MINIMAL DEGREE LIFTINGS IN CHARACTERISTIC 2

LUÍS R. A. FINOTTI

Abstract. In this paper we analyze liftings of hyperelliptic curves over perfect fields in characteristic 2 to curves over rings of Witt vectors. This theory can be applied to construct error-correcting codes, and lifts of points with minimal degrees are likely to yield the best codes, and these are the main focus of the paper. We find upper and lower bounds for their degrees, give conditions to achieve the lower bounds and analyze the existence of lifts of the Frobenius. Finally, we exhibit explicit computations for genus 2 and show codes obtained using this theory.

1. Introduction

In this paper we analyze liftings of hyperelliptic curves over perfect fields of characteristic 2 to curves over rings of Witt vectors (characteristic 0). As we shall discuss later, these liftings can be used to construct error-correcting codes, and we will focus on particular kinds of liftings that are likely to yield good codes. For reasons that will become quite clear later on, we shall refer to those particular liftings as minimal degree liftings.

The case of characteristic p > 2 was dealt with in [Fin04], and we shall prove here similar results for p = 2. We observe that for concrete applications to coding theory, this case is especially important, since we can obtain binary codes, which can be effectively implemented.

As with the case p > 2, these minimal degree liftings are also of independent interest, and although we shall keep their applications to coding theory in mind, we will not restrict ourselves only to results that are relevant to this particular aspect. We shall also study, for instance, liftings of the Frobenius and the relations of minimal degree liftings with canonical liftings.

This paper is organized as follows: in section 2 we give a brief introduction to algebraic geometric codes over rings, which was first introduced by Walker in [Wal99]. Our goal in this session is to only give a rough idea of how such codes are obtained and to motivate the introduction of minimal degree liftings.

1991 Mathematics Subject Classification. Primary 11G20; Secondary 11T71.

Key words and phrases. hyperelliptic curves, canonical liftings, minimal degree liftings, algebraic geometric codes over rings.

2 LUÍS R. A. FINOTTI

In section 3 we introduce some definitions and notation that will be used throughout this paper and in section 4 we give a precise definition of minimal degree liftings.

In section 5 we discuss liftings of the Frobenius and define the notion of a lift of the Frobenius associated to a lift of points.

With the background established by the previous sections, we are able to state in section 6 the main results of this paper, while leaving the proofs to the later sections.

In section 7 we prove some technical results about Witt vectors and valuations which are then used in the proofs of the main theorems.

Sections 8 and 9 contain the proofs of the upper and lower bounds, respectively, for the degrees of the minimal degree liftings, while section 10 has the proofs of the statements about achieving degrees exactly equal to the lower bounds.

Section 11 contains the proofs of the main results about lifting the Frobenius map to characteristic zero. In particular, it proves that minimal degree liftings satisfying the established lower bounds always have a lift of the Frobenius modulo 8 associated to the corresponding lift of points.

In section 12, we exhibit explicit examples of lifts (modulo 16) of hyperelliptic curves of genus 2 whose degrees are equal to the lower bounds.

Finally, section 13 contains examples of error-correcting codes.

2. Algebraic Geometric Codes over Rings

In this section we follow sections 2 and 3 of [VW00], adapting the notation and some results to the particular cases in which we are interested here.

Let k be a finite field of characteristic p (not necessarily equal to 2) and W_{l}(k) be the ring of Witt vectors of length l over k. Also, let C/W_{l}(k) be a projective curve with good reduction modulo p, C/k be its reduction modulo p, 𝒫 =^{def} {P_{1}, …, P_{n}} be a set of W_{l}(k)- rational points of C with distinct reductions modulo p, say {P_{1}, …, P_{n}}, D be a Cartier divisor of C such that no P_{i} is in the support of D, and ℒ =^{def} ℒ(D) be the sheaf associated to D (as in section II.6 of [Har77]). Observe that one can think of ℒ as a set of functions on C which are regular on the support of D.

Definition 2.1. Let C, 𝒫 and ℒ be as above. We define the algebraic geometric code over W_{l}(k) associated to C, 𝒫 and ℒ, denoted by C_{W_{l}(k)}(C, 𝒫, ℒ), as

C_{W_{l}(k)}(C, 𝒫, ℒ) =^{def} {(f(P_{1}), …, (f(P_{n})) : f ∈ ℒ} ⊂ (W_{l}(k))^{n}.

The next theorem summarizes some of the main results of [Wal99]:

Theorem 2.2 (Walker). Let C, 𝒫 =^{def} {P_{1}, …, P_{n}}, D, ℒ and C =^{def} C_{W_{l}(k)}(C, 𝒫, ℒ) be as above and let g denote the genus of the curve C. If (2g − 2) < deg D < n, then C is a linear code over W_{l}(k) which is free as a W_{l}(k)-module. Moreover, C has rank deg D + 1 − g and minimum Hamming distance at least n − deg D.

We are particularly interested in codes over ℤ/p^{l}ℤ, where p is a prime. If k = 𝔽_{p}, the field of p elements, then W_{l}(k) ≅ ℤ/p^{l}ℤ, and the algebraic geometric codes above are then codes over ℤ/p^{l}ℤ. On the other hand, if k is a finite field of characteristic p, but k ≠ 𝔽_{p}, one can apply the trace tr : W_{l}(k) → W_{l}(𝔽_{p}) to each coordinate of all codewords to obtain codes over ℤ/p^{l}ℤ

When p = 2 we can obtain non-linear binary codes from codes over ℤ/2^{l}ℤ by using the generalized Gray map, defined by Carlet in [Car98], which is a map G : (ℤ/2^{l}ℤ) → 𝔽_{2}^{2^{l−1}} such that the Hamming weight of G(x − y) is equal to the Hamming distance between G(x) and G(y), i.e., G is distance preserving. The binary codes are obtained by applying G coordinatewise to the codewords of the ℤ/2^{l}ℤ-codes.

Since we are mainly interested in these binary codes, whenever l = 2 (and p = 2) the Lee weight for the codes over ℤ/4ℤ is of interest, instead of the the Hamming weight, since in this case the Lee weight of a codeword over ℤ/4ℤ is equal to the Hamming weight of the image of this codeword under the generalized Gray map applied coordinatewise. (Observe that for l = 2, G is equal to the original Gray map, defined by G(0) = (0, 0), G(1) = (0, 1), G(2) = (1, 1) and G(3) = (1, 0).)

A computation (see [VW00]) shows that if x⃗ = (x_{1}, …, x_{n}) is a codeword of length n over ℤ/4ℤ, then the Lee weight of x⃗, denoted by w_{L}(x⃗), satisfies

w_{L}(x⃗) ≥ n − |∑_{j=1}^{n} e^{2πix_{j}/4}| .  (2.1)

Hence, to find lower bounds to the minimum Lee weight of a code over ℤ/4ℤ, it suffices to find an upper bound for the exponential sum above, and thus, in the case of algebraic geometric codes (with l = 2), where the codewords are of the form (tr(f(P_{1}), …, tr(f(P_{n})), with f ∈ ℒ and 𝒫 = {P_{1}, …, P_{n}}, we need to find upper bounds for the absolute value of the exponential sums

S_{f} =^{def} ∑_{j=1}^{n} e^{2πi tr(f(P_{j})/4},

for f ∈ ℒ.

In fact, more generally, if p^{l} ≠ 4, one can use Euclidean weights instead of the Lee weight: given x ∈ ℤ/p^{l}ℤ, the Euclidean weight of x is given by

w_{E}(x) =^{def} \sqrt{2 − 2 cos (\frac{2πx}{p^{l}})},

4 LUÍS R. A. FINOTTI

and if x⃗ = (x_{1}, …, x_{n}), then

w_{E}^{2}(x⃗) =^{def} ∑_{j=1}^{n} w_{E}^{2}(x_{j}).

When p^{l} = 4, ones has that w_{L}(x) = 1/2 w_{E}^{2}(x). But, in general, we have that

w_{E}^{2}(x⃗) ≥ 2n − 2 |∑_{j=1}^{n} e^{2πix_{j}/p^{l}} | ,

and if x⃗ is a codeword of the form (tr(f(P_{1}), …, tr(f(P_{n})), as above, but with arbitrary p and l, then to find a lower bound to the Euclidean weight of the code, it suffices to find an upper bound for the absolute value of

S_{f} =^{def} ∑_{j=1}^{n} e^{2πi tr(f(P_{j})/p^{l}},  (2.2)

for f ∈ ℒ.

Let then k be a finite field of characteristic p, C/k and C/W_{l}(k) be curves as before and U and U be open subsets of C and C, respectively, such that U is the reduction modulo p of U. Also, let ν be a lift of points on U, i.e., a map ν : U(k̄) → U(W_{l}(k̄)) that is a section of the reduction modulo p. So, if ν(P) = P and f is a function regular in P, then f(P) = f ∘ ν(P), and f ∘ ν = (f_{0}, f_{1}, …, f_{l−1}), where each f_{i} ∈ k(C), for i = 0, …, l − 1, and k(C) denotes the function field of C. Since f is regular at P, all the f_{i}’s are regular at P.

So, fix a function f and let

𝒫 = {P_{1}, …, P_{n}} = {ν(P) : P ∈ U(k) and f is regular at ν(P)}.  (2.3)

In this situation, Voloch and Walker found an upper bound for the exponential sum (2.2), more precisely, Theorem 3.1 of [VW00] states:

Theorem 2.3 (Voloch-Walker). Let k = 𝔽_{q} be the field of q elements with characteristic p, f be a function on the curve C/W_{l}(k) and 𝒫 be as in equation (2.3). Let also {Q_{1}, …, Q_{s}} be the set of poles (in C(k̄)) of the coordinates of f ∘ ν, and v_{Q_{i}} be the valuation of k(C) given by the order of vanishing at Q_{i}. Finally, let g be the genus of the curve C and assume that f ∘ ν is not of the form σ(g) − g + c for any g ∈ W_{l}(k(C)) and c ∈ W_{l}(k), where

σ(g_{0}, …, g_{l}) =^{def} (g_{0}^{p}, …, g_{l}^{p}). Then

|S_{f}| ≤ (2g − 1 + ∑_{j=1}^{s} max_{0≤i≤(l−1)}{−p^{l−1−i} v_{Q_{j}}(f_{i})} [k(Q_{j}) : k]) q^{1/2},  (2.4)

where k(Q_{j}) is the minimal field of definition of Q_{j}.

MIN. DEG. LIFTS IN CHAR. 2 5

Hence to have a larger bound for the Euclidean weight of the algebraic geometric codes over ℤ/p^{l}ℤ, and hence larger Lee weight when p^{l} = 4, we need to have a smaller value for |S_{f}|, and thus we want the coordinate functions of f ∘ ν to have small order of poles at the Q_{i}’s, and in order to have the best possible bounds, we look for lifts of points ν that yield minimal order of poles. This will be the motivation for our definition of minimal degree liftings.

3. Conventions and Definitions

Before we state the main results, we need to establish some notation and review a few previous results.

Throughout this paper, k will be a perfect field of characteristic 2. (For the applications to coding theory k will be a finite field, but the theoretical results hold in the more general case of perfect fields.) Also let

C/k : y_{0}^{2} + g(x_{0}) y_{0} = f(x_{0}),  (3.1)

be a (non-singular and projective) hyperelliptic curve over k, where f(x_{0}) is a monic polynomial of odd degree (as a polynomial in x_{0}), which we shall denote by d, and g(x_{0}) has degree (as a polynomial in x_{0}) less than or equal to (d − 1)/2. Therefore, C has genus (d − 1)/2, there is only one point at infinity, which we shall denote by P_{∞}, and the polynomials g(x_{0}) and (f′(x_{0}) + g′(x_{0}) y_{0}) have no common zeros on C (since C is non-singular).

Let ω denote the holomorphic differential

ω =^{def} \frac{dy_{0}}{f′(x_{0}) + g′(x_{0}) y_{0}} = \frac{dx_{0}}{g(x_{0})}  (3.2)

which has no zeros at the affine part of C and a zero of order (2d − 2) − (d + 1) = (d − 3) at P_{∞}.

Also observe that, by the Riemann-Roch Theorem, or more precisely, by [Sil85] Corollary II.5.5(b), every hyperelliptic curve over k of genus (d − 1)/2, for any odd number d, can be put in that form.

Also, let

C/W(k) : y^{2} + g(x) y = f(x)  (3.3)

be a hyperelliptic curve over the ring of Witt vectors W(k), where f is a monic polynomial of degree d (i.e., same degree as f), the degree of g is less than or equal to (d − 1)/2, and such that the reductions of f and g modulo 2 are f and g respectively (i.e, C is the reduction modulo 2 of C).

Note that since C is non-singular, C is also non-singular, and so 2y + g(x) and f′(x) − g′(x) y have no common zeros on C. And, similarly as done with C, let P_{∞} denote the point at infinity of C and let ω denote the holomorphic differential

ω =^{def} \frac{dy}{f′(x) − g′(x) y} = \frac{dx}{2y + g(x)}.

We shall often identify C with its Greenberg transform G(C), which is an infinite dimensional scheme over k, defined in the following manner: writing x = (x_{0}, x_{1}, …) and y = (y_{0}, y_{1}, …), where the x_{i}’s and y_{i}’s are variables, one can expand both sides of equation (3.3) using the addition and multiplication of Witt vectors. The equations (on the x_{i}’s and y_{i}’s) obtained by comparing the coordinates of the Witt vectors in both sides of this expansion are the equations that define G(C).

As mentioned in section 2, in our construction of codes we shall use lifts of points between open sets of C and C. For us these open sets will always be the affine parts of C and C. So, let U and U denote the affine parts of C and C respectively. We then define:

Definition 3.1. A lift of points from C/k to C/W(k) is a regular map

ν : U(k̄) → U(W(k̄)) ≈ G(U)(k̄)

which is a section of the reduction modulo 2. (Hence, in this paper, we consider lifts of points between the affine parts only.)

In terms of the Greenberg transform, a lift of points

ν : U(k̄) → U(W(k̄))

can be written as

ν(x_{0}, y_{0}) = (x_{0}, F_{1}(x_{0}, y_{0}), F_{2}(x_{0}, y_{0}), …, y_{0}, G_{1}(x_{0}, y_{0}), G_{2}(x_{0}, y_{0}), …),  (3.4)

where, since this map cannot have any poles in the affine part of C, we have in fact that F_{i}, G_{i} ∈ k[x_{0}, y_{0}]. Also, we will write F_{0} =^{def} x_{0} and G_{0} =^{def} y_{0}.

