NONEXISTENCE OF PSEUDO-CANONICAL LIFTINGS

LUÍS R. A. FINOTTI

Abstract. In this paper we show that pseudo-canonical liftings do not exist, by showing that if j_{0} ↦ (j_{0}, J_{1}(j_{0}), J_{2}(j_{0}), …) is the map that gives canonical liftings for ordinary j_{0}, then J_{2} has a pole at j_{0} = 1728 if p ≡ 3 (mod 4) and J_{3} has a pole at j_{0} = 0 if p ≡ 5 (mod 6). Moreover, precise descriptions of J_{2} and J_{3} are given.

1. Introduction

Let 𝕜 be a perfect field of characteristic p > 0, W(𝕜) be the ring of Witt vectors over 𝕜, and W_{n}(𝕜) denote the ring of Witt vectors of length n, which in this case can be seen as the quotient of W(𝕜) modulo the principal ideal generated by p^{n}. Then, given an ordinary elliptic curve E/𝕜, there is a unique elliptic curve (up to isomorphism), say E/W(𝕜), which reduces to E modulo p and for which we can lift the Frobenius. E is then called the canonical lifting of E. (See, for instance, [Deu41] or [LST64].) Hence, given an ordinary j-invariant j_{0} ∈ 𝕜, the canonical lifting gives us a unique j ∈ W(𝕜). Therefore, if 𝕜^{ord} denotes the set of ordinary values of j-invariants in 𝕜, then we have functions J_{i} : 𝕜^{ord} → 𝕜, for i = 1, 2, 3, …, such that the j-invariant of the canonical lifting of an elliptic curve with j-invariant j_{0} ∈ 𝕜^{ord} is (j_{0}, J_{1}(j_{0}), J_{2}(j_{0}), …).

B. Mazur asked about the nature of these functions J_{i} and J. Tate asked about the possibility of extending them to supersingular values.

We’ve proved that the functions J_{i} are rational functions over 𝔽_{p} in [Fin10]. Tate’s question motivates the following definition:

Definition 1.1. Suppose that j_{0} ∉ 𝕜^{ord} and J_{i} is regular at j_{0} for all i ≤ n. Then, we call an elliptic curve over W(𝕜) whose j-invariant reduces to (j_{0}, J_{1}(j_{0}), …, J_{n}(j_{0})) modulo p^{n+1} a pseudo-canonical lifting modulo p^{n+1} (or over W_{n+1}(𝕜)) of the elliptic curve associated to j_{0}.

If J_{i} is regular for all i, we call the elliptic curve with j-invariant (j_{0}, J_{1}(j_{0}), J_{2}(j_{0}), …) the pseudo-canonical lifting of the elliptic curve associated to j_{0}.

2000 Mathematics Subject Classification. Primary 11G07; Secondary 11Y99.

Key words and phrases. elliptic curves, canonical lifting, pseudo-canonical lifting, modular polynomial.

Hence, Tate asks about the existence of such pseudo-canonical liftings. One would not expect pseudo-canonical liftings to exist, as they would yield curves which although are not canonical liftings, as those do not exist in the supersingular case, are obtained by the same formulas. On the other hand, we’ve proved that pseudo-canonical liftings modulo p^{2} and p^{3} do exist for specific supersingular values. More precisely, we’ve studied J_{1} and J_{2} in detail in [Fin10] (using many results from [KZ98]) and [Fin11b] respectively, proving the following:

Theorem 1.2. With the notation above and p ≥ 5:

(1) J_{1}(X) is always regular at X = 0 and X = 1728, even when those values are supersingular, and (0, J_{1}(0)) ≡ 0 (mod p^{2}) and (1728, J_{1}(1728)) ≡ 1728 (mod p^{2}).

(2) If j_{0} ∉ 𝕜^{ord} ∪ {0, 1728}, then J_{1} has a simple pole at j_{0}.

(3) J_{2}(X) is always regular at X = 0, even if 0 is supersingular, and (0, J_{1}(0), J_{2}(0)) ≡ 0 (mod p^{3}).

(4) If j_{0} ∉ 𝕜^{ord} ∪ {0, 1728}, then J_{2} has a pole of order 2p + 1 at j_{0}.

As one can see, this statement does not give any information modulo p^{3} in the case of 1728 being supersingular. We will prove here the following theorem, which was stated as a conjecture in [Fin10], more precisely, item (1) of Conjecture 9.3.

Theorem 1.3. If 1728 ∉ 𝕜^{ord} (i.e., if p ≡ 3 (mod 4)), then J_{2} has a pole of order p at 1728.

So, this would tell us 1728 never yields pseudo-canonical liftings, leaving 0 as the only possibility. On the other hand, we will also show here that 0 also fails. This again was a conjecture of [Fin10], more precisely, Conjecture 10.1. (In fact, we prove here that Conjecture 9.7 from [Fin11b], which is equivalent to item (2) of Conjecture 9.3 from the same reference, is equivalent to Conjecture 10.1, and therefore all conjectures of [Fin11b] are proved here.)

Theorem 1.4. If 0 ∉ 𝕜^{ord} (i.e., if p ≡ 5 (mod 6)), then J_{3} has a pole of order p^{2} at 0.

This gives a complete answer to Tate’s question, showing that, as expected, no pseudocanonical lifting exist, and the only possible ones modulo p^{2} are given by 0 and 1728, and modulo p^{3}, only by 0.

We will heavily rely on results and techniques from the author’s [Fin10] and [Fin11b], although we will restate most of the necessary results. It should also be observed that Kaneko and Zagier’s [KZ98], from which many results from [Fin10] are derived, provided many of the necessary tools, although we may refer to [Fin10] instead, as the results are phrased in a more compatible way. Finally, we will also need results from [Fin11a], which will be the main tool to analyze the properties of J_{3}.

We now give a brief description of the next sections. In Section 2 we review the concept of the Greenberg transform of a polynomial and recall the formulas for those which were derived in [Fin11b] and [Fin11a]. In Section 3 we introduce some alternatives to the jinvariant which will help us deal with the pole of J_{2} at 1728, similarly to what was done in [Fin10]. In Section 4 we use these invariants to prove Theorem 1.3. In Section 5 we give a formula for J_{3}, similar to the formula for J_{2} given in [Fin11b], while in Section 6 we use this formula to prove Theorem 1.4. Finally, on Section 7 we give some more information on the formulas for J_{2} and J_{3}.

2. The Greenberg Transform

In this section we briefly review the definition of the Greenberg transform. (See also [Lan52] and [Gre61].)

Definition 2.1. Let f(x, y) ∈ W(𝕜)[x, y]. If we replace x and y by (x_{0}, x_{1}, …) and (y_{0}, y_{1}, …), seen as Witt vectors of unknowns, and expand the resulting expression using sums and products of Witt vectors, we obtain a Witt vector (f_{0}, f_{1}, …), with f_{i} ∈ 𝕜[x_{0}, …, x_{i}, y_{0}, …, y_{i}]. This resulting vector is called the Greenberg transform of f and will be denoted by 𝒢 (f).

Moreover, if

C/W(𝕜) : f(x, y) = 0,

we define the Greenberg transform 𝒢 (C) of C to be the (infinite dimensional) variety over 𝕜 defined by the common zeros of the coordinates of 𝒢 (f).

