Quadratic and differential forms over function fields of Pfister quadrics in characteristic two^{∗}

R. Aravire Departamento de Cs. Físicas y Matemáticas

Universidad Arturo Prat, Casilla 121, Iquique, Chile e-mail: raravire@unap.cl

R. Baeza Instituto de Matemáticas, Universidad de Talca, Casilla 721, Talca, Chile e-mail: rbaeza@inst-mat.utalca.cl

Abstract

Let F be a field of characteristic 2. Let Ω_{F}^{n} be the F -space of differential forms over F. There is a homomorphism ℘ : Ω_{F}^{n} ⟶ Ω_{F}^{n} / d Ω_{F}^{n−1} given by ℘ (x \frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}} ) = (x^{2} −x) \frac{d x_{1}}{x_{1}} ∧⋯∧ \frac{d x_{n}}{x_{n}} mod d Ω_{F}^{n−1}. Let

H^{n+1}(F) = coker(℘). If p =≪ a_{1}, …, a_{n}; b]] is an anisotropic quadratic Pfister form over F and F (p) the function field of the Pfister quadric {p = 0}, we compute the kernel H^{n+1}(F (p)/F) = ker [H^{n+1}(F) → H^{n+1}(F (p))] for all m. Using Kato’s correspondence between differential and quadratic forms we compute the kernels I^{m}W_{q}(F (p)/F) = ker [I^{m}W_{q}(F) → I^{m}W_{q}(F (p))], where W_{q}(F) denotes the Witt group of quadratic forms over F and I_{F} is the maximal ideal of the Witt ring W (F) of symmetric bilinear forms over F.

Keywords: Quadratic Forms, Bilinear forms, Pfister forms, Witt ring, Differential forms.

1 Introduction

We continue in this paper our previous work [Ar-Ba_{1}] on the behavior of quadratic and differential forms under function field of Pfister quadrics over fields with 2 = 0. In [Ar-Ba_{1}] we considered bilinear Pfister quadrics, and in this paper we will treat the case of quadratic Pfister quadrics. If F is a field with 2 = 0, we will denote by W (F) the Witt ring of non singular symmetric bilinear forms over F and by W_{q}(F) the W (F)- module of non singular quadratic forms over F. Let I_{F} ⊂ W (F) be the maximal ideal of even dimensional forms (see [Sa], [Ba], [Mi] for details). For a_{i} ∈ F^{∗} = F \{0}, 1 ≤ i ≤ n, we will denote by < a_{1}, …, a_{n} > the bilinear form with diagonal Gram-matrix and entries a_{1}, …, a_{n} on the diagonal. The quadratic form x^{2} + xy + ay^{2} will be denoted by [1, a], a ∈ F. The maximal ideal I_{F} ⊂ W (F) is additively generated by the forms ≪ a ≫=< 1, a >, a ∈ F^{∗}, so that the powers I_{F}^{n} are generated by the nfold bilinear Pfister forms ≪ a_{1}, …, a_{n} ≫=≪ a_{1} ≫ ⋯ ≪ a_{n} ≫, a_{i} ∈ F^{∗}. The submodules I^{n}W_{q}(F) of W_{q}(F) are generated by the n-fold quadratic Pfister forms ≪ a_{1}, …, a_{n}; a]] =≪ a_{1}, …, a_{n} ≫ ·[1, a]. The graded objects I_{F}^{n}/I_{F}^{n+1}, resp. I^{n}W_{q}(F)/I^{n+1}W_{q}(F) will be denoted by \overline{I_{F}^{n}}, resp. \overline{I^{n}W_{q}(F)}.

∗Corresponding author E-mail address: rbaeza@inst-math.utalca.cl (R. Baeza).

Let p be an anisotropic bilinear or quadratic n-fold Pfister form. Let F (p) denote the function field of the quadric {p = 0} over F. In [Ar-Ba_{1}] we computed the kernels ker [\overline{I^{m}W_{q}(F)} ⟶ \overline{I^{m}W_{q}(F (p))}] if p is a bilinear n-fold Pfister form for all m ≥ 0. In this paper we will compute these kernels if p is a quadratic n-fold Pfister form. The bilinear case implies the quadratic case for m ≤ n, as it is shown in [Ar-Ba_{1}]. Although the methods are similar, the arguments we need in the latter case are much more delicate. Actually we will compute the absolute kernels ker [I^{m}W_{q}(F) ⟶ I^{m}W_{q}(F (p))] by a trick used in [Ar-Ba_{2}] (see the statements below).

As in [Ar-Ba_{1}] we will use Kato’s correspondence between differential forms and quadratic forms. We will work with differential forms and then translate the results into the language of quadratic forms. Let us first introduce briefly some notations and results. If Ω_{F}^{1} denotes the F -vector space generated by the symbols d a, a ∈ F, with the relations d(a + b) = d a + d b, d(ab) = b d a +

a d b, then let Ω_{F}^{n} = ⋀^{n} Ω_{F}^{1} and d : Ω_{F}^{n} ⟶ Ω_{F}^{n+1} be the differential operator d(x d x_{1} ∧ ⋯ ∧ d x_{n}) = d x ∧ d x_{1} ∧ ⋯ ∧ d x_{n}. Then d^{2} = 0 and d extends the

derivation d : F ⟶ Ω_{F}^{1} . The space Ω_{F}^{∗} = ⨁_{n=0}^{∞} Ω_{F}^{n} (Ω_{F}^{0} = F) is a ℤ-graded

algebra with the exterior multiplication ∧ : Ω_{F}^{n} × Ω_{F}^{m} ⟶ Ω_{F}^{n+m}. The usual Artin-Schreier operator ℘ : F ⟶ F, ℘(x) = x^{2} − x extends to a well defined homomorphism

℘ : Ω_{F}^{n} ⟶ Ω_{F}^{n} / d Ω_{F}^{n−1}

through

℘ (x\frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}} ) = \overline{(x^{2} − x)\frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}}} .

Let ν_{F} (m) = ker(℘), H^{m+1}(F) = coker(℘), so that

0 ⟶ ν_{F} (m) ⟶ Ω_{F}^{m} ⟶ Ω_{F}^{m}/ d Ω_{F}^{m−1} ⟶ H^{m+1}(F) ⟶ 0

is exact. The groups ν_{F} (n) act on the groups H^{m+1}(F) by exterior multiplication of forms

⋀ : ν_{F} (n) × H^{m+1} ⟶ H^{n+m+1}(F).

These groups behave functiorially with respect to field extensions. If F ↪ L is a field extension we will denote by ν_{L/F} (m), resp. H^{m+1}(L/F), the kernels ker(ν_{F} (m) ⟶ ν_{L}(m)), resp. ker(H^{m+1}(F) ⟶ H^{m+1}(L)). A basic lemma due to Kato and stated below shows that ν_{F} (m) is additively generated by the pure logarithmic differential forms \frac{d x_{1}}{x_{1}} ∧ ⋯ ∧ \frac{d x_{n}}{x_{n}}, x_{i} ∈ F^{∗}. Recall that a 2-basis of F is a subset B = {a_{i}, i ∈ I} such that the elements {a^{ε} = ∏_{i∈I} a^{ε_{i}}, ε =

(ε_{i})_{i∈I}, ε_{i} ∈ {0, 1}, ε_{i} = 0 for almost all i ∈ I} form a F^{2}-basis of F. Then ordering I, it is easy to see that for any n ≥ 1, \{\frac{d a_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{d a_{in}}{a_{in}} , i_{1} < ⋯ < i_{n}\} is a F -basis of Ω_{F}^{n}. Using such a basis we can define a B-depending Artin-Schreier operator ℘ : Ω_{F}^{n} ⟶ Ω_{F}^{n}, ℘ (∑_{i_{1}<···<i_{n}} c_{i_{1},··· ,i_{n}} \frac{d a_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{d a_{in}}{a_{in}} ) =

∑_{i_{1}<···<i_{n}} ℘ (c_{i_{1},··· ,i_{n}}) \frac{d a_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{d a_{in}}{a_{in}}. This operator ℘ is modulo d Ω_{F}^{n−1}, well defined, and we will use it some times when the 2-basis B is given. Moreover if we set for η = ∑_{i_{1}<···<i_{n}} c_{i_{1},··· ,i_{n}} \frac{d a_{i_{1}}}{a_{i_{1}}} ∧ ⋯ ∧ \frac{d a_{in}}{a_{in}}, η^{[2]} = ∑_{i_{1}<···<i_{n}} c_{i_{1},··· ,i_{n}}^{2} \frac{d a_{i_{1}}}{a_{i_{1}}} ∧

⋯ ∧ \frac{d a_{in}}{a_{in}}, we get ℘(η) = η^{[2]} − η. In fact the square operation η^{[2]} is well defined modulo d Ω_{F}^{n−1}. Using this notation we can write H^{m+1}(F) = Ω_{F}^{m}/ (℘Ω_{F}^{m} + d Ω_{F}^{m−1}). Another important operator on differential forms is the Cartier operator. Given a 2-basis B ={a_{i}, i ∈ I} of F we define

[Ω_{F}^{n}]^{2} = {η^{[2]}| η ∈ Ω_{F}^{n} }

which is the subgroup of squares of Ω_{F}^{n} (with respect to B). In [Ca] it is shown that the space Z_{F}^{n} = ker(d) ⊂ Ω_{F}^{n} of closed n-forms satisfies Z_{F}^{n} = [Ω_{F}^{n}]^{2} ⨁ d Ω_{F}^{n−1}. Then we define the Cartier operator

C : Z_{F}^{n} ⟶ Ω_{F}^{n}

by

C(η^{[2]} + d ω) = η

C is well defined, and independent of the 2-basis B. This operator satisfies the following rules:

• C (a^{2} \frac{d b}{b}) = a\frac{d b}{b}  for all a, b ∈ F^{∗}
• C (d η) = 0  for all η ∈ Ω_{F}^{n−1}
• C (a^{2}w) = aw  for all w ∈ Z_{F}^{n}
• C (w ∧ λ) = C (w) ∧ C (λ)  for all w ∈ Z_{F}^{n}, λ ∈ Z_{F}^{m}.

The elements of ν_{F} (n) are characterized by the following properties: for any η ∈ Ω_{F}^{n} it holds

η ∈ ν_{F} (n) if and only if η ∈ Z_{F}^{n} and C(η) = η.

Moreover for η ∈ Ω_{F}^{n}

η ∈ d Ω_{F}^{n−1} if and only if η ∈ Z_{F}^{n} and C(η) = 0.

We will use these properties thoroughly in this paper without mentioning any source. Let us state now Kato’s result (see [Ka]). To this end we need some notation. Let B be a fixed 2-basis and take any fixed ordering on I. For any n ≥ 1, let Σ_{n,F} denote the set of functions σ : {1, …, n} ⟶ I with σ(i) < σ(j) whenever i < j. We order Σ_{n,F} lexicographically. For any α ∈ Σ_{n,F} we denote by \frac{d a_{α}}{a_{α}} the form \frac{d a_{α(1)}}{a_{α(1)}} ∧ ⋯ ∧ \frac{d a_{α(n)}}{a_{α(n)}} . Thus \{\frac{d a_{α}}{a_{α}} , α ∈ Σ_{n,F} \} is a F -basis of

Ω_{F}^{n}. Let Ω_{F,α}^{n} be the subspace generated by the forms \frac{d a_{β}}{a_{β}}, β ≤ α and Ω_{F,<α}^{n}

the subspace generated by \frac{d a_{β}}{a_{β}} with β < α. For any i ∈ I we will also denote by F_{i} the subfield F^{2}(a_{j}| j ≤ i) and by F_{<i} the subfield F^{2}(a_{j}| j < i). Thus F_{i} = F_{<i}(a_{i}). (If i_{0} is the first element in I, then F_{<i_{0}} = F^{2}). Kato’s lemma (see [Ka]) can be stated as follows:

Lemma (Kato [Ka]) Fix α ∈ Σ_{n,F}, y ∈ F and assume ℘(y \frac{d a_{α}}{a_{α}}) ∈ Ω_{F,<α}^{n} + d Ω_{F}^{n−1}. Then there exist v ∈ Ω_{F,<α}^{n} and c_{i} ∈ F_{α(i)}^{∗}, 1 ≤ i ≤ n, such that

y \frac{d a_{α}}{a_{α}} = v + \frac{d c_{1}}{c_{1}} ∧ ⋯ ∧ \frac{d c_{n}}{c_{n}} .

This implies in particular, that any η ∈ ν_{F} (n) can be written as

η = ∑_{β≤α} ε_{β} \frac{d f_{β}}{f_{β}}

for some α ∈ Σ_{n,F}, f_{β(i)} ∈ F_{β(i)}^{∗}, ε_{β} ∈ \{0, 1\} and \frac{d f_{β}}{f_{β}} = \frac{d f_{β(1)}}{f_{β(1)}} ∧ ⋯ ∧ \frac{d f_{β(n)}}{f_{β(n)}} .

