ISOGENY COVARIANT DIFFERENTIAL MODULAR FORMS MODULO p

CHRIS HURLBURT

Abstract. An interesting theory arises when the classical theory of modular forms is expanded to include differential analogs of modular forms. One of the main motivations for expanding the theory of modular forms is the existence of differential modular forms with a remarkable property, called isogeny covariance, that classical modular forms cannot possess. Among isogeny covariant differential modular forms there exists a particular modular form that plays a central role in the theory. The main result presented in the article will be the explicit computation modulo p of this fundamental isogeny covariant differential modular form.

1. Introduction

The classical theory of modular forms can be extended to include certain interesting new functions of elliptic curves called differential modular forms. Some of these possess a remarkable property called isogeny covariance that does not have an analog in the classical theory. Among isogeny covariant differential modular forms there is one that plays a central role. In this paper we compute the reduction modulo p of this form explicitly and deduce a few of the unexpected corollaries.

Differential modular forms, introduced in A. Buium’s paper entitled Differential Modular Forms [1], involve either arithmetic analogs of derivations called pderivations or usual derivations. In providing the relevent definitions we define everything for the case of p-derivations and then note later in the introduction how the definitions vary from those for p-derivations in the case of usual derivations.

Let p be a prime number greater than three. In general a p-derivation is as follows:

Definition 1.1. A p-derivation is a set theoretic map, δ : A → B, from a ring A to an A-algebra B such that

(1.1)  δ(x + y) = δx + δy + C_{p}(x, y)
(1.2)  δ(xy) = y^{p}δx + x^{p}δy + pδxδy

for all x, y ∈ A where C_{p}(X, Y) = \frac{X^{p}+Y^{p}−(X+Y)^{p}}{p}.

If we let R be any complete discrete valuation ring where R has maximal ideal generated by p and an algebraically closed residue field k, then there exists a unique lifting of the Frobenius morphism to R which we will refer to as ϕ and there exists a unique p-derivation when both A and B are R given by δ(x) = (ϕ(x) − x^{p})/p.

1991 Mathematics Subject Classification. 11, 14.

Key words and phrases. differential modular forms, p-derivations.

.

Let M(R) be the set

M(R) = {(a, b) ∈ R^{2}|4a^{3} + 27b^{2} ∈ R^{∗}}.

The set M(R) is therefore in one-to-one correspondence with the set of pairs consisting of an elliptic curve over R and an invertible 1-form. Set M^{0} = ℤ_{p}[c_{1}, c_{2}, Δ^{−1}]ˆ and M^{1} = ℤ_{p}[c_{1}, c_{2}, δc_{1}, δc_{2}, Δ^{−1}]ˆ where Δ = −4c_{1}^{3} − 27c_{2}^{2}. Here c_{1}, c_{2}, δc_{1}, δc_{2} are variables over ℤ_{p}, and ˆ represents the p-adic completion. The elements of M^{1} are called δ-modular functions of order 1. Any δ-modular function f ∈ M^{1} defines a map (still denoted by f) from M(R) to R, by substituting a, b, δa, δb in for c_{1}, c_{2}, δc_{1}, δc_{2}. The element f ∈ M^{1} is uniquely determined by the associated map from M(R) to R. As such every δ-modular function is a function of elliptic curves. There is a canonical reduction modulo p map from M^{0} to M_{0}^{0} = 𝔽_{p}[c_{1}, c_{2}, Δ^{−1}] and from M^{1} to M_{0}^{1} = 𝔽_{p}[c_{1}, c_{2}, δc_{1}, δc_{2}, Δ^{−1}]. The reduction modulo p of a δ-modular function f is its image under the canonical map from M^{1} to M_{0}^{1}.

We define a δ-character of order ≤ 1 to be a group homomorphism χ : R^{∗} → R^{∗} of the form χ = χ_{m,n} with (m, n) ∈ ℤ × ℤ_{p} where

χ_{m,n}(λ) = λ^{m} (\frac{ϕ(λ)}{λ^{p}})^{n} .

Then a δ-modular function has weight χ if for any λ ∈ R^{∗}

f(λ^{4}a, λ^{6}b) = χ(λ)f(a, b)

for all (a, b) ∈ M(R). A δ-modular form is a δ-modular function with a weight. This definition of δ-modular forms introduced in Buium’s paper [1] represents an extension of the definitions given by Deligne and Katz [2].

A special collection these δ-modular forms posses the following property that does not have a classical analog.

Definition 1.2. A δ-modular form is isogeny covariant if for any two pairs (a, b) and (ã, b̃) with an etale isogeny of degree N between the corresponding elliptic curves that pulls back \frac{dx}{y} to \frac{dx}{y}

f(a, b) = N^{−k/2}f(ã, b̃)

where k is a constant that depends solely on the weight.

Note that for χ = χ_{m,n} the constant is k = m + n(1 − p). Buium proves in Corollary 7.24 in [1] that up to multiplication by a constant in ℤ_{p} there is a unique isogeny covariant δ-modular form of order one and weight χ_{−p−1,−1}. (From now on unique will mean up to multiplication by an invertible constant of ℤ_{p}.) We can now state our main result.

Theorem 1.3. The reduction modulo p of the unique, isogeny covariant δ-modular form of weight χ_{−p−1,−1} and order 1 is

E_{p−1} \frac{(−9c_{2}^{p}δc_{1} + 6c_{1}^{p}δc_{2})}{Δ^{p}} + f_{0}

where E_{p−1} ∈ M_{0}^{0} is the Hasse invariant and f_{0} ∈ M_{0}^{0}.

Deligne’s congruence tells us the Hasse invariant is the reduction modulo p of the normalized Eisenstein form of weight p − 1 [2], hence the choice of notation here. (Later, in Theorem 3.6 we will give an explicit formula for f_{0} which does have weight χ_{−p−1,0} and is a modular form of order 0.)

