ZERO LOCI OF DIFFERENTIAL MODULAR FORMS

CHRIS HURLBURT

Abstract. The theory of p-adic modular forms initiated by Serre, Dwork, and Katz [7] “lives” on the complement (in the p-adic completion of the appropriate modular curve) of the zero locus of the Eisenstein form E_{p−1}. On the other hand most of the interesting phenomena in the theory of differential modular forms [5] take place on the complement of the zero locus of a fundamental differential modular form called f_{jet}. We establish that the zero locus of the reduction modulo p for p not congruent to one modulo 12 of the Eisenstein form E_{p−1} is not contained in the zero locus of the reduction modulo p of the differential modular form f_{jet} implying that the theory of differential modular forms is applicable in certain situations not covered by the theory of p-adic modular forms.

1. Introduction

Differential Modular Forms, as introduced by A. Buium in his article [5], are sections of appropriate line bundles on modular curves in an “arithmetic differential algebraic geometry” where the latter is a geometry obtained by “adjoining” an arithmetic analog of usual derivations to traditional algebraic geometry [3], [4]. The existence of non-trivial maps from modular curves to projective spaces that are constant on “isogeny classes” in the larger arithmetic differential algebraic geometry serves as one of the primary motivations for introducing this extension of modular forms to differential modular forms. In particular outside of the zero locus of the reduction modulo p, f̄_{jet}, of a fundamental differential modular form called f_{jet}, M. Barcau and Buium give a rather nice picture of the geometry of modular curves modulo isogeny [2], [1].

On the other hand the theory of Serre-Tate coordinates developed in [9] and Dwork’s p-adic theory of the trace of Frobenius [8] are wellbehaved only at ordinary points, namely outside the zero locus of the reduction modulo p, Ē_{p−1}, of the Eisenstein form E_{p−1}. (Recall that Ē_{p−1} coincides with the Hasse invariant.) The aim of this paper is to show that the set of points where the theory of differential modular forms behaves nicely (i.e. f̄_{jet} ≠ 0) is not entirely contained in the set of ordinary points (i.e. Ē_{p−1} ≠ 0). We shall succeed to prove this for primes p greater than three and p ≢ 1 modulo 12.

1991 Mathematics Subject Classification. 11,14.

Theorem 1.1. Let p be a prime greater than 3. If p ≡ 2 modulo 3 or if p ≡ 3 modulo 4, then the zero locus of Ē_{p−1} is not contained in the zero locus of f̄_{jet}.

Before we prove this theorem we remind the reader of the relevant definitions for the theory of differential modular forms. Let p be a prime number greater than three. In general a p-derivation is as follows:

Definition 1.2. 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.

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}[a_{4}, a_{6}, Δ^{−1}]ˆ and 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}). Here a_{4}, a_{6}, δa_{4}, δa_{6} are variables over ℤ_{p}, and ˆ represents the p-adic completion. The elements of M^{1} are called δmodular 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 a_{4}, a_{6}, δa_{4}, δa_{6}. 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}[a_{4}, a_{6}, Δ^{−1}] and from M^{1} to M_{0}^{1} = 𝔽_{p}[a_{4}, a_{6}, δa_{4}, δa_{6}, Δ^{−1}]. The reduction modulo p of a δ-modular function f is its image, f̄, under the canonical map from M^{1} to M_{0}^{1}. By the zero locus of f̄ we understand the set of all (a, b) ∈ M(R) such that f(a, b) ≡ 0 modulo p.

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 [5] represents an extension of the definitions given by Deligne and Katz [7]. With respect to this definition, the Eisenstein form, E_{p−1}, is an element of M_{0}.

A special collection of these δ-modular forms posses the following property that does not have a classical analog. Definition 1.3. 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 and for χ = χ_{m,n} the constant is k = m + n(1 − p).

The fundamental differential modular form f_{jet} is the unique (up to a p−adic unit) isogeny covariant differential modular form of weight χ_{−p−1,−1} and f̄_{jet} is its reduction modulo p.

