LIFTING THE j-INVARIANT: QUESTIONS OF MAZUR AND TATE

LUÍS R. A. FINOTTI

Abstract. In this paper we analyze the j-invariant of the canonical lifting of an elliptic curve as a Witt vector. We show that its coordinates are rational functions on the jinvariant of the elliptic curve in characteristic p. In particular, we prove that the second coordinate is always regular at j = 0 and j = 1728, even when those correspond to supersingular values. A proof is given which yields a new proof for some results of Kaneko and Zagier about the modular polynomial.

1. Introduction

Let k be a perfect field of characteristic p > 0. We say that an elliptic curve E/k is ordinary if the p-torsion subgroup of E is isomorphic to ℤ/pℤ. Associated to an ordinary elliptic curve E, there exists a unique (up to isomorphisms) elliptic curve E over W(k), called the canonical lifting of E, and a map τ : E(k̄) → E(W(k̄)), i.e., a lift of points, called the elliptic Teichmüller lift, characterized by the following properties:

(1) the reduction modulo p of E is E;

(2) if σ denotes the Frobenius of both k and W(k), then the canonical lifting of E^{σ} (the elliptic curve obtained by applying σ to the coefficients of the equation that defines E) is E^{σ};

(3) τ is an injective group homomorphism and a section of the reduction modulo p;

(4) let ϕ : E → E^{σ} denote the p-th power Frobenius; then there exists a map ϕ : E → E^{σ}, such that the diagram

E(W(k̄)) ⟶^{ϕ} E^{σ}(W(k̄))
τ↑  ↑^{τ^{σ}}
E(k̄) ⟶^{ϕ} E^{σ}(k̄)

commutes. (In other words, there exists a lift of the Frobenius.) This concept of canonical lifting of elliptic curves was first introduced by Deuring in

[Deu41] and then generalized to Abelian varieties by Serre and Tate (see [LST64]). Apart from being of independent interest, this theory has been used in many interesting applications, such as counting rational points in ordinary elliptic curves, as in Satoh’s [Sat00], and counting torsion points of curves of genus g ≥ 2, as in Poonen’s [Poo01].

2000 Mathematics Subject Classification. Primary 11G07; Secondary 11F11.

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

The j-invariant of the canonical lifting E, say j, depends only on the j-invariant of E, say j_{0}. Hence, as a Witt vector, we have

j = (j_{0}, j_{1}, j_{2}, …),

and the j_{i}’s can be seen as functions of j_{0}, say j_{n} = J_{n}(j_{0}). B. Mazur asked J. Tate about the nature of these functions. Tate used some of the author’s previous computations of canonical liftings of general elliptic curves of small fixed characteristic to explicitly compute these functions in a few cases. More precisely, he found that:

j_{1} = J_{1}(j_{0}) = \begin{cases}3j_{0}^{3} + j_{0}^{4}, & if p = 5, \\ 3j_{0}^{5} + 5j_{0}^{6}, & if p = 7. \end{cases}

(Remember that p denotes the characteristic of the base field.) At this point, Tate asked the author for some more computations. In particular, he was surprised that these functions were polynomials (over 𝔽_{p}), as they are then defined for supersingular values of j_{0}, such as j_{0} = 0 for p = 5 and j_{0} = −1 for p = 7.

Hence, if all j_{n}’s would turn out to be polynomial functions on j_{0}, i.e., if J_{i} ∈ 𝔽_{p}[X], then we could use the same functions J_{n}’s to lift supersingular elliptic curves.

More generally, if the functions J_{i}’s are all regular at some supersingular j_{0}, we shall call the elliptic curve over W(k) associated to the j-invariant j = (J_{0}(j_{0}), J_{1}(j_{0}), …) a pseudocanonical liftings of the curve associated to j_{0}. If the functions J_{i}’s are regular for i ≤ n, then we shall call a pseudo-canonical liftings modulo p^{n+1} any curve over W(k) having its j-invariant j congruent to (J_{0}(j_{0}), J_{1}(j_{0}), …, J_{n}(j_{0}), …) modulo p^{n+1}. Tate asked when these pseudo-canonical liftings exist.

Some further computations have shown that in characteristic 5 we have:

J_{2}(X) = 3X^{5} + 2X^{10} + 2X^{13} + 4X^{14} + 4X^{15}+ 4X^{16} + X^{17} + 4X^{18} + X^{19} + X^{20} + 3X^{23} + X^{24}, while in characteristic 7, we have:

J_{2}(X) = (3X^{21} + 6X^{28} + 3X^{33} + 5X^{34} + 4X^{35} + 2X^{36} + 3X^{37}

+ 6X^{38} + 3X^{39} + 5X^{40} + 5X^{41} + 5X^{42} + 2X^{43} + 3X^{44} + 6X^{45} + 3X^{46} + 5X^{47} + 5X^{48} + 3X^{49} + 3X^{54} + 5X^{55})/(1 + X^{7}). (1.1)

So, although for p = 5 we have that J_{2}(X) ∈ 𝔽_{5}[X], for p = 7 we have that J_{2}(X) ∈ 𝔽_{7}(X) and has a pole of order 7 at −1. This is a bit more consistent with what was expected, as j_{0} = −1 is supersingular in characteristic 7.

So, one sees that indeed it was too much to expect that J_{n}(X) ∈ 𝔽_{p}[X], but we will show that J_{n}(X) ∈ 𝔽_{p}(X). This is a very superficial answer to Mazur’s question, but one can easily get more specific information on J_{1}. As we shall see, de Shalit’s [dS94] and Kaneko and Zagier’s [KZ98], which study of the (classical) modular polynomial modulo p^{2}, easily gives us the following theorems:

Theorem 1.1. With the notation above, we have that J_{1}(X) is regular at X = 0 and X = 1728, even if those values are supersingular, and that (0, J_{1}(0)) ≡ 0 (mod p^{2}) and (1728, J_{1}(1728)) ≡ 1728 (mod p^{2}). In other words, j_{0} = 0 and j_{0} = 1728 yield pseudocanonical liftings modulo p^{2} whenever those values are supersingular, and these have jinvariants j such that j ≡ 0 (mod p^{2}) and j ≡ 1728 (mod p^{2}) respectively.

Theorem 1.2. If j_{0} ≠ 0, 1728 is supersingular, then J_{1} has a pole at j_{0}. In other words, no value of j_{0} other than 0 and 1728 can yield pseudo-canonical liftings.

Before realizing how the above references give the desired answers (modulo p^{2}), the author came up with a different proof of Theorem 1.1. On the other hand, this proof can be used to obtain a few results from [KZ98], with a very different approach. While the orginal proofs use modular forms, and hence are more analytic in nature, the proof given here is more elementary and algebraic, relying almost exclusively on the existence of the canonical lifting.

2. Rationality of J_{i}(X)

The main goal of this section is to prove the following proposition:

Proposition 2.1. For any p ≥ 5, we have that J_{n}(X) ∈ 𝔽_{p}(X).

We shall assume henceforth that the characteristic, still denoted by p, is greater than or equal to 5. To compute the general form of J_{i}(X) for this fixed p, one can use the base field k_{0} =^{def} 𝔽_{p}(a_{0}, b_{0}), where a_{0} and b_{0} are algebraically independent transcendental elements, and compute the canonical lifting of

E/k_{0} : y_{0}^{2} = f(x_{0}) =^{def} x_{0}^{3} + a_{0}x_{0} + b_{0}.  (2.1)

The curve E/k_{0} is an ordinary elliptic curve, since its Hasse invariant, i.e., the coefficient of x_{0}^{p−1} in f(x_{0})^{(p−1)/2}, say A, is non-zero in k_{0}. More explicitly, we have

A = ∑_{i=r_{1}}^{r_{2}} \binom{r}{i}\binom{i}{3i − r}a_{0}^{3i−r}b_{0}^{r−2i},  (2.2)

where r =^{def} (p − 1)/2, r_{1} =^{def} ⌈(p − 1)/6⌉, and r_{2} =^{def} ⌊(p − 1)/4⌋.