It is clear from the definition that there is a bijection between C(W(𝕜)) and 𝒢 (C)(𝕜). We will need the formula for the second coordinate of the Greenberg transform of a

polynomial. This is given by Theorem 6.1 from [Fin11b], restated below as Theorem 2.4. But before we can state it, we need some extra notation:

Definition 2.2. Let p be a prime. Define η_{0}(X_{1}, …, X_{r}) =^{def} X_{1} + ⋯ + X_{r}, and recursively for k ≥ 1

η_{k}(X_{1}, …, X_{r}) =^{def} \frac{X_{1}^{p^{k}} + ⋯ + X_{r}^{p^{k}}}{p^{k}} − ∑_{i=0}^{k−1} \frac{η_{i}(X_{1}, …, X_{r})^{p^{k−i}}}{p^{k−i}} .  (2.1)

Also, define η_{k}(X_{1}) = 0 for k ≥ 1.

If R is a ring of characteristic p and v = (a_{1}, …, a_{r}) ∈ R^{r}, we define η_{k}(v) = η_{k}(a_{1}, …, a_{r}) as the evaluation of η_{k}(X_{1}, …, X_{r}) at v. (This makes sense as η_{k}(X_{1}, …, X_{r}) ∈ ℤ[X_{1}, …, X_{r}]. See [Fin11a].)

Moreover, if f is a polynomial (possibly in many variables) with coefficients in R, we write vec (f) for the vector that contains the terms of f (after some choice of order for the monomials). We then may write η_{k}(f) for η_{k}(vec (f)). (It is important to observe that we are assuming that the terms are reduced, i.e., if f = 1 + X + 2X, then vec (f) = (1, 3X), not (1, X, 2X).)

Sometimes it will be useful to use the following notation:

Definition 2.3. Given f = ∑_{i,j} a_{i,j}x^{i}y^{j} ∈ W(𝕜)[x, y] and a positive integer n, define

f^{[p^{n}]} =^{def} ∑_{i,j} a_{i,j}^{p^{n}}x^{ip^{n}}y^{jp^{n}}.

We also define η_{k}(f) to be the reduction modulo p of

η_{k}(f) = η_{k}(vec (f)) = \frac{f^{[p^{k}]} − f^{p^{k}}}{p^{k}} − \frac{η_{1}(vec (f))^{p^{k−1}}}{p^{k−1}} − ⋯ − \frac{η_{k−1}(vec (f))^{p}}{p} .  (2.2)

Then, if f reduces to f modulo p, we have that η_{k}(f) = η_{k}(f).

With the notation above, we can give a formula for the third coordinate of the Greenberg transform of f.

Theorem 2.4. Let f ∈ W(𝕜)[x, y] be given by

f(x, y) = ∑_{i,j} a_{i,j}x^{i}y^{j},

with partial derivatives with respect to x and y

f_{x}(x, y) = ∑_{i,j} b_{i,j}x^{i}y^{j}  and  f_{y}(x, y) = ∑_{i,j} c_{i,j}x^{i}y^{j},

respectively. Also, let f be the reduction modulo p of f (and use subscripts x_{0} and y_{0} to denote its partial derivatives), and

a_{i,j} ≡ (a_{i,j,0}, a_{i,j,1}, a_{i,j,2}) (mod p^{3}),
b_{i,j} ≡ (b_{i,j,0}, b_{i,j,1}, b_{i,j,2}) (mod p^{3}),
c_{i,j} ≡ (c_{i,j,0}, c_{i,j,1}, c_{i,j,2}) (mod p^{3}).

Then, the third coordinate of the Greenberg transform of f is given by

f_{x_{0}}^{p^{2}}x_{2} + f_{y_{0}}^{p^{2}}y_{2} + (∑_{i,j} b_{i,j,1}x_{0}^{ip}y_{0}^{jp})^{p} x_{1}^{p} + (∑_{i,j} c_{i,j,1}x_{0}^{ip}y_{0}^{jp})^{p} y_{1}^{p}

+ (f_{x_{0}x_{0}}/2)^{p^{2}}x_{1}^{2p} + f_{x_{0}y_{0}}^{p^{2}}x_{1}^{p}y_{1}^{p} + (f_{y_{0}y_{0}}/2)^{p^{2}}y_{1}^{2p} + ∑_{i,j} a_{i,j,2}x_{0}^{ip^{2}}y_{0}^{jp^{2}}

+ η_{1}(f_{x_{0}}^{p}x_{1} + f_{y_{0}}^{p}y_{1} + ∑_{i,j} a_{i,j,1}x_{0}^{ip}y_{0}^{jp})

+ η_{1}(f_{x_{0}}^{p}x_{1} + f_{y_{0}}^{p}y_{1} + ∑_{i,j} a_{i,j,1}x_{0}^{ip}y_{0}^{jp}, η_{1}(f)) + η_{2}(f). (2.3)

We also need a formula for the fourth coordinate of the Greenberg transform. In [Fin11a] we give a general formula (Theorem 5.4). Since this general formula is too convoluted in the general setting, we will give here only the particular case of the fourth coordinate here. The formula is still quite involved, and we need to introduce some extra notation in addition to the notation from Theorem 2.4.

Let

\frac{1}{2}f_{xx}(x, y) = ∑_{i,j} d_{i,j}x^{i}y^{j},  f_{xy}(x, y) = ∑_{i,j} e_{i,j}x^{i}y^{j},  \frac{1}{2}f_{yy}(x, y) = ∑_{i,j} f_{i,j}x^{i}y^{j},

and

d_{i,j} ≡ (d_{i,j,0}, d_{i,j,1}) (mod p^{2}),
e_{i,j} ≡ (e_{i,j,0}, e_{i,j,1}) (mod p^{2}),
f_{i,j} ≡ (f_{i,j,0}, f_{i,j,1}) (mod p^{2}).

Moreover, let 𝒢_{1} be the vector

vec ((f_{x})^{p}x_{1} + f_{y}^{p}y_{1} + ∑_{i,j} a_{i,j,1}x_{0}^{ip}y_{0}^{jp})

with η_{1}(f) appended (at the last entry) to it, and 𝒢_{2} be

vec (f_{x_{0}}^{p^{2}}x_{2} + f_{y_{0}}^{p^{2}}y_{2} + (∑_{i,j} b_{i,j,1}x_{0}^{ip}y_{0}^{jp})^{p} x_{1}^{p} + (∑_{i,j} c_{i,j,1}x_{0}^{ip}y_{0}^{jp})^{p} y_{1}^{p}

+(f_{x_{0}x_{0}}/2)^{p^{2}}x_{1}^{2p} + f_{x_{0}y_{0}}^{p^{2}}x_{1}^{p}y_{1}^{p} + (f_{y_{0}y_{0}}/2)^{p^{2}}y_{1}^{2p} + ∑_{i,j} a_{i,j,2}x_{0}^{ip^{2}}y_{0}^{jp^{2}} )

with η_{1}(𝒢_{1}) and η_{2}(f) appended (at the last two entries) to it.

