New York Journal of Mathematics

New York J. Math. 6 (2000) 21–54.

The Chromatic E_{1}-term H^{0}M_{1}^{2} for p > 3

Hirofumi Nakai

Abstract. In this paper, study the module structure of

Ext_{BP_{∗}BP}^{0} (BP_{∗}, BP_{∗}/(p, v_{1}^{∞}, v_{2}^{∞})), which is regarded as the chromatic E_{1}-term converging to the second line of the Adams-Novikov E_{2}-term for the Moore spectrum. The main difficulty here is to construct elements x(sp^{r}/j; k) from the Miller-Ravenel-Wilson elements (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} ∈ H^{0}M_{2}^{1}. We achieve this by developing some inductive methods of constructing x(sp^{r}/j; k) on k.

Contents

1. Introduction 21 2. BP_{∗}-homology and Bockstein spectral sequence 24 3. Definitions of some elements 28 4. Preliminary calculations 34 5. Proof of the main theorem 46 References 54

1. Introduction

Let BP be the Brown-Peterson spectrum for a fixed prime p. As is well known, the pair of homotopy groups BP_{∗} and the BP_{∗}-homology BP_{∗}BP forms a Hopfalgebroid

( BP_{∗} , BP_{∗}BP ) = ( ℤ_{(p)}[v_{1}, v_{2}, …] , BP_{∗}[t_{1}, t_{2}, …] ).

The Adams-Novikov spectral sequence (ANSS) is one of the fundamental tools to compute the p-component of the stable homotopy groups π_{∗}^{S}X_{(p)} for a spectrum X:

E_{2}^{∗,∗} = Ext_{BP_{∗}BP}^{∗} (BP_{∗}, BP_{∗}X) ⟹ π_{∗}^{S}X_{(p)}.

Here, for any BP_{∗}BP -comodule M, Ext_{BP_{∗}BP}^{∗} (BP_{∗}, M) is regarded as the right derived functor of Hom_{BP_{∗}BP} (BP_{∗}, M). We abbreviate Ext_{BP_{∗}BP}^{s} (BP_{∗}, M) to H^{s}M as usual.

21^{ISSN 1076-9803/00}

Received May 5, 1999.

Mathematics Subject Classification. 55Q45; secondary 55Q51, 55T15, 55N22.

Key words and phrases. Stable homotopy of spheres, Adams-Novikov spectral sequence, chromatic spectral sequence.

As an important example of finite spectra, we have the Smith-Toda spectrum V (n) for each prime p with BP_{∗}V (n) ≅ BP_{∗}/(p, v_{1}, ⋯, v_{n}), although its existence is verified only for 0 ≤ n ≤ 3 and 2n + 1 ≤ p so far. (Recently, Lee S. Nave has shown the non-existence of V ((p + 3)/2) for p > 5.) Then the E_{2}-term of the ANSS for V (n) is H^{∗}BP_{∗}V (n). Miller, Ravenel and Wilson [2] have constructed an algebraic spectral sequence converging to H^{∗}BP_{∗}V (n) as follows:

Denote the BP_{∗}BP -comodules BP_{∗}/(p, v_{1}, ⋯, v_{n−1}) by N_{n}^{0}. Then, define N_{n}^{m} (m ≥ 1) inductively on m by short exact sequences

0 ⟶ N_{n}^{m} ⟶ v_{m+n}^{−1}N_{n}^{m} ⟶ N_{n}^{m+1} ⟶ 0.

We also define M_{n}^{m} by M_{n}^{m} = v_{m+n}^{−1}N_{n}^{m}. Indeed, they can be described directly as

\begin{aligned}N_{n}^{m} & = & BP_{∗}/(p, ⋯, v_{n−1}, v_{n}^{∞}, ⋯, v_{n+m−1}^{∞}), \\ M_{n}^{m} & = & v_{n+m}^{−1}BP_{∗}/(p, ⋯, v_{n−1}, v_{n}^{∞}, ⋯, v_{n+m−1}^{∞}). \end{aligned}

Splicing the above short exact sequences, we get a long exact sequence:

0 ⟶ N_{n}^{0} ⟶ M_{n}^{0} ⟶ M_{n}^{1} ⟶ M_{n}^{2} ⟶ ⋯,

called the chromatic resolution of N_{n}^{0}. Applying H^{∗}(−) to the above long exact sequence, we obtain a spectral sequence converging to H^{∗}N_{n}^{0} with E_{1}^{s,t} = H^{t}M_{n}^{s} and d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t−r+1}, called the chromatic spectral sequence.