So, let

E/W(k̄_{0}) : y^{2} = x^{3} + ax + b,  (2.3)

with a = (a_{0}, a_{1}, …) and b = (b_{0}, b_{1}, …), be the canonical lifting of E. We shall identify E with its Greenberg transform G(E), which is the infinite dimensional scheme over k_{0} defined by the equations that one obtains when comparing the coordinates (as Witt vectors) of Eq. (2.3) when x and y are replaced by Witt vectors of variables (x_{0}, x_{1}, …) and (y_{0}, y_{1}, …) respectively.

As seen in Section 1, associated with the canonical lifting E we have the elliptic Teichmüller. In [Fin02] it is shown that τ(x_{0}, y_{0}) = ((x_{0}, F_{1}, F_{2}, …), (y_{0}, y_{0}H_{1}, y_{0}H_{2}, …)), where F_{i}, H_{i} ∈ k_{0}[x_{0}]. (Remember that τ is a section of the reduction modulo p.)

Lemma 2.2. With the notation above, we have that a_{n}, b_{n} ∈ k_{0} and F_{n}, H_{n} ∈ k_{0}[x_{0}] for all n ≥ 0, i.e., the canonical lifting E is defined over W(k_{0}) and the elliptic Teichmüller lift τ : E → G(E) is defined over k_{0}.

Proof. Applying Lemma 7.4 in [Fin06], we see that at the (n + 1)-th equation of G(E) we have:

2y_{0}^{p^{n}}y_{n} + ⋯ = (3x_{0}^{2} + a_{0})^{p^{n}}x_{n} + a_{n}x_{0}^{p^{n}} + b_{n} + ⋯,

where no omitted term involves x_{n}, y_{n}, a_{n} or b_{n}. Pulling this back via τ^{∗} gives an equality modulo the ideal I =^{def} (y_{0}^{2} − (x_{0}^{3} − a_{0}x_{0} + b_{0})). But since τ^{∗}(y_{i}) = y_{0}H_{i}, with H_{i} ∈ k_{0}[x_{0}], we have that this pullback gives us

2f(x_{0})^{(p^{n}−1)/2}H_{n} + ⋯ ≡ (3x_{0}^{2} + a_{0})^{p^{n}}F_{n} + a_{n}x_{0}^{p^{n}} + b_{n} + ⋯ (mod I).

(Remember that f is the cubic from Eq. (2.1).) Since the expression above can be simplified so that it does not involve y_{0} (see the proof of Proposition 5.2 in [Fin04]), it must be an actual equality of polynomials in k_{0}[x_{0}], i.e.,

2f(x_{0})^{(p^{n}−1)/2}H_{n} + ⋯ = (3x_{0}^{2} + a_{0})^{p^{n}}F_{n} + a_{n}x_{0}^{p^{n}} + b_{n} + · · · .  (2.4)

We now proceed by induction, assuming that F_{i}, H_{i} ∈ k_{0}[x_{0}] and a_{i}, b_{i} ∈ k_{0} for all i < n. From [Fin04], we know that

\frac{dF_{n}}{dx_{0}} = A^{−(p^{n}−1)/(p−1)} f(x_{0})^{(p^{n}−1)/2} − x_{0}^{p^{n}−1} − ∑_{i=1}^{n−1} F_{i}^{(p^{n−i}−1)} \frac{dF_{i}}{dx_{0}} ,  (2.5)

and from [Fin02], we know that

deg_{x_{0}} F_{n} ≤ N(n) =^{def} \frac{(n + 2)p^{n} − np^{n−1}}{2}

and

deg_{x_{0}} H_{n} ≤ M(n) =^{def} \frac{(n + 3)p^{n} − np^{n−1} − 3}{2}.

Hence, if

F_{n} = ∑_{i=0}^{N(n)} c_{i} x_{0}^{i},

the terms c_{0}, c_{p}, c_{2p}, …, c_{N(n)} are unknown if n > 1, and for n = 1, the terms c_{0} and c_{p} are unknown. But all other c_{i}’s are clearly in k_{0}, by Eq. (2.5) and our induction hypothesis.

We shall also denote

H_{n} = ∑_{i=0}^{M(n)} d_{i} x_{0}^{i}.

So, by looking at Eq. (2.4) with the d_{i}’s, a_{n}, b_{n}, and the c_{i}’s singled out above as unknowns, we obtain a linear system on those variables with coefficients in k_{0}.

Also, by Proposition 5.1 of [Fin02], we have that

τ = ((F_{0}, …, F_{n}, …), (y_{0}H_{0}, …, y_{0}H_{n}, …))

is the elliptic Teichmüller lift if, and only if, τ^{∗}(x/y) is regular at the point at infinity. This implies that (n + 1)-th coordinate of its expansion, namely,

(\frac{1}{y_{0}} )^{p^{n}}F_{n} + (−\frac{x_{0}}{y_{0}^{2}} )^{p^{n}}y_{0}H_{n} + ⋯,

must be regular at the point at infinity. Note that the division of Witt vectors gives us that the denominators appearing above contain only powers of y_{0}. (In fact, it is not hard to prove that the largest power is y_{0}^{2p^{n}}.) To have this to be regular at the point at infinity, terms in the numerator (after collecting all terms by means of a common denominator) having order of poles higher than the order of the denominator must cancel out. Imposing such cancellations on the numerator gives us another linear system on the c_{i}’s and d_{i}’s with coefficient in k_{0}.

So, any solution of these two linear systems put together yields the canonical lifting (modulo isomorphism) and elliptic Teichmüller lift. (The first system guarantees that we have a well defined lift and the second guarantees that this lift is the elliptic Teichmüller lift.) We know that there is a solution over k̄_{0}, since the elliptic curve is ordinary. But, since the system is linear and over k_{0}, there is also a solution over k_{0}. □

Before we can prove Proposition 2.1, we also need the following basic lemma:

Lemma 2.3. Let k be a field and g, h ∈ k[a_{0}, b_{0}], with g and h non-zero and relatively prime. If

\frac{g(a_{0}X^{2}, b_{0}X^{3})}{h(a_{0}X^{2}, b_{0}X^{3})} = \frac{g(a_{0}, b_{0})}{h(a_{0}, b_{0})}

(in k(a_{0}, b_{0}, X)), then there is a positive integer s such that

g(a_{0}, b_{0}) = ∑_{ℓ=0}^{s} α_{ℓ} a_{0}^{3ℓ}b_{0}^{2(s−ℓ)}, h(a_{0}, b_{0}) = ∑_{ℓ=0}^{s} β_{ℓ} a_{0}^{3ℓ}b_{0}^{2(s−ℓ)},

with α_{ℓ}, β_{ℓ} ∈ k. (Hence, if a_{0} has weight 2 and b_{0} has weight 3, then g and h are homogeneous of degree 6s.)

Proof. If we write

g(a_{0}X^{2}, b_{0}X^{3}) = g_{0}(a_{0}, b_{0}) + g_{1}(a_{0}, b_{0})X + g_{2}(a_{0}, b_{0})X^{2} + ⋯
h(a_{0}X^{2}, b_{0}X^{3}) = h_{0}(a_{0}, b_{0}) + h_{1}(a_{0}, b_{0})X + h_{2}(a_{0}, b_{0})X^{2} + ⋯,

with g_{i}, h_{i} ∈ k[a_{0}, b_{0}], then we must have that g h_{i} = h g_{i} for all i. Since g and h are relatively prime, we must have that there is some d_{i} ∈ k[a_{0}, b_{0}] such that g_{i} = g d_{i} and h_{i} = h d_{i}. Thus, g(a_{0}X^{2}, b_{0}X^{3}) = g(a_{0}, b_{0}) d(a_{0}, b_{0}, X) and h(a_{0}X^{2}, b_{0}X^{3}) = h(a_{0}, b_{0}) d(a_{0}, b_{0}, X), where d = ∑_{i} d_{i}X^{i}.

Since g(a_{0}, b_{0}) and g(a_{0}X^{2}, b_{0}X^{3}) have the same number of monomials (in k[a_{0}, b_{0}, X]), we must have that d(a_{0}, b_{0}, X) has a single monomial, say d(a_{0}, b_{0}, X) = λ(a_{0}, b_{0})X^{r}. Since also g(a_{0}, b_{0}) and g(a_{0}X^{2}, b_{0}X^{3}) have the same degrees in a_{0} and b_{0}, we must have that λ ∈ k.