This theorem has interesting corollaries. First we provide the formula for a class of polynomials that occur frequently in the first corollary and later in the explicit formula for f_{0}. For two integers a and b with b odd, write a = 3m + n where n ∈ {0, 1, 2} and let

γ_{a,b} = (1.3)

∑_{k=0}^{∞} \binom{m + k}{3k + 2 − n}\binom{m − 2k − 2 + n}{\frac{b−1}{2}}(−1)^{m+k− \frac{b−1}{2}} (c_{1})^{3k+2−n}(c_{2})^{m−2k−2+n− \frac{b−1}{2}}

where \binom{j}{k} denotes the binomial coefficient with the convention that \binom{j}{k} = 0 if k > j. Next we clarify the concept of a Fourier expansion of a δ-modular form which is also part of our first corollary. Indeed there is a natural ring homomorphism M^{1} → ℤ_{p}((q))ˆ called the Fourier expansion or q expansion that sends F to F (q) by sending

c_{1} ↦ c_{4}(q) = −2^{−4}3^{−1}E_{4}(q)
c_{2} ↦ c_{6}(q) = −2^{−5}3^{−3}E_{6}(q)
δc_{1} ↦ δ_{0}c_{4}(q) = \frac{c_{4}(q^{p}) − (c_{4}(q))^{p}}{p}
δc_{2} ↦ δ_{0}c_{6}(q) = \frac{c_{6}(q^{p}) − (c_{6}(q))^{p}}{p}

where E_{4}(q) and E_{6}(q) are the Eisenstein series of weights 4 and 6 respectively. This homomorphism from M^{1} → ℤ_{p}((q))ˆ induces a homomorphism M_{0}^{1} → 𝔽_{p}((q))ˆ that takes the reduction of F modulo p denoted \overline{F} to the reduction of F (q) modulo p denoted \overline{F} (q). Buium proves in Remark 7.5 [1] that the unique isogeny covariant δ-modular form of order one and weight χ_{−p−1,−1} is in the kernel of this homomorphism. We also know that E_{p−1}(q) ≡ 1 modulo p. As an immediate consequence

Corollary 1.4. The q expansions, c_{4}(q), c_{6}(q), δ_{0}c_{4}(q), and δ_{0}c_{6}(q) are related by the identity

9c_{6}(q)^{p}δ_{0}c_{4}(q) − 6c_{4}(q)^{p}δ_{0}c_{6}(q) =

∑_{k=1}^{p−1} \frac{(−1)^{k}}{k} [6c_{4}(q)^{p+k}c_{6}(q)^{p−k}γ_{2p+k,p}(q) − 9c_{4}(q)^{k}c_{6}(q)^{2p−k}γ_{p+k,p}(q)]

+ ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k} c_{4}(q)^{i}c_{6}(q)^{p−k−i} [4c_{4}(q)^{2p}γ_{3k+i,p}(q)]

+ ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k} c_{4}(q)^{i}c_{6}(q)^{p−k−i} [6c_{4}(q)^{p}γ_{2p+3k+i,p}(q)]

− ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k} c_{4}(q)^{i}c_{6}(q)^{p−k−i} [9c_{6}(q)^{p}γ_{p+3k+i,p}(q)]

+Δ(q)^{p} ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k} c_{4}(q)^{i}c_{6}(q)^{p−k−i}γ_{p+3k+i,3p}(q)

where γ_{a,b}(q) is the q expansion of γ_{a,b}, a polynomial in c_{1} and c_{2} which is explicitly given in formula 1.3.

Note that even though δ_{0}c_{4}(q) and δ_{0}c_{6}(q) are not modular forms, the combination 9c_{6}(q)^{p}δ_{0}c_{4}(q) − 6c_{4}(q)^{p}δ_{0}c_{6}(q) is a modular form. This phenomenon of combining c_{4}(q), c_{6}(q), δ_{0}c_{4}(q), and δ_{0}c_{6}(q) to get the q expansion of a modular form should be viewed as an arithmetic analogue of the fact that

2c_{4}(q)θc_{6}(q) − 3c_{6}(q)θc_{4}(q) = Δ(q)

where θ = q(d/dq) [3]. The latter fact is a direct consequence of Ramanujan’s formulas for θc_{6}(q) and θc_{4}(q).

There exists a relationship between the formula in Theorem 1.3 and the coefficients of the modular polynomial. The relationship is based on the following congruence proved by Buium in Corollary 6.3 and Remark 6.4 [1]

E_{p−1} [(j^{p^{2}} − j)j′ + ∑_{μ=0}^{p} ∑_{ν=0}^{p} c_{μν}j^{μ+νp}]

≡ a(j^{p^{2}} − j)\frac{c_{1}^{2p}c_{2}^{p}}{Δ^{p}} [E_{p−1} \frac{−9c_{2}^{p}δc_{1} + 6c_{1}^{p}δc_{2}}{Δ^{p}} + f_{0}]

modulo p where j = \frac{c_{1}^{3}}{Δ} is the j − invariant, j′ is its p-derivative δ(j) and a ∈ ℤ_{p}. The c_{μν} are constants such that

± ∑_{μ=0}^{p} ∑_{ν=0}^{p} c_{μν}j^{μ+νp} = \frac{1}{p} [Φ − (X^{p} − Y)(Y^{p} − X)]

modulo p where Φ is the modular polynomial. We can manipulate this formula as follows

E_{p−1} ∑_{μ=0}^{p} ∑_{ν=0}^{p} c_{μν}j^{μ+νp} ≡

(j^{p^{2}}−j) [a\frac{c_{1}^{2p}c_{2}^{p}}{Δ^{p}} f_{0} − E_{p−1} \frac{c_{1}^{3p}}{Δ^{2p}} (c_{1}^{3p}δ(4) + c_{2}^{2p}δ(27) + ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k} (4c_{1}^{3})^{k}(27c_{2}^{2})^{p−k})]

modulo p or

±E_{p−1} \frac{1}{p} [Φ − (X^{p} − Y)(Y^{p} − X)] ≡

(j^{p^{2}}−j) [a\frac{c_{1}^{2p}c_{2}^{p}}{Δ^{p}} f_{0} − E_{p−1} \frac{c_{1}^{3p}}{Δ^{2p}} (c_{1}^{3p}δ(4) + c_{2}^{2p}δ(27) + ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k} (4c_{1}^{3})^{k}(27c_{2}^{2})^{p−k})]

which relates Theorem 1.3 directly to coefficients of the modular polynomial modulo p^{2}.

The proof of Corollary 6.3 in [1] uses the following corollary of Theorem 1.3.

Corollary 1.5. Let f be the unique isogeny covariant δ-modular form of weight χ_{−p−1,−1} and order 1. Then the ring (M^{1}/(f))_{E_{p−1}} is regular.

Proof. It is enough to show that the ring (M_{0}^{1}/(\overline{f}))_{\overline{E}_{p−1}} is regular since (M^{1}/(f))_{E_{p−1}} is p-adically complete. We can cover the spectrum of the ring (M_{0}^{1}/(\overline{f}))_{\overline{E}_{p−1}} with the two opens sets given by the spectra of (M_{0}^{1}/(\overline{f}))_{_{\overline{E}_{p−1}}_{\overline{c}_{1}}} = 𝔽_{p}[\overline{c}_{1}, \overline{c}_{2}, δ\overline{c}_{1}]_{Δ} and (M_{0}^{1}/(\overline{f}))_{_{\overline{E}_{p−1}}_{\overline{c}_{2}}} = 𝔽_{p}[\overline{c}_{1}, \overline{c}_{2}, δ\overline{c}_{2}]_{Δ} which are regular.

