield in [12], [13]. A recent paper of F. Clarke and K. Johnson [14] has also considered the analogous cooperation algebra for the level 2 version of elliptic cohomology, and we in effect prove their conjecture on the structure of their analogue of Eℓℓ_{∗}Eℓℓ.

A particular ingredient is our identification of the cooperation algebra Eℓℓ_{∗}Eℓℓ with a ring of ‘generalized modular forms’. The most significant aspect of this involves understanding certain integrality conditions, and this is closely related to the calculation by N. Katz in [23] of the ring of all ‘divided congruences’ amongst modular forms (in 1 variable). Indeed, Katz’s work amounts to a calculation of the topological gadget KU_{∗}Eℓℓ rather than Eℓℓ_{∗}Eℓℓ; however, we use his results to determine the latter. We also wish to point out that the construction by G. Nishida [32] of Hecke operators appears to be closely related to the ideas of the present work.

We will assume the reader is familiar with the apparatus of algebraic topology contained in [1] and [33], to which the reader is referred for all basic ideas on complex oriented cohomology theories and their associated formal group laws. As basic references on elliptic cohomology theories, P. S. Landweber’s two articles [28] and [29] are highly recommended although their main emphasis is on level 2 theories. A more recent reference is that of J. Francke [15]. A convenient source for all the basic notions of Hecke algebras is [26].

In detail, Part I is structured as follows. §1 contains a brief resumé of modular forms and elliptic cohomology. §2 gives details of the formal group law associated to elliptic curves in Weierstrass form and the canonical complex orientation of elliptic cohomology. §3 introduces the cooperation Hopf algebroid Eℓℓ_{∗}Eℓℓ. §4 introduces our notion of generalized modular form. In §5 and §6 we describe certain categories of isogenies and their realisation as stable operations on elliptic cohomology. §7 recalls the properties of the classical rings of stably numerical polynomials, familiar in the context of the stable cooperation Hopf algebroid for K-theory, KU_{∗}KU. In §8 and §9 we describe a major result of N. Katz and apply it to the calculation of our ring of generalized modular forms which is isomorphic to Eℓℓ_{∗}Eℓℓ. In §10 and §11 we complete the description of Eℓℓ_{∗}Eℓℓ by considering its coproduct structure and use duality to construct stable operations, particularly operations which generalize the classical Hecke operators. Finally, in §12 we discuss an important operation ∂ on modular forms which is a derivation and plays a major rôle in the arithmetic theory of Swinnerton-Dyer, Serre and Katz; we show this cannot extend to a stable operation in elliptic cohomology.

I would like to thank the following for help and advice on this work and related topics over many years: Francis Clarke, Mark Hovey, John Hunton, Keith Johnson, Peter Landweber, Jack Morava, Goro Nishida, Serge Ochanine, Doug Ravenel, Nigel Ray, Robert Stong and Charles Thomas.

1. Modular forms and elliptic cohomology

Let L denote the set of all oriented lattices in ℂ, i.e., discrete free subgroups L ⊆ ℂ such ℝ ⊗ L = ℂ as oriented real vector spaces. This set can be identified with the coset space

SL_{2}(ℤ)\V,

where V is the set of all oriented bases {ω_{1}, ω_{2}} in the real vector space ℂ and we use the convention that for an oriented (ordered) basis {ω_{1}, ω_{2}},

ω_{1}/ω_{2} ∈ ℌ = {τ ∈ ℂ : im τ > 0}.

The action of SL_{2}(ℤ) is the obvious one,

(\begin{matrix}a & b \\ c & d \end{matrix}) · \{ω_{1}, ω_{2}\} = \{aω_{1} + bω_{2}, cω_{1} + dω_{2}\}.

This of course induces the usual action on the upper half plane ℌ on passage from {ω_{1}, ω_{2}} to ω_{1}/ω_{2}. Thus L possesses a natural 2-dimensional complex analytic structure.

Notice that the group of non-zero complex numbers ℂ^{×} acts compatibly on both V and L by

λ · {ω_{1}, ω_{2}} = {λω_{1}, λω_{2}}

and

λ · ⟨ω_{1}, ω_{2}⟩ = ⟨λω_{1}, λω_{2}⟩ ,

where ⟨ω_{1}, ω_{2}⟩ denotes the lattice spanned by the basis {ω_{1}, ω_{2}}.

We will follow [22] and [25] in defining a modular form of weight k to be a holomorphic function F: L ⟶ ℂ which satisfies the functional equation

F (λ · L) = λ^{−k}F (L)

whenever λ ∈ ℂ^{×}. To avoid excessively elaborate notation, we will sometimes regard such a function as having as its domain V and being invariant under the action of SL_{2}(ℤ). We can associate to such an F a function f: ℌ ⟶ ℂ defined by f(τ) = F (⟨τ, 1⟩) for τ ∈ ℌ the upper half plane. After setting q = e^{2πiτ}, we say that F is holomorphic at infinity (i.e., at i∞) if the Fourier series expansion

f(τ) = ∑_{−∞<n<∞} a_{n}q^{n}

has a_{n} = 0 for n < 0; if also a_{0} = 0, then F is a cusp form. We say that F is meromorphic at infinity if the Fourier series of F has a_{n} = 0 for n ≪ 0. If the coefficients a_{n} lie in some subring K ⊆ ℂ, then we say that F is defined over K. Throughout this paper we will assume as we did in [6] that ℤ[1/6] ⊆ K, the reader is referred to [22] and [42] for details on the reasons for this. We will denote by S(K)_{k} the set of all weight k modular forms holomorphic at infinity and by M(K)_{k} the set of all weight k modular forms meromorphic at infinity; of course we have S(K)_{k} ⊆ M(K)_{k}. Thus there are two strictly commutative graded rings S(K)_{∗} and M(K)_{∗} with a homomorphism of graded rings S(K)_{∗} ⟶ M(K)_{∗}. The following classical result describes the structure of such rings. Elementary accounts of this result can be found in [25, 39]; for a discussion of rigidity under base change, see [22].

Theorem 1.1. If 1/6 ∈ K, then as graded rings we have

S(K)_{∗} = K[E_{4}, E_{6}],

and

M(K)_{∗} = S(K)_{∗}[Δ^{−1}] = K[E_{4}, E_{6}, Δ^{−1}],

where E_{2n} ∈ S(K)_{2n} ⊆ M(K)_{2n} is the 2nth Eisenstein function and

Δ = \frac{1}{1728} (E_{4}^{3} − E_{6}^{2})

is the discriminant function.

We recall the following q-expansions defined over ℚ:

(1.1)  E_{2n}(q) = 1 − \frac{4n}{B_{2n}} ∑_{k⩾1} σ_{2n−1}(k)q^{k} for n ⩾ 1
(1.2)  Δ = q ∏_{n⩾1}(1 − q^{n})^{24}

where σ_{m}(k) = ∑_{d|k} d^{m}. Whenever n > 1, the q-expansion E_{2n}(q) corresponds to a modular form of weight 2n, which we will denote by E_{2n}. Notice that for any subring K ⊆ ℂ, we have E_{4}, E_{6} ∈ S(K)_{∗} ⊆ M(K)_{∗}. Following [40, 41], we will use the notation Q = E_{4} and R = E_{6}.

For each n ⩾ 0, define a basis {F_{n,a}} of S(K)_{n} over K as follows. For 0 ⩽ n ⩽ 14, set

F_{0,0} = 1,
F_{4,0} = Q = E_{4},
F_{6,0} = R = E_{6},
F_{8,0} = Q^{2},
F_{10,0} = QR,
F_{12,0} = Q^{3},
F_{12,1} = Δ,
F_{14,0} = Q^{2}R.

For n ⩾ 16, inductively define the basis so that F_{n,0} = Q^{3}F_{n−12,0}, and if a ⩾ 1, F_{n,a} = ΔF_{n−12,a−1}. Notice that we have

(1.3) F_{m,a}F_{n,b} = \begin{cases}F_{m+n,0} + (cusp form) & if a = b = 0, \\ (cusp form) & otherwise. \end{cases}

We will refer to the basis {F_{n,a}} as the standard basis of the graded K-module S(K)_{∗}. We can lexicographically order this basis by the index (n, a).

We next introduce the following topologically motivated notation:

eℓℓ_{2n} = S(ℤ[1/6])_{n},
Eℓℓ_{2n} = M(ℤ[1/6])_{n}.

We define elliptic cohomology to be the functor (on the category of finite CW complexes or spectra)

Eℓℓ^{∗}( ) = Eℓℓ_{∗} ⊗_{MU_{∗}} MU^{∗}( ). (1.4)

In Landweber’s papers [28, 29] and also [6], it is shown that this a cohomology theory. There is also a connective theory eℓℓ^{∗}( ) whose coefficient ring is eℓℓ_{∗}, although we make no use of it in this paper. However, its representing spectrum eℓℓ is not the connective covering of Eℓℓ, even if the notation may suggest this.

We end this section with some further remarks on elliptic cohomology, intended to highlight its properties as a cohomology theory. In [8] we observed that after a suitable completion, the spectrum Eℓℓ carries a unique topological A_{∞} ring structure (in unpublished work we have also shown that this is true for Eℓℓ itself). An important consequence of this is that for any A_{∞} module spectrum M over Eℓℓ and any spectrum X, there are Künneth and Universal Coefficient spectral sequences for M_{∗}(X) and M^{∗}(X), This depends upon work of C. A. Robinson [35, 36, 37]. An alternative approach to such spectral sequences comes from recent work of M. J. Hopkins and J. R. Hunton [20, 21], whose methods yield the following theorem.

Theorem 1.2. For any d ∈ ℤ, let Ω^{∞−d}Eℓℓ denote the term in the Ω-spectrum Eℓℓ which represents the elliptic cohomology group Eℓℓ^{d}( ). Then the ordinary homology H_{∗}(Ω^{∞−d}Eℓℓ; ℤ[1/6]) is torsion free. Similarly, Eℓℓ_{∗}(Ω^{∞−d}Eℓℓ) is free over Eℓℓ_{∗}. Consequently, the spectrum Eℓℓ is a colimit of finite CW spectra E_{α} each having the property that both Eℓℓ_{∗}(E_{α}) and Eℓℓ_{∗}(DE_{α}) are free over Eℓℓ_{∗}.

Recall the conditions for Adams’ universal coefficient spectral sequence of [1], Part III.

Corollary 1.3. The conditions for Adams’ universal coefficient spectral sequence are satisfied by the spectrum Eℓℓ. Hence the Künneth and Universal Coefficient spectral sequences exist for any module spectrum over Eℓℓ and any spectrum X, and have the usual forms:

\begin{cases}E_{∗,∗}^{2}(X) & ⟹ M_{∗}(X) \\ E_{∗,∗}^{2}(X) & = Tor_{Eℓℓ_{∗}}^{∗,∗} (Eℓℓ_{∗}(X), M_{∗}) \end{cases}

and

\begin{cases}E_{2}^{∗,∗}(X) & ⟹ M^{∗}(X) \\ E_{2}^{∗,∗}(X) & = Ext_{Eℓℓ_{∗}}^{∗,∗} (Eℓℓ_{∗}(X), M_{∗}) \end{cases}

Thus, elliptic homology and cohomology possess the usual battery of computational technology. However, the fact that the coefficient ring Eℓℓ_{∗} is not a principal ideal domain suggests that serious calculations will usually be of greater difficulty than they would in say K-theory. For reductions modulo invariant ideals and relations with Morava K(1) and K(2), see [8, 9, 10].