If

g(a_{0}, b_{0}) = ∑_{i,j} α_{i,j} a_{0}^{i}b_{0}^{j}

with α_{i,j} ∈ k, then

(λX^{r}) (∑_{i,j} α_{i,j} a_{0}^{i}b_{0}^{j}) = g(a_{0}X^{2}, b_{0}X^{3}) = ∑_{i,j} α_{i,j} a_{0}^{i}b_{0}^{j}X^{2i+3j}

and hence λ = 1 and 2i + 3j = r for all i and j such that α_{i,j} ≠ 0. Similarly we obtain the analogous result for h.

Now, observe that since g and h are relatively prime, if a_{0} | g, then a_{0} ∤ h. So, we must always have that either g or h is not divisible by a_{0}, and hence one of these has a term with i = 0, and hence 3 | r. The analogous argument for b_{0} gives us that 2 | r, and hence 6 | r.

So, for each pair (i, j) appearing in a term of either g or h, we must have that r ≡ 2i ≡ 0 (mod 3) and r ≡ 3j ≡ 0 (mod 2), and hence 3 | i and 2 | j. Thus, taking s = r/6 and remembering that 2i + 3j = 6s, one obtains the desired formulas for g and h. □

Now we are ready to prove Proposition 2.1.

Proof of Proposition 2.1. Since a_{n}, b_{n} ∈ k_{0} = 𝔽_{p}(a_{0}, b_{0}) for all n, i.e., a = (a_{0}, a_{1}, …), b = (b_{0}, b_{1}, …) ∈ W(𝔽_{p}(a_{0}, b_{0})), we have that

j = 1728\frac{4a^{3}}{4a^{3} + 27b^{2}} = (j_{0}, j_{1}, …)

with j_{n} ∈ 𝔽_{p}(a_{0}, b_{0}). So, let g, h ∈ 𝔽_{p}[X, Y] relatively prime polynomials such that j_{n} = g(a_{0}, b_{0})/h(a_{0}, b_{0}). Now, since j_{n} depends only on j_{0} = 1728 (4a_{0}^{3}/(4a_{0}^{3} +b_{0}^{2})), for any λ ∈ k̄_{0} we must have

j_{n} = g(λ^{2}a_{0}, λ^{3}b_{0})/h(λ^{2}a_{0}, λ^{3}b_{0}) = g(a_{0}, b_{0})/h(a_{0}, b_{0}).

By Lemma 2.3, there is a positive integer s such that if

g(a_{0}, b_{0}) = ∑_{ℓ=0}^{s} α_{ℓ} a_{0}^{3ℓ}b_{0}^{2(s−ℓ)}, h(a_{0}, b_{0}) = ∑_{ℓ=0}^{s} β_{ℓ} a_{0}^{3ℓ}b_{0}^{2(s−ℓ)},

Hence,

j_{n} = \frac{g(a_{0}, b_{0})}{h(a_{0}, b_{0})} = \frac{g(a_{0}, b_{0})/b_{0}^{2s}}{h(a_{0}, b_{0})/b_{0}^{2s}} = \frac{∑_{ℓ=0}^{s} α_{ℓ} (a_{0}^{3}/b_{0}^{2})^{ℓ}}{∑_{ℓ=0}^{s} β_{ℓ} (a_{0}^{3}/b_{0}^{2})^{ℓ}} .

Since

\frac{a_{0}^{3}}{b_{0}^{2}} = \frac{27j_{0}}{4(1728 − j_{0})},

we have that j_{n} is a rational function of j_{0}. □

3. Pseudo-Canonical Liftings Modulo p^{2}

We will need the following definition (from [Fin04]):

Definition 3.1. Let g(x_{0}, y_{0}) ∈ k[x_{0}, y_{0}] and g(x, y) ∈ W_{2}(k) be the lift of g defined by applying the Teichmüller lift to the coefficients of g, i.e., if λ is a coefficient of some monomial of g, then the corresponding monomial of g has coefficient (λ, 0). We define

ψ(g) =^{def} ψ(g) =^{def} reduction modulo p of \frac{g^{σ}(x^{p}, y^{p}) − g(x, y)^{p}}{p} ,

where σ denotes the Frobenius of Witt vectors.

Let Φ_{p}(X, Y) ∈ ℤ[X, Y] denote the classical modular polynomial. Then, by Theorem 3 of [LST64], we have that

Φ_{p}((J_{0}, J_{1}, …), (J_{0}^{p}, J_{1}^{p}, …)) = 0.  (3.1)

Since, by Kronecker’s congruence relation we have that

Φ_{p}(X, Y) ≡ (X − Y^{p})(X^{p} − Y) (mod p),

Lemma 8.1 of [Fin04] gives us that the second coordinate of Φ_{p}((X_{0}, X_{1}), (Y_{0}, Y_{1})) when expanded as Witt vectors is

(X_{0}^{p} − Y_{0})^{p}X_{1} + (Y_{0}^{p} − X_{0})^{p}Y_{1} + ψ(Φ_{p}) + ∑_{i,j} β_{i,j}X_{0}^{ip}Y_{0}^{pj},  (3.2)

where

Φ_{p}(X, Y) ≡ ∑_{i,j}(α_{i,j}, β_{i,j})X^{i}Y^{j} (mod p^{2}).

Also, Kronecker’s congruence relation tells us that

ψ(Φ_{p}(X, Y)) = ∑_{i=1}^{p−1} \frac{1}{p}\binom{p}{i}X_{0}^{i}(X_{0}^{p} − Y_{0})^{i}Y_{0}^{p−i}(Y_{0}^{p} − X_{0})^{p−i},

and hence,

ψ(Φ_{p}(X, Y))|_{(X_{0},Y_{0})=(j_{0},j_{0}^{p})} = 0.

Therefore, Eqs. (3.1) and (3.2) give

J_{1} = −\frac{∑_{i,j} β_{i,j}^{1/p}j_{0}^{i+pj}}{j_{0}^{p^{2}} − j_{0}} .

Now, since p(a_{0}, a_{1}, …) = (0, a_{0}^{0}, a_{1}^{p}, …), we have that this numerator is exactly the reduction modulo p of Φ_{p}(X, X^{p})/p evaluated at X = j_{0}. Kaneko and Zagier denoted

H_{p}(X) =^{def} \frac{Φ_{p}(X, X^{p})}{p},  and  φ_{p}(X) =^{def} \frac{H_{p}(X)}{X^{p^{2}} − X} ,

(this H_{p} should not to be confused with our H_{n} ∈ k[x_{0}] coming from the elliptic Teichmüller lift) and studied their properties in [KZ98]. (It should also be mentioned that Buium’s theory of differential modular forms also yield H_{p} modulo p explicitly. See [Bui00] and [Hur01].) Note then, that J_{1}(X) is simply the reduction modulo p of −φ_{p}(X).

Let’s denote

ss_{p}(X) =^{def} ∏_{j_{0} supersing.}(X − j_{0}),

i.e., ss_{p}(X) is the supersingular polynomial (in characteristic p). Observe that X = 0 is a root of ss_{p}(X) if, and only if, p ≡ 5 (mod 6) and X = 1728 is a root of ss_{p}(X) if, and only if, p ≡ 3 (mod 4). (See, for instance, [Fin08].)

In [KZ98] we have:

Theorem 3.2 (de Shalit, Kaneko, Zagier). Let H̄_{p}(X) and φ̄_{p}(X) denote the reductions modulo p of H_{p}(X) and φ_{p}(X) respectively.

(1) H̄_{p}(j_{0}) = 0 for j_{0} = 0, 1728 and all ordinary j_{0} ∈ 𝔽_{p^{2}}. (This was originally proved by de Shalit in [dS94], but was deduced again in [KZ98].)