We then have:

Theorem 2.5. With the notation above and p ≥ 3, the fourth coordinate of the Greenberg transform of f is given by

∑_{i,j} a_{i,j,3}x_{0}^{ip^{3}}y_{0}^{jp^{3}} + f_{x_{0}}^{p^{3}}x_{3} + f_{y_{0}}^{p^{3}}y_{3}+

+ (∑_{i,j} b_{i,j,1}^{p^{2}}x_{0}^{ip^{3}}y_{0}^{jp^{3}} ) x_{2}^{p} + (∑_{i,j} c_{i,j,1}^{p^{2}}x_{0}^{ip^{3}}y_{0}^{jp^{3}} ) y_{2}^{p}

+ (∑_{i,j} b_{i,j,2}^{p}x_{0}^{ip^{3}}y_{0}^{jp^{3}} ) x_{1}^{p^{2}} + (∑_{i,j} c_{i,j,2}^{p}x_{0}^{ip^{3}}y_{0}^{jp^{3}} ) y_{1}^{p^{2}}

+ (f_{x_{0}x_{0}})^{p^{3}}x_{1}^{p^{2}}x_{2}^{p} + f_{x_{0}y_{0}}^{p^{3}}(x_{1}^{p^{2}}y_{2}^{p} + x_{2}^{p}y_{1}^{p^{2}}) + (f_{y_{0}y_{0}})^{p^{3}}y_{1}^{p^{2}}y_{2}^{p}

+ (∑_{i,j} d_{i,j,1}^{p^{2}}x_{0}^{ip^{3}}y_{0}^{jp^{3}} ) x_{1}^{2p^{2}} + (∑_{i,j} e_{i,j,1}^{p^{2}}x_{0}^{ip^{3}}y_{0}^{jp^{3}} ) x_{1}^{p^{2}}y_{1}^{p^{2}} + (∑_{i,j} f_{i,j,1}^{p^{2}}x_{0}^{ip^{3}}y_{0}^{jp^{3}} ) y_{1}^{2p^{2}}

+ (f_{x_{0}x_{0}x_{0}}/6)^{p^{3}}x_{1}^{3p^{2}} + (f_{x_{0}x_{0}y_{0}}/2)^{p^{3}}x_{1}^{2p^{2}}y_{1}^{p^{2}} + (f_{x_{0}y_{0}y_{0}}/2)^{p^{3}}x_{1}^{p^{2}}y_{1}^{2p^{2}} + (f_{y_{0}y_{0}y_{0}}/6)^{p^{3}}y_{1}^{3p^{2}} + η_{1}(𝒢_{2}) + η_{2}(𝒢_{1}) + η_{3}(f). (2.4)

3. Alternative Invariants

To prove Theorem 1.3 we will use a couple of different invariants.

Definition 3.1. If j is the j-invariant of an elliptic curve, we shall denote ĵ =^{def} j − 1728. We may refer to this alternative invariant as the ĵ-invariant of the elliptic curve.

Let also Φ̂_{p}(X, Y) = Φ_{p}(X + 1728, Y + 1728), where Φ_{p} is the (classical) modular polynomial. Hence, two curves with ĵ-invariants ĵ_{1} and ĵ_{2} have an isogeny of degree p between them if, and only if, Φ̂_{p}(ĵ_{1}, ĵ_{2}) = 0.

Now, if ĵ_{0} is the ĵ-invariant of an ordinary elliptic curve in characteristic p, then, as with the original j-invariant (see [Fin10]), the ĵ-invariant of its canonical lifting is given by

ĵ = (ĵ_{0}, Ĵ_{1}(ĵ_{0}), Ĵ_{2}(ĵ_{0}), …) = (j_{0}, J_{1}(j_{0}), J_{2}(j_{0}), …) − 1728,  (3.1)

where Ĵ_{i}(X) ∈ 𝔽_{p}(X) and j_{0} = ĵ_{0} + 1728 is the usual j-invariant of the curve. The other invariant that we need was studied in [Fin10].

Definition 3.2. We define the ˜ j̃ of an elliptic curve with j ≠ 0 to be

˜j̃ = \frac{4(1728 − j)}{27j} = −\frac{4ĵ}{27(ĵ+ 1728)}.  (3.2)

This other invariant also has its own corresponding rational functions giving the canonical lifting, say ˜ J̃_{i}(X), which can be obtained from the Ĵ_{i}(X) (or J_{i}(X)) using Eq. (3.2).

The first step in proving Theorem 1.3 is to obtain the proper formula for Ĵ_{2} from Φ̂_{p}, in the same way we’ve obtained a formula for J_{2} from Φ_{p} in [Fin11b]. In fact, the computation is quite similar, as Φ̂_{p}(X, Y) ≡ Φ_{p}(X, Y) (mod p).

Applying Theorem 2.4, we obtain the following proposition, which is the analogue of Theorem 9.1 from [Fin11b].

Proposition 3.3. Let

Φ̂_{p} = ∑_{i,j} â_{i,j}X^{i}Y^{j},  (Φ̂_{p})_{X} = ∑_{i,j} b̂_{i,j}X^{i}Y^{j},  and  (Φ̂_{p})_{Y} = ∑_{i,j} ĉ_{i,j}X^{i}Y^{j},

respectively, with â_{i,j} = (â_{i,j,0}, â_{i,j,1}, …), b̂_{i,j} = (b̂_{i,j,0}, b̂_{i,j,1}, …), ĉ_{i,j} = (ĉ_{i,j,0}, ĉ_{i,j,1}, …). Also, let

ĝ_{2}(X_{0}, Y_{0}, Y_{1}) =^{def} η_{1}((Y_{0}^{p} − X_{0})^{p}Y_{1} + ∑_{i,j} â_{i,j,1}X_{0}^{ip}Y_{0}^{jp}).

Then, ĝ_{2}(X, X^{p}, Ĵ_{1}(X)^{p}) is a p-power and

Ĵ_{2}(X) = \frac{−1}{(X^{p^{2}} − X)^{p}} [(∑_{i,j} b̂_{i,j,1}X^{ip+jp^{2}} ) Ĵ_{1}(X) + (∑_{i,j} ĉ_{i,j,1}X^{ip+jp^{2}} ) Ĵ_{1}(X)^{p}

−Ĵ_{1}(X)^{p+1} + ∑_{i,j} â_{i,j,2}X^{ip+jp^{2}} + ĝ_{2}(X, X^{p}, Ĵ_{1}(X)^{p})^{1/p}] . (3.3)

Proof. By Theorem 3 of [LST64], we have that if (j_{0}, J_{1}, …) is the j-invariant of the canonical lifting of the curve with j-invariant j_{0}, then

Φ_{p}((j_{0}, J_{1}, …), (j_{0}^{p}, J_{1}^{p}, …)) = 0.  (3.4)

So, if j = (j_{0}, J_{1}, J_{2}, …) and ĵ = (ĵ_{0}, Ĵ_{1}, Ĵ_{2}, …) are the j and ĵ-invariants of the canonical lifting of the curves with j and ĵ-invariants j_{0} and ĵ_{0} respectively, then

Φ̂_{p}((ĵ_{0}, Ĵ_{1}, Ĵ_{2}), (ĵ^{p}, Ĵ_{1}^{p}, Ĵ_{2}^{p})) ≡ Φ_{p}((j_{0}, J_{1}, J_{2}), (j^{p}, J_{1}^{p}, J_{2}^{p})) ≡ 0 (mod p^{3}),

and thus the proof is virtually the same as the proof of Theorem 9.1 from [Fin11b]. □

We shall keep the notation of Proposition 3.3 throughout the next section.

4. Pole of J_{2} at 1728

We shall prove Theorem 1.3 in this section. We will need some preliminary results.

Lemma 4.1. Let K be a field and v a valuation on K, u = (u_{0}, u_{1}), v = (v_{0}, v_{1}) ∈ W_{2}(K) with v(u_{0}) = 0, v(v_{0}) = 1, and v(u_{1}), v(v_{1}) ≥ 0. If w = (w_{0}, w_{1}) = u · v, then v(w_{1}) ≥ min{v(v_{1}), p}.

Proof. This is a simple application of the formulas for products of Witt vectors. Since

w_{1} = u_{0}^{p}v_{1} + u_{1}v_{0}^{p},

clearly the statement about v(w_{1}) holds. □

Let v_{0} = ord_{X=0}, the order of zero at X = 0. (We shall keep this notation.) We then have:

Lemma 4.2. With the previous notation, we have v_{0}(Ĵ_{1}) ≥ min{v_{0}( ˜J̃_{i}), p}, and v_{0}( ˜J̃_{1}) ≥ min{v_{0}(Ĵ_{1}), p}.