Remember that we are after lifts ν that will yield minimal order of poles for the coordinates of compositions of ν with functions in C. In our applications our divisor D (as in section 2) will be a positive multiple of P_{∞}, and hence the functions in ℒ(D) are polynomials in W(k)[x, y]. Therefore, to have the best lower bound for the minimum Euclidean (or Lee weight, if p^{l} = 4), we need to have lifts ν such that the coordinate functions of

x ∘ ν = (x_{0}, F_{1}, F_{2}, …)

and

y ∘ ν = (y_{0}, G_{1}, G_{2}, …)

have minimal order of poles at P_{∞}.

MIN. DEG. LIFTS IN CHAR. 2 7

Observe that in both [Fin02] and [Fin04] we referred to “degrees” as degrees as polynomials in x_{0}. To say that the F_{i}’s and G_{i}’s have minimal degrees as polynomials is the same as to say that these functions have minimal order of poles at P_{∞}. This was convenient in those papers since we mostly dealt with polynomials in a single variable, namely with k[x_{0}]. The case of characteristic 2 is rather different and we will have to deal with polynomials in two variables, and so we will adopt a different convention here.

Definition 3.2. Let h be a function in the k(C). Then the degree of h, denoted by deg h, is defined as the number of poles of h counted with multiplicity. (Note that this is the same as to define deg h =^{def} [k(C) : k(h)], by [Sil85] Proposition II.2.6(a).)

Hence, deg x_{0} = 2 and deg y_{0} = d. Whenever we need to refer to the degree of some function as a polynomial, we shall explicitly say so.

So our goal is to obtain lifts of points ν whose coordinate functions F_{i}’s and G_{i}’s have minimal degrees.

4. Minimal Degree Liftings

As in [Fin04], we have a few different choices when dealing with minimal degree liftings. First, one can choose which coordinate is to have its degrees minimized, i.e., one can either minimize the degrees of the F_{i}’s (as in equation (3.4)) or of the G_{i}’s. In general, one cannot minimize both at the same time. Secondly, one can either assume that the curves C and C are fixed a priori, and then find lifts of points between (the affine parts of) those curves with minimal possible degrees (for the chosen coordinate), or we can assume that only C is fixed, and then find a lift C of C which has a lift of points with minimal degrees (for the chosen coordinate) among all lifts of points from C to any other lift C̃.

We will make these notions precise in the following two definitions below.

Definition 4.1. Let C and C be curves given by equations (3.1) and (3.3) respectively. A minimal degree lifting from C to C/W_{2}(k) with respect to y (resp., x) is a lift of points ν : U(k̄) → U(W_{2}(k̄)), with

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

where deg G_{1} (resp., deg F_{1}) is minimal.

Inductively, a minimal degree lifting from C to C/W_{n+1}(k) with respect to y (resp., x) is a lift of points ν : U(k̄) → U(W_{n+1}(k̄)), with

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

where the reduction modulo 2^{n} is a minimal degree lifting from C to C/W_{n}(k), and deg G_{n} (resp., deg F_{n}) is minimal.

8 LUÍS R. A. FINOTTI

Definition 4.2. Let C be a hyperelliptic curve given by (3.1). An absolute minimal degree curve modulo 4 over C with respect to y is a curve C/W_{2}(k) (given by (3.3)) which reduces to C modulo 2, and which satisfies the following property. Let

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

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

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

from C to C̃, we have deg G̃_{1} ≥ deg G_{1}.

Inductively, an absolute minimal degree curve modulo 2^{n+1} over C with respect to y is a curve C/W_{n+1}(k) whose reduction modulo 2^{n} is an absolute minimal degree curve modulo 2^{n} over C with respect to y, 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 lifting with respect to y from C to C, and let C̃/W_{n+1}(k) be any curve whose reduction modulo 2^{n} is equal to the reduction modulo 2^{n} of C. Then, for a minimal degree lifting with respect to y

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 G̃_{n} ≥ deg G_{n}. In this case we call the minimal degree lift ν from C to C an absolute minimal degree lift (of points) with respect to y (modulo 2^{n+1}).

We also have the analogous definitions with respect to x, rather than y.

In contrast with the case of p > 2, where liftings with respect to x and y had very similar properties, here these two liftings are quite different. One will notice that the lifting with respect to y in this case has properties very similar to the properties we have when p > 2, but for the x coordinates, the cases p = 2 and p > 2 are quite different.

One of the author’s first motivations to deal with minimal degree liftings was to try to lower the degrees of the elliptic Teichmüller lift, which is a special lift of points in the case of ordinary elliptic curves (i.e., genus 1 or d = 3), which we shall now briefly describe.

Let k be a perfect field of characteristic p > 0. (We do not assume that p = 2 for this part.) 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)) (not only between the affine parts, but for the whole projective curves), called the elliptic Teichmüller lift, characterized by the following properties:

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

MIN. DEG. LIFTS IN CHAR. 2 9

(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]. Also, one can clearly use ordinary elliptic curves and the elliptic Teichmüller lift in the construction of codes described in section 2, and, in fact, the codes constructed by Voloch and Walker in [VW00] were obtained this way.

In order to improve the bounds for the codes that would be obtained using the Teichmüller lift

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

we would need to reduce the degrees of the F_{i}’s and G_{i}’s. Proposition 4.2 in [VW00] states that the degrees of F_{1} and G_{1} cannot be improved. On the other hand, the degrees of the F_{i}’s and G_{i}’s for i ≥ 2 can almost always be improved. If p ≥ 3, then one can choose to either reduce the degrees of the F_{i}’s without increasing the degrees of the G_{i}’s, or the other way around. On the other hand, for p = 2, we can still reduce the degrees of the G_{i}’s without increasing the degrees of the F_{i}’s, but we cannot reduce the degrees of the F_{i}’s without increasing the degrees of the G_{i}’s. In fact, in this case, the degrees of the G_{i}’s increase considerably. Hence, it seemed more natural to first consider minimal degree liftings in characteristic 2 with respect to y instead of x.

5. Lifts of the Frobenius

In this section we briefly discuss lifts of the Frobenius. We probably should start by observing that the question of whether or not such lift exists is of purely theoretical interest

10 LUÍS R. A. FINOTTI

and has no application to the explicitly construction of codes or apparent relation with how good the obtained codes are.

Let C be a hyperelliptic curve given by equation (3.1). Let, as before, σ denote the Frobenius of k (now of characteristic 2), and C^{σ} be the curve given by the zeros of the equation defined by applying σ to the coefficients of the equation that defines C. Then, the relative Frobenius is a morphism

ϕ : C → C^{σ}

defined by ϕ(x_{0}, y_{0}) =^{def} (x_{0}^{2}, y_{0}^{2}).

Certainly in characteristic zero one does not have a naturally defined analogue of the Frobenius map. On the other hand, we can look for liftings of C for which we have a lift of the Frobenius. More precisely, if σ denote also the Frobenius in W(k), can one find a curve C, given by an equation as in (3.3) for which there is a morphism

ϕ : C → C^{σ},

such that the reduction modulo 2 gives the Frobenius of C (as in the case of ordinary elliptic curves)? In general, the answer is no. Raynaud showed in [Ray83] that curves of genus greater than one have no lift of the Frobenius. Moreover, only ordinary elliptic curves have lifts of the Frobenius. On the other hand, for (smooth) affine curves over W_{n}(k) with good reduction, there is always a lifting of the Frobenius. (Although not entirely immediate, a proof of this result in the case of plane curves can be obtained by using Lemma 8.1 and a slightly modified version of Lemma 11.1.)

As one can see from [Bui96] and [Fin04], lifts of points and lifts of the Frobenius are somewhat related. (For example, Theorem 4.1 in [Fin04] states that having a lift of points modulo p^{n+1} gives a lift of the n-th power of the Frobenius modulo p^{n+1}.) To make this connection between lifts of points and lifts of the Frobenius more precise, we introduce the following definition:

Definition 5.1. Let C/k and C/W_{n}(k) be curves given by equations (3.1) and (3.3) respectively and ν : U(k̄) → U(W(k̄)) be a lift of points 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)) ⟶^{ϕ} U^{σ}(W_{n}(k̄))
ν↑  ↑^{ν^{σ}}
U(k̄) ⟶^{ϕ} U^{σ}(k̄)

commute.

MIN. DEG. LIFTS IN CHAR. 2 11

Note that a lift of the Frobenius associated to a lift of points is, in principle, only a map between the affine parts of C and C^{σ}. Clearly, in the case of ordinary elliptic curves, the lift of the Frobenius associated to the elliptic Teichmüller lift can be extended to the whole curve.

So, one can ask about the existence of lifts of the Frobenius that are associated to particular lifts of points. For instance, for (projective) curves of genus greater than one in characteristic p > 2, Mochizuki showed in [Moc96] that (in most cases) there is a lift of the Frobenius in an open subset of the curve associated to a lift of points which has “small” degrees. In section 12, we show that some special examples of minimal degree liftings (in characteristic 2) have lifts of the Frobenius associated to them, at least modulo 8. Also, it is worth noting that in characteristic p > 2 we also often have lifts of the Frobenius associated to minimal degree liftings, and for p = 3, the Mochizuki liftings are in fact minimal degree liftings. (See [Fin04].)

6. Statements of Main Results

In this section we state the main results of this paper, while leaving the corresponding proofs for the later sections.

This first proposition, proved in section 8, gives upper bounds for the minimal degrees of the lifts of points with respect to y in the most general situation.

Proposition 6.1. Let C and C be as in equations (3.1) and (3.3) respectively. Then, the minimal degree lifting from C to C with respect to y,

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

satisfies

deg G_{1} ≤ 4(d − 1) + (d − 2),
deg F_{1} ≤ max{4, 2 deg g + (d − 2)},

and for n ≥ 2,

deg G_{n} ≤ 2^{n+1}(d − 1) + (d − 2),
deg F_{n} ≤ 2^{n} (2 + \frac{n 3(d − 2)}{2}) .

The next proposition, also proved in section 8, improves the upper bounds for the degrees of the F_{n}’s in Proposition 6.1, by the addition of extra conditions on the degrees of F_{1} and G_{1}, which, as we shall soon see in Proposition 6.9, are often satisfied.

Proposition 6.2. Let C and C be as in equations (3.1) and (3.3) respectively, and assume that the minimal degree lift of points with respect to y modulo 4 is such that

deg G_{1} ≤ 2 (d + \frac{5(d − 2)}{8}) ,
deg F_{1} ≤ 2 (2 + \frac{5(d − 2)}{8}) .

Then, the minimal degree lift of points from C to C with respect to y,

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

satisfies, for n ≥ 2,

deg G_{n} ≤ 2^{n−1}(d − 1) + (d − 2),
deg F_{n} ≤ 2^{n} (2 + \frac{n 5(d − 2)}{8}) .

As mentioned in section 4, if one now tries to minimize the degrees with respect to x, the degrees in the y coordinate tend to increase considerably.

Although this seems to ruin the possible obtained codes, one should notice that if the Cartier divisor in question, as in section 2, is of the form D = nP_{∞}, with n < d, then the functions in ℒ(D) are all powers of x, and hence the G_{i}’s are irrelevant to the code. Hence, it does make sense to consider such lifts.

Proposition 6.3. Let C and C be as in equations (3.1) and (3.3) respectively. Then, the minimal degree lift of points from C to C with respect to x,

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

satisfies, for n ≥ 1,

deg F_{n} ≤ 2^{n} deg g + d − 2,
deg G_{n} ≤ 2^{n}(d + ne_{n}),

where

e_{n} =^{def} max \{0, deg g + \frac{d − 6}{2}\} + (d − deg g) ∑_{j=1}^{n} \frac{1}{j}.

In the same spirit as Proposition 6.2, Proposition 6.4 improves the bounds on the degrees of the G_{n}’s in Proposition 6.3 by adding assumptions to the degrees of F_{1} and G_{1}, which again will often be satisfied as a consequence of Proposition 6.9.

Proposition 6.4. Let C and C be as in equations (3.1) and (3.3) respectively, with deg g ≤ (3d + 2)/4. If the minimal degree lift of points from C to C with respect to x,

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

is such that

deg F_{1} ≤ d + 2 and deg G_{1} ≤ 3d − 2,

then, for n ≥ 2,

deg F_{n} ≤ 2^{n} deg g + d − 2,
deg G_{n} ≤ 2^{n}(d + ne_{n}),

where

e_{n} =^{def} \frac{d − 2}{2} − (d − deg g) + (d − deg g) ∑_{j=1}^{n} \frac{1}{j}.

The next theorem gives lower bounds for the degrees of the G_{n}’s.

Theorem 6.5. Let C and C be as in equations (3.1) and (3.3), and ν, as in equation (3.4), be a lift of points between U and U, the affine parts of C and C respectively. Assume that

deg G_{i} = 2^{i+1}(d − 1) − (d − 2)

for i = 0, …, (n − 1). Then,

deg G_{n} ≥ 2^{n+1}(d − 1) − (d − 2).  (6.1)

Moreover, if the equality holds, then:

(1) the coefficient of x_{0} in g, say λ, is non-zero;

(2) g′ = λ;

(3) dG_{n} = λ^{−(2^{n}−1)}(f′ + g′y_{0})^{2^{n}−1} dy_{0} + ∑_{i=0}^{n−1} G_{i}^{2^{n−i}−1} dG_{i}.

Thus, the above theorem implies that when trying to minimize the degrees of the G_{n}’s, the best one can expect is to obtain deg G_{n} = 2^{n+1}(d − 1) − (d − 2) for n ≥ 1.

Theorem 6.5 above is somewhat similar to Theorem 2.4 in [Fin04]: both give lower bounds for the degrees of lifts of points and a necessary condition on the equation of the curve to achieve those bounds. (In fact, the ideas behind the proofs are the same.) In the case of [Fin04], computations show that the obtained condition seems to be also sufficient, at least for n = 1, 2. The next proposition, proved in section 10, shows that this is not the case here.

14 LUÍS R. A. FINOTTI

Proposition 6.6. Let C and C be curves defined by equations (3.1) and (3.3) and

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

be a lift of points between their affine parts such that deg G_{1} = (3d − 2). Then deg g = 2 (i.e., as polynomial in x_{0}, g(x_{0}) has degree one) and F_{1} ∈ k[x_{0}] with deg F_{1} ≤ 2 (i.e., as polynomial in x_{0}, F has degree at most one).