(2) If j_{0} is supersingular, then

H̄_{p}(j_{0}) = −j_{0}^{r}(j_{0} − 1728)^{s}/ss′_{p}(j_{0})^{2},

where

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

and

s =^{def} \begin{cases}(p − 1)/2, & if p ≡ 1 (mod 4), \\ (p + 1)/2, & if p ≡ 3 (mod 4). \end{cases}

Hence, if j_{0} is supersingular and different from 0 and 1728, we have that φ̄_{p}(X) has a pole at X = j_{0}.

(3) We have that

φ̄′_{p}(X) = X^{p−1} − X^{r} \frac{(X − 1728)^{s}}{ss_{p}(X)^{2}} ,

with r and s defined as above. In particular, if

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

and

s′ =^{def} \begin{cases}(p − 1)/2, & if p ≡ 1 (mod 4), \\ (p − 3)/2, & if p ≡ 3 (mod 4), \end{cases}

then ord_{X=0} φ̄_{p}(X) = r′ and φ̄_{p}^{(t)}(1728) = (t − 1)! (−1728)^{(1−t)} for 1 ≤ t ≤ s′.

As observed in [KZ98], this theorem allows us to compute H_{p}(X), and hence also J_{1}(X), explicitly.

Observe that Theorem 3.2 allows us to prove Theorem 1.1 almost completely. Since X^{p^{2}} − X have simple zeros for X = 0 and X = 1728 and H̄_{p}(X) also has zeros at those values (by item 1), we have that J_{1}(X) is regular at those values. Moreover, item 3 gives us that (0, J_{1}(0)) = (0, 0). The only piece missing then is that (1728, J_{1}(1782)) ≡ 1728 (mod p^{2}). But note also that item 2 proves Theorem 1.2.

It’s also worth noticing that our observation that J_{1}(X) = −φ̄_{p}(X) can be used to prove the second part of item 1 immediately: since J_{1}(j_{0}) must be regular at all ordinary values, if j_{0} is ordinary and in 𝔽_{p^{2}}, then we must have that H̄_{p}(j_{0}) = 0.

The goal now is to give an alternative proof of Theorem 1.1 without using the modular polynomial. (This proof will also include the missing piece.) Then, the identification J_{1}(X) = −φ̄_{p}(X) will give alternative proofs of some of the facts in Theorem 3.2. More precisely, it will prove item 1 (which follows immediately from Theorem 1.1), ord_{X=0} φ̄_{p}(X) ≥ r′, and the formula for φ̄_{p}^{(t)}(1728) from item 3.

4. Valuations and Alternative Invariants

As in the previous section, let k_{0} =^{def} 𝔽_{p}(a_{0}, b_{0}), where p ≥ 5 and a_{0}, b_{0} are algebraically independent transcendental elements. To simplify our computation, it will be easier to avoid the usual j-invariant and use instead:

j̃_{0} =^{def} \frac{a_{0}^{3}}{b_{0}^{2}}  and ˜  j̃_{0} =^{def} \frac{b_{0}^{2}}{a_{0}^{3}} .

Those are certainly invariant under isomorphisms of elliptic curves as long as b_{0} ≠ 0 (i.e., j_{0} ≠ 1728) for the former and a_{0} ≠ 0 (i.e., j_{0} ≠ 0) for the latter. We will also use the invariants j̃ and ˜ j̃, defined in the same way, for curves over W(k_{0}).

Then, clearly,

j̃_{0} = \frac{27j_{0}}{4(1728 − j_{0})},  and ˜  j̃_{0} = \frac{4(1728 − j_{0})}{27j_{0}},  (4.1)

and

j_{0} = 1728\frac{4j̃_{0}}{4j̃_{0} + 27} = 1728\frac{4}{27˜j̃_{0} + 4},  (4.2)

and the analogous formulas hold for j̃ and ˜ j̃ in terms of j. Hence, since products and sums of Witt vectors are given by polynomial formulas, Proposition 2.1 tells us that there are J̃_{i}, ˜ J̃_{i} ∈ 𝔽_{p}(X) such that

j̃ = (J̃_{0}(j̃_{0}), J̃_{1}(j̃_{0}), J̃_{2}(j̃_{0}), …) and ˜ j̃ = ( ˜J̃_{0}(˜j̃_{0}), ˜ J̃_{1}(˜j̃_{0}), ˜ J̃_{2}(˜j̃_{0}), …),

where j̃ and ˜ j̃ are the invariants associated to the canonical lifting of the curve with invariants j̃_{0} and ˜ j̃_{0}. More precisely,

(J̃_{0}(X̃), J̃_{1}(X̃), …) = \frac{27(J_{0} (1728\frac{4X̃}{4X̃+27}) , J_{1} (1728\frac{4X̃}{4X̃+27}) , …)}{4 (1728 − (J_{0} (1728\frac{4X̃}{4X̃+27}) , J_{1} (1728\frac{4X̃}{4X̃+27}) , …))}  (4.3)

and

( ˜J̃_{0}( ˜X̃), ˜ J̃_{1}( ˜X̃), …) = \frac{4 (1728 − (J_{0} (1728\frac{4}{27 ˜X̃+4}) , J_{1} (1728\frac{4}{27 ˜X̃+4}) , …))}{27(J_{0} (1728\frac{4}{27 ˜X̃+4}) , J_{1} (1728\frac{4}{27 ˜X̃+4}) , …)}.  (4.4)

We then have the following proposition:

Proposition 4.1. The rational function J_{i}(X) is regular at X = 0 (resp., at X = 1728) for all i ≤ n if, and only if, J̃_{i}(X̃) (resp., ˜ J̃_{i}( ˜X̃)) is regular at X̃ = 0 (resp., ˜ X̃ = 0) for all i ≤ n. Moreover, we have that j(j_{0}) ≡ 0 (mod p^{n}) (resp., at j(j_{0}) ≡ 1728 (mod p^{n})) if, and only if, j̃(j̃_{0}) ≡ 0 (mod p^{n}) (resp., ˜ j̃(˜j̃_{0}) ≡ 0 (mod p^{n})).

Proof. We first observe that for any ring R, an element c = (c_{0}, c_{1}, …) ∈ W(R) is invertible if, and only if, c_{0} ∈ R^{×}.

For n = 0, the question is trivial, as J_{0}(X) = X, J̃_{0}(X̃) = X̃, and J̃_{0}( ˜X̃) = ˜ X̃.

Assume then that J_{i}(X) is regular at X = 0 (resp., at X = 1728) for all i ≤ n. We will work then in W_{n+1}(k_{0}), i.e., with Witt vectors of length (n+1). Since J_{0}(X) = X, we have that (1728 − (J_{0}(0), …, J_{n}(0))) ≠ 0 (resp., 27 (J_{0}(1728), …, J_{n}(1728)) ≠ 0), as 0 ≢ 1728 (mod p). Hence the denominators of Eq. (4.3) (resp., Eq. (4.4)) when evaluated at X̃ = 0 (resp., ˜ X̃ = 0) are invertible in W(𝔽_{p}) (and in W_{n+1}(𝔽_{p})), by our remark in the beginning of the proof.

Hence, formulas (4.3) and (4.4) show that if the J_{i}’s are regular at X = 0 (resp., X = 1728) for i ≤ n, then J̃_{i}’s are regular at X̃ = 0 (resp., ˜ X̃ = 0) for i ≤ n.

The converse is similar, using

(J_{0}(X), J_{1}(X), …) = 1728\frac{4 (J̃_{0} (\frac{27X}{4(1728−X)}) , J̃_{1} (\frac{27X}{4(1728−X)}) , …)}{4 (J̃_{0} (\frac{27X}{4(1728−X)}) , J̃_{1} (\frac{27X}{4(1728−X)}) , …) + 27}  (4.5)

= 1728\frac{4}{27 ( ˜J̃_{0} (\frac{4(1728−X)}{27X}) , ˜ J̃_{1} (\frac{4(1728−X)}{27X}) , …) + 4}  (4.6)

instead of Eqs. (4.3) and (4.4).

The final statement follows immediately from the transition formulas. □

Proposition 4.1 then allows us to use the invariants j̃ and ˜ j̃ to prove Theorem 1.1, i.e., it suffices to prove that J̃_{1}(0) = ˜ J̃_{1}(0) = 0.