2

We use the explicit formula for f̄_{jet}, given in [6]. For the purpose of this paper we define γ_{a,b} as the result computed by the author in [6] rather than the object computed there. For a and b positive integers, we let m and n be integers such that a = 3m + n for n ∈ {0, 1, 2}. We also assume the following convention for binomial coefficients that if j and k are integers such that j > k, then \binom{k}{j} = 0. If b is even we define γ_{a,b} = 0 and if b is odd we define

γ_{a,b} = (2.1)

∑_{k=0}^{∞} \binom{m + k}{3k + 2 − n}\binom{m − 2k − 2 + n}{\frac{b−1}{2}}(−1)^{m+k− \frac{b−1}{2}} (a_{4})^{3k+2−n}(a_{6})^{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.

The formula γ_{a,b} is necessary to describe the isogeny covariant differential modular form f_{jet} modulo p. From theorems 1.3 and 3.6 in [6] Theorem 2.1. The following equality holds

f̄_{jet} = \frac{−Ē_{p−1}9a_{6}^{p}δa_{4} + Ē_{p−1}6a_{4}^{p}δa_{6}}{2^{−4}Δ^{p}} + f_{0}

where f_{0} ∈ 𝔽_{p}[a_{4}, a_{6}, Δ^{−1}] with Δ = −2^{4}(4a_{4}^{3} + 27a_{6}^{2}) 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 Ē_{p−1} = γ_{2p,p} is the Hasse invariant. For an explicit expression for γ_{a,b} see equation 2.1.

We recall that Deligne’s congruence cf. [7], p. 98 tells us the Hasse invariant is the reduction modulo p of the normalized Eisenstein form of weight p − 1, and hence the choice of Ē_{p−1} for notation.

Using these formulas we can now state the following lemmas which lead to examples where the Eisenstein form is zero modulo p but f̄_{jet} is not for an infinite number of primes. Lemma 2.2. 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 2.3. 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}

Proof. If a_{6} = 0, then γ_{a,b} is not zero if and only if (1) m + n − \frac{b−1}{2} − 2 ≥ 0

(2) m + n − \frac{b−1}{2} − 2 is even.

Note that the second criterion is equivalent to 3m + n − \frac{b−1}{2} = a − \frac{b−1}{2} being even. Since b is odd we only have to consider two cases of b modulo 4 to determine if \frac{b−1}{2} is even or odd. If b ≡ 1 modulo 4, then \frac{b−1}{2} is even. If b ≡ 3 modulo 4, then \frac{b−1}{2} is odd. The rest is computation. □

Remark 2.4. If a_{4} = 0, then the Hasse invariant Ē_{p−1} = γ_{2p,p} = 0 if and only if p ≡ 2 modulo 3. If a_{6} = 0, then Ē_{p−1} = γ_{2p,p} = 0 if and only if p ≡ 3 modulo 4.

Theorem 2.5. Suppose p ≡ 2 modulo 3 and let q be an integer such that p = 3q + 2. Then if a_{4} = 0, the Hasse invariant is 0 and

f̄_{jet} = \frac{(−1)^{\frac{p−1}{2}}}{2^{−4}Δ^{p}} 9a6^{\frac{11q+7}{2}} ∑_{k=0}^{\frac{p−1}{2}} \frac{(−1)^{k}}{k + \frac{p+1}{6}} \binom{\frac{p−1}{2}}{k}.

Proof. When a_{4} = 0 and p ≡ 1 modulo 3, f̄_{jet} = f_{0} and f_{0} becomes

f_{0} = ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k2^{−4}Δ^{p}} a_{6}^{p−k} [−9a_{6}^{p}γ_{p+3k,p}] + ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k} a_{6}^{p−k}γ_{p+3k,3p}.

For 1 ≤ k ≤ p−1, remember that p = 3q+2 giving p+3k = 3(q+k)+2 and

γ_{p+3k,b} = \binom{q + k}{\frac{b−1}{2}}(−a_{6})^{q+k− \frac{b−1}{2}} .