Hence, to achieve the lower bound, besides having g′ ∈ k^{×}, it is also necessary to have deg g = 2. On the other hand, those two conditions do seem to be sufficient, as we shall see in Theorem 6.13 and in the computation of section 12.

One can also use the same approach used to prove Theorem 6.5 to find lower bounds for the degrees of the F_{n}’s. But, in this case, one obtains the trivial bound deg F_{n} ≥ 0. Although the bound itself is useless, the proof again gives us a necessary conditions to have the equality, and, as we shall see in Proposition 6.8, one can indeed often obtain equality.

Theorem 6.7. Let C and C be as in equations (3.1) and (3.3), and ν, as in equation (3.4), be a lift of points between U and U, the affine parts of C and C respectively. Then, if dF_{1} = 0, then either g = λ or g = λx_{0}, for some λ ∈ k^{×}. In particular, this restriction on g has to hold if deg F_{1} = 0.

Furthermore, one can prove:

Proposition 6.8. Let C be a curve given by equation (3.1) with g ∈ k^{×}. Then, for any lifting C of C, there exists a lift of points

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

such that

deg F_{n} = 0,
deg G_{n} ≤ 2^{n} (d + n (−\frac{1}{2} + d · ∑_{j=1}^{n} \frac{1}{j})) .

(The proof of Theorem 6.7 is given in section 9, while the proof of Proposition 6.8 is given in section 10.)

Also, as we shall see in Proposition 6.9 below, if g = x_{0}, one can also obtain deg F_{1} = 0, but in this case, not always one can have deg F_{2} = 0: for example, as we shall state in Theorem 6.13, the ordinary elliptic curve y_{0}^{2} + x_{0}y_{0} = x_{0}^{3} + a_{0} has no lift modulo 8 with deg F_{2} = 0. So, the condition g = λx_{0} in Theorem 6.7 is not sufficient.

The next proposition, also proved in section 10, shows that if we have g = x_{0}, one can achieve the lower bounds for both F_{1} and G_{1} modulo 4. Observe that, by Theorem 6.7, it

MIN. DEG. LIFTS IN CHAR. 2 15

is not enough to have deg g = 2, even though such a curve is isomorphic to a curve with g = x_{0}: the degrees of a lift of points depends on the equation of the curve in question, not on its isomorphism class. On the other hand, having g = λx_{0}, for any λ ∈ k^{×}, instead of g = x_{0} does yield same degrees.

Proposition 6.9. Let C be a curve defined by equation (3.1). If g = x_{0}, then there exists a lifting of C, say C (defined by (3.3)), for which we have a lift of points

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

between the affine parts of C and C with deg G_{1} = (3d − 2) and F_{1} ∈ k. Hence, C is an absolute minimal degree curve over C with respect to both x and y, and ν is an absolute minimal degree lift of points.

It is worth noticing that the proof is actually constructive, giving us a method to obtain the curve C and lift of points ν (modulo 4), which can then be used in the construction of the error-correcting codes.

Also note that, with the above proposition, when g = x_{0} one can use Propositions 6.2 and 6.4 instead of 6.1 and 6.3 to bound the degrees of the later F_{i}’s and G_{i}’s.

Observe that, in the case of minimal degree liftings with respect to y, although we have deg F_{1} = 0 < 2(2 + 5(d − 2)/8) and deg G_{1} = 3d − 2 < 2(d + 5(d − 2)/8), the bounds from Proposition 6.2 cannot be improved. The reason for that comes from the condition e ≥ (d − 2)(1 + 1/2^{r})/r in Lemma 8.2, which is, in fact, the key part of the proof of Propositions 6.1 and 6.2. Very roughly, the reason for that is that the degrees of G_{2} and F_{2} might have to reach the stated upper bounds just to match the degrees of the terms that do not involve F_{1} or G_{1} in the third coordinate of the Greenberg transform.

In the case of minimal degree liftings with respect to x, note that, since we have deg g = 2, Proposition 6.3 gives, for n ≥ 2,

deg G_{n} ≤ 2^{n} (d + n (\frac{d − 2}{2} + (d − 2) ∑_{j=1}^{n} \frac{1}{j})) ,

while Proposition 6.4 gives

deg G_{n} ≤ 2^{n} (d + n (−\frac{(d − 2)}{2} + (d − 2) ∑_{j=1}^{n} \frac{1}{j})) ,

The next proposition, proved in section 11, will give a necessary and sufficient condition for the existence of a lift of the Frobenius (between affine parts of hyperelliptic curves, as in section 5) associated to a lift of points modulo 8 (for a hyperelliptic curve C given by equation (3.1)). Note, though, that the existence of a lift modulo 4 is guaranteed by Theorem 4.1 in [Fin04]. Before we can give a precise statement, we need the following definition.

Definition 6.10. Let h(x_{0}, y_{0}) ∈ k[x_{0}, y_{0}] and h(x, y) ∈ W_{2}(k) be the lift of h defined by applying the Teichmüller lift to the coefficients of h, i.e., if λ is a coefficient of some monomial of h, then the corresponding monomial of h has coefficient (λ, 0). (We shall refer to such lift as the Teichmüller lift of the polynomial h.) We define

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

Note that when p = 2, Ψ(h(x_{0}, y_{0})) is just the sum of all possible products of pairs of distinct monomials of h.

Proposition 6.11. Let C/k and C/W_{3}(k) be curves given by equations (3.1) and (3.3), and let

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

be a lift of points. There is a lift of the Frobenius between the affine parts associated to ν if, and only if,

F_{2} + x_{0}^{2} F_{1} + F_{1}^{2} + Ψ(F_{1}) + (\frac{∂F_{1}}{∂x_{0}} )^{2} F_{1} + (\frac{∂F_{1}}{∂y_{0}} )^{2} G_{1}

and

G_{2} + y_{0}^{2} G_{1} + G_{1}^{2} + Ψ(G_{1}) + (\frac{∂G_{1}}{∂x_{0}} )^{2} F_{1} + (\frac{∂G_{1}}{∂y_{0}} )^{2} G_{1}

are both squares, say P (x_{0}, y_{0})^{2} and Q(x_{0}, y_{0})^{2}, respectively. In this case, the lift of the Frobenius given by

ϕ(x, y) = (x^{2} + 2F_{1} + 4P , y^{2} + 2G_{1} + 4Q),

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, respectively, to W_{3}[x, y].

Observe that Proposition 6.11 is similar to Proposition 2.7 in [Fin04], which deals with the case p > 2. On the other hand Proposition 6.11 is more general, since here there are no restrictions on the lift ν as there was in [Fin04]. But, in fact, one could easily make Proposition 2.7 of [Fin04] more general following the same ideas from the proof of the above proposition.

The next theorem, also proved in section 11, shows that if a lift has the degrees of either the F_{n}’s or the G_{n}’s satisfying the lower bounds given by Theorems 6.7 and 6.5, then there is a lift of the Frobenius modulo 8.

MIN. DEG. LIFTS IN CHAR. 2 17

Theorem 6.12. Let C and C be given by equations (3.1) and (3.3) respectively. Let

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

be a lift of points such that

dF_{1} = (h g + x_{0}) dx_{0},
dF_{2} = (h^{3} g^{3} + x_{0}^{3}) dx_{0} + F_{1} dF_{1},

for some h ∈ k(C). Then there is a lift of the Frobenius (modulo 8) between the affine parts of C and C associated to ν. In particular, if deg F_{n} = 0 or if deg G_{n} = 2^{n+1}(d−1)−(d−2) (i.e., the degrees of either the F_{n}’s or the G_{n}’s are equal to the lower bounds) for n = 1, 2, then there is a lift of the Frobenius modulo 8.

We note that the last sentence of the theorem, at least for the case when deg G_{n} = 2^{n+1}(d − 1) − (d − 2), is not totally trivial at this points, but will follow from some later results.

Finally, it might also be worth mentioning the case of elliptic curves (i.e., d = 3), to relate minimal degrees liftings and canonical liftings. One has:

Theorem 6.13. Let C be an elliptic curve, i.e., a curve given by equation (3.1) with d = 3. Then:

(1) C is ordinary if, and only if, deg g = 2.

(2) If C is not ordinary, then there is no lift of points satisfying the lower bounds for the G_{i}’s from Theorem 6.5.

(3) If C is ordinary, then, modulo 16, the absolute minimal degree curve over C with respect to y is the canonical lifting of C (hence it’s unique), the minimal degree lift of points is also unique and it satisfies the lower bounds of Theorem 6.5. (Hence, by Theorem 6.12, there is a lift of the Frobenius modulo 8 associated to the lift of points.)

(4) If C is ordinary, then there is no lift of points with deg F_{2} = 0. But, modulo 16, its absolute minimal degree curve over C with respect to x is the canonical lifting, and, although deg F_{2} > 0 and deg G_{2} > 15, there is still a lift of the Frobenius modulo 8 associated to this lift.

So, as in the case of p > 2, it seems that minimal degree curves and canonical liftings are closely related.

Item 1 of the theorem is a well known fact: see, for instance, section V.4 of [Sil85]. Item 2 is an immediate consequence of Theorem 6.5 (more precisely, the condition g′ ∈ k^{×}) and item 1.

Items 3 and 4 can be proved by explicit computations, which are too long to be presented here, but are perfectly feasible with the use of a computer.

7. Witt Vectors and Valuations

In this section we obtain the results that we shall need to deal with Witt vectors. Although the proofs are rather technical and sometimes tedious, the results are necessary for the proofs of the main results.

Let p be a prime, and for any 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})

and

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

(See [Ser79].)

Thus, if s = (s_{0}, s_{1}, …) and t = (t_{0}, t_{1}, …) are Witt vectors, we have by definition

s + t =^{def} (S_{0}(s_{0}, t_{0}), S_{1}(s_{0}, s_{1}, t_{0}, t_{1}), …)

and

s · t =^{def} (P_{0}(s_{0}, t_{0}), P_{1}(s_{0}, s_{1}, t_{0}, t_{1}), …).

We may write, to simplify the notation,

S_{n}(s, t) =^{def} S_{n}(s_{0}, …, s_{n}, t_{0}, …, t_{n})

and

P_{n}(s, t) =^{def} P_{n}(s_{0}, …, s_{n}, t_{0}, …, t_{n}).

Since the entries of our Witt vectors are in characteristic p, we can use the polynomials S̄_{n}, P̄_{n} ∈ 𝔽_{p}[X_{0}, …, X_{n}, Y_{0}, …, Y_{n}], that are the reductions of S_{n}, P_{n} modulo p, to give us the sum and product of Witt vectors.

We now introduced some technical lemmas that we shall need later on.

Lemma 7.1. The monomials ∏ X_{i}^{a_{i}} ∏ Y_{j}^{b_{j}} (disregarding the coefficient) occurring in P̄_{n} satisfy

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

Proof. This is Lemma 2.1 in [Fin02]. □

MIN. DEG. LIFTS IN CHAR. 2 19

Let K be a field of characteristic p > 0 and v : K → ℝ ∪ {∞} be a valuation on K. (In the applications, K will be the function field of our curve and v = ord_{P_{∞}}.) Then, for any positive integer r and positive real number e, we 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).) Also, for any v_{0} ∈ ℝ, let

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

and M(v_{0}, e) =^{def} ⋂_{r≥0} M_{r}(v_{0}, e)

We then have the following lemmas:

Lemma 7.2. The sets U(e) and U_{r}(e) are subgroups of W(K)^{×}.

Proof. This is Lemma 3.1 in [Fin04]. □

Lemma 7.3. Let k be a subfield of K for which every non zero element has valuation zero. (E.g., 𝔽_{p} or the field of constants in the case where K is a function field of a curve.) Then, the set M_{r}(v_{0}, e) is a W(k)-submodule of W(K). In particular, so is M(v_{0}, e).

Proof. We first show that M_{r}(v_{0}, e) is closed under addition for all non-negative integers r. Let s = (s_{0}, s_{1}, …), t = (t_{0}, t_{1}, …) ∈ M_{r}(v_{0}, e). Then,

s + t = (S_{0}(s, t), S_{1}(s, t), …),

where S_{n} ∈ ℤ[X_{0}, …, X_{n}, Y_{0}, …, Y_{n}] is defined recursively by

S_{n} = (X_{n} + Y_{n}) + \frac{1}{p}(X_{n−1}^{p} + Y_{n−1}^{p} − S_{n−1}^{p}) + ⋯ + \frac{1}{p^{n}} (X_{0}^{p^{n}} + Y_{0}^{p^{n}} − S_{0}^{p^{n}}).  (7.1)

It is enough to prove that every monomial in S_{n}, for n = 0, …, r, has valuation, when computed at (s, t), greater than or equal to p^{n}(v_{0} − ne).

Observe that the valuations of the coefficients of the monomials of S_{n} are zero, since, in characteristic p > 0, they are roots of unity. Hence, we shall disregard the coefficients of the monomials.

We proceed by induction on n. For n = 0, S_{0}(s, t) = s_{0} + t_{0}, and since v(s_{0}), v(t_{0}) ≥ v_{0}, the statement clearly holds.

So, assume that, for 0 ≤ m < n ≤ r, the valuations of the monomials of S_{m} when evaluated at (s, t) are less than or equal to p^{m}(v_{0} −me) and let ∏ X_{i}^{a_{i}} ∏ Y_{j}^{b_{j}} be a monomial (with the coefficient dropped) that appears in S_{n}.

By equation (7.1), this monomial comes from S_{m}^{p^{n−m}}, or it is either of the form X_{i}^{p^{n−i}} or Y_{j}^{p^{n−j}}.

The monomials of S_{m}^{p^{n−m}} are products of p^{n−m} monomials of S_{m}, and therefore, by the induction hypothesis, they have valuation, when computed at (s, t), greater or equal to p^{n−m}p^{m}(v_{0} − me) ≥ p^{n}(v_{0} − ne).

If the monomial is either X_{i}^{p^{n−i}} or Y_{j}^{p^{n−j}}, then the corresponding valuations are p^{n−i}v(s_{i}) ≥ p^{n}(v_{0} − ie) and p^{n−j}v(t_{j}) ≥ p^{n}(v_{0} − je), respectively. In either case, the monomials have valuation greater than or equal to p^{n}(v_{0}−ne). Therefore, M_{r}(v_{0}, e) is closed under addition.