The key to proving Theorem 1.3 is the fact that the unique isogeny covariant δmodular form of weight χ_{−p−1,−1} is congruent modulo p to a specific ”deformation class” of an elliptic curve called \overline{f}_{def} in Buium’s paper, Differential Modular Forms [1]. This ”deformation class” is a p-derivation analog of the Kodaira-Spencer class for a family of elliptic curves for usual derivations. The construction of \overline{f}_{def} is the construction of a Kodaira-Spencer map for p-derivations in an analogous manner to the construction of a Kodaira-Spencer map for usual derivations.

In order to describe the p-derivation analog of the Kodaira-Spencer map, we must first establish the setting in which this map is defined. Let E be the elliptic curve over M^{0} defined by the homogeneous equation

f(X, Y, Z) = ZY^{2} − X^{3} − c_{1}XZ^{2} − c_{2}Z^{3}.

Then let U and V be the affine open sets of E given by the equations f(x, y, 1) and f(u, 1, z), respectively. The unique p-derivation δ : M^{0} → M^{1} that sends c_{1}, c_{2} into δc_{1}, δc_{2} can then be lifted to p-derivations δ_{U} : O(U) → O(U) ⊗ M_{0}^{1} and δ_{V} : O(V) → O(V) ⊗ M_{0}^{1}. Their difference δ_{U} − δ_{V} induces a map (δ_{U} − δ_{V}) : O(U ∩ V) ⊗ M_{0}^{0} → O(U ∩ V) ⊗ M_{0}^{1} that is additive, vanishes on M_{0}^{0}, and satisfies

(δ_{U} − δ_{V})(fg) = f^{p}(δ_{U} − δ_{V})(g) + g^{p}(δ_{U} − δ_{V})(f).

Consequently if F : E_{0} → E_{0} for E_{0} = E ⊗ M_{0}^{0} is the absolute Frobenius on E_{0} that takes x ↦ x^{p} and F^{∗}T is the pull-back of the tangent sheaf by F, then δ_{U} − δ_{V} defines a cocycle in F^{∗}T and hence an element in H^{1}(E_{0} ⊗ M_{0}^{1}, F^{∗}T). (If the δ_{U} and δ_{V} were liftings of a usual derivation then their difference would be a cocycle in H^{1}(E, T) that represents the image of δ under the Kodaira-Spencer map.) In the case of p-derivations this difference represents a natural analog to the Kodaira-Spencer map.

We know that the M_{0}^{1} module H^{1}(E_{0} ⊗ M_{0}^{1}, F^{∗}T) is free with a basis consisting of the class [θ] of θ = y^{p} (F ∘ \frac{∂}{∂x}) where θ is viewed as a Čech cocyle in F^{∗}T with respect to the covering {U, V }. The mapping from H^{1}(E_{0} ⊗ M_{0}^{1}, O) to H^{1}(E_{0} ⊗ M_{0}^{1}, F^{∗}T) that sends the equivalence class of [a] ∈ H^{1}(E_{0} ⊗ M_{0}^{1}, O) for a ∈ O(U ∩ V) to the equivalence class of [aθ] ∈ H^{1}(E_{0} ⊗ M_{0}^{1}, F^{∗}T) is therefore a module isomorphism. In a similar vein, the class [x^{2}/y] of the cocycle x^{2}/y generates H^{1}(E_{0} ⊗ M_{0}^{1}, O) as a rank one free M_{0}^{1}-module, and thus the map that sends λ ∈ M_{0}^{1} to the equivalence class [\frac{λx^{2}}{y} ] is also a module isomorphism. We obtain isomorphisms

\begin{aligned}H^{1}(E_{0} ⊗ M_{0}^{1}, F^{∗}T) & ≃ & H^{1}(E_{0} ⊗ M_{0}^{1}, O) & ≃ & M_{0}^{1} \\ [aθ] & ⟷ & [a] = [\frac{λx^{2}}{y} ] & ⟷ & λ \end{aligned}

We claim that under the above isomorphisms [δ_{U} − δ_{V}] ∈ H^{1}(E_{0} ⊗ M_{0}^{1}, F^{∗}T) corresponds to [\frac{(δ_{U} −δ_{V} )(x)}{y^{p}}] ∈ H^{1}(E_{0} ⊗ M_{0}^{1}, O). Indeed if δ_{U} − δ_{V} = aθ for some

a ∈ O(U ∩ V) then (δ_{U} − δ_{V})(x) = ay^{p} (\frac{∂x}{∂x})^{p}. Hence a = \frac{(δ_{U} −δ_{V} )(x)}{y^{p}}. Now we uniquely write a = ∑ a_{n}y^{n} + x ∑ b_{n}y^{n} + x^{2} ∑ e_{n}y^{n} with a_{n}, b_{n}, e_{n} ∈ M_{0}^{1}. Due to the fact that all monomials x^{i}y^{j} except for x^{2}/y are coboundaries with respect to {U, V }, the class [a] ∈ H^{1}(E_{0} ⊗ M_{0}^{1}, O) equals the class [e_{−1}x^{2}/y]. Under the isomorphism H^{1}(E_{0} ⊗ M_{0}^{1}, O) ≃ M_{0}^{1}, the image of this class, [e_{−1}x^{2}/y], is e_{−1}. From now on we will refer to the image of the class [e_{−1}x^{2}/y] which is e_{−1} as \overline{f}_{def} and to the expression ∑ a_{n}y^{n} + x ∑ b_{n}y^{n} + x^{2} ∑ e_{n}y^{n} as the normal form of a.

With certain modifications the same computation can be applied to usual derivations. In order to do this, first set M^{0} = ℚ[c_{1}, c_{2}, Δ^{−1}], and M^{1} = M_{0}^{1} = ℚ[c_{1}, c_{2}, δc_{1}, δc_{2}, Δ^{−1}]. The elements of M^{1} in this case are still called δ-modular functions of order ≤ 1. Now let R be an algebraically closed field of characteristic zero and δ : R → R a non-zero derivation. Any f ∈ M^{1} still defines a map f : M(R) → R by substitution. We must next set the character to be χ = χ_{m} : R^{∗} → R^{∗}, χ_{m}(λ) = λ^{m}. Under this definition of χ, the δ-modular function f ∈ M^{1} then has weight χ if f(λ^{4}a, λ^{6}b) = χ(λ)f(a, b). In this case a δ-modular form is still a δ-modular function with weight. If f ∈ M^{1} of weight χ_{m} where m an even integer is isogeny covariant then for any (a, b), (ã, b̃) where there is an isogeny of degree N between the corresponding elliptic curves,

f(a, b) = N^{−m/2}f(ã, b̃).