To further simplify our computations note that if a = (a_{0}, a_{1}) and b = (b_{0}, b_{1}) are the coefficients of the canonical lifting of the curve with coefficients a_{0} and b_{0}, then we can assume that either a_{1} = 0, if a_{0} ≠ 0, or b_{1} = 0, if b_{0} ≠ 0. Indeed, it suffices to take the curve with coefficients λ^{2}a and λ^{3}b instead, where λ =^{def} (1, −a_{1}/(2a_{0}^{p})) for the former, and λ =^{def} (1, −b_{1}/(3b_{0}^{p})) for the latter.

So, over k_{0} (where a_{0}, b_{0} ≠ 0), we can assume that the canonical lifting of the curve given by Eq. (2.1) is such that a = (a_{0}, a_{1}) and b = (b_{0}, 0). In this case, we have

j̃ = \frac{(a_{0}, a_{1})^{3}}{(b_{0}, 0)^{2}} = (\frac{a_{0}^{3}}{b_{0}^{2}} , \frac{3a_{0}^{2p}a_{1}}{b_{0}^{2p}} ) .  (4.7)

Assuming that the canonical lifting has a = (a_{0}, 0) and b = (b_{0}, b_{1}), we get that

˜j̃ = \frac{(b_{0}, b_{1})^{2}}{(a_{0}, 0)^{3}} = (\frac{b_{0}^{2}}{a_{0}^{3}} , \frac{2b_{0}^{p}b_{1}}{a_{0}^{3p}} ) .  (4.8)

Let now v =^{def} ord_{a_{0}=0} and w =^{def} ord_{b_{0}=0} denote the orders of zeros of elements of k_{0} at a_{0} = 0 (seeing k_{0} as (𝔽_{p}(b_{0}))(a_{0})) and b_{0} = 0 (seeing k_{0} as (𝔽_{p}(a_{0}))(b_{0})) respectively. We have:

Proposition 4.2. Let v(a_{1}) = r in the case of b_{1} = 0 and w(b_{1}) = s in the case a_{1} = 0. Then

ord_{j̃=0}(J̃_{1}) = \frac{2p + r}{3} and ord_{˜j̃=0}( ˜J̃_{1}) = \frac{p + s}{2}.

Hence, by Proposition 4.1, to prove Theorem 1.1, it suffices to prove that w(a_{1}) > −2p in Eq. (4.7) and that v(b_{1}) > −p in Eq. (4.8).

Proof. Since v(a_{1}) = r, we can write a_{1} = a_{0}^{r}a′_{1}, with v(a′_{1}) = 0. Then,

J̃_{1}(j̃) = \frac{3a_{0}^{2p}a_{1}}{b_{0}^{2p}} = 3j̃^{p}a_{0}^{r−p}a′_{1}.

Then, clearly, a_{0}^{r−p}a′_{1} is a rational function on j̃. Thus,

ord_{j̃=0}(a_{0}^{r−p}a′_{1}) = \frac{1}{3}v(a_{0}^{r−p}a′_{1}) = \frac{r − p}{3}.

So, ord_{j̃=0}(J̃_{1}) = p + (r − p)/3 = (2p + r)/3 > 0 if, and only if, r > −2p.

In the same way, if w(b_{1}) = s, and b_{1} = b_{0}^{s}b′_{1} with w(b′_{1}) = 0, then

˜J̃_{1}(˜j̃) = \frac{2b_{0}^{p}b_{1}}{a_{0}^{3p}} = 2˜j̃^{p}b_{0}^{s−p}b′_{1}.

Then, clearly, b_{0}^{s−p}b′_{1} is a rational function on ˜ j̃. Thus,

ord_{˜j̃=0}(b_{0}^{s−p}b′_{1}) = \frac{1}{2}w(b_{0}^{s−p}b′_{1}) = \frac{s − p}{2}.

So, ord_{˜j̃=0}( ˜J̃_{1}) = p + (s − p)/2 = (p + s)/2 > 0 if, and only if, s > −p. □

We shall prove a stronger result:

Proposition 4.3. Let A denote the Hasse invariant of the curve (2.1), v_{0} =^{def} v(A), and

w_{0} =^{def} w(A). Then, v(a_{1}) ≥ −v_{0} and w(b_{1}) ≥ −w_{0}.

Most of the remaining of this paper will be dedicated to the proof of Proposition 4.3. Note that

v_{0} = \begin{cases}0, & if p ≡ 1 (mod 6), \\ 1, & if p ≡ 5 (mod 6), \end{cases} and w_{0} = \begin{cases}0, & if p ≡ 1 (mod 4), \\ 1, & if p ≡ 3 (mod 4), \end{cases}  (4.9)

as if we let r =^{def} (p − 1)/2, r_{1} =^{def} ⌈r/3⌉, and r_{2} =^{def} ⌊r/2⌋, then, an easy computation (see [Fin08]) shows that

A = ∑_{i=r_{1}}^{r_{2}} \binom{r}{i}\binom{i}{3i − r}a_{0}^{3i−r}b_{0}^{r−2i}.

5. Proof of Proposition 4.3

Let a = (a_{0}, a_{1}), b = (b_{0}, b_{1}) be the coefficients of the canonical lifting E of the curve E, given by Eq. (2.1), and τ = ((x_{0}, F_{1}), (y_{0}, y_{0}H_{1})), with F_{1}, H_{1} ∈ k_{0}[x_{0}], be the elliptic Teichmüller lift (modulo p^{2}).

Then, Lemma 8.1 of [Fin04] gives us that the pullback of second coordinate of the equation of E by τ is given by

2(x_{0}^{3} + a_{0}x_{0} + b_{0})^{(p+1)/2}H_{1} = (3x_{0}^{2} + a_{0})^{p}F_{1} + a_{1}x_{0}^{p} + b_{1} + ψ(f),  (5.1)

where ψ is the function given in Definition 3.1. Since by Eq. (2.5) we have

F′_{1} = A^{−1}(x_{0}^{3} + a_{0}x_{0} + b_{0})^{(p−1)/2} − x_{0}^{p−1},

taking derivatives with respect to x_{0} in Eq. (5.1) and dividing by (x_{0}^{3} + a_{0}x_{0} + b_{0})^{(p−1)/2}, we obtain

(3x_{0}^{2} + a_{0})H_{1} + 2(x_{0}^{3} + a_{0}x_{0} + b_{0})H′_{1} = A^{−1}(3x_{0}^{2p} + a_{0}^{p}) − (x_{0}^{3} + a_{0}x_{0} + b_{0})^{(p−1)/2}(3x_{0}^{2} + a_{0}). (5.2)

We will analyze these two equations to estimate the wanted valuations.

Let’s denote

F_{1} = ∑_{i=0}^{(3p−1)/2}c_{i}x_{0}^{i}  and  H_{1} = ∑_{i=0}^{2p−2} d_{i}x_{0}^{i}.

(Note that since we know F′_{1}, we know all c_{i}’s except c_{0} and c_{p}.) Hence, looking at the terms of degree (r + 2) in Eq. (5.2), we obtain

(2r + 3)d_{r} + (2r + 5)a_{0}d_{r+2} + (2r + 6)b_{0}d_{r+3} = ⋯,  (5.3)

where the omitted terms in the right hand side do not contain d_{i}’s.

From now on, let’s fix the notation v_{0} =^{def} v(A), w_{0} =^{def} w(A).

Proposition 5.1. Let (3p − 1)/2 ≤ r ≤ 2p − 2.

(1) Case b_{1} = 0: We have that v(d_{r}) > −v_{0} if 2p − r ≢ 2 (mod 3) and v(d_{r}) = −v_{0} otherwise.

(2) Case a_{1} = 0: We have that w(d_{r}) > −w_{0} if r is odd and w(d_{r}) = −w_{0} otherwise.

Proof. We start by observing that the right hand side of Eq. (5.2) has a term of degree 2p, namely 3A^{−1}x_{0}^{2p}, but no terms of degrees between 2p and (3p + 3)/2. So, the term of degree 2p gives us that d_{2p−2} = −3A^{−1}. For (3p − 1)/2 < r < 2p − 2, the omitted terms on Eq. (5.3) are then zero. We can then determine the valuations of d_{i}’s inductively.