In [Ar-Ba_{2}] we have shown the following result, which we quote here for the sake of completeness, since we will use it several times in the future.

Lemma 1.1 Let B = {a_{i}, i ∈ I} be a 2-basis of F with a given ordering on I. Let α ∈ Σ_{n,F} and let ω = ∑_{γ≤α} c_{γ} \frac{d a_{γ}}{a_{γ}} ∈ Ω_{F}^{n} be a differential form with c_{α} ≠ 0. If ω ∈ d Ω_{F}^{n−1}, then there exist elements M_{i} ∈ F_{<α(i)}, 1 ≤ i ≤ n such that

c_{α} = a_{α(1)}M_{1} + ⋯ + a_{α(n)}M_{n}.

After these preliminaries we proceed now to formulate our main result. Let p =≪ a_{1}, …, a_{n}; b]] be an anisotropic quadratic n-fold Pfister form over F. Since we can alter b modulo ℘(F) we may assume b ∈ F^{2}. Moreover since ≪ a_{1}, …, a_{n} ≫ is anisotropic, a_{1}, …,a_{n} are part of a 2-basis of F. Let F (p) be the function field of the quadric {p = 0} over F. Then the main result in this paper is

Theorem 1.2 For all m ≥ 0 it holds

H^{m+1}(F (p)/F) = \{ \begin{matrix}0 & if m < n \\ ν_{F} (m − n) ∧ \overline{b\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}}} & if m ≥ n. \end{matrix}

This result follows easily from (4.1) (5.4) in [Ar-Ba_{1}] for n = 0 and for m ≤ n, n ≥ 1. Thus we can assume n ≥ 1 and m ≥ n. Using Kato’s correspondence (1.2) reads

Theorem 1.3 For all m, n ≥ 0

\overline{I^{m}W_{q}(F (p)/F)} = \{ \begin{matrix}0 & if m < n \\ \overline{I_{F}^{m−n} · p} & if m ≥ n. \end{matrix}

We give now the full graded version of this result

Theorem 1.4 Let p =≪ a_{1}, …, a_{n}; b]] be an anisotropic quadratic n-fold Pfister form over F. Then for all m ≥ 0

I^{m}W_{q}(F (p)/F) = \{ \begin{matrix}W (F) · p & if m < n \\ I_{F}^{m−n} · p & if m ≥ n \end{matrix}

(here we write W (F) instead of I_{F}^{0} ).

Remark 1.5 In [Ar-Ba_{1}] we computed \overline{I^{m}W_{q}(F (ϕ)/F)} for ϕ =≪ a_{1}, …, a_{n} ≫. The corresponding full graded version of the main result of [Ar-Ba_{1}] is now

I^{m}W_{q}(F (ϕ)/F) = \{ \begin{matrix}ϕ · W_{q}(F) & if m < n \\ ϕ · I^{m−n}W_{q}(F) & if m ≥ n \end{matrix}  (1.6)

Since the proofs of (1.4) and (1.6) are very similar and rely on the same principle, we will derive only (1.4) from (1.3).

Proof of (1.4). Let p =≪ a_{1}, …, a_{n}; b]] be anisotropic. Let us first assume that the field F has a finite 2-basis B = {a_{1}, …, a_{N}}. Then (see [Mi]) I_{F}^{N+1} = 0. Take q ∈ I^{m}W_{q}(F (p)/F) i.e. q ∈ I^{m}W_{q}(F) and q ⊗ F (p) ∼ 0. Assume q ≠ 0. Then \overline{q} ∈ \overline{I^{m}W_{q}(F (p)/F)} and hence if m ≥ n, \overline{q} ∈ \overline{I^{m−n} · p}, i.e. q = ϕ_{1}p + q_{1} with ϕ_{1} ∈ I_{F}^{m−n} and q_{1} ∈ I^{m+1}W_{q}(F). Since q⊗F (p) = 0, p⊗F (p) = 0 it follows q_{1} ⊗ F (p) = 0. Thus q_{1} = ϕ_{2}p + q_{2} with ϕ_{2} ∈ I_{F}^{m+1−n} and q_{2} ∈ I^{m+2}W_{q}(F). Iterating this procedure M ≥ N + 1 − m times and using I^{N+1} = 0 we arrive at an equation q = (ϕ_{1} + ϕ_{2} + ⋯ + ϕ_{M})p ∈ I_{F}^{m−n} · p. This proves the claim in this case. In the general case we have the relation q ⊗ F (p) = 0 and this relation involves finitely many elements a_{1}, …, a_{N} ∈ B. Let B_{0} = {a_{1}, …, a_{N}} ⊂ B and set F_{0} = F^{2}(a_{1}, …, a_{N}). Then there exist forms q_{0}, p_{0} over F_{0}, p_{0} =≪ a_{1}, …, a_{n}; b]], such that q = q_{0} ⊗ F, p = p_{0} ⊗ F and q_{0} ⊗ F (p_{0}) = 0. From the first part we conclude q_{0} ∈ I_{F_{0}}^{m−n} · p_{0}, and extending scalars to F we obtain q ∈ I_{F}^{m−n} · p. This proves the claim in the case m ≥ n. If m < n, then in the first case we would obtain q ∈ I^{m+1}W_{q}(F) and iterating, we may assume m = n, i.e. q ∈ W (F) · p. The rest of the proof is the same. This concludes the proof of (1.4) and also of (1.6).

We proceed now to describe briefly the plan of the paper. In section 2 we reduce the computation of H^{m+1}(F (p)/F) to study for a form w ∈ Ω_{F}^{m} the equation w = ℘u + d v + T η, where u ∈ Ω_{F}^{m} ⊗ M, v ∈ Ω_{F}^{m−1} ⊗ M, η ∈ ν_{M}(m) and T is the polynomial (2.1). Here M denotes the field F (X_{μ}^{2}, Y_{μ}^{2}, μ ∈ S_{n}), X_{μ}, Y_{μ} are variables for μ ∈ S_{n} = set of maps μ : {1, …, n} → {0, 1}, μ ≠ 0 (see (2.8)).

The main idea is now to reduce this equation to one without variables, so that during this process the form η changes and successively it gets factors of

the form \frac{d a_{i}}{a_{i}}, thereby at the end of the reduction process we get an equation

w = ℘u_{0} + d v_{0} + b\frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ ξ

with ξ ∈ ν_{F} (m − n), u_{0}, v_{0} ∈ Ω_{F}, which is the content of theorem (1.2). This reduction process is achieved in section 4 and 6. Section 3 is of preparatory character (see also [Ar-Ba_{1}]), and we analyze here the decomposition in partial fractions (= p-components) of forms contained in ν_{N(Y^{2})}(m) for some field N, where Y is a variable over N (see (3.6)). We show that the p−components are also forms contained in groups ν_{•}(m). In section 5, we characterize the forms η ∈ ν_{N(Y^{2})}(m) with the property that a^{σ}η ∈ d Ω_{M(Y^{2})}^{m−1} for all σ ≤ τ ∈ S_{n}, where a_{1}, …, a_{n} ∈ M are part of a 2-basis of N (see (5.1)). This property, called good shape, plays a crucial role during the reduction process done in section 6 (see main lemma (6.1)).

2 First reduction step in the proof of (1.2)

Let F be a field with 2 = 0 and let p =≪ a_{1}, …, a_{n}; b]] be an anisotropic quadratic n-fold Pfister form over F with function field F (p). We may assume without restriction b ∈ F^{2}. Let S_{n} be the set of maps μ : {1, …, n} ⟶ {0, 1}, μ ≠ 0, and let B be a 2-basis of F, which we may assume contains a_{1}, …, a_{n} as the first elements in some ordering of the index set I of B. Let Σ_{n,F} be the set of maps σ : {1, …, n} ⟶ I with σ(i) < σ(j) whenever i < j. For any μ ∈ S_{n} we take a pair of independent variables X_{μ}, Y_{μ} and we form the field L = F (X_{μ}, Y_{μ}, μ ∈ S_{n}). For m < n we will identify S_{m} ⊂ S_{n} in the obvious way. Let T_{0} = b+∑_{μ∈S_{n}} a^{μ}Y_{μ}^{2}(b+X_{μ} +X_{μ}^{2}) be the generic polynomial

representing the pure part of p. Here a^{μ} = a_{1}^{μ(1)} ⋯ a_{n}^{μ(n)}. Then F (p) = L(z_{0}) with z_{0}^{2} + z_{0} = T_{0}. We can change T_{0} modulo ℘F [X_{μ}, Y_{μ}, μ ∈ S_{n}], so that we will work in this paper with the polynomial

T = b + ∑_{μ∈S_{n}} a^{μ}Y_{μ}^{2}(b + X_{μ}^{2}) + ∑_{μ∈S_{n}} (a^{μ}Y_{μ}^{2})^{2} X_{μ}^{2}
= ∏_{μ∈S_{n}} (1 + a^{μ}Y_{μ}^{2}) (b + a^{μ}Y_{μ}^{2}X_{μ}^{2})  (2.1)

and F (p) = L(z), z^{2} + z = T.

The 2-basis B of F extends to the 2-basis B_{L} = B ∪{X_{μ}, Y_{μ}, μ ∈ S_{n}} of L, where we order the X′s, Y′s lexicographically. Since F (p)/L is a separable quadratic extension then B_{L} remains a 2-basis of F (p). This is consequence of the following

Lemma 2.2 If E/F is a finite separable extension, then any 2-basis of F is also a 2-basis of E. Thus Ω_{E}^{∗} = Ω_{F}^{∗} ⊗_{F} E.

In particular we obtain Ω_{F (p)}^{n} = Ω_{L}^{n} ⊗_{L} F (p) for all n ≥ 1, and B_{F} ∪{X_{μ}, Y_{μ}, μ ∈ S_{n}} is 2-basis of F (p).

Take now w ∈ H^{m+1}(F (p)/F), i.e. w ∈ Ω_{F}^{m} satisfies

w = ℘(u) + d v  (2.3)

with u ∈ Ω_{F (p)}^{m}, v ∈ Ω_{F (p)}^{m−1}. Thus u, v are forms generated over L(z) = L + Lz by the differentials d a_{i}, i ∈ I, d X_{μ}, d Y_{μ}, μ ∈ S_{n}.

Lemma 2.4 The forms u, v in (2.3) can be chosen in Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}, μ ∈ S_{n})(z).

Proof. Let us represent w, u and d v in the 2-basis B_{L} = {b_{j}, j ∈ J} of F (p) introduced above, i.e. they are generated by the differentials \frac{d a_{i}}{a_{i}}, a_{i} ∈ B_{F}, \frac{d X_{μ}}{X_{μ}},

\frac{d Y_{μ}}{Y_{μ}} over F (p). Let γ ∈ Σ_{m,F} be the maximal multi-index occurring in w, and δ ∈ Σ_{m,L} the maximal one occurring in u. Then the maximal multi-index of d v is ≤ max{γ, δ}. If γ < δ then we have from (2.3)

℘ (u_{δ} \frac{d b_{δ}}{b_{δ}} ) ≡ 0 mod (d Ω_{F (p)}^{m−1} + Ω_{F (p),<δ}^{m})  (2.5)

where u_{δ} \frac{d b_{δ}}{b_{δ}} is the maximal (≠ 0) element in the expansion of u. Using Kato’s lemma we can write

u_{δ} \frac{d b_{δ}}{b_{δ}} = \frac{d f_{1}}{f_{1}} ∧ ⋯ ∧ \frac{d f_{m}}{f_{m}} + u′

with f_{i} ∈ F (p)_{δ(i)}, 1 ≤ i ≤ m, and u′ ∈ Ω_{F (p),<δ}^{m}. Thus we get

w = ℘\overline{u} + d \overline{v}

with \overline{u} ∈ Ω_{F (p),<δ}^{m}. Hence we can assume γ = δ. In particular u belongs to

Ω_{F}^{m} ⊗ F (p) and let w_{γ} \frac{d a_{γ}}{a_{γ}} be the maximal term in w as well as u_{γ} \frac{d a_{γ}}{a_{γ}} the maximal term of u. Using lemma ( 3.3) of [Ar-Ba_{2}] we conclude

w_{γ} = ℘u_{γ} + ∑_{i=1}^{m} a_{γ(i)}M_{γ(i)}  (2.6)

with M_{γ(i)} ∈ F (p)_{<γ(i)}, 1≤ i ≤ m. In particular M_{γ(i)} is contained in F (X_{μ}^{2}, Y_{μ}^{2})(z) for all 1 ≤ i ≤ m. Then (2.6) implies ℘u_{γ} ∈ F (X_{μ}^{2}, Y_{μ}^{2})(z) and hence also u_{γ} ∈ F (X_{μ}^{2}, Y_{μ}^{2})(z), i.e. u_{γ} does not contain odd powers of X_{μ}, Y_{μ} in its 2-basis expansion. Thus ℘(u) = ℘(u_{γ})\frac{d a_{γ}}{a_{γ}} + ℘(u_{<γ}) with u_{<γ} ∈ Ω_{F (p),<γ}^{m} and ℘(u_{γ}) ∈ F (X_{μ}^{2}, Y_{μ}^{2})(z). We claim now that there is a decomposition v = v_{γ} + v_{<γ}, where d v_{γ} is free from d X_{μ}, d Y_{μ}’s and its coefficients lie in F (X_{μ}^{2}, Y_{μ}^{2})(z), and d v_{<γ} ∈ Ω_{F (p),<γ}^{m−1}. To see this we set d v = h_{γ} \frac{d a_{γ}}{a_{γ}} + v′ with v′ ∈ Ω_{F (p),<γ}^{m}. Using lemma (3.2) of [Ar-Ba_{2}] again, we have h_{γ} = ∑_{i=1}^{m} a_{γ(i)}M_{γ(i)} with M_{γ(i)} ∈ F (X_{μ}^{2}, Y_{μ}^{2})(z) for 1 ≤ i ≤ m (see above remarks). Thus

d v = (∑_{i=1}^{m} a_{γ(i)}M_{γ(i)}) \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ \frac{d a_{γ(m)}}{a_{γ(m)}} + v′

= ∑_{i=1}^{m} a_{γ(i)}M_{γ(i)} \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ \frac{d a_{γ(i)}}{a_{γ(i)}} ∧ ⋯ ∧ \frac{d a_{γ(m)}}{a_{γ(m)}} + v′

= ∑_{i=1}^{m} d (a_{γ(i)}M_{γ(i)} \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ ∨^{i} ∧ ⋯ ∧ \frac{d a_{γ(m)}}{a_{γ(m)}} )

+ a_{γ(i)}M_{γ(i)} \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ \frac{d M_{γ(i)}}{M_{γ(i)}} ∧ ⋯ ∧ \frac{d a_{γ(m)}}{a_{γ(m)}} + v′.