Proof. Let

X̂ =^{def} −1728\frac{27X}{27X + 4},  ˜  X̃ =^{def} −\frac{4X}{27(X + 1728)},

and

ĵ(X) = (X, Ĵ_{1}(X)), ˜ j̃(X) = (X, ˜ J̃_{1}(X)).

Then, by Eq. (3.2), we have that

ĵ(X) = −\frac{1728 · 27}{27˜j̃( ˜X̃) + 4} · ˜ j̃( ˜X̃).

Let

−\frac{1728 · 27}{27˜j̃( ˜X̃) + 4} = (α_{0}(X), α_{1}(X)).

By Propositions 4.2 and 4.3 of [Fin10], we have that v_{0}( ˜J̃_{1}(X)) ≥ (p − 1)/2, and hence the left hand side of this equation is regular at X = 0. Moreover, it is different from zero when evaluated at X = 0 (or ˜ X̃ = 0), and therefore we have that v_{0}(α_{0}) = 0, and v_{0}(α_{1}) ≥ 0. We can then apply Lemma 4.1, and thus v_{0}(Ĵ_{1}) ≥ min{v_{0}( ˜J̃_{1}), p}, and thus v_{0}(Ĵ_{1}) ≥ (p − 1)/2.

We also have, again by Eq. (3.2), that

˜j̃(X) = −\frac{4}{27(ĵ(X̂) + 1728)} · ĵ(X̂).

Let

−\frac{4}{27(ĵ(X̂) + 1728)} = (β_{0}(X), β_{1}(X)).

Again, since v_{0}(Ĵ_{1}) ≥ (p − 1)/2, the left hand side of this equation is regular at X = 0, and different from zero when evaluated at X = 0 (or X̂ = 0), and hence v_{0}(β_{0}) = 0, and v_{0}(β_{1}) ≥ 0. Lemma 4.1 then gives us that v_{0}( ˜J̃_{1}) ≥ min{v_{0}(Ĵ_{1}), p}. □

Lemma 4.3. The following are equivalent:

(1) J_{2}(X) has a pole of order p at X = 1728.

(2) Ĵ_{2}(X) has a pole of order p at X = 0.

(3) â_{0,0,2} ≠ 0.

Proof. The equivalence of the first two items is an immediate consequence of Eq. (3.1) and arithmetic of Witt vectors. More precisely, if 1728 = (γ_{0}, γ_{1}, γ_{2}), then

Ĵ_{1}(X − 1728) = J_{1}(X) + γ_{1} + η_{1}(X, γ_{0}).

Since, η_{1}(X, Y) is a polynomial and J_{1} is regular at X = 1728 (by Theorem 1.2), we have that Ĵ_{1} is regular at X = 0. In the same way,

Ĵ_{2}(X − 1728) = J_{2}(X) + γ_{2} + f(X, J_{1}, γ_{0}, γ_{1})

for some polynomial f and hence γ_{2} + f(X, J_{1}, γ_{0}, γ_{1}) is regular at X = 1728. The equivalence of the first two items then follows immediately.

The equivalence of the last two items follows from Eq. (3.3). Indeed, as observed in the proof of Lemma 4.2, Ĵ_{1}(X) has a zero at X = 0. This also implies that ĝ_{2}(X, X^{p}, Ĵ_{1}(X)^{p})^{1/p} has a zero at X = 0. Thus, Eq. (3.3) gives us that Ĵ_{2}(X) has a pole of order p at X = 0 if, and only if, â_{0,0,2} ≠ 0. □

We shall prove then that â_{0,0,2} ≠ 0. To do this, we follow the same idea used in [Fin11b] to show that the corresponding a_{0,0,2} for the usual modular polynomial Φ_{p} is zero.

Proposition 4.4. If â_{0,0,2} = 0, then v_{0}( ˜J̃_{1}) ≥ (p + 1)/2.

Proof. We use square roots of ĵto obtain a simplified polynomial Ψ̂_{p} such that Ψ̂_{p}(ĵ_{1}^{1/2}, ĵ_{2}^{1/2}) = 0 if the elliptic curves associated to ĵ_{1} and ĵ_{2} have an isogeny of degree p. This is the analogue of the polynomial Ψ_{p} from [Fin11b] (which we will use again in Section 6), and satisfies the analogous property:

Φ̂_{p}(X^{2}, Y^{2}) = Ψ̂_{p}(X, Y)Ψ̂_{p}(X, −Y).  (4.1)

(This corresponds to Eq. (23) of [Elk98] for Φ̂_{p}.) This clearly implies that

Φ̂_{p}(X^{2}, 0) = (Ψ̂_{p}(X, 0))^{2}  (4.2)

and hence â_{0,0} is a square. By Kronecker’s relation,

Ψ̂_{p}(X, 0) ≡ X^{p+1} (mod p)  (4.3)

(as Φ̂_{p}(X, 0) ≡ X^{p+1} (mod p)). Then, Eq. (4.3) implies that all coefficients of Ψ̂_{p}(X, 0) are divisible by p, except for the coefficient of X^{p+1}. Thus, by Eq. (4.2), we have that v_{p}(â_{i,0}) ≥ 2 for all i < (p + 1)/2, where v_{p} denotes the valuation at p.

Since â_{0,0} is a square, if â_{0,0,2} = 0, then p^{4} | â_{0,0}, and thus p^{2} | Ψ̂_{p}(0, 0). This last condition implies that p^{2} | â_{(p+1)/2,0}, and we then have that a_{i,0,1} = 0 for i ∈ {0, …, (p + 1)/2}.

Now, the same proof from [Fin10] that gives

J_{1}(X) ≡ −\frac{Φ_{p}(X, X^{p})}{p(X^{p^{2}} − X)} (mod p)

also gives the equivalent formula for Ĵ_{1}, namely,

Ĵ_{1}(X) ≡ −\frac{Φ̂_{p}(X, X^{p})}{p(X^{p^{2}} − X)} ≡ −\frac{∑ â_{i,j,1}X^{i+jp}}{X^{p^{2}} − X} (mod p).

But, by the computation above, this implies that v_{0}(Ĵ_{1}) ≥ (p + 1)/2. And by Lemma 4.2, we have that v_{0}( ˜J̃_{1}) ≥ (p + 1)/2. □

We finally can prove Theorem 1.3.

Proof of Theorem 1.3. By Lemma 4.3, it suffices to prove that â_{0,0,2} ≠ 0. Assume that â_{0,0,2} = 0. In the proof of Proposition 5.6 of [Fin10], it is shown if ˜ J̃_{1}(X) has a zero of order greater than or equal to s + 1 at 0, then the t-th derivative of J_{1} at X = 1728 is given by J_{1}^{(t)}(1728) = −(t − 1)!(−1728)^{1−t} for 1 ≤ t ≤ s. Hence, by Proposition 4.4, we obtain that

J_{1}^{((p−1)/2)}(1728) = −((p − 3)/2)! (−1728)^{−(p−3)/2}  (4.4)
= (−1)^{(p−1)/2}((p − 3)/2)! 1728^{(p+1)/2}.

Now, since p ≡ 3 (mod 4), we have that Eq. (38) from [KZ98] (or from Theorem 3.2(3) from [Fin10]) reduces to

J′_{1}(X) = −X^{p−1} + X^{r} \frac{(X − 1728)^{(p+1)/2}}{ss_{p}(X)^{2}}  (4.5)

= −X^{p−1} + (X − 1728)^{(p−3)/2} \frac{X^{r}}{f(X)},

where

r =^{def} \begin{cases}(2p − 2)/3, & if p ≡ 1 (mod 6), \\ (2p + 2)/3, & if p ≡ 5 (mod 6), \end{cases}

and f(1728) ≠ 0.

