MINIMAL DEGREE LIFTINGS OF HYPERELLIPTIC CURVES

LUÍS R. A. FINOTTI

Abstract. The main goal of this paper is to analyze the properties of lifts of hyperelliptic curves y_{0}^{2} = f(x_{0}) over perfect fields of characteristic p > 2 (to hyperelliptic curves over the ring of Witt vectors) that have lifts of points whose coordinate functions have minimal degrees. It is shown that, when trying to minimize the degrees of the x-coordinate, the (n + 1)-th entry, say F_{n}, can be taken to be a polynomial in x_{0} such that (dp^{n} − (d − 2))/2 ≤ deg F_{n} ≤ (dp^{n} + (d − 2))/2, where d = deg f(x_{0}). Besides upper and lower bounds for the degrees, other topics discussed include a necessary condition to achieve the lower bounds and lifting the Frobenius. Computational aspects are also considered and the case of elliptic curves is analyzed in more detail. An explicit formula for derivatives of coordinate functions of the elliptic Teichmüller lift is proved, namely dF_{n}/dx_{0} = 0, if p = 2, and dF_{n}/dx_{0} = A^{(p^{n}−1)/(p−1)} y_{0}^{p^{n}−1} − ∑_{i=0}^{n−1} F_{i}^{(p^{n−i}−1)} dF_{i}/dx_{0}, if p ≥ 3, where A is the Hasse invariant of the curve. Finally, we establish a connection between minimal degree liftings and Mochizuki’s theory of “canonical liftings” in the case of genus 2 curves.

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 the ring W(k) of Witt vectors over k, called the canonical lift of E, and a map τ : E(k̄) → E(W(k̄)), 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 lift 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;

1991 Mathematics Subject Classification. Primary 11G20; Secondary 11T71. Key words and phrases. elliptic curves, canonical liftings, elliptic Teichmüller lift, Witt vectors, minimal degree

liftings, Mochizuki liftings.

This work was partly funded by CAPES (an agency of the Brazilian government).

2 LUÍS R. A. FINOTTI

(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)) \underrightarrow{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 applications, such as counting rational points in ordinary elliptic curves, as in Satoh’s [Sat00], counting torsion points of curves of genus g ≥ 2, as in Poonen’s [Poo01] and Voloch’s [Vol97], and coding theory, as in Voloch/Walker’s [VW00]. This last reference, together with [VW99] (by the same authors) and Mochizuki’s [Moc96], are the main motivation for this paper.

Before we make it clearer what we are going to pursue here, we need to introduce some more notation. We can identify E/W(k) with its Greenberg transform G(E)/k, which is an (infinite dimensional) scheme over k, and we can then view τ as a morphism of schemes over k. Thus,

τ(x_{0}, y_{0}) = (x_{0}, F_{1}, F_{2}, …, y_{0}, G_{1}, G_{2}, …),

where F_{i}, G_{i} ∈ k(x_{0}, y_{0}). One can prove that the F_{i}’s are in fact in k[x_{0}], and the G_{i}’s, for p ≠ 2, can be written as G_{i} = y_{0} H_{i}, with H_{i} ∈ k[x_{0}].

The error-correcting codes constructed by Voloch and Walker in [VW00] (using canonical liftings) have parameters that depend on the order of poles of the F_{i}’s and G_{i}’s at the point at infinity O (or equivalently, on the degrees of the F_{i}’s and H_{i}’s). In [Fin02] precise bounds for the orders were found: Theorem 1.1. We have

ord_{O} F_{n} ≥ −(n + 2)p^{n} + np^{n−1}, ord_{O} G_{n} ≥ −(n + 3)p^{n} + np^{n−1}.

for all n ≥ 0. For p > 2, those bounds may be written

deg F_{n} ≤ \frac{(n + 2)p^{n} − np^{n−1}}{2} , deg H_{n} ≤ \frac{(n + 3)p^{n} − np^{n−1} − 3}{2}.

Moreover, equality does not occur for ord_{O} F_{n} (or for deg F_{n}) if, and only if, p divides (n + 1), and the equality does not occur for ord_{O} G_{n} (or for deg H_{n}) if, and only if, p divides (n + 1)(n + 2)/2.

On the other hand, one can also use other lifts, different from the canonical, to construct codes. When estimating the paramenter of these codes, lifts with poles of smaller order, or equivalently, smaller degrees, give better bounds. In the cases dealt with in [VW00], only the reduction modulo p^{2} of E/W(k) was considered. In that case, using the canonical lift instead of any other, was the

MIN. DEG. LIFTS OF HYPERELL. CURVES 3

best possible choice to keep the degrees small: Proposition 4.2 in [VW00] tells us that if we have any lift of points ν : (x_{0}, y_{0}) ↦ ((x_{0}, F_{1}), (y_{0}, G_{1})) from the affine part of the elliptic curve E/k (possibly non-ordinary) to the affine part of the elliptic curve E/W_{2}(k), satisfying ord_{O} F_{1} ≥ (−3p + 1) or ord_{O} G_{1} ≥ (−4p + 1), then E is ordinary, E is the canonical lift of E (modulo p^{2}) and ν is the elliptic Teichmüller lift.

But one can also use lifts modulo higher powers of p to construct codes. Moreover, only the affine part of the curve is really relevant. Although one might have expected that the canonical lift and elliptic Teichmüller lift would be again the best choices, it turns out that there are other lifts which yield even smaller degrees. Section 5 of [Fin02] presents a lift of points

ν : (x_{0}, y_{0}) ↦ ((x_{0}, F_{1}, F̃_{2}), (y_{0}, G_{1}, G̃_{2}))

from the affine part of the elliptic curve

E/𝔽_{5} : y_{0}^{2} = x_{0}^{3} + x_{0}

to the affine part of its canonical lift

E/W_{3}(𝔽_{5}) : y^{2} = x^{3} + x

with deg F̃_{2} = 37 < deg F_{2} = 45 (and deg H̃_{2} = deg H_{2} = 56), where

τ : (x_{0}, y_{0}) ↦ ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2}))

denotes the elliptic Teichmüller lift. (Observe that, in contrast with τ, the lift ν cannot be extended to the point at infinity.)

We thus see that there are lifts which yield better bounds for the parameters of codes associated to them, and which therefore possibly give better codes. Since lifts with minimal degrees give us the best bounds, in this paper we study the properties of such lifts.

Similar ideas to those used in the case of elliptic curves can be applied to obtain lifts of hyperelliptic curves, and these curves may also be used in the construction of codes. (See [VW99]). Moreover, the results obtained here can be used to develop an algorithm to find the liftings of the curves as well as the lifts of points, and then the associated codes can be effectively constructed.

Throughout this paper, we will restrict ourselves to the case p > 2. The case p = 2 (for which similar results hold, and more concrete applications to coding theory might exist) requires different techniques and will be dealt with elsewhere.

Since we will be using hyperelliptic curves, throughout this paper we will use the following notation: let k be a perfect field of characteristic p > 2 and C be a hyperelliptic curve over k, given by

C/k : y_{0}^{2} = f(x_{0}),  (1.1)

4 LUÍS R. A. FINOTTI

where f is a monic polynomial of degree d ≥ 3, with (f, f′) = 1. (Here f′ denotes the derivative of f.) Note that although we write only the affine equation for C, we think of C as a projective curve, i.e., the unique smooth compactification of the affine curve given by equation (1.1). We shall refer to the affine curve as the affine part of C.

Let also

C/W(k) : y^{2} = f(x),  (1.2)

where f is a monic polynomial that reduces to f modulo p, i.e., C is a lift of C. (Again, C is a projective curve.) Also, let U denote the affine part of C and U the affine part of C. Definition 1.2. Let C and C be as above. A hyperelliptic lift (of points) from C to C is a regular map ν : U(k̄) → U(W(k̄)) (so it is a map between the affine parts only) given by

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}, F_{2}, …), (y_{0}, G_{1}, G_{2}, …)),  (1.3)

such that G_{i} = y_{0}H_{i} and F_{i}, H_{i} ∈ k[x_{0}] for all i. We also write F_{0} =^{def} x_{0}, G_{0} =^{def} y_{0}, H_{0} =^{def} 1.

Observe that saying that ν is hyperelliptic is equivalent to saying that it commutes with the hyperelliptic involutions of C and C.

We will be looking for lifts of points whose coordinate functions have poles of small order at infinity. For simplicity we shall refer to “degrees” instead of orders of poles, since we will be dealing with polynomials. But since we have polynomials in x_{0} and y_{0}, to be precise we have to define deg x_{0} =^{def} 1 and deg y_{0} =^{def} d/2, since y_{0}^{2} = f(x_{0}). (We will deal mostly with polynomials in x_{0}.)

It also seems that the best way to obtain these small degrees is to proceed coordinate by coordinate, i.e., to first obtain the minimal degree for the second coordinate, and then, with the second coordinate fixed, to obtain minimal degree for the third, and so on.

We will consider here only hyperelliptic lifts of points. Although this assumption greatly simplifies our analysis, it might seem to defeat the purpose of obtaining minimal possible degrees. But, in this spirit of obtaining minimal degrees coordinate by coordinate, Proposition 5.2 will partly justify why this is not entirely bad. Also, the last few paragraphs of section 6 further clarifies this choice.

We may consider two different notions of minimal degree: one in which the curves C and C are fixed a priori, and another in which we only fix C and want to find a curve C which has a hyperelliptic lift of points having minimal degrees among all other choices of curves that reduce to C modulo p. We will make these two notions precise in the following two definitions below. Definition 1.3. Let C and C be curves given by equations (1.1) and (1.2) respectively. A minimal degree lift from C to C/W_{2}(k) with respect to x (resp., y) is a hyperelliptic lift of points

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}), (y_{0}, G_{1})),

MIN. DEG. LIFTS OF HYPERELL. CURVES 5

where deg F_{1} (resp., deg H_{1}) is minimal. Inductively, a minimal degree lift from C to C/W_{n+1}(k) with respect to x (resp., y) is a hyperelliptic lift of points

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}, …, F_{n}), (y_{0}, G_{1}, … G_{n})),

whose reduction modulo p^{n} is a minimal degree lift from C to C/W_{n}(k), and deg F_{n} (resp., deg H_{n}) is minimal. Also, if we say “minimal degree lift from C to C/W_{n+1}(k)” without specifying with respect to what coordinate, we will be referring to the minimal degree lift with respect to x. Definition 1.4. Let C be a hyperelliptic curve given by (1.1). An absolute minimal degree curve modulo p^{2} over C (with respect to x) is a curve C/W_{2}(k) (given by (1.2)) which reduces to C modulo p, and which satisfies the following property. Let

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}), (y_{0}, G_{1})),

be a minimal degree lift from C to C, and let C̃/W_{2}(k) be any curve that reduces to C modulo p. Then for any minimal degree lift

ν̃(x_{0}, y_{0}) = ((x_{0}, F̃_{1}), (y_{0}, G̃_{1})),

from C to C̃, we have deg F̃_{1} ≥ deg F_{1}.

Inductively, an absolute minimal degree curve modulo p^{n+1} over C (with respect to x) is a curve C/W_{n+1}(k) whose reduction modulo p^{n} is an absolute minimal degree curve modulo p^{n} over C, satisfying the following property. Let

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}, …, F_{n−1}, F_{n}), (y_{0}, G_{1}, …, G_{n−1}, G_{n})),

be a minimal degree lift from C to C, and let C̃/W_{n+1}(k) be any curve whose reduction modulo p^{n} is equal to the reduction modulo p^{n} of C. Then, for a minimal degree lift

ν̃(x_{0}, y_{0}) = ((x_{0}, F_{1}, …, F_{n−1}, F̃_{n}), (y_{0}, G_{1}, …, G_{n−1}, G̃_{n})),

from C to C̃, we have deg F̃_{n} ≥ deg F_{n}. In this case we call the minimal degree lift ν from C to C an absolute minimal degree lift of points (modulo p^{n+1}).

We also have the analogous definitions with respect to y, rather than x. Remark. Note that in Definitions 1.3 and 1.4, the lift of points ν is hyperelliptic, and is therefore only a lift from the affine part of C to the affine part of C.

2. Statement of Main Results

We will now describe how this paper is organized and state its main results.

In section 3 we introduce the notation and state some results that will be used in the following sections. In section 4 we deal with liftings of powers of the Frobenius and, as a corollary, we

6 LUÍS R. A. FINOTTI

establish a formula for the derivatives of the entries of the x-coordinate of the elliptic Teichmüller lift. Namely, we prove: Theorem 2.1. Let E be an ordinary elliptic curve and E be its canonical lift. Let τ : E(k̄) → E(W(k̄)) be the elliptic Teichmüller lift

τ(x_{0}, y_{0}) = ((x_{0}, F_{1}, …), (y_{0}, G_{1}, …)).

Then, for n ≥ 1,

\frac{dF_{n}}{dx_{0}} = 0

if p = 2, and

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

if p > 2, where A is the Hasse invariant of E.