Buium proves in [1] that up to multiplication by a constant in ℚ there is a unique isogeny covariant δ-modular form of weight χ_{−2} and order 1.

Theorem 1.6. The unique, isogeny covariant δ-modular form of weight χ_{−2} and order 1 in the case of usual derivations is

\frac{−9c_{2}δc_{1} + 6c_{1}δc_{2}}{Δ} .

2. Computation of \overline{f}_{def} for usual derivations

This section is dedicated to proving Theorem 1.6 as an illustration of the process used to compute \overline{f}_{def} in the case of p-derivations. Let δ be the usual derivation we just described. Similarly let E, U, V, δ_{U}, and δ_{V} be as described above. In this case of usual derivations \overline{f}_{def} is the image of δ_{U} − δ_{V} in M_{0}^{1} under the module isomorphisms H^{1}(E, T) ≃ H^{1}(E, O) ≃ M^{1}. Then to start with liftings must be found for δ.

2.1. Any lifting is defined by its behavior on the generators. In the case of δ_{U} these are x and y and δ_{U} is subject to the constraint that δ_{U}(f(x, y, 1)) = 0. Let U_{x} be the open subset of U defined by ∂f/∂x ≠ 0 and let U_{y} be the open subset of U defined by ∂f/∂y ≠ 0. Then let δ_{U_{x}} be a lifting of δ to U_{x} and δ_{U_{y}} a lifting of δ to U_{y}. If δ_{U} is applied ’formally’ to f(x, y, 1) the result is

(−3x^{2} − c_{1})δ_{U}x + 2yδ_{U}y − xδc_{1} − δc_{2} = 0

and any lifting of δ must satisfy this result.

Specifically let δ_{U_{x}} be the lifting on U_{x} obtained by letting δ_{U_{x}}(y) = 0 and then

δ_{U_{x}}(x) = \frac{xδc_{1} + δc_{2}}{−3x^{2} − c_{1}} .

Let δ_{U_{y}} be the lifting on U_{y} obtained by letting δ_{U_{y}}(x) = 0 and then

δ_{U_{y}}(y) = \frac{xδc_{1} + δc_{2}}{2y} .

We use the following trivial lemma to combine δ_{U_{x}} and δ_{U_{y}} into a derivation δ_{U} on O(U_{x} ∩ U_{y}) that is a linear combination of δ_{U_{x}} and δ_{U_{y}}. Then we note that the resulting derivation is defined over all of U and as such is a lifting over all of U.

Lemma 2.1. If δ_{1} δ_{2} : C → D are two derivations lifting δ : ℚ[c_{1}, c_{2}, Δ^{−1}] → ℚ[c_{1}, c_{2}, δc_{1}, δc_{2}, Δ^{−1}] where

ℚ[c_{1}, c_{2}, Δ^{−1}] ⟶ ℚ[c_{1}, c_{2}, δc_{1}, δc_{2}, Δ^{−1}]
↓  ↓
C  ⟶  D

is a commutative ring diagram and if b_{1}, b_{2} ∈ D are such that b_{1} + b_{2} = 1, then b_{1}δ_{1} + b_{2}δ_{2} : C → D is a derivation that lifts δ.

Let A and B be polynomials such that A∂f/∂x + B∂f/∂y = 1. If we let b_{1} = A∂f/∂x and b_{2} = B∂f/∂y then by the lemma δ_{U} = b_{1}δ_{U_{x}} + b_{2}δ_{U_{y}} is a derivation on O(U_{x} ∩ U_{y}) that lifts δ and is in fact defined over all of U. Explicitly the polynomials

A = \frac{4c_{1}^{2} + 6x^{2}c_{1} − 9xc_{2}}{Δ}
B = \frac{9y(2xc_{1} − 3c_{2})}{2Δ}

work. Then the linear combination δ_{U} = b_{1}δ_{U_{x}} + b_{2}δ_{U_{y}} is

δ_{U}(x) = \frac{(xδc_{1} + δc_{2})(4c_{1}^{2} + 6x^{2}c_{1} − 9xc_{2})}{Δ}
δ_{U}(y) = \frac{(xδc_{1} + δc_{2})9y(2xc_{1} − 3c_{2})}{2Δ}

which is defined over all of U.

The method for finding δ_{V} is the same. Specifically the generators are u and z and δ_{V} is subject to the constraint δ_{V} (f(u, 1, z)) = 0. Let V_{u} and V_{z} be the open subsets of V defined by ∂f/∂u ≠ 0 and ∂f/∂z ≠ 0 respectively. Let δ_{V_{u}} be a lifting of δ to V_{u} and δ_{V_{z}} a lifting of δ to V_{z}. Formally applying δ_{V} to f(u, 1, z) the constraint is

(−3u^{2} − c_{1}z^{2})δ_{V} (u) + (1 − 2uzc_{1} − 3z^{2}c_{2})δ_{V} (z) − uz^{2}δc_{1} − z^{3}δc_{2} = 0.

Explicitly let δ_{V_{u}} be the lifting on V_{u} arising from setting δ_{V_{u}}(z) = 0 and solving for

δ_{V_{u}}(u) = \frac{uz^{2}δc_{1} + z^{3}δc_{2}}{(−3u^{2} − c_{1}z^{2})} .

Let δ_{V_{z}} be the lifting on V_{z} arising from setting δ_{V_{z}}(u) = 0 and solving for

δ_{V_{z}}(z) = \frac{uz^{2}δc_{1} + z^{3}δc_{2}}{(1 − 2uzc_{1} − 3z^{2}c_{2})}.