Of the two possible cases for b, if b = 3p then \frac{b−1}{2} > (q + p − 1) resulting in γ_{p+3k,3p} = 0 for 1 ≤ k ≤ p − 1. Hence

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

= ∑_{k=1}^{p−1} \frac{(−1)^{\frac{p−1}{2}}}{k2^{−4}Δ^{p}} a6^{\frac{11q+7}{2}} [9\binom{q + k}{\frac{p−1}{2}}]

= \frac{(−1)^{\frac{p−1}{2}}}{2^{−4}Δ^{p}} 9a6^{\frac{11q+7}{2}} ∑_{k= \frac{p+1}{6}}^{p−1} \frac{1}{k} \binom{q + k}{\frac{p−1}{2}}

= \frac{(−1)^{\frac{p−1}{2}}}{2^{−4}Δ^{p}} 9a6^{\frac{11q+7}{2}} ∑_{k=0}^{\frac{p−1}{2}} \frac{1}{k + \frac{p+1}{6}} \binom{\frac{p−1}{2} + k}{\frac{p−1}{2}}

= \frac{(−1)^{\frac{p−1}{2}}}{2^{−4}Δ^{p}} 9a6^{\frac{11q+7}{2}} ∑_{k=0}^{\frac{p−1}{2}} \frac{(−1)^{k}}{k + \frac{p+1}{6}} \binom{\frac{p−1}{2}}{k}.

□

Theorem 2.6. Suppose p ≡ 3 modulo 4. Then if a_{6} = 0, the Hasse invariant is 0 and

f̄_{jet} = \frac{(−1)^{\frac{p+1}{4}}}{2^{−4}Δ^{p}} (a_{4})^{\frac{11p−1}{4}} (−2 ∑_{k=0}^{\frac{p−1}{2}} \frac{(−1)^{k}}{k + \frac{p+1}{4}} \binom{\frac{p−1}{2}}{k}) .

Proof. When a_{6} = 0 and p ≡ 3 modulo 4, f̄_{jet} = f_{0} and f_{0} becomes

f_{0} = ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k2^{−4}Δ^{p}} a_{4}^{p−k} [4a_{4}^{2p}γ_{p+2k,p} + 6a_{4}^{p}γ_{3p+2k,p}]

+ ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k} a_{4}^{p−k}γ_{2p+2k,3p}.

Finding f̄_{jet} is simply a matter of computing γ_{p+2k,p}, γ_{3p+2k,p}, and γ_{2p+2k,3p}.

Recall that m + n − \frac{b−1}{2} − 2 ≥ 0 is necessary for γ_{a,b} ≠ 0 when a_{6} = 0. This implies that \frac{2a−1−3b}{4} ≥ 0. To check that this condition is not satisfied by γ_{2p+2k,3p}, for any k it is enough to check it for k = p − 1. In this case \frac{2a−1−3b}{4} = \frac{8p−4−1−9p}{4} ≤ 0 meaning γ_{2p+2k,3p} = 0 for 1 ≤ k ≤ p − 1.

Next consider γ_{p+2k,p}. The condition \frac{2a−1−3b}{4} ≥ 0 in this case implies k ≥ \frac{p+1}{4}. When this is true

γ_{p+2k,p} = \binom{k + \frac{p−3}{4}}{k − \frac{p+1}{4}}(−a_{4})^{k− \frac{p+1}{4}} .

In the case of γ_{3p+2k,p}, \frac{2a−1−3b}{4} ≥ 0 for all k and

γ_{3p+2k,p} = \binom{k + \frac{5p−3}{4}}{k + \frac{3p−1}{4}}(−a_{4})^{k+ \frac{3p−1}{4}} .

Hence

f_{0} = ∑_{k= \frac{p+1}{4}}^{p−1} \frac{(−1)^{k}}{k2^{−4}Δ^{p}} a_{4}^{p−k} [4a_{4}^{2p}\binom{k + \frac{p−3}{4}}{k − \frac{p+1}{4}}(−a_{4})^{k− \frac{p+1}{4}} ]

+ ∑_{k=1}^{p−1} \frac{(−1)^{k}}{k2^{−4}Δ^{p}} a_{4}^{p−k} [6a_{4}^{p}\binom{k + \frac{5p−3}{4}}{k + \frac{3p−1}{4}}(−a_{4})^{k+ \frac{3p−1}{4}} ]