Since λ = a_{γ(i)}M_{γ(i)} \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ \frac{d M_{γ(i)}}{M_{γ(i)}} ∧ ⋯ ∧ \frac{d a_{γ(m)}}{a_{γ(m)}} + v′ ∈ Ω_{F (p),<γ}^{m−1} and from the last relation we see that λ is exact, we conclude λ = d(v_{<γ}) for some v_{<γ} ∈ Ω_{F (p)}^{m−1}. Hence we have d v = d v_{γ} + d(v_{<γ}), where

v_{γ} = ∑_{i=1}^{m} a_{γ(i)}M_{γ(i)} \frac{d a_{γ(1)}}{a_{γ(1)}} ∧ ⋯ ∧ \frac{d a_{γ(i−1)}}{a_{γ(i−1)}} ∧ \frac{d a_{γ(i+1)}}{a_{γ(i+1)}} ∧ ⋯ ∧ \frac{d a_{γ(m)}}{a_{γ(m)}}

belongs to Ω_{F}^{m−1} ⊗ F (X_{μ}^{2}, Y_{μ}^{2})(z). Inserting all of this in (2.3) it follows

w + ℘(u_{γ}) + d v_{γ} = ℘(u_{<γ}) + d (v_{<γ})

where the left hand side of this equation is in Ω_{F}^{m}⊗F (X_{μ}^{2}, Y_{μ}^{2})(z). Repeating the above arguments for this relation we finally arrive at an equation w = ℘(u)+d v in Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2})(z) which proves the lemma.

Let us now consider the equation (2.3) with u, v ∈ Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2})(z). We write u = u_{0} + u_{1}z, v = v_{0} + v_{1}z with u_{0}, u_{1} ∈ Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}), v_{0}, v_{1} ∈ Ω_{F}^{m−1} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}). Then

w = ℘(u_{0} + u_{1}z) + d(v_{0} + v_{1}z)
= ℘u_{0} + T u_{1}^{[2]} + zu_{1}^{[2]} + zu_{1} + d v_{0} + z d v_{1} + v_{1} ∧ d T

i.e.

w = ℘u_{0} + d v_{0} + T u_{1}^{[2]} + v_{1} ∧ d T
0 = ℘u_{1} + d v_{1}.

The second equation implies (with L = F (X_{μ}, Y_{μ}))

u_{1} ∈ ν_{L}(m) ∩ Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2})

i.e.

u_{1} ∈ ν_{F (X_{μ}^{2},Y_{μ}^{2})}(m)

according to lemma (2.9) below. Inserting u_{1}^{[2]} = u_{1} + d v_{1} in the first equation, we get

w = ℘u_{0} + d v_{0} + T u_{1} + T d v_{1} + v_{1} ∧ d T

and we finally obtain an equation for w ∈ Ω_{F}^{m}

w = ℘u + d v + T η  (2.7)

with u ∈ Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}), v ∈ Ω_{F}^{m−1} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}) and η ∈ ν_{F (X_{μ}^{2},Y_{μ}^{2})}(m). Here we have set u = u_{0}, v = v_{0} + T v_{1}, η = u_{1}. Equation (2.7) will be the basic relation which will be used in the proof of theorem (1.2). Thus we summarize the above results in

Proposition 2.8 Let w ∈ H^{m+1}(F). Then w ∈ H^{m+1}(F (p)/F) if and only if there exist u ∈ Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}), v ∈ Ω_{F}^{m−1} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}) and η ∈ ν_{F (X_{μ}^{2},Y_{μ}^{2})}(m) such that

w = ℘u + d v + T η.

In the proof of the above result we have used the following

Lemma 2.9

ν_{F (X_{μ},Y_{μ})}(m) ∩ Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}) = ν_{F (X_{μ}^{2},Y_{μ}^{2})}(m).

Proof. The contention ⊇ is trivial. Write L = F (X_{μ}, Y_{μ}) and M = F (X_{μ}^{2}, Y_{μ}^{2}). Take now λ ∈ ν_{L}(m) ∩ Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}), and write

λ = c_{γ} \frac{d a_{γ}}{a_{γ}} + λ_{<γ}

with λ_{<γ} ∈ Ω_{F,<γ}^{m} ⊗ M. Since ℘λ ∈ d Ω_{L}, it follows

℘(c_{γ})\frac{d a_{γ}}{a_{γ}} ∈ d Ω_{L}^{m−1} + Ω_{L,<γ}^{m}.

Applying now Kato’s lemma, we obtain

c_{γ} \frac{d a_{γ}}{a_{γ}} = \frac{d f_{γ}}{f_{γ}} + λ′_{<γ}

where f_{γ(i)} ∈ L^{2}(a_{j}, j ≤ γ(i)), 1 ≤ i ≤ m and λ′_{<γ} ∈ Ω_{L,<γ}^{m}. Thus

λ = \frac{d f_{γ}}{f_{γ}} + λ_{<γ} + λ′_{<γ}.

In particular, since \frac{d f_{γ}}{f_{γ}} ∈ ν_{M}(m), we get λ′ = λ_{<γ} + λ′_{<γ} ∈ ν_{L}(m) ∩ Ω_{F} ⊗ M, but now λ′ ∈ Ω_{L,<γ} is in a lower filtration. We proceed with λ′ as above and after finitely many steps we obtain λ ∈ ν_{F (X_{μ}^{2},Y_{μ}^{2})}(m). This proves the lemma.

Similarly we can prove the following result, which we will need later.

Lemma 2.10 Let F ⊆ M be fields and Y a variable over M. Assume d M ⊆ Ω_{F}^{1} ⊗ M. Then,

ν_{M(Y^{2})}(m) ∩ Ω_{F}^{m}M[Y^{2}] = ν_{M}(m)

(we will apply this result for M = F (X_{μ}^{2}, Y_{μ}^{2}) for certain variables X_{μ}, Y_{μ}, μ ∈ S_{n}).

Proof. Since d M ⊆ Ω_{F}^{1} ⊗ M, the contention ⊇ is clear. Let us show now ⊆. Let η ∈ ν_{M(Y^{2})}(m) ∩ Ω_{F}^{m}M[Y^{2}] and write

η = ∑_{γ} \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} = ∑_{α} c_{α} \frac{d a_{α}}{a_{α}}

with c_{α} ∈ M[Y^{2}], f_{γ(i)} ∈ M_{γ(i)}(Y^{2}), 1 ≤ i ≤ m (according to Kato’s lemma). Choose γ to be the maximal multi-index on the left hand side. Expanding the

term d f_{γ}/f_{γ}, we see that d f_{γ}/f_{γ} = ( ∏_{i=1}^{m} a_{γ(i)}D_{γ(i)}(f_{γ(i)})) / ( ∏_{i=1}^{m} f_{γ(i)}) d a_{γ}/a_{γ}+terms

of lower filtration. On the right the maximal term is c_{γ} d a_{γ}/a_{γ}, thus we obtain

c_{γ} = \frac{∏_{i=1}^{m} a_{γ(i)}D_{γ(i)}(f_{γ(i)})}{∏_{i=1}^{m} f_{γ(i)}}

with c_{γ} ∈ M[Y^{2}] and all f_{γ(i)} ∈ M[Y^{2}]. Moreover we can assume that the f_{γ(i)} have only irreducible factors of multiplicity one. We want to show f_{γ(i)} ∈ M for all i. Let k = max{i | f_{γ(i)} ∉ M} and consider an irreducible monic factor p of f_{γ(k)} (which has without restriction multiplicity one). From the lemma (2.11) below, we know that filtration(p)=γ(k) =filtration of f_{γ(k)}. In particular (loc.cit.) p ∤ f_{γ(i)} for all i < k. Thus (2.11) implies p|D_{γ(k)}(f_{γ(k)}), because filtration(D_{γ(i)}f_{γ(i)}) ≤ filtration(f_{γ(i)}) and for i > k, all f_{γ(i)} ∈ M. Let us write f_{γ(k)} = p · h with (p, h) = 1 in M[Y^{2}]. Then D_{γ(k)}(f_{γ(k)}) = pD_{γ(k)}(h) + hD_{γ(k)}(p) and hence p|D_{γ(k)}(p) which is a contradiction since p being monic, deg(D_{γ(k)}(p)) < deg(p). This proves f_{γ(i)} ∈ M for all i and the lemma follows.

Lemma 2.11 Let f(Y) ∈ M_{l}[Y] be a polynomial with filtration l and whose irreducible monic factors have multiplicity 1. Assume that all of these factors have filtration ≥ l. Then all such factors have exactly filtration = l.

Proof. Let p|f be a monic irreducible factor such that filtration(p) = k is maximal among all such factors. Write p = p_{0} + a_{k}p_{1} with fil(p_{0}), fil(p_{1}) < k. By assumption k ≥ l. We have (p_{0} + a_{k}p_{1})h = f in M[Y], where fil(f) = l ≤ k. Then fil(h) ≤ k, otherwise we get a contradiction. Let r = fil(h) ≤ k, and set h = h_{0} + a_{r}h_{1}, fil(h_{i}) < r, i = 1, 2. Obviously p_{1}, h_{1} ≠ 0. Assume now k > l. We have

f = (p_{0} + a_{k}p_{1})(h_{0} + a_{r}h_{1})
= p_{0}h_{0} + a_{k}p_{1}h_{0} + a_{r}h_{1}(p_{0} + a_{k}p_{1}).

If r < k, then the term a_{k}p_{1}(h_{0} + a_{r}h_{1}) has filtration = k, because fil(p_{1}) < k, and hence fil(f) = k > l, contradiction. Then necessarily r = k and we can write

f = p_{0}h_{0} + a_{k}^{2}p_{1}h_{1} + a_{k}(p_{1}h_{0} + h_{1}p_{0}).

Since fil(p_{1}h_{1}) < k, the last term dominates if it is ≠ 0. But l < k implies

p_{1}h_{0} + h_{1}p_{0} = 0

and this means, since (p_{0}, p_{1}) = 1, that p_{1}|h_{1}, p_{0}|h_{0} and

\frac{h_{1}}{p_{1}} = \frac{h_{0}}{p_{0}} = t ∈ M[Y]

is a certain polynomial. Thus h = t · p and hence p^{2}|f, contradicting our assumption on the multiplicity. This shows k = l.