Now let c = (c_{0}, c_{1}, …) ∈ W(k) and s ∈ M_{r}(v_{0}, e). Note that either v(c_{i}) = 0 or c_{i} = 0. The (n + 1)-th coordinate of c · s, for 0 ≤ n ≤ r, is given by P̄_{n}(c, s). Let ∏ X_{i}^{a_{i}} ∏ Y_{j}^{b_{j}}

be a monomial (disregarding the coefficient) in P̄_{n}. If c_{i} = 0, for some fixed i for which a_{i} ≠ 0, then

v (∏ c_{i}^{a_{i}} ∏ s_{j}^{b_{j}}) = v(0) > p^{n}(v_{0} − ne).

If c_{i} ≠ 0 for all i such that a_{i} ≠ 0, then, by Lemma 7.1,

v (∏ c_{i}^{a_{i}} ∏ s_{j}^{b_{j}}) = ∑ a_{i}v(c_{i}) + ∑ b_{j}v(s_{j})
= ∑ b_{j}v(s_{j}) ≥ ∑ b_{j}p^{j}(v_{0} − je)
≥ p^{n}(v_{0} − ne).

Therefore, v(P̄_{n}(c, s)) ≥ p^{n}(v_{0} − ne), and so c · s ∈ M_{r}(v_{0}, e).

□

Lemma 7.4. Let k be a field of characteristic p > 0 and h(x, y) ∈ W(k)[x, y]. If we let x = (x_{0}, x_{1}, …) and y = (y_{0}, y_{1}, …) and expand h(x, y) as Witt vector (as we do with Greenberg transforms), then the (n + 1)-th coordinate of this expansion is of the form

x_{n} (\frac{∂h}{∂x_{0}} (x_{0}, y_{0}))^{p^{n}} + y_{n} (\frac{∂h}{∂y_{0}} (x_{0}, y_{0}))^{p^{n}} + …,

where h(x_{0}, y_{0}) is the reduction modulo p of h(x, y) and the omitted terms depend only on x_{i} and y_{i}, for i = 0, … (n − 1).

Proof. The lemma clearly holds for h(x, y) equal to either x, y or a constant (in W(k)). So, it suffices to show that if the lemma holds for h_{1} and h_{2}, then it must also hold for their sum and product.

We first show that it holds for h_{1} + h_{2}: just observed that the (n + 1)-th coordinate is given by the polynomial S_{n} (as in equation (7.1)), and so, in the (n + 1)-th coordinate of h_{1} + h_{2}, we have

(x_{n} (\frac{∂h_{1}}{∂x_{0}} )^{p^{n}} + y_{n} (\frac{∂h_{1}}{∂y_{0}} )^{p^{n}} + …) + (x_{n} (\frac{∂h_{2}}{∂x_{0}} )^{p^{n}} + y_{n} (\frac{∂h_{2}}{∂y_{0}} )^{p^{n}} + …) + …

= (x_{n} (\frac{∂(h_{1} + h_{2})}{∂x_{0}})^{p^{n}} + y_{n} (\frac{∂(h_{1} + h_{2})}{∂y_{0}})^{p^{n}} + …) + …,

where no omitted term involves x_{n} or y_{n}. Thus, it works for the sum.

For the product, remember that

P̄_{n} = (X_{0}^{p^{n}}Y_{n} + X_{1}^{p^{n−1}}Y_{n−1}^{p} + ⋯ + X_{n}Y_{0}^{p^{n}}) + …,

where no omitted term involves either X_{n} or Y_{n}. (See, for instance, formula (3) in [Fin02].) Hence, the (n + 1)-th coordinate of h_{1} · h_{2} is given by

h_{2}^{p^{n}} (x_{n} (\frac{∂h_{1}}{∂x_{0}} )^{p^{n}} + y_{n} (\frac{∂h_{1}}{∂y_{0}} )^{p^{n}} + …) + h_{1}^{p^{n}} (x_{n} (\frac{∂h_{2}}{∂x_{0}} )^{p^{n}} + y_{n} (\frac{∂h_{2}}{∂y_{0}} )^{p^{n}} + …) + …

= (x_{n} (\frac{∂(h_{1} · h_{2})}{∂x_{0}})^{p^{n}} + y_{n} (\frac{∂(h_{1} · h_{2})}{∂y_{0}})^{p^{n}} + …) + …,

where no omitted term involves x_{n} or y_{n}. Therefore, it also works for the product.

□

The next lemma will be quite useful when analyzing the equations that define the Greenberg transform of a curve over W(k), in particular in the construction of lifts of points.

Lemma 7.5. Let K and k be as in Lemma 7.3, s, t ∈ U_{r−1}(e) and h(x, y) ∈ W(k)[x, y]. Furthermore, let −v_{0} be the weighted degree of h, where the weight of x is defined to be −v(s_{0}) and the weight of y is defined to be −v(t_{0}). Then, on the (r + 1)-th coordinate of h(s, t), we have

s_{r} (\frac{∂h}{∂x_{0}} (s_{0}, t_{0}))^{p^{r}} + t_{r} (\frac{∂h}{∂y_{0}} (s_{0}, t_{0}))^{p^{r}} + …,  (7.2)

where all the omitted terms do not involve s_{n} or t_{n} and have valuation less than or equal to p^{r}(v_{0} − re).

Proof. By Lemma 7.4, the (n + 1)-th coordinate of h(s, t) indeed is as in equation (7.2). So, all that is left to do is to prove the statement about the valuation of the omitted terms.

For every monomial x^{i}y^{j} of h, disregarding the coefficient, the term s^{i}t^{j} ∈ U_{r−1}(e). But since the weighted degree of h is −v_{0}, which implies that i v(s_{0}) + j v(t_{0}) ≥ v_{0}, we have, by Lemma 7.3, that h(s, t) ∈ M_{r−1}(v_{0}, e).

By the same reasoning, if s and t were in U_{r}(e) instead of in U_{r−1}(e), then h(s, t) would be in M_{r}(v_{0}, e), and the bounds would clearly hold. But the proof of Lemma 7.3 bounds

22 LUÍS R. A. FINOTTI

the valuation of each monomial appearing in the sum of two elements of M_{n}(v_{0}, e) and in the product of an element of W(k) with an element of M_{n}(v_{0}, e). So, this would bound the valuation of every monomial appearing in the (r + 1)-th coordinate of h(s, t) by p^{r}(v_{0} − e). In our case, we are only missing the bounds for the valuations of s_{r} and t_{r}, but we could still bound the valuation of every monomial that does not involve those two terms in exactly the same way. Hence, since by Lemma 7.4 no omitted term in equation (7.2) involves t_{n} or s_{n}, we can bound the valuations of the omitted terms by p^{r}(v_{0} − e).

□

The following lemma will be helpful when dealing with the second coordinate of the Greenberg transforms of curves.

Lemma 7.6. Let

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

and suppose that

h((x_{0}, x_{1}), (y_{0}, y_{1})) = (h_{0}(x_{0}, y_{0}), h_{1}(x_{0}, x_{1}, y_{0}, y_{1})).

Then, if

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

we have

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

where Ψ is as in Definition 6.10.

Proof. This is Lemma 8.1 in [Fin04]. □

8. Proofs of the Upper Bounds

In this section we prove the upper bounds for the minimal degree lifting with respect to both x and y. More precisely, we prove here Propositions 6.1, 6.2, 6.3 and 6.4.

The following lemma is an adaptation of Lemma IV.1 from [VW99] to our case, and it is the main tool to obtain the upper bounds for the degrees of the lifts of points.

Lemma 8.1. Let C be a curve given by equation (3.1), and R =^{def} O(U) be the ring of regular functions on the affine part of C. Let also a, b, c ∈ R with (a, b) = 1, deg a ≤ n, deg b = m and deg c ≤ r. Also, assume that n + m + (d − 2) ≤ r. Then, there exist u, v ∈ R such that au + bv = c with deg u ≤ (m + (d − 2)) and deg v ≤ (r − m).

MIN. DEG. LIFTS IN CHAR. 2 23

Proof. Since m + n + (d − 2) ≤ r, the map

ψ : L((m + (d − 2))P_{∞}) ⊕ L((r − m)P_{∞}) → L(rP_{∞}),

defined by ψ(u, v) =^{def} au + bv is well defined. Hence, to prove the lemma, it suffices to show that ψ is surjective.

Since the genus of C is (d − 1)/2 and (m + d − 2), (r − m) and r are all greater than (d − 2), by the Riemann-Roch Theorem, or more precisely, Corollary II.5.5(c) of [Sil85], the dimensions of the k-vector spaces above are

ℓ((m + d − 2)P_{∞}) = m + (d − 2) − (d − 1)/2 + 1,
ℓ((r − m)P_{∞}) = (r − m) − (d − 1)/2 + 1,
ℓ(rP_{∞}) = r − (d − 1)/2 + 1.

On the other hand, since a and b are relatively prime, (u, v) ∈ ker ψ if, and only if, u = bz and v = −az, for some z ∈ R.

Therefore, with the restrictions on the degrees of u and v, and since n + m + (d − 2) ≤ r, we have that ker(ψ) ≅ L((d − 2)P_{∞}). Observing that ℓ((d − 2)P_{∞}) = d − 2 − (d − 1)/2 + 1, the dimension of our domain minus the dimension of the kernel is equal the dimension of the co-domain, and hence our map is surjective.

□

The next lemma is the main step for the proofs of Propositions 6.1 and 6.2.

Lemma 8.2. Let C and C be as in equations (3.1) and (3.3) respectively, and let

ν̄(x_{0}, y_{0}) = ((x_{0}, F_{1}, …, F_{r−1}), (y_{0}, G_{1}, …, G_{r−1}))

be a lift of points such that ν̄^{∗}x, ν̄^{∗}y ∈ U_{r−1}(e), with e ≥ (d − 2)(1 + 1/2^{r})/r. Then, ν̄ can be completed to a lift

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

where ν^{∗}x, ν^{∗}y ∈ U(e). Moreover,

deg G_{n} ≤ 2^{n+1}(d − 1) + (d − 2),

for all n ≥ r.

Proof. We shall inductively construct the F_{n}’s and G_{n}’s for n ≥ r. So, assume we have (x_{0}, F_{1}, …, F_{n−1}), (y_{0}, G_{1}, …, G_{n−1}) ∈ U_{n−1}(e) for some n ≥ r. We shall find F_{n}, G_{n} ∈ O(U) such that

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

is a lift of points with (x_{0}, F_{1}, …, F_{n−1}, F_{n}), (y_{0}, G_{1}, …, G_{n−1}, G_{n}) ∈ U_{n}(e).

By Lemma 7.4, in the (n + 1)-th coordinate of the Greenberg transform, one has

y_{n}g^{2^{n}} + x_{n}(f′ + g′y_{0})^{2^{n}} = …,  (8.1)

where no omitted term depends on x_{n} or y_{n}. Thus, to produce a lift modulo p^{n+1}, we need to find F_{n} and G_{n} such that

G_{n}g^{2^{n}} + F_{n}(f′ + g′y_{0})^{2^{n}} = …,  (8.2)

where the omitted terms here are obtained by substituting x_{i} by F_{i} and y_{i} by G_{i}, for i = 1, …, (n − 1), in equation (8.1). (Note that the equality in equation (8.2) is equality in k(C), not in k[x_{0}, y_{0}].)

We shall apply Lemma 8.1 with a =^{def} g^{2^{n}}, b =^{def} (f′ + g′y_{0})^{2^{n}} and c as the omitted terms in (8.2). By Lemma 7.5, c has degree less than or equal to 2^{n}(2d + ne). By assumption C is non-singular, and hence ((f′ + g′y_{0}), g) = 1. Furthermore, since e ≥ (d − 2)(1 + 1/2^{r})/r ≥ (d − 2)(1 + 1/2^{n})/n and deg g ≤ (d − 1), we have 2^{n}(2d + ne) ≥ deg a + deg b + (d − 2), and thus we can indeed apply Lemma 8.1, which gives us G_{n} and F_{n} such that

deg G_{n} ≤ 2^{n+1}(d − 1) + (d − 2), deg F_{n} ≤ 2^{n}(2 + ne).

Therefore, clearly (x_{0}, F_{1}, …, F_{n}) ∈ U_{n}(e), and using again the bound on e, one can easily verify that 2^{n+1}(d − 1) + (d − 2) ≤ 2^{n}(d + ne), yielding that (y_{0}, G_{1}, …, G_{n}) ∈ U_{n}(e). □

With the above lemma in hand, we can now prove Proposition 6.1.

Proof of Proposition 6.1. If one applies Lemma 8.2 with r = 1, one obtains the existence of a lift such that

deg G_{n} ≤ 2^{n+1}(d − 1) + (d − 2), deg F_{n} ≤ 2^{n}(2 + n3(d − 2)/2)

for all n ≥ 1. So, we need to show that we can improve that bound on the degree of F_{1} to have it as in the statement.

We take, then, a closer look at the construction of F_{1} and G_{1}. We need in this case F_{1} and G_{1} to satisfy

G_{1}g^{2} + F_{1}(f′ + g′y_{0})^{2} = …,

where the omitted terms are the terms independent of x_{1} and y_{1} in the second coordinate of the Greenberg transform of C. We apply Lemma 8.1 again, with a = g^{2}, b = (f′ + g′y_{0})^{2} and c as the omitted terms. In this case, observe that, by Lemma 7.5, deg c ≤ 2(2d + e) for any e > 0, and hence deg c ≤ 4d. So, to apply the lemma, we take r =^{def} max{4d, 2 deg g + 4(d − 1) + (d − 2)}.

This way we obtain F_{1} and G_{1} such that deg G_{1} ≤ 4(d − 1) + (d − 2) and

deg F_{1} ≤ r − 4(d − 1) = max{4, 2 deg g + (d − 2)} < 2(2 + 3(d − 2)/2),

and hence, (x_{0}, F_{1}), (y_{0}, G_{1}) ∈ U_{1}(3(d − 2)/2).

Finally, Lemma 8.2, with r = 2 and e = 3(d − 2)/2, finishes the proof.

□

Example 8.3. Consider the ordinary elliptic curve

E/k : y_{0}^{2} + x_{0}y_{0} = x_{0}^{3} + 1.