The let C and D be polynomials such that C∂f/∂u + D∂f/∂z = 1 on V. Precisely

C = u(−\frac{3}{2}c_{2}z − c_{1}u)
D = −\frac{3}{2}c_{2}z^{2} − zc_{1}u + 1

Using the lemma above, if b_{1} = C∂f/∂u and b_{2} = D∂f/∂z, then δ_{V} = b_{1}δ_{V_{u}} +b_{2}δV_{z} is a lifting of δ. Applied to u and v,

δ_{V} (u) = (uz^{2}δc_{1} + z^{3}δc_{2})u (−\frac{3}{2}c_{2}z − c_{1}u)
δ_{V} (z) = (uz^{2}δc_{1} + z^{3}δc_{2}) (−\frac{3}{2}c_{2}z^{2} − zc_{1}u + 1)

2.2. In order to take (δ_{U} − δ_{V})(x) on U ∩ V it is necessary to find δ_{V} (x). On U ∩ V, x = u/z, y = 1/z, u = x/y, and z = 1/y. So δ_{V} (x) = δ_{V} (u/z) which is equal to

\frac{zδ_{V} (u) − uδ_{V} (z)}{z^{2}} = −(uδc_{1} + zδc_{2})u

Then substituting gives

δ_{V} (x) = −\frac{(xδc_{1} + δc_{2})x}{y^{2}}

Therefore the difference (δ_{U} − δ_{V})(x) is

\frac{(xδc_{1} + δc_{2})(4c_{1}^{2} + 6x^{2}c_{1} − 9xc_{2})}{Δ} + \frac{(xδc_{1} + δc_{2})x}{y^{2}}

and the corresponding element in H^{1}(E, O) is the class of the function

\frac{1}{y} (\frac{(xδc_{1} + δc_{2})(4c_{1}^{2} + 6x^{2}c_{1} − 9xc_{2})}{Δ} + \frac{(xδc_{1} + δc_{2})x}{y^{2}}) .

The normal form of this function is

(\frac{4c_{1}^{2}δc_{2} − 6c_{1}c_{2}δc_{1}}{yΔ} + \frac{6c_{1}δc_{1}y}{Δ} )

+x (\frac{δc_{2}}{y^{3}} + \frac{−2c_{1}^{2}δc_{1} − 9c_{2}δc_{2}}{yΔ} )

+x^{2} (\frac{δc_{1}}{y^{2}} + \frac{−9c_{2}δc_{1} + 6c_{1}δc_{2}}{yΔ} )

and \overline{f}_{def}, the coefficient of x^{2}/y, in this normal form is then exactly as given in Theorem 1.6.

As seen in this section the computation of \overline{f}_{def} for usual derivations is relatively straightforward. The basic procedure is the same for p-derivations, but many of the steps are more complicated due to the more complex axioms of p-derivations.

3. Computation of \overline{f}_{def} for p-derivations

Let δ : M^{0} → M^{1} be the unique p-derivation that sends c_{1}, c_{2} into δc_{1}, δc_{2} where p is a prime number greater than three. Then by definition \overline{f}_{def} is the image of δ_{U} − δ_{V} in M_{0}^{1} under the module isomorphisms H^{1}(E_{0} ⊗ M_{0}^{1}, F^{∗}T) ≃ H^{1}(E_{0}⊗M_{0}^{1}, O) ≃ M_{0}^{1}. Also recall that the normal form of a function a ∈ O(U∩V) is the expression of a as ∑ a_{n}y^{n} + x ∑ b_{n}y^{n} + x^{2} ∑ e_{n}y^{n}. In the previous section it was possible to write explicitly the normal form of δ_{U} −δ_{V} for usual derivations. In this section it will be necessary to find the normal form of various parts of δ_{U} − δ_{V} and then sum together the parts. To this end, Υ_{a,b} will denote the normal form of \frac{x^{a}}{y^{b}} and γ_{a,b} will be the coefficient of x^{2}/y in Υ_{a,b}.

3.1. Let U_{x}, and U_{y} be the open sets defined in the previous section. Then any lifting of δ to δ_{U} : O(U) → O(U) ⊗ M_{0}^{1} is defined by its behavior on the generators subject to the constraint δ_{U}(f(x, y, 1)) = 0. This constraint means

2y^{p}δ_{U}(y) + pδ_{U}(y)^{2} − 3x^{2p}δ_{U}(x) − 3px^{p}δ_{U}(x)^{2} − p^{2}δ_{U}(x)^{3} − x^{p}δc_{1} − c_{1}^{p}δ_{U}(x)

−pδ_{U}(x)δc_{1} − δc_{2} + \frac{y^{2p} − x^{3p} − (c_{1}x)^{p} − c_{2}^{p} − (y^{2} − x^{3} − c_{1}x − c_{2})^{p}}{p} = 0

and is derived by applying the p-derivation axioms to the equation f(x, y, 1). Since we are working modulo p, this constraint on the lifting of δ simplifies to

2y^{p}δ_{U}(y) + (−3x^{2p} − c_{1}^{p})δ_{U}(x) − x^{p}δc_{1} − δc_{2} + C_{p}^{poly}(f_{U})

where

C_{p}^{poly}(f_{U}) = \frac{y^{2p} − x^{3p} − (c_{1}x)^{p} − c_{2}^{p} − (y^{2} − x^{3} − c_{1}x − c_{2})^{p}}{p} .

Now to find a specific lifting, first define explicitly δ_{U_{x}} on U_{x} by letting δ_{U_{x}}(y) = 0 and then

δ_{U_{x}}(x) = \frac{x^{p}δc_{1} + δc_{2} − C_{p}^{poly}(f_{U})}{−3x^{2p} − c_{1}^{p}}.

Then define δ_{U_{y}} on U_{y} by letting δ_{U_{y}}(x) = 0 and hence

δ_{U_{y}}(y) = \frac{x^{p}δc_{1} + δ(c_{2}) − C_{p}^{poly}(f_{U})}{2y^{p}}.

To combine δ_{U_{x}} and δ_{U_{y}} into δ_{U}, the following analog of Lemma 2.1 for p-derivations is used whose proof is again straightforward. Namely,

Lemma 3.1. If δ_{1} δ_{2} : C → D are two p-derivations lifting δ : ℤ_{p}[c_{1}, c_{2}, Δ^{−1}] → ℤ_{p}[c_{1}, c_{2}, δc_{1}, δc_{2}, Δ^{−1}] where

ℤ_{p}[c_{1}, c_{2}, Δ^{−1}] ⟶ ℤ_{p}[c_{1}, c_{2}, δc_{1}, δc_{2}, Δ^{−1}]
↓  ↓
C  ⟶  D

is a commutative ring diagram, if char D = p, and if b_{1}, b_{2} ∈ D are such that b_{1} + b_{2} = 1, then b_{1}δ_{1} + b_{2}δ_{2} : C → D is a p-derivation that lifts δ.

Here the condition that char D = p is essential. To combine δ_{U_{x}} and δ_{U_{y}} let b_{1} = A^{p}(∂f/∂x)^{p} and b_{2} = B^{p}(∂f/∂y)^{p} where A and B are as given in the previous section. Then δ_{U} = b_{1}δ_{U_{x}} + b_{2}δU_{y} is a p-derivation lifting δ defined on U. Specifically

δ_{U}(x) = \frac{(x^{p}δc_{1} + δc_{2} − C_{p}^{poly}(f_{U}))(4c_{1}^{2} + 6x^{2}c_{1} − 9xc_{2})^{p}}{Δ^{p}}
δ_{U}(y) = \frac{(x^{p}δc_{1} + δc_{2} − C_{p}^{poly}(f_{U}))9y^{p}(2xc_{1} − 3c_{2})^{p}}{2Δ^{p}} .