We observe that these derivatives allow us to create an algorithm, similar to the one described in the last section of [Fin02], to compute the canonical lift and elliptic Teichmüller lift modulo any power of p, although a generalization of Theorem 5.3 in [Fin02] (restated here as Theorem 3.3), would greatly improve such algorithm.

In section 5 we find upper bounds for the lifts (of hyperelliptic curves) with minimal degrees. We prove the following result: Proposition 2.2. Let C/k and C/W(k) be curves given by equations (1.1) and (1.2). Then there exists a unique minimal degree lift

ν = ((x_{0}, F_{1}, F_{2}, …), (y_{0}, y_{0}H_{1}, y_{0}H_{2}, …)),

from C to C with respect to x, and we have

deg F_{n} ≤ \frac{dp^{n} + (d − 2)}{2}

and

deg H_{n} ≤ \frac{(n(d − 2) + d)p^{n} + n(d − 2)p^{n−1} − d}{2},

for all n > 0.

We also prove the corresponding result for lifts with respect to y: Proposition 2.3. Let the hypotheses and notation be as in Proposition 2.2, and suppose in addition that p does not divide (d − 1). Then, there exists a unique minimal degree lift

ν = ((x_{0}, F_{1}, F_{2}, …), (y_{0}, y_{0}H_{1}, y_{0}H_{2}, …)),

from C to C with respect to y, and we have

deg H_{n} ≤ (d − 1)p^{n} − 1

MIN. DEG. LIFTS OF HYPERELL. CURVES 7

and

deg F_{n} ≤ \frac{(n(d − 2) + 2)p^{n} + n(d − 2)p^{n−1}}{2} ,

for all n > 0.

In section 6 we prove lower bounds for the absolute minimal degree lifts: Theorem 2.4. Let

ν = ((x_{0}, F_{1}, …, F_{n}), (y_{0}, G_{1}, …, G_{n})),

where n ≥ 1, be a lift of points (not necessarily hyperelliptic) from the affine part of C, given by equation (1.1), to the affine part of a lift C, given by equation (1.2), where F_{i} ∈ k[x_{0}] with

deg F_{i} = \frac{dp^{i} − (d − 2)}{2},

for i = 0, …, (n − 1). Then, deg F_{n} ≥ (dp^{n} − (d − 2))/2. If equality holds, then

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

where A is the (necessarily non-zero) coefficient of x_{0}^{p−1} in f(x_{0})^{(p−1)/2}.

Thus, Theorem 2.4 gives lower bounds for the absolute minimal degree lifts with respect to x. The best one can expect is to have

deg F_{n} = \frac{dp^{n} − (d − 2)}{2}  (2.1)

for all n ≥ 1. It also gives us a necessary condition to achieve these lower bounds: in order for the lower bound to be attained when n = 1, it is necessary for A^{−1}f(x_{0})^{(p−1)/2} − x_{0}^{p−1} to be a derivative. For this to occur, the coefficient of x_{0}^{rp−1} in f(x_{0})^{(p−1)/2} must be equal to zero for r ≠ 1, and the coefficient of x_{0}^{p−1} must be non-zero. Also the formula for the derivative of F_{n} helps us to explicitly compute this lift when it exists.

In section 7 we analyze minimal degree lifts in the case of elliptic curves, and prove that, modulo p^{3}, we can achieve the lower bounds above. (Theorem 7.2.) We also relate the minimal degree lift with the elliptic Teichmüller lift.

In order to describe the results of section 8, we require the following definition: Definition 2.5. Let C/k and C/W_{n}(k) be curves such that the reduction modulo p of C is C and for which we have a lift of points ν : U(k̄) → U(W(k̄)) between the affine parts. Let also ϕ : C → C^{σ} denote the Frobenius map in characteristic p. We say that ϕ : U → U^{σ} is a lift of the Frobenius associated to ν if it is a map that makes the diagram

U(W_{n}(k̄)) \underrightarrow{ϕ} U^{σ}(W_{n}(k̄))
ν↑  ↑^{ν^{σ}}
U(k̄) ⟶^{ϕ} U^{σ}(k̄)

8 LUÍS R. A. FINOTTI

commute.

It is shown in section 4 that, modulo p^{2}, any lift of points has an associated lift of the Frobenius. The main result of section 8 gives a necessary and sufficient condition for a lift of the Frobenius modulo p^{3} associated to a lift of points to exist. In order to be more precise, we need to establish some further notation. Definition 2.6. 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 shall refer to such lift as the Teichmüller lift of the polynomial g.) We define

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

Remark. One can define ψ(g) without lifting g, with a recursive definition: if g is a monomial, define ψ(g) = 0; if not let m(x_{0}, y_{0}) = λx_{0}^{i}y_{0}^{j} be a monomial of g, so that g − m has one term less than g. If

b(r) =^{def} \frac{1}{p}\binom{p}{r}, for r ∈ \{1, …, (p − 1)\},

(and hence the b(r)’s are integers) then we can define

ψ(g) =^{def} ψ(g − m) − ∑_{r=1}^{p−1} b(r)(g − m)^{r} m^{p−r}.

The main result of section 8 may be stated as follows: Proposition 2.7. Let C/k and C/W_{3}(k) be curves given by equations (1.1) and (1.2), and let

ν = ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2}))

be a hyperelliptic lift with

\frac{dF_{1}}{dx_{0}} = A^{−1}f(x_{0})^{(p−1)/2} − x_{0}^{p−1},

where A is the coefficient of x_{0}^{p−1} in f(x_{0})^{(p−1)/2}. Then, there is a lift of the Frobenius associated to ν if, and only if,

F_{2} − x_{0}^{p(p−1)} F_{1} − ψ(F_{1}) − (F′_{1})^{p} F_{1}

and

G_{2} − y_{0}^{p(p−1)} G_{1} − ψ(G_{1}) − (\frac{∂G_{1}}{∂x_{0}} )^{p} F_{1} − (\frac{∂G_{1}}{∂y_{0}} )^{p} G_{1}

are both p-th powers, say P (x_{0})^{p} and Q(x_{0}, y_{0})^{p}, respectively. In this case, the lift of the Frobenius is given by

ϕ(x, y) = (x^{p} + pF_{1} + p^{2}P , y^{p} + pG_{1} + p^{2}Q),

where F_{1} and G_{1} are the Teichmüller lifts of F_{1} and G_{1} respectively, and P and Q are lifts of P and Q to W_{3}(k)[x] and W_{3}[x, y] respectively.

MIN. DEG. LIFTS OF HYPERELL. CURVES 9

We then use this proposition in section 9 to prove that minimal degree lifts satisfying the lower bounds in section 6, or more precisely, lifts for which equation (2.1) holds for i = 1, 2, have a lift of the Frobenius modulo p^{3}. (Theorem 9.2.)

In section 10 we use this theory of minimal degree lifts to give examples of Mochizuki lifts for curves of genus 2 in characteristic 3: we observe that if the genus of the curve is greater than 1, there is no “canonical lift”, i.e., we cannot lift the Frobenius. (See [Ray83].) On the other hand, Mochizuki has developed a theory of canonical liftings of higher genus curves (in [Moc96]), where there exists liftings of the Frobenius with certain singularities at finitely many points, which are referred to as the supersingular points. In this paper, a Mochizuki lift (modulo p^{2}) will satisfy the condition of the statement of Proposition 4.10 on pg. 1114 of [Moc96], i.e., the height of the lift of the Frobenius will be less than or equal to one minus the genus of the curve. (We define the height of the lift of the Frobenius precisely on Definition 10.2.) Note that a Mochizuki lift is not unique.

Mochizuki’s theory does not have many known examples, and the results obtained in this paper allows us to explicitly find examples for curves of genus 2. More precisely, in sections 10 and 11 we prove: Theorem 2.8. Let k be a perfect field of characteristic 3 and C a smooth, proper, geometrically connected curve of genus 2 over k.

(1) Any Mochizuki lift of C to W_{2}(k) determines an absolute minimal degree curve modulo 9 over the affine curve U obtained by removing the supersingular points associated to the Mochizuki lift from C.

(2) Suppose that k is algebraically closed. Then if C admits a Mochizuki lift over W_{2}(k), then

C can be given by

y_{0}^{2} = x_{0}^{6} + a_{0}x_{0}^{4} + x_{0}^{2} + b_{0}x_{0} + c_{0},

where a_{0}, b_{0}, c_{0} ∈ k, and the supersingular points associated to the Mochizuki lift are the two points at infinity. In particular, if the moduli of C are sufficiently general, then C may be written in that form.

(3) Suppose that C is given by the equation

y_{0}^{2} = x_{0}^{6} + a_{0}x_{0}^{4} + x_{0}^{2} + b_{0}x_{0} + c_{0},

where a_{0}, b_{0}, c_{0} ∈ k. Then one can compute an explicit example of a Mochizuki lift of C modulo 9 (by computing a minimal degree lift). Moreover, the following three conditions are equivalent:

(a) this Mochizuki lift is Mochizuki-ordinary;

(b) the Jacobian of the curve C is ordinary;

(c) a_{0} ≠ 0.

10 LUÍS R. A. FINOTTI

3. Definitions and Previous Results

In this section we will introduce notation and results from [Fin02] that will be used in the rest of the paper.

First we recall that, for p ≠ 2, an elliptic curve given by

E/k : y_{0}^{2} = f(x_{0}),

where f(x) is a monic cubic, is ordinary if, and only if, the coefficient of x_{0}^{p−1} of f^{(p−1)/2} is non-zero. We shall denote this coefficient by A and call it the Hasse invariant of E.

We now review some facts about Witt vectors. Let p be a prime, and and for each non-negative integer n consider

W_{n}(X_{0}, …, X_{n}) =^{def} X_{0}^{p^{n}} + pX_{1}^{p^{n−1}} + ⋯ + p^{n−1}X_{n−1}^{p} + p^{n}X_{n},

the corresponding Witt polynomial. Then, there exist polynomials S_{n}, P_{n} ∈ ℤ[X_{0}, …, X_{n}, Y_{0}, …, Y_{n}] satisfying:

W_{n}(S_{0}, …, S_{n}) = W_{n}(X_{0}, …, X_{n}) + W_{n}(Y_{0}, …, Y_{n})  (3.1)

and

W_{n}(P_{0}, …, P_{n}) = W_{n}(X_{0}, …, X_{n}) · W_{n}(Y_{0}, …, Y_{n}).  (3.2)

(See [Ser79].)

If a = (a_{0}, a_{1}, …) and b = (b_{0}, b_{1}, …) are Witt vectors,

a + b =^{def} (S_{0}(a_{0}, b_{0}), S_{1}(a_{0}, a_{1}, b_{0}, b_{1}), …)

and

a · b =^{def} (P_{0}(a_{0}, b_{0}), P_{1}(a_{0}, a_{1}, b_{0}, b_{1}), …).

Since we will deal with Witt vectors over fields of characteristic p, we may use S̄_{n}, P̄_{n} ∈ 𝔽_{p}[X_{0}, …, X_{n}, Y_{0}, …, Y_{n}] respectively, defined to be the reductions modulo p of S_{n}, P_{n} ∈ ℤ[X_{0}, …, X_{n}, Y_{0}, …, Y_{n}], to define the addition and the product of Witt vectors.

Now, let K be a field of characteristic p > 0 and let v : K → ℝ ∪ {∞} be a valuation of the field K. (In the applications below, we will choose K to be the function field of a curve over k and v to be the order of vanishing at a point.) For e, r ≥ 0, define:

U_{r}(e) =^{def} {s = (s_{0}, s_{1}, …) ∈ W(K)^{×} | v(s_{n}) ≥ p^{n}(v(s_{0}) − ne), for n ≤ r} .

and

U(e) =^{def} {s = (s_{0}, s_{1}, …) ∈ W(K)^{×} | v(s_{n}) ≥ p^{n}(v(s_{0}) − ne), ∀n > 0} .

(So, U(e) = ⋂_{r≥0} U_{r}(e).) Lemma 3.1. The sets U(e) and U_{r}(e) are subgroups of W(K)^{×}.

Proof. The proof that U(e) is a group can be found in [Fin02] (Lemma 2.2) and can be easily adapted to prove that the same is true of U_{r}(e). □

A careful examination of the proof of Lemma 2.2 in [Fin02] also gives us the following lemma: Lemma 3.2. Let s, t ∈ U_{r−1}(e). Then, the (r + 1)-th coordinate of s · t is given by

t_{0}^{p^{r}}s_{r} + s_{0}^{p^{r}}t_{r} + …,

where all the omitted terms have valuation greater than or equal to p^{n}(v(s_{0} t_{0}) − en).

Proof. The (r + 1)-th coordinate of s · t is given by

P̄_{r}(s_{0}, …, s_{r}, t_{0}, …, t_{r}) = t_{0}^{p^{r}}s_{r} + s_{0}^{p^{r}}t_{r} + …,

where the omitted terms are monomials in s_{0}, …, s_{r−1}, t_{0}, …, t_{r−1}.

The proof of Lemma 2.2 in [Fin02] bounds each monomial appearing in

P̄_{r}(s_{0}, …, s_{r}, t_{0}, …, t_{r}).