Then, the map

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

with

F_{1} =^{def} x_{0}y_{0}
F_{2} =^{def} 1 + x_{0}^{3} + x_{0}^{5} + x_{0}^{7} + x_{0}^{8} + x_{0}^{9} + x_{0}^{10} + (x_{0} + x_{0}^{5} + x_{0}^{7})y_{0}
G_{1} =^{def} x_{0} + x_{0}^{3}y_{0}
G_{2} =^{def} 1 + x_{0} + x_{0}^{2} + x_{0}^{3} + x_{0}^{6} + (1 + x_{0}^{5} + x_{0}^{6} + x_{0}^{7})y_{0}

is a minimal degree lifting from E to

E/W_{3}(k) : y^{2} + xy = x^{3} + (1, 0, 1)

with respect to y. Notice that, in contrast with the case of characteristic p > 2, we cannot obtain a minimal degree lift of points that is hyperelliptic, i.e., that commutes with the hyperelliptic involutions of C and C. Also, observe the the degrees are equal to the upper bounds given by Proposition 6.1.

On the other hand, E is not the absolute minimal degree curve over E, since the curve is ordinary and the canonical lifting with the elliptic Teichmüller lift would give us smaller degrees.

Again, in contrast with the case of p > 2, or, more precisely, with Proposition 2.3 in [Fin04], note that if we have one particular lift of points ν satisfying the bound in Proposition 6.1, this is not necessarily a minimal degree lift of points with respect to y. This happens because we have a non-trivial kernel for the map ψ defined in the proof of Lemma 8.1.

26 LUÍS R. A. FINOTTI

On the other hand, since we know exactly what that kernel is, it is fairly easy to obtain a minimal degree lift of points with respect to y in every step of the procedure of finding the G_{n}’s and F_{n}’s: after a pair (G_{n}, F_{n}) is found, any other pair

G̃_{n} =^{def} G_{n} + z (f′ + g′y_{0})^{2^{n}},
F̃_{n} =^{def} F_{n} + z g^{2^{n}},

with deg z ≤ (d − 2), defines another lift also satisfying the bounds from Proposition 6.1. Observe that since deg z < d, we must have z ∈ k[x_{0}], and so deg z is even. Hence if deg G_{n} is even and greater than or equal to 2^{n} deg g, then the degree of G_{n} can always be lowered, and if either deg G_{n} is odd or less than or equal to 2^{n} deg g, then it is a minimal degree lift with respect to y.

Something else worth noting is that the smaller deg g is, the smaller is the bound for deg F_{1}. Also, as stated in Theorem 6.5, having g = x_{0} (and so deg g = 2) in fact yields smaller degrees also for G_{1}.

Lemma 8.2 tells us that having smaller degrees for first coordinates of a lift of points allows us to obtains lifts with smaller degrees for the remaining F_{n}’s. Proposition 6.2, which is the analogue to Proposition 5.4 in [Fin04] and an immediate consequence of Lemma 8.2, illustrates this point in a particular case.

Proof of Proposition 6.2. Just apply Lemma 8.2 with e = 5(d − 2)/8 and r = 2.

□

To prove Propositions 6.3 and 6.4, we first introduce a lemma analogous to Lemma 8.2. As we can see in its statement, in contrast with the case with Lemma 8.2, the degrees of the G_{n}’s increase in such a way that there is no positive e for which the obtained (y_{0}, G_{1}, G_{2}, …) is in U(e).

Lemma 8.4. Let C and C be as in equations (3.1) and (3.3) respectively, and let

ν̄(x_{0}, y_{0}) = ((x_{0}, F_{1}, …, F_{r−1}), (y_{0}, G_{1}, …, G_{r−1}))

be a lift of points such that ν̄^{∗}x, ν̄^{∗}y ∈ U_{r−1}(e_{r−1}), with e_{r−1} ≥ (deg g + (d − 2)/2^{r} − 2)/r. Then, ν̄ can be extended to a lift

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

with ν^{∗}x, ν^{∗}y ∈ U(e_{n}), where e_{n} = e_{r−1} + (d − deg g) ∑_{j=r}^{n} 1/j. Moreover,

deg F_{n} ≤ 2^{n} deg g + (d − 2),

for all n ≥ r.

MIN. DEG. LIFTS IN CHAR. 2 27

Proof. We again construct inductively the desired F_{i}’s and G_{i}’s. So, assume we have (F_{0}, …, F_{n−1}), (G_{0}, …, G_{n−1}) ∈ U_{n−1}(e_{n−1}). By Lemma 7.5, to extend this map to the next coordinate, we have to find F_{n} and G_{n} such that

F_{n}(f′ + g′y_{0})^{2^{n}} + G_{n}g^{2^{n}} = …,

where the omitted terms (coming from the (n+1)-th coordinate of the Greenberg transform) have degree less than or equal to 2^{n}(2d + ne_{n−1}).

Since

e_{n−1} ≥ e_{r−1} ≥ (deg g + (d − 2)/2^{r} − 2)/r ≥ (deg g + (d − 2)/2^{n} − 2)/n,  (8.3)

we have 2^{n}(2d + ne_{n−1}) ≥ 2^{n+1}(d − 1) + 2^{n} deg g + (d − 2), and thus we can apply Lemma 8.1. We then obtain F_{n} and G_{n} such that

deg F_{n} ≤ 2^{n} deg g + (d − 2)
deg G_{n} ≤ 2^{n}(2d + ne_{n−1}) − 2^{n} deg g = 2^{n} (d + n (\frac{d − deg g}{n} + e_{n−1}))
= 2^{n}(d + ne_{n}).

Finally, inequality (8.3) implies that deg F_{n} ≤ 2^{n} deg g +(d−2) ≤ 2^{n}(2+ne_{n}), and hence ν^{∗}x, ν^{∗}y ∈ U(e_{n}).

□

We can now prove Propositions 6.3 and 6.4.

Proof of Proposition 6.3. We just apply Lemma 8.4, with r = 1 and e_{0} = max{0, deg g + (d − 6)/2}.

□

Proof of Proposition 6.4. The bounds on deg F_{1} and deg G_{1} imply that (x_{0}, F_{1}), (y_{0}, G_{1}) ∈ U_{1}((d − 2)/2). Since deg g ≤ (3d + 2)/4, we have e_{1} ≥ (deg g + (d − 2)/4 − 2)/2. The proposition then immediately follows from Lemma 8.4.

□

9. Proofs of the Lower Bounds

In this section we prove the lower bounds (and the necessary conditions to achieve those lower bounds) for the minimal degree liftings with respect to both x and y. More precisely, we prove Theorems 6.5 and 6.7 .

28 LUÍS R. A. FINOTTI

Proof of Theorem 6.5. If we work modulo 2^{n+1}, or equivalently, truncate the lift ν at the (n + 1)-th coordinate, Theorem 4.1 in [Fin04] tells us that there exists a lift of ϕ^{n}, the 2^{n}-th power Frobenius of C associated to ν. We shall denote such lift by (bold-face) ϕ^{n}.

Furthermore, Theorem 4.1 in [Fin04] also tells us that ϕ^{n} can be written as

ϕ^{n}(x, y) = (∑_{i=0}^{n} 2^{i}F _{i}^{2^{n−i}}, ∑_{i=0}^{n} 2^{i}G_{i}^{2^{n−i}}) ,  (9.1)

where F_{i}, G_{i} ∈ W_{n+1}(k)[x, y] are lifts of F_{i}, G_{i} ∈ k[x_{0}, y_{0}]. Hence, if

ω_{n} =^{def} \frac{dx}{2y + g^{σ^{n}}(x)} = \frac{dy}{(f′)^{σ^{n}}(x) − (g′)^{σ^{n}}(x) y} ,  (9.2)

then (1/2^{n} ϕ^{n})^{∗}ω_{n} reduces, modulo 2, to

ω_{n} =^{def} \frac{∑_{i=0}^{n} G_{i}^{2^{n−i}−1}dG_{i}/dy_{0}}{(f′ + g′ y_{0})^{2^{n}}} dy_{0} = \frac{∑_{i=0}^{n} G_{i}^{2^{n−i}−1}dG_{i}/dy_{0}}{(f′ + g′ y_{0})^{2^{n}−1}} \frac{dx_{0}}{g} .  (9.3)

Observe that the differential dx_{0}/g does not vanish on U, and since ω_{n} is regular on U^{σ^{n}}, ω_{n} must also be regular on U. Thus,

\frac{∑_{i=0}^{n} G_{i}^{2^{n−i}−1} dG_{i}/dy_{0}}{(f′ + g′ y_{0})^{2^{n}−1}} ∈ O(U),

and so it can be written as a polynomial, say h_{n}(x_{0}, y_{0}). Hence

ω_{n} = h_{n}(x_{0}, y_{0})\frac{dx_{0}}{g}  (9.4)

and

\frac{dG_{n}}{dy_{0}} = h_{n}(x_{0}, y_{0})(f′ + g′ y_{0})^{2^{n}−1} + ∑_{i=0}^{n−1} G_{i}^{2^{n−i}−1} \frac{dG_{i}}{dy_{0}} .  (9.5)

Note that h_{n}(x_{0}, y_{0}) is not equal to zero, since (1/2^{n}ϕ^{n})^{∗} is the “inverse” of the n-th power of the Cartier operator, which we shall denote by C^{n}, and therefore, C^{n}(ω_{n}) = dx_{0}/g(x_{0}) ≠ 0, which implies that ω_{n} cannot be equal to zero.

Now, for i = 0, …, (n − 1),

deg (G_{i}^{2^{n−i}−1} \frac{dG_{i}}{dy_{0}} ) = 2^{n+1}(d − 1) − 2^{n−i}(d − 2) − d < (2^{n} − 1)(2d − 2),

and hence,

deg (\frac{dG_{n}}{dy_{0}} ) = deg h_{n} + (2^{n} − 1)(2d − 2) ≥ (2^{n} − 1)(2d − 2).

Thus,

deg G_{n} ≥ 2^{n+1}(d − 1) − (d − 2).

Moreover, we can only have equality if h_{n}(x_{0}, y_{0}) is a constant.

Let us assume now that

deg G_{i} = 2^{i+1}(d − 1) − (d − 2),

MIN. DEG. LIFTS IN CHAR. 2 29

for i = 0, …, n. By equation (9.4) and the conclusion of the previous paragraph, we must have

ω_{i} = λ_{i} \frac{dx_{0}}{g} = λ_{i} \frac{dy_{0}}{f′ + g′y_{0}} ,  (9.6)

where λ_{i} ∈ k.

On the other hand, for i = 1, ω_{1} is defined as the reduction modulo 2 of

\frac{1}{2}ϕ^{∗}(ω_{1}) = \frac{1}{2}ϕ^{∗} (\frac{dx}{2y + g^{σ}} ) ,

and thus, by equation (9.1) with n = 1,

ω_{1} = \frac{dF_{1} + x_{0}dx_{0}}{g(x_{0})^{2}} .  (9.7)

Comparing equations (9.7) and (9.6), with i = 1, we obtain

(λ_{1}g + x_{0}) = \frac{dF_{1}}{dx_{0}} .

By taking differentials in the above equation, and since we are in characteristic 2, we obtain

(λ_{1}g′ + 1)dx_{0} = d (\frac{dF_{1}}{dx_{0}} ) = 0.

Thus, λ_{1} = λ^{−1}, where λ is then the necessarily non-zero coefficient of x_{0} in g, and we have established items (1) and (2) of the theorem. Also, note that, by equation (9.5), we also proved item (3) for n = 1 in the theorem.

With the case i = 1 done, we can now finish the proof of item (3) by induction on i. So, assume that

dG_{i−1} = λ^{−(2^{i−1}−1)}(f′ + g′y_{0})^{2^{i−1}−1} dy_{0} + ∑_{j=0}^{i−2} G_{j}^{2^{i−1−j}−1} dG_{j}

for some i ≤ n.

The properties of the Cartier divisor, together with our induction hypothesis, give us

C (\frac{∑_{j=0}^{i} G_{j}^{2^{i−j}−1}dG_{j}}{(f′ + g′ y_{0})^{2^{i}}} ) = \frac{∑_{j=0}^{i−1} G_{j}^{2^{i−1−j}−1}dG_{j}}{(f′ + g′ y_{0})^{2^{i−1}}} = λ^{−(2^{i−1}−1)} \frac{dy_{0}}{f′ + g′y_{0}} .  (9.8)

On the other hand, since by assumption deg G_{i} = 2^{i+1}(d−1)−(d−2), equation (9.6) holds, and thus we just need to show that λ_{i} = λ^{−(2^{i}−1)}. But, since g′ = λ,

C (\frac{∑_{j=0}^{i} G_{j}^{2^{i−j}−1}dG_{j}}{(f′ + g′ y_{0})^{2^{i}}} ) = C (λ_{i} \frac{dy_{0}}{f′ + g′y_{0}} ) = λ_{i}^{1/2} λ^{1/2} \frac{dy_{0}}{f′ + g′y_{0}} .  (9.9)

Comparing equations (9.8) and (9.9), we obtain λ_{i} = λ^{−(2^{i}−1)}.

□

30 LUÍS R. A. FINOTTI

The next proposition, which follows from an analysis of the proof of Theorem 6.5 above, establishes a relation between the formulas for the dF_{n}’s and formulas for the dG_{n}’s in general and will be used in the proof of Theorem 6.12.

Proposition 9.1. Let C and C be as in equations (3.1) and (3.3), and ν, as in equation (3.4), be a lift of points between the affine parts of C and C. Then,

dF_{n} = h_{n}g^{2^{n}−1} dx_{0} + ∑_{i=0}^{n−1} F_{i}^{2^{n−i}−1} dF_{i},

and

dG_{n} = h_{n}(f′ + g′y_{0})^{2^{n}−1} dy_{0} + ∑_{i=0}^{n−1} G_{i}^{2^{n−i}−1} dG_{i}

for some h_{n} ∈ k[x_{0}, y_{0}].

Proof. In the proof of Theorem 6.5 one sees that, by using the two different forms of ω_{n} in equation (9.2), we must have

ω_{n} = \frac{∑_{i=0}^{n} F_{i}^{2^{n−i}−1}dF_{i}}{g^{2^{n}}} = \frac{∑_{i=0}^{n} G_{i}^{2^{n−i}−1}dG_{i}}{(f′ + g′ y_{0})^{2^{n}}} .