For the lifting δ_{V} of δ on V, the constraint modulo p is

(−2c_{1}^{p}u^{p}z^{p} + 1 − 3c_{2}^{p}z^{2p})δ_{V} (z) + (−3u^{2p} − c_{1}^{p}z^{2p})δ_{V} (u) − u^{p}z^{2p}δc_{1} − z^{3p}δc_{2}

+C_{p}^{poly}(f_{V}).

where

C_{p}^{poly}(f_{V}) = \frac{(z^{p} − u^{3p} − (z − u^{3} − uz^{2}c_{1} − z^{3}c_{2})^{p} − u^{p}z^{2p}c_{1}^{p} − z^{3p}c_{2}^{p})}{p}.

Then δ can be lifted to δ_{V_{u}} on V_{u} by letting δ_{V_{u}}(z) = 0 and

δ_{V_{u}}(u) = \frac{u^{p}z^{2p}δc_{1} + z^{3p}δc_{2} − C_{p}^{poly}(f_{V})}{(−3u^{2p} − c_{1}^{p}z^{2p})}.

Similarly δ can be lifted to δ_{V_{z}} on V_{z} by letting δ_{V_{z}}(u) = 0 and

δ_{V_{z}}(z) = \frac{u^{p}z^{2p}δc_{1} + z^{3p}δc_{2} − C_{p}^{poly}(f_{V})}{(1 − 2c_{1}^{p}u^{p}z^{p} − 3c_{2}^{p}z^{2p})}.

Then if b_{1} = C^{p}(∂f/∂u)^{p} and b_{2} = D^{p}(∂f/∂z)^{p}, δ_{V} = b_{1}δ_{V_{u}}+b_{2}δ_{V_{z}} is a p-derivation on V lifting δ. On u and z,

δ_{V} (u) = (u^{p}z^{2p}δc_{1} + z^{3p}δc_{2} − C_{p}^{poly}(f_{V}))u^{p}(−\frac{3}{2}c_{2}z − c_{1}u)^{p}
δ_{V} (z) = (u^{p}z^{2p}δc_{1} + z^{3p}δc_{2} − C_{p}^{poly}(f_{V}))(−\frac{3}{2}c_{2}z^{2} − zc_{1}u + 1)^{p}

3.2. The next step in determining \overline{f}_{def} for p-derivations lies in computing δ_{V} (x) = δ_{V} (u/z) on U ∩ V. The product rule for p-derivations implies

δ_{V} (u/z) = \frac{z^{p}δ_{V} (u) − u^{p}δ_{V} (z)}{z^{2p}}

modulo p. So in fact

δ_{V} (u/z) = −\frac{u^{p} (−\frac{3}{2}c_{2}z^{2} − zc_{1}u + 1)^{p} (u^{p}z^{2p}δc_{1} + z^{3p}δc_{2} − C_{p}^{poly}(f_{V}))}{z^{2p}}

+\frac{(−\frac{3}{2}c_{2}uz − c_{1}u^{2})^{p} (u^{p}z^{2p}δc_{1} + z^{3p}δc_{2} − C_{p}^{poly}(f_{V}))}{z^{p}}

which simplifies to

δ_{V} (u/z) = −u^{2p}δc_{1} − u^{p}z^{p}δc_{2} + \frac{u^{p}}{z^{2p}} C_{p}^{poly}(f_{V}).

Then noting that when written in terms of x and y, C_{p}^{poly}(f_{V}) is the same as

\frac{C_{p}^{poly}(f_{U})}{y^{3p}},

δ_{V} (x) = \frac{−x^{2p}δc_{1} − x^{p}δc_{2} + x^{p}C_{p}^{poly}(f_{U})}{y^{2p}}.

Proposition 3.2. The equivalence class [a] = [\frac{(δ_{U} −δ_{V} )(x)}{y^{p}}] in H^{1}(E_{0} ⊗ M_{0}^{1}, O)

mapped to M_{0}^{1} is given by

a = \frac{(x^{p}δc_{1} + δc_{2} − C_{p}^{poly}(f_{U}))(4c_{1}^{2} + 6x^{2}c_{1} − 9xc_{2})^{p}}{y^{p}Δ^{p}} −\frac{−x^{2p}δc_{1} − x^{p}δc_{2} + x^{p}C_{p}^{poly}(f_{U})}{y^{3p}}.

Proof. Both δ_{U}(x) and δ_{V} (x) have been computed above. Simply take their difference and divide out by y^{p}.

At this point things become more complicated. Unlike in the case of usual derivations, a simple substitution of y^{2}−c_{1}x−c_{2} in for x^{3} won’t suffice to determine the normal form because of the ubiquitous p powers. Therefore the next section is devoted to developing tools for determining what the coefficient of x^{2}/y is in the normal form.

3.3. Recall that the normal form of \frac{x^{a}}{y^{b}} will be denoted Υ_{a,b} and that γ_{a,b} is defined as the coefficient of x^{2}/y in Υ_{a,b}. Also we let \binom{n}{k} denote the binomial coefficient with the convention that \binom{n}{k} = 0 if k > n.

Proposition 3.3. Let α = y^{2} − c_{2} and β = −c_{1}, and suppose x^{3} = α + βx. Then the normal form of x^{a} is

Υ_{a,0} = ∑_{k=0}^{⌊ \frac{a−1}{2} ⌋} \binom{⌊\frac{a+k−1}{3}⌋}{k}α^{r_{a,k}}β^{k}x^{s_{a,k}}

where 3r_{a,k} + 2k + s_{a,k} = a with s_{a,k} ∈ {0, 1, 2} for all k.

Proof. Induction.