3 p-components of differential forms

Our basic equation (2.7) holds in Ω_{F}^{m} ⊗ F (X_{μ}^{2}, Y_{μ}^{2}, μ ∈ S_{n}). More generally fix some index μ ∈ S_{n}, and let M = F (X_{ν}^{2}, Y_{λ}^{2}| ν > μ, λ ≥ μ) and set X = X_{μ}. Assume that we have a form u ∈ Ω_{F}^{m} ⊗ M(X^{2}). The p-component of this form u arise from the partial fraction decomposition in M(X) of the coefficients of the form. We will call the subgroup Ω_{F}^{m}M[X^{2}] the integral forms and for any irreducible monic polynomial p ∈ M[X] set

p^{−∞}Ω_{F}^{m}M[X^{2}] = \{\frac{v}{p^{s}} | v ∈ Ω_{F}^{m}M[X^{2}], s ≥ 1, deg_{X} v < s deg_{X} p\}  (3.1)

if p ∈ M[X^{2}] and

p^{−∞}Ω_{F}^{m}M[X^{2}] = \{\frac{w}{p^{2s}} | w ∈ Ω_{F}^{m}M[X^{2}], s ≥ 1, deg_{X} v < 2s deg_{X} p\}

if p ∉ M[X^{2}]. Here we set deg_{X} w = 2t whenever we have w = w_{0} + w_{1}X^{2} + ⋯ + w_{t}X^{2t}, w_{t} ≠ 0 and w_{i} ∈ Ω_{F}^{m}. Then using partial fraction decomposition in M(X) we obtain

Ω_{F}^{m}M(X^{2}) = Ω_{F}^{m}M[X^{2}] ⊕ ⨁_{p} p^{−∞}Ω_{F}^{m}M[X^{2}].  (3.2)

It is shown in [Ar-Ba_{1}] that this decomposition is compatible with the operators d and ℘. If u ∈ Ω_{F}^{m}M(X^{2}), then u = u_{0} + ∑_{p} u_{p}, and the forms u_{p} (including u_{0}) are called the p-components of u.

Let us consider the p-decomposition of the form η ∈ ν_{L}(m) where L = F (X_{μ}, Y_{μ}, μ ∈ S_{n}). Assume η ∈ ν_{M(X^{2})}(m) where M and X are as above. Then η = η_{E} + ∑_{p} η_{p} is the p-decomposition of η, η_{E} ∈ Ω_{F}^{m}M[X^{2}] and η_{p} ∈ p^{−∞}Ω_{F}^{m}M[X^{2}]. We claim

Lemma 3.3 (i) η_{E} ∈ ν_{M(X^{2})}(m) ∩ Ω_{F}^{m}M[X^{2}] = ν_{M}(m)

(ii) η_{p} ∈ ν_{M(X^{2})}(m) ∩ p^{−∞}Ω_{F}^{m}M[X^{2}].

Proof. We use d η = 0 (since η ∈ ν(m)) and the compatibility of the pdecomposition with d to get from d η_{E} + ∑_{p} d η_{p} = 0 necessarily

d η_{E} = 0, d η_{p} = 0 for all p.

Now we apply the Cartier operator to η to obtain from η = C(η)

η_{E} + ∑_{p} η_{p} = C(η_{E}) + ∑_{p} C(η_{p})

and hence

η_{E} = C(η_{E})
η_{p} = C(η_{p})

i.e. η_{E} ∈ ν_{M}(m), η_{p} ∈ ν_{M(X^{2})}(m). This proves (i) and (ii).

The equality

ν_{M(X^{2})}(m) ∩ Ω_{F}^{m}M[X^{2}] = ν_{M}(m)

has been shown in (2.10). We give here a shorter proof of it. The contention ⊇ is clear. Let α ∈ ν_{M(X^{2})}(m)∩Ω_{F}^{m}M[X^{2}] and write α = α_{0} +α_{1}X^{2} +⋯+α_{s}X^{2s} with α_{i} ∈ Ω_{F}^{m}M. From d α = 0 it follows d α_{i} = 0 for all 0 ≤ i ≤ s and the relation C(α) = α implies

α_{0} + α_{1}X^{2} + ⋯ + α_{s}X^{2s} = C(α_{0}) + C(α_{1})X + ⋯ + C(α_{s})X^{s}

which holds only if s = 0, α = α_{0} ∈ Ω_{F}^{m}M. But d α_{0} = 0 and C(α_{0}) = α_{0} imply α_{0} ∈ ν_{M}(m). This shows (2.10) again.

Let us now study the intersection

ν_{M(X^{2})}(m) ∩ p^{−∞}Ω_{F}^{m}M[X^{2}]  (3.4)

and take η_{p} contained in it.

We can write η_{p} = \frac{ξ}{p^{2t}} or η_{p} = \frac{ξ}{p^{2t+1}} where ξ ∈ Ω_{F}^{m}M[X^{2}], p ∤ ξ and deg ξ < 2t deg p, resp. deg ξ < (2t + 1) deg p, with t ≥ 1, resp. t ≥ 0. We set

ξ = ξ_{0} + ξ_{1}X^{2} + ⋯ + ξ_{s}X^{2s}

with ξ_{i} ∈ Ω_{F}^{m}M, 0 ≤ i ≤ s, and we choose s and t in the above representations to be minimal. We have d η_{p} = 0, C(η_{p}) = η_{p} because η_{p} ∈ ν_{M(X^{2})}. The first equation implies d ξ_{i} = 0 for all i and the second one means

\frac{C(ξ_{0}) + C(ξ_{1})X + ⋯ + C(ξ_{s})X^{s}}{p^{t}} = \frac{ξ_{0} + ξ_{1}X^{2} + ⋯ + ξ_{s}X^{2s}}{p^{2t}}

and this implies s = 0, t = 0 i.e. η_{p} = 0. Thus only the second case can occur.

We have η_{p} = \frac{pξ}{p^{2t+2}} and from d η_{p} = 0, C(η_{p}) = η_{p} it follows d (pξ_{i}) = 0 for all

i and

\frac{C(pξ)}{p^{t+1}} = \frac{ξ}{p^{2t+1}} .

Thus ξ = p^{t}C(pξ). Since p ∤ ξ we conclude t = 0, i.e.

η_{p} = \frac{ξ}{p}  (3.5)

with ξ ∈ Ω_{F}^{m}M[X^{2}], degξ < deg p. Thus we have shown that the p-component η_{p} of a form η ∈ ν_{M(X^{2})}(m) is of the form (3.5). Furthermore we claim that η_{p} ≠ 0 only if p ∈ M[X^{2}]. Assume p ∉ M[X^{2}] and set p = p_{0} + Xp_{1} with p_{0}, p_{1} ∈ M[X^{2}], p_{1} ≠ 0. Then ξ = p_{0}η_{p} + Xp_{1}η_{p}. Since η_{p} ∈ Ω_{F} M(X^{2}), it follows Xp_{1}η_{p} ∈ Ω_{F} M(X^{2}) with p_{1} ∈ M[X^{2}]. This is impossible if p_{1} ≠ 0. Therefore p = p_{0} ∈ M[X^{2}]. Summing up these results we have shown:

Lemma 3.6 Let η ∈ ν_{M(X^{2})}(m). Then the p-decomposition η = η_{E} + ∑ η_{p} of η satisfies

(i) η_{E} ∈ ν_{M}(m)

(ii) Only for p ∈ M[X^{2}] we may have η_{p} ≠ 0, and then

η_{p} = \frac{ξ}{p}

with ξ ∈ Ω_{F}^{m}M[X^{2}], deg ξ < deg p. Moreover η_{p} ∈ ν_{M(X^{2})}(m), d(pξ) = 0 and C(pξ) = ξ.

Let us return to the basic equation

w = ℘u + d v + T η

in Ω_{F}^{m}F (X_{μ}^{2}, Y_{μ}^{2}) = Ω_{F}^{m}M(X^{2}), where X = X_{1} and M = F (X_{ν}^{2}, Y_{μ}^{2}, μ, ν ∈ S_{n}, ν > 1). We insert the p-decomposition of the forms involved in this equation and we obtain as in [Ar-Ba_{1}]

w = ℘u_{E} + d v_{E} + T η_{E} + ∑_{p} E_{p}  (3.7)
E_{p} = ℘u_{p} + d v_{p} + T η_{p}  (3.8)

where the forms E_{p} belong to Ω_{F} M[X^{2}] and arise from the multiplication of T with η_{p}. For all irreducible polynomials p ∈ M[X]\M[X^{2}] we have shown η_{p} = 0 and hence E_{p} = ℘u_{p} + d v_{p}. Since ℘, d are compatible with the pdecomposition, we have in this case E_{p} = 0. Thus in (3.7) only polynomials p ∈ M[X^{2}] do appear. The main idea is now to get rid of the forms E_{p} in the equation (3.7), and this will be accomplished in the last section of this paper.

4 Second reduction step in the proof of the main theorem

We keep the same notation as in the last section. Thus in the basic equations (3.7), (3.8) only polynomials p ∈ M[X^{2}] occur. Since p is monic and irreducible, we have p(0) ≠ 0 for all such p’s. Therefore we can specialize in these equations X ⟶ 0 and we obtain

w = ℘u_{0} + d v_{0} + T_{1}η_{0}  (4.1)

where u_{0}, v_{0} ∈ Ω_{F} M and η_{0} ∈ ν_{M}(m). Here T_{1} is now the polynomial b + a_{1}bY_{1}^{2} + ∑_{μ>1} a^{μ}Y_{μ}^{2}(b + X_{μ}^{2} + a^{μ}Y_{μ}^{2}X_{μ}^{2}). Notice that the specialization X ⟶ 0 is compatible with the operators ℘ and d. Proceeding with the other variables X_{μ} in the same way as above, we finally obtain an equation

w = ℘u + d v + T η  (4.2)

with u ∈ Ω_{F}^{m}F (Y_{μ}^{2}, μ ∈ S_{n}), v ∈ Ω_{F}^{m−1}F (Y_{μ}^{2}, μ ∈ S_{n}), η ∈ ν_{F (Y_{μ}^{2})}(m) and

T = b (1 + ∑_{μ∈S_{n}} a^{μ}Y_{μ}^{2})  (4.3)

This equation will be from now on our basic equation and our goal is now to eliminate in an appropriate way the variables Y_{μ}. Thus we have shown

Proposition 4.4 If w ∈ H^{m+1}(F (p)/F), then there exist u ∈ Ω_{F}^{m}F (Y_{μ}^{2}, μ ∈ S_{n}), v ∈ Ω_{F}^{m−1}F (Y_{μ}^{2}, μ ∈ S_{n}) and η ∈ ν_{F (Y_{μ}^{2})}(m) such that (4.2) holds, where T is given by (4.3).

During the process of elimination of the variables Y_{μ}, we will see that the following type of conditions on the forms η ∈ ν(m) appear:

For some τ ∈ S_{n} and E ⊂ F (Y_{μ}^{2}, μ ∈ S_{n}) intermediate field (to be later specified) one has η ∈ ν_{E}(m) and a^{σ}η ∈ d Ω_{E}^{m−1} for all σ ≤ τ. We will in the next section study this property and characterize these forms for some specific subfields of F (Y_{μ}^{2}, μ ∈ S_{n}).

5 Forms in good shape

Let N be some field (with 2 = 0) and a_{1}, …, a_{n} ∈ N elements contained in a 2-basis of N. Let Y be a variable over N and fix some τ ∈ S_{n}. Set t_{τ} = t = max{i| τ(i) = 1}. Then

Proposition 5.1 Let η ∈ ν_{N(Y^{2})}(m) be such that a^{σ}η ∈ d Ω_{N(Y^{2})}^{m−1} for all σ ≤ τ. Then η has the form

η = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t−1}}{a_{t−1}} ∧ ∑_{γ} \frac{d (a_{t} + d_{γ})}{a_{t} + d_{γ}} ∧ \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

where d_{γ} = ∑_{α} a^{α}c_{α,γ}^{2}, c_{α,γ} ∈ N(Y), α ∈ S_{t−1} such that α + t > τ, f_{γ(i)} ∈ N_{γ(i)}[Y^{2}], i ≥ t + 1, γ ∈ Σ_{n} with γ(1) = 1, …, γ(t) = t. (Here α + t means that one adds 1 at the t-th place of α). Conversely, each form of the above type satisfies a^{σ}η ∈ d Ω for all σ ≤ τ.

As a special case τ = max{σ| σ ∈ S_{n}} we get t = n and the set α ∈ S_{n−1} with α + n > τ is empty. Thus we get the useful

Corollary 5.2 If η ∈ ν_{N(Y^{2})}(m) satisfies

a^{σ}η ∈ d Ω_{N(Y^{2})}^{m−1}

for all σ ∈ S_{n}, then

η = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ ξ

with some ξ ∈ ν(m − n).

Proof of (5.1). Let first consider the case τ = (1, 0, …, 0) = first element in S_{n}. Then t = 1. By assumption a_{1}η ∈ d Ω, which implies d a_{1} ∧ η = 0 (since d η = 0) and hence η = \frac{d a_{1}}{a_{1}} ∧ ξ with some ξ ∈ Ω^{m−1}. From Kato’s lemma we can write

η = ∑_{γ} \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

with γ(1) = 1 for all γ, i.e. f_{γ(1)} = n_{γ}^{2}a_{1}+m_{γ}^{2} and n_{γ} ≠ 0 in N[Y]. For simplicity replace f_{γ(1)} by n_{γ}^{−2}f_{γ(1)} so that we can assume n_{γ} = 1 and m_{γ} ∈ N(Y). Then

η = ∑_{γ} \frac{d (a_{1} + m_{γ}^{2})}{a_{1} + m_{γ}^{2}} ∧ ξ_{γ}

where ξ_{γ} = \frac{d f_{γ(2)}}{f_{γ(2)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} . Hence

a_{1}η = ∑_{γ} \frac{(a_{1}^{3} + a_{1}^{2}m_{γ}^{2})}{(a_{1} + m_{γ}^{2})^{2}} \frac{d a_{1}}{a_{1}} ∧ ξ_{γ}.