In this case, we don’t have the bounds for the s_{r} and t_{r}, but the bounds for s_{0}, …, s_{r−1} and t_{0}, …, t_{r−1} are enough to bound the valuations of the omitted terms.

□

The next theorem gives an important characterization of the canonical lift modulo p^{3}. It is also helpful in the explicit computation of such lifts. Theorem 3.3. Suppose that p ≠ 2, 3 and that E/W_{3}(k) is an elliptic curve whose reduction modulo p is E/k. Suppose also that there exists a lift of points

τ(x_{0}, y_{0}) = ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2})),

between the affine parts of E and E, such that, modulo p^{2}, E and τ are the canonical lift and elliptic Teichmüller lift respectively. Then E and τ are also the canonical lift and the elliptic Teichmüller lift modulo p^{3} if, and only if,

deg (x_{0}^{p^{2}}F_{2} − \frac{3}{4} F_{1}^{2p}) ≤ \frac{5p^{2} − 1}{2}.

If this inequality holds, then it is in fact an equality.

Proof. This is Theorem 5.3 from [Fin02]. □

12 LUÍS R. A. FINOTTI

4. Liftings of the Frobenius

In [VW00] and [Fin02], the derivatives of F_{1} and F_{2} (from the elliptic Teichmüller lift) were computed. In order to do this, the reduction modulo p of 1/p ϕ^{∗}(dx/y) and (1/p ϕ^{∗})^{2}(dx/y) were computed, where ϕ denotes the lift of the Frobenius. The main goal of this section is to obtain more general results that will allow us to deduce the general formula for the derivative of F_{n} from the elliptic Teichmüller lift stated in Theorem 2.1, and to give similar results for hyperelliptic curves. Theorem 4.1. Let k be a perfect field of characteristic p > 0 and

U/k : g(x_{0}, y_{0}) = 0

be an affine curve in 𝔸^{2}. Let

U/W_{n+1}(k) : g(x, y) = 0

be an affine curve with reduction U, i.e., g reduces to g modulo p. If we have a lift of points ν : U(k̄) → U(W_{n+1}(k̄)) given by

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}, …, F_{n}), (y_{0}, G_{1}, …, G_{n})),

with F_{i}, G_{i} ∈ k[x_{0}, y_{0}], then we have a lift ϕ^{n} : U → U^{σ^{n}} of the p^{n}-th power Frobenius ϕ^{n} : U → U^{σ^{n}} associated to ν (as in Definition 2.5) given by

ϕ^{n}(x, y) =^{def} (x^{p^{n}} + pF _{1}^{p^{n−1}} + ⋯ + p^{n}F_{n}, y^{p^{n}} + pG_{1}^{p^{n−1}} + ⋯ + p^{n}G_{n}),

where F_{i}, G_{i} ∈ W_{n+1}(k)[x, y], and F_{i} (resp., G_{i}) is a lift of F_{i} ∈ k[x_{0}, y_{0}] (resp., G_{i}).

Proof. We need to prove that the map ϕ^{n} above is well defined. It suffices to show that the map is well defined for the Greenberg transforms G(U) and G(U^{σ^{n}}). Writing x = (x_{0}, …, x_{n}) and y = (y_{0}, …, y_{n}),

g(x, y) = (g_{0}(x_{0}, y_{0}), g_{1}(x_{0}, x_{1}, y_{0}, y_{1}), …, g_{n}(x_{0}, …, x_{n}, y_{0}, …, y_{n})).

So, G(U) is the defined as the common zeros of the equations g_{0}, …, g_{n} in 𝔸^{2n+2}. (Note that g_{0} = g.) Since ν is a (well defined) lift, we have that

ν^{∗}g_{1}, …, ν^{∗}g_{n} ≡ 0 (mod (g_{0})).  (4.1)

(Observe that,

ν^{∗}g_{i} = g_{i}(x_{0}, F_{1}, … F_{i}, y_{0}, G_{1}, …, G_{i}),

for i = 0, … n.)

If v = (v_{0}, …, v_{n}) is a Witt vector of length n + 1, then p^{i}v^{p^{n−i}} is the Witt vector whose the (i + 1)-th coordinate is v_{0}^{p^{n}} and whose other coordinates are zero. So, the map ϕ^{n} defined in the statement is such that

ϕ^{n} = ((x_{0}^{p^{n}}, F_{1}^{p^{n}}, …, F_{n}^{p^{n}}), (y_{0}^{p^{n}}, G_{1}^{p^{n}}, …, G_{n}^{p^{n}})).  (4.2)

To prove that ϕ^{n} is well defined, it suffices to prove that the coordinates of g^{σ^{n}}(ϕ^{n}(x, y)) are congruent to zero modulo I, where I =^{def} (g_{0}, …, g_{n}). But, by equation (4.2),

g^{σ^{n}}(ϕ^{n}(x, y)) = ((ν^{∗}g_{0})^{p^{n}}, (ν^{∗}g_{1})^{p^{n}}, …, (ν^{∗}g_{n})^{p^{n}}),

and so, by equation (4.1), ϕ^{n} is well defined.

Therefore, for any point (x, y) = ((x_{0}, …, x_{n}), (y_{0}, …, y_{n})) ∈ U(W_{n+1}(k̄)), ϕ^{n}(x, y) = ν^{σ^{n}} ∘ ϕ^{n}(x_{0}, y_{0}), and so the diagram

U(W_{n+1}(k̄)) \underrightarrow{ϕ^{n}} U^{σ^{n}}(W_{n+1}(k̄))
ν↑  ↑_{ν^{σ^{n}}}
U(k̄)  ⟶^{ϕ^{n}}  U^{σ^{n}}(k̄)

commutes. □

Corollary 4.2. With the same hypotheses as Theorem 4.1, we have that the reduction modulo p of

(\frac{1}{p^{n}} ϕ^{n})^{∗} dx

is dF_{n} + F_{n−1}^{p−1}dF_{n−1} + ⋯ + F_{1}^{p^{n−1}−1}dF_{1} + x_{0}^{p^{n}−1}dx_{0}.

Proof. Using the formula for ϕ^{n} from the Theorem 4.1, we have

(\frac{1}{p^{n}} ϕ^{n})^{∗} dx = \frac{1}{p^{n}} d (x^{p^{n}} + pF _{1}^{p^{n−1}} + ⋯ + p^{n}F_{n}) =

= dF_{n} + F _{n−1}^{p−1}dF_{n−1} + ⋯ + F _{1}^{p^{n−1}−1}dF_{1} + x^{p^{n}−1}dx,

which reduces to dF_{n} + F_{n−1}^{p−1}dF_{n−1} + ⋯ + F_{1}^{p^{n−1}−1}dF_{1} + x_{0}^{p^{n}−1}dx_{0}. □

We now want to apply the previous corollary in the case of canonical liftings of elliptic curves to prove Theorem 2.1. But, in order to do use the corollary, we need to show that, modulo p^{n}, the n-th power of the lift of the Frobenius of the canonical lift is equal to the lift defined in Theorem 4.1. We need the following lemma: Lemma 4.3. Let C be a curve, C be a lift of C and ϕ : U → U^{σ} be any lift of the Frobenius between the affine parts. Then, if we set x = (x_{0}, x_{1}, …) and y = (y_{0}, y_{1}, …), the reductions of (ϕ^{∗})^{n}(x) and (ϕ^{∗})^{n}(y) modulo p^{n+1} depend only on x_{0} and y_{0}.

Proof. We prove the lemma by induction on n.

For any lift of the Frobenius ϕ,

ϕ^{∗}(x) = x^{p} + pF (x, y)

14 LUÍS R. A. FINOTTI

for some polynomial F (x, y) ∈ W(k)[x, y]. Modulo p^{2}, pF (x, y) = (0, F (x_{0}, y_{0})^{p}), where F ∈ k[x_{0}, y_{0}] is the reduction modulo p of F, and x^{p} = (x_{0}^{p}, 0). Thus, the lemma holds for n = 1 and ϕ^{∗}(x), and the same method may be use to show that it also holds for ϕ^{∗}(y).

Now assume that the lemma holds for (ϕ^{∗})^{n}(x) and (ϕ^{∗})^{n}(y). We have

(ϕ^{∗})^{n+1}(x) = (ϕ^{∗})^{n}((ϕ^{∗})(x)) = ((ϕ^{∗})^{n}(x))^{p} + pF ((ϕ^{∗})^{n}(x), (ϕ^{∗})^{n}(y)).

Since modulo p^{n+2}, both a^{p} and pa depend only on a modulo p^{n+1}, using the induction hypothesis one easily sees that the lemma holds for (ϕ^{∗})^{n+1}(x), and in the analogous way, for (ϕ^{∗})^{n+1}(y). □

Proposition 4.4. Let C be a curve, and suppose that C is a lift of C. Let

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}, …), (y_{0}, G_{1}, …))

be a lift of points between the affine parts and assume that there exists a lift of the Frobenius between affine parts associated to ν, say ϕ : U → U^{σ}. Then, modulo p^{n+1}, ϕ^{n} is equal to the map defined in Theorem 4.1, i.e.,

ϕ^{n}(x, y) = (x^{p^{n}} + pF _{1}^{p^{n−1}} + ⋯ + p^{n}F_{n}, y^{p^{n}} + pG_{1}^{p^{n−1}} + ⋯ + p^{n}G_{n}),

where F_{i}, G_{i} ∈ W_{n+1}(k)[x, y], and F_{i} (resp., G_{i}) is a lift of F_{i} ∈ k[x_{0}, y_{0}] (resp., G_{i}).

Proof. We again work with the Greenberg transforms. Let π : G(C)(k̄) → C(k̄) be the reduction modulo p (or the projection in the first coordinates). By Lemma 4.3, ϕ^{n}(x, y) modulo p^{n+1} depends only on x_{0} and y_{0}, and thus ϕ^{n} ≡ ϕ^{n} ∘ ν ∘ π (mod p^{n+1}). Since the diagram

U(W(k̄)) \underrightarrow{ϕ^{n}} U^{σ^{n}}(W(k̄))
ν↑  ↑_{ν^{σ^{n}}}
U(k̄) ⟶^{ϕ^{n}} U^{σ^{n}}(k̄)

commutes, ϕ^{n} ∘ ν = ν^{σ^{n}} ∘ ϕ^{n}, and so ϕ^{n} ≡ ν^{σ^{n}} ∘ ϕ^{n} ∘ π (mod p^{n+1}), or

ϕ^{n} ≡ ((x_{0}^{p}, F_{1}^{p}, …, F_{n}^{p}), (y_{0}^{p}, G_{1}^{p}, …, G_{n}^{p})) (mod p^{n+1}).

By looking at the proof of Theorem 4.1 (more precisely, equation (4.2)), one easily sees that the reduction modulo p^{n+1} of ϕ^{n} is this case coincides with the map defined in that theorem. □

Now we can prove Theorem 2.1.

Proof of Theorem 2.1. Assume first that p > 2. The reduction modulo p of

\frac{1}{p}ϕ^{∗} (\frac{dx}{y}) ,

is a holomorphic differential. Thus, if ω denotes the reduction modulo p of

(\frac{1}{p}ϕ^{∗})^{n} (\frac{dx}{y}) = (\frac{1}{p^{n}} ϕ^{n})^{∗} (\frac{dx}{y}) ,

then ω is also holomorphic, and so, there is a λ ∈ k such that ω = λ dx_{0}/y_{0}. Applying the n-th power of the Cartier operator C to ω, we get

C^{n}(ω) = C^{n}(λ dx_{0}/y_{0}) = λ^{p^{−n}}A^{(p^{−1}+···+p^{−n})} \frac{dx_{0}}{y_{0}} .

On the other hand, since (1/p ϕ)^{∗} is the “inverse” of the Cartier operator, C^{n}(ω) = dx_{0}/y_{0}. Therefore, λ = A^{−(p^{n}−1)/(p−1)}, and

ω = A^{−(p^{n}−1)/(p−1)} \frac{dx_{0}}{y_{0}} .

But Proposition 4.4 tells us that we can use the Corollary 4.2 in this case, and we then get

ω = \frac{∑_{i=0}^{n} F_{i}^{p^{n−i}−1}dF_{i}}{y_{0}^{p^{n}}}

(with F_{0} =^{def} x_{0}). Comparing these two expressions for ω we obtain the formula for dF_{n}/dx_{0}.

The case p = 2 is analogous. We just need to use the invariant differential

\frac{dx}{2y + x}

instead of dx/y. Then we obtain

C^{n}(ω) = C^{n} (λ\frac{dx_{0}}{x_{0}} ) = λ^{p^{−n}} \frac{dx_{0}}{x_{0}} ,

and thus λ = 1. So,

ω = \frac{dx_{0}}{x_{0}} .

On the other hand, by Corollary 4.2,

ω = \frac{∑_{i=0}^{n} F_{i}^{p^{n−i}−1}dF_{i}}{x_{0}^{p}} .