Corollary 3.4. Let m and n be integers such that a = 3m + n for n ∈ {0, 1, 2}. Then the coefficient of x^{2} in Υ_{a,0} is

∑_{k=0}^{∞} \binom{m + k}{3k + 2 − n}(−c_{1})^{3k+2−n} ∑_{i=0}^{m−2k−2+n} \binom{m − 2k − 2 + n}{i}(−c_{2})^{m−2k−2+n−i}y^{2i}

By construction, Υ_{a,b} = \frac{1}{y^{b}} Υ_{a,0}. In particular, this means that an x^{2}/y term exists only if b is odd, because in the coefficient of x^{2} in Υ_{a,0}, all of the powers of y are even. Therefore if b is even, γ_{a,b} = 0. In the case that b is odd, there is a formula. The formula is

γ_{a,b} = (3.1)

∑_{k=0}^{∞} \binom{m + k}{3k + 2 − n}\binom{m − 2k − 2 + n}{\frac{b−1}{2}}(−1)^{m+k− \frac{b−1}{2}} (c_{1})^{3k+2−n}(c_{2})^{m−2k−2+n− \frac{b−1}{2}}

Obviously because of the convention for binomial coefficients there will be integers a and b with b odd for which γ_{a,b} is 0. For example γ_{p,p}. To see this write p = 3m+n as prescribed. Then the question becomes are there any non-negative values of k for which 3m + n − 1 ≤ 2(m + n) − 4(k + 1)? Since the answer is no, γ_{p,p} = 0. In fact if b ≥ a, γ_{a,b} = 0 because of the binomial coefficient \binom{m−2k−2+n}{\frac{b−1}{2}}.

For a complete analysis of the equivalence class in Proposition 3.2 to be possible using this formula, it is necessary to write C_{p}^{poly}(f_{U}) in terms of an expression not containing p in the denominator. Recall that for all 1 ≤ n ≤ p,

\binom{p}{n} = \frac{p!}{n!(p − n)!} ≡ (−1)^{n−1} \frac{p}{n} mod p^{2}.

Proposition 3.5.

C_{p}^{poly}(f_{U}) = ∑_{k=1}^{p−1} (\frac{−y^{2p}}{k} )+∑_{k=1}^{p−1} \frac{(−1)^{k−1}}{k} (c_{2}^{p−k}c_{1}^{k}x^{k})+∑_{k=1}^{p−1} \frac{(−1)^{k−1}}{k} (x^{3k}(c_{1}x + c_{x})^{p−k})

On the open set U,

0 =(−y^{2} + x^{3} + c_{1}x + c_{2})^{p}
= − y^{2p} + x^{3p} + (c_{1}x)^{p} + c_{2}^{p} + ∑_{k=1}^{p−1} \frac{−py^{2p}}{k}

+ ∑_{k=1}^{p−1} \frac{p(−1)^{k−1}}{k} (x^{3k}(c_{1}x + c_{2})^{p−k}) + ∑_{k=1}^{p−1} \frac{p(−1)^{k−1}}{k} (c_{1}x)^{k}c_{2}^{p−k}.

Therefore

C_{p}^{poly}(f_{U}) = ∑_{k=1}^{p−1} \frac{−y^{2p}}{k} + ∑_{k=1}^{p−1} \frac{(−1)^{k−1}}{k} (x^{3k}(c_{1}x + c_{2})^{p−k}) + ∑_{k=1}^{p−1} \frac{(−1)^{k−1}}{k} (c_{1}x)^{k}c_{2}^{p−k}.

3.4. The preparations in the previous section can now be applied to prove

Theorem 3.6. The class \overline{f}_{def} is

\overline{f}_{def} = \frac{−γ_{2p,p}9c_{2}^{p}δc_{1} + γ_{2p,p}6c_{1}^{p}δc_{2}}{Δ^{p}} + f_{0}

where f_{0} ∈ M_{0}^{0} is given by

f_{0} = ∑_{k=1}^{p−1} \frac{(−1)^{k}}{kΔ^{p}} [6c_{1}^{p+k}c_{2}^{p−k}γ_{2p+k,p} − 9c_{1}^{k}c_{2}^{2p−k}γ_{p+k,p}]

+ ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{kΔ^{p}} c_{1}^{i}c_{2}^{p−k−i} [4c_{1}^{2p}γ_{3k+i,p} + 6c_{1}^{p}γ_{2p+3k+i,p} − 9c_{2}^{p}γ_{p+3k+i,p}]

+ ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k} c_{1}^{i}c_{2}^{p−k−i}γ_{p+3k+i,3p}

For an explicit expression for γ_{a,b} see equation 3.1.

Proof. The proof consists of two steps. The first step is the analysis of the terms containing either δc_{1} or δc_{2} and the second step is the analysis of the terms containing C_{p}^{poly}(f_{U}). There are eight terms containing either δc_{1} or δc_{2} from Proposition 3.2 that need to be analyzed. They are

\frac{4c_{1}^{2p}δc_{2}}{y^{p}Δ^{p}} , \frac{6x^{2p}c_{1}^{p}δc_{2}}{y^{p}Δ^{p}} , \frac{−9x^{p}c_{2}^{p}δc_{2}}{y^{p}Δ^{p}} , \frac{4x^{p}c_{1}^{2p}δc_{1}}{y^{p}Δ^{p}} ,

\frac{6x^{3p}c_{1}^{p}δc_{1}}{y^{p}Δ^{p}} , \frac{−9x^{2p}c_{2}^{p}δc_{1}}{y^{p}Δ^{p}} , \frac{x^{2p}δc_{1}}{y^{3p}} , \frac{x^{p}δc_{2}}{y^{3p}}

The first term is in normal form and contains no \frac{x^{2}}{y} terms. The second term

contributes a \frac{γ_{2p,p}6c_{1}^{p}δc_{2}}{Δ^{p}} to \overline{f}_{def}. The third term is \frac{−9c_{2}^{p}δc_{2}}{Δ^{p}} times \frac{x^{p}}{y^{p}} which doesn’t contribute because γ_{p,p} = 0. The fourth term doesn’t contribute for the same reason. The fifth term is \frac{6c_{1}^{p}δc_{1}}{Δ^{p}} times \frac{x^{3p}}{y^{p}} which since we are working modulo p and

x^{3p} = y^{2p} − c_{1}^{p}x^{p} − c_{2}^{p},

γ_{3p,p} = γ_{0,−p} − c_{1}^{p}γ_{p,p} − c_{2}^{p}γ_{0,p}, meaning γ_{3p,p} = 0. So the fifth term doesn’t contribute. The sixth term contributes a \frac{−γ2p,p9c_{2}^{p}δc_{1}}{Δ^{p}} to \overline{f}_{def}. The seventh term doesn’t contribute because the power of y in the denominator is greater than the power of x in the numerator. Similarly the eighth term doesn’t contribute. Hence the two contributing terms are exactly those mentioned in the theorem.