The form ∑_{γ} \frac{a_{1}^{3}}{(a_{1}+m_{γ}^{2})^{2}} \frac{d a_{1}}{a_{1}} ∧ ξ_{γ} is exact, so that our hypothesis implies

∑_{γ} \frac{a_{1}^{2}m_{γ}^{2}}{(a_{1} + m_{γ}^{2})^{2}} \frac{d a_{1}}{a_{1}} ∧ ξ_{γ} ∈ d Ω_{N(Y^{2})}^{m−1}.

Applying the Cartier operator to this form, we obtain

∑_{γ} \frac{a_{1}m_{γ}}{(a_{1} + m_{γ}^{2})} \frac{d a_{1}}{a_{1}} ∧ ξ_{γ} = 0.

But the elements \frac{d a_{1}}{a_{1}} ∧ ξ_{γ} belong to different filtrations for each γ, thus we obtain

m_{γ} = 0 for all γ

and hence η = \frac{d a_{1}}{a_{1}} ∧ ξ with ξ = ∑_{γ} ξ_{γ} ∈ ν(m − 1). This proves our assertion in the case τ = (1, 0, …, 0).

Assume now our assertion for all σ ≤ τ, τ ∈ S_{n}. Let δ be the next element in S_{n}. We want to show the assertion for δ. Assume t = t_{τ} = max{i| τ(i) = 1} and let t_{δ} = max{i| δ(i) = 1}. Then t_{δ} = t or t_{δ} = t + 1. First case: t_{δ} = t. Then δ = (…, 1, 0, …) with 1 at the t-th place. By induction we have η = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t−1}}{a_{t−1}} ∧ ∑_{γ} \frac{d(a_{t}+d_{γ})}{a_{t}+d_{γ}} ∧ ξ_{γ} with d_{γ} = ∑_{{σ∈S_{t−1} \atop σ+t>τ}} a^{σ}m_{σ,γ}^{2}, m_{σ,γ} ∈ N(Y),

and ξ_{γ} = \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} . Thus we can write

d_{γ} = ∑_{{σ∈S_{t−1} \atop σ+t≥δ}} a^{σ}m_{σ,γ}^{2} = a^{δ−t}m_{δ−t,γ}^{2} + ∑_{{σ∈S_{t−1} \atop σ+t>δ}} a^{σ}m_{σ,γ}^{2}

where δ − t is the element σ ∈ S_{t−1} with σ + t = δ. Now we use a^{δ}η ∈ d Ω_{N(Y^{2})}^{m−1}. The form

a^{δ}η = ∑_{γ} \frac{d a}{a} ∧ \frac{a^{δ} d (a_{t} + d_{γ})}{a_{t} + d_{γ}} ∧ ξ_{γ}

= ∑_{γ} \frac{d a}{a} ∧ \frac{a^{δ} (a_{t} + d_{γ}) a_{t}}{(a_{t} + d_{γ})^{2}} \frac{d a_{t}}{a_{t}} ∧ ξ_{γ} ∈ d Ω_{N(Y^{2})}^{m−1}

because d a ∧ d a^{δ−t} = d a ∧ d a^{σ} = 0 for all σ ∈ S_{t−1}. Now the terms

\frac{a^{δ}a_{t}^{2}}{(a_{t} + d_{γ})^{2}} \frac{d a}{a} ∧ \frac{d a_{t}}{a_{t}} ∧ ξ_{γ}

are all exact, as well as the forms

\frac{a^{δ}a_{t} ∑_{σ+t>δ} a^{σ}m_{σ,γ}^{2}}{(a_{t} + d_{γ})^{2}} \frac{d a}{a} ∧ \frac{d a_{t}}{a_{t}} ∧ ξ_{γ}

because each summand of these expressions has the form

\frac{a^{δ+t+σ}m_{σ,γ}^{2}}{(a_{t} + d_{γ})^{2}} \frac{d a}{a} ∧ \frac{d a_{t}}{a_{t}}

which is indeed exact, since δ + t + σ > 0 and δ + t + σ ∈ S_{t−1}. Therefore we obtain

∑_{γ} \frac{a^{2δ}m_{δ−t,γ}^{2}}{(a_{t} + d_{γ})^{2}} \frac{d a}{a} ∧ \frac{d a_{t}}{a_{t}} ∧ ξ_{γ} ∈ d Ω_{N(Y^{2})}^{m−1}.

Applying the Cartier operator to this form we obtain

∑_{γ} \frac{a^{δ}m_{δ−t,γ}}{(a_{t} + d_{γ})} \frac{d a}{a} ∧ \frac{d a_{t}}{a_{t}} ∧ ξ_{γ} ∈ 0.

Since the forms \frac{d a}{a} ∧ \frac{d a_{t}}{a_{t}} ∧ ξ_{γ} are independent, we conclude m_{δ−t,γ} = 0 for all γ and hence d_{γ} = ∑_{{σ∈S_{t−1} \atop σ+t>δ}} a^{σ}m_{σ,γ}^{2}. This concludes the proof in this case.

We have to look at the case t_{δ} = t + 1. This case is only possible for τ = (1, 1, …, 1, 0, …, 0) (the last 1 at the t-th place) and δ = (0, …, 0, 1, 0, …, 0) with 1 at the (t + 1)-th place. In this case there is no α ∈ S_{t−1} with α + t > τ, hence we have by induction, because of a^{σ}η ∈ d Ω for σ ≤ τ,

η = ∑_{γ} \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t−1}}{a_{t−1}} ∧ \frac{d a_{t}}{a_{t}} ∧ \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} .

Since a^{δ} = a_{t+1}, we have still the condition a_{t+1}η ∈ d Ω. In particular d a_{t+1} ∧ η = 0. It follows η ∈ d a_{1} ∧ ⋯ ∧ d a_{t+1} ∧ Ω^{m−t−1}. Comparing this with the above expression of η, we conclude γ(t + 1) = t + 1 for all γ in the sum. Thus f_{γ(t+1)} ∈ M_{t+1}[Y^{2}] for all γ, i.e. f_{γ(t+1)} = A_{γ}(a_{t+1}+B_{γ}) with A_{γ}, B_{γ} ∈ M_{t}[Y^{2}]. Since d a_{1} ∧ ⋯ ∧ d a_{t} ∧ d A_{γ} = 0, we may assume f_{γ(t+1)} = a_{t+1} + B_{γ}. Write now B_{γ} = n_{γ}^{2} + ∑ a^{α}n_{α,γ}^{2}, where α runs over S_{t} with α + (t + 1) > δ. We want to show n_{γ} = 0 for all γ. The assumption a_{t+1}η ∈ d Ω means

∑_{γ} \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t}}{a_{t}} ∧ \frac{(a_{t+1} + B_{γ}) a_{t+1}^{2}}{(a_{t+1} + B_{γ})^{2}} \frac{d a_{t+1}}{a_{t+1}} ∧ ξ_{γ} ∈ d Ω

with ξ_{γ} = \frac{d f_{γ(t+2)}}{f_{γ(t+2)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} . A similar argument as before shows that

∑_{γ} \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t}}{a_{t}} ∧ \frac{a_{t+1}^{2}n_{γ}^{2}}{(a_{t+1} + B_{γ})^{2}} \frac{d a_{t+1}}{a_{t+1}} ∧ ξ_{γ}

is exact. Applying the Cartier operator, we obtain

∑_{γ} \frac{a_{t+1}n_{γ}}{(a_{t+1} + B_{γ})} \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t}}{a_{t}} ∧ \frac{d a_{t+1}}{a_{t+1}} ∧ ξ_{γ} = 0

and again it follows n_{γ} = 0 for all γ. This completes the proof of our proposition.

The property assumed in the last proposition for forms η ∈ ν(m) will play a crucial role in the proof of our main theorem. We will state this property in the following definition.

Definition 5.3 Let E be any field with 2 = 0 and a_{1}, …, a_{n} ∈ E be elements contained in a 2-basis of E. Fix some τ ∈ S_{n}. Let η ∈ ν_{E}(m). We say that η is in good shape with respect to τ (and a_{1}, …, a_{n}) if

a^{σ}η ∈ d Ω_{E}^{m−1}

for all σ ≤ τ.

Thus proposition (5.1) characterizes forms in good shape for certain fields.

6 Last reduction step in the proof of the main theorem

We fix the lexicographic ordering in the set S_{n}, and for any μ ∈ S_{n} let μ^{+} be the next element in this ordering. Let T = T_{0} = b(1 + ∑_{μ} a^{μ}Y_{μ}^{2}) and we define T_{μ} inductively by

T_{μ} = ba^{μ}Y_{μ}^{2} + T_{μ^{+}}

Let M = F (Y_{ν}^{2})_{ν>μ} (for a given μ) and let w ∈ Ω_{F}^{n}. We consider the following assertions on w and μ:

(A_{μ}) There exist u ∈ Ω_{F}^{m}M(Y_{μ}^{2}), v ∈ Ω_{F}^{m−1}M(Y_{μ}^{2}) and η ∈ ν_{M(Y_{μ}^{2})}(m) ∩

Ω_{F}^{m}M(Y_{μ}^{2}) such that

w = ℘(u) + d v + T_{μ}η
and a^{σ}η ∈ d Ω for all σ < μ.

(B_{μ}) There exist u ∈ Ω_{F}^{m}M[Y_{μ}^{2}], v ∈ Ω_{F}^{m−1}M[Y_{μ}^{2}] and η ∈ ν_{M}(m) ∩ Ω_{F}^{m}M such

that

w = ℘(u) + d v + T_{μ}η

and a^{σ}η ∈ d Ω for all σ < μ.

Lemma 6.1 (Main lemma) For any μ ∈ S_{n}

(A_{μ}) ⟹ (B_{μ}) ⟹ (A_{μ^{+}}).

Clearly (A_{0}) is the basic equation (4.2). Thus (6.1) implies that starting with (A_{0}), i.e. (4.2), we arrive at an equation (for the last element in S_{n})

w = ℘u + d v + bη  (6.2)

with u ∈ Ω_{F}^{m}, v ∈ Ω_{F}^{m−1} and η ∈ ν_{F} (m) such that a^{σ}η ∈ d Ω_{F} for all σ ∈ S_{n}. Applying now corollary (5.2) we conclude η = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{n}}{a_{n}} ∧ ξ with ξ ∈ ν_{F} (m − n). This implies the main theorem.

The rest of this section is devoted to the proof of (6.1). Let us first show the easy part, namely (B_{μ})⟹(A_{μ^{+}}). Assuming (B_{μ}) one easily checks that, without restriction, u = u_{0} ∈ Ω_{F}^{m}M and v = v_{0} + v_{2}Y_{μ}^{2} with v_{0}, v_{2} ∈ Ω_{F}^{m−1}M, as well as η ∈ ν_{M}(m) ∩ Ω_{F}^{m}M with a^{σ}η ∈ d Ω for all σ < μ.

From

w = ℘u_{0} + d(v_{0} + v_{2}Y_{μ}^{2}) + (ba^{μ}Y_{μ}^{2} + T_{μ^{+}})η

we obtain

w = ℘u_{0} + d(v_{0}) + T_{μ^{+}}η

and

d(v_{2}) = ba^{μ}η.

Since b ∈ F^{2}, we get a^{μ}η = b^{−1} d(v_{2}) ∈ d Ω, i.e. it holds a^{σ}η ∈ d Ω for all σ < μ^{+}. This proves (A_{μ^{+}}).

The hard part of the lemma is the implication (A_{μ})⟹(B_{μ}). Let us start with the basic equation (i.e. (A_{μ}))

w = ℘u + d v + T η

with w ∈ Ω_{F}^{m}, u ∈ Ω_{F}^{m}M(Y^{2}), v ∈ Ω_{F}^{m−1}M(Y^{2}), η ∈ ν_{M(Y^{2})}(m) ∩ Ω_{F}^{m}M(Y^{2}), T = T_{μ}, Y = Y_{μ} for some μ ∈ S_{n}, where a^{σ}η ∈ d Ω for all σ < μ. For μ = 0, this is the equation (4.2). Let t = max{i | μ^{−}(i) = 1}, where μ^{−} denotes the antecessor of μ. We consider the p-decomposition (see §3 and [Ar-Ba_{1}]) of the form η

η = η_{E} + ∑_{p} η_{p}  (6.3)

with η_{E} ∈ ν_{M(Y^{2})}(m) ∩ Ω_{F}^{m}M[Y^{2}] = ν_{M}(m), η_{p} ∈ \frac{1}{p}Ω_{F} M[Y^{2}] ∩ ν_{M(Y^{2})}(m) (see (2.10)). Then we have

Lemma 6.4 If η in ( 6.3) satisfies a^{σ}η ∈ d Ω for all σ < μ, then η_{E}, η_{p}, for all p, have the same property.

