Complement of the Zero Locus

of f_{jet}

Chris Hurlburt January 10, 2001 • The theory of p-adic modular forms initi-

ated by Serre, Dwork, and Katz ”lives” on the complement (in the in the p-adic completion of the appropriate modular curve)of the zero locus of the Eisenstein form E_{p−1}.

• The most interesting phenomena in the theory of differential modular forms initiated by Buium take place on the complement of the zero locus of a fundamental differential modular form called f_{jet}.

QUESTION: Are the cases where both theories are applicable the same?

Theorem. Let p be a prime greater than 3. If p ≡ 2 modulo 3 or if p ≡ 3 modulo 4, then the zero locus of E_{p−1} is not contained in the zero locus of f_{jet} An Arithmetic Analog of a Derivation

Assume that p > 3 is a prime number.

Definition. If A is a ring and B is an Aalgebra, then p-derivation is a set theoretic map δ : A → B such that

δ(x + y) = δx + δy + C_{p}(x, y)
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} .

This definition is analogous to the well known definition:

Definition. If A is a ring and B is an Aalgebra, then a derivation is a map d : A → B such that

d(x + y) = dx + dy
d(xy) = ydx + xdy.

Example of a p-derivation

Let

M^{0} = ℤ_{p}[a_{4}, a_{6}, Δ^{−1}]ˆ
M^{1} = ℤ_{p}[a_{4}, a_{6}, δa_{4}, δa_{6}, Δ^{−1}]ˆ

where Δ = −2^{4}(4a_{4}^{3} + 27a_{6}^{2}) and the δa_{i} are new variables. Then δ : M^{0} → M^{1} is the unique p-derivation such that

δ(a_{4}) = δa_{4}
δ(a_{6}) = δa_{6}
δ(α) = \frac{α − α^{p}}{p} for α ∈ ℤ_{p}.

Let R have the following properties • Complete, discrete valuation ring • Maximal ideal generated by p

• Algebraically closed residue field k Let

• ϕ be the unique lifting of the Frobenius on k.

• δ(x) = \frac{ϕ(x) − x^{p}}{p} be a (unique) p-derivation on R.

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

Remark Recall that to any (a, b) ∈ M(R) we can associate the elliptic curve y^{2} = x^{3} +ax+b Differential Modular Functions

Definition. A δ modular function of order ≤ n is a function f : M(R) → R such that

f(a, b) = F (a, b, δa, δb, ..., δ^{n}a, δ^{n}b, (4a^{3}+27b^{2})^{−1})

where F is a restricted power series in 2n + 3 variables with ℤ_{p} coefficients.

Remark In the case of order 1, any element f ∈ M^{1} = ℤ_{p}[a_{4}, a_{6}, δa_{4}, δa_{6}, Δ^{−1}]ˆ defines a δ modular function (still denoted by) f : M(R) → R by substituting a, b, δa, δb in for a_{4}, a_{6}, δa_{4}, δa_{6}.

Definition. A δ character of order ≤ 1 is a group homomorphism χ : R^{∗} → R^{∗} of the form χ = χ_{m,n} where

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

Remark We can define δ characters of any order in a similar way.

Differential Modular Forms

Definition. A modular δ function of order 1 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 of order 1 is a modular δ function of order 1 with a weight.

Isogeny Covariance

Definition. A δ modular form is isogeny-covariant if for any two pairs (a, b) and (ã,b̃) such that there is 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 = m + n(1 − p) if χ = χ_{m,n}. For χ = χ_{−p−1,−1} the constant is k = −2.

Theorem. (Buium) For all δ-characters, χ, of order ≤ 1 there is up to multiplication by a constant in ℤ_{p} a unique isogeny covariant δ modular form of weight χ.

From now on ’unique’ will mean ’up to multiplication of a non-zero constant in ℤ_{p}’.

The ’reduction modulo p’ of a δ modular form f ∈ M^{1} = ℤ_{p}[a_{4}, a_{6}, δa_{4}, δa_{6}, Δ^{−1}]ˆ is the image of f in M_{0}^{1} = 𝔽_{p}[a_{4}, a_{6}, δa_{4}, δa_{6}, Δ^{−1}].

The fundamental differential modular form f_{jet} is the unique isogeny covariant differential modular form of weight χ_{−p−1,−1} and \overline{f}_{jet} is its reduction modulo p.