One can easily check the cases r = 2p − 2, 2p − 3, 2p − 4 directly, using Eq. (5.2). So, we assume the statement to hold for all i > r ≥ 2p − 5.

If b_{1} = 0 and 2p − r ≢ 2 (mod 3), then, by the induction hypothesis, v((2r + 5)a_{0}d_{r+2}) and v((2r + 6)b_{0}d_{r+3}) are greater than −v_{0}. Hence, by Eq. (5.3), so is v(d_{r}).

If b_{1} = 0 and 2p − r ≡ 2 (mod 3), then, by the induction hypothesis, v((2r + 5)a_{0}d_{r+2}) > −v_{0} and v((2r + 6)b_{0}d_{r+3}) = −v_{0}. Then, Eq. (5.3) gives us v(d_{r}) = −v_{0}.

In the same way, if a_{1} = 0 and r is odd, then, by the induction hypothesis, w((2r + 5)a_{0}d_{r+2}) and w((2r + 6)b_{0}d_{r+3}) are greater than −w_{0}. Hence, by Eq. (5.3), so is w(d_{r}).

If a_{1} = 0 and r is even, then, by the induction hypothesis, w((2r + 5)a_{0}d_{r+2}) = −w_{0} and w((2r + 6)b_{0}d_{r+3}) > −w_{0}. Then, Eq. (5.3) gives us w(d_{r}) = −w_{0}. □

Observe that the proof actually gives us an algorithm to find d_{r}’s for (3p − 1)/2 ≤ r ≤ 2p−2. But, for r = (3p−3)/2 we encounter a difficulty, as (2r +3) = 0, and hence Eq. (5.3) does not give us d_{(3p−3)/2}. So, let’s denote v_{1} =^{def} v(d_{(3p−3)/2}), w_{1} =^{def} w(d_{(3p−3)/2}), and let δ_{1} be the initial coefficient of d_{(3p−3)/2} with respect to w, i.e.,

d_{(3p−3)/2} = δ_{1}b_{0}^{w_{1}} + ⋯,

where all omitted terms have valuation greater than w_{1}.

Proposition 5.2. If b_{1} = 0 (resp., a_{1} = 0) and v_{1} ≥ −v_{0} (resp., w_{1} ≥ −w_{0}), then v(d_{r}) ≥ −v_{0} (resp., w(d_{r}) ≥ −w_{0}) for (p − 1)/2 ≤ r ≤ (3p − 3)/2.

Proof. Clearly only the terms of degree 0 and 2p on the right hand side of Eq. (5.2) have possibly negative valuations for v (resp., w), and these have valuation exactly equal to −v_{0} (resp., −w_{0}). Hence all omitted terms on the right hand side of Eq. (5.3) have valuations greater than or equal to −v_{0} (resp., −w_{0}). Thus, using Proposition 5.1, one then can easily see that if v_{1} ≥ −v_{0} (resp., w_{1} ≥ −w_{0}), then we can continue using Eq. (5.3), as in the proof of Proposition 5.1, to guarantee that v(d_{r}) ≥ −v_{0} (resp., w(d_{r}) ≥ −w_{0}) for (p − 1)/2 ≤ r ≤ (3p − 3)/2. □

We want to show that indeed v_{1} ≥ −v_{0} and w_{1} ≥ −w_{0}. We will proceed by assuming otherwise and deriving a contradiction. We first need the following proposition:

Proposition 5.3. Let (p − 1)/2 ≤ r ≤ (3p − 3)/2.

(1) Case b_{1} = 0: Assume that v_{1} < −v_{0}. We have that v(d_{r}) > v_{1} if r ≢ 0 (mod 3) and v(d_{r}) = v_{1} otherwise.

(2) Case a_{1} = 0: Assume that w_{1} < −w_{0}. We have that w(d_{r}) > w_{1} if r ≢ (3p − 3)/2 (mod 2) and w(d_{r}) = w_{1} otherwise. In the latter case, the initial coefficient of d_{r} is

( ∏_{i=0}^{(3p−7−2r)/4} \frac{2i + 1}{2i + 2}) (−a_{0})^{(3p−3−2r)/4}δ_{1}.

Proof. We first observe that, by Proposition 5.1, we have that v(d_{r}) ≥ −v_{0} and w(d_{r}) ≥ −w_{0} for r > (3p − 3)/2. Also, as observed above, for (p − 1)/2 ≤ r ≤ (3p − 3)/2, the omitted terms in the right hand side of Eq. (5.3) have all valuations greater than or equal to 0. (Note that if v_{1} < −v_{0}, then v_{1} < 0, and if w_{1} < −w_{0}, then w_{1} < 0.) We again determine the valuations inductively using Eq. (5.3).

One can easily check the cases r = (3p − 3)/2, (3p − 5)/2, (3p − 7)/2 directly. So, we assume the statement to hold for i > r ≥ (3p − 9)/2.

If b_{1} = 0 and r ≢ 0 (mod 3), then, by the induction hypothesis, v((2r + 5)a_{0}d_{r+2}) and v((2r + 6)b_{0}d_{r+3}) are greater than v_{1}. Since the omitted terms also have valuation greater than v_{1} (as they have positive valuation), Eq. (5.3) gives us that v(d_{r}) > v_{1}.

If b_{1} = 0 and r ≡ 0 (mod 3), then, by the induction hypothesis, v((2r + 5)a_{0}d_{r+2}) > v_{1} and v((2r + 6)b_{0}d_{r+3}) = v_{1}. Then, since the omitted terms have valuation greater than v_{1}, Eq. (5.3) gives us v(d_{r}) = v_{1}.

In the same way, if a_{1} = 0 and r ≢ (3p−3)/2 (mod 2), then, by the induction hypothesis, w((2r + 5)a_{0}d_{r+2}) and w((2r + 6)b_{0}d_{r+3}) are greater than w_{1}, and so w(d_{r}) > w_{1}.

If a_{1} = 0 and r ≡ (3p − 3)/2 (mod 2), then, by the induction hypothesis, w((2r + 5)a_{0}d_{r+2}) = w_{1} and w((2r + 6)b_{0}d_{r+3}) > w_{1}. Then, Eq. (5.3) gives us

d_{r} = −\frac{2r + 5}{2r + 3} a_{0} [(  ∏_{i=0}^{(3p−7−2r)/4−1} \frac{2i + 1}{2i + 2}) (−a_{0})^{(3p−3−2r)/4−1}δ_{1} b_{0}^{w_{1}} + ⋯]

= ( ∏_{i=0}^{(3p−7−2r)/4} \frac{2i + 1}{2i + 2}) (−a_{0})^{(3p−3−2r)/4}δ_{1} b_{0}^{w_{1}} + ⋯

□

For r = (p − 3)/2, formula (5.3) cannot determine d_{r}. So, we yet again, need to deal with a term of unknown valuation. Similarly as before, we define v_{2} =^{def} v(d_{(p−3)/2}), w_{2} =^{def} w(d_{(p−3)/2}).

Proposition 5.4. If b_{1} = 0 (resp., a_{1} = 0) and v_{1}, v_{2} ≥ −v_{0} (resp., w_{1}, w_{2} ≥ −w_{0}), then v(d_{r}) ≥ −v_{0} (resp., w(d_{r}) ≥ −w_{0}) for all r.

Proof. The proof follows the same idea as the proof of Proposition 5.2. □

We still want to show that the assumption that neither v_{1} < −v_{0} nor w_{1} < −w_{0} can occur. But now we have to deal with another unknown valuation. We will show that also v_{2} ≥ −v_{0} and w_{2} ≥ −w_{0}. We need a new proposition to derive a contradiction in the many possible cases.

Proposition 5.5. Let 0 ≤ r ≤ (p − 3)/2.

(1) Case b_{1} = 0:

(a) Subcase v_{1} < min{−v_{0}, v_{2}}: If p ≡ 1 (mod 6), then v(d_{r}) > v_{1} if r ≢ (p − 1)/2 (mod 3), and v(d_{r}) = v_{1} otherwise. If p ≡ 5 (mod 6), then v(d_{r}) > v_{1} if r ≢ (p + 1)/2 (mod 3), and v(d_{r}) = v_{1} otherwise.