Also, still following the proof of Theorem 6.5, we see that, as in equation (9.4),

ω_{n} = h_{n} \frac{dx_{0}}{g} = h_{n} \frac{dy_{0}}{f′ + g′y_{0}}

for some non-zero h_{n} ∈ k[x_{0}, y_{0}]. Comparing the two formulas for ω_{n} finishes the proof. □

As observed in section 6, the same idea used to prove Theorem 6.5 gives the analogous result for the F_{n}’s.

Proof of Theorem 6.7. We work modulo 4, and hence there is a lift of ϕ, which we denote by ϕ, given by equation (9.1) (with n = 1). With ω_{1} as in equation (9.2), (1/2 ϕ)^{∗}ω_{1} reduces, modulo 2, to

ω_{1} =^{def} \frac{dF_{1} + x_{0} dx_{0}}{g^{2}} = \frac{x_{0} dx_{0}}{g^{2}} .

Since dx_{0}/g does not vanish on U, and since ω_{1} is regular on U^{σ}, ω_{1} must also be regular on U. Thus,

ω_{1} = h_{1} \frac{dx_{0}}{g}

for some h_{1} ∈ k[x_{0}, y_{0}]. Hence

x_{0} = h_{1} g.

Therefore, either g = λ or g = λx_{0} for some λ ∈ k. Since the curve is non-singular, we must have λ ≠ 0.

□

MIN. DEG. LIFTS IN CHAR. 2 31

10. Proofs of Propositions About Achieving the Lower Bound

In this section we prove Propositions 6.6, 6.8 and 6.9.

Proof of Proposition 6.6. By Theorem 6.5, we have that g′ = λ, with λ ∈ k^{×}. This says that the coefficient of x_{0} in g is λ, and this, with a change of variables x̃_{0} = λx_{0} and ỹ_{0} = y_{0}/λ^{d/2}, we may assume, without loss of generality, that λ = 1.

Write

f(x) = ∑_{i=0}^{d} a_{i}x^{i},

and

g(x) = ∑_{j=0}^{(d−1)/2} b_{j}x^{j},

where

a_{i} = (a_{i,0}, a_{i,1}), for i = 0, …, d,
b_{j} = (b_{j,0}, b_{j,1}), for j = 0, …, (d − 1)/2.

By Lemma 7.6, we have on the second coordinate of equation the Greenberg transform of equation (3.3)

y_{1}g^{2} + x_{1}(g′y_{0})^{2} + Ψ(y_{0}^{2} + g y_{0}) + ( ∑_{j=0}^{(d−1)/2} b_{j,1}x_{0}^{2j}) y_{0}^{2} = x_{1}(f′)^{2} + Ψ(f) + ∑_{i=0}^{d−1} a_{i,1}x_{0}^{2i},

For ν to be well defined, it is necessary that

G_{1} g^{2} + F_{1}(g′y_{0})^{2} + Ψ(y_{0}^{2} + g y_{0}) + ( ∑_{j=0}^{(d−1)/2} b_{j,1}x_{0}^{2j}) y_{0}^{2} = F_{1}(f′)^{2} + Ψ(f) + ∑_{i=0}^{d−1} a_{i,1}x_{0}^{2i},

in k(C), or

G_{1} g^{2} + F_{1}(f′ + g′y_{0})^{2} = Ψ(f) + ∑_{i=0}^{d−1} a_{i,1}x_{0}^{2i} + Ψ(y_{0}^{2} + g y_{0}) + ( ∑_{j=0}^{(d−1)/2} b_{j,1}x_{0}^{2j}) y_{0}^{2}. (10.1)

One can easily check that all the terms in the right-hand-side of the equation above have degrees less than or equal to (4d − 1). Therefore, all terms from G_{1} g^{2} and F_{1}(f′ + g′y_{0})^{2} of degree greater than (4d − 1), if any, have to cancel each other out.

Assume now that deg g ≥ (d + 1)/2, so that we have deg(G_{1} g^{2}) ≥ (4d − 1). Since deg G_{1} = (3d − 2), we must then have

deg F_{1} = 2 deg g − d + 2 ≤ d.  (10.2)

Hence, if we write

F_{1} = h_{1} + h_{2} y_{0}, h_{1}, h_{2} ∈ k[x_{0}],

then h_{2} ∈ k. Therefore, by Proposition 9.1 and Theorem 6.5,

dF_{1} = (h′_{1} + \frac{f′h_{2}}{g} + \frac{h_{2}}{g} y_{0}) dx_{0} = (g + x_{0})dx_{0}.

Thus, h_{2} = 0 and F′_{1} = h′_{1} = g+x_{0}. Then, deg F_{1} ≥ deg g+2, unless deg g = 2, in which case the proposition automatically holds. But if deg F_{1} ≥ deg g+2, then the equality in equation (10.2) would imply that deg g ≥ d, a contradiction, since the degree of deg g is always less than or equal to (d − 1). Therefore, if deg g ≠ 2, then we must have deg g < (d + 1)/2.

But, if deg g < (d + 1)/2, then all terms in the right-hand-side of (10.1) have degrees less than or equal to (4d − 2) and since G_{1} g^{2} also has degree less than or equal to this bound, so does F_{1}(f′ + g′y_{0})^{2}. Hence, deg F_{1} ≤ 2 and therefore F_{1} is a polynomial in x_{0} of degree (as a function in k(C)) at most 2.

Now, all terms of odd degree in equation (10.1) come from either G_{1}g^{2} or Ψ(y_{0}^{2} + g y_{0}) = g y_{0}^{3} + Ψ(g y_{0}). Observe that every term of Ψ(gy_{0}) has even degree, since every term of gy_{0} has odd degree. So, the terms of highest odd degree in G_{1}g^{2} and gy_{0}^{3} have to cancel out each other. Hence,

(3d − 2) + 2 deg g = 3d + deg g.

Thus, deg g = 2.

□

We now proceed to prove Proposition 6.8.

Proof of Proposition 6.8. We prove the proposition by induction on n. For n = 1, in order to have a lift modulo 4, F_{1} and G_{1} have to satisfy

G_{1} λ^{2} + F_{1} (f′)^{2} = …,

where λ =^{def} g ∈ k^{×} and the omitted terms are the terms in the right hand side of equation (10.1), and therefore, have degrees less than or equal to 4d − 1. Since deg(f′)^{2} = 4d − 4, we can take F_{1} to be any constant and obtain, from the above equation, G_{1} with deg G_{1} ≤ 4d−1. Thus, the bounds hold for n = 1.

Now suppose that the bounds for the G_{i}’s in the statement hold for i = 1, …, n. Then (y_{0}, G_{1}, …, G_{n}) ∈ U_{n}(e_{n}), where

e_{n} =^{def} d (∑_{j=1}^{n} \frac{1}{j}) − \frac{1}{2},

since, for i = 1, …, n,

deg G_{i} ≤ 2^{i} (d + i (−\frac{1}{2} + d · ∑_{j=1}^{i} \frac{1}{j})) ≤ 2^{n} (d + n (−\frac{1}{2} + d · ∑_{j=1}^{n} \frac{1}{j})) = 2^{n}(d + n e_{n}).

Since, clearly (x_{0}, F_{1}, …, F_{n}) ∈ U_{n}(e_{n}), by Lemma 7.5, to obtain a lift modulo 2^{n+2} we need to find F_{n+1} and G_{n+1} such that

G_{n+1} λ^{2^{n+1}} + F_{n+1} (f′)^{2^{n+1}} = …

where the omitted terms have degrees less then or equal to 2^{n+1}(2d + (n + 1) e_{n}). Hence, we can take F_{n+1} to be any constant and find G_{n+1} with

deg G_{n+1} ≤ 2^{n+1}(2d + (n + 1) e_{n}) = 2^{n+1} (2d + (n + 1) (−\frac{1}{2} + d ∑_{j=1}^{n} \frac{1}{j}))

= 2^{n+1} (d + (n + 1) (−\frac{1}{2} + d ∑_{j=1}^{n+1} \frac{1}{j})) ,

which finishes the proof.

□

Finally, we prove Proposition 6.9.

Proof of Proposition 6.9. We shall use the same notation for f(x) and g(x) as in the proof of Proposition 6.6. We will also assume that b_{0,1} = b_{1,1} = 0, since we can always make a change of variables for C to have it that way.

Write,

G_{1} = h_{1} + h_{2} y_{0}, h_{1}, h_{2} ∈ k[x_{0}].

To have deg G_{1} ≤ (3d − 2), as in the statement, we need to find h_{1} and h_{2} such that deg h_{1} ≤ (3d − 3) and deg h_{2} = (2d − 2).

Since y_{0}^{2} = x_{0}y_{0} + f(x_{0}) (in k(C)), we can rewrite equation (10.1) as

(x_{0}^{2} h_{1} + F_{1} f + x_{0}^{2} f + f ( ∑_{j=2}^{(d−1)/2} b_{j,1}x_{0}^{2j}) + F_{1} (f′)^{2} + Ψ(f) + ∑_{i=0}^{d−1} a_{i,1}x_{0}^{2i}) +

(x_{0}^{2} h_{2} + x_{0} F_{1} + x_{0}^{3} + x_{0} f + x_{0} ( ∑_{j=2}^{(d−1)/2} b_{j,1}x_{0}^{2j})) y_{0} = 0,

and thus our goal is to find a_{i,1}’s, b_{j,1}’s (which will define C) and F_{1}, h_{1}, h_{2} (which will define ν) that satisfy this equation (and the requirements on the degrees of h_{1}, h_{2} and F_{1}). Hence we need:

x_{0}^{2} h_{1} + F_{1} f + x_{0}^{2} f + f ( ∑_{j=2}^{(d−1)/2} b_{j,1}x_{0}^{2j}) + F_{1} (f′)^{2} + Ψ(f) + ∑_{i=0}^{d−1} a_{i,1}x_{0}^{2i} = 0  (10.3)

34 LUÍS R. A. FINOTTI

and

x_{0}^{2} h_{2} + x_{0} F_{1} + x_{0}^{3} + x_{0} f + x_{0} ( ∑_{j=2}^{(d−1)/2} b_{j,1}x_{0}^{2j}) = 0  (10.4)

From the term in x_{0} in equation (10.4), and since we want F_{1} ∈ k, one immediately sees that we need F_{1} = a_{0,0}, i.e., F_{1} has to be the constant term of f. Hence, equation (10.4) has now only terms of power 2 or higher in x_{0}, and we can define

h_{2} =^{def} \frac{f + a_{0,0}}{x_{0}} + x_{0} + ∑_{j=2}^{(d−1)/2} b_{j,1}x_{0}^{2j−1}

for any choice of b_{j,1}’s that we might make. (Note that deg h_{2} = (2d − 2).)

We now try to find a suitable h_{1}. The constant term of equation (10.3) is a_{0,0}^{2} +a_{0,0} a_{1,0}^{2} + a_{0,1}, and thus we need to have a_{0,1} =^{def} a_{0,0}^{2} + a_{0,0} a_{1,0}^{2}. Also, since the coefficient of the term in x_{0} of Ψ(f(x_{0})) is a_{0,0} a_{1,0}, our choice of F_{1} makes the coefficient of the term in x_{0} in (10.3) zero. Hence, let

f_{1}(x_{0}) =^{def} ∑_{j=2}^{(d−1)/2} b_{j,1}x_{0}^{2j}
f_{2}(x_{0}) =^{def} ∑_{i=2}^{2d−1} α_{i} x_{0}^{i} =^{def} a_{0,0}f + x_{0}^{2}f + a_{0,0}(f′)^{2} + Ψ(f).

and one can now always have h_{1} ∈ k[x_{0}] that will satisfy equation (10.3) by taking

h_{1} =^{def} ∑_{l=0}^{2d−3} β_{l} x_{0}^{l} =^{def} \frac{1}{x_{0}^{2}} (∑_{i=0}^{d−1} a_{i,1}x_{0}^{2i} + f f_{1} + f_{2}) .  (10.5)

On the other hand, we need deg h_{1} ≤ (3d − 3), and thus we must choose the a_{i,1}’s, for i ≥ 1, and b_{j,1}’s, for j ≥ 2, in such a way that β_{l} = 0 for all l ≥ (3d − 1)/2.

By choosing the b_{j,1}’s such that

( \begin{matrix}1 & 0 & 0 & … & 0 \\ a_{d−2,0} & 1 & 0 & … & 0 \\ a_{d−4,0} & a_{d−2,0} & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{5,0} & a_{7,0} & a_{9,0} & … & 1 \end{matrix} ) · ( b_{(d−1)/2,1} ; b_{(d−3)/2,1} ; b_{(d−5)/2,1} ; ⋮ ; b_{2,1} ) = ( α_{2d−1} ; α_{2d−3} ; α_{2d−5} ; ⋮ ; α_{d+4} ) ,

we obtain f_{1} such that (ff_{1} + f_{2}) has no odd power of x_{0} greater than (d + 2) ≤ (3d + 1)/2. Since ∑_{i=0}^{d−1} a_{i,1}x_{0}^{2i} has only even powers of x_{0}, this implies that β_{l} = 0 for all odd l ≥ (d+2).

If d = 3, then (d + 2) = (3d + 1)/2 = 5, and hence we have we have β_{l} = 0 for all odd l ≥ (3d − 1)/2 = 4. If d > 3, then (d + 2) ≤ (3d − 1)/2, and we also have β_{l} = 0 for all odd l ≥ (3d − 1)/2.

MIN. DEG. LIFTS IN CHAR. 2 35

Now, if we let

a_{i,1} =^{def} coefficient of x_{0}^{2i} in (f f_{1} + f_{2}),

(with f_{1} chosen as above) for i = ⌊(3d−1)/4⌋, …, (d−1) (note that ⌊(3d−1)/4⌋ ≥ 2), since (2⌊(3d − 1)/4⌋ − 2) ≤ (3d − 1)/2, we obtain β_{l} = 0, for all even l such that (3d − 1)/2 ≤ l ≤ (2d − 4).

Hence β_{l} = 0 for all l ≥ (3d − 1)/2, and thus deg h_{1} ≤ (3d − 3).

□

If d = 3, i.e., if C is an elliptic curve, then the requirement that g′ = λ ∈ k^{×} is equivalent to saying that deg g = 2, which, by item 1 of Theorem 6.13, is a necessary and sufficient condition for elliptic curve to be ordinary. In this case, by Proposition 4.2 in [VW00], the construction described above actually gives us the elliptic Teichmüller lift and the canonical lifting of the elliptic curve (modulo 4). Hence, by following this procedure, one can easily reproduce the explicit formulas for the canonical lifting and elliptic Teichmüller lift (in characteristic 2) exhibited in [VW00].