Comparing the two expressions for ω in the case n = 1, gives that dF_{1}/dx_{0} = 0. Inductively, one can deduce then that dF_{n}/dx_{0} = 0 for all n. □

5. Minimal Degrees

We first try to justify why assuming that our lifts of points are all hyperelliptic lifts does not greatly compromise our goal of finding lifts of points with small order of poles at infinity (i.e., small degrees).

We will need the following lemma: Lemma 5.1. Every monomial of P̄_{n} (as defined in section 3) has even degree.

16 LUÍS R. A. FINOTTI

Proof. Lemma 2.1 of [Fin02] states that the monomials ∏ X_{i}^{a_{i}} ∏ Y_{j}^{b_{j}} of P̄_{n} are such that

∑ a_{i} p^{i} = ∑ b_{j} p^{j} = p^{n}.

Since p ≡ 1 (mod 2), we have

∑ a_{i} ≡ ∑ b_{j} ≡ 1 (mod 2),

and hence, the degree of a such monomial, namely ∑ a_{i} + ∑ b_{j}, is even. □

Proposition 5.2. Suppose that we have a lift of points

ν = ((x_{0}, F_{1}, …, F_{n}), (y_{0}, G_{1}, …, G_{n})),

where n ≥ 1, from the affine part of the hyperelliptic curve C/k, given by equation (1.1), to the affine part of some lift C/W_{n+1}(k), given by equation (1.2). Also, assume that ν is hyperelliptic modulo p^{n} and write F_{n} = F_{n,1}+y_{0} F_{n,2} and G_{n} = G_{n,1}+y_{0} G_{n,2}, with F_{n,1}, F_{n,2}, G_{n,1}, G_{n,2} ∈ k[x_{0}]. Then, the map

ν̃ = ((x_{0}, F_{1}, …, F_{n−1}, F_{n,1}), (y_{0}, G_{1}, …, G_{n−1}, y_{0} G_{n,2})),

defines a hyperelliptic lift whose degrees are not larger than those of ν.

Proof. Writing x = (x_{0}, x_{1}, …) and y = (y_{0}, y_{1}, …), one can expand (1.2) using the multiplication and addition of Witt vectors. The equality of the coordinates on both sides of this expansion gives the equations that determine the Greenberg transform. In particular, comparing the (n + 1)-th coordinates, we have:

2y_{0}^{p^{n}}y_{n} + ⋯ = f′(x_{0})^{p^{n}}x_{n} + …

where neither x_{n} nor y_{n} appear in any of the omitted terms. Since ν defines a lift, we have:

2y_{0}^{p^{n}}G_{n} + ⋯ = f′(x_{0})^{p^{n}}F_{n} + ….  (5.1)

The omitted terms on the right hand side of (5.1) involve only x_{0}, F_{1}, …, F_{n−1} ∈ k[x_{0}], and therefore form a polynomial in k[x_{0}]. The omitted terms on the left hand side of (5.1) come from P̄_{n}(y_{0}, G_{1}, …, G_{n}, y_{0}, G_{1}, …, G_{n}), and by Lemma 5.1, each one has an even number of G_{i}’s. Replacing each G_{i} by y_{0} H_{i}, we can factor an even power of y_{0} in each monomial, and it will be multiplying a polynomial in k[x_{0}]. But since y_{0}^{2} = f(x_{0}), this factored term is also a polynomial in k[x_{0}]. Hence, all the omitted terms of (5.1) are polynomials in k[x_{0}].

Thus, equation (5.1) implies that

2y_{0}^{p^{n}}(y_{0} G_{n,2}) + ⋯ = f′(x_{0})^{p^{n}}F_{n,1} + …

(with the same omitted terms as equation (5.1)) and

2f(x_{0})^{(p^{n}−1)/2}G_{n,1} = f′(x_{0})^{p^{n}}F_{n,2}.

Thus, taking F_{n,2} = G_{n,1} = 0 also gives us a well defined hyperelliptic lift of points, with degrees not greater than the degrees of F_{n} and G_{n}. □

So, remembering that we want to obtain minimal degrees coordinate by coordinate, this shows that the last coordinate of the lift of points can always be made hyperelliptic if the previous coordinates already are, and in particular, modulo p^{2}, we can always have a hyperelliptic lift with minimal degrees.

Note also that in principle we could have lifts ν and ν̃, say ν = ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2})) and ν̃ = (x_{0}, F̃_{1}, F̃_{2}), (y_{0}, G̃_{1}, G̃_{2})) such that deg F_{1} < deg F̃_{1} but deg F_{2} > deg F̃_{2}, and so our minimal lift (or even absolute minimal lift) might not have the minimal degree among all possible F_{2}’s if we don’t impose restrictions on the degree of F_{1}. But in general one expects that if deg F_{1} < deg F̃_{1}, then also deg F_{2} < deg F̃_{2}.

We now introduce a useful lemma, that will be essential for the proofs of Propositions 2.2 and 2.3: Lemma 5.3. Let a, b, c ∈ k[x_{0}], with deg(a) = n, deg(b) = m, deg(c) = r. Also, let s =^{def} max{r, n+m−1} and assume (a, b) = 1. Then, there exists a unique pair of polynomials u, v ∈ k[x_{0}] with deg(u) ≤ m − 1 and deg(v) ≤ s − m such that au + bv = c.

Proof. We follow the same idea of Lemma IV.1 in [VW99]. Let L(i) denote the vector space of polynomials in k[x_{0}] with degrees less than or equal to i. Consider the linear map

ψ : L(m − 1) ⊕ L(s − m) → L(s),

given by ψ(u, v) =^{def} au + bv. Since (a, b) = 1, ψ(u, v) = 0 if, and only if, u = bz and v = −az, for some polynomial z ∈ k[x_{0}]. But deg(u) ≤ m − 1 < deg(b), which implies u = z = 0. Thus ker ψ = {0}. Since dim L(i) = i + 1, comparing dimensions, we have that ψ is an isomorphism, and since c ∈ L(s), there exist a unique pair u, v as in the statement. □

We can now prove the main results of this section.

Proof of Proposition 2.2. We will work in U_{r}((d − 2)(p + 1)/p), where the valuation is defined by v(x_{0}) =^{def} −2, v(y_{0}) =^{def} −d, v(α) =^{def} 0, for α ∈ k^{×}, v(0) =^{def} ∞, and extended in the natural way to k[x_{0}, y_{0}]. (In other words, if d is odd, v is the order of vanishing at the point at infinity, and if d is even, v is twice the order of vanishing at one of the points at infinity.)

We prove the theorem by constructing the F_{n}’s and H_{n}’s inductively. The case n = 0 is trivial. Now suppose we have constructed ν up to the n-th coordinate. We construct F_{n} and H_{n} in the following way: observe that (x_{0}, F_{1}, …, F_{n−1}) and (y_{0}, y_{0}H_{1}, …, y_{0}H_{n−1}) are both in the group U_{n−1}((d − 2)(p + 1)/p), by the induction hypothesis.

We have in the (n + 1)-th coordinate of the equation of G(C)/k,

−f′(x_{0})^{p^{n}}x_{n} + 2 y_{0}^{p^{n}} y_{n} = …,

and, since ν will be a lift, we need

−f′(x_{0})^{p^{n}}F_{n} + 2 f(x_{0})^{(p^{n}+1)/2}H_{n} = …,  (5.2)

where no omitted term involves either F_{n} or H_{n}. (Here we view F_{n} and H_{n} as “unknowns”, rather than as polynomials.)

By Lemmas 3.1 and 3.2, all the omitted terms in (5.2) have valuations greater than or equal to −(n(d − 2) + 2d)p^{n} − n(d − 2)p^{n−1}. Let c denote these omitted terms. By the same argument as the one used in the proof of Proposition 5.2, c is a polynomial in x_{0}. Let a =^{def} −f′(x_{0})^{p^{n}} and b =^{def} 2f(x_{0})^{(p^{n}+1)/2}. Then, by Lemma 5.3, there are polynomials u and v, with valuations greater than or equal to −dp^{n} − (d − 2) and −(n(d − 2) + d)p^{n} − n(d − 2)p^{n−1} + d respectively, such that au + bv = c. Thus, we can define F_{n} =^{def} u, and H_{n} =^{def} v.

The fact that deg F_{n} is minimal comes from the uniqueness in Lemma 5.3. We cannot have a F̃_{n} with degree less than the degree of F_{n}, unless we allow deg H̃_{n} > ((n(d − 2) + d)p^{n} + n(d − 2)p^{n−1} − d)/2. But in this case the degree of the left hand side of the equation

−f′(x_{0})^{p^{n}} F̃_{n} + 2f(x_{0})^{(p^{n}+1)/2}H̃_{n} = …,

would have degree larger than the upper bound for the degree of the right hand side. Therefore, there can be no such pair F̃_{n}, H̃_{n}. □

Observe that if we have a supersingular elliptic curve E, by Proposition 4.2 in [VW00], the minimal lift from E to any lift E is such that deg F_{1} ≥ (3p+1)/2, i.e., in this case the upper bound in Proposition 2.2 cannot be improved.

With the same approach, one can prove Proposition 2.3.

Proof of Proposition 2.3. The proof follows the exact same idea as the proof of Proposition 2.2: again we will work in U((d − 2)(p + 1)/p) and we just apply Lemma 5.3 with a =^{def} 2f(x_{0})^{(p^{n}+1)/2}, b =^{def} −f′(x_{0})^{p^{n}}, and c as before. □

Propositions 2.2 and 2.3 have obvious applications to elliptic curves, by taking d = 3. But by Theorem 1.1, we can see that taking E ordinary and E its canonical lift, we can have deg F_{1} ≤ (3p−1)/2, deg H_{1} ≤ 2p−2, and so have degrees smaller than the upper bounds found in Proposition 2.2. This (together with the results to be obtained in section 6) gives the motivation for the following proposition, with better bounds for deg H_{n}:

MIN. DEG. LIFTS OF HYPERELL. CURVES 19

Proposition 5.4. Let C/k and C/W(k) be curves given by equations (1.1) and (1.2), and suppose that the minimal degree lift of points

ν = ((x_{0}, F_{1}, F_{2}, …), (y_{0}, y_{0}H_{1}, y_{0}H_{2}, …)),

is such that

deg F_{1} ≤ \frac{dp − (d − 2)}{2}

and

deg H_{1} ≤ \frac{(2d − 2)p − (d − 2) − d}{2}.

Then, we must have

deg F_{n} ≤ \frac{dp^{n} + (d − 2)}{2}

and

deg H_{n} ≤ \frac{(n(d − 2) + d)p^{n} − n(d − 2)p^{n−1} − d}{2},

for all n > 0.

Proof. The idea is that the restrictions on F_{1} and H_{1} allow us to work on U_{r}((d − 2)(p − 1)/p) instead of U_{r}((d − 2)(p + 1)/p). Inductively, the term c (as in the proof of Proposition 2.2) will have degree less than or equal to (n(d − 2) + 2d)p^{n} − n(d − 2)p^{n−1})/2, and we just apply Lemma 5.3 again. □

6. Lower Bounds

The main goal of this section is to prove Theorem 2.4. F. Voloch proved the particular case of this theorem where p = 3, d = 6 and n = 1, and it is possible to generalize his proof. The proof given below is somewhat shorter.

Proof of Theorem 2.4. By Theorem 4.1, we have a lift ϕ^{n} of the p^{n}-th power Frobenius ϕ^{n}. Let U denote the affine part of C and U denote the affine part of C . Then, by Corollary 4.2, the reduction modulo p of (1/p^{n} ϕ^{n})^{∗}(dx/y) is given by

ω =^{def} \frac{1}{y_{0}^{p^{n}}} (∑_{i=0}^{n} F_{i}^{(p^{n−i}−1)} \frac{dF_{i}}{dx_{0}} ) dx_{0}.

Since dx/y is regular in U, ω must be regular on U. In particular, it is regular at the points with y_{0} = 0, and since

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

we have that

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

20 LUÍS R. A. FINOTTI

for some g(x_{0}) ∈ k[x_{0}]. (Remember that since the curve is non-singular, (f, f′) = 1.) Therefore, deg F_{n} ≥ (dp^{n} − (d − 2))/2 and if we have equality, necessarily λ_{n} =^{def} g(x_{0}) ∈ k^{×}.

We now need to prove that λ_{n} = A^{−(p^{n}−1)/(p−1)}. First observe that, whether n = 1 or n > 1, we must have the equality for the degree of F_{1}, and thus we must have that

\frac{dF_{1}}{dx_{0}} = λ_{1}f(x_{0})^{(p−1)/2} − x_{0}^{p−1}.

for some λ_{1} ∈ k^{×}. But since the right hand side is a derivative in characteristic p, it cannot have a term in x_{0}^{p−1}, and hence λ_{1} = A^{−1}, where A is the (necessarily non-zero) coefficient of x_{0}^{p−1} in f(x_{0})^{(p−1)/2}.