We end this section by describing a modified version of elliptic cohomology which is 2-periodic. We take as its coefficient ring

Eℓℓ_{∗} = Eℓℓ_{∗}[Λ]/(Λ^{12} − Δ),

where Λ ∈ Eℓℓ_{2}. Then the natural homomorphism Eℓℓ_{∗} ⟶ Eℓℓ_{∗} allows us to define the functors (on finite CW complexes or spectra)

(1.5)  Eℓℓ^{∗}( ) = Eℓℓ_{∗} ⊗_{MU_{∗}} MU^{∗}( ) ≅ Eℓℓ_{∗} ⊗_{Eℓℓ_{∗}} Eℓℓ^{∗}( ),
(1.6)  Eℓℓ_{∗}( ) = Eℓℓ_{∗} ⊗_{MU_{∗}} MU_{∗}( ) ≅ Eℓℓ_{∗} ⊗_{Eℓℓ_{∗}} Eℓℓ_{∗}( ),

This ring Eℓℓ_{∗} can be interpreted as a ring of meromorphic modular forms with character in the finite cyclic group Hom(SL_{2}(ℤ), ℚ/ℤ) ≅ ℤ/12. In other words, the usual modularity conditions on a holomorphic function F: V ⟶ ℂ are replaced by

(1.7)  F (λ · {ω_{1}, ω_{2}}) = λ^{−k}F ({ω_{1}, ω_{2}}),

F (\{aω_{1} + bω_{2}, cω_{1} + dω_{2}\}) = χ_{F} (A)F (\{ω_{1}, ω_{2}\})∀ A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ), (1.8)

for some character χ_{F} : SL_{2}(ℤ) ⟶ ℚ/ℤ. Then Λ = η^{2} is the square of Dedekind’s η-function [25] and has character of order 12 which generates the finite cyclic group Hom(SL_{2}(ℤ), ℚ/ℤ). Because of this, we may identify such a ring of ‘twisted’ modular forms with the extension Eℓℓ_{∗} of Eℓℓ_{∗}. Although we make no use of this here, there are advantages in having a 2-periodic cohomology theory rather than one of period 24.

2. Elliptic curves, Weierstrass formal group laws and complex orientations in elliptic cohomology

Given an analytic torus ℂ/L, we can construct a Weierstrass cubic (elliptic curve) (thought of as a projective cubic curve)

ℭ_{W}(L): Y^{2}Z = 4X^{3} − \frac{1}{12}E_{4}(L)XZ^{2} + \frac{1}{216}E_{6}(L)Z^{3},

where the function E_{2n} is the 2nth Eisenstein function of Section 1, regarded as a function of the lattice L. The classical theory of the Weierstrass function gives us an explicit uniformisation of this curve. We define an analytic isomorphism

Φ: ℂ/L ⟶ ℭ_{W}(L)
z + L ⟼ \begin{cases}[℘(z, L), ℘′(z, L), 1], & if z ∉ L, \\ [0, 1, 0], & otherwise. \end{cases}

Here the Weierstrass function is normalised as in [6], so that for the lattice L = 2πi ⟨τ, 1⟩ with τ ∈ ℌ, we have

℘(z, L) = \frac{1}{(ez/2 − e−z/2)^{2}} + ∑_{n⩾1} [\frac{qnez}{(1 − qnez)^{2}} + \frac{qne−z}{(1 − qne−z)^{2}} ] .

The local parameter

T (z, L) = \frac{−2℘(z, L)}{℘′(z, L)}

is an elliptic function on ℭ_{W}(L) which has a simple zero at each lattice point. The multiplication on ℭ_{W}(L) gives rise to a formal group law

F_{L}^{Eℓℓ}(T_{1}, T_{2}) ∈ ℤ[1/6][E_{4}(L), E_{6}(L)][[T_{1}, T_{2}]]

which we call the Weierstrass formal group law associated to the lattice L, and is determined by the relation

T (z_{1} + z_{2}, L) = F_{L}^{Eℓℓ}(T (z_{1}, L), T (z_{2}, L)).

Of course, the universal example for such formal group laws is furnished by the power series

F^{Eℓℓ}(T_{1}, T_{2}) ∈ ℤ[1/6][Q, R, Δ^{−1}][[T_{1}, T_{2}]] = Eℓℓ_{∗}[[T_{1}, T_{2}]]

which is the canonical formal group law in elliptic cohomology. The natural choice of orientation for the canonical complex line bundle η ⟶ ℂP^{∞} then corresponds to T ∈ Eℓℓ_{∗}[[T ]] ≅ Eℓℓ^{∗}(ℂP^{∞}). See [6] for further details on these points. Evaluation of q-expansions gives rise to a homomorphism

Eℓℓ_{∗} = ℤ[1/6][Q, R, Δ^{−1}] ⟶ KU[1/6]_{∗}((q)) = ℤ[1/6][t, t^{−1}]((q)),

in which we use the Bott generator t ∈ KU_{2} to keep track of the weight which is half the topological grading. This an analogue of the classical Chern character, essentially discussed as such in [30], which focuses on modular forms of level 2 and uses the ring KO[1/2]_{∗}. One major advantage to the use of level 2 modular forms and the original definition of elliptic cohomology is that the formal group law and its logarithm can be displayed more explicitly in terms of natural algebra generators of the coefficient ring; see [14] for some calculational observations.

3. The Hopf algebroid Eℓℓ_{∗}Eℓℓ

In this section we will give some algebraic results on the cooperation algebra Eℓℓ_{∗}Eℓℓ = Eℓℓ_{∗}(Eℓℓ). The construction of the functors Eℓℓ^{∗}( ) and Eℓℓ_{∗}( ) depends crucially on the following consequence of the Landweber Exact Functor Theorem [27] (the last statement follows from an argument similar to one for E(n) in [31]).

Theorem 3.1. There is an isomorphism of bimodules over Eℓℓ_{∗}

Eℓℓ_{∗}Eℓℓ ≅ Eℓℓ_{∗} ⊗_{MU_{∗}} MU_{∗}MU ⊗_{MU_{∗}} Eℓℓ_{∗}

where we use the natural genus MU_{∗} ⟶ Eℓℓ_{∗} associated to the formal group law F^{Eℓℓ} to form tensor products. Moreover, Eℓℓ_{∗}Eℓℓ is flat as both a left and right module over Eℓℓ_{∗}.

Corollary 3.2. The pair (Eℓℓ_{∗}Eℓℓ, Eℓℓ_{∗}) is a Hopf algebroid over ℤ[1/6]. More generally, for any subring R of ℚ containing ℤ[1/6], the pair

(Eℓℓ_{∗}Eℓℓ ⊗_{ℤ[1/6]} R, Eℓℓ_{∗} ⊗_{ℤ[1/6]} R)

is a Hopf algebroid over R.

The term Hopf algebroid is thoroughly explained in [33]. The structure maps of Eℓℓ_{∗}Eℓℓ are derived ultimately from those of the ‘universal’ Hopf algebroid (MU_{∗}MU, MU_{∗}). Let η_{L}, η_{R}: Eℓℓ_{∗} ⟶ Eℓℓ_{∗}Eℓℓ be the left and right units; we will often abuse notation and write X = η_{L}(X).

Working over the rational numbers ℚ we have a simple description. First we note a consequence of the Landweber Exact Functor Theorem, which implies that multiplication by a prime p is a monomorphism on Eℓℓ_{∗}Eℓℓ; this was also noted in [14] for example.

Proposition 3.3. The rationalisation map Eℓℓ_{∗}Eℓℓ ⟶ Eℓℓ_{∗}Eℓℓ ⊗ ℚ is injective.

Proposition 3.4. As graded ℚ algebras we have

Eℓℓ_{∗}Eℓℓ ⊗ ℚ = ℚ[Q, R, Δ^{−1}, η_{R}(Q), η_{R}(R), η_{R}(Δ)^{−1}].

We also have a well known relationship between the two natural formal group laws over Eℓℓ_{∗}Eℓℓ and Eℓℓ_{∗}Eℓℓ ⊗ ℚ. Let log^{Eℓℓ} T and log^{Eℓℓ′} T denote the logarithms of the images over Eℓℓ_{∗}Eℓℓ ⊗ ℚ of the canonical formal group law induced by η_{L} and η_{R}.

Proposition 3.5. Let B(T) = ∑_{k⩾0} B_{k}T^{k+1} denote the strict isomorphism from the formal group law on Eℓℓ_{∗}Eℓℓ ⊗ ℚ induced from η_{L} to that induced from η_{R}. Then we have:

(1) as algebras over Eℓℓ_{∗} ⊗ ℚ = η_{L}(Eℓℓ_{∗} ⊗ ℚ),

Eℓℓ_{∗}Eℓℓ ⊗ ℚ = Eℓℓ_{∗} ⊗ ℚ[η_{R}(Q), η_{R}(R), η_{R}(Δ)^{−1}];

(2) log^{Eℓℓ} T = log^{Eℓℓ′}(B(T));

(3) for each n ⩾ 0, we have B_{n} ∈ Eℓℓ_{2n}Eℓℓ;

(4) as an Eℓℓ_{∗} = η_{L}(Eℓℓ_{∗}) algebra, Eℓℓ_{∗}Eℓℓ is generated by the elements B_{n}

with n ⩾ 1 together with η_{R}(Δ^{−1}).

We can describe (Eℓℓ_{∗}Eℓℓ, Eℓℓ_{∗}) as a universal object.

Proposition 3.6. Let R_{∗} be any graded commutative ring, let F_{1}, F_{2} be formal group laws over R_{∗} induced from Eℓℓ_{∗} by the ring homomorphisms θ_{1}, θ_{2} : Eℓℓ_{∗} ⟶ R_{∗}, and let H: F_{1} ≅ F_{2} be a strict isomorphism over R_{∗}. Then there is a unique ring homomorphism Θ: Eℓℓ_{∗}Eℓℓ ⟶ R_{∗} such that

Θ ∘ η_{L} = θ_{1} and Θ ∘ η_{R} = θ_{2}

and the series Θ(B(X)) = ∑_{n⩾0} Θ(B_{n})X^{n+1} satisfies

H(X) = Θ(B(X)).

This follows from the analogous universality of (MU_{∗}MU, MU_{∗}).

4. Generalized modular forms

We continue to use the notation established in Section 1. Recall the left principal bundle

V ⟶ L;
{ω_{1}, ω_{2}} ⟼ ⟨ω_{1}, ω_{2}⟩

with structure group SL_{2}(ℤ).

For any natural number N > 0, we denote by M_{2}(N) the set of 2 × 2 integer matrices with determinant N and set

(1/N)M_{2}(N) = {(1/N)A : A ∈ M_{2}(N)} .

Of course, these are isomorphic as right and left SL_{2}(ℤ) sets. The associated bundle

π_{V(N)} : V(N) = (1/N)M_{2}(N) ×_{SL_{2}(ℤ)} V ⟶ L

has fibre (1/N)M_{2}(N). Given an oriented basis {ω_{1}, ω_{2}} for a lattice L and A ∈ M_{2}(N) with

A = (\begin{matrix}a & b \\ c & d \end{matrix})

we have an oriented basis

\{\frac{aω_{1} + bω_{2}}{N} , \frac{cω_{1} + dω_{2}}{N} \}

for the lattice

L′ = ⟨\frac{aω_{1} + bω_{2}}{N} , \frac{cω_{1} + dω_{2}}{N} ⟩

which contains L with index N. Notice that each of the projection maps

V(N) ⟶^{π_{V(N)}} L

is an infinite covering, with fibre isomorphic to the set (1/N)M_{2}(N) ≅ M_{2}(N).

Factoring out by the left action of any subgroup G ⩽ SL_{2}(ℤ) on V(N), and we obtain a covering V(N) ⟶ G\V(N). If the subgroup G contains the congruence subgroup Γ(N), then this is a finite covering. We will be particularly interested in the two extreme cases G = SL_{2}(ℤ) and G = Γ(N). We set

L(N) = SL_{2}(ℤ)\V(N),
F(N) = Γ(N)\V(N),

which admit finite covering maps

L(N)π: L(N) ⟶ L,
F(N)π: F(N) ⟶ L,

whose fibres are the sets

SL_{2}(ℤ)\(1/N)M_{2}(N) ≅ SL_{2}(ℤ)\M_{2}(N),
Γ(N)\(1/N)M_{2}(N) ≅ Γ(N)\M_{2}(N).

Of course, these maps are holomorphic maps of complex analytic manifolds. The projection maps are also equivariant with respect to the obvious action of the complex units ℂ^{×} by multiplication.

The space L(N) can be viewed as the space of pairs of lattices L ⊆ L′ with index N. Similarly, we can interpret F(N) as the space of pairs (L, {ω′_{1} + L, ω′_{2} + L}), where

ω′_{1} = \frac{aω_{1} + bω_{2}}{N} ,
ω′_{2} = \frac{cω_{1} + dω_{2}}{N}

for an oriented basis {ω_{1}, ω_{2}} of L and

A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(N).

Thus {ω′_{1}, ω′_{2}} is an oriented basis of the module L′/L over the ring ℤ/N. We will frequently make use of these interpretations without further comment.

Now we can make the following definition of a notion of level N modular forms.