The simplest example in these E_{1}-terms is the 0-th cohomology of the n-th Morava stabilizer group H^{0}M_{n}^{0}, which is isomorphic to ℤ/p [v_{n}^{±1}]. Moreover, H^{t}M_{n}^{0} (1 ≤ t ≤ 2) has been computed by Ravenel [5]. In general the calculation of H^{t}M_{n}^{s} becomes terribly difficult as s + t increases, except for the following case:

Theorem (Morava’s vanishing theorem). If (p − 1) ∤ n and t > n^{2}, then

H^{t}M_{n}^{0} = 0.

Many chromatic E_{1}-terms are computed so far (cf. [12]), most of them are due to Miller-Ravenel-Wilson [2], Ravenel [5] and Shimomura’s works. In particular, H^{0}M_{n}^{1} is computed in [2, Theorem 5.10] and H^{0}M_{n}^{2} (n ≥ 2, p > 2) is also done in [10, Theorem 1.2]. The purpose of this paper is to prove the following theorem about the k(1)_{∗}-module structure of unknown H^{0}M_{1}^{2} for p > 3.

Theorem. For each Miller-Ravenel-Wilson element x_{3,r+k}^{s}/v_{2}^{jp^{k}} ∈ H^{0}M_{2}^{1} (p ∤ j and p ∤ s) and 1/v_{2}^{mp^{r}} ∈ H^{0}M_{2}^{1} (p ∤ m), there exists an element

x(sp^{r}/j; k) ∈ v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞})

which is congruent to (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} mod (v_{1}), and

1/X_{2,r}^{m} ∈ BP_{∗}/(p, v_{2}^{∞})

congruent to 1/v_{2}^{mp^{r}} mod (v_{1}), so that as a k(1)_{∗}-module

H^{0}M_{1}^{2} ≅

k(1)_{∗} {x(sp^{r}/j; k)/v_{1}^{N(s,r,j;k)} | k ≥ 0, r ≥ 0, p ∤ s ∈ ℤ and p ∤ j ≤ a_{3,r}}

⊕ k(1)_{∗} {1/v_{1}^{a_{2,r}}X_{2,r}^{m} | r ≥ 0 and p ∤ m ≥ 1} ,

The Chromatic E_{1}-term H^{0}M_{1}^{2} for p > 3 23

where the integers a_{2,r} and a_{3,r} are defined in 2.3.1 and 3.3.1, and the integers N(s, r, j; k) are given as follows:

(1)  [(p^{3} + p^{2} − p − 1)p^{k−3}] + [p^{k−3}] for r = 0 and p ∤ (s − 1),
(2)  (p^{k+1} − 1)/(p − 1) for r = 0, p | (s − 1) and s ∉ ℕ_{1},
(3)  [(2p^{2} − 1)p^{k−2}] + [p^{k−2}] for r = 0 and s ∈ ℕ_{1},
(4)  (p^{k+1} − 1)/(p − 1) − [p^{k−1}] for odd r ≥ 1, s ∉ ℕ_{0} and j = a_{3,r} − 1,
(5)  p^{k} + p^{k−1} − 1 ( = a_{2,k} ) for odd r ≥ 1, s ∉ ℕ_{0} and j ≤ a_{3,r} − 2
or even r ≥ 2 and s ∉ ℕ_{0},
(6)  p^{k+1} + p^{k} − 1 ( = a_{2,k+1} ) for r = 1, s ∈ ℕ_{0} and j = p − 1,
(7) [(2p^{3} + p^{2} − p − 1)p^{k−3}] + [p^{k−3}] for r = 1, s ∈ ℕ_{0} and j ≤ p − 2,
(8)  [(p^{2} + p − 1)p^{k−2}] + [p^{k−2}] for even r ≥ 2 and s ∈ ℕ_{0},
(9)  [(p^{4} + p^{2} + p − 1)p^{k−4}] + [p^{k−4}] for odd r ≥ 3 and j = a_{3,r},

and for odd r ≥ 3, s ∈ ℕ_{0} and j ≤ a_{3,r} − 1, we have

(10)  [(p^{3} + p^{2} − 1)p^{k−3}] + [p^{k−3}] for j = a_{3,r} − 1,
(11)  same as the case (1) for a_{3,r} − p ≤ j ≤ a_{3,r} − 2,
(12)  2p^{k} for j = a_{3,r} − p − 1,
(13)  same as the case (7) for j = a_{3,r} − p + 2,