Proof. Since d is compatible with the p-decomposition (see [Ar-Ba_{1}]) we obtain

0 = d(a^{σ}η) = d(a^{σ}η_{E}) + ∑_{p} d(a^{σ}η_{p})

and hence d(a^{σ}η_{E}) = 0, d(a^{σ}η_{p}) = 0 for all p and σ < μ. Also C is compatible with p-decomposition (loc. cit.), so that from C(a^{σ}η) = 0 it follows

C(a^{σ}η_{E}) = 0
C(a^{σ}η_{p}) = 0.

This implies a^{σ}η_{E}, a^{σ}η_{p} ∈ d Ω for all p, σ < μ.

Let us now insert the p-decompositions u = u_{E} + ∑ u_{p}, v = v_{E} + ∑ v_{p}, η = η_{E} + ∑ η_{p} into the equation (A_{μ}). Here we have u_{p} ∈ \frac{1}{p}Ω_{F}^{m}M[Y^{2}], v_{p} ∈ \frac{1}{p^{2}} Ω_{F}^{m−1}M[Y^{2}], η_{E} ∈ ν_{M(Y^{2})}(m) ∩ Ω_{F}^{m}M[Y^{2}] = ν_{M}(m) and η_{p} ∈ \frac{1}{p}Ω_{F} M[Y^{2}] ∩ ν_{M(Y^{2})}(m). We obtain

w = ℘u_{E} + d v_{E} + T η_{E} + ∑_{p} E_{p}  (6.5)
E_{p} = ℘u_{p} + d v_{p} + T η_{p}  (6.6)

with E_{p} ∈ Ω_{F}^{m}M[Y^{2}] for all p (see [Ar-Ba_{1}]). We have seen in (3.6) that the irreducible polynomials appearing in the p-decomposition of η are in M[Y^{2}]. Thus for the others polynomials p we have E_{p} = ℘u_{p} + d v_{p} and it follows immediately that E_{p} = 0 since E_{p} is integral and ℘u_{p}, d v_{p} are in the p-component so these terms can be discarded in the above expression (6.5). Thus we assume in (6.5) and (6.6) that all irreducible polynomials p are in M[Y^{2}]. Moreover, according to (6.4) we know that the forms η_{E}, η_{p} satisfy a^{σ}η_{E}, a^{σ}η_{p} ∈ d Ω for all σ < μ.

In the next lemma we will use the following notation. Given a polynomial f ∈ M[Y^{2}], we denote by fil(f) the maximal filtration of the coefficients of f (with respect to the given 2-basis). For f, g ∈ M[Y^{2}] we write f < g if deg f < deg g or deg f = deg g and fil f < fil g.

Lemma 6.7 For any p and η_{p} as above it holds

η_{p} = η_{0} + η_{1} ∧ \frac{d p}{p}

with η_{0}, η_{1} ∈ ν_{M(Y^{2})} ∩ ⨁_{q<p} \frac{1}{q}Ω_{F} M[Y^{2}] and a^{σ}η_{0}, a^{σ}η_{1} ∈ d Ω for all σ < μ.

Proof. From the characterization of forms in good shape given in (5.1) we get

η_{p} = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t−1}}{a_{t−1}} ∧ ∑_{γ} \frac{d (n_{γ}^{2}a_{t} + m_{γ})}{n_{γ}^{2}a_{t} + m_{γ}} ∧ \frac{d f_{γ}}{f_{γ}}

where m_{γ} = ∑_{σ} a^{σ}m_{σ,γ}^{2}, m_{σ,γ}, n_{γ} ∈ M[Y] and σ runs over those σ ∈ S_{t−1} with σ + t ≥ μ, and \frac{d f_{γ}}{f_{γ}} = \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} with t = max\{i | μ^{−}(i) = 1\}. Moreover for all γ ∈ Σ_{m} in this sum we can write γ(i) = i for i ≤ t, f_{γ(i)} = a_{i} , i ≤ t − 1 and f_{γ(t)} = n_{γ}^{2}a_{t} + m_{γ} and f_{γ(j)} ∈ M_{γ(j)}[Y^{2}] for all j. Let γ be the maximal multi-index in the above sum, and set

η_{p,γ} = \frac{d f_{γ}}{f_{γ}} = \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t−1}}{a_{1}} ∧ \frac{d (n_{γ}^{2}a_{t} + m_{γ})}{n_{γ}^{2}a_{t} + m_{γ}} ∧ \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} (6.8)

and

η_{p,<γ} = \frac{d a}{a} ∧ ∑_{λ<γ} \frac{d (n_{λ}^{2}a_{t} + m_{λ})}{n_{λ}^{2}a_{t} + m_{λ}} ∧ \frac{d f_{λ(t+1)}}{f_{λ(t+1)}} ∧ ⋯ ∧ \frac{d f_{λ(m)}}{f_{λ(m)}} .

Thus η_{p} = η_{p,γ} + η_{p,<γ}. We use now the fact that η_{p} ∈ \frac{1}{p}Ω_{F} M[Y^{2}] to obtain the 2−basis expansion

η_{p} = ∑_{λ≤γ} c_{λ} \frac{d a_{1}}{a_{1}} ∧ ⋯ ∧ \frac{d a_{t−1}}{a_{t−1}} ∧ \frac{d a_{t}}{a_{t}} ∧ \frac{d a_{λ(t+1)}}{a_{λ(t+1)}} ∧ ⋯ ∧ \frac{d a_{λ(m)}}{a_{λ(m)}}

with c_{λ} ∈ \frac{1}{p}M[Y^{2}]. Let c_{λ} = \frac{h_{λ}}{p}, where h_{λ} ∈ M[Y^{2}], deg_{Y} h_{λ} < deg_{Y} p. On the other hand only the term γ_{p,γ} contributes to the maximal term in this expansion, and its contribution is (see the proof of (2.10))

\frac{d a}{a} ∧ \frac{a_{t}n_{γ}^{2}}{n_{γ}^{2}a_{t} + m_{γ}} \frac{d a_{t}}{a_{t}} ∧ \frac{a_{γ(t+1)}D_{γ(t+1)} (f_{γ(t+1)})}{f_{γ(t+1)}} \frac{d a_{γ(t+1)}}{a_{γ(t+1)}} ∧

⋯ ∧ \frac{a_{γ(m)}D_{γ(m)} (f_{γ(m)})}{f_{γ(m)}} \frac{d a_{γ(m)}}{a_{γ(m)}}

= ∏_{j=t}^{m} (\frac{a_{γ(j)}D_{γ(j)} (f_{γ(j)})}{f_{γ(j)}}) \frac{d a_{γ}}{a_{γ}} .

Thus we obtain

\frac{h_{λ}}{p} = ∏_{j=t}^{m} (\frac{a_{γ(j)}D_{γ(j)} (f_{γ(j)})}{f_{γ(j)}})

where deg h_{γ} < deg p, deg (D_{γ(i)} (f_{γ(i)})) ≤ deg f_{γ(i)}, fil (D_{γ(i)} (f_{γ(i)})) < γ(i) = fil (f_{γ(i)}). This last remark follows from the definition of the operator D_{γ(i)}. Therefore

h_{λ} ∏_{j=t}^{m} f_{γ(j)} = ap ∏_{j=t}^{m} D_{γ(j)} (f_{γ(j)})  (6.9)

with a = ∏_{j=t}^{m} a_{γ(j)} ∈ F. In particular p| ∏_{j=t}^{m} f_{γ(j)} and there is some t ≤ k ≤ m with p|f_{γ(k)}. This k is unique and it follows, since p is irreducible and monic (see (2.11)), that fil(p) = γ(k). We can assume without restriction that f_{γ(j)} are products of irreducible polynomials q with multiplicity 1 and filtration γ(j).

Now we distinguish three cases:

(a) j > k. Then f_{γ(j)} ∈ M is constant.

To see this, we choose j > k maximal with f_{γ(j)} ∈ M[Y^{2}]\M. Let q|f_{γ(j)} be a monic irreducible factor. Then fil(q) = γ(j). From (6.7) we get q| ∏_{i=t}^{m} D_{γ(i)} (f_{γ(i)}), because it can not be p since fil(p) = γ(k) ≠ γ(j). Hence q|D_{γ(i)} (f_{γ(i)}) for some i ≥ t. If l > j, by the choice of j, f_{γ(l)} ∈ M. Hence i ≤ j. But i ≤ j implies fil (D_{γ(i)} (f_{γ(i)})) < γ(i) ≤ γ(j) and this contradicts q|D_{γ(i)} (f_{γ(i)}), since fil(q) = γ(j). This implies f_{γ(j)} ∈ M for all j > k.

(b) j = k. We claim f_{γ(k)} = cp with c ∈ M_{γ(k)}.

Assume q|f_{γ(k)} is an irreducible monic factor different from p. We have fil(q) = γ(k). But from (6.9) it follows q|D_{γ(i)} (f_{γ(i)}) for some i ≥ t, and this implies fil(q) < γ(i). Since f_{γ(l)} ∈ M for l > k, it follows i ≤ k, and hence fil(q) < γ(i) ≤ γ(k), which is a contradiction. Thus q = p and since its multiplicity is one, we have f_{γ(k)} = cp. Since fil(f_{γ(k)}) = γ(k), it follows c ∈ M_{γ(k)}.

(c) j < k. We claim deg(f_{γ(j)}) ≤ deg(p) for all j < k.

Since fil(f_{γ(j)}) = γ(j) < γ(k) = fil(p), we see that any monic irreducible factor of f_{γ(j)}, say q, has fil(q) = γ(j) < γ(k) and hence q ≠ p. Inserting (a) and (b) in the relation (6.9) we obtain an equation of the form

dh_{γ}f_{γ(t)} ⋯ f_{γ(k−1)} = ∏_{t≤j≤k−1} D_{γ(j)} (f_{γ(j)}) · D_{γ(k)} (cp)  (6.10)

with some d ∈ M. If q|f_{γ(k−1)} is a monic irreducible factor, we have fil(q) = γ(k − 1) and since fil(D_{γ(j)} (f_{γ(j)})) < γ(j) for all j ≤ k − 1 and therefore q|D_{γ(k)} (cp). Since all irreducible factors of f_{γ(k−1)} have multiplicity 1 we obtain f_{γ(k−1)}|D_{γ(k)} (cp) and hence deg(f_{γ(k−1)}) ≤ deg(p). We set D_{γ(k)} (cp) = l_{γ(k)}f_{γ(k−1)} with fil(l_{γ(k)}) < γ(k) and deg(l_{γ(k)}) = deg(p) − deg(f_{γ(k−1)}). From (6.10) it follows

dh_{γ}f_{γ(t)} ⋯ f_{γ(k−2)} = l_{γ(k)} ∏_{t≤j≤k−1} D_{γ(j)} (f_{γ(j)}) .  (6.11)

Let now q be a monic irreducible factor of f_{γ(k−2)}. Then fil(q) = γ(k − 2) and from (6.10) we conclude q|l_{γ(k)}D_{γ(k−1)} (f_{γ(k−1)}). Since all factors of f_{γ(k−2)} have multiplicity 1, it follows f_{γ(k−2)}|l_{γ(k)}D_{γ(k−1)} (f_{γ(k−1)}). Hence

deg(f_{γ(k−2)}) ≤ deg(l_{γ(k)}) + deg(D_{γ(k−1)} (f_{γ(k−1)}))
≤ deg(p) − deg(f_{γ(k−1)}) + deg(D_{γ(k−1)} (f_{γ(k−1)}))
≤ deg(p).

Iterating this argument we obtain our claim.

Summing up, we have in (6.8) the following situation. There is an integer k, t ≤ k ≤ m, such that f_{γ(k)} = cp with c ∈ M_{γ(k)}, f_{γ(j)} ∈ M for all j > k and if j < k, deg f_{γ(j)} ≤ deg p.

If γ(k) > t, then

η_{p,γ} = \frac{d a}{a} ∧ \frac{d(n_{t}^{2}a_{t} + m_{t})}{n_{t}^{2}a_{t} + m_{t}} ∧ ⋯ ∧ \frac{d f_{γ(k−1)}}{f_{γ(k−1)}} ∧ \frac{d c}{c} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

+ \frac{d a}{a} ∧ \frac{d(n_{t}^{2}a_{t} + m_{t})}{n_{t}^{2}a_{t} + m_{t}} ∧ ⋯ ∧ \frac{d f_{γ(k−1)}}{f_{γ(k−1)}} ∧ \frac{d p}{p} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

where both summands are in good shape with respect to μ^{−} and satisfy the requirements of the lemma, since all f_{γ(j)} (j ≠ k), have irreducible factors < p.