Formula for γ_{a,b}

Let a = 3m + n for m and n integers with n ∈ {0, 1, 2}. Then if b is even, γ_{a,b} = 0. If b is odd,

γ_{a,b} =

∞ ; ∑ ; k=0 \binom{α_{1}}{α_{2}}\binom{α_{3}}{α_{4}}(−1)^{α_{1}−α_{4}}(a_{4})^{α_{2}}(a_{6})^{α_{3}−α_{4}}

where α_{1} = m + k, α_{2} = 3k + 2 − n, α_{3} = m − 2k − 2 + n, and α_{4} = \frac{b−1}{2}.

Remark The γ_{2p,p} is the Hasse invariant and Deligne’s congruence tells us the Hasse invariant is the reduction modulo p of E_{p−1}, the normalized Eisenstein form of weight p − 1.

Formula for \overline{f}_{jet}

Theorem. (H) The class \overline{f}_{jet} is

\overline{f}_{jet} = \frac{−E_{p−1}9a_{6}^{p}δa_{4} + E_{p−1}6a_{4}^{p}δa_{6}}{2^{−4}Δ^{p}} + f_{0}

where f_{0} ∈ 𝔽_{p}[a_{4}, a_{6}, Δ^{−1}] is given by

f_{0} = ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k2^{−4}Δ^{p}} [6a_{4}^{p+k} a_{6}^{p−k}γ_{2p+k,p} − 9a_{4}^{k}a_{6}^{2p−k}γ_{p+k,p} ]

+ ∑_{k=1}^{p−1} ∑_{i=0}^{p−k} \binom{p − k}{i}\frac{(−1)^{k}}{k2^{−4}Δ^{p}} a_{4}^{i}a_{6}^{p−k−i}

[4a_{4}^{2p}γ_{3k+i,p} + 6a_{4}^{p}γ_{2p+3k+i,p} − 9a_{6}^{p}γ_{p+3k+i,p} ]

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

and E_{p−1} = γ_{2p,p} is the Hasse invariant.

Lemma. If a_{4} = 0, then

γ_{a,b} = \begin{cases}0  if n = 0 \\ 0  if n = 1 \\ \binom{m}{\frac{b−1}{2}}(−a_{6})^{m−\frac{b−1}{2}} if n = 2 \end{cases}

Lemma. If a_{6} = 0 and m + n − \frac{b−1}{2} − 2 ≥ 0, then

γ_{a,b} = \begin{cases}0 \\ if a is odd and b ≡ 1 modulo 4 \\ ( \frac{2a−3−b}{4} ; \frac{2a−1−3b}{4} )(−a_{4})^{\frac{2a−1−3b}{4}} \\ if a is odd and b ≡ 3 modulo 4 \\ ( \frac{2a−3−b}{4} ; \frac{2a−1−3b}{4} )(−a_{4})^{\frac{2a−1−3b}{4}} \\ if a is even and b ≡ 1 modulo 4 \\ 0 \\ if a is even and b ≡ 3 modulo 4 \end{cases}

Remark If a_{4} = 0, then the Hasse invariant E_{p−1} = γ_{2p,p} = 0 if and only if p ≡ 2 modulo 3. If a_{6} = 0, then E_{p−1} = γ_{2p,p} = 0 if and only if p ≡ 3 modulo 4.

Theorem. (H) Suppose p ≡ 2 modulo 3 and let q be an integer such that p = 3q + 2. Then if a_{4} = 0,

\overline{f}_{jet} = \frac{(−1)^{\frac{p−1}{2}}}{2^{−4}Δ^{p}} 9a6^{\frac{11q+7}{2}} p−1 ; ∑ ; k=1 \frac{1}{k} \binom{q + k}{\frac{p−1}{2}}

Theorem. (H) Suppose p ≡ 3 modulo 4. Then if a_{6} = 0, the Hasse invariant is 0 and

\overline{f}_{jet} = \frac{(−1)^{\frac{p+1}{4}}}{2^{−4}Δ^{p}} (a_{4})^{\frac{11p−1}{4}} (−2 p−1 ; ∑ ; k=\frac{p+1}{4} \frac{1}{k}\binom{k + \frac{p−3}{4}}{\frac{p−1}{2}})