or p ∤ (j + 1) and j ≤ a_{3,r} − p − 2,

(14)  p^{k} + (p^{k+1} − 1)/(p − 1) for p | (j + 1)
and a_{3,r} − p^{2} ≤ j ≤ a_{3,r} − 2p,
(15)  [(p^{2} + p − 1)p^{k−1}] + [p^{k−1}] for j = a_{3,r} − p^{2} − p,

or p | (j + 1), p^{3} ∤ (j + 1)(j + p + 1)

and j ≤ a_{3,r} − p^{2} − 3p,
(16) [(p^{4} + p^{3} − p^{2} + 1)p^{k−3}] + [2p^{k−1}] for j ≤ a_{3,r} − 2p^{2} − p and p^{2} | (j + 1),
(17)  p^{k+1} + p^{k} for j ≤ a_{3,r} − p^{2} − 2p
and p^{2} | (j + p + 1),

Here [x] is the greatest integer which does not exceed x, and

\begin{aligned}ℕ_{0} & = & \{ ap − 1 | p ∤ a \} , \\ ℕ_{1} & = & \{ (ap^{2} − p − 1)p^{r} + 1 | p ∤ a, r ≥ 1 : odd \} . & □ \end{aligned}

In Section 2 we shall review BP -theory and the Bockstein spectral sequence and recall the structure of the chromatic E_{1}-term H^{0}M_{2}^{1} computed in [2]. Then we change the 𝔽_{p}-module basis of H^{0}M_{2}^{1} and state the method of getting the structure of H^{0}M_{1}^{2} (originally due to Miller-Ravenel-Wilson). It is enough to read this section for an idea of the theoretical part. In Section 3 we give the fundamental elements u_{n,k}, w_{n,k} and X_{3,r}, construct the new element 1/X_{2,r}, and give the differentials on these elements (some of them are introduced in [1] and [8]). In Section 4 we set up the elements X_{0}(v_{2}^{j}, X_{3,r}^{s}) and X(v_{2}^{j}, X_{3,r}^{s}), each of which is congruent to X_{3,r}^{s}/v_{2}^{j} mod (v_{1}), and compute the differentials. We also introduce some inductive methods of constructing x(k) for large k. This is the hardest part of this paper. Using these results, we construct the series of elements x(sp^{r}/j; k) and complete the proof of the main theorem in Section 5.

We can deduce some applications to H^{2}BP_{∗}V (0) from this result. These will appear in the forthcoming paper [3].

Acknowledgment. I wish to thank Professor Katsumi Shimomura for leading me to this field, and giving guidance and much helpful advice. I also would like to thank Professor Mikiya Masuda and Professor Zen-ichi Yosimura for their carefully reading the draft version, and for their encouragement. I could not have finished these hard calculations without stimulation from them.

2. BP_{∗}-homology and Bockstein spectral sequence

In this section we review several basic facts in BP -theory and explain how to determine the structure of H^{0}M_{1}^{2}.

2.1. Summary of BP_{∗}-homology and related maps. For a fixed prime p, there is a spectrum BP called the Brown-Peterson spectrum, which is characterized by

\begin{aligned}BP_{∗} & = & ℤ_{(p)}[v_{1}, v_{2}, ⋯, v_{n}, ⋯], \\ BP_{∗}BP & = & BP_{∗}[t_{1}, t_{2}, ⋯, t_{n}, ⋯], \end{aligned}

where |v_{n}| = |t_{n}| = 2(p^{n} − 1). The pair (A, Γ) = (BP_{∗}, BP_{∗}BP) form a Hopf algebroid with the following structure maps:

η_{L} (resp.η_{R}) : A → Γ (left (resp. right) unit), c : Γ → Γ (conjugation),

ϵ : Γ → A (augmentation), Δ : Γ → Γ ⊗_{A} Γ (coproduct).