Definition 4.1. Let G ⩽ SL_{2}(ℤ) be a subgroup containing Γ(N). Then a holomorphic map

F: G\V(N) ⟶ ℂ

is a modular form of level N for G of weight k if for λ ∈ ℂ^{×},

F (G[(1/N)A, {λω_{1}, λω_{2}}]) = λ^{−k}F (G[(1/N)A, {ω_{1}, ω_{2}}]).

If G = Γ(N), then we frequently refer to such a modular form as a modular form of level N.

Notice that for such a G and a subgroup G′ containing Γ(N), a modular form of weight k for G is also one for G′. Holomorphic functions L(N) ⟶ ℂ for which the composite

Γ(N)\V(N) ⟶ L(N) ⟶ ℂ

is a modular form of level N will often be met in this work; we will loosely refer to these as level N modular forms on L(N).

Given such a modular form F of level N, we can evaluate F on the fibres over the lattices of the form ⟨τ, 1⟩, where τ ∈ ℌ. For each pair (r, s) with 0 < r, s and rs = N, there is a function

f_{F,r,s}: τ ⟼ F (G [(\begin{matrix}r/N & 0 \\ 0 & s/N \end{matrix}) , ⟨τ, 1⟩]) ,

with Fourier expansion of the form

∑_{−∞<n<∞} a_{n}^{F,r,s}qn/N where q1/N = e2πiτ/N.

We will refer to these q-expansions as the q-expansions of F along the fibres. For each coset BG ∈ SL_{2}(ℤ)/G, we also have the holomorphic function

F_{|_{B}}(G[(1/N)A, {ω_{1}, ω_{2}}]) = F (BGB^{−1}[(1/N)BA, {ω_{1}, ω_{2}}]).

Definition 4.2. The modular form F for G is holomorphic at infinity if for each coset GB ∈ G\SL_{2}(ℤ) and (r, s) as above, the functions

τ ⟼ F_{|_{B}} (G [(\begin{matrix}r/N & 0 \\ 0 & s/N \end{matrix}) , ⟨τ, 1⟩])

have q-expansions

∑_{−∞<n<∞} a_{n}^{F,r,s,B}qn/N

with a_{n}^{F,r,s,B} = 0 for n < 0; similarly, it is meromorphic at infinity if its q-expansions have a_{n}^{F,r,s,B} = 0 for n ≪ 0.

We will refer to the collection of q-expansions along the fibres of the functions F_{|_{B}} as the q-expansions of F at the cusps.

Now let K ⊆ ℂ be a subring which contains 1/6, and let ζ_{N} be a primitive Nth root of 1.

Definition 4.3. The modular form F for G is defined over the ring K if all the q-expansion coefficients of all the functions F_{|_{B}}, with BG ∈ SL_{2}(ℤ)/G, are in the ring K[1/N, ζ_{N}].

We now want to define a generalized modular form as a function on all of the spaces L(N) simultaneously in such a way that the restriction to each L(N) depends upon N in a controlled fashion. To do this we require that for each N we have a holomorphic function F_{N}: L(N) ⟶ ℂ which is simultaneously a modular form for each of the two lattices associated to each point of L. Thus we will require that our function is induced from a suitable type of function upon the product space L × L via the product _{V(N)}π_{V(N)} of the two projection maps to L. Finally, we will do this uniformly by requiring that these functions L × L ⟶ ℂ are independent of N.

Remark 4.4. The following definitions may appear somewhat forced in that we need to work with certain proper subsets of Map(X ×Y, ℂ). In fact, in the examples we consider, the spaces X and Y can be given the structures of complex analytic spaces X_{h}, Y_{h} as discussed in [38] and also more briefly in [17], Appendix B (in fact they are obtained as the analytic spaces associate to algebraic varieties over ℂ). Hence, we could characterise these sets of functions as analytic functions on the product X_{h} ×Y_{h}. The case of L itself follows since there is an analytic isomorphism between L and the affine variety

{(x, y) ∈ ℂ^{2} : x^{3} − y^{2} ≠ 0} ⊆ ℂ^{2}.

In order to avoid excessive technicalities, we proceed along the route below even though it may seem somewhat laboured to those well versed in algebraic geometry.

Recall that given two spaces X, Y, there is an embedding

Map(X, ℂ) ⊗_{ℂ} Map(Y, ℂ) ⟶ Map(X × Y, ℂ),

which sends the element f ⊗ g to the pointwise product function

f · g: (x, y) ⟼ f(x)g(y).

We will identify Map(X, ℂ) ⊗_{ℂ} Map(Y, ℂ) with its image in Map(X × Y, ℂ). More generally, given two vector subspaces A ⊆ Map(X, ℂ) and B ⊆ Map(Y, ℂ), we may identify the subspace A ⊗_{ℂ} B ⊆ Map(X, ℂ) ⊗_{ℂ} Map(Y, ℂ) with a subpace of Map(X × Y, ℂ).

Let MF(ℂ)_{k} denote the set of all weight k modular forms, i.e., holomorphic functions L ⟶ ℂ satisfying the modularity condition

F (λ · L) = λ^{−k}F (L) ∀λ ∈ ℂ^{×}.

Given a subring K ⊆ ℂ, let MF(K)_{k} denote the set of all modular forms whose associated q-expansions have coefficients in K.

We now make a series of definitions.

Definition 4.5. A modular form of weight k on L × L is a holomorphic map F: L × L ⟶ ℂ such that for λ ∈ ℂ^{×},

F (λ · L_{1}, λ · L_{2}) = λ^{−k}F (L_{1}, L_{2}),

and

F ∈ ∑_{r∈ℤ} MF(ℂ)_{r} ⊗_{ℂ} MF(ℂ)_{k−r} ⊆ Map(L × L, ℂ).

We can now give our definition of a generalized modular form.

Definition 4.6. A generalized modular form of level 1 and weight k is the coproduct F_{•} = ∐_{N⩾1} F_{N} of a family of holomorphic maps of the form

F_{N}: L(N) ⟶^{_{V(N)}π×π_{V(N)}} L × L ⟶^{F} ℂ

where F: L × L ⟶ ℂ is a fixed modular form of weight k on L × L .

Notice that for each N ⩾ 1, F_{N} is a modular form of level N for SL_{2}(ℤ) of weight k.

Definition 4.7. The generalized modular form F_{•} is defined over K if for each N ⩾ 1, the modular form F_{N} of level N for SL_{2}(ℤ) is defined over K.

Definition 4.8. The generalized modular form F_{•} is holomorphic at infinity if for each N ⩾ 1, the modular form F_{N} of level N is holomorphic at ∞; similarly, F_{•} is meromorphic at infinity if each F_{N} is meromorphic at ∞.

Now let us consider the groups of all holomorphic (resp. meromorphic) generalized modular forms of weight k and defined over K, which we will denote by ^{Gen}S(K)_{k} (resp. ^{Gen}M(K)_{k}). These can be combined into two graded rings GenS(K)_{∗} and ^{Gen}M(K)_{∗} which are algebras over the rings S(K)_{∗} and M(K)_{∗} of Section 1. Since both of these rings are torsion free, we have

GenS(K)_{∗} ⊆ ^{Gen}S(Kℚ)_{∗}
GenM(K)_{∗} ⊆ ^{Gen}M(Kℚ)_{∗},

where Kℚ is the smallest subring of ℂ containing both K and ℚ. We can easily prove the next result.

Theorem 4.9. As graded algebras over the rings S(ℚ)_{∗} and M(ℚ)_{∗} we have

GenS(ℚ)_{∗} = S(ℚ)_{∗}[E′_{4}, E′_{6}]
GenM(ℚ)_{∗} = M(ℚ)_{∗}[E′_{4}, E′_{6}, Δ′^{−1}],

where for each N ⩾ 1,

E′_{2n} = E_{2n} ∘_{L(N)}π,
Δ′ = Δ ∘_{L(N)}π

as functions L(N) ⟶ ℂ.

Recall from the definition of elliptic cohomology that M(ℤ[1/6])_{∗} = Eℓℓ_{∗}. By Proposition 3.4, we obtain the following.

Corollary 4.10. As graded algebras over Eℓℓℚ_{∗} ≅ Eℓℓ_{∗} ⊗ ℚ,

GenM(ℚ)_{∗} ≅ Eℓℓℚ_{∗}Eℓℓ ≅ Eℓℓ_{∗}Eℓℓ ⊗ ℚ.

This suggests that we ought to be able to describe Eℓℓ_{∗}Eℓℓ in terms of the ring GenM(ℤ[1/6])_{∗}. The crucial question is of course what effect integrality conditions have on the existence of generalized modular forms. The complete algebraic calculations of ^{Gen}S(ℤ[1/6])_{∗} and ^{Gen}M(ℤ[1/6])_{∗} will be given later, using work of N. Katz [23].

We will now discuss a multiplicative structure on the space ∐_{n⩾1} L(N), which induces coproducts on the rings of generalized modular forms.

For M, N ⩾ 1, there is a partial product map

(1/M)M_{2}(M) ×_{SL_{2}(ℤ)} V ×_{L} (1/N)M_{2}(N) ×_{SL_{2}(ℤ)} V ⟶ (1/MN)M_{2}(MN) ×_{SL_{2}(ℤ)} V (4.1)

which is defined on elements by the formula

([A, {ω′_{1}, ω′_{2}}] , [B, {ω_{1}, ω_{2}}]) ⟼ [AT B, {ω_{1}, ω_{2}}] , (4.2)

where we have

{ω′_{1}, ω′_{2}} = T B{ω_{1}, ω_{2}}

for some T ∈ SL_{2}(ℤ). Here the symbol ×_{L} implies that we form the pullback of the

diagram

(1/M)M_{2}(M) ×_{SL_{2}(ℤ)} V ⟶^{π_{V}} L ⟵^{_{V} π} (1/N)M_{2}(N) ×_{SL_{2}(ℤ)} V.

It is easily verified that this partial product is then compatible with the action of SL_{2}(ℤ) in the sense that it passes down to a partial product

L(M)×_{L}L(N) ⟶ L(MN).

This product can be viewed as making the space

L^{•} = ∐_{N⩾1} L(N)

into a ‘monoid over L’. It is clearly associative and the space L(1) acts via the identity. Taking functions on this space we obtain a coproduct which sends the function F: L^{•} ⟶ ℂ to the function

L^{•}×_{L}L^{•} ⟶ L^{•} ⟶^{F} ℂ.

The space L^{•} over L appears to play a rôle in elliptic cohomology analogous to that of the non-zero integers in K-theory, where they index (stable) Adams operations. We will make this more explicit in future work, but in this paper we will only demonstrate its connections with stable operations. In Section 5, we will describe the space L^{•} in a more algebraic fashion.

5. Isogenies of elliptic curves and cooperation algebras

By an elliptic curve ℭ over the complex numbers ℂ we will mean a non-singular Riemann surface of genus 1 with a distinguished basepoint O_{ℭ}. It is known that this can be uniformised, i.e., there is an analytic isomorphism

Φ: ℭ ≅ ℂ/L
Φ(O_{ℭ}) = 0 + L,

where L ⊆ ℂ is a lattice. Particular examples are furnished by the Weierstrass cubics of Section 5. Moreover, the torus ℂ/L is unique up to an analytic isomorphism of the form

[λ]: ℂ/L ⟶ ℂ/L′

where [λ] is induced by multiplication by λ and λ · L = L′. We can scale L so that it has the form L = ⟨τ, 1⟩ for some τ ∈ ℌ (the upper half plane); then Φ is unique up to analytic automorphism of ℂ/L. Of course, there is a canonical abelian group structure on ℂ/L which is transferred to ℭ by Φ, and ℭ is an analytic group with O_{ℭ} as its zero element.

Given two elliptic curves ℭ_{1}, ℭ_{2} over ℂ, an isogeny from ℭ_{1} to ℭ_{2} is an analytic homomorphism of groups Θ: ℭ_{1} ⟶ ℭ_{2} such that ker Θ is finite (it is then necessarily surjective). Let deg Θ = |ker Θ|, the degree of Θ, and K_{Θ} ⊆ ℂ be the unique lattice such that K_{Θ}/L_{1} = ker Θ. For ℭ_{1} = ℂ/L_{1} and ℭ_{2} = ℂ/L_{2} such an isogeny has to be of the form [λ] with

λ · K_{Θ} = L_{2},

and thus there is a unique factorisation

ℂ/L_{1} ⟶ ℂ/K_{Θ} ⟶^{[λ]} ℂ/L_{2} (5.1)

where the first map is induced by the canonical inclusion L_{1} ⟶ K_{Θ}. We will say that an isogeny is strict if λ = 1. Notice that for a strict isogeny,

L_{1} ⊆ L_{2}

has finite index and also

L_{2}/L_{1} = ker([1]: ℂ/L_{1} ⟶ ℂ/L_{2}).