Now, to prove that in general λ_{n} = A^{−(p^{n}−1)/(p−1)}, we proceed by induction: suppose that λ_{n−1} = A^{−(p^{n−1}−1)/(p−1)}, i.e., (with a computation analogous to the one above for ω in the case n − 1)

\frac{1}{y_{0}^{p^{n−1}}} (∑_{i=0}^{n−1} F_{i}^{(p^{n−1−i}−1)} \frac{dF_{i}}{dx_{0}} ) dx_{0} = A^{−(p^{n−1}−1)/(p−1)} \frac{dx_{0}}{y_{0}} .  (6.1)

Since

ω = \frac{1}{y_{0}^{p^{n}}} (∑_{i=0}^{n} F_{i}^{(p^{n−i}−1)} \frac{dF_{i}}{dx_{0}} ) dx_{0} = λ_{n} \frac{dx_{0}}{y_{0}} ,

applying the Cartier operator to both sides of the second equality of the equation above, we obtain

\frac{1}{y_{0}^{p^{n−1}}} (∑_{i=0}^{n−1} F_{i}^{(p^{n−1−i}−1)} \frac{dF_{i}}{dx_{0}} ) dx_{0} = λ_{n}^{1/p} A^{1/p} \frac{dx_{0}}{y_{0}} .

Comparing with equation (6.1), we obtain the formula for λ_{n}. □

We observe here that the condition that A^{−1}f(x_{0})^{p−1} − x_{0}^{p−1} is a derivative, which is necessary to achieve these lower bounds, also seems to be sufficient to obtain deg F_{1} = (dp − (d − 2))/2, i.e., if A^{−1}f(x_{0})^{(p−1)/2} − x_{0}^{p−1} is a derivative, then there is some lift C of C (modulo p^{2}) for which we can obtain a hyperelliptic lift of points satisfying deg F_{1} = (dp − (d − 2))/2. (In another words, the absolute minimal degree lift modulo p^{2} of C satisfying the above condition, has deg F_{1} equal to the lower bound of Theorem 2.4.) Note that if C is fixed from the beginning, one might not be able to obtain F_{1} with such small degree.

For the case d = 3 (i.e., elliptic curves), one can always find ν and C such that deg F_{1} = (dp − (d − 2))/2: the condition for this case is equivalent to saying that the elliptic curve is ordinary, and choosing C to be the canonical lift of C, we have the elliptic Teichmüller lift with deg F_{1} satisfying the lower bound. Also we have done several computations with hyperelliptic curves and they always had liftings of points satisfying the lower bound.

The condition also seems to be sufficient to obtain the lower bound for deg F_{2} as well, but not as many examples were tried in this case. But again, it is true for elliptic curves, as we will prove in section 7.

MIN. DEG. LIFTS OF HYPERELL. CURVES 21

Also, observe that if we achieve the lower bound for n = 1, we can use Proposition 5.4 to bound the degrees instead of Proposition 2.2, getting better bounds for deg H_{n}.

If these lower bounds can be achieved, the assumption that ν is hyperelliptic does not affect the fact that the degrees are minimal, meaning that we cannot have smaller degrees even if we drop this assumption. Indeed, by induction assume we can obtain a lift with F_{n} (not necessarily a polynomial in x_{0}) having degree less than or equal to (dp^{n} − (d − 2))/2, while assuming also that, for i = 0, …, (n − 1), F_{i} ∈ k[x_{0}] with deg F_{i} = (dp^{i} − (d − 2))/2 and G_{i} = y_{0} H_{i}, with H_{i} ∈ k[x_{0}]. Let us write F_{n} = F_{n,1} + y_{0} F_{n,2} and G_{n} = G_{n,1} + y_{0} G_{n,2}, with F_{n,1}, F_{n,2}, G_{n,1}, G_{n,2} ∈ k[x_{0}]. As in the proof of Proposition 5.2, we have in the (n + 1)-th coordinate of the Greenberg transform:

2y_{0}^{p^{n}}y_{n} + ⋯ = f′(x_{0})^{p^{n}}x_{n} + …,

with no x_{n} or y_{n} in the omitted terms, and this implies that

2f(x_{0})^{(p^{n}−1)/2}G_{n,1} = f′(x_{0})^{p^{n}}F_{n,2}.

Hence, if F_{n,2} ≠ 0, then it is a multiple of f(x_{0})^{(p^{n}−1)/2}, and thus the term y_{0} F_{n,2} has degree greater than or equal to dp^{n}/2 (remember that we defined deg y_{0} = d/2) and then so does F_{n}, what is a contradiction to the initial assumption on the degree of F_{n}.

7. Minimal Degrees for Elliptic Curves

In this section we will study absolute minimal degree lifts and curves modulo p^{3} of ordinary elliptic curves only and in characteristic p ≠ 2, 3. (For p = 2, 3, the elliptic Teichmüller lift is also the (we have uniqueness) absolute minimal lift modulo p^{3}.)

As observed before, modulo p^{2}, Proposition 4.2 of [VW00] tells us that the choice of curve that gives the minimal possible degree for F_{1} is the canonical lift itself, the minimal degree map is the elliptic Teichmüller lift and the degree of F_{1} is exactly (3p − 1)/2. Moreover, if E is not ordinary, then necessarily deg F_{1} = (3p + 1)/2.

The next proposition is the key step in obtaining the results modulo p^{3}. It was stated as a conjecture (and proved, with the help of a computer, for p ≤ 877) in the author’s doctoral thesis [Fin01] and was later proved in general by J. Tate in [Tat02]. Proposition 7.1 (Tate). Let τ = ((x_{0}, F_{1}), (y_{0}, G_{1})) be the elliptic Teichmüller lift from an ordinary elliptic curve

E/k : y_{0}^{2} = f(x_{0})

to its canonical lift. Let q(x_{0}) and r(x_{0}) be the quotient and remainder of the division of F_{1}^{2} by x_{0}^{p} f(x_{0})^{(p+1)/2}, i.e.,

F_{1}^{2} = (x_{0}^{p} f(x_{0})^{(p+1)/2})q(x_{0}) + r(x_{0}), deg r(x_{0}) ≤ \frac{5p + 1}{2}.

Then, in fact deg r(x_{0}) ≤ (5p − 1)/2.

Proof. See [Tat02] for a proof of this proposition. □

In this section, since we are dealing with elliptic curves, we will write

τ(x_{0}, y_{0}) = ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2}))

for the elliptic Teichmüller lift and

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}, F̃_{2}), (y_{0}, G_{1}, G̃_{2}))

for the minimal lift. Theorem 7.2. An absolute minimal degree lift

ν = ((x_{0}, F_{1}, F̃_{2}), (y_{0}, G_{1}, G̃_{2}))

of an ordinary elliptic curve E is such that deg F̃_{2} = (3p^{2} − 1)/2 and the corresponding absolute minimal degree curve E is its canonical lift (modulo p^{3}). (So, up to isomorphism, the absolute minimal degree curve in this case is unique and satisfies the lower bound of section 6.) Moreover,

\frac{dF̃_{2}}{dx_{0}} = \frac{dF_{2}}{dx_{0}} .

Proof. We first prove that the minimal degree lift from E to its canonical lift E is such that deg F̃_{2} = (3p^{2} − 1)/2. We will actually give a way to construct the absolute minimal degree lift from the elliptic Teichmüller lift: if

τ = ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2}))

is the elliptic Teichmüller lift, we write, using the division algorithm,

F_{2} = f(x_{0})^{(p^{2}+1)/2} q_{1}(x_{0}) + r_{1}(x_{0}) (deg r_{1} ≤ (3p^{2} + 1)/2).

Now define F̃_{2} =^{def} r_{1}(x_{0}) and G̃_{2} =^{def} G_{2} − y_{0}(f′(x_{0})^{p^{2}}q_{1}(x_{0}))/2. Then, we have

2y_{0}^{p^{2}} G̃_{2} − f′(x_{0})^{p^{2}} F̃_{2} = 2y_{0}^{p^{2}}G_{2} − f′(x_{0})^{p^{2}}F_{2},

and thus, by equation (5.1),

ν =^{def} ((x_{0}, F_{1}, F̃_{2}), (y_{0}, G_{1}, G̃_{2}))

is another lift from E to its canonical lift, and since deg F̃_{2} ≤ (3p^{2} + 1)/2, by Proposition 2.2, it is the minimal lift.

We now have to prove that deg F̃_{2} = (3p^{2} − 1)/2. Let d(x_{0}) =^{def} x_{0}^{p^{2}}F_{2} − 3/4 F_{1}^{2p}. Theorem 3.3 tells us that deg d(x_{0}) = (5p^{2} − 1)/2. Also, by Proposition 7.1,

\frac{3}{4}F_{1}^{2} = (x_{0}^{p} f(x_{0})^{(p+1)/2})q(x_{0}) + r(x_{0}), with deg r(x_{0}) ≤ (5p − 1)/2.  (7.1)

We can then write

\begin{aligned}x_{0}^{p^{2}}F_{2} & = x_{0}^{p^{2}}f(x_{0})^{(p^{2}+p)/2}q(x_{0})^{p} + (r(x_{0})^{p} + d(x_{0})) \\ & = x_{0}^{p^{2}} f(x_{0})^{(p^{2}+1)/2} (f(x_{0})^{(p−1)/2}q(x_{0})^{p}) + (r(x_{0})^{p} + d(x_{0})) , \end{aligned}  (7.2)

with deg (r(x_{0})^{p} + d(x_{0})) = (5p^{2} − 1)/2, and thus it is the remainder of the division of x_{0}^{p^{2}}F_{2} by x_{0}^{p^{2}}f(x_{0})^{(p^{2}+1)/2}. We see then that x_{0}^{p^{2}} divides this remainder and F̃_{2} = r_{1}(x_{0}) = (r(x_{0})^{p} + d(x_{0})) /x_{0}^{p^{2}}, which implies that deg F̃_{2} = (3p^{2} − 1)/2.

Hence, by Theorem 2.4, this has to be the absolute minimal degree lift, and

\frac{dF̃_{2}}{dx_{0}} = A^{−p−1}f(x_{0})^{(p^{2}−1)/2} − x_{0}^{p^{2}−1} − F_{1}^{p−1} \frac{dF_{1}}{dx_{0}} = \frac{dF_{2}}{dx_{0}} .

(One could also deduce that dF̃_{2}/dx_{0} = dF_{2}/dx_{0} from equation (7.2), since it implies that

\begin{aligned}F_{2} & = f(x_{0})^{(p^{2}+p)/2}q(x_{0})^{p} + \frac{r(x_{0})^{p} + d(x_{0})}{x_{0}^{p^{2}}} \\ & = f(x_{0})^{(p^{2}+p)/2}q(x_{0})^{p} + F̃_{2}, \end{aligned}

and taking derivatives would give us the result.)

Now, we prove that if we can obtain deg F̃_{2} = (3p^{2} − 1)/2, then E must be the canonical lift of E. So assume we have a lift of E to some curve E (not necessarily the canonical lift) with deg F̃_{2} = (3p^{2} − 1)/2. Let

˜F̃_{2} =^{def} F̃_{2} + (q(x_{0})f(x_{0})^{(p+1)/2})^{p}

(with the q(x_{0}) from equation (7.1)) and

˜G̃_{2} =^{def} G̃_{2} + \frac{y_{0}}{2} (f′(x_{0})^{p^{2}}f(x_{0})^{(p−1)/2}q(x_{0})^{p}).

Then,

ν̃ : (x_{0}, y_{0}) ↦ ((x_{0}, F_{1}, ˜ F̃_{2}), (y_{0}, G_{1}, ˜ G̃_{2})),

is another lift, since

2y_{0}^{p^{2}} ˜G̃_{2} − f′(x_{0})^{p^{2}} ˜F̃_{2} = 2y_{0}^{p^{2}} G̃_{2} − f′(x_{0})^{p^{2}} F̃_{2}.

But then, by hypothesis,

x_{0}^{p^{2}} ˜F̃_{2} − \frac{3}{4}F_{1}^{2p} = x_{0}^{p^{2}} F̃_{2} + (x_{0}^{p} q(x_{0}) f(x_{0})^{(p+1)/2})^{p} − \frac{3}{4}F_{1}^{2p} = x_{0}^{p^{2}} F̃_{2} − r(x_{0})^{p}

has degree less than or equal to (5p^{2} − 1)/2. Theorem 3.3 then tells us that E is the canonical lift modulo p^{3}, ˜ F̃_{2} and ˜ G̃_{2} are F_{2} and G_{2} from the elliptic Teichmüller lift.

□

24 LUÍS R. A. FINOTTI

Thus, modulo p^{3}, the absolute minimal degree curve over an ordinary elliptic curve is the same as its canonical lift.

So, we can modify the algorithm described in the section 6 of [Fin02] to compute this absolute minimal lift and the canonical lift of an elliptic curve over a perfect field of characteristic p ≥ 5. The function (written for the package Magma) available at

http://www.math.ucsb.edu/~finotti/can_lifts.html

that computes the canonical lift and elliptic Teichmüller lift, also gives you an option to compute this absolute minimal degree lift instead of the elliptic Teichmüller lift.