Given a BP_{∗}BP -comodule M and its BP_{∗}BP -comodule structure ϕ : M → M⊗_{A}Γ, we use the following notation as usual:

H^{∗}M = Ext_{Γ}^{∗}(A, M) ≅ H^{∗}(Ω^{∗}M, d),

where the cobar complex (Ω^{∗}M, d) is the double graded ℤ_{(p)}-module with

Ω^{n}M = M ⊗_{A} Γ ⊗_{A} ⋯ ⊗_{A} Γ (n factors of Γ),
d_{n}(m ⊗ γ_{1} ⊗ ⋯ γ_{n}) = ϕ(m) ⊗ γ_{1} ⊗ ⋯ ⊗ γ_{n}
+ ∑ (−1)^{k} m ⊗ γ_{1} ⊗ ⋯ ⊗ Δ(γ_{k}) ⊗ ⋯ ⊗ γ_{n}
+ (−1)^{n+1} m ⊗ γ_{1} ⊗ ⋯ ⊗ γ_{n} ⊗ 1.

In this paper we compute only the 0-th differential d_{0} = η_{R} − η_{L} : M_{1}^{2} → M_{1}^{2} ⊗_{A} Γ. By definition, d_{0} satisfies

d_{0}(xy) = d_{0}(x) y + η_{R}(x) d_{0}(y)

for any x, y ∈ A. This formula is frequently used for computations in this paper. When d_{0}(x) ≡ z mod (p, J) for an element z ∈ Γ and an ideal J ⊂ A, we also have

d_{0}(x^{p}) ≡ z^{p} mod (p, J^{p}).

See also [2, Observation 5.8].

To compute d_{0}, we summarize some known results about η_{R}. Ravenel [4] has shown the following congruence of formal group laws:

∑_{i,j≥0}^{F} t_{i}η_{R}(v_{j}^{p^{i}}) ≡ ∑_{i,j≥0}^{F} v_{i}t_{j}^{p^{i}} mod p.

In general η_{R}(v_{n}) ≡ v_{n} +v_{n−1}t_{1}^{p^{n−1}} −v_{n−1}^{p}t_{1} mod I_{n−1} for the invariant prime ideal I_{n−1} = (p, v_{1}, ⋯, v_{n−2}). Then, for instance, direct calculation shows that

(2.1.1) η_{R}(v_{2}^{i}) ≡ (v_{2} + v_{1}t_{1}^{p})^{i} − i v_{1}^{p}v_{2}^{i−1}t_{1} − i(i − 1) v_{1}^{p+1}v_{2}^{i−2}t_{1}^{p+1} mod p, v_{1}^{p+2}.

More precisely, η_{R}(v_{3}) and η_{R}(v_{4}) satisfy the following congruences (cf. [6](4.3.21)):

(2.1.2) η_{R}(v_{3}) ≡ v_{3} + v_{2}t_{1}^{p^{2}} − v_{2}^{p}t_{1} + v_{1}t_{2}^{p} − v_{1}^{2}v_{2}^{p−1}t_{1}^{p} mod p, v_{1}^{3},
(2.1.3) η_{R}(v_{4}) ≡ v_{4} + v_{3}t_{1}^{p^{3}} − v_{3}^{p}t_{1} + v_{2}t_{2}^{p^{2}} − v_{2}^{p}t_{1}^{p^{3}+1} mod p, v_{1}, v_{2}^{p+1}.

2.2. Bockstein spectral sequence. In this paper will deduce the k(1)_{∗}-module structure of H^{0}M_{1}^{2} from H^{0}M_{2}^{1}, which has already been computed in [2]. By definitions of comodules M_{2}^{1} and M_{1}^{2}, we obtain the short exact sequence

0 ⟶ M_{2}^{1} ⟶^{1/v_{1}} M_{1}^{2} ⟶^{v_{1}} M_{1}^{2} ⟶ 0.

Applying H^{∗}(−) to this sequence, we get the long exact sequence

0 ⟶ H^{0}M_{2}^{1} ⟶^{1/v_{1}} H^{0}M_{1}^{2} ⟶^{v_{1}} H^{0}M_{1}^{2} ⟶^{δ} H^{1}M_{2}^{1} ⟶^{1/v_{1}} · · · .

Regarding this long sequence as an exact couple, we get a Bockstein type spectral sequence in the usual way, leading from H^{∗}M_{2}^{1} to H^{∗}M_{1}^{2}. But as in [2] we compute H^{0}M_{1}^{2} directly by making use of the following lemma.