Now, since

\frac{d^{(p−3)/2}}{dX^{(p−3)/2}} (−X^{p−1})|_{X=1728} = (−1)^{(p−1)/2}((p − 3)/2)! 1728^{(p+1)/2},

while

\frac{d^{(p−3)/2}}{dX^{(p−3)/2}} ((X − 1728)^{(p−3)/2} \frac{X^{r}}{f(X)})|_{X=1728} = \frac{1728^{r}}{f(1728)} ≠ 0,

Eq. (4.5) implies that J_{1}^{(p−1)/2}(1728) ≠ (−1)^{(p−1)/2}((p − 3)/2)! 1728^{(p+1)/2}, contradicting Eq. (4.4). Thus, we must have that â_{0,0,2} ≠ 0. □

5. Formula for J_{3}

We will now deduce the formula for J_{3} from Eq. (2.4).

As with proof of the formula for J_{2} from [Fin11b] (the analogous to Eq. (3.3) above), the main idea is again to use Eq. (3.4).

We will use the notation of Theorems 2.4 and 2.5 for f = Φ_{p}. So, in particular, f = (x_{0} − y_{0}^{p})(x_{0}^{p} − y_{0}) (by Kronecker’s relation) and f_{x_{0}} = x_{0}^{p} − y_{0}. Thus, we can obtain J_{3} by evaluating Eq. (2.4) at ((x_{0}, …, x_{3}), (y_{0}, …, y_{3})) = ((j_{0}, J_{1}, J_{2}, J_{3}), (j_{0}^{p}, J_{1}^{p}, J_{2}^{p}, J_{3}^{p})). Thus, we can find an expression for J_{3}^{p}.

On the other hand, terms from Eq. (2.4) that are divisible by (x_{0}^{p} − y_{0}) will vanish when evaluating, and hence can be discarded from the formula for J_{3}.

Since we will often lift to characteristic 0 and use Eq. (2.2), we should note that since (x^{p} − y) is primitive, we have that if g ∈ ℤ[x, y] and g = (x^{p} − y)g_{1}, with g_{1} ∈ ℚ[x_{0}, x_{1}, …, y_{0}, y_{1}, …], then in fact g_{1} has integral coefficients.

Lemma 5.1. If p ≠ 2, then η_{k}(f) ≡ 0 (mod (x_{0}^{p} − y_{0})) for all k ≥ 1.

Proof. We have that η_{k}(f) = η_{k}(f_{1}), where f_{1} = (x − y^{p})(x^{p} − y). Then, if p ≠ 2, we have

f_{1}^{[p^{k}]} = (x^{p^{k}} − y^{p^{k+1}})(x^{p^{k+1}} − y^{p^{k}}) = (x^{p} − y)f_{1,k},

for some f_{1,k} ∈ ℤ[x, y]. Thus,

\frac{f_{1}^{[p^{k}]} − f_{1}^{p^{k}}}{p^{k}} = (x^{p} − y)f_{2,k}.

Hence, with k = 1 we have that η_{1}(f_{1}) ≡ 0 (mod (x_{0}^{p} − y_{0})).

Inductively, we get

\frac{f^{[p^{k}]} − f^{p^{k}}}{p^{k}} − \frac{η_{1}(f)^{p^{k−1}}}{p^{k−1}} − ⋯ − \frac{η_{k−1}(f)^{p}}{p} ≡ 0 (mod (x^{p} − y)),

and hence η_{k}(f) ≡ 0 (mod (x_{0}^{p} − y_{0})). □

Lemma 5.2. If g_{1} ≡ 0 (mod (x_{0}^{p} − y_{0})), then η_{k}(g_{1}, g_{2}) ≡ 0 (mod (x_{0}^{p} − y_{0})).

Proof. Let g_{1}, g_{2} ∈ ℤ[x, y] be liftings of g_{1} and g_{2}. Since g_{1} ≡ 0 (mod (x_{0}^{p} − y_{0})), we can assume that g_{1} ≡ 0 (mod (x^{p} − y)).

We clearly have that

∑_{i=1}^{p^{k}−1} \frac{1}{p^{n}} \binom{p^{k}}{i}g_{1}^{i}g_{2}^{p^{k}−i} ≡ 0 (mod (x^{p} − y))

(in ℚ[x, y]). If k = 1, then we obtain η_{1}(g_{1}, g_{2}) = η_{1}(g_{1}, g_{2}) ≡ 0 (mod (x_{0}^{p} − y_{0})).

Inductively, we obtain that η_{k}(g_{1}, g_{2}) = η_{k}(g_{1}, g_{2}) ≡ 0 (mod (x_{0}^{p} − y_{0})), as it is the reduction modulo p of

∑_{i=1}^{p^{k}−1} \frac{1}{p^{n}} \binom{p^{k}}{i}g_{1}^{i}g_{2}^{p^{k}−i} − \frac{η_{1}(g_{1}, g_{2})^{p^{k−1}}}{p^{k−1}} − ⋯ − \frac{η_{k−1}(g_{1}, g_{2})^{p}}{p} ≡ 0 (mod (x^{p} − y)).

□

The following Lemma gives particular cases of Proposition 4.4 from [Fin11a]:

Lemma 5.3. Let

ℳ_{i,1} =^{def} η_{i}(X_{1}, …, X_{n})
ℳ_{i,2} =^{def} η_{i}(X_{n+1}, …, X_{n+m})
ℳ_{i,3} =^{def} η_{i}(X_{1} + ⋯ + X_{n}, X_{n+1} + ⋯ + X_{n+m}).

Then, we have

η_{1}(X_{1}, …, X_{n+m}) = ℳ_{1,1} + ℳ_{1,2} + ℳ_{1,3}.

and

η_{2}(X_{1}, …, X_{n+m}) = ℳ_{2,1} + ℳ_{2,2} + ℳ_{2,3} + η_{1}(ℳ_{1,1}, ℳ_{1,2}, ℳ_{1,3}).

In particular, if m = 1, we get

η_{1}(X_{1}, …, X_{n+1}) = ℳ_{1,1} + ℳ_{1,3}

and

η_{2}(X_{1}, …, X_{n+1}) = ℳ_{2,1} + ℳ_{2,3} + η_{1}(ℳ_{1,1}, ℳ_{1,3}).

We then have:

Proposition 5.4. Let p ≥ 3 and

ℋ_{1} =^{def} vec ((f_{x})^{p}x_{1} + (f_{y})^{p}y_{1} + ∑_{i,j} a_{i,j,1}x_{0}^{ip}y_{0}^{ip}) ,

h = f_{x_{0}}^{p^{2}}x_{2} + f_{y_{0}}^{p^{2}}y_{2} + (∑_{i,j} b_{i,j,1}x_{0}^{ip}y_{0}^{jp})^{p} x_{1}^{p} + (∑_{i,j} c_{i,j,1}x_{0}^{ip}y_{0}^{jp})^{p} y_{1}^{p}

+ (f_{x_{0}x_{0}}/2)^{p^{2}}x_{1}^{2p} + f_{x_{0}y_{0}}^{p^{2}}x_{1}^{p}y_{1}^{p} + (f_{y_{0}y_{0}}/2)^{p^{2}}y_{1}^{2p} + ∑_{i,j} a_{i,j,2}x_{0}^{ip^{2}}y_{0}^{jp^{2}},

and ℋ_{2} = vec (h). Then (still with the notation from Theorem 2.5),

η_{2}(𝒢_{1}) ≡ η_{2}(ℋ_{1}) (mod (x_{0}^{p} − y_{0}))

and

η_{1}(𝒢_{2}) ≡ η_{1}(ℋ_{2}) + η_{1}(h, η_{1}(ℋ_{1})) (mod (x_{0}^{p} − y_{0})).