From the above discussion, we see that the category of elliptic curves over ℂ with isogenies as morphisms, is naturally equivalent to the category of tori ℂ/L and isogenies, which will denote by Iso_{ℂ}. We will restrict attention to elliptic curves of the form ℂ/L and work with the category SIso_{ℂ} of all such elliptic curves together with their strict isogenies as morphisms.

We can decompose these categories Iso_{ℂ} and SIso_{ℂ} into disjoint unions

Iso_{ℂ} = ∐_{N⩾1} Iso_{ℂ}(N)
SIso_{ℂ} = ∐_{N⩾1} SIso_{ℂ}(N)

where Iso_{ℂ}(N) consists of isogenies with degree N and we have the equation SIso_{ℂ}(N) = Iso_{ℂ}(N) ∩ SIso_{ℂ}. Of course, the set SIso_{ℂ}(1) can be viewed as the set of objects in the categories Iso_{ℂ} and SIso_{ℂ}.

We can identify the morphism sets SIso_{ℂ}(N) with the underlying sets of the spaces L(N) defined in Section 4, since by construction a point of L(N) is equivalent to an inclusion of lattices L ⊆ L′ of index N. Moreover, the two projections π_{L(N)},_{L(N)}π: L(N) ⟶ L simply pick out these two lattices, which are the domain and codomain of a unique morphism in SIso_{ℂ}(N) ⊆ SIso_{ℂ}. Thus, we have the following result.

Proposition 5.1. There is an isomorphism of small categories

SIso_{ℂ} ≅ L^{•}

under which

SIso_{ℂ}(N) ≅ L(N)

for each N ⩾ 1. The category SIso_{ℂ} is therefore naturally topologised and is the union of countably infinitely many complex manifolds SIso_{ℂ}(N).

This result together with the ideas of Section 4 gives us an interesting class of functions on SIso_{ℂ}, which are analytic when restricted to the spaces SIso_{ℂ}(N) ≅ L(N). We will freely interpret generalized modular forms as functions on the category SIso_{ℂ}. Of course, the structure maps of the category SIso_{ℂ} correspond to the partial monoid structure on L^{•}; thus there will be a coproduct structure on the ring of generalized modular forms. This structure becomes interesting when we tensor up with a subring R ⊆ ℚ and force morphisms to become invertible; we then obtain the structure of a Hopf algebroid on an appropriate ring of generalized modular forms.

Now most of the morphisms in Iso_{ℂ} and SIso_{ℂ} are not invertible and we will need to form various categories of fractions for these. Let R ⊆ ℚ be a subring of the rational numbers and let R_{+}^{×} denote the subgroup R^{×} ∩ ℚ_{+} of all positive units in R. We wish to invert the strict isogenies [1]: ℂ/L_{1} ⟶ ℂ/L_{2} with |L_{2}/L_{1}| ∈ R_{+}^{×}. To do this we replace the ℤ lattices L_{1} and L_{2} by the R lattices RL_{1} ≅ R ⊗_{ℤ} L_{1} and RL_{2} ≅ R ⊗_{ℤ} L_{2}, and consider ‘isogenies’ of the form

[u]: ℂ/RL_{1} ⟶ ℂ/RL_{2}

where u ∈ R_{+}^{×}. Notice that such an isogeny has trivial kernel, and has inverse

[u^{−1}]: ℂ/RL_{2} ⟶ ℂ/RL_{1}.

Such morphisms lie in a category Iso_{ℂ}^{R_{+}^{×}} whose objects are those of Iso_{ℂ} and where for any two lattices L_{1} and L_{2} for which RL_{1} = RL_{2}, there is unique morphism [u]: ℂ/L_{1} ⟶ ℂ/L_{2} whenever u ∈ R_{+}^{×}. We will call such a morphism an R-isogeny; furthermore, if u = 1, then we say that it is a strict R-isogeny. The strict R-

isogenies form a subcategory SIso_{ℂ}^{R_{+}^{×}} of Iso_{ℂ}^{R_{+}^{×}}. If two lattices L_{1} and L_{2} satisfy RL_{1} = RL_{2}, then we will say that they are R-commensurable. It is easy to see that the notion of being R-commensurable is an equivalence relation. Notice that if L_{1} and L_{2} are R-commensurable, then the lattice L_{1} ∩ L_{2} is R-commensurable with both L_{1} and L_{2}; moreover, the unique diagram

ℂ/L_{1} ⟵ ℂ/L_{1} ∩ L_{2} ⟶ ℂ/L_{2}

in SIso_{ℂ} gives rise to a unique diagram

ℂ/L_{1} ⟶ ℂ/L_{1} ∩ L_{2} ⟶ ℂ/L_{2}

in SIso_{ℂ}^{R_{+}^{×}}.

Theorem 5.2. The functor SIso_{ℂ} ⟶ SIso_{ℂ}^{R_{+}^{×}} which is the identity on objects and sends the strict isogeny ℂ/L_{1} ⟶ ℂ/L_{2} to the strict R isogeny [1]: ℂ/L_{1} ⟶ ℂ/L_{2} is the localization of SIso_{ℂ} with respect to all morphisms [1]: ℂ/L′ ⟶ ℂ/L′ for which | ker[1]| ∈ R_{+}^{×}.

Notice that in particular this means that a strict isogeny ℂ/L ⟶ ℂ/(1/N)L for

which N ∈ R_{+}^{×} ∩ ℕ always has an inverse in SIso_{ℂ}^{R_{+}^{×}}.

In practise, we will work with rings R for which ℤ[1/6] ⊆ R, although this restriction is only important when we consider rings of modular forms as rings of

functions on Iso_{ℂ} and Iso_{ℂ}^{R_{+}^{×}}.

We end this section by introducing another collection of categories. Let SL Iso_{ℂ}^{R_{+}^{×}} denote the category whose objects are lattices L ∈ L, and where whenever RL_{1} = RL_{2}, the morphisms from L_{1} to L_{2} are the orientation preserving monomorphisms L_{1} ⟶ L_{2} which induce R-linear isomorphisms RL_{1} ≅ RL_{2}. In particular, when R = ℤ, there are morphisms L_{1} ⟶ L_{2} if and only if L_{1} = L_{2}; on the other hand, when R = ℚ, there are morphisms L_{1} ⟶ L_{2} if and only if ℚL_{1} = ℚL_{2}. In the case

R = ℤ, we may identify SL Iso_{ℂ} = SL Iso_{ℂ}^{R_{+}^{×}} with the space

V^{•} = ∐_{N⩾1} V(N).

6. The action of isogenies on Weierstrass formal groups and operations in elliptic cohomology

Given a strict isogeny [1]: ℂ/L_{1} ⟶ ℂ/L_{2} of degree N, together with a modular form F of level 1, the function

([1]: ℂ/L_{1} ⟶ ℂ/L_{2}) ⟼ F (L_{2})

is a modular form in the variable L_{2}. If we choose an oriented basis for L_{1} and use this to make the identifications

(6.1)  SL(L_{1}) ≅ SL_{2}(ℤ)

and

(6.2)  SL((1/N)L_{1}) ≅ SL_{2}(ℤ),

then we can interpret this function as a modular form for the subgroup of SL_{2}(ℤ) corresponding to

SL(L_{1}) ∩ SL(L_{2}) ⊆ SL((1/N)L_{1}) ≅ SL_{2}(ℤ)

under the isomorphism of (6.2). The proof of the following key result is similar to arguments of [6, 9].

Proposition 6.1. The formal group laws F_{L_{1}}^{Eℓℓ} and F_{L_{2}}^{Eℓℓ} are strictly isomorphic over the ring of level N modular forms on L(N) defined over the ring ℤ[1/6].

Proof. The coefficients of F_{L_{1}}^{Eℓℓ} and F_{L_{2}}^{Eℓℓ} considered as functions of the pair L_{1} ⊆ L_{2} are level N modular forms. In fact they lie in the rational subalgebra ℚ[E_{4}(L_{1}), E_{6}(L_{1}), E_{4}(L_{2}), E_{6}(L_{2})] ⊆ ℂ generated by the complex numbers E_{r}(L_{s}). The series T (X, L_{1}) and T (X, L_{2}) provide strict isomorphisms from the additive group law to F_{L_{1}}^{Eℓℓ} and F_{L_{2}}^{Eℓℓ}, hence there is a strict isomorphism

φ_{L_{1},L_{2}} : F_{L_{1}}^{Eℓℓ} ⟶ F_{L_{2}}^{Eℓℓ}

with coefficients in the latter ring. Now by specialising to the case where L_{1} = ⟨τ, 1⟩ (τ ∈ ℌ) the series φ_{L_{1},L_{2}}(X) gives a q-expansion

φ_{⟨τ,1⟩,L_{2}}(X) ∈ ℚ[ζ_{N}][[q^{1/N}]][[X]].

Following [6], we can use the theory of Tate curves described in [22] to deduce that the coefficients of F_{⟨τ,1⟩}^{Eℓℓ} and F_{L_{2}}^{Eℓℓ} actually lie in the rings ℤ[1/6][[q]] and ℤ[1/6N, ζ_{N}][[q^{1/N}]]. Hence

φ_{⟨τ,1⟩,L_{2}}(X) ∈ ℤ[1/6N, ζ_{N}][[q^{1/N}]][[X]],

showing that the coefficients of φ_{L_{1},L_{2}}(X) are level N modular forms on L(N) defined over ℤ[1/6N].

Let φ_{L_{1},L_{2}}(X) be the unique strict isomorphism from F_{L_{1}}^{Eℓℓ} to F_{L_{2}}^{Eℓℓ} used in the proof of this result; we will write φ when the isogeny is understood. The following Corollary makes use of the fact that the considerations of the above proof are essentially independent of L ⊆ L′. Indeed, the coefficients of φ_{L_{1},L_{2}}(X) are rational polynomials in the coefficients of the formal group laws F_{L_{1}}^{Eℓℓ} and F_{L_{2}}^{Eℓℓ}, independently of the lattices L_{1} ⊆ L_{2} and the index N.

Corollary 6.2. The coefficient of X^{n+1} in φ_{L_{1},L_{2}}(X) when considered as a function of pairs L_{1} ⊆ L_{2} for arbitrary N ⩾ 1, is a holomorphic generalized modular form of weight n, i.e., is contained in ^{Gen}S(ℤ[1/6])_{n} ⊆ ^{Gen}M(ℤ[1/6])_{n}.

Now given any R isogeny [u]: ℂ/L_{1} ⟶ ℂ/L_{2}, we can assume that L_{2} ⊆ (1/N)L_{1} for some N ∈ R_{+}^{×} and then an easy calculation gives

(6.3)  T (uz, L_{2}) = [u]_{F_{L_{2}}^{Eℓℓ}}(T (z, L_{2}))
= φ ([u]_{F_{(1/N)L_{1}}^{Eℓℓ}} T (z, (1/N)L_{1}))
= φ ((1/N)[uN]_{F_{L_{1}}^{Eℓℓ}}T (z, L_{1})) .

But this is a power series in T (z, L_{1}) with coefficients in the ring of level N modular forms on L(N) defined over R. Hence, any strict isogeny [u] as above induces an isomorphism between the formal group law associated with the elliptic curve ℂ/L_{1} and a ‘twisted version’ of that associated to ℂ/L_{2}. In the case where [u] = [1] is strict, so is the induced isomorphism of formal group laws. Notice that this implies that for each strict R-isogeny [1]: ℂ/L_{1} ⟶ L_{2}, there is a ring homomorphism Ψ_{L_{1},L_{2}} with domain MU_{∗}MU and extending the two homomorphisms

MU_{∗} ⟶^{φ_{L_{1}}} R[1/6][E_{4}(L_{1}), E_{6}(L_{1})],
MU_{∗} ⟶^{φ_{L_{2}}} R[1/6][E_{4}(L_{2}), E_{6}(L_{2})]

which classify the formal group laws F_{L_{1}}^{Eℓℓ} and F_{L_{2}}^{Eℓℓ}; this takes values which are level N modular forms when considered as functions of L_{1} ⊆ L_{2}.