If γ(k) = t, then f_{γ(k)} = f_{t} = cp is the polynomial n_{t}^{2}a_{t} + m_{t}, and obviously η_{p,γ} has the decomposition stated in the lemma, and is in good shape with respect to μ.

Let us now consider η_{p,<γ}. We have

η_{p,<γ} = \frac{d a}{a} ∧ ∑_{α<γ} \frac{d(n_{α}^{2}a_{t} + m_{α})}{n_{α}^{2}a_{t} + m_{α}} ∧ \frac{d f_{α}}{f_{α}}

which is in good shape with respect to μ^{−}. Decomposing η_{p,<γ} in p-components we get

η_{p,<γ} = η_{p,γ,p} + η_{p,γ,0},

where η_{p,γ,p} belongs to \frac{1}{p}Ω_{F} M[Y^{2}]∩ν_{M(Y^{2})}(m) and η_{p,γ,0} ∈ ⨁_{q<p} \frac{1}{q}Ω_{F} M[Y^{2}]∩ ν_{M(Y^{2})}(m). From (6.4) we know that all these summands are in good shape with respect to μ^{−}. Thus η_{p,γ,0} contributes to h_{0}. We apply now the above procedure to η_{p,γ,p} considering the highest multi-index λ < γ appearing in η_{p,γ,p}. We continue with this procedure to finally get the desired decomposition of η_{p}. This proves the lemma.

Lemma 6.12 For any p appearing in (6.6) it holds

u_{p} = u_{0} + u_{1} ∧ \frac{d p}{p}
v_{p} = v_{0} + v_{1} ∧ \frac{d p}{p}

with u_{0}, u_{1}, v_{0}, v_{1} ∈ Ω_{F} M[Y^{2}], deg u_{i}, deg v_{i} < deg(p).

Proof. Set u_{p} = \frac{u′}{p}, v_{p} = \frac{v′}{p^{2}} with u′, v′ ∈ Ω_{F} M[Y^{2}], deg u′ < deg p, deg v′ < 2 deg p. Inserting in (6.6) we get

p^{2}E_{p} = u^{′[2]} + pu′ + d v′ + pT (pη_{p})

with pη_{p} ∈ Ω_{F} M[Y^{2}]. Set M(p) = M[Y]/ < p > and consider the rest class homomorphism Ω_{F} M[Y^{2}] ⟶ Ω_{M(p)}. The last equation induces the relation

u^{′[2]} + d v′ = 0

in Ω_{M(p)}. Using the Cartier-operator we conclude from d v′ = u^{′[2]} that u′ = 0 and d v′ = 0 in Ω_{M(p)}. From lemma (3.2) in [Ar-Ba_{1}] we get

u′ = pu_{0} + u_{1} ∧ d p
d v′ = p^{2} d v_{0} + p d v_{1} ∧ d p

with u_{0}, u_{1}, v_{0}, v_{1} ∈ Ω_{F} M[Y^{2}], and deg u_{i}, v_{i} < deg p. Thus

u_{p} = u_{0} + u_{1} ∧ \frac{d p}{p}

and

v_{p} = v_{0} + v_{1} ∧ \frac{d p}{p}.

This proves the lemma.

We insert now the above relations in (6.6). It follows

E_{p} + ℘u_{0} + d v_{0} + T η_{0} = (℘u_{1} + d v_{1} + T η_{1}) ∧ \frac{d p}{p}.  (6.13)

Here η_{0} is in good shape with respect to μ^{−} as well as η_{1} ∧ \frac{d p}{p}. Actually if fil(p) > t, then η_{1} is in good shape with respect to μ^{−}, and if fil(p) = t, then η_{1} ∧ \frac{d cp}{cp} with some c ∈ M_{t}, is in good shape with respect to μ^{−}.

In what follows we will need the following remark concerning 2-basis of M(p). Assume p = p_{0} + a_{k}p_{1} with p_{0}, p_{1} ∈ M_{<k}[Y^{2}], fil(p_{0}), fil(p_{1}) < t, and t = fil(p). Then we obtain the following 2-basis of M(p) = M[y] where y ≡ Y mod(p): {a_{1}, …, a_{t−1}, a_{t+1}, …} = B − {a_{t}} ∪ {y}. We keep the same ordering in B −{a_{t}}as in B, but we set y as the maximal element in this 2-basis.

If f ∈ M[Y^{2}] has filtration s < t = fil(p), then it follows easily that fil(\overline{f}) = s in M(p). If f ∈ M[Y^{2}] has degree < deg p and filtration fil(f) = s > t, then fil(\overline{f}) ≥ s in M(p) too (and in fact fil(\overline{f}) = s). To see this, let us assume fil(\overline{f}) < s and write in M[Y^{2}]

f = n_{0} + a_{t}n_{1} + a_{s}(n_{2} + a_{t}n_{3})

where n_{i} has degree < p and do not contain a_{t} or a_{s} in the 2-basis expansion, 0 ≤ i ≤ 3. In particular fil(n_{i}) < s, 0 ≤ i ≤ 3. In M(p) we get

\overline{f} = \overline{n}_{0} + a_{t}\overline{n}_{1} + a_{s}(\overline{n}_{2} + a_{t}\overline{n}_{3})

where a_{t} = \frac{p_{0}p_{1}}{p_{1}^{2}} has filtration < t in M(p). If fil(\overline{f}) < s, then we have

\overline{n}_{2} + a_{t}\overline{n}_{3} = 0 in M(p) and this implies n_{2} = a_{t}n_{3} + lp with some polynomial l ∈ M[Y^{2}]. Since deg n_{2}, deg n_{3} < deg p, it follows l = 0, i.e. n_{2} = a_{t}n_{3}, and this is a contradiction.

From (6.13) we obtain

℘u_{1} + d v_{1} + T η_{1} = 0

in Ω_{M(p)} (see [Ar-Ba_{1}]). In this relation we know that η_{1} is in good shape with respect to μ^{−} if γ(k) > t or if γ(k) = t, then η_{1} ∧ \frac{d p}{p} is in good shape with respect to μ^{−}. Moreover since for all γ(j) > t, the elements f_{γ(i)} appears in η_{1} are constants (see (a) in the proof of (6.7)), we get from the last remark that the multi-indices of η_{1} keep the same ordering in M(p) with respect to the 2-basis of M(p) introduced in this remark. Now the equation

T η_{1} = ℘u_{1} + d v_{1}

leads to the relation

T = ℘c + ∑_{i≠k} A_{γ(i)}f_{γ(i)}

with A_{γ(i)} ∈ M(p)_{γ(i)} and deg A_{γ(i)} < deg p. This follows from lemma (3.3) in [Ar-Ba_{2}], and the fact that we can change the 2-basis of M(p) through the replacement f_{γ(i)} ⟷ a_{γ(i)} for i ≠ k. Here γ is the maximal multi-index in η_{1}. Thus in M[Y^{2}] we obtain

T = ℘c + ∑_{i≠k} A_{γ(i)}f_{γ(i)} + ph  (6.14)

with c ∈ M[Y^{2}], deg c < deg p, A_{γ(i)} ∈ M[Y^{2}], deg A_{γ(i)} < deg p and hence also deg h < deg p. We also have fil A_{γ(i)} < γ(i). If γ(k) = t, recall that f_{γ(k)} = f_{t} = cp with c ∈ M_{t}, and in this case we write cp instead of p in the above relation. Changing the notation a little, we will write f_{γ(k)} = p if γ(k) > t and f_{γ(k)} = cp if γ(k) = t, and h = A_{γ(k)} in both cases. Thus we have

T = ℘c + ∑_{i} A_{γ(i)}f_{γ(i)}.  (6.15)

We will now use this relation to get rid of the term T η_{1} in the equation (6.13).

To this end we distinguish two cases: γ(k) = fil p for k > t and γ(k) = t for k = t.

First Case : γ(k) = fil p, k > t. We can write

η_{1} = \frac{d a}{a} ∧ \frac{d(n_{t}^{2}a_{t} + m_{t})}{n_{t}^{2}a_{t} + m_{t}} ∧ \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ k ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}
+ η_{1,<γ}
= η_{1,γ} + η_{1,<γ}

for the form η_{1} in lemma (6.7). Here γ is the maximal multi-index appearing in η_{1}. Let us now use (6.15) with k > t, to compute T η_{1,γ} ∧ \frac{d p}{p}. Notice that in this case we have written f_{γ(k)} = p, A_{γ(k)} = h, where deg h < deg p, and f_{γ(t)} = f_{t} = n_{t}^{2}a_{t} + m_{t}. We get

T η_{1,γ} ∧ \frac{d p}{p} = (℘c + ∑_{i} A_{γ(i)}f_{γ(i)}) \frac{d a}{a} ∧ \frac{d f_{γ(t)}}{f_{γ(t)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} .

It is convenient to write f_{γ(i)} = a_{i}, γ(i) = i, for i < t, so that

T η_{1,γ} ∧ \frac{d p}{p} = (℘c + ∑_{i=1}^{m} A_{γ(i)}f_{γ(i)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} .

Let ν_{<p,gs}(m) denote the subgroup of ν_{M(Y^{2})}(m) of forms in good shape with respect to μ^{−} and where the generators are logarithmic differential forms d q/q with q < p. We will compute the above expression modulo the group ℘Ω^{m} + d Ω^{m−1} + T ν_{<p,gs}(m). Since

A_{γ(i)}f_{γ(i)} \frac{d f_{γ(i)}}{f_{γ(i)}} ≡ A_{γ(i)}f_{γ(i)} \frac{d A_{γ(i)}}{A_{γ(i)}}

we have

T η_{1,γ} ∧ \frac{d p}{p} ≡ ∑_{i=1}^{m} A_{γ(i)}f_{γ(i)} \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

≡ ∑_{i=1}^{m} (T + ∑_{j≠i} A_{γ(j)}f_{γ(j)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

≡ T ∑_{i=1}^{m} \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

+ ∑_{i=1}^{m} (∑_{j≠i} A_{γ(j)}f_{γ(j)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} .

Since fil(A_{γ(i)}) < γ(i), it follows that d a ∧ d A_{γ(i)} = 0 for all i < t. Hence

the above sums can be taken for i > t. Moreover the term \frac{d a}{a} ∧ \frac{d(n_{t}^{2}a_{t}+m_{t})}{n_{t}^{2}a_{t}+m_{t}}

appears in the first summand, so that T ∑_{i=1}^{m} \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} is contained in T ν_{<p,gs}, and can be dropped. Hence

T η_{1,γ} ∧ \frac{d p}{p} ≡ ∑_{i=1}^{m} (∑_{j≠i} A_{γ(j)}f_{γ(j)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

≡ ∑_{1≤j<i≤m}^{m} (A_{γ(j)}f_{γ(j)} + A_{γ(i)}f_{γ(i)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(j)}}{A_{γ(j)}} ∧ ⋯

⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

Replacing A_{γ(j)}f_{γ(j)}+A_{γ(i)}f_{γ(i)} by T +∑_{l≠i,j} A_{γ(l)}f_{γ(l)} mod ℘ we obtain

T η_{1,γ} ∧ \frac{d p}{p} ≡ ∑_{1≤j<i≤m}^{m} (T + ∑_{l≠i,j} A_{γ(l)}f_{γ(l)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(j)}}{A_{γ(j)}} ∧ ⋯

⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

≡ ∑_{1≤j<i≤m}^{m} ∑_{l≠i,j} A_{γ(l)}f_{γ(l)} \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(j)}}{A_{γ(j)}} ∧ ⋯ ∧

\frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} .

Iterating this reasoning we obtain

T η_{1,γ} ∧ \frac{d p}{p} ≡ ∑_{i=1}^{m} A_{γ(i)}f_{γ(i)} \frac{d A_{γ(1)}}{A_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(m)}}{A_{γ(m)}}

≡ T \frac{d A_{γ(1)}}{A_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(m)}}{A_{γ(m)}}

≡ 0

i.e.

T η_{1,γ} ∧ \frac{d p}{p} ∈ ℘Ω^{m} + d Ω^{m−1} + T ν_{<p,gs}(m)

Thus we get rid of the term T η_{1,γ} ∧ \frac{d p}{p} in the equation (6.13). We can now continue with T η_{1,<γ} which has lower filtration, and this shows that we can assume η_{1} = 0 in (6.13).