(b) Subcase v_{2} < min{−v_{0}, v_{1}}: We have that v(d_{r}) > v_{2} if r ≢ (p − 3)/2 (mod 3), and v(d_{r}) = v_{2} otherwise.

(c) Subcase v_{1} = v_{2} < −v_{0}: If p ≡ 1 (mod 6), then v(d_{r}) > v_{1} if r ≡ (p + 1)/2 (mod 3), and v(d_{r}) = v_{1} otherwise. If p ≡ 5 (mod 6), then v(d_{r}) > v_{1} if r ≡ (p − 1)/2 (mod 3), and v(d_{r}) = v_{1} otherwise.

(2) Case a_{1} = 0:

(a) Subcase w_{1} < min{−w_{0}, w_{2}}: We have that w(d_{r}) > w_{1}.

(b) Subcase w_{2} ≤ min{−w_{0} − 1, w_{1}}: We have that w(d_{r}) > w_{2} if r ≡ (p − 1)/2 (mod 2), and w(d_{r}) = w_{2} otherwise.

Proof. We again proceed by induction, and use Proposition 5.3 to check the cases when r = (p − 3)/2, (p − 5)/2, (p − 7)/2 directly.

So, now assume b_{1} = 0 and v_{1} ≤ min{−v_{0}−1, v_{2}}. If p ≡ 1 (mod 6), then for r ≡ (p−1)/2 (mod 3), the induction hypothesis and Eq. (5.3) give us that v(d_{r}) = v_{1}.

In the same way, if p ≡ 5 (mod 6), then for r ≡ (p + 1)/2 (mod 3) the induction hypothesis and Eq. (5.3) give us that v(d_{r}) = v_{1}.

If now we assume b_{1} = 0 and v_{2} ≤ min{−v_{0} − 1, v_{1}}, then for r ≡ (p − 3)/2 (mod 3) the induction hypothesis and Eq. (5.3) give us that v(d_{r}) = v_{2}.

The case when a_{1} = 0 and w_{1} < min{−w_{0}, w_{2}} is straight forward, as the omitted terms in the right hand side of Eq. (5.3) have non-negative valuations.

Finally, the case when a_{1} = 0 and w_{2} ≤ min{−w_{0} −1, w_{1}}, if r ≡ (p−1)/2 (mod 2), then the induction hypothesis gives us that w((2r + 5)a_{0}d_{r+2}) > w_{2} and w((2r + 6)b_{0}d_{r+3}) = w_{2} + 1. Thus, Eq. (5.3) guarantees that w(d_{r}) > w_{2}. If r ≡ (p − 3)/2 (mod 2), then the induction hypothesis and Eq. (5.3) give us that w(d_{r}) = w_{2}. □

To derive a contradiction from the assumptions that either v_{1} < −v_{0}, w_{1} < −w_{0}, v_{2} < −v_{0}, or w_{2} < −w_{0}, we need some extra equations.

Taking the terms of degrees 1 and 0 of Eq. (5.2), we have

3a_{0}d_{1} + 4b_{0}d_{2} = − (\frac{p − 1}{2}) a_{0}^{2}b_{0}^{(p−3)/2},  (5.4)
a_{0}d_{0} + 2b_{0}d_{1} = A^{−1}a_{0}^{p} − a_{0}b_{0}^{(p−1)/2}.  (5.5)

Now, let

(x_{0}^{3} + a_{0}x_{0} + b_{0})^{(p+1)/2} = ∑_{i=0}^{(3p+3)/2} α_{i} x_{0}^{i}.

Then, taking the terms of degrees 0, p, 2p, and 3p from Eq. (5.1), we have

2b_{0}^{(p+1)/2}d_{0} = a_{0}^{p}c_{0} + b_{1},  (5.6)
2 ∑_{i=0}^{p} α_{i} d_{p−i} = a_{0}^{p}c_{p} + a_{1} + terms from ψ(f),  (5.7)
d_{(p−3)/2} + ∑_{i=2}^{(3p−1)/2} α_{i} d_{2p−i} = \frac{3}{2}c_{0} + terms from ψ(f),  (5.8)
d_{(3p−3)/2} + ∑_{i=p+2}^{(3p−1)/2} α_{i} d_{3p−i} = \frac{3}{2}c_{p}.  (5.9)

Assume first that b_{1} = 0 and v_{2} ≤ min{−v_{0} − 1, v_{1}}. If (p − 3)/2 ≡ 2 (mod 3), then Proposition 5.5 tells us that v(d_{2}) = v_{2} and v(d_{1}) > v_{2}. On the other hand, by Eq. (5.4), we have that

v(d_{2}) ≥ min{v(d_{1}) + 1, 2} > v_{2},

a contradiction.

If (p − 3)/2 ≡ 1 (mod 3), then Proposition 5.5 tells us that v(d_{1}) = v_{2} and v(d_{0}) > v_{2}. On the other hand, by Eq. (5.5), we have that

v(d_{1}) ≥ min{v(d_{0}) + 1, 1, p − v_{0}} > v_{2},

a contradiction.

So, assume now that b_{1} = 0 and v_{1} < min{−v_{0}, v_{2}}. Note that, by Proposition 5.5, independently of the congruence class of p modulo 6, we have that v(d_{0}) = v_{1}. Eq. (5.6) then gives us that v(d_{0}) = v(c_{0}) + p, i.e., v(c_{0}) = v_{1} − p. But this would imply that while Propositions 5.3 and 5.5 tells us that the left hand side of Eq. (5.8) has valuation greater than or equal to v_{1}, the right hand side has valuation strictly smaller than that, a contradiction.

So, assume now that a_{1} = 0 and w_{2} ≤ min{−w_{0} − 1, w_{1}}. If p ≡ 1 (mod 4), then by Proposition 5.5 we have that w(d_{2}) > w_{2} and w(d_{1}) = w_{2}. Hence, the valuation of the left hand side of Eq. (5.4) is w_{2}, while the valuation of its right hand side is (p − 3)/2, a contradiction.

If p ≡ 3 (mod 4), then by Proposition 5.5 we have that w(d_{1}) > w_{2} and w(d_{0}) = w_{2}. Hence, the valuation of the left hand side of Eq. (5.5) is w_{2}, while the valuation of its right hand side is −w_{0}, a contradiction.

Finally, assume that a_{1} = 0 and w_{1} < min{−w_{0}, w_{2}}. Then, by Proposition 5.1, we have that w(d_{3p−i}) ≥ −w_{0} for p + 2 ≤ i ≤ (3p − 1)/2. Hence, Eq. (5.9) tells us that w(c_{p}) = w_{1}, and its initial coefficient is 2/3 δ_{1}. Also, Eq. (5.7) gives us that

c_{p} = \frac{2}{a_{0}^{p}} [( ∑_{i=0}^{(p−1)/2} α_{i} d_{p−i}) + α_{(p+1)/2} d_{(p−1)/2} + ( ∑_{i=(p+3)/2}^{p} α_{i} d_{p−i})] .  (5.10)

By Proposition 5.5, the terms inside the second parentheses in the equation above have valuations greater than w_{1}. Observe that w(α_{i}) > 0 for i = 0, …, (p − 1)/2, and thus Proposition 5.3 tells us that the valuations of the terms inside the first parentheses also have valuations greater than w_{1}.

Observing that w(α_{(p+1)/2}) = 0, and its initial coefficient is a_{0}^{(p+1)/2}, Proposition 5.3 gives us that the initial term of c_{p} is

(−1)^{(p−1)/2} 2 ( ∏_{i=0}^{(p−3)/2} \frac{2i + 1}{2i + 2}) δ_{1}.

But,

∏_{i=0}^{(p−3)/2} \frac{2i + 1}{2i + 2} = \frac{1}{2} · \frac{3}{4} ⋯ \frac{p − 4}{p − 3} · \frac{p − 2}{p − 1}

= \frac{1}{p − 1} · \frac{3}{p − 3} ⋯ \frac{p − 4}{4} · \frac{p − 2}{2}

= (−1)^{(p−1)/2}.