Proof. Let

h_{1} = ∑_{t∈ℋ_{1}} t = (f_{x})^{p}x_{1} + f_{y}^{p}y_{1} + ∑_{i,j} a_{i,j,1}x_{0}^{ip}y_{0}^{ip}.

By Lemma 5.3, we have

η_{2}(𝒢_{1}) = η_{2}(ℋ_{1}) + η_{2} (h_{1}, η_{1}(f)) + η_{1} (η_{1}(ℋ_{1}), η_{1} (h_{1}, η_{1}(f))) .

By Lemma 5.1, η_{1}(f) ≡ 0 (mod (x_{0}^{p} − y_{0})) and thus, by Lemma 5.2, we get the first desired congruence.

Also, again by Lemma 5.3,

η_{1}(𝒢_{2}) = η_{1}(ℋ_{2}) + η_{1}(η_{1}(𝒢_{1}), η_{2}(f)) + η_{1} (h, η_{1}(𝒢_{1}) + η_{2}(f)) .  (5.1)

By Lemmas 5.1, 5.2, and 5.3, we get that

η_{1}(𝒢_{1}) = η_{1}(ℋ_{1}) + η_{1} (h_{1}, η_{1}(f)) ≡ η_{1}(ℋ_{1}) (mod (x_{0}^{p} − y_{0}))

and

η_{1}(η_{1}(𝒢_{1}), η_{2}(f)) ≡ 0 (mod (x_{0}^{p} − y_{0})).  (5.2)

Moreover, since \frac{1}{p} \binom{p}{i} ∈ ℤ for i ∈ \{1, …, p − 1\}, we get

η_{1}(h, η_{1}(𝒢_{1}) + η_{2}(f)) = ∑_{i=1}^{p−1} \frac{1}{p}\binom{p}{i}h^{i}(η_{1}(𝒢_{1}) + η_{2}(f))^{p−i}
≡ ∑_{i=1}^{p−1} \frac{1}{p}\binom{p}{i}h^{i}(η_{1}(ℋ_{1}))^{p−1}  (5.3)
= η_{1}(h, η_{1}(ℋ_{1})) (mod (x_{0}^{p} − y_{0})).

Thus, Eqs. (5.1), (5.2), and (5.3) give the desired formula. □

Finally, we can give the simplified formula for J_{3}.

Theorem 5.5. Let p ≥ 3. With the notation above, we have that if g_{3}(x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}) =^{def} η_{2}(ℋ_{1}) + η_{1}(ℋ_{2}) + η_{1}(h, η_{1}(ℋ_{1})), then g_{3}(X, J_{1}(X), J_{2}(X), X^{p}, J_{1}(X)^{p}, J_{2}(X)^{p}) is a p-th power and

J_{3}(X) = −\frac{1}{(X^{p^{2}} − X)^{p^{2}}} [∑_{i,j} a_{i,j,3}X^{ip^{2}+jp^{3}}

+ (∑_{i,j} b_{i,j,1}X^{ip^{2}+jp^{3}} ) J_{2} + (∑_{i,j} c_{i,j,1}X^{ip^{2}+jp^{3}} ) J_{2}^{p}

+ (∑_{i,j} b_{i,j,2}X^{ip^{2}+jp^{3}} ) J_{1}^{p} + (∑_{i,j} c_{i,j,2}X^{ip^{2}+jp^{3}} ) J_{1}^{p^{2}}

− (J_{1}^{p}J_{2}^{p} + J_{1}^{p^{2}}J_{2}) + (∑_{i,j} d_{i,j,1}X^{ip^{2}+jp^{3}} ) J_{1}^{2p}

+ (∑_{i,j} e_{i,j,1}X^{ip^{2}+jp^{3}} ) J_{1}^{p+p^{2}} + (∑_{i,j} f_{i,j,1}X^{ip^{2}+jp^{3}} ) J_{1}^{2p^{2}}

+ g_{3}(X, J_{1}, J_{2}, X^{p}, J_{1}^{p}, J_{2}^{p})^{1/p}] (5.4)

Proof. Applying the formula for the Greenberg transform from Theorem 2.5 to Φ_{p}(x, y) and evaluating at (x_{0}, x_{1}, x_{2}, x_{2}, y_{0}, y_{1}, y_{2}, y_{3}) = (X, J_{1}, J_{2}, J_{3}, X^{p}, J_{1}^{p}, J_{2}^{p}, J_{3}^{p}) should give zero by Eq. (3.4).

We then obtain the desired formula applying Proposition 5.4 to simplify the terms involving the η_{i}’s, solving for J_{3}^{p}, and taking p-th roots. Observe that since Φ_{p} has integral coefficients, we have that the coefficients of the sums above are in 𝔽_{p}, and so invariant under p-th powers.

Finally, the fact that g_{3} is a p-th power follows from the fact that J_{3}(X) ∈ 𝔽_{p}(X). [See [Fin10].] □

We have computed J_{3} before in [Fin11a] by using general methods to compute the Greenberg transform of a polynomial (by means of Theorem 2.5 above). The simplification given by Proposition 5.4 above gives significant improvements in memory usage. Table 5.1 below shows differences in times and memory usages with (“New”) and without (“Old”) using Proposition 5.4. The tests were performed using MAGMA (version 2.16-1) on a Dell Precision 690 server with two dual-core 64 bit 3.2 gigahertz Inter Xeon processors, 16 gigabytes of RAM, and 8 gigabytes of swap, running Fedora Core 11 (GNU/Linux) with kernel 2.6.30.

Old  New
Char. time (sec.) memory (MB) time (sec.) memory (MB)
7  7.300  40.97  5.089  33.22
11  421.090  1010.03  289.439  103.94
13  6542.590  4175.28  7496.840  356.16
17  −−  −−  45967.959  1982.28
19  −−  −−  267733.840  3650.62
23  −−  −−  1574171.979  13647.28

Table 5.1. Computations of J_{3}

6. Pole of J_{3} at 0

In this section we prove Theorem 1.4. We shall keep the notation from the previous sections and assume p ≥ 5.

Lemma 6.1. Let p ≥ 5. The function g_{3}(X, J_{1}, J_{2}, X^{p}, J_{1}^{p}, J_{2}^{p}) has a zero at X = 0. Thus, we have that, with notation of Theorem 5.5, J_{3}(X) has a pole at X = 0 of order p^{2} if, and only if, a_{0,0,3} ≠ 0.

Proof. By Theorem 1.2, we have that J_{1} and J_{2} have zeros at X = 0. In particular, we have that a_{0,0,k} = 0 for k ∈ {0, 1, 2}. (See Proposition 9.4 from [Fin11b].) Thus, every entry of the vectors ℋ_{1} and ℋ_{2} when evaluated at (X, J_{1}, J_{2}, X^{p}, J_{1}^{p}, J_{2}^{p}) are divisible by X. Clearly, also h evaluated at (X, J_{1}, J_{2}, X^{p}, J_{1}^{p}, J_{2}^{p}) is divisible by X. Thus, this must be the case also for all η_{1}(ℋ_{2}), η_{2}(ℋ_{1}), and η_{1}(h, η_{1}(ℋ_{1})), and hence for g_{3}. □

Thus, we need to show that if p ≡ 5 (mod 6), then a_{0,0,3} ≠ 0, i.e., v_{p}(a_{0,0}) = 3. It turns out that this is related to Conjecture 9.10 of [Fin11b], which is equivalent to the second item of Conjecture 9.3 in the same reference. More precisely, these conjectures state the following:

Theorem 6.2. Let p ≥ 5. We have that J_{2}(X) has a zero of order (exactly) sp, where s = (2 ⌊(p − 1)/6⌋ + 1), at X = 0, or equivalently, that v_{p}(a_{s+1,0}) = 2.