Thus we have

E_{p} + ℘u_{0} + d v_{0} + T η_{0} = (℘u_{1} + d v_{1}) ∧ \frac{d p}{p}  (6.16)

where η_{0} ∈ ν_{M(Y^{2})}(m) is in good shape with respect to μ and is in ν_{<p,gs}(m), i.e. its q-components are in lower filtration than p. Now a simple argument, see [Ar-Ba_{1}], shows that we can get rid of the term (℘u_{1} + d v_{1}) ∧ \frac{d p}{p} and hence

E_{p} = ℘u_{0} + d v_{0} + T η_{0}

and therefore E_{p} disappears in the basic equation (6.5), because the u_{0}, v_{0}, η_{0} are in lower components. Thus we are done in the case k > t. Let us now consider the

Second case: k = t i.e. γ(k) = k = t = fil(p) = fil(f_{t}) where f_{γ(t)} = n_{t}^{2}a_{t} + m_{t} = c_{γ}p with c_{γ} ∈ M_{t}, m_{t} = ∑_{σ} a^{σ}m_{σ}^{2}, σ ∈ S_{t−1}, σ + t ≥ μ and n_{t}, m_{t} ∈ M[Y]. From the proof of (6.7), case (b) we know that f_{γ(i)} ∈ M_{γ(i)} for all i > t and f_{t} = f_{γ(t)} = c_{γ}p, fil(c_{γ}) ≤ t. Let us at this place introduce the following M^{2}-vector space

W = {x^{2}a_{t} + ∑_{σ} a^{σ}y_{σ} | x, y_{σ} ∈ M, σ ∈ S_{t−1}, σ + t ≥ μ}  (6.17)

so that f_{γ(t)} ∈ W [Y^{2}]. We will say that the polynomials in W [Y^{2}] are in good shape with respect to μ^{−}. In particular c_{γ} ∈ W. We have in this case

η_{p} = \frac{d a}{a} ∧ ∑_{γ} \frac{d c_{γ}p}{c_{γ}p} ∧ \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

and f_{γ(t)} = c_{γ}p for all γ. We write f_{γ(i)} = a_{i}, γ(i) = i for i < t, so that we have η_{p} = ∑_{γ} \frac{d f_{γ}}{f_{γ}}. Let γ be the maximal multi-index in this summand, set

η_{γ,p} = \frac{d f_{γ}}{f_{γ}} = \frac{d a}{a} ∧ \frac{d c_{γ}p}{c_{γ}p} ∧ \frac{d f_{γ(t+1)}}{f_{γ(t+1)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

with f_{γ(j)} ∈ M for all j > t. Set η_{<γ,p} = ∑_{λ<γ}(d f_{λ}/f_{λ}) so that η_{p} = η_{γ,p} + η_{<γ,p}. The equation

E_{p} + ℘u_{0} + d v_{0} + T η_{0} = (℘u_{1} + d v_{1} + T η_{1}) ∧ \frac{d p}{p}

implies ℘u_{1} + d v_{1} + T η_{1} = 0 in Ω_{M(p)}. Notice in this case η_{p} = η_{1} ∧ \frac{d p}{p}, η_{0} = 0, since k = t. Since the ordering of the multi-index appearing in η_{p} remains the same in the 2-basis of M(p), we obtain in M[Y^{2}] (see remark after the proof of (6.12))

T = ℘c + ∑_{i≠t} A_{γ(i)}f_{γ(i)} + (c_{γ}p)h  (6.18)

with h ∈ M[Y^{2}]. Here all f_{γ(i)} are constants, c_{γ}, A_{γ(i)}, h ∈ M[Y^{2}] with deg c_{γ}, deg A_{γ(i)}, deg h < deg p and A_{γ(i)} ∈ M_{<γ(i)}[Y^{2}]. Moreover f_{γ(t)} = c_{γ}p and A_{γ(t)} = h. Then

T = ℘c + ∑_{i} A_{γ(i)}f_{γ(i)}

Lemma 6.19 If deg p > 2, then h ∈ W [Y^{2}]

We postpone the proof of (6.19) to the end of the section. We look at the basic equation for p of degree > 2.

E_{p} + ℘u_{0} + d v_{0} = (℘u_{1} + d v_{1} + T η_{1}) ∧ \frac{d p}{p}.

We want to get rid of the term T η_{1} ∧ \frac{d p}{p} = T η_{p}. We have for γ = maximal multi-index in η_{p}

T η_{γ,p} = (℘c + ∑_{i=1}^{m} A_{γ(i)}f_{γ(i)}) \frac{d a}{a} ∧ \frac{d c_{γ}p}{c_{γ}p} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

and we compute again modulo the subgroup ℘Ω^{m} +d Ω^{m−1} +T ν_{<p,gs}(m). Then

T η_{γ,p} ≡ (∑_{i=1}^{m} A_{γ(i)}f_{γ(i)}) \frac{d f_{γ}}{f_{γ}}

≡ ∑_{i=1}^{m} A_{γ(i)}f_{γ(i)} \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}}

≡ ∑_{i=1}^{m} (T + ∑_{j≠i} A_{γ(j)}f_{γ(j)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} .

Whenever i < t, the terms \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} are zero because A_{γ(i)} ∈ M_{<γ(i)}, thus only for i ≥ t there are contribution to the sum. Moreover, all the terms T \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} for i ≥ t belong to T ν_{<p,gs}(m). Only the case i = t in non trivial, and this assertion follows from lemma (6.19) since A_{t} = h ∈ W [Y^{2}]. Thus we have

T η_{γ,p} ≡ ∑_{i=1}^{m} (∑_{j≠i} A_{γ(j)}f_{γ(j)}) \frac{d f_{γ(1)}}{f_{γ(1)}} ∧ ⋯ ∧ \frac{d A_{γ(i)}}{A_{γ(i)}} ∧ ⋯ ∧ \frac{d f_{γ(m)}}{f_{γ(m)}} .

Now we continue as in the case k > t to conclude T η_{γ,p} ∈ ℘Ω^{m} + d Ω^{m−1} + T ν_{<p,gs}(m). Doing the same with T η_{<γ,p} we finally obtain the desired result T η_{p} ∈ ℘Ω_{<p}^{m} + d Ω_{<p}^{m−1} + T ν_{<p,gs}(m) and we are done in this case.

Thus we are left with the case deg p = 2, i.e. f_{γ(k)} = f_{γ(t)} = c_{γ}p = c_{γ}Y^{2} + d_{γ} ∈ W [Y^{2}], c_{γ} ∈ W, d_{γ} ∈ W. From the equation

T = ℘c + ∑_{i≠t} A_{γ(i)}f_{γ(i)} + (c_{γ}p)h

we see deg h = 0,i.e. h ∈ M, as well as A_{γ(i)} ∈ M. Hence comparing coefficients we obtain ba^{μ} = c_{γ}h, i.e.

h = \frac{ba^{σ}a_{t}}{c_{γ}} = \frac{ba^{σ}a_{t}}{c_{γ}^{2}} c_{γ}.

But b ∈ F^{2} and σ < μ, and

\frac{d h}{h} = \frac{d a^{σ}}{a^{σ}} + \frac{d a_{t}}{a_{t}} + \frac{d c_{γ}}{c_{γ}}
\frac{d a}{a} ∧ \frac{d h}{h} = \frac{d a}{a} ∧ \frac{d a^{σ}}{a^{σ}} + \frac{d a}{a} ∧ \frac{d a_{t}}{a_{t}} + \frac{d a}{a} ∧ \frac{d c_{γ}}{c_{γ}}

and we see immediately that the terms \frac{d a}{a} ∧ \frac{d h}{h} ∧ \frac{d f_{λ}}{f_{λ}} are in good shape with respect to μ^{−} and belongs to lower degree or filtrations (in fact they are constants). Thus we also get rid of the terms T η_{p} = T η_{1} ∧ \frac{d p}{p} in the basic equation.

It remains

E_{p} + ℘u_{0} + d v_{0} + T η_{0} = (℘u_{1} + d v_{1}) ∧ \frac{d p}{p}

with u_{0}, v_{0}, η_{0} having at most denominators divisible by q’s with q < p, and η_{0} in good shape respect to μ^{−}. Again ℘u_{1} + d v_{1} = 0 in M(p) enable us to eliminate E_{p} from the basic equation (6.5).

Proceeding in the same way we finally arrive at a relation

w = ℘u_{E} + d v_{E} + T η_{E}

with u_{E}, v_{E}, η_{E} ∈ Ω_{F} M[Y^{2}] and η_{E} ∈ ν_{M(Y^{2})}(m) in good shape with respect to μ, i.e. a^{σ}η_{E} ∈ d Ω for all σ < μ. But since (see (2.10)) Ω_{F}^{m}M[Y^{2}]∩ν_{M[Y^{2}]}(m) = ν_{M}(m), we get η_{E} ∈ ν_{M}(m). This conclude the proof of the basic lemma: (A_{μ})⟹(B_{μ}). □

Now we return to the proof of lemma(6.19).

Proof of lemma 6.19. To this end, we look at the relation

T = ℘c + ∑_{i≠t} A_{γ(i)}f_{γ(i)} + (c_{γ}p)h

with c_{γ} ∈ W, deg c, deg A_{γ(i)}, deg h < deg p and also deg f_{γ(i)} = 0 for i ≠ t, i.e. f_{γ(i)} ∈ M for all i ≠ t. Here T = ba^{μ}Y^{2} +T_{μ^{+}}. We write c_{γ}p = p_{0} +p_{1}Y^{2} +⋯+ p_{s}Y^{2s} with p_{0}, …, p_{s} ∈ W and p_{s} = c_{γ}. Set also h = h_{0}+h_{1}Y^{2}+⋯+h_{s−1}Y^{2s−2} and c = c_{0} + c_{1}Y^{2} + ⋯ + c_{s−1}Y^{2s−2}, c_{i}, h_{i} ∈ M. Then the equation above implies h_{s−1} = 0 and the other coefficients satisfy the following system of linear equations

( \begin{matrix}p_{s} & p_{s−1} & p_{s−2} & ⋯ & p_{3} & p_{2} \\ 0 & p_{s} & p_{s−1} & ⋯ & p_{4} & p_{3} \\ 0 & 0 & p_{s} & ⋯ & & p_{4} \\ ⋮ & ⋮ & & ... & & ⋮ \\ 0 & 0 & 0 & & p_{s} & p_{s−1} \\ 0 & 0 & 0 & ⋯ & 0 & p_{s} \end{matrix} ) ( h_{0} ; h_{1} ; h_{2} ; ⋮ ; h_{s−3} ; h_{s−2} ) = ( ∗ ; ⋮ ; 0 ; c_{s−2}^{2} ; 0 ; c_{s−1}^{2} )

where ∗ = c_{s/2} if s is even and ∗ = 0 otherwise. Since the matrix of this system has entries in W, it suffices to show that the inverse matrix also has entries in W. This will imply h_{i} ∈ W for all i. Let I_{n} denote the identity matrix and consider the n × n-matrix E = (e_{i,j}) with e_{i,i+1} = 1 and e_{i,j} = 0 otherwise. Then E^{n} = 0. Fix n elements q_{i} ∈ M and define the M^{2}-vector space W = M^{2}q_{1} + ⋯ + M^{2}q_{n}. Assume q_{n} ≠ 0. Let M_{n×n}(W) be the abelian group of n × n-matrices with entries in W and let

A = ( \begin{matrix}q_{n} & q_{n−1} & ⋯ & q_{1} \\ 0 & q_{n} & ⋯ & q_{2} \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & q_{n} \end{matrix} )

i.e. A = q_{n}I_{n} + q_{n−1}E + ⋯ + q_{1}E^{n−1} = q_{n} (I_{n} + ∑_{j=1}^{n−1} \frac{q_{n−j}}{q_{n}} E^{j}) = q_{n} (I_{n} + B)

where B^{n} = 0. Then A^{−1} = q_{n}^{−1} (I_{n} + B + ⋯ + B^{n−1}).

We claim A^{−1} ∈ M_{n×n}(W). Obviously this claim proves our lemma (6.19). We have

A^{−1} = \frac{q_{n}}{q_{n}^{2}} I_{n} + \frac{q_{n}}{q_{n}^{2}} B + ⋯ + \frac{q_{n}}{q_{n}^{2}} B^{n−1}

and it suffices to show \frac{q_{n}}{q_{n}^{2}} B^{i} ∈ M_{n×n}(W).

If i = 0 this is clear. Assume i ≥ 1.

If i = 2t is even, then

\frac{q_{n}}{q_{n}^{2}} B^{2t} = \frac{q_{n}}{q_{n}^{2}} ∑_{j=1}^{n−1} (\frac{q_{n−j}}{q_{n}} )^{2t} E^{2tj} = q_{n} ∑_{j=1}^{n−1} (\frac{q_{n−j}^{t}}{q_{n}^{t+1}} )^{2} E^{2tj}

is clearly in M_{n×n}(W).

If i = 2t + 1 is odd, then

\frac{q_{n}}{q_{n}^{2}} B^{i} = (∑_{j=1}^{n−1} (\frac{q_{n−j}}{q_{n}^{2}} ) E^{j}) (∑_{j=1}^{n−1} (\frac{q_{n−j}}{q_{n}^{2}} )^{2t} E^{2tj})

which also belongs to M_{n×n}(W) since q_{n−j}/q_{n}^{2} ∈ W.

This proves the claim.

Acknowledgments

This work has been supported by Proyecto Fondecyt #1030218 and Programa Reticulados, Universidad de Talca (the second author), Proyecto Fondecyt #102 0516 (the first author).

References

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