8. A Necessary and Sufficient Condition to Lift the Frobenius

In sections 9 and 10 we will deal with liftings of the Frobenius modulo p^{3}, and so in this section we prove Proposition 2.7, which will allow us to obtain such liftings. Lemma 8.1. Let

g(x, y) = ∑_{i,j} a_{i,j} x^{i}y^{j} ∈ W_{2}(k)[x, y],

and suppose that

g(x, y) = (g_{0}(x_{0}, y_{0}), g_{1}(x_{0}, x_{1}, y_{0}, y_{1})),

Then, if

a_{i,j} = (a_{i,j,0}, a_{i,j,1})

we have

g_{1}(x_{0}, x_{1}, y_{0}, y_{1}) = x_{1} (\frac{∂g_{0}}{∂x_{0}} )^{p} + y_{1} (\frac{∂g_{0}}{∂y_{0}} )^{p} + ψ(g_{0}) + ∑_{i,j} a_{i,j,1} x_{0}^{pi}y_{0}^{pj}.

(Here ψ is the function defined in Definition 2.6.)

Sketch of the Proof. First we observe that, with the same notation as section 3,

S̄_{1}(X_{0}, X_{1}, Y_{0}, Y_{1}) = X_{1} + Y_{1} + ψ(X_{0} + Y_{0})

and

P̄_{1}(X_{0}, X_{1}, Y_{0}, Y_{1}) = X_{1} Y_{0}^{p} + X_{0}^{p} Y_{1}.

Then, one can easily prove that the formula is true for g equal to powers of x and y, then for monomials, and finally for an arbitrary polynomial, by an induction on the number of terms. □

Lemma 8.2. Let C be a hyperelliptic curve with reduction C (given by equations (1.2) and (1.1) respectively) and

ν = ((x_{0}, F_{1}), (y_{0}, y_{0}H_{1}))

a hyperelliptic lift of points. If

\frac{dF_{1}}{dx_{0}} = A^{−1}f(x_{0})^{(p−1)/2} − x_{0}^{p−1},

then

2 f H′_{1} + f′ H_{1} − A^{−1} (f′)^{p} + f^{(p−1)/2} f′ = 0.

Proof. By Lemma 8.1, when expanding the equation (1.2) as Witt vectors and comparing the second coordinates we have

2 y_{0}^{p} y_{1} = x_{1} (f′)^{p} + ψ(f) + …,

where the omitted terms are p-th powers. Since ν is a lift, we have,

2 f^{(p+1)/2} H_{1} = F_{1}(f′)^{p} + ψ(f) + …,

and so this is an equality of polynomials in x_{0}. Taking derivatives, one obtains

f′ f^{(p−1)/2} H_{1} + 2 f^{(p+1)/2} H′_{1} = A^{−1} f^{(p−1)/2} (f′)^{p} − f^{p−1} f′,

and dividing both terms by the common factor f^{(p−1)/2}, we obtain the differential equation for H_{1} from the statement. □

We also need the following simple lemma: Lemma 8.3. Let P (X, Y) be a polynomial in two variables. Then

\begin{matrix}P (X_{0} + p & X_{1}, Y_{0} + pY_{1}) \\ ≡ P (X_{0}, Y_{0}) + p (\frac{∂P}{∂X} (X_{0}, Y_{0})X_{1} + \frac{∂P}{∂Y} (X_{0}, Y_{0})Y_{1}) (mod p^{2}). \end{matrix}

Proof. This is an easy application of Taylor’s formula for P (X, Y). □

We finally prove Proposition 2.7, which can be quite useful when dealing with explicit computations.

Proof of Proposition 2.7. We first prove that the condition is necessary. Assume we have a lift of ϕ associated to ν. By Theorem 4.1, it must have the form

ϕ(x, y) = (x^{p} + pF_{1} + p^{2}P , y^{p} + pG_{1} + p^{2}Q),

for some P , Q ∈ W_{3}(k)[x, y].

Let δ be the p-derivation associated to ϕ (as in [Bui96]):

δu =^{def} \frac{ϕ^{∗}u^{σ} − u^{p}}{p} .

We then have

δx = F_{1} + pP

and, using Lemma 8.3,

\begin{aligned}δ^{2}x & = \frac{(F_{1} + pP)^{σ} ∘ ϕ − (F_{1} + pP)^{p}}{p} \\ & = \frac{F _{1}^{σ}(x^{p}) − F _{1}^{p}}{p} + \frac{dF_{1}}{dx}^{σ}(x^{p}) · F_{1} + P^{σ}(x^{p}) + p · (…). \end{aligned}  (8.1)

But, by Lemma 2.6 of [Bui96], the reduction modulo p of δ^{2}x must be equal to F_{2} − x_{0}^{p(p−1)}F_{1}. Since the reduction modulo p of P^{σ}(x^{p}) is clearly a p-th power, say P^{p}, and F_{1} is the Teichmüller lift of F_{1},

F_{2} − x_{0}^{p(p−1)} F_{1} = ψ(F_{1}) + (F′_{1})^{p} F_{1} + P^{p}.  (8.2)

An analogous computation with δ^{2}y, gives

G_{2} − y_{0}^{p(p−1)} G_{1} = ψ(G_{1}) + (\frac{∂G_{1}}{∂x_{0}} )^{p} F_{1} + (\frac{∂G_{1}}{∂y_{0}} )^{p} G_{1} + Q^{p},

and hence, the condition is necessary.

We now prove the converse, more precisely, that ϕ is well defined and that the diagram

\begin{matrix}U(W_{3}(k̄)) & \underrightarrow{ϕ} & U^{σ}(W_{3}(k̄)) \\ ^{ν}↑ & & ↑^{ν^{σ}} \\ U(k̄) & ⟶^{ϕ} & U^{σ}(k̄) \end{matrix}  (8.3)

commutes, where U and U are the affine parts of C and C respectively. It suffices to prove it for the Greenberg transform. Defining

g =^{def} y^{2} − f(x),

we write

g(x, y) = (g_{0}(x_{0}, y_{0}), g_{1}(x_{0}, x_{1}, y_{0}, y_{1}), g_{2}(x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2})).

Then, to prove that ϕ is well defined, it suffices to prove that ϕ^{∗}g_{i}^{σ} ∈ I, for i = 0, 1, 2, where

I =^{def} (g_{0}, g_{1}, g_{2}).

By Theorem 4.1, we have that ϕ^{∗}g_{0}^{σ}, ϕ^{∗}g_{1}^{σ} ∈ I. So we just need to show that ϕ^{∗}g_{2}^{σ} ∈ I.

One has

x^{p} = (x_{0}, x_{1}, x_{2})^{p} = (x_{0}^{p}, 0, x_{0}^{p^{2}(p−1)}x_{1}^{p}).

Also, by Lemma 8.1

pF_{1} = (0, F_{1}^{p}, (x_{1}(F′_{1})^{p} + ψ(F_{1}))^{p}).

Hence,

x^{p} + pF_{1} + p^{2}P = (x_{0}^{p}, F_{1}^{p}, F_{2}^{p} + 𝒳_{2}^{p}) ,  (8.4)

where,

𝒳_{2} =^{def} (x_{1} − F_{1}) (x_{0}^{p−1} + F′_{1})^{p} ,

MIN. DEG. LIFTS OF HYPERELL. CURVES 27

and in a similar manner,

y^{p} + pG_{1} + p^{2}Q = (y_{0}^{p}, G_{1}^{p}, G_{2}^{p} + 𝒴_{2}^{p})  (8.5)

where

𝒴_{2} =^{def} (x_{1} − F_{1}) (y_{0} H′_{1})^{p} + (y_{1} − G_{1}) (y_{0}^{p−1} + H_{1})^{p} .

Note that

g_{2}^{σ} = 2y_{0}^{p^{2}}y_{2} − ((f′)^{σ}(x_{0}))^{p^{2}}x_{2} + …,

where no omitted term depends on either x_{2} or y_{2}. Hence

ϕ^{∗}g_{2}^{σ} = 2y_{0}^{p^{3}}(G_{2} + 𝒴_{2})^{p} − f′(x_{0})^{p^{3}}(F_{2} + 𝒳_{2})^{p} + ….  (8.6)

Since

ν = ((x_{0}, F_{1}, F_{2}), (x_{0}, G_{1}, G_{2})),

is a lift,

2y_{0}^{p^{3}}G_{2}^{p} − f′(x_{0})^{p^{3}}F_{2}^{p} + ⋯ ≡ 0 (mod (g_{0})),

where the omitted terms are the same as the ones in formula (8.6). Therefore

ϕ^{∗}g_{2}^{σ} ≡ 2y_{0}^{p^{3}}𝒴_{2}^{p} − f′(x_{0})^{p^{3}}𝒳_{2}^{p} (mod I),

and it suffices to prove that

2y_{0}^{p^{2}}𝒴_{2} − f′(x_{0})^{p^{2}}𝒳_{2} ≡ 0 (mod I),

or,

(x_{1} − F_{1}) (−A^{−1} (f′)^{p} f^{(p−1)/2} + 2 f^{(p+1)/2} H′_{1})^{p}

+ (y_{1} − G_{1}) 2 y_{0}^{p} (f^{p−1} + f^{(p−1)/2} H_{1})^{p} ≡ 0 (mod I).

Factoring f^{(p^{2}−p)/2}, it is enough to prove

(x_{1} − F_{1}) (−A^{−1} (f′)^{p} + 2 f H′_{1})^{p} + (y_{1} − G_{1}) 2 y_{0}^{p} (f^{(p−1)/2} + H_{1})^{p} ≡ 0 (mod I).  (8.7)

Using Lemma 8.2, equation (8.7) becomes

(H_{1} + f^{(p−1)/2})^{p} ((x_{1} − F_{1})(−f′)^{p} + (y_{1} − G_{1})(2 y_{0})^{p}) ≡ 0 (mod I).

But g_{1}, ν^{∗}g_{1} ≡ 0 (mod I), and then

g_{1}(x_{0}, x_{1}, y_{0}, y_{1}) − g_{1}(x_{0}, F_{1}, y_{0}, G_{1}) = (x_{1} − F_{1})(−f′)^{p} + (y_{1} − G_{1})(2 y_{0})^{p} ≡ 0 (mod I), what finishes the proof that ϕ is well defined.

Finally, equations (8.4) and (8.5) show the diagram (8.3) commutes. □ The proof of the proposition above also gives us the following corollary:

28 LUÍS R. A. FINOTTI

Corollary 8.4. Let C/k and C/W_{3}(k) be curves given by equations (1.1) and (1.2), and suppose that we can lift the Frobenius for the affine part of C. Let

ν =^{def} ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2}))

be the lift of points, and also assume that F_{i} ∈ k[x_{0}]. Then,

\frac{dF_{2}}{dx_{0}} = (\frac{dF_{1}}{dx_{0}} )^{p+1} + (\frac{dF_{1}}{dx_{0}} )^{p} x_{0}^{p−1} + (x_{0}^{p(p−1)} − F_{1}^{p−1}) \frac{dF_{1}}{dx_{0}} .

In particular, if dF_{1}/dx_{0} = λ y_{0}^{p−1} − x_{0}^{p−1} for some λ ∈ k^{×}, then

\frac{dF_{2}}{dx_{0}} = λ^{p+1} y_{0}^{p^{2}−1} − x_{0}^{p^{2}−1} − F_{1}^{p−1} \frac{dF_{1}}{dx_{0}} .

Proof. By Proposition 2.7, the first coordinate of the lift of the Frobenius has the form

x^{p} + pF_{1} + p^{2}P

where F_{1} is a lift of F_{1} to W_{3}(k)[x] and P is some polynomial. Equation (8.2) also holds in this case. Observing that

\frac{dψ(F_{1})}{dx_{0}} = x_{0}^{p−1} (\frac{dF_{1}}{dx_{0}} )^{p} − F_{1}^{p−1} \frac{dF_{1}}{dx_{0}} ,  (8.8)

if we take derivatives of equation (8.2) we obtain the formula for dF_{2}/dx_{0} in the statement. □

9. Minimal Lifts and the Frobenius

By Theorem 4.1, the existence of a lift from C to C/W_{2}(k) is enough to give a lift of the Frobenius ϕ on the affine part of C. On the other hand, the existence of a lift from C to C/W_{3}(k) merely guarantees a lift of ϕ^{2}. Of course, in the case of elliptic curves, the canonical lift always has a lift of ϕ associated to τ for any power of p. So, one could ask if there is also a lift of the Frobenius (between the affine parts) associated to the minimal lift, at least modulo p^{3}.

Theorem 9.2 below gives a precise answer to this question. But we first need the following lemma: Lemma 9.1. Let

ν = ((x_{0}, F_{1}, …, F_{n}), (y_{0}, G_{1}, …, G_{n})),

be a lift of points from the affine part of C, given by equation (1.1), to the affine part of a lift C, given by equation (1.2), with

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

for i = 1, …, n, where A denotes the coefficient of x_{0}^{p−1} in f^{(p−1)/2}. Then

dG_{i} = (A^{−(p^{i}−1)/(p−1)}(f′)^{p^{i}−1} − y_{0}^{p^{i}−1}) dy_{0} − ∑_{j=1}^{i−1} G_{j}^{p^{i−j}−1}dG_{j}.