It is now immediate that there is a unique homomorphism

Eℓℓ_{∗} ⊗_{MU_{∗}} MU_{∗}MU ⊗_{MU_{∗}} Eℓℓ_{∗} ⟶ Eℓℓ_{∗}Eℓℓ ⟶ ^{Gen}M(R)_{∗}

which specialises for each pair L_{1} ⊆ L_{2} to give Ψ_{L_{1},L_{2}}, using Corollary 6.2. In the case of a strict R-isogeny of the form [1]: L ⟶ (1/N)L, we find that the left unit on an element F ∈ Eℓℓ_{2n} is sent to N^{n}F by this homomorphism; in this case we can produce a multiplicative stable operation in elliptic cohomology:

(6.4)  ψ^{N}: Eℓℓ^{∗}( ) ≅ (S^{0} ∧ Eℓℓ)^{∗}( ) ⟶ (Eℓℓ ∧ Eℓℓ)^{∗}( )

⟶^{≅} Eℓℓ_{∗}Eℓℓ ⊗_{Eℓℓ_{∗}} Eℓℓ^{∗}( )

⟶^{Ψ_{L_{1},L_{2}}} EℓℓR_{∗} ⊗_{Eℓℓ_{∗}} Eℓℓ^{∗}( )

⟶ EℓℓR^{∗}( ),

which makes use of the above homomorphism Eℓℓ_{∗}EℓℓR ⟶ EℓℓR_{∗}. This is the Adams operation ψ^{N} mentioned in [6], and has a unique extension to a stable operation

ψ^{N}: EℓℓR^{∗}( ) ⟶ EℓℓR^{∗}( ).

For a fixed L and N ∈ R_{+}^{×}, we can take all of the induced ring homomorphisms Eℓℓ_{∗} ⟶ EℓℓR_{∗} and average them (i.e., sum up and divide by N). This gives rise to a left Eℓℓ_{∗}-linear homomorphism

T̃: Eℓℓ_{∗}Eℓℓ ⟶ EℓℓR_{∗}

which yields a stable operation

(6.5)  \overline{T}_{N}: Eℓℓ^{∗}( ) ≅ (S^{0} ∧ Eℓℓ)^{∗}( ) ≅ Eℓℓ_{∗}Eℓℓ ⊗_{Eℓℓ_{∗}} Eℓℓ^{∗}( )

\underrightarrow{_{T}} EℓℓR^{∗} ⊗_{Eℓℓ_{∗}} Eℓℓ^{∗}( )

≅ EℓℓR^{∗}( )

that is merely additive; again there is a unique extension to a stable operation \overline{T}_{N}: EℓℓR^{∗}( ) ⟶ EℓℓR^{∗}( ). This is the extension of the Nth Hecke operator constructed in [6]. This type of operation requires that we use not just the ring EℓℓR_{∗} but the larger ring of modular forms of level N to build enough multiplicative operations over which we symmetrise to get an operation within the theory EℓℓR^{∗}( ) itself. This sort of consideration is not necessary in K-theory, and represents a considerable complication in understanding the operations in elliptic cohomology.

Of course, the above discussion can also be interpreted in the light of the observation in Section 5 that the rings of generalized modular forms may be viewed as func-

tions on the categories Iso_{ℂ}^{R_{+}^{×}} and SIso_{ℂ}^{R_{+}^{×}}. Indeed, given an R-isogeny [u]: ℂ/L_{1} ⟶ ℂ/L_{2}, the coefficients of the power series discussed above can be viewed as functions

on the category Iso_{ℂ}^{R_{+}^{×}} and hence as elements of ^{Gen}S(ℂ)_{∗} ⊆ ^{Gen}M(ℂ)_{∗}. A careful consideration of q-expansions actually shows that they lie in ^{Gen}S(R)_{∗} ⊆ ^{Gen}M(R)_{∗} provided that we make the standard assumption that ℤ[1/6] ⊆ R. This provides us with a natural homomorphism Eℓℓ_{∗}Eℓℓ ⟶ ^{Gen}M(R)_{∗}. Later we will demonstrate the following theorem.

Theorem 6.3. For each subring R ⊆ ℚ containing 1/6, there is an isomorphism of graded rings

Eℓℓ_{∗}EℓℓR ≅ Eℓℓ_{∗}Eℓℓ ⊗ R ⟶ ^{Gen}M(R)_{∗},

and moreover this is an isomorphism of Hopf algebroids over R.

The antipode in ^{Gen}M(R)_{∗} is induced by the inverse map in the category Iso_{ℂ}^{R_{+}^{×}}, and corresponds under this isomorphism to the antipode in Eℓℓ_{∗}EℓℓR.

7. Some rings of numerical Laurent polynomials and K-theory cooperations

In this section we review the properties of some rings of numerical (Laurent) polynomials in sufficient detail for our purposes in calculating the rings of generalized modular forms contained in Section 9. The present section owes much to previous joint work with Francis Clarke, see [11] and [4]; for more on the topological connections, see [3, 2].

Let K ⊆ ℚ be a subring. Then we define the ring of numerical polynomials over K to be

A(w; K) = {f(w) ∈ ℚ[w] : f(r) ∈ K ∀r ∈ ℤ}.

Similarly, we define the ring of stably numerical (Laurent) polynomials over K to be

A^{S}(w; K) = {f(w) ∈ ℚ[w, w^{−1}] : f(r) ∈ K[1/r] ∀r ∈ ℤ, 0 ≠ r}.

Finally we define the subring of semistable numerical polynomials over K by

A_{0}^{S}(w; K) = A^{S}(w; K) ∩ ℚ[w].

We set A(w) = A(w; ℤ), A^{S}(w) = A^{S}(w; ℤ) and A_{0}^{S}(w) = A_{0}^{S}(w; ℤ).

Proposition 7.1. As a module over K, A(w; K) has a basis consisting of the bi-

nomial coefficient polynomials C_{n}(w) = \binom{w}{n} for n ⩾ 0. Hence we have an iso-

morphism of algebras over K,

A(w; K) ≅ A(w) ⊗_{ℤ} K.

As algebras over K,

A^{S}(w; K) = A(w; K)[w^{−1}].

Proofs of these results are given in [11].

Let us now assume that K = ℤ_{(p)}, the ring of p-local integers for a prime p. Let ord_{p}(h(w)) be the minimum value of ord_{p} on the coefficients of a Laurent polynomial h(w), or equivalently

ord_{p}(h(w)) = min{ord_{p}(h(a)) : a ∈ ℤ_{(p)}^{×}}.

We define increasing filtrations on A^{S}(w; ℤ_{(p)}) and A_{0}^{S}(w; ℤ_{(p)}) as follows. Let

M^{k} = {f(w) ∈ A^{S}(w; ℤ_{(p)}) : p^{k}f(w) ∈ ℤ_{(p)}[w, w^{−1}]},
M_{0}^{k} = {f(w) ∈ A_{0}^{S}(w; ℤ_{(p)}) : p^{k}f(w) ∈ ℤ_{(p)}[w]} = M^{k} ∩ A_{0}^{S}(w; ℤ_{(p)}).

Clearly we have M^{0} = ℤ_{(p)}[w, w^{−1}] and M_{0}^{0} = ℤ_{(p)}[w]; also the two filtrations

M^{0} ⊆ M^{1} ⊆ ⋯ ⊆ M^{k} ⊆ ⋯ ⊆ M^{∞} = A^{S}(w; ℤ_{(p)})
M_{0}^{0} ⊆ M_{0}^{1} ⊆ ⋯ ⊆ M_{0}^{k} ⊆ ⋯ ⊆ M_{0}^{∞} = A_{0}^{S}(w; ℤ_{(p)})

are exhaustive. Let us investigate the successive quotients M^{k}/M^{k−1} and M_{0}^{k}/M_{0}^{k−1} for k ⩾ 1.

By Proposition (7.1), any element f(w) ∈ A^{S}(w; ℤ_{(p)}) has the form

f(w) = ∑_{0⩽i⩽d(f)} h_{i}(w)C_{i}(w) (7.1)

where h_{i}(w) ∈ ℤ_{(p)}[w, w^{−1}] and we assume that h_{d(f)}(w) ≠ 0. The p-adic ordinal of n! is given by

ord_{p}(n!) = \frac{n − α_{p}(n)}{p − 1}, (7.2)

where α_{p}(n) is the sum of the p-adic digits of n. In particular,

ord_{p}(p^{r}!) = \frac{p^{r} − 1}{p − 1} = 1 + p + ⋯ + p^{r−1}. (7.3)

Now C_{n}(w) represents non-zero elements in the quotients

M^{ord_{p}(n!)}/M^{ord_{p}(n!)−1} and M_{0}^{ord_{p}(n!)}/M_{0}^{ord_{p}(n!)−1}.

Thus for a general element f(w), we see that f(w) ∈ M^{k} if and only if

k ⩾ max{ord_{p}(n!) − ord_{p}(h_{n}(w)) : 0 ⩽ n ⩽ d(f)}

and moreover it represents a non-zero element in M^{k}/M^{k−1} if and only if the last inequality is actually an equality.

It will be convenient to use a different basis for the p-local numerical polynomial ring A(w; ℤ_{(p)}). We require the following results taken from [4].

Proposition 7.2. Define the following sequence of polynomials in ℚ[w]:

θ_{0}(w) = w,
θ_{1}(w) = \frac{(θ_{0}(w) − θ_{0}(w)^{p})}{p},
θ_{2}(w) = \frac{(θ_{0}(w) − pθ_{1}(w)^{p} − θ_{0}(w)^{p^{2}})}{p^{2}},
⋮
θ_{r}(w) = \frac{θ_{0}(w) − p^{r−1}θ_{r−1}(w)^{p} − p^{r−2}θ_{r−2}(w)^{p^{2}} − ⋯ − θ_{0}(w)^{p^{r}}}{p^{r}} ,
⋮

Then

(1) for each r ⩾ 0, θ_{r}(w) ∈ A(w; ℤ_{(p)}) and moreover defines a function θ_{r}: ℤ_{(p)} ⟶ ℤ_{(p)};

(2) we have

ord_{p}(θ_{r}(w)) = \frac{p^{r} − 1}{p − 1} = 1 + p + p^{2} + ⋯ + p^{r−1};

(3) the monomials

θ_{0}(w)^{s_{0}}θ_{1}(w)^{s_{1}} ⋯ θ_{d}(w)^{s_{d}} for 0 ⩽ s_{r} < p

form a ℤ_{(p)}-basis for A(w; ℤ_{(p)});

(4) the monomials

θ_{1}(w)^{s_{1}} ⋯ θ_{d}(w)^{s_{d}} for 0 ⩽ s_{r} < p

span A^{S}(w; ℤ_{(p)}) as a module over ℤ_{(p)}[w, w^{−1}];

(5) for each k ⩾ 1, the monomials

θ_{0}(w)^{s_{0}}θ_{1}(w)^{s_{1}} ⋯ θ_{d}(w)^{s_{d}} for 0 ⩽ s_{r} < p and 0 ⩽ s_{0} < p − 1

form a ℤ/p^{k}-basis for A^{S}(w; ℤ_{(p)})/(p^{k}), which can also be identified with the ring of functions ℤ_{(p)}^{×} ⟶ ℤ/p^{k} which are continuous with respect the p-adic norm on the domain and the discrete topology on the range.

We will set \overline{θ}_{r}(w) = w^{−1}θ_{r}(w) ∈ A_{0}^{S}(w; ℤ_{(p)}) for r ⩾ 1.

Now consider an element of A_{0}^{S}(w; ℤ_{(p)}) of the form

f(w) = cθ_{0}(w)^{s_{0}} \overline{θ}_{1}(w)^{s_{1}} ⋯ \overline{θ}_{d}(w)^{s_{d}}

where 0 ⩽ s_{r} < p, 0 ⩽ s_{0} < p − 1 and c ∈ ℤ_{(p)}. Then the p-adic ordinal of this polynomial is

ord_{p}(f(w)) = ord_{p}(c) − ∑_{1⩽j⩽d} s_{j} \frac{(p^{j} − 1)}{(p − 1)},

and so f(w) ∈ M_{0}^{k} if and only if

k ⩾ ∑_{1⩽j⩽d} s_{j} \frac{(p^{j} − 1)}{(p − 1)} − ord_{p}(c),

and represents a non-zero element of M_{0}^{k}/M_{0}^{k−1} if and only if this is actually an equality.