The f_{0} term arises from the analysis of the terms containing C_{p}^{poly}(f_{U}). The procedure is similar, except that in this case ranges of terms are thrown out when either the y exponent in the denominator is greater than any of the x exponents in the numerator or there aren’t any powers of y in the denominator. So parsing the terms in Proposition 3.2,

(4c_{1}^{2p} + 6x^{2p}c_{1}^{p} − 9x^{p}c_{2}^{p})\frac{1}{y^{p}Δ^{p}} ∑_{k=1}^{p−1} \frac{−y^{2p}}{k}

doesn’t contribute to f_{0} because there aren’t any powers of y in the denominator when this expression is simplified. From the term

(4c_{1}^{2p} + 6x^{2p}c_{1}^{p} − 9x^{p}c_{2}^{p})\frac{1}{y^{p}Δ^{p}} ∑_{k=1}^{p−1} \frac{(−1)^{k−1}}{k} (c_{1}x)^{k}c_{2}^{p−k}

comes the

∑_{k=1}^{p−1} \frac{(−1)^{k}}{kΔ^{p}} [6c_{1}^{p+k}c_{2}^{p−k}γ_{2p+k,p} − 9c_{1}^{k}c_{2}^{2p−k}γ_{p+k,p}]

part of f_{0}. The term

(4c_{1}^{2p} + 6x^{2p}c_{1}^{p} − 9x^{p}c_{2}^{p})\frac{1}{y^{p}Δ^{p}} ∑_{k=1}^{p−1} \frac{(−1)^{k−1}}{k} (x^{3k}(c_{1}x + c_{2})^{p−k})

which is equal to

(4c_{1}^{2p} + 6x^{2p}c_{1}^{p} − 9x^{p}c_{2}^{p})\frac{1}{y^{p}Δ^{p}} ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k} c_{1}^{i}c_{2}^{p−k−i}x^{3k+i}

contributes

∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{kΔ^{p}} c_{1}^{i}c_{2}^{p−k−i} [4c_{1}^{2p}γ_{3k+i,p} + 6c_{1}^{p}γ_{2p+3k+i,p} − 9c_{2}^{p}γ_{p+3k+i,p}]

to f_{0}. Lastly

\frac{x^{p}C_{p}^{poly}(f_{U})}{y^{3p}}

contributes the term

∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k} c_{1}^{i}c_{2}^{p−k−i}γ_{p+3k+i,3p}

to f_{0}. This completes the analysis of all of the terms in Proposition 3.2 concluding the second step.

The theorem gives an explicit expression for \overline{f}_{def} of p-derivations and hence by Proposition 4.6 in [1] an explicit formula for the reduction modulo p of the unique δ-modular form of weight χ_{−p−1,−1}.

4. γ_{2p,p} is the Hasse Invariant

The next step is to show that γ_{2p,p} appearing in Theorem 3.6 equals the Hasse invariant.

Proposition 4.1. The term γ_{2p,p} is isobaric of weight p − 1.

Proof. Using the standard weights 4, 6, 2, and 3 given to c_{1}, c_{2}, x, and y respectively on an elliptic curve, x^{2p}/y^{p} has weight 2(2p) − 3p. The procedure for finding the normal form simply involves substituting an expression of equal weight in for x^{3} repeatedly. This does not however change the weight. Therefore the weight of γ_{2p,p} is simply the weight of the normal form minus the weight of x^{2}/y which is 2(2) − 3 = 1. Hence γ_{2p,p} has weight of p − 1.

Proposition 4.2. The term γ_{2p,p} is the coefficient of (xyz)^{p−1} in f(x, y, z)^{p−1} and hence equals the Hasse invariant E_{p−1}.

Proof. For the elliptic curve E_{0} over M_{0}^{0} there is the following commutative diagram:

M_{0}^{0} [\frac{x^{2}}{y} ] ≃ H^{1}(E_{0}, O) ⟶^{σ} H^{2}(ℙ^{2}, O(−3)) ≃ M_{0}^{0} [\frac{1}{XY Z} ]
F^{∗}↓  ↓F^{∗}
H^{1}(E_{0}^{p}, O) ⟶ H^{2}(ℙ^{2}, O(−3p))
↓  ↓f(X, Y, Z)^{p−1}
M_{0}^{0} [\frac{x^{2}}{y} ] ≃ H^{1}(E_{0}, O) ⟶^{σ} H^{2}(ℙ^{2}, O(−3)) ≃ M_{0}^{0} [\frac{1}{XY Z} ]

where F^{∗} is the Frobenius homomorphism that maps ( ) ↦ ( )^{p}, f(X, Y, Z)^{p−1} is the map that is multiplication by f(X, Y, Z)^{p−1}, and σ is the homomorphism that takes [\frac{x^{2}}{y} ] to b [\frac{1}{XY Z} ] for b an invertible element in M_{0}^{0}.

On the left side of the diagram [\frac{x^{2}}{y} ] is mapped to [\frac{x^{2p}}{y^{p}} ] which is then mapped to

γ_{2p,p} [\frac{x^{2}}{y} ]. Then mapping this across gives γ_{2p,p}b [\frac{1}{XY Z} ]. Tracing down the right

side [\frac{x^{2}}{y} ] is mapped to b [\frac{1}{XY Z} ] which is then mapped by F^{∗} to b^{p} [\frac{1}{(XY Z)^{p}} ]. The

next step is multiplication by f(X, Y, Z)^{p−1} and the image is simply b^{p}H [\frac{1}{XY Z} ] where H is the coefficient of (xyz)^{p−1} in f(x, y, z)^{p−1}. Therefore by the commutativity of the diagram γ_{2p,p}b = b^{p}H.

Now (M_{0}^{0})^{∗} = {λΔ^{n}|λ ∈ 𝔽_{p}^{∗}, n ∈ ℤ} meaning that b = λΔ^{n}. However, since both H and γ_{2p,p} have the same weight, it follows b and b^{p} must have the same weight. This means that n must be zero since Δ^{n} has weight of 12n and Δ^{np} has weight 12np. Therefore b = λ ∈ 𝔽_{p}^{∗} meaning b^{p} = b. Ergo γ_{2p,p} = H.

The theorem in the introduction immediately follows from this latest proposition, Proposition 4.6 in [1] which says that \overline{f}_{def} equals the isogeny covariant differential modular form of weight χ_{−p−1,−1} times an invertible constant modulo p, and Theorem 3.6.

References

1. A. Buium, Differential Modular Forms, To appear in Crelle J.

2. N. Katz, p-adic Properties of Modular Schemes and Modular Forms, Lecture Notes in Mathematics, vol. 350, Springer Verlag, 1973.

3. S. Lang, Introduction to Modular Forms, Grundlehren der mathematischen Wissenschaften, vol. 222, Springer Verlag, 1976.

University of New Mexico, Albuquerque, New Mexico 87131

E-mail address: hurlburt@math.unm.edu