The equivalence of the two statements in the theorem above is proved in [Fin11b]. Before we can explicitly show the connection between Theorems 1.4 and 6.2, we need a little more notation.

Let Ψ_{p}(X, Y) be as in [Fin11b], i.e., Ψ_{p}(X, Y) is the polynomial proposed by Atkin such that Ψ_{p}(j^{1/3}, (j′)^{1/3}) = 0 if the elliptic curves associated to j and j′ have an isogeny of degree p. (See, for instance, [Elk98].) This polynomial also satisfies:

Φ_{p}(X^{3}, Y^{3}) = Ψ_{p}(X, Y)Ψ_{p}(X, ωY)Ψ_{p}(X, ω^{2}Y),  (6.1)

where ω =^{def} e^{2πi/3}. (This is Eq. (23) of [Elk98].) This clearly implies that

Φ_{p}(X^{3}, 0) = (Ψ_{p}(X, 0))^{3}  (6.2)

and, by Kronecker’s relation,

Ψ_{p}(X, 0) ≡ X^{p+1} (mod p)  (6.3)

(as Φ_{p}(X, 0) ≡ X^{p+1} (mod p)). Let’s write

Φ_{p}(X, 0) = ∑_{i=0}^{p+1} a_{i}X^{i}  and  Ψ_{p}(X, 0) = ∑_{i=0}^{p+1} b_{i}X^{i}.

(So, a_{i} = a_{i,0}.) Then, Eq. (6.3) implies that p | b_{i} for i ∈ {0, …, p} and p ∤ b_{p+1}. Thus, by Eq. (6.2), we have that v_{p}(a_{i,0}) ≥ 3 for all i < (p + 1)/3.

Proposition 6.3. Let p ≥ 5 and

i_{0} =^{def} \begin{cases}2, & if p ≡ 1 (mod 6), \\ 0, & if p ≡ 5 (mod 6). \end{cases}

Then, with the notation above, we have that the following are equivalent:

(1) v_{p}(a_{i_{0}}) = 3;

(2) v_{p}(b_{i_{0}}) = 1;

(3) v_{p}(a_{s+1}) = 2, where s =^{def} (2 ⌊(p − 1)/6⌋ + 1).

Proof. We have that if p ≡ 1 (mod 6), then a_{0} = a_{1} = 0. (See, for instance, Proposition 9.4 from [Fin11b].) Thus, by Eq. (6.2), we have also that b_{0} = b_{1} = 0.

Then, by Eq. (6.2) again, we have in general that a_{i_{0}} = b_{i_{0}}^{3}, and hence v_{p}(a_{i_{0}}) = 3 if, and only if, v_{p}(b_{i_{0}}) = 1.

Observe now that 3(s + 1) = p + 1 + 2i_{0}. So, the coefficient on X^{p+1+2i_{0}} of the left hand side of Eq. (6.2) is a_{s+1}, while on the left hand side is ∑_{i+j+k=p+1+2i_{0}} b_{i} b_{j} b_{k}. In other words,

a_{s+1} = 3b_{i_{0}}^{2}b_{p+1} + ∑_{\begin{matrix}i+j+k=p+1+2i_{0} \\ i,j,k≠p+1 \end{matrix}} b_{i} b_{j} b_{k}.

Thus, by our remarks above on the valuations of the b_{i}’s, we have that v_{p}(a_{s+1}) = 2 if, and only if, v_{p}(b_{i_{0}}) = 1. □

The next proposition then proves Theorems 1.4 and 6.2.

Proposition 6.4. With the previous notation (and still p ≥ 5), we have that v_{p}(b_{i_{0}}) = 1, and hence v_{p}(a_{s+1,0}) = 2 and v_{p}(a_{i_{0},0}) = 3.

Proof. As observed in [Fin10], we have that J_{1} is the reduction modulo p of

−\frac{Φ_{p}(X, X^{p})}{p(X^{p^{2}} − X)} = −\frac{1}{X^{p^{2}} − X} [∑_{i=0}^{p−1} \frac{a_{i,0}}{p} X^{i} + \frac{X^{p}}{p} ∑′ a_{i,j}X^{i+jp−p}] ,

where ∑′ is a sum over (i, j) such that either j ≠ 0 or i ≥ p. Moreover, by Theorem 3.2 from this reference, we have that J_{1} has a zero of order exactly r =^{def} ⌊(2p + 1)/3⌋ at X = 0. Therefore, v_{p}(a_{i,0}) ≥ 2, for i ∈ {0, …, r}, and v_{p}(a_{r+1,0}) = 1. Using the notation above, the coefficient of 3r + 3 = 2p + 2 + i_{0} in the left hand side of Eq. (6.2) is a_{r+1}, while on the right hand side is

∑_{i+j+k=2p+2+i_{0}} b_{i} b_{j} b_{k} = 3b_{i_{0}}b_{p+1}^{2} + ∑^{∗} b_{i} b_{j} b_{k},

where ∑^{∗} is the sum over (i, j, k) such that i + j + k = 2p + 2 + i_{0} and at most one of them is equal to (p + 1). Since p | b_{i} for i ∈ {0, …, p} and p ∤ b_{p+1}, as observed above, we have that p^{2} | ∑^{∗} b_{i} b_{j} b_{k}, and since v_{p}(a_{r+1}) = 1, we must have that v_{p}(b_{i_{0}}) = 1. □

7. Refinements on the Formulas for J_{2} and J_{3}

With the results from the previous sections, we are able to give more precise descriptions of J_{2} and J_{3}.

We need some notation. Let

S_{p}(X) =^{def} \frac{ss_{p}(X)}{X^{δ}(X − 1728)^{ϵ}} ,

where

ss_{p}(X) =^{def} ∏_{j supersing.}(X − j)

is the supersingular polynomial (as in, for instance, [Fin09]),

δ =^{def} \begin{cases}0, & if p ≡ 1 (mod 6); \\ 1, & if p ≡ 5 (mod 6); \end{cases}  and  ϵ =^{def} \begin{cases}0, & if p ≡ 1 (mod 4); \\ 1, & if p ≡ 3 (mod 4). \end{cases}

Hence, S_{p}(X) ∈ 𝔽_{p}[X], and S_{p}(0), S_{p}(1728) ≠ 0. (See, for instance, [Fin09].) Also, let

ι = \begin{cases}1, & if p ≠ 31; \\ 2, & if p = 31. \end{cases}

With this notation, [Fin10] (heavily relying on results from [KZ98] and [dS94]) gives a very precise description of J_{1}:

Theorem 7.1. Let p ≥ 5 and

J_{1}(X) = F_{1}(X)/G_{1}(X), with F_{1}, G_{1} ∈ 𝔽_{p}[X] and (F_{1}, G_{1}) = 1.

Then, (1) deg F_{1} − deg G_{1} = p − ι.

(2) F_{1} (and hence J_{1}) has a zero at 0 of order (exactly) r =^{def} ⌊(2p + 1)/3⌋.

(3) Assuming G_{1} is monic, we have G_{1}(X) = S_{p}(X).

(4) deg F_{1} = p − ι + S_{p}(X). (Note that deg S_{p}(X) = ⌊(p − 1)/4⌋ − ⌈(p − 1)/6⌉. See, for instance, [Fin09].)