We end this section by recalling the topological significance of the ring of stably numerical polynomials. This involves the determination of the cooperation algebra for complex K-theory, KU_{∗}KU, discussed in [3, 11].

Theorem 7.3. Let u = η_{L}(t) and v = η_{R}(t) be the images of the Bott generator t ∈ KU_{2} under the left and right units KU_{∗} ⟶ KU_{∗}KU, and let w = vu^{−1} ∈ KU_{0}KU. Then the image of the (monomorphic) rationalisation map

KU_{0}KU ⟶ KU_{0}KUℚ ≅ KU_{0}KU ⊗ ℚ

is equal to the ring of stably numerical polynomials A^{S}(w). More generally, if R ⊆ ℚ is any subring, then the image of the localization KU_{0}KUR ≅ KU_{0}KU ⊗ R under the rationalisation map

KU_{0}KUR ⟶ KU_{0}KU ⊗ ℚ

is equal to the ring of stably R-numerical polynomials A^{S}(w; R).

The natural Hopf algebroid structure on the pair (KU_{∗}KU, KU_{∗}) is then induced by the left and right units together with the maps

(coproduct)  u ⟼ u ⊗ 1; v ⟼ 1 ⊗ v; w ⟼ w ⊗ w;
(antipode)  u ⟼ v; v ⟼ u; w ⟼ w^{−1},

where the coproduct is a ring homomorphism

KU_{∗}KU ⟶ KU_{∗}KU ⊗_{KU_{∗}} KU_{∗}KU

into the tensor product of bimodules obtain using the right-left KU_{∗} module structures.

This result provides a model for our description of the cooperation algebra Eℓℓ_{∗}Eℓℓ.

8. Katz’s work on divided congruences amongst modular forms

In this section we will describe briefly results from N. Katz’s paper [23], especially section 5. These will be applied to determine the rings of generalized modular forms.

Let p > 3 be a prime. We will work with the ring S(ℤ_{(p)})_{∗} of holomorphic modular forms defined over the ring of p-local integers ℤ_{(p)}. For the remainder of this section we let S(ℤ_{(p)})_{⊕} denote the subring of ℤ_{(p)}[[q]] generated by the images of all of the individual gradings S(ℤ_{(p)})_{k} under the homomorphism

S(ℤ_{(p)})_{k} ⟶^{eval} ℤ_{(p)}[[q]]; F ⟼ F̃(q)

which assigns to each modular form its q-expansion. Clearly this is a polynomial subring ℤ_{(p)}[Q̃, R̃] of ℤ_{(p)}[[q]]. However, it is not a direct summand as a ℤ_{(p)}- module, as the congruence 1 − Ẽ_{p−1} ≡ 0 (mod p) shows. For each k ⩾ 1, we will describe the kernel of the composition

eval_{p^{k}}: S(ℤ_{(p)})_{⊕} ⟶^{eval} ℤ_{(p)}[[q]] ⟶^{red} ℤ/p^{k}[[q]].

Definition 8.1. Define the numerical function h by

h(r) = \frac{p^{r} − 1}{p − 1} if r ⩾ 1,

and h(0) = 0.

Theorem 8.2. There is sequence of elements R_{0} = p, R_{1}, …, R_{k}, … in S(ℤ_{(p)})_{⊕} such that

(1) each R_{k} is a sum of the q-expansions of modular forms of weight at most p^{k} − 1;

(2) for each k ⩾ 1 there is a element R′_{k} ∈ ℤ_{(p)}[[q]] such that

R_{k} = p^{h(k)}R′_{k}

in ℤ_{(p)}[[q]];

(3) the evaluation modulo p^{k} map, eval_{p^{k}}, has as its the kernel the ideal I_{k} ◁ S(ℤ_{(p)})_{⊕} generated by the elements

R_{0}^{r_{0}}R_{1}^{r_{1}} ⋯ R_{d}^{r_{d}}

for which

r_{0} + ∑_{1⩽j⩽d} r_{j}h(j) ⩾ k.

In fact, in his theorem 5.5, Katz gives gives an explicit construction for the elements R′_{k} and R_{k}, and we will make use of this in Section 9. We define the ring of (p-local) divided congruences to be

DC_{p} = {Θ ∈ ℚ[Q̃, R̃] : Θ(q) ∈ ℤ_{(p)}[[q]]} .

Theorem 8.3. The ℤ_{(p)}-algebra DC_{p} is generated by the elements Q̃, R̃ and the R′_{k} (k > 0). As a ℤ_{(p)}[Q̃, R̃]-module, it is spanned by the elements

R_{0}^{′r_{0}}R_{1}^{′r_{1}} ⋯ R_{d}^{′r_{d}}

for which

r_{0} + ∑_{1⩽j⩽d} r_{j}h(j) ⩾ k.

There is an action of the p-local units ℤ_{(p)}^{×} on the ungraded ring of modular

forms ℚ[Q̃, R̃] ⊆ ℚ[[q]], namely that given by

a · (∑_{k} F̃_{k}) = ∑_{k} a^{k}F̃_{k},

where F_{k} has weight k. This action ultimately comes from the operation of including each lattice L into (1/N)L, for any natural number N, and is related to the elliptic cohomology Adams operations of [6].

Proposition 8.4. The action of ℤ_{(p)}^{×} on ℚ[Q̃, R̃] restricts to an action on the subring DC_{p}. Moreover, the eigenspaces of this action are the submodules of homogeneous weight modular forms.

This is implicitly demonstrated by Katz in [23]. The second statement means that for X ∈ ℚ[Q̃, R̃],

∀a ∈ ℤ_{(p)}^{×}, a · X = a^{k}X ⟺ X is the image of a weight k modular form over ℚ.

We may view each element Θ ∈ DC_{p} as defining a function

ℤ_{(p)}^{×} ⟶ ℤ_{(p)}[[q]],

and thus we have

(a · Θ)(q) = ∑_{n⩾0} c_{n}(a)q^{n},

where the coefficient functions c_{n} are rational polynomial functions in a taking values in ℤ_{(p)}, i.e., each c_{n} lies in the ring of semi-numerical polynomials A_{0}^{S}(w; ℤ_{(p)}). One interpretation of this is in terms of the embedding DC_{p} ⟶ ℚ[w][[q]] which sends Θ to ∑_{n⩾1} c_{n}(w)q^{n}, and has image in the subring A_{0}^{S}(w; ℤ_{(p)})[[q]]. Thus there is an embedding of rings

DC_{p} ⟶ A_{0}^{S}(w; ℤ_{(p)})[[q]]. (8.1)

Notice that we can modify the definition of the ring of divided congruences to give a global version, namely

DC = {Θ ∈ ℚ[Q̃, R̃] : ∀a ∈ ℤ − {0}, (a · Θ)(q) ∈ ℤ[1/6a][[q]]} ,

where we define the action of a ∈ ℤ − {0} similarly to the above action of ℤ_{(p)}^{×}. By viewing each Θ as a function ℤ − {0} ⟶ ℚ[[q]], we see that there is an embedding of rings

DC ⟶ A_{0}^{S}(w; ℤ[1/6])[[q]]. (8.2)

Of course, for either of rings DC_{p} and DC, we can get back from subrings of A_{0}^{S}(w; ℚ)[[q]] to subrings of ℚ[[q]] by evaluating w at 1.

We end this section by remarking that although the element \overline{θ}_{1}(w) = (1−w^{p−1})/p can appear as the constant term of an element of DC_{p}, there is no element whose constant term is

w^{−1} \frac{(w − w^{p})/p − ((w − w^{p})/p)^{p}}{p} .

This is related to the fact that

\frac{(1 − E_{p−1})/p − ((1 − E_{p−1})/p)^{p}}{p}

is not a modular form modulo p in the sense of Serre, see [40].

We suspect that a direct proof of Katz’s results (and equivalently of ours) should be possible making use of the ring of stably numerical polynomials, however at present this eludes us.

9. Calculation of the rings of generalized modular forms

In this section we determine the algebraic structure of the two rings of generalized modular forms

GenS(ℤ[1/6])_{∗} and ^{Gen}M(ℤ[1/6])_{∗}.

Our approach to this makes use of Katz’s work which we have described in Section 8.

We are primarily interested in the (graded) ring ^{Gen}M(ℤ[1/6])_{∗}, but it clearly suffices to consider the subring ^{Gen}S(ℤ[1/6])_{∗} consisting of holomorphic generalized modular forms. Now by Corollary 4.10, it suffices to determine the subring of ℚ[Q, R, Q′, R′] consisting of those homogeneous elements whose q-expansions lie in ℤ[1/6N, ζ_{N}][[q, q′]] whenever we evaluate on a pair of the form L = ⟨τ, 1⟩ ⊆ L′ with index N. Here Q(L ⊆ L′) = E_{4}(L), R(L ⊆ L′) = E_{6}(L) (a modular form in L alone), Q′(L ⊆ L′)) = E_{4}(L′) and R′(L ⊆ L′)) = E_{6}(L′) (a modular form in L′ alone). Let us examine these conditions in more detail.

Now let Φ ∈ ^{Gen}S(ℤ[1/6])_{n} and N ⩾ 1. Let us evaluate Φ at a pair of lattices L = ⟨τ, 1⟩ ⊆ L′ with [L; L′] = N and τ ∈ ℌ; notice that L′ ⊆ ⟨τ/N, 1/N⟩. Our data gives rise to an element of ℤ[1/6N, ζ_{N}]((q^{1/N})). It is easily seen that

L′ = ⟨\frac{rτ + t}{N}, \frac{s}{N} ⟩

for 0 ⩽ r, s, t ∈ ℤ satisfying rs = N and 0 ⩽ t < s. Notice that given L, τ is unique to within an integer summand, and hence the element τ′ = (rτ + t)/N ∈ ℌ is unique up to a summand of the form kr/N. Now suppose that we have the following expression for Φ ∈ ℚ[Q, R, Q′, R′]_{n},

Φ = ∑_{m,a,b} c_{m,a,b}F_{m,a}F′_{n−m,b}, (9.1)

with c_{m,a,b} ∈ ℚ, and F_{m,a} ∈ M(ℤ[1/6])_{m}, F′_{n−m,b} ∈ S(ℤ[1/6])_{n−m} being taken from the standard basis of Section 1 evaluated on L and L′. Then we have

Φ(L ⊆ L′) = ∑_{m,a,b} c_{m,a,b}r^{n−m}eval_{q}(F_{m,a})eval_{q′}(F′_{n−m,b}),

where

q′ = e2πiτ′.

Thus our integrality condition on Φ amounts to the requirement that this series in q, q′ has coefficients in ℤ[1/6N] for all r|N.

For a modular form F: L ⟶ ℂ, let F̃: ℌ ⟶ ℂ denote the q-series of the corresponding function on the upper half plane. Thus we have

Φ(⟨τ, 1⟩ ⊆ L′) = ∑_{m,a,b} c_{m,a,b}r^{n−m}F̃_{m,a}(q)F̃_{n−m,b}(q′). (9.2)

Notice that τ and q vary over infinite sets, and given τ, we may vary τ′, and hence q′, over infinite sets. Thus we can view Φ(L, L′) as an element of ℤ[1/6N][[q, q′]]. The coefficients of monomials q^{i}q′^{j} are rational polynomials g_{i,j}(r) in r which also live in ℤ[1/6N] for all r|N. Since N (hence r) ranges over an infinite set, the polynomials g_{i,j}(w) ∈ ℚ[w] are uniquely determined by Φ; in fact they are in A_{0}^{S}(w; ℤ[1/6]) (consider the case r = N). We have established the next theorem.

Theorem 9.1. Evaluation at pairs L ⊆ L′ of index N and having the form

L = ⟨τ, 1⟩ ⊆ L′ = ⟨\frac{rτ + t}{N}, \frac{s}{N} ⟩ ⊆ ⟨τ/N, 1/N⟩ (0 ⩽ r, s, t, rs = N, 0 ⩽ t < s),

induces embeddings of (ungraded) rings

GenS(ℤ[1/6])_{∗} ⟶ A_{0}^{S}(w; ℤ[1/6])[[q, q′]],
GenM(ℤ[1/6])_{∗} ⟶ A^{S}(w; ℤ[1/6])((q, q′)),

which in weight n yield embeddings

GenS(ℤ[1/6])_{n} ⟶ A_{0}^{S}(w; ℤ[1/6])[[q, q′]],
GenM(ℤ[1/6])_{n} ⟶ A^{S}(w; ℤ[1/6])((q, q′)).