Proof. We fix some i ∈ {1, …, n}, and work modulo p^{i+1}. Then, by Corollary 4.2, the reduction modulo p of (1/p^{i}ϕ^{i})^{∗}(dx/y) is given by

ω =^{def} \frac{1}{y_{0}^{p^{i}}} (∑_{j=0}^{i} F_{j}^{(p^{i−j}−1)} \frac{dF_{j}}{dx_{0}} ) dx_{0} = A^{−(p^{i}−1)/(p−1)} \frac{dx_{0}}{y_{0}} .

On the other hand, since dx/y = 2dy/f′, and by the analogous to Corollary 4.2 for dy, we have that

ω = 2 (\frac{dG_{i} + ∑_{j=0}^{i−1} G_{j}^{p^{i−j}−1}dG_{j}}{(f′)^{p^{i}}} ) .

Since dx_{0}/y_{0} = 2dy_{0}/(f′), comparing the two expressions for ω we obtain the formula for dG_{i} in the statement. □

Theorem 9.2. Let

ν = ((x_{0}, F_{1}, F_{2}), (y_{0}, G_{1}, G_{2})),

be a hyperelliptic lift of points from C, given by equation (1.1), to a lift C, given by equation (1.2), where again we write G_{i} = y_{0} H_{i} with H_{i} ∈ k[x_{0}]. If

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

for i = 1, 2, where A denotes the coefficient of x_{0}^{p−1} in f^{(p−1)/2}, then there is a lift of the Frobenius modulo p^{3}. In particular, if the minimal degree lift of C satisfies the lower bounds in Theorem 2.4, namely deg F_{1} = (dp − (d − 2))/2 and deg F_{2} = (dp^{2} − (d − 2))/2, there is a lift of the Frobenius modulo p^{3}.

Proof. We use Proposition 2.7. So, it suffices to prove that

\frac{d}{dx_{0}} (F_{2} − x_{0}^{p(p−1)} F_{1} − ψ(F_{1}) − (F′_{1})^{p} F_{1}) = 0  (9.1)

and

d (G_{2} − y_{0}^{p(p−1)} G_{1} − ψ(G_{1}) − (\frac{∂G_{1}}{∂x_{0}} )^{p} F_{1} − (\frac{∂G_{1}}{∂y_{0}} )^{p} G_{1}) = 0  (9.2)

By using equation (8.8), the equality in (9.1) is easily verified.

The proof of equation (9.2) is also a straightforward calculation, but requires a little more work. First we observe that

dψ(G_{1}) = (\frac{∂G_{1}}{∂x_{0}} )^{p} x_{0}^{p−1}dx_{0} + (\frac{∂G_{1}}{∂y_{0}} )^{p} y_{0}^{p−1}dy_{0} − G_{1}^{p−1}dG_{1}.

So, the left-hand-side of equation (9.2) becomes

dG_{2} − y_{0}^{p(p−1)}dG_{1} − (\frac{∂G_{1}}{∂x_{0}} )^{p} x_{0}^{p−1}dx_{0} + (\frac{∂G_{1}}{∂y_{0}} )^{p} y_{0}^{p−1}dy_{0}

− G_{1}^{p−1}dG_{1} − (\frac{∂G_{1}}{∂x_{0}} )^{p} (A^{−1}f^{(p−1)/2} − x_{0}^{p−1})dx_{0} − (\frac{∂G_{1}}{∂y_{0}} )^{p} dG_{1}.

Using Lemma 9.1, dx_{0} = (2y_{0}/f′) dy_{0} and G_{1} = y_{0} H_{1}, one sees that the expression above is equal to

(A^{−1}(f′)^{p} − f^{(p−1)/2} f′ − f′ H_{1} − 2 f H′_{1})^{p} \frac{A^{−1}}{f′} dy_{0},

which, by Lemma 8.2, is equal to zero. □

With the help of this theorem, we can prove the following proposition that deals with the case of minimal degree lifts of elliptic curves. Proposition 9.3. Let

ν(x_{0}, y_{0}) = (x_{0}, F_{1}, F̃_{2}, y_{0}, G_{1}, G̃_{2})

be the minimal lift from E (ordinary) to E/W_{3}(k), such that, modulo p^{2}, E is the canonical lift and ν gives us the Teichmüller lift. We have a lift of the Frobenius associated to ν if, and only if, deg F̃_{2} = (3p^{2} − 1)/2 (and then ν is the absolute minimal degree lift and E is the canonical lift also modulo p^{3}).

Proof. Assume that we have a lift of ϕ associated to ν. So

F̃_{2} − x_{0}^{p(p−1)} F_{1} − ψ(F_{1}) − (F′_{1})^{p} F_{1}

is a p-th power by Proposition 2.7, and thus cannot have a term of degree (3p^{2} + 1)/2. Since all terms in the above equation, except possibly F̃_{2}, have degrees less than or equal to (3p^{2} − 1)/2, deg F̃_{2} ≤ (3p^{2} − 1)/2, and hence ν is the absolute minimal degree lift.

The converse is a trivial consequence of the Theorem 9.2.

□

10. Mochizuki Lifts

As mentioned in section 2, curves of genus g > 1 do not have lifts of the Frobenius (see [Ray83]), but Mochizuki showed in [Moc96] that a Mochizuki-ordinary (defined in section 11) curve of genus g admits a lift of the Frobenius with certain singularities. Although the theory is completely developed (over almost two hundred pages), few examples are known of Mochizuki lifts. In this section we give an example of a minimal degree lift that is also a Mochizuki lift. (Observe that Mochizuki lifts are supposed to have a lift of points with “small” degrees.)

MIN. DEG. LIFTS OF HYPERELL. CURVES 31

We first observe that the number of singularities of Mochizuki lifts has to be equal to (g−1)(p−1): indeed, Corollary 4.9, Proposition 4.10 and Definition 4.11 of [Moc96] on pg. 1116 and 1117, tell us that for the lifting of the Frobenius ϕ,

ht(ϕ) = −p(g − 1) + deg(C − U) = (1 − g)

where ht(ϕ) denotes the height of ϕ (which we will discuss in more detail below) and U is the open set on which we can lift the Frobenius. (Note that we have no marked points!) Thus, deg(C − U) = (p − 1)(g − 1).

Finding an explicit example of a Mochizuki lift was Voloch’s motivation for proving Theorem 2.4 for p = 3, d = 6 and n = 1: genus 2 curves are necessarily hyperelliptic, and if the two supersingular points form a set that is invariant under the hyperelliptic involution, we can assume that those points are at infinity. But Mochizuki’s theory is indeed invariant with respect to the hyperelliptic involution. Roughly, this is true because the hyperelliptic involution can be deformed along with an arbitrary deformation of the curve. Hence, in the genus 2 case, the supersingular points can always be put at infinity.

Note that the genus 2 case is the only one for which we can try to relate Mochizuki’s theory and minimal degree lifts, since for the former we have (p − 1)(g − 1) singularities and for the latter either 1 (if d is odd) or 2 (if d is even).

We will now establish the connection with Mochizuki’s theory, as outlined to the author by the referee of this paper. We will consider a genus 2 curve

C/k : y_{0}^{2} = f(x_{0}),

where k is a perfect field of characteristic 3 and deg f = 6, and assume it admits a Mochizuki lift

C/W_{2}(k) : y^{2} = f(x),

with supersingular points at infinity. As in [Bui96], the lift of the Frobenius defines a lift of points (between the affine parts) ν(x_{0}, y_{0}) = ((x_{0}, F_{1}), (y_{0}, G_{1})). (The lift of the Frobenius is then a lift of the Frobenius associated to ν.) Definition 10.1. Let ξ_{P} be a local Frobenius defined in a neighborhood of the point in P ∈ C with reduction P. Let t be a local parameter at P and let δ_{P} be the reduction modulo p of the rational function

\frac{1}{p}(ξ_{P}^{∗} (t^{σ}) − ϕ^{∗}(t^{σ})).

Then, the local height of ϕ at P, denoted by ht_{P} (ϕ), is zero if δ_{P} is regular at P, and equal to the order of the pole of δ_{P} at P otherwise. (As in [Moc96], pg. 1116.)

By Definition 4.7 and Proposition 4.8 in [Moc96], pg. 1116, we can define:

Definition 10.2. The height of the lift of the Frobenius, denoted by ht(ϕ), is given by

ht(ϕ) =^{def} ( ∑_{P ∈C}[k(P) : k] ht_{P} (ϕ)) − p(g − 1),  (10.1)

where ht_{P} (ϕ) is the local height at P and k(P) is the minimal field of definition of P over k.

We now compute the local heights. Since ϕ is regular on the affine part of C, the non-zero local heights can only occur at the points at infinity, say P_{1} and P_{2}. So, for i = 1, 2, we have that t = 1/x is a local parameter at P_{i}, the points at infinity of C. Hence

ξ_{P_{i}}^{∗}(t^{σ}) = t^{3} + 3 ⋯ = \frac{1}{x^{3}} + 3 · . . . ,

where the omitted terms are regular at P_{i}. On the other hand,

ϕ^{∗}(t^{σ}) = \frac{1}{ϕ^{∗}(x)} = \frac{1}{x^{3} + 3F_{1} + 9 · . . .},

where F_{1} is a lift of F_{1}. So,

\frac{1}{3}(ξ_{P_{i}}^{∗}(t^{σ}) − ϕ^{∗}(t^{σ})) = \frac{F_{1} + 3 · . . .}{x^{3}(x^{3} + 3F_{1} + 9 · . . .)} + …

where the omitted terms after the plus sign are regular at P_{i}. So

δ_{P_{i}} = \frac{F_{1}}{x_{0}^{6}} + …,

where the omitted terms are regular at P_{i}, and hence

ht_{P_{i}}(ϕ) = deg F_{1} − 6.

Thus, equation (10.1), in this case (p = 3 and g = 2), gives us

ht(ϕ) = 2 deg F_{1} − 15.

Remember that for us a Mochizuki lift has height less than or equal to (1 − g) = −1, and in this case this implies that deg F_{1} ≤ 7. Hence by Theorem 2.4, it determines an absolute minimal degree lifting, which proves item 1 of Theorem 2.8.

Theorem 2.4 also tells us that if deg F_{1} = 7 then the coefficient of x_{0}^{2} in f(x_{0}), say A, is non-zero. If we assume that k is algebraically closed (or work on some finite extension of k) we may assume that A = 1, and so we will consider f given by equations of the form

f(x_{0}) = x_{0}^{6} + α_{0}x_{0}^{4} + β_{0}x_{0}^{3} + x_{0}^{2} + γ_{0}x_{0} + δ_{0}.

But, with the linear change of variables

(x_{0}, y_{0}) ↦ (x_{0} + ϵ_{0}, y_{0}),

MIN. DEG. LIFTS OF HYPERELL. CURVES 33

with ϵ_{0} satisfying 2ϵ_{0}^{3}+α_{0} ϵ_{0}+β_{0} = 0 (again, using the fact the k is algebraically closed, or extending k), allows us to consider f given by

f(x_{0}) = x_{0}^{6} + a_{0}x_{0}^{4} + x_{0}^{2} + b_{0}x_{0} + c_{0},

which proves all but the last sentence of item 2 of 2.8. The last sentence follows from Corollary 3.8 on pg. 1048 of [Moc96], which implies that curves whose moduli are sufficiently general admit Mochizuki lifts.

Now, let C be given by an equation of the form

y_{0}^{2} = f(x_{0}) = x_{0}^{6} + a_{0}x_{0}^{4} + x_{0}^{2} + b_{0}x_{0} + c_{0}.