= \frac{(−1)^{\frac{p+1}{4}}}{2^{−4}Δ^{p}} (a_{4})^{\frac{11p−1}{4}} (4 ∑_{k= \frac{p+1}{4}}^{p−1} \frac{1}{k}\binom{k + \frac{p−3}{4}}{k − \frac{p+1}{4}} − 6 ∑_{k=1}^{p−1} \frac{1}{k}\binom{k + \frac{5p−3}{4}}{k + \frac{3p−1}{4}})

= \frac{(−1)^{\frac{p+1}{4}}}{2^{−4}Δ^{p}} (a_{4})^{\frac{11p−1}{4}} (4 ∑_{k= \frac{p+1}{4}}^{p−1} \frac{1}{k}\binom{k + \frac{p−3}{4}}{\frac{p−1}{2}} − 6 ∑_{k=1}^{p−1} \frac{1}{k}\binom{k + \frac{5p−3}{4}}{\frac{p−1}{2}})

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

= \frac{(−1)^{\frac{p+1}{4}}}{2^{−4}Δ^{p}} (a_{4})^{\frac{11p−1}{4}} (−2 ∑_{k=0}^{\frac{p−1}{2}} \frac{1}{k + \frac{p+1}{4}} \binom{k + \frac{p−1}{2}}{\frac{p−1}{2}})

= \frac{(−1)^{\frac{p+1}{4}}}{2^{−4}Δ^{p}} (a_{4})^{\frac{11p−1}{4}} (−2 ∑_{k=0}^{\frac{p−1}{2}} \frac{(−1)^{k}}{k + \frac{p+1}{4}} \binom{\frac{p−1}{2}}{k}) .

□

In conclusion we show that in general sums like those in the theorems are not zero modulo p. Indeed if we define the function

g_{m}(t) = ∑_{j=0}^{m} \frac{(−1)^{j}}{t + j} \binom{m}{j}t

then g_{m}(t) = g_{m−1}(t) − \frac{t}{t+1}g_{m−1}(t + 1) and

g_{m}(t) = \frac{m!}{(t + 1)(t + 2)...(t + m)}.

So for example

\frac{g_{\frac{p−1}{2}} (\frac{p+1}{4})}{\frac{p+1}{4}} = ∑_{k=0}^{\frac{p−1}{2}} \frac{(−1)^{k}}{k + \frac{p+1}{4}} \binom{\frac{p−1}{2}}{k}

= \frac{1}{\frac{p+1}{4}} \frac{(\frac{p−1}{2})!}{(\frac{p+1}{4} + 1) (\frac{p+1}{4} + 2) ... (\frac{p+1}{4} + \frac{p−1}{2})}

is a product of invertible elements in 𝔽_{p} and is therefore not zero modulo p. We would like to thank William Blair for pointing this recursion relation out to us. Our final remark is that since f̄_{jet}(0, a_{6}) ≠ 0 and f̄_{jet}(a_{4}, 0) ≠ 0 one can find a, b ∈ R such that f_{jet}(0, b) ≢ 0 modulo p and f_{jet}(a, 0) ≢ 0 modulo p; cf. [5], p. 108. Hence together these two theorems and this last fact imply the theorem in the introduction.

References

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2. M. Barcau and A. Buium, Siegel Differential Modular Forms, Internat. Math. Res. Notices 2002 (2002), no. 28, 1457–1503.

3. A. Buium, Differential Characters of Abelian Varieties over p-adic Fields, Invent. Math. 122 (1995), no. 2, 309–340.

4. , Geometry of p-jets, Duke Math. J. 82 (1996), no. 2, 349–367.

5. , Differential Modular Forms, J. reine angew. Math. (2000), no. 520, 95– 167.

6. C. Hurlburt, Isogeny Covariant Differential Modular Forms modulo p, Compositio Mathematica 128 (2001), no. 1, 17–34.

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

8. , Travaux de Dwork, Lecture Notes in Mathematics, vol. 317, Springer Verlag, 1973.

9. , Serre-Tate Local Moduli, Lecture Notes in Mathematics, vol. 868, Springer Verlag, 1981.

Department of Mathematical Sciences, Northern Illinois University,

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