Setting the variable q equal to zero gives homomorphisms into the ring of divided congruences DC of Section 8. After localizing at a prime p > 3, we obtain ℤ_{(p)}- module homomorphisms

(9.3)  ^{Gen}S(ℤ_{(p)})_{n} ⟶ DC_{p} ⊆ A_{0}^{S}(w; ℤ_{(p)})[[q′]],
(9.4)  ^{Gen}M(ℤ_{(p)})_{n} ⟶ DC_{p}[Δ̃^{−1}] ⊆ A^{S}(w; ℤ_{(p)})((q′)).

The rings generated by the images of all these maps are equal to the ring of divided congruences and its localization at powers of Δ̃^{−1}, as we shall see.

Now from Section 2, we see that there is a unique ring homomorphism

MU_{∗}MU ⟶ ^{Gen}S(ℤ[1/6])_{∗} ⊆ ^{Gen}M(ℤ[1/6])_{∗}

extending the two homomorphisms

MU_{∗} ⟶ S(ℤ[1/6])_{∗} ⟶^{η_{L}} ; ⟶_{η_{R}}^{Gen}S(ℤ[1/6])_{∗} ⊆ ^{Gen}M(ℤ[1/6])_{∗}

and classifying the universal isomorphism H(T) ∈ ^{Gen}S(ℤ[1/6])_{∗}[[T]] between the two Weierstrass formal group laws induced by the latter. Let

log^{Eℓℓ} T = ∑_{n⩾1} \frac{L_{n}}{n + 1}T^{n+1},

log^{Eℓℓ′} T = ∑_{n⩾1} \frac{L′_{n}}{n + 1}T^{n+1}

be the logarithms of these two formal group laws over ^{Gen}S(ℤ[1/6])_{∗}. It is wellknown that L_{n} and L′_{n} lie in ^{Gen}S(ℤ[1/6])_{n}.

Now there is a unique expression

B(T) = ∑_{k⩾1}^{Eℓℓ′} H_{k}T^{k+1} ∈ ^{Gen}S(ℤ[1/6])_{∗}[[T]], (9.5)

with H_{k} having weight k. The H_{k} can be determined inductively using the equation

log^{Eℓℓ′} H(T) = log^{Eℓℓ} T,

which yields

L_{n−1} = ∑_{m|n} \frac{n}{m}L′_{m−1}H_{n/m−1}^{m}. (9.6)

In particular, if p is a prime, we have

L_{p^{r}−1} = ∑_{0⩽s⩽r} p^{r−s}L′_{p^{s}−1}H_{p^{r−s}−1}^{p^{s}} (9.7)

in ^{Gen}S(ℤ[1/6])_{∗} ⊆ ^{Gen}S(ℤ_{(p)})_{∗}.

This motivates us to define (for given prime p > 3)

A_{r} = L_{p^{r}−1},
A′_{r} = L′_{p^{r}−1},
D_{r} = H_{p^{r}−1}.

Thus we have in ^{Gen}S(ℤ[1/6])_{∗} ⊆ ^{Gen}S(ℚ)_{∗},

D_{0} = 1,
D_{1} = \frac{A_{1} − A′_{1}}{p} ,
D_{2} = \frac{A_{2} − pA′_{1}D_{1}^{p} − A′_{2}}{p^{2}} ,
⋮
D_{r} = \frac{A_{r} − p^{r−1}A′_{1}D_{r−1}^{p} − p^{r−2}A′_{2}D_{r−2}^{p^{2}} − ⋯ − A′_{r}}{p^{r}} ,

⋮ (9.8)

The following is closely related to Katz [23], theorem 5.5, and is easily established by induction on r.

Proposition 9.2. For r ⩾ 1,

D′_{r} = p^{h(r)}D_{r} ∈ ℤ[1/6][Q, R, Q′, R′] ⊆ ^{Gen}S(ℤ[1/6])_{∗}.

Notice also that if we expand D_{r}(⟨τ, 1⟩ ⊆ L′) where ⟨τ, 1⟩ ⊆ L′ with index N as above, in the form of Equation 9.2, we obtain

∑_{0⩽n⩽p^{r}−1} c_{p^{r}−1−n,0,0}w^{n} = \overline{θ}_{r}(w). (9.9)

and we also obtain a series in ℤ[1/6N][[q, q′]] which on setting q = 0 yields an element of ℤ[1/6N][[q′]]. This maps each D_{r} to an element R′_{r} which is Katz’s choice of generator as explained in Section 8 (with q replaced by q′).

We will prove the following theorem.

Theorem 9.3. For each prime p > 3, the ring ^{Gen}S(ℤ_{(p)})_{∗} is generated as an algebra over S(ℤ_{(p)})_{∗} by the elements D_{r}, r ⩾ 1, together with Q′ and R′. Similarly, as an algebra over M(ℤ_{(p)})_{∗}, ^{Gen}M(ℤ_{(p)})_{∗} is generated by the elements D_{r}, r ⩾ 1 together with Q′, R′ and Δ′^{−1}, i.e.,

GenM(ℤ_{(p)})_{∗} = ^{Gen}S(ℤ_{(p)})_{∗}[Q′, R′, Δ^{−1}, Δ′^{−1}].

Proof. We will prove Theorem 9.3 for ^{Gen}S(ℤ_{(p)})_{∗} by induction upon the weight wt Φ of an element. Clearly the weight 0 case is true, so assume that whenever wt Φ < n, Φ is expressible as a polynomial in the generators indicated.

Now assume that wt Φ = n. Then Φ can be expressed in the form indicated in Equation 9.1 and 9.2:

Φ = ∑_{m,a,b} c_{m,a,b}F_{m,a}F′_{n−m,b}.

On taking q-expansions in the manner of Theorem 9.1, we have

Φ̃ = ∑_{m,a,b} c_{m,a,b}w^{n−m}F̃_{m,a}F̃′_{n−m,b}.

By setting q = 0, we obtain a q′-expansion

Φ̃(0, q′) = ∑_{m,a,b} c_{m,a,b}F̃′_{n−m,b}

lying in DC_{p} ⊆ A_{0}^{S}(w; ℤ_{(p)})[[q′]]. Now by Theorem 8.3, this can be expressed as a polynomial in the elements Q̃, R̃ and R̃′_{k} (k ⩾ 1) (evaluated at q′ rather than q). Now construct a (non-homogeneous) element of ^{Gen}S(ℤ_{(p)})_{∗} as follows.

First replace each occurrence of R′_{k} in Φ̃(0, q′) by the element D_{k} ∈ ^{Gen}S(ℤ_{(p)})_{∗} defined in Equation 9.8. This will be a sum of homogeneous terms Θ_{d} of weights d in the range 0 ⩽ d ⩽ n. Now multiply Θ_{d} by the basis element F_{n−d,0} to get an element F_{n−d,0}Θ_{d} which has weight n. Let

Φ_{0} = ∑_{0⩽d⩽n} F_{n−d,0}Θ_{d}.

Notice that we have

Φ̃(0, q′) − Φ̃_{0}(0, q′) = 0,

and hence we have

Φ = Φ_{0} + ΔΦ′

in the ring ^{Gen}S(ℤ_{(p)})_{∗}. Hence, we can appeal to the inductive assumption to express Φ′ ∈ ^{Gen}S(ℤ_{(p)})_{n−12} in the required form. Thus, Φ is also of the required form and we have completed the inductive step.

This completes the proof of Theorem 9.3.

As an immediate consequence we obtain our desired global result.

Theorem 9.4. As an algebra over S(ℤ[1/6])_{∗}, the ring ^{Gen}S(ℤ[1/6])_{∗} is generated by the elements H_{n}, n ⩾ 1; similarly, as an algebra over M(ℤ[1/6])_{∗}, ^{Gen}M(ℤ[1/6])_{∗} is generated by the elements H_{n}, n ⩾ 1 together with Δ′^{−1}, i.e.,

GenM(ℤ[1/6])_{∗} = ^{Gen}S(ℤ[1/6])_{∗}[Δ^{−1}, Δ′^{−1}].

Hence there is an isomorphism of algebras over Eℓℓ_{∗} ≅ M(ℤ[1/6])_{∗},

Eℓℓ_{∗}Eℓℓ ≅ ^{Gen}M(ℤ[1/6])_{∗}.

The proof of Theorem 9.3 actually shows the following, which should be compared with the result of Katz, Theorem 8.2. Recall the element D′_{r} = p^{h(r)}D_{r} of Proposition 9.2.

Theorem 9.5. An element Φ ∈ Eℓℓ_{∗}Eℓℓ_{(p)} has q, q′-expansion in p^{k}ℤ((q, q′)) if and only if Φ is in the ideal generated by

p^{r_{0}}D′_{1}^{r_{1}} ⋯ D′_{d}^{r_{d}}

for which

r_{0} + ∑_{1⩽j⩽d} r_{j}h(j) ⩾ k.

10. The cooperation algebra as a Hopf algebroid

In this section we complete our description of the cooperation algebra by describing the Hopf algebroid structure in terms of generalized modular forms. The existence of the Hopf algebroid structure over ℤ[1/6] follows the topological result for Eℓℓ_{∗}Eℓℓ. An element Φ ∈ Eℓℓ_{2n}Eℓℓ is equivalent to a generalized modular form

(F_{•}: L^{•} ⟶ ℂ) ∈ ^{Gen}M(ℤ[1/6])_{n}

with certain properties. At the end of Section 4, a partial monoid structure

μ: L^{•} ×_{L} L^{•} ⟶ L^{•}

was described. This induces a coproduct

F_{•} ⟼ F_{•} ∘ μ

which is actually a ring homomorphism

ψ: ^{Gen}M(ℤ[1/6])_{∗} ⟶ ^{Gen}M(ℤ[1/6])_{∗} ⊗_{M(ℤ[1/6])_{∗}} ^{Gen}M(ℤ[1/6])_{∗},

where the tensor product involves the right and left M(ℤ[1/6])_{∗}-module structures. This is derived ultimately from the composition of lattice inclusions L ⊆ L′ and L′ ⊆ L′′ to give L ⊆ L′′; then

F_{•}μ(L′ ⊆ L′′, L ⊆ L′) = F_{•}(L ⊆ L′′).

There is also an antipode map, which arises as follows. Let L ⊆ L′ with index N. Then there is a dual isogeny L′ ⊆ (1/N)L, also of index N, and this can be scaled to give the inclusion N · L′ ⊆ L. We can evaluate a generalized modular form F_{•} of weight n on this inclusion to obtain a function of the form

(L ⊆ L′) ⟼ F_{•}(N · L′ ⊆ L).

Writing

F_{•} = ∑_{r} F_{r}F′_{n−r},

where

F_{•}(L, L′) = ∑_{r} F_{r}(L)F′_{n−r}(L′),

this is the same as the function

(L ⊆ L′) ⟼ ∑_{r} N^{−r}F_{r}(L′)F′_{n−r}(L).

We then define action of the antipode χ on F_{•} by

χF_{•}(L, L′) = F_{•}(N · L′ ⊆ L).

Thus we may loosely say that the antipode is induced by inverting each inclusion L ⊆ L′ and evaluating on its inverse.

It would be interesting to give a purely algebraic proof that the coproduct ψ actually lands in the tensor product over M(ℤ[1/6])_{∗}, since although it is clear that the rationalisation behaves correctly, the arithmetic conditions appear subtle. Of course, we can appeal to the topological fact that Eℓℓ_{∗}Eℓℓ is a Hopf algebroid to obtain this. A similar problem occurs with the ring of stably numerical polynomials A^{S}(w; ℤ), which is a Hopf algebra over ℤ, but the easiest proof of this uses the topological gadget KU_{∗}KU.

We can interpret this Hopf algebroid as a Hopf algebroid of functions on the category Iso_{ℂ}, see Proposition 5.1. More generally, we have the following (see Theorem 5.2 for the categorical localization result).