We want to compute a Mochizuki lift of such curve. But as remarked above, this has to have deg F_{1} = 7 and hence is an absolute minimal degree lift, and thus Theorem 2.4 gives the a formula for the derivative of F_{1}. Using an algorithm similar to the one described in section 6 of [Fin02], we obtain an absolute minimal degree curve

C/W_{2}(k) : y^{2} = x^{6} + ax^{4} + x^{2} + bx + c,

and an absolute minimal degree lift ν(x_{0}, y_{0}) = ((x_{0}, F_{1}), (y_{0}, y_{0}H_{1})). If a_{0} ≠ 0 (the case when the curve is not Mochizuki-ordinary, as we shall see in the next section), we obtain

F_{1} = x_{0}^{7} + \frac{b_{0}}{a_{0}^{2}} x_{0}^{6} + 2 a_{0} x_{0}^{5}+

\frac{a_{0}^{4} c_{0}^{2} + 2 a_{0}^{4} + 2 a_{0}^{3} b_{0}^{2} + 2 a_{0}^{3} c_{0} + 2 a_{0}^{2} b_{0}^{2} c_{0} + 2 a_{0}^{2} c_{0}^{2} + a_{0}^{2} + b_{0}^{4}}{a_{0}^{2}} x_{0}^{3}

+ 2 b_{0} x_{0}^{2} + c_{0} x_{0} + \frac{a_{0}^{4} b_{0} + 2 a_{0}^{3} b_{0} c_{0} + a_{0}^{2} b_{0} + b_{0}}{a_{0}^{5}}

H_{1} = (2 a_{0}^{3} + a_{0}) x_{0}^{4} + 2 a_{0} b_{0} x_{0}^{3} + (2 a_{0}^{2} + 1) x_{0}^{2} + (2 a_{0}^{2} b_{0} + b_{0}) x_{0} + 2 a_{0}^{2} c_{0} + b_{0}^{2}

a_{1} = 2 a_{0}^{5} c_{0}^{2} + 2 a_{0}^{5} + a_{0}^{4} b_{0}^{2} + a_{0}^{4} c_{0} + a_{0}^{3} b_{0}^{2} c_{0} + a_{0}^{3} c_{0}^{2} + 2 a_{0}^{3} + a_{0}^{2} c_{0} + 2 a_{0} b_{0}^{4} + a_{0} + 2 b_{0}^{2} + c_{0}

b_{1} = (2 a_{0}^{7} b_{0}^{3} c_{0}^{2} + 2 a_{0}^{7} b_{0}^{3} + a_{0}^{6} b_{0}^{5} + a_{0}^{6} b_{0}^{3} c_{0} + a_{0}^{6} b_{0} c_{0}^{2} + a_{0}^{5} b_{0}^{5} c_{0} + a_{0}^{5} b_{0}^{3} c_{0}^{2} +2 a_{0}^{5} b_{0}^{3} + a_{0}^{5} b_{0} c_{0} + a_{0}^{4} b_{0} + 2 a_{0}^{3} b_{0}^{7} + 2 a_{0}^{3} b_{0} c_{0} + a_{0}^{2} b_{0} + b_{0}) a_{0}^{−5}

c_{1} = \frac{a_{0}^{7} c_{0}^{3} + 2 a_{0}^{5} b_{0}^{2} c_{0}^{2} + 2 a_{0}^{4} b_{0}^{4} + a_{0}^{3} b_{0}^{4} c_{0} + 2 a_{0}^{2} b_{0}^{4} + 2 b_{0}^{4}}{a_{0}^{5}}

For a_{0} = 0 we have:

F_{1} = x_{0}^{7} + 2 b_{0} x_{0}^{6} + (2 b_{0}^{4} + 2 b_{0}^{2} c_{0} + 2 c_{0}^{2} + 1) x_{0}^{3} + 2 b_{0} x_{0}^{2} + c_{0} x_{0}
H_{1} = x_{0}^{2} + b_{0} x_{0} + b_{0}^{2}
a_{1} = 2 b_{0}^{2} + c_{0}
b_{1} = b_{0}^{7} + b_{0}^{5} c_{0} + b_{0}^{3} c_{0}^{2} + 2 b_{0}^{3} + b_{0} c_{0}

34 LUÍS R. A. FINOTTI

c_{1} = 2 b_{0}^{2} c_{0}^{2}

Since we could obtain deg F_{1} = 7, Proposition 4.10 on pg. 1117 of [Moc96] tells us that we indeed computed a Mochizuki lift. (Note that Theorem 4.1 allows us to explicitly compute the lift of the Frobenius.) Hence we proved all but the last sentence of item 3 of Theorem 2.8, which we prove in the next section.

One can now proceed to compute a minimal degree lifting modulo 27. (Note we do not have an analogue to Proposition 4.10 on pg. 1117 of [Moc96] modulo p^{3}.) We again can achieve the lower bound given by Theorem 2.4, namely deg F_{2} = 25. So, Theorem 9.2 tells us that there exists a lift of the Frobenius also modulo 27. The formulas for F_{2}, H_{2}, P and Q (the latter two as in the statement of Proposition 2.7) are too long to be given here, but can be easily computed with the help of a computer using the analogue to the modified algorithm for elliptic curves described in the end of section 6 of [Fin02] and Proposition 2.7.

11. Ordinariness

We finally prove the last sentence of item 3 in Theorem 2.8, namely that given a curve

C/k : y_{0}^{2} = x_{0}^{6} + a_{0}x_{0}^{4} + x_{0}^{2} + b_{0}x_{0} + c_{0},  (11.1)

where k is a perfect field of characteristic 3, then its Mochizuki lift being Mochizuki-ordinary (or hyperbolic-ordinary, as Mochizuki refers to it in [Moc96]), and its Jacobian being an ordinary Abelian variety, are both equivalent to a_{0} being non-zero. Usually, when one simply says that C is ordinary, we understand that its Jacobian is ordinary. (Mochizuki refers to this usual notion of ordinariness as parabolic-ordinariness.) It is not true in general that these two notions are the same, as happens in this particular case.

We first show that a_{0} ≠ 0 if, and only if, then the Jacobian of C is an ordinary Abelian variety. We recall that the Jacobian of C is ordinary if, and only if, the restriction of the Cartier operator to global differentials on the curve

C|_{Γ(C,Ω_{C/k})} : Γ(C, Ω_{C/k}) → Γ(C, Ω_{C/k})

is surjective. Let ω_{0} =^{def} dx_{0}/y_{0} and ω_{1} =^{def} x_{0} ω_{0}. One has

C(ω_{1}) = a_{0}^{1/3}ω_{1} + b_{0}^{1/3}ω_{0},
C(ω_{0}) = ω_{0},

and so, C|_{Γ(C,Ω_{C/k})} is surjective if, and only if, a_{0} ≠ 0.

We now briefly recall what it means to say that a Mochizuki lift of C is Mochizuki-ordinary. Let T_{C^{σ}/k} and T_{C/k} denote the relative tangent bundles of C^{σ}/k and C/k respectively, and let

H_{E} : ϕ^{∗}(T_{C^{σ}/k}) → T_{C/k}

MIN. DEG. LIFTS OF HYPERELL. CURVES 35

be the square Hasse invariant of an indigenous bundle (E, ∇_{E}) ([Moc96], Proposition 2.6(1), pg. 1032).

Dualizing H_{E} yields

H_{E}^{∨} : Ω_{C/k} → ϕ^{∗}(Ω_{C^{σ}/k}) ≅ Ω_{C/k}^{⊗3}.

Tensoring with Ω_{C/k} gives a map

id ⊗H_{E}^{∨} : Ω_{C/k}^{⊗2} → Ω_{C/k} ⊗ ϕ^{∗}(Ω_{C^{σ}/k}),

and now pushing forward by ϕ yields

ϕ_{∗}(id ⊗H_{E}^{∨}) : ϕ_{∗}(Ω_{C/k}^{⊗2}) → ϕ_{∗}(Ω_{C/k}) ⊗ Ω_{C^{σ}/k}.  (11.2)

Next we recall that there is a Cartier isomorphism (see [Kat70])

C̃: H^{1}(ϕ_{∗}Ω_{C/k}^{•}) → Ω_{C^{σ}/k},

related to the Cartier operator C as follows: if we also denote by C̃ the map between ϕ_{∗}(Ω_{C/k}) and Ω_{C^{σ}/k} (having the exact differentials as its kernel) that induces C̃: H^{1}(ϕ_{∗}Ω_{C/k}^{•}) → Ω_{C^{σ}/k}, then for all ω ∈ Γ(C, ϕ_{∗}(Ω_{C/k})) one has C̃(ω) = C(ω)^{σ}. The Cartier isomorphism induces a map

C̃⊗ id : ϕ_{∗}(Ω_{C/k}) ⊗ Ω_{C^{σ}/k} → Ω_{C^{σ}/k}^{⊗2}  (11.3)

The composition of the maps given by equations (11.2) and (11.3) induces a map on global sections

V : Γ(C, Ω_{C^{/}k}^{⊗2}) → Γ(C^{σ}, Ω_{C^{σ}/k}^{⊗2}),

called the Verschiebung. (This is the analogue of the map Φ_{E}^{ω} on pg. 1037 of [Moc96].) Now, by Definition 3.1 on pg. 1044 and Proposition 2.12 on pg. 1037 of [Moc96], the curve the Mochizuki lift of C/k is Mochizuki-ordinary if V is surjective.

We now prove that if C/k is given by (11.1), then V is surjective if, and only if, a_{0} ≠ 0. This proof was also outlined by the referee (and was further clarified to the author by S. Mochizuki).

To understand V, we first look at the square Hasse invariant H_{E}. This map (and its dual) is determined by the “multiplication” by a quadratic differential θ_{0} on C/k. By Proposition 2.6 on pg. 1032 of [Moc96], the divisor of zeros of such a quadratic differential is the double supersingular locus of C/k, i.e., its support is the set of points where the lift of the Frobenius is not defined. In our case, these points are the points at infinity, and hence θ_{0} has to be a non-zero constant multiple of ω_{0}^{2}, and we can assume that θ_{0} = ω_{0}^{2}. Hence,

H_{E}^{∨}((λ_{1}x_{0} + λ_{0})ω_{0}) = (λ_{1}x_{0} + λ_{0})ω_{0}^{σ} ≈ (λ_{1}x_{0} + λ_{0})ω_{0}^{3},  (11.4)

where “≈” is the identification via the isomorphism between ϕ^{∗}(Ω_{C^{σ}/k}) and Ω_{C/k}^{⊗3}.

Let θ_{1} =^{def} x_{0} θ_{0} and θ_{2} =^{def} x_{0}^{2} θ_{0}. Then, equation (11.4) implies that the map induced by (11.2) on global sections is given by

λ_{2}θ_{2} + λ_{1}θ_{1} + λ_{0}θ_{0} ↦ (λ_{2}x_{0}^{2} + λ_{1}x_{0} + λ_{0})ω_{0} ⊗ ω_{0}^{σ}.

Hence

V(λ_{2}θ_{2} + λ_{1}θ_{1} + λ_{0}θ_{0}) = C̃((λ_{2}x_{0}^{2} + λ_{1}x_{0} + λ_{0})ω_{0}) ⊗ ω_{0}^{σ}.

Therefore,

V(θ_{0}) = θ_{0}^{σ},
V(θ_{1}) = a_{0}θ_{1}^{σ} + b_{0}θ_{0}^{σ},
V(θ_{2}) = θ_{2}^{σ} + c_{0}θ_{0}^{σ},

which clearly is surjective if, and only if, a_{0} ≠ 0.

This shows that the two notions of ordinariness are the same in the present situation and finishes the proof of Theorem 2.8.

Acknowledgement. The author would like to thank J. F. Voloch, J. Tate and S. Mochizuki for most valuable discussions and A. Agboola for his insightful comments. Also, the referee’s comments were of great importance to the connections with Mochizuki’s theory. The author acknowledges the use of the packages Magma and Mathematica in computations mentioned in the text. Finally, part of this work was done with the financial support from CAPES (Brazil).

References

[Bui96] A. Buium. Geometry of p-jets. Duke Math. Journal, 82:349–367, 1996.

[Deu41] M. Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hansischen Univ., 14:197–272, 1941.

[Fin01] L. R. A. Finotti. Canonical and Minimal Degree Liftings of Curves. PhD thesis, University of Texas at Austin, 2001.

[Fin02] L. R. A. Finotti. Degrees of the elliptic Teichmüller lift. J. Number Theory, 95(2):123–141, 2002.

[Kat70] N. M. Katz. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Études Sci. Publ. Math., (39):175–232, 1970.

[LST64] J. Lubin, J-P. Serre, and J. Tate. Elliptic curves and formal groups. Proc. of Woods Hole summer institute in algebraic geometry, 1964. Unpublished. Available at http://www.ma.utexas.edu/users/voloch/lst.html.

[Moc96] S. Mochizuki. A theory of ordinary p-adic curves. Publ. Res. Inst. Math. Sci., 32:957–1152, 1996.

[Poo01] B. Poonen. Computing torsion points on curves. Experiment. Math., 10(3):449–465, 2001.

[Ray83] M. Raynaud. Around the Mordell conjecture for function fields and a conjecture of Serge Lang. Lecture Notes in Math., 1016:1–19, 1983.

[Sat00] T. Satoh. The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc., 15(4):247–270, 2000.

[Ser79] J-P. Serre. Local Fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979.

[Tat02] J. Tate. On a conjecture of Finotti. Bull. Braz. Math. Soc. (N.S.), 33(2):225–229, 2002.

[Vol97] J. F. Voloch. Torsion points of y^{2} = x^{6} + 1. unpublished manuscript, 1997. available at http://www.ma.utexas.edu/users/voloch/oldpreprint.html.

[VW99] J. F. Voloch and J. L. Walker. Codes over rings from curves of higher genus. IEEE Trans. Inform. Theory, 45:1768–1776, 1999.

[VW00] J. F. Voloch and J. L. Walker. Euclidean weights of codes from elliptic curves over rings. Trans. Amer. Math. Soc., 352(11):5063–5076, 2000.

University of California, Department of Mathematics, Santa Barbara, CA – 93106

   E-mail address: finotti@math.ucsb.edu