Hence, the initial coefficient of c_{p} is 2δ_{1}. On the other hand, we had previously established that this initial term was 2/3 δ_{1}, and hence we have a contradiction, as δ_{1} ≠ 0 by definition.

Thus, we have established that, for all r, we have v(d_{r}) ≥ −v_{0} when b_{1} = 0, and w(d_{r}) ≥ −w_{0} when a_{1} = 0.

We now can prove Proposition 4.3.

Proof of Proposition 4.3. If b_{1} = 0, then the Eq. (5.9) gives us that v(c_{p}) ≥ −v_{0}, and then Eq. (5.7) gives us that v(a_{1}) ≥ −v_{0}.

In the same way, if a_{1} = 0, then the Eq. (5.8) gives us that w(c_{0}) ≥ −w_{0}, and then Eq. (5.6) gives us that w(b_{1}) ≥ −w_{0}. □

This conclude the proof of Theorem 1.1. Note that by Proposition 4.2 we have that ord_{j̃_{0}=0}(J̃_{1}) = (2p + v(a_{1}))/3 and ord_{˜j̃_{0}=0}( ˜J̃_{1}) = (p + w(b_{1}))/2. Since v(a_{1}) ≥ −v_{0} and w(b_{1}) ≥ −w_{0}, Eq. (4.9) implies that

v(a_{1}) ≥ \begin{cases}1, & if p ≡ 1 (mod 6), \\ −1, & if p ≡ 5 (mod 6), \end{cases} and w(b_{1}) ≥ \begin{cases}1, & if p ≡ 1 (mod 4), \\ −1, & if p ≡ 3 (mod 4). \end{cases}

Hence,

ord_{j̃_{0}=0}(J̃_{1}) ≥ \begin{cases}(2p + 1)/3, & if p ≡ 1 (mod 6), \\ (2p − 1)/3, & if p ≡ 5 (mod 6), \end{cases}

and

ord_{˜j̃_{0}=0}( ˜J̃_{1}) ≥ \begin{cases}(p + 1)/2, & if p ≡ 1 (mod 4), \\ (p − 1)/2, & if p ≡ 3 (mod 4). \end{cases}

We can obtain then the analogue result for the usual invariant.

Proposition 5.6. Let r′ and s′ be as in Theorem 3.2. Then, ord_{j_{0}=0}(J_{1}) ≥ r′, J_{1}(1728) = (1728 − 1728^{p})/p, and J_{1}^{(t)}(1728) = −(t − 1)! (−1728)^{1−t}, for 1 ≤ t ≤ s′.

Proof. This is a simple computation with Witt vectors. Let

4 = (α_{0}, α_{1}), 27 = (β_{0}, β_{1}), 1728 = (γ_{0}, γ_{1}),

and F (X, Y) =^{def} ψ(X + Y) (with ψ as in Definition 3.1). (Note that α_{0}, β_{0}, γ_{0} ≠ 0.) Then, the second coordinate of 1728(4j̃/(4j̃ + 27)) is equal to

γ_{0} (−α_{0}J̃_{0}^{p} \frac{α_{0}J̃_{1} + α_{1}J̃_{0}^{p} + β_{1} + F (α_{0}J̃_{0}, β_{0})}{(α_{0}J̃_{0}^{p} + β_{0})^{p}} + \frac{α_{0}J̃_{1} + α_{1}J̃_{0}^{p}}{α_{0}J̃_{0}^{p} + β_{0}} ) + γ_{1} \frac{α_{0}J̃_{0}^{p}}{α_{0}J̃_{0}^{p} + β_{0}} . (5.11)

Thus, by Eq. (4.5), if J̃_{1}(X̃) has a zero of order less than p at X̃ = 0, then J_{1}(X) has a zero of the same order at X = 0. If J̃_{1}(X̃) has a zero of order greater than or equal to p at X̃ = 0, then J_{1}(X) has a zero of order greater than or equal to p at X = 0. Thus Proposition 4.3 gives the desired lower bound for ord_{j_{0}=0}(J_{1}).

Now, by Eq. (4.6), we have that

j − 1728 = 1728\frac{27˜j̃}{27˜j̃ + 4}.

Similarly to the computation above, one sees that the second coordinate of the right hand side of this equation is

γ_{0} (−β_{0}J̃_{0}^{p} \frac{β_{0}J̃_{1} + β_{1}J̃_{0}^{p} + α_{1} + F (β_{0}J̃_{0}, α_{0})}{(β_{0}J̃_{0}^{p} + α_{0})^{p}} + \frac{β_{0}J̃_{1} + β_{1}J̃_{0}^{p}}{β_{0}J̃_{0}^{p} + α_{0}} ) + γ_{1} \frac{β_{0}J̃_{0}^{p}}{β_{0}J̃_{0}^{p} + α_{0}} ,

while the left hand side is

J_{1}(X) − γ_{1} + ψ(X − γ_{0}).

Now, the order of the zero of the right hand side at X = γ_{0} = 1728 is the same as the order of the zero at ˜ X̃ = 0, and hence, by our estimates on ord_{˜j̃_{0}=0}( ˜J̃_{1}) above, it is greater than or equal to s′.

Since

ψ(X − γ_{0})|_{X=γ_{0}} = −γ_{0}^{p} ∑_{i=1}^{p−1} (\frac{1}{p}\binom{i}{p}) (−1)^{i} = 0,

we have that J_{1}(γ_{0}) = γ_{1} = (1728 − 1728^{p})/p.

Also, since the t-th derivative (with respect to X) of the right hand since is zero at X = γ_{0} for 1 ≤ t ≤ s′ − 1, we have that J_{1}^{(t)}(γ_{0}) = −(ψ(X − γ_{0}))^{(t)}|_{X=γ_{0}}. But, for 1 ≤ t ≤ (p − 1), we have

ψ(X − γ_{0}))^{(t)} = (p − 1)(p − 2) ⋯ (p − t + 1)(X^{p−t} − (X − γ_{0})^{p−t})
= (−1)^{t−1}(t − 1)! (X^{p−t} − (X − γ_{0})^{p−t},

which gives the desired formula for J_{1}^{(t)}(1728). □

Note that Proposition 5.6 is just a restatement of the second part of item 3 of Theorem 3.2, except that we only proved a lower bound for ord_{X=0} φ̄_{p}(X).

6. Experimental Data and Further Questions

A question that naturally arises is what happens modulo p^{3}. For instance, is J_{2} regular at j_{0} = 0 and j_{0} = 1728? Unfortunately the author’s MAGMA program to compute the general formulas (i.e., over 𝔽_{p}(a_{0}, b_{0}) with a_{0} and b_{0} algebraically independent transcendental elements) of canonical liftings modulo p^{3} seem to require a lot of computer power. Assuming there is no bug in the authors code (or in MAGMA), a computer with 16 gigabytes of memory cannot compute the general formula for the canonical lifting (and elliptic Teichmüller lift) modulo 17^{3}. The problem lies in the computation of the Greenberg transform of the elliptic curve, computed using the polynomial formulas for the sum and products of Witt vectors. So the data in this case is quite restricted, and so far we only have it for p ≤ 13. On the other hand, as seen in Eq. (1.1), for p = 7 we have that j_{0} = 1728 = −1 is a pole of J_{2}. Hence, in contrast with the case modulo p, we have j_{0} = 1728 might not yield pseudo-canonical liftings modulo p^{3}. Also, the same holds for p = 11, i.e., j_{0} = 1728 is a pole of J_{2}.

On the other hand, we have that j_{0} = 0 is supersingular for p = 5, 11, but J_{2} has zeros at that value (of orders 5 and 33 respectively). So, it seems that it could be the case that j_{0} = 0 still yield pseudo-canonical liftings modulo p^{3}. But even if that turns out to be the case, the failure of j_{0} = 1728 makes one wonder if j_{0} = 0 will also fail for large enough power of p.

More data would certainly be helpful, and the author is current working in improving his algorithms.

Acknowledgement. The author would like to thank F. Voloch for the invaluable discussions on the subject of this paper. Also, the computations mentioned were done with MAGMA and Sage.

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

   Email address: finotti@math.utk.edu