Theorem 10.1. Let R be a subring of ℚ containing 1/6. Then we may identify GenM(R)_{∗} with the ring of generalized modular forms on L^{•} which extend to func-

tions on SIso_{ℂ}^{R_{+}^{×}} ⊇ SIso_{ℂ} which have q-expansions defined over R. Morover, com-

position and inversion in SIso_{ℂ}^{R_{+}^{×}} give rise to the natural Hopf algebroid structure on ^{Gen}M(R)_{∗}.

It is interesting to compare this with the corresponding situation for stably numerical polynomials; there we have

Proposition 10.2. For any subring K ⊆ ℚ,

A^{S}(w, K) ⊆ {f(w) ∈ ℚ[w, w^{−1}] : ∀u ∈ K^{×}, f(u) ∈ K}.

In particular, for any prime p,

A^{S}(w, ℤ_{(p)}) = {f(w) ∈ ℚ[w, w^{−1}] : ∀u ∈ ℤ_{(p)}^{×}, f(u) ∈ ℤ_{(p)}}.

Of course, Theorem 10.1 gives a similar interpretation for the Hopf algebroid Eℓℓ_{∗}EℓℓR.

11. Operations dual to cooperations

In this section we will briefly describe how our knowledge of Eℓℓ_{∗}Eℓℓ gives information about stable operations in elliptic cohomology. For any subring R ⊆ ℚ containing 1/6, the Universal Coefficient spectral sequence described in Equation 1.3 applied to the case where M = EℓℓR and X = Eℓℓ gives

E_{2}^{∗,∗}(Eℓℓ) = Ext_{Eℓℓ_{∗}}^{∗,∗} (Eℓℓ_{∗}(Eℓℓ), EℓℓR_{∗}) ⟹ EℓℓR^{∗}(Eℓℓ). (11.1)

As Eℓℓ_{∗} is a ring of dimension 2, we know that

Ext_{Eℓℓ_{∗}}^{k,∗} = 0 if k > 2.

Hopkins and Hunton’s work as described in 1.2 together with the Milnor exact sequence yields

EℓℓR^{∗}(Eℓℓ) ≅ lim_{α} EℓℓR^{∗}(E_{α}),

where the E_{α} form a cofinal collection of finite CW subspectra of Eℓℓ. Thus stable operations Eℓℓ^{∗}( ) ⟶ EℓℓR^{∗}( ) determine unique morphisms of spectra Eℓℓ ⟶ EℓℓR from their values on finite CW spectra.

Now to construct stable operations it suffices to write down natural transformations Eℓℓ^{∗}( ) ⟶ EℓℓR^{∗}( ) defined on the category of finite CW spectra; the most accessible type of these arise as follows. We use the coaction map

ψ: Eℓℓ_{∗}Eℓℓ ⟶ Eℓℓ_{∗}Eℓℓ ⊗_{Eℓℓ_{∗}} Eℓℓ_{∗}Eℓℓ

which is left Eℓℓ_{∗} linear. Given any left Eℓℓ_{∗}-linear mapping

Θ: Eℓℓ_{∗}Eℓℓ ⟶ EℓℓR_{∗}

we obtain an operation as the composite

(11.2)  \overline{Θ}: Eℓℓ^{∗}( ) ≅ (S^{0} ∧ Eℓℓ)^{∗}( ) ⟶ (Eℓℓ ∧ Eℓℓ)^{∗}( )

⟶^{≅} Eℓℓ_{∗}Eℓℓ ⊗_{Eℓℓ_{∗}} Eℓℓ^{∗}( )

⟶^{Θ} EℓℓR_{∗} ⊗_{Eℓℓ_{∗}} Eℓℓ^{∗}( )

⟶ EℓℓR^{∗}( ).

This is the construction underlying the Adams and Hecke operations described in Equations 6.4 and 6.5, based on [6]. We will return to this in Part II, where we will view Eℓℓ_{∗}Eℓℓ as a kind of dual object to a Hecke algebra. Once again, this closely follows the situation for KU_{∗}KU, which can be thought of as a sort of dual to the monoid ring ℤ[ℤ − {0}].

This approach to stable operations in elliptic cohomology becomes more manageable if we reduce modulo an invariant ideal in the coefficient ring Eℓℓ_{∗}. Such ideals were considered in [7]. The most interesting examples are of the form

J_{p,1} = (p) and J_{p,1}^{r}, r ⩾ 1
(11.3)  J_{p,2} = (p, E_{p−1}) and J_{p,2}^{s}, s ⩾ 1,

where p > 3 is a prime. Actually the second example consists of ideals in the p-localization (Eℓℓ_{∗})_{(p)} since E_{(p−1)} may only exist p-locally. We can form completions with respect to such ideals, and the reductions Eℓℓ_{∗}Eℓℓ/I and their completions Eℓℓ_{∗}Eℓℓ_{I} have interpretations as rings of continuous functions on completions of Hecke algebras and their underlying monoids. Again, this is parallel to known constructions for reduction modulo p^{k} and p-adic completion of KU_{∗}KU which gives spaces of continuous functions on the group of p-adic units ℤ_{(p)}^{×} and its progroup ring ℤ_{p}[ℤ_{(p)}^{×}].

We end this section with some remarks on the Adams spectral sequence in elliptic homology. As usual for good homology theories, there is a spectral sequence of the form

\begin{cases}E_{2}^{∗,∗}(X) & ⟹ π_{∗}(L_{Eℓℓ}X); \\ E_{2}^{∗,∗}(X) & = Ext_{Eℓℓ_{∗}Eℓℓ}^{∗,∗} (Eℓℓ_{∗}, Eℓℓ_{∗}(X)) , \end{cases} (11.4)

where the Ext functor is defined on the category of comodules over Eℓℓ_{∗}Eℓℓ. Using the above families of ideals there are various ‘chromatic’ approaches to calculating this E_{2}-term and these may be interesting to pursue. For example, in [14], Clarke and Johnson have made some observations on the K-theoretic part of the 1-line E_{2}^{1,∗}, using Serre’s theory of p-adic modular forms. This p-adic theory is discussed in [41] and its elliptic cohomology version in [5]. For the supersingular theory of modular forms, see [34], and also [7] for the topological version.

12. The operator of Halphen–Fricke–Ramanujan–Swinnerton-Dyer–Serre

The operator of the title has an interesting history; it plays a central rôle in the algebraic theory of the ring of modular forms. For our present purposes, it is an operator ∂ on the ring of modular forms which raises weight by 2, is a derivation and annihilates the discriminant Δ. For an early reference see [16], and for more recent descriptions see [24, 40, 22]. The congruence conditions in Section 9 ultimately rely upon arguments making use of ∂.

We have the following formulæ for the action of ∂:

(12.1)  ∂(Q) = R,
(12.2)  ∂(R) = \frac{3}{2}Q^{2},
(12.3)  ∂(Δ) = 0,
(12.4)  ∂(AB) = ∂(A)B + A∂(B) if A, B ∈ Eℓℓ_{∗}.

Notice that multiplication by Δ (the periodicity operator in elliptic cohomology) commutes with ∂. Thus the following conjecture may seem reasonable.

Conjecture 12.1. The derivation ∂ extends to a stable operation on elliptic cohomology Eℓℓ^{∗}( ).

The fact that ∂ plays a major rôle in the algebraic theory of the rings eℓℓ_{∗} and Eℓℓ_{∗} also make this conjecture interesting. However, Conjecture 12.1 is actually false.

Theorem 12.2. Let p > 3 be a prime. Then there is no stable operation Eℓℓ^{∗}( ) ⟶ Eℓℓ_{(p)}^{∗}( ) raising degree by 4 and extending ∂ on the coefficient ring Eℓℓ_{−∗}. Hence there is no stable operation Eℓℓ^{∗}( ) ⟶ Eℓℓ^{∗}( ) raising degree by 4 and extending ∂ on Eℓℓ_{−∗}.

Proof. Suppose that such a stable operation \overline{∂} exists; then there is a corresponding morphism of spectra ∂: Eℓℓ ⟶ Σ^{−4}Eℓℓ_{(p)} inducing \overline{∂} as a natural transformation of representable functors Eℓℓ^{∗}( ) ⟶ Eℓℓ_{(p)}^{∗}( ). We can extend ∂ to a morphism of Eℓℓ module spectra

∂^{†}: Eℓℓ ∧ Eℓℓ ⟶^{1∧∂} Eℓℓ ∧ Σ^{−4}Eℓℓ_{(p)} ⟶^{μ_{Eℓℓ}} Eℓℓ_{(p)}

where μ_{Eℓℓ}: Eℓℓ ∧ Eℓℓ ⟶ Eℓℓ is the product map and its localization. Hence, we obtain a homomorphism of Eℓℓ_{∗} modules

∂_{∗}^{†}: Eℓℓ_{∗}(Eℓℓ) ⟶ (Eℓℓ_{∗})_{(p)}.

Notice that we also have a commutative diagram

S^{0} ∧ Eℓℓ ⟶^{≃} Eℓℓ
η∧1↓  ^{∂}↓
Eℓℓ ∧ Eℓℓ ⟶^{∂^{†}} Eℓℓ_{(p)}

where η: S^{0} ⟶ Eℓℓ is the unit for the ring spectrum Eℓℓ. But this means that the composite

Eℓℓ_{∗} ⟶^{η_{R}} Eℓℓ_{∗}(Eℓℓ) ⟶^{∂_{∗}^{†}} (Eℓℓ_{∗})_{(p)}

agrees with ∂.

Now in the ring Eℓℓ_{∗}(Eℓℓ)_{(p)} we have an element of the form

(\frac{E_{p−1} − E′_{p−1}}{p} ) ∈ Eℓℓ_{2(p−1)}(Eℓℓ)_{(p)} ⊆ Eℓℓ_{2(p−1)}(Eℓℓ) ⊗ ℚ,

where E′_{p−1} = η_{R}(E_{p−1}). This follows from the well known fact that working modulo p in (Eℓℓ_{∗})_{(p)}, E_{p−1} agrees with the image of Hazewinkel generator v_{1} ∈ (MU_{2(p−1)})_{(p)} under the elliptic genus (MU_{∗})_{(p)} ⟶ (Eℓℓ_{∗})_{(p)} (see [29] for example). But now applying the homomorphism ∂^{†}_{∗} and we see that

∂_{∗}^{†} (\frac{E_{p−1} − E′_{p−1}}{p} ) = \frac{∂(E_{p−1})}{p} ∈ (Eℓℓ_{∗})_{(p)} ⊆ Eℓℓ_{∗} ⊗ ℚ,

since

∂_{∗}^{†}(E_{p−1}) = E_{p−1}∂_{∗}^{†}(1) = 0.

However, from [41] we have

∂(E_{p−1}) ≡ \frac{1}{12}E_{p+1} mod p
≢ 0 mod p.

Hence, this is an element of Eℓℓ_{∗} ⊗ ℚ which is not in (Eℓℓ_{∗})_{(p)}.

However, there is still the possibility of unstable extensions and we make the modified conjecture:

Conjecture 12.3. There are extensions of ∂ to unstable operations in elliptic cohomology Eℓℓ^{∗}( ).

What is really meant here is that for a given n ∈ ℤ, there might be a map

Ω^{∞−n}Eℓℓ ⟶ Ω^{∞−n+4}Eℓℓ

inducing the operator ∂ in homotopy; however, such a map need not deloop.

An alternative approach is to try to construct a suitable stable operation locally at each prime. An obvious candidate would be an extension of the derivation ∂_{p} which raises weight by p + 1 and is given by

∂_{p}(F) = E_{p−1}∂(F) − \frac{wt(F)}{p − 1}∂(E_{p−1})F,

which has the property that ∂_{p}(E_{p−1}) = 0 and avoids the difficulties encountered with ∂. In fact, on q-expansions taken modulo p, ∂_{p} agrees with the action of qd/dq; it is thus the same as the operation θ studied by Serre and Swinnerton-Dyer on modular forms modulo p. However, it is still not clear if this extends to an operation taking values in elliptic cohomology modulo p; it also fails to commute with multiplication by Δ.

Finally, we note that in [18, 19], Gross and Hopkins have explored deformation theory for Lubin–Tate formal group laws; in particular they consider certain Gauss– Manin connections. Now it is known from [22, 24] that ∂ is also a Gauss–Manin connection, so there may well be some relationship between their work and the above discussion. We hope to return to these matters in future work.

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Department of Mathematics, Glasgow University, Glasgow G12 8QW, Scotland. e-mail: andy@maths.gla.ac.uk