Now we can give the corresponding result for J_{2}:

Theorem 7.2. Let p ≥ 5 and

J_{2}(X) = F_{2}(X)/G_{2}(X), with F_{2}, G_{2} ∈ 𝔽_{p}[X] and (F_{2}, G_{2}) = 1.

Then, (1) deg F_{2} − deg G_{2} = p^{2} − ι.

(2) F_{2} (and hence J_{2}) has a zero at 0 of order (exactly) sp, where s =^{def} (2 ⌊(p − 1)/6⌋+1).

(3) Assuming G_{2} is monic, we have G_{2}(X) = (X − 1728)^{ϵp}S_{p}(X)^{2p+1}.

(4) deg F_{2} = p^{2} − ι + (2p + 1) deg S_{p}(X) + pϵ.

Proof. Most of these properties are given by Theorem 9.6 from [Fin10]. The missing ones are given by Theorem 6.2 above. □

We will now deal with J_{3}, although we will not be able to be as precise as Theorems 7.1 and 7.2 above. To deal with g_{3} from Eq. (5.4), we need the following lemma:

Lemma 7.3. Let K be a field of rational functions over 𝕜 and v be a valuation on K such that v(a) = 0 for all a ∈ 𝕜^{×}. Let (α_{1}, …, α_{n}) be a vector with coefficients in K, and assume that min{v(α_{i}) : i ∈ {1, …, n}} = v_{0}. Then, v(η_{k}(α_{1}, …, α_{n})) ≥ p^{k}v_{0}.

Proof. By a simple induction, we see that η_{k}(X_{1}, …, X_{n}) is a homogeneous polynomial (with coefficients in ℤ) of degree p^{k}. The lemma then immediately follows. □

Theorem 7.4. Let p ≥ 5 and

J_{3}(X) = F_{3}(X)/G_{3}(X), with F_{3}, G_{3} ∈ 𝔽_{p}[X] and (F_{3}, G_{3}) = 1.

Then, (1) deg F_{3} − deg G_{3} = p^{3} − ι.

(2) F_{3} (and hence J_{3}) does not have a zero at X = 0 unless p ≡ 1 (mod 6), but in this case it has a zero at 0 of order (exactly) p^{2}.

(3) Assuming G_{3} is monic, we have G_{3}(X) = X^{δp^{2}}(X − 1728)^{ϵi}S_{p}(X)^{3p^{2}+2p} for some i ∈ {0, …, 2p^{2}}.

(4) With i as above, deg F_{3} = (p^{3} − ι) + δp^{2} + ϵi + (3p^{2} + 2p) deg(S_{p}).

Proof. This theorem follows directly from Eq. (5.4) and Lemma 7.3.

For the first item, we have that deg F_{3} − deg G_{3} is the order of pole at infinity of J_{3}. By Theorems 7.1 and 7.2, we have that J_{1} and J_{2} have poles of order p − ι and p^{2} − ι, respectively. Also, as seen [Fin10], we have that

deg (∑_{i,j} b_{i,j,1}X^{ip+jp^{2}} ) = p^{3} + p^{2} − p  and  deg (∑_{i,j} c_{i,j,1}X^{ip+jp^{2}} ) ≤ p^{3}.

In a similar way, we have

deg (∑_{i,j} b_{i,j,1}X^{ip^{2}+jp^{3}} ) = p^{4} + p^{3} − p^{2}, deg (∑_{i,j} c_{i,j,1}X^{ip^{2}+jp^{3}} ) ≤ p^{4},

deg (∑_{i,j} b_{i,j,2}X^{ip^{2}+jp^{3}} ) ≤ p^{4} + p^{3} − p^{2}, deg (∑_{i,j} d_{i,j,1}X^{ip^{2}+jp^{3}} ) ≤ p^{4} + p^{3} − 2p^{2},

deg (∑_{i,j} e_{i,j,1}X^{ip^{2}+jp^{3}} ) ≤ p^{4} − p^{2},  deg (∑_{i,j} f_{i,j,1}X^{ip^{2}+jp^{3}} ) ≤ p^{4} − p^{3}.

Now, applying Lemma 7.3 with v as the order of zero at infinity to the definition of g_{3} gives that g_{3}(X, …, J_{2}^{p})^{1/p} has a pole of order at most p^{4} + p^{3} − p. Comparing with the order of poles of the other terms we obtain the desired result.

The first observation of the second item is an immediate consequence of Theorem 1.4. Now, in case a_{0,0,3} = 0, i.e., p ≡ 1 (mod 6), we need to analyze the orders of zeros at 0 of the terms in Eq. (5.4).

So, suppose that p ≡ 1 (mod 6) and let v_{0} denote again the order of zero at X = 0. Then, from Theorems 7.1 and 7.2, we get v_{0}(J_{1}) = r and v_{0}(J_{2}) = sp, where r =^{def} ⌊(2p + 1)/3⌋ and s =^{def} (2 ⌊(p − 1)/6⌋ + 1). This is in fact a consequence of Proposition 9.4 from [Fin11b], which states that a_{0,0} = a_{1,0} = 0 (if p ≡ 1 (mod 6)), a_{i,0} ≡ 0 (mod p^{2}) for i ∈ {0, …, r}, and a_{i,0} ≡ 0 (mod p^{3}) for i ∈ {0, …, s}. Since in this case r ≥ 5 and s ≥ 3, this implies that a_{0,0,n} = a_{1,0,n} = 0 for all n, and that a_{2,0,2}, b_{0,0,1}, b_{1,0,1}, b_{0,0,2}, b_{1,0,2}, d_{0,0,1} are all equal to zero.

So, all terms inside the brackets of Eq. (5.4), except possibly g_{3}, has order at zero at least 2p^{2}. Among those, we see that only ∑ a_{i,j,3}X^{ip^{3}+jp^{3}} has order exactly 2p^{2} (by Proposition 6.4).

We also have v_{0}(g_{3}(X, J_{1}, J_{2}, X^{p}, J_{1}^{p}, J_{2}^{p})^{1/p}) > 2p^{2}, again by using its definition and Lemma 7.3, which finishes the proof of the second item.

For the third item, we need to find the order of poles at X = 0 for all j_{0} ∉ 𝕜^{ord}, as no ordinary value can give a pole. If 0 ∉ 𝕜^{ord}, we have seen J_{3} has a pole of order p^{2} at X = 0.

For j_{0} ∉ 𝕜^{ord} and j_{0} ≠ 0, 1728, i.e., for the zeros of S_{p}, we have that the term inside the brackets of Eq. (5.4) with highest order of pole is J_{1}^{p}J_{2}^{p}, having order of pole equal to 2p^{2} + 2p. This includes the pole of g_{3}(X, J_{1}, J_{2}, X^{p}, J_{1}^{p}, J_{2}^{p})^{1/p}, which, by Lemma 7.3, is at most 2p^{2}+p. Hence, J_{3} must have a pole of order 3p^{3}+2p at those values, as S_{p} | (X^{p^{2}} −X).

Finally, for j_{0} = 1728, if ϵ ≠ 0, then only J_{2} can introduce poles, and thus the pole inside the bracket of Eq. (5.4) is of order at most p^{2} (again using Lemma 7.3), giving the desired bound.

Finally, the last item is a trivial consequence of first and the third.

□

Acknowledgement. The author would like to thank A. V. Sutherland for suggesting the use of Ψ_{p}, which was essential to the proofs presented here. Also, the author would like to thank S. Mulay, for very valuable discussions, and the Centre de Recherches Mathématique, for providing an stimulating and productive environment in which the main ideas of this paper were born.

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Department of Mathematics, University of Tennessee, Knoxville, TN 37996

   E-mail address: finotti@math.utk.edu
   URL: http://www.math.utk.edu/~finotti/