11. Proofs About Lifting the Frobenius

In this section we prove Proposition 6.11 and Theorem 6.12. But, before we can prove Proposition 6.11, we shall need two simple lemmas.

Lemma 11.1. 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 will also use the following lemma:

Lemma 11.2. Let C/k and C/W(k) be curves given by equation (3.1) and (3.3) respectively and

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

be a lift of points between their affine parts. Then

(f′ + g′y_{0})^{2} ((x_{0} + \frac{∂F_{1}}{∂x_{0}} ) + \frac{f′ + g′y_{0}}{g} \frac{∂F_{1}}{∂y_{0}} ) +

g^{2} (\frac{∂G_{1}}{∂x_{0}} + \frac{f′ + g′y_{0}}{g} (\frac{∂G_{1}}{∂y_{0}} + y_{0})) = 0 (11.1)

36 LUÍS R. A. FINOTTI

Proof. By Theorem 4.1 in [Fin04],

ϕ̄(x, y) =^{def} (x^{2} + 2F_{1}, y^{2} + 2G_{1})  (11.2)

is a well defined lift of the Frobenius modulo 4. Since

\frac{1}{2}ϕ̄^{∗}(ω^{σ}) = \frac{1}{2}ϕ̄^{∗} (\frac{dx}{2y + g^{σ}} ) = \frac{1}{2}ϕ̄^{∗} (\frac{dy}{(f′)^{σ} − (g′)^{σ}y} ) ,

using equation (11.2), the reduction the equation above modulo 2 gives us

\frac{dF_{1} + x_{0}dx_{0}}{g^{2}} = \frac{dG_{1} + y_{0}dy_{0}}{(f′ + g′y_{0})^{2}} .

Since dy_{0} = (f′ + g′y_{0})dx_{0}/g, equation (11.1) follows. □

We now can prove Proposition 6.11.

Proof of Proposition 6.11. We first prove that the condition is necessary. Assume we have a lift of the Frobenius associated to ν. By Theorem 4.1 in [Fin04], it must have the form

ϕ(x, y) = (x^{2} + 2F_{1} + 4P , y^{2} + 2G_{1} + 4Q),

for some P , Q ∈ W_{3}(k)[x, y].

Let δ be the 2-derivation associated to ϕ (as in [Bui96]):

δu =^{def} \frac{ϕ^{∗}u^{σ} − u^{2}}{2} .

We then have

δx = F_{1} + 2P

and, using Lemma 11.1,

\begin{aligned}δ^{2}x & = \frac{(F_{1} + 2P)^{σ} ∘ ϕ − (F_{1} + 2P)^{2}}{2} \\ & = \frac{F _{1}^{σ}(x^{2}, y^{2}) − F _{1}^{2}}{2} + \frac{∂F_{1}}{∂x}^{σ}(x^{2}, y^{2}) · F_{1} + \frac{∂F_{1}}{∂y}^{σ}(x^{2}, y^{2}) · G_{1}+ \\ & P^{σ}(x^{2}, y^{2}) + 2 · (…). \end{aligned}  (11.3)

But, by Lemma 2.6 of [Bui96], the reduction modulo 2 of δ^{2}x must be equal to F_{2} + x_{0}^{2} F_{1} + F_{1}^{2}. Also, since the reduction modulo 2 of P^{σ}(x^{2}, y^{2}) is clearly a square, say P^{2}, and F_{1} is the Teichmüller lift of F_{1}, reducing equation (11.3) modulo 2, we obtain

F_{2} + x_{0}^{2} F_{1} + F_{1}^{2} = Ψ(F_{1}) + (\frac{∂F_{1}}{∂x_{0}} )^{2} F_{1} + (\frac{∂F_{1}}{∂y_{0}} )^{2} G_{1} + P^{2}.  (11.4)

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

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

and hence, the condition is necessary.

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

\begin{matrix}U(W_{3}(k̄)) & ⟶^{ϕ} & U^{σ}(W_{3}(k̄)) \\ ^{ν}↑ & & ↑^{ν^{σ}} \\ U(k̄) & ⟶^{ϕ} & U^{σ}(k̄) \end{matrix}  (11.5)

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

h =^{def} y^{2} + g(x)y − f(x),

we write

h(x, y) = (h_{0}(x_{0}, y_{0}), h_{1}(x_{0}, x_{1}, y_{0}, y_{1}), h_{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^{∗}h_{i}^{σ} ∈ I, for i = 0, 1, 2, where I =^{def} (h_{0}, h_{1}, h_{2}).

By Theorem 4.1 of [Fin04], we have that ϕ^{∗}h_{0}^{σ}, ϕ^{∗}h_{1}^{σ} ∈ I. So we just need to show that ϕ^{∗}h_{2}^{σ} ∈ I.

One has

x^{2} = (x_{0}, x_{1}, x_{2})^{2} = (x_{0}^{2}, 0, x_{0}^{4} x_{1}^{2} + x_{1}^{4}).

Also, by Lemma 7.6, and noticing that F_{1} is the Teichmüller lift of F_{1},

2F_{1} = (0, F_{1}^{2}, x_{1}^{2} (\frac{∂F_{1}}{∂x_{0}} )^{4} + y_{1}^{2} (\frac{∂F_{1}}{∂y_{0}} )^{4} + Ψ(F_{1})^{2}) .

Hence, if P is as in the statement,

x^{2} + 2F_{1} + 4P = (x_{0}^{2}, F_{1}^{2}, F_{2}^{2} + 𝒳_{2}^{2}) ,  (11.6)

where,

𝒳_{2} =^{def} (x_{1} + F_{1}) (x_{0} + \frac{∂F_{1}}{∂x_{0}} )^{2} + (y_{1} + G_{1}) (\frac{∂G_{1}}{∂y_{0}} )^{2} + (x_{1} + F_{1})^{2},

and in a similar manner,

y^{2} + 2G_{1} + 4Q = (y_{0}^{2}, G_{1}^{2}, G_{2}^{2} + 𝒴_{2}^{2}) ,  (11.7)

where

𝒴_{2} =^{def} (x_{1} + F_{1}) (\frac{∂G_{1}}{∂x_{0}} )^{2} + (y_{1} + G_{1}) (y_{0} + \frac{∂G_{1}}{∂y_{0}} )^{2} + (y_{1} + G_{1})^{2}.

Note that, by Lemma 7.4,

h_{2}^{σ} = (g^{σ})^{4} y_{2} + ((f′)^{σ} + (g′)^{σ}y_{0})^{4} x_{2} + …,

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

ϕ^{∗}h_{2}^{σ} = g^{8} (G_{2} + 𝒴_{2})^{2} + (f′ + g′y_{0})^{8} (F_{2} + 𝒳_{2})^{2} + ….  (11.8)

38 LUÍS R. A. FINOTTI

Since

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

is a (well defined) lift,

g^{8} G_{2}^{2} + (f′ + g′y_{0})^{8} F_{2}^{2} + ⋯ ≡ 0 (mod (h_{0})),

where the omitted terms are the same as the ones in formula (11.8). Therefore

ϕ^{∗}h_{2}^{σ} ≡ g^{8} 𝒴_{2}^{2} + (f′ + g′y_{0})^{8} 𝒳_{2}^{2} (mod I),

and it suffices to prove that

g^{4} 𝒴_{2} + (f′ + g′y_{0})^{4} 𝒳_{2} ≡ 0 (mod I),

or,

(x_{1} + F_{1}) ((f′ + g′y_{0})^{2} (x_{0} + \frac{∂F_{1}}{∂x_{0}} ) + g^{2} \frac{∂G_{1}}{∂x_{0}} )^{2} +

(y_{1} + G_{1}) ((f′ + g′y_{0})^{2} \frac{∂F_{1}}{∂y_{0}} + g^{2} (y_{0} + \frac{∂G_{1}}{∂y_{0}} ))^{2} +

((x_{1} + F_{1})(f + g′y_{0})^{2} + (y_{1} + G_{1}) g^{2})^{2} ≡ 0 (mod I).

Applying Lemma 11.2, this equation becomes

((x_{1} + F_{1})\frac{(f′ + g′y_{0})^{2}}{g^{2}} + (y_{1} + G_{1})) ((f′ + g′y_{0})^{2} \frac{∂F_{1}}{∂y_{0}} + g^{2} (y_{0} + \frac{∂G_{1}}{∂y_{0}} ))^{2} +

((x_{1} + F_{1})(f + g′y_{0})^{2} + (y_{1} + G_{1}) g^{2})^{2} ≡ 0 (mod I).

Thus, it is enough to prove that

(x_{1} + F_{1})(f′ + g′y_{0})^{2} + (y_{1} + G_{1}) g^{2} ≡ 0 (mod I).

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

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

(x_{1} + F_{1})(f′ + g′y_{0})^{2} + (y_{1} + G_{1}) g^{2} ≡ 0 (mod I),

which finishes the proof that ϕ is well defined.

Finally, equations (11.6) and (11.7) show the diagram (11.5) commutes, since ν^{∗}𝒳_{2} = ν^{∗}𝒴_{2} = 0.

□

With Proposition 6.11 we can now prove Theorem 6.12.

Proof of Theorem 6.12. By Proposition 9.1, we have

dG_{1} = (h (f′ + g′y_{0}) + y_{0}) dy_{0},  (11.9)
dG_{2} = (h^{3} (f′ + g′y_{0})^{3} + y_{0}^{3}) dy_{0} + G_{1} dG_{1}.  (11.10)

Moreover, by equation (3.2), dF_{1} = (h g + x_{0}) dx_{0} implies that

\frac{∂F_{1}}{∂x_{0}} + \frac{∂F_{1}}{∂y_{0}} \frac{f′ + g′y_{0}}{g} = h g + x_{0},  (11.11)

and equation (11.9) implies that

\frac{∂G_{1}}{∂x_{0}} \frac{g}{f′ + g′y_{0}} + \frac{∂G_{1}}{∂y_{0}} = h (f′ + g′y_{0}) + y_{0}.  (11.12)

Now, to prove the theorem, by Proposition 6.11, it suffices to prove that

d (F_{2} + x_{0}^{2} F_{1} + Ψ(F_{1}) + (\frac{∂F_{1}}{∂x_{0}} )^{2} F_{1} + (\frac{∂F_{1}}{∂y_{0}} )^{2} G_{1}) =

dF_{2} + x_{0}^{2} dF_{1} + d(Ψ(F_{1})) + (\frac{∂F_{1}}{∂x_{0}} )^{2} dF_{1} + (\frac{∂F_{1}}{∂y_{0}} )^{2} dG_{1} = 0 (11.13)

and

d (G_{2} + y_{0}^{2} G_{1} + Ψ(G_{1}) + (\frac{∂G_{1}}{∂x_{0}} )^{2} F_{1} + (\frac{∂G_{1}}{∂y_{0}} )^{2} G_{1}) =

dG_{2} + y_{0}^{2} dG_{1} + d(Ψ(G_{1})) + (\frac{∂G_{1}}{∂x_{0}} )^{2} dF_{1} + (\frac{∂G_{1}}{∂y_{0}} )^{2} dG_{1} = 0 (11.14)

Since

d(Ψ(F_{1})) = x_{0} (\frac{∂F_{1}}{∂x_{0}} )^{2} dx_{0} + y_{0} (\frac{∂F_{1}}{∂y_{0}} )^{2} dy_{0} + F_{1}dF_{1},  (11.15)

using the formulas for dF_{1} and dF_{2} (in the statement), together with equations (11.9) and (3.2), equation (11.13) reduces to

h g (h g + x_{0} + \frac{∂F_{1}}{∂x_{0}} + \frac{∂F_{1}}{∂y_{0}} \frac{f′ + g′y_{0}}{g} )^{2} dx_{0} = 0,  (11.16)

which immediately follows from equation (11.11).

We prove that equation (11.14) holds in a similar fashion. Using the formula for dF_{1}, equations (11.9), (11.10) and (3.2), and the analogue to equation (11.15) for Ψ(G_{1}), a tedious computation shows that equation (11.14) reduces to

h (f′ + g′y_{0}) (h (f′ + g′y_{0}) + y_{0} + \frac{∂G_{1}}{∂x_{0}} \frac{g}{f′ + g′y_{0}} + \frac{∂G_{1}}{∂y_{0}} )^{2} dy_{0} = 0,

which immediately follows from equation (11.12). Hence (11.14) also holds.

40 LUÍS R. A. FINOTTI

To finish the proof, it suffices to show now that if either deg F_{n} = 0 or if deg G_{n} = 2^{n+1}(d − 1) − (d − 2), for n = 1, 2, then there exists h ∈ k(C) such that

dF_{1} = (h g + x_{0}) dx_{0},
dF_{2} = (h^{3} g^{3} + x_{0}^{3}) dx_{0} + F_{1} dF_{1}.

If deg F_{1} = deg F_{2} = 0, then dF_{1} = dF_{2} = 0. Moreover, by Theorem 6.7, either g = λ or g = λ x_{0}, for some λ ∈ k^{×}. Thus, take h = λ^{−1}x_{0} if g = λ, or h = λ^{−1} if g = λ x_{0}.

Finally, if deg G_{n} = 2^{n+1}(d − 1) − (d − 2), for n = 1, 2, then, by Theorem 6.5,

dG_{1} = λ^{−1}(f′ + g′y_{0}) dy_{0} + y_{0}dy_{0},
dG_{2} = (λ^{−3}(f′ + g′y_{0})^{3} + y_{0}^{3})dy_{0} + G_{1} dG_{1},

where λ is the (non-zero) coefficient of x_{0} in g. By Proposition 9.1,

dF_{1} = (λ^{−1} g + x_{0}) dx_{0},
dF_{2} = (λ^{−3} g^{3} + x_{0}^{3}) dx_{0} + F_{1} dF_{1}.

Hence, we just take h = λ^{−1}.

□

12. The Genus 2 Case

In this section we will do some explicit calculations with curves of genus two (i.e., d = 5). Besides illustrating the theory with a concrete example, another goal here is to show how the condition that g = x_{0} seems to be sufficient to obtain a lift of points with the degrees of the G_{n}’s equal to the lower bounds. Note that, by Proposition 6.6, it is necessary only to have deg g = 2, but we further choose to have g = x_{0} so that we can also obtain minimal degree for F_{1}, as in Proposition 6.9. (Observe that the condition g = x_{0} makes the Jacobian of C non-ordinary.)

Let

C/k : y_{0}^{2} + x_{0}y_{0} = x_{0}^{5} + d_{0}x_{0}^{4} + e_{0}x_{0}^{3} + f_{0}x_{0}^{2} + g_{0}x_{0} + h_{0}.  (12.1)

Since computations in such generality would be too long, we first simplify, via isomorphisms, equation (12.1). Note that, as observed earlier, the degrees are not necessarily invariant under isomorphisms!

An isomorphism of curves of genus 2 that preserve the point at infinity is given by a change of variables of the form

x̃_{0} ↦ α^{2}x_{0} + β,  (12.2)
ỹ_{0} ↦ α^{5}y_{0} + γx_{0}^{2} + δx_{0} + ϵ,  (12.3)

where α, β, γ, δ, ϵ ∈ k and α ≠ 0. Therefore, clearly if

C̃/k : ỹ_{0}^{2} + (ã_{0}x_{0}^{2} + b̃_{0}x_{0} + c̃_{0})ỹ_{0} = x̃_{0}^{5} + d̃_{0}x̃_{0}^{4} + ẽ_{0}x̃_{0}^{3} + f̃_{0}x̃_{0}^{2} + g̃_{0}x̃_{0} + h̃_{0}  (12.4)

is isomorphic to C, then ã_{0} = 0, and again we can make b̃_{0} = 1 and c̃_{0} = 0. Hence, in order for two isomorphic curves have this same form (i.e., g = x_{0}), we must have β = 0 and α must be a cubic root of unity. A tedious computation gives us:

d̃_{0} = αγ^{2} + α^{2}d_{0};
ẽ_{0} = α^{2}γ + αe_{0};
f̃_{0} = αδ + α^{2}δ^{2} + f_{0};
g̃_{0} = ϵ + α^{2}g_{0};
h̃_{0} = ϵ^{2} + αh_{0}.

Comparing the equations for d̃_{0} and ẽ_{0}, we obtain

γ = (αd_{0} + α^{2}d̃_{0})^{1/2} = α^{2}e_{0} + αẽ_{0},  (12.5)

and comparing the equations for g̃_{0} and h̃_{0}, we obtain

ϵ = (αh_{0} + h̃_{0})^{1/2} = α^{2}g_{0} + g̃_{0}.  (12.6)

Observe that the curves are non-singular if, and only if, Δ =^{def} g_{0}^{2} + h_{0} and Δ̃ =^{def} g̃_{0}^{2} + h̃_{0} are non-zero. But equations (12.5) and (12.6) imply

d̃_{0} + ẽ_{0}^{2} = α^{2}(d_{0} + e_{0}^{2}),
Δ̃ = αΔ

Therefore, clearly

j_{1} =^{def} \frac{d_{0} + e_{0}^{2}}{Δ^{2}} = \frac{d̃_{0} + ẽ_{0}^{2}}{Δ̃^{2}}

and

j_{2} =^{def} Δ^{3} = Δ̃^{3}

(remember that α is cubic root of unity) are invariants for curves of this particular form.

Conversely, if we have curves C and C̃ given by equations (12.1) and (12.4), with ã_{0} = 0, b̃_{0} = 1 and c̃_{0} = 0, such that

\frac{d_{0} + e_{0}^{2}}{Δ^{2}} = \frac{d̃_{0} + ẽ_{0}^{2}}{Δ̃^{2}}

and

Δ^{3} = Δ̃^{3},

42 LUÍS R. A. FINOTTI

then the curves are isomorphic, either over k itself or over a degree two extension of k: just let (as in equations (12.2) and (12.3)):

α =^{def} Δ̃/Δ
β =^{def} 0
γ =^{def} α^{2}e_{0} + αẽ_{0}
δ =^{def} a solution of α^{2}X^{2} + αX + (f_{0} + f̃_{0}) = 0
ϵ =^{def} α^{2}g_{0} + g̃_{0}.

Hence, the invariants j_{1}, j_{2} ∈ k, with j_{2} ≠ 0, determine a curve of genus 2 with g = x_{0} up to isomorphism, and given such any j_{1}, j_{2} ∈ k, with j_{2} ≠ 0, the curve

y_{0}^{2} + x_{0}y_{0} = x_{0}^{5} + ((j_{1}j_{2}^{2/3})^{1/2})x_{0}^{3} + j_{2}^{1/3}

is isomorphic (over k̄) to a curve with invariants j_{1} and j_{2}. So, we will consider here only curves given by equations of the form

y_{0}^{2} + x_{0}y_{0} = x_{0}^{5} + e_{0}x_{0}^{3} + h_{0}

(with h_{0} ≠ 0). The computation, up to the fourth coordinate of the Witt vectors, of the absolute minimal degree lifting with respect to y for a curve given by the above equation yields a lifting (over W_{4}(k)) given by

C/W_{4}(k) : y^{2} + (ax^{2} + x)y = x^{5} + dx^{4} + ex^{3} + fx^{2} + gx + h,

where a = (a_{0}, a_{1}, a_{2}, a_{3}), …, h = (h_{0}, h_{1}, h_{2}, h_{3}), are given by

a = (0, 0, e_{0} + e_{1}^{2} + h_{0}, a_{3}),
d = (0, e_{0} + h_{0}, d_{2}, d_{3}),
e = (e_{0}, e_{1}, e_{2}, e_{3}),
f = (0, f_{1}, f_{2}, f_{3}),
g = (0, g_{1}, g_{2}, g_{3}),
h = (h_{0}, h_{0}^{2}, g_{1} + h_{0}^{4}, g_{1}^{4}h_{0}^{4} + h_{0}^{8} + f_{1}^{4}h_{0}^{8}),

with a_{3}, d_{2}, d_{3}, e_{1}, e_{2}, e_{3}, f_{1}, f_{2}, f_{3}, g_{1}, g_{2}, g_{3} ∈ k arbitrary, and a lift of points is given by

ν(x_{0}, y_{0}) = ((x_{0}, F_{1}, F_{2}, F_{3}), (y_{0}, G_{1}, G_{2}, G_{3})),

where F_{1} ∈ k (in fact, F_{1} = h_{0}, as in section 10), deg G_{1} = 13, deg F_{2} = 20, deg G_{2} = 29, deg F_{3} = 28 and deg G_{3} = 61, and hence, for i = 0, …, 3, deg G_{i} = (2^{i+3} − 3), i.e., the G_{i}’s have the minimal possible degrees.

MIN. DEG. LIFTS IN CHAR. 2 43

It seems worth mentioning that if we just want the lift with minimal degrees modulo 4, then you do not need d_{1} = e_{0} + h_{0} as in the formulas above. In this case d_{1} could be arbitrary. The condition on d_{1} is imposed so that we are able to obtain G_{2} with order 29. In the same way, if you only want the minimal degrees modulo 8, you can have any a_{2} ∈ k, but if you want deg G_{3} = 61, then you have to impose the condition that a_{2} = a_{0} + e_{1}^{2} + h_{0} as above. So, to obtain deg G_{4} = 125, if at all possible, one might have to impose conditions on the constants left as arbitrary here.

Also, observe that, without extra conditions, the lifting of the curve is not unique up to isomorphisms!

As stated in Theorem 6.13, similar computations with ordinary elliptic curves (genus 1) show we can also obtain lifts of points with the degrees of G_{i}’s equal to the lower bound. It seems then that the condition g = x_{0} might also to be sufficient to have a lift with G_{i}’s having degrees equal to the lower bounds.

13. Examples of Codes

In this section we exhibit some codes constructed with liftings of hyperelliptic curves. We will deal only with the case when the characteristic of the field and the length of the Witt vectors are both 2, i.e., when, with the notation of section 2, p^{l} = 4. (Also, as observed before, we consider only divisors of the form nP_{∞}.) It is worth noticing that a few examples for genus 1 were computed by Voloch and Walker in [VW00].

Let q = 8 and 𝔽_{8} = 𝔽_{2}[α], where α^{3} + α + 1 = 0. Consider the curve

C : y_{0}^{2} + y_{0} = x_{0}^{5} + x_{0}^{3} + x_{0},

with lifting

C : y^{2} + ((0, 1)x^{2} + (0, 1)x + (1, 1))y = x^{5} + (1, 1)x^{3} + (1, 0)x,

and the minimal degree lift of points from C to C with respect to y

F_{1} = x_{0} + 1,
G_{1} = x_{0}^{5} y_{0} + x_{0}^{4} y_{0} + x_{0}^{3} y_{0} + x_{0}^{2} y_{0} + x_{0} y_{0} + x_{0}^{5} + x_{0}^{3} + x_{0} + 1.

(Observe that, in this case, C is not an absolute minimal degree curve over C, so ν is not an absolute minimal degree lift of points.)

Then, considering functions in ℒ(5P_{∞}) and following the procedure described in section 2, one obtains a code with the following generating matrix (over ℤ/4):

( \begin{matrix}1 & 0 & 0 & 0 & 1 & 3 & 0 & 3 \\ 0 & 1 & 0 & 0 & 3 & 1 & 1 & 2 \\ 0 & 0 & 1 & 0 & 3 & 0 & 3 & 1 \\ 0 & 0 & 0 & 1 & 0 & 3 & 3 & 1 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 \end{matrix} ) ,

The corresponding binary code has length 16, 2^{11} codewords and minimum weight 4. According to N. Sloane’s site,

               http://www.research.att.com/~njas/codes/And/,

this is the size of the largest possible non-linear binary code with this length and minimum distance. Note also that, in this particular case, there is a linear code with the same parameters.

One can also use only a few functions in ℒ(nP_{∞}) (for some fixed n) to generate the code instead of all functions. We were then able to find a few codes matching the best known linear code for a fixed length and size. For example, again over 𝔽_{8}, consider the curve

C : y_{0}^{2} + α^{6} x_{0} y_{0} = x_{0}^{5} + α^{2} x_{0}^{4} + α^{4} x_{0}^{2} + α^{5}

with lifting

C : y^{2} + (α^{6}, 0)x y = x^{5} + (α^{2}, α)x^{4} + (α^{4}, 0)x^{2} + (α^{5}, α^{5}),

and the minimal degree lift of points from C to C with respect to y (which turns out to be also a minimal degree lift with respect to x in this case) given by

F_{1} = x_{0} y_{0} + α^{3} x_{0}^{3} + x_{0}^{2} + α^{2},
G_{1} = (α^{2} x_{0}^{7} + α^{3} x_{0}^{4} + α^{5} x_{0}^{3} + α^{2} x_{0}^{2} + α^{4} x_{0}) y_{0}+
α^{5} x_{0}^{9} + α^{2} x_{0}^{8} + α^{4} x_{0}^{7} + α^{4} x_{0}^{6} + α^{5} x_{0}^{5} + α^{2} x_{0}^{3} + x_{0}^{2} + α x_{0} + α^{6}.

If we now use only the functions of the form

h = c_{0} + c_{1} x + c_{2} y, c_{0}, c_{1}, c_{2} ∈ W_{2}(𝔽_{8})

in the construction of the code (so we discard functions in ℒ(5P_{∞}) involving x^{2}), then we obtain the generating matrix

( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 & 1 & 2 & 1 & 2 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 2 & 0 & 1 & 3 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 3 & 1 & 0 & 3 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 1 & 0 & 3 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 3 & 2 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 3 \end{matrix} ) ,

and the resulting binary code has length 26, 2^{14} codewords and minimum weight 6, which according to A. Browuer’s web site

                http://www.win.tue.nl/~aeb/voorlincod.html,

is the minimum weight of the best known linear code with that size and length. (The size of the best non-linear code with this length and minimum weight is 2^{15}.)

Finally, one can also use the results from the previous section, where we computed absolute minimal degree liftings, to construct codes. If we have, again over 𝔽_{8},

C : y_{0}^{2} + x_{0} y_{0} = x_{0}^{5} + α^{2} x_{0}^{3} + α^{6},

then an absolute minimal degree curve modulo 4 over C, with respect to both x and y, is given by (as in the previous section)

C : y^{2} + x y = x^{5} + (0, α^{5})x^{4} + (α^{2}, 1)x^{3} + (0, 1)x^{2} + (0, 1)x + (α^{6}, α^{5}),

and we have an absolute minimal degree lift of points

F_{1} = α^{6},
G_{1} = (x_{0}^{4} + α^{2} x_{0}^{2} + x_{0}) y_{0} + α^{4} x_{0}^{6} + x_{0}^{5} + x_{0}^{4} + α^{2} x_{0}^{3} + α x_{0}^{2} + α^{2}.

(Observe that the proof of Proposition 6.9 gives us a method to explicitly find C, F_{1} and G_{1}.) Using again only functions generated by {1, x, y}, one obtains the code given by the generating matrix

( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 3 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 3 & 2 & 2 \end{matrix} ) .

This code has then length 22, 2^{14} codewords and minimum weight 4, which is the minimum weight for the best linear code with this length and size. (The largest non-linear code with the same length and minimum weight is 9 · 2^{13}.)

Acknowledgement. The computations mentioned in this paper were made with either Magma or Mathematica, and the author would like to thank J. Cannon for his invaluable help with the use of Magma in the computations of the parameters of the obtained codes and J. Walker for her helpful comments on the results.

References

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

[Car98] Claude Carlet. ℤ_{2^{k}}-linear codes. IEEE Trans. Inform. Theory, 44(4):1543–1547, 1998.

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

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

[Fin04] L. R. A. Finotti. Minimal degree liftings of hyperelliptic curves. J. Math. Sci. Univ. Tokyo, 11(1):1– 47, 2004.

[Har77] R. Hartshorne. Algebriac Geometry, volume 52 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1977.

[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.

[Sil85] J. H. Silverman. The Arithmetic of Elliptic Curves, volume 106 of Graduate Texts in Mathematics. Springer-Verlag, 1985.

[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.

[Wal99] J. L. Walker. Algebraic geometric codes over rings. J. Pure Appl. Algebra, 144(1):91–110, 1999.

MIN. DEG. LIFTS IN CHAR. 2 47

Department of Mathematics, Ohio State University, Columbus, OH – 43210

E-mail address: finotti@math.ohio-state.edu