Lemma 2.2.1 (cf. [2], Remark 3.11). Assume that there exists a k(1)_{∗}-submodule B^{t} of H^{t}M_{1}^{2} for each t < N, such that the following sequence is exact:

0 → H^{0}M_{2}^{1} ⟶^{1/v_{1}} B^{0} ⟶^{v_{1}} B^{0} ⟶^{δ} H^{1}M_{2}^{1} ⟶^{1/v_{1}} ⋯ ⟶^{1/v_{1}} B^{N−1} ⟶^{v_{1}} B^{N−1} ⟶^{δ} H^{N}M_{2}^{1},

where δ : B^{t} → H^{t+1}M_{2}^{1} is the restriction of the coboundary map δ : H^{t}M_{1}^{2} → H^{t+1}M_{2}^{1}. Then the inclusion map i_{t} : B^{t} → H^{t}M_{1}^{2} is an isomorphism between k(1)_{∗}-modules for each t < N.

Sketch of the proof. Because H^{t}M_{1}^{2} is a v_{1}-torsion module, we can filter B^{t} and H^{t}M_{1}^{2} as

P_{t}(m) = {x ∈ B^{t} | v_{1}^{m}x = 0} and Q_{t}(m) = {x ∈ H^{t}M_{1}^{2} | v_{1}^{m}x = 0}.

Assume that the inclusion i_{k} is an isomorphism for k ≤ t − 1 (the t = 0 case is obvious), and consider the following commutative ladder diagram:

B^{t−1}  ⟶^{δ} H^{t}M_{2}^{1} ⟶^{1/v_{1}} P_{t}(m) ⟶^{v_{1}} P_{t}(m − 1) ⟶^{δ} H^{t+1}M_{2}^{1}
≅↓ i_{t−1}  ∥  ↓ i_{t}  ↓ i_{t}  ∥
H^{t−1}M_{1}^{2} ⟶^{δ} H^{t}M_{2}^{1} ⟶^{1/v_{1}} Q_{t}(m) ⟶^{v_{1}} Q_{t}(m − 1) ⟶^{δ} H^{t+1}M_{2}^{1}.

Using the Five Lemma, we can show that P_{t}(m) ≅ Q_{t}(m) (m ≥ 1) by induction on m. □

We shall construct B^{0} satisfying the above lemma to determine the k(1)_{∗}-module structure of H^{0}M_{1}^{2}. In order to construct a k(1)_{∗}-module basis of B^{0}, it is natural to push each element of H^{0}M_{2}^{1} to H^{0}M_{1}^{2} and to divide it by v_{1} as many times as possible. So we need to review the module structure of H^{0}M_{2}^{1}.

2.3. H^{0}M_{2}^{1} and changing its basis. We first recall some notations defined in [2] to write down a k(2)_{∗}-module basis of H^{0}M_{2}^{1}. Hereafter we assume that p > 2.

Definition 2.3.1 ([2], (5.11) and (5.13)). Define integers a_{3,k} by

a_{3,0} = 1, a_{3,1} = p, a_{3,2t} = p a_{3,2t−1}, a_{3,2t+1} = p a_{3,2t} + p − 1 for t ≥ 1

and elements x_{3,k} ∈ v_{3}^{−1}BP_{∗} by

x_{3,0} = v_{3},  x_{3,1} = v_{3}^{p} − v_{2}^{p}v_{3}^{−1}v_{4},  x_{3,2t} = x_{3,2t−1}^{p},
x_{3,2t+1} = x_{3,2t}^{p} − v_{2}^{a_{3,2t+1}−p}v_{3}^{(p−1)p^{2t}+1}

for t ≥ 1.

Using these notations, Miller-Ravenel-Wilson ([2, Theorem 5.10]) have shown the following:

Theorem 2.3.2. As a k(2)_{∗}-module,

H^{0}M_{2}^{1} ≅ k(2)_{∗} {x_{3,k}^{m}/v_{2}^{a_{3,k}} | k ≥ 0, p ∤ m ∈ ℤ} ⊕ 𝔽_{p} {1/v_{2}^{i} | i ≥ 1} .

In this paper we will consider an analogue to Miller-Ravenel-Wilson construction of the elements x_{3,k}^{m} ∈ v_{3}^{−1}BP_{∗} ([2, Section 5]): The elements x_{3,k} have been defined

inductively on k with x_{3,0} = v_{3}, and each of them has the relation x_{3,k} ≡ v_{3}^{p^{k}} mod (p, v_{2}^{N}) for a small enough integer N. Motivated by this, we shall construct elements x(k) ∈ v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}) inductively on k with x(0) = x_{3,r}^{s}/v_{2}^{j} so that x(k) ≡ (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} mod (p, v_{1}^{N}) for a small enough N. Keeping it in mind, we now change the 𝔽_{p}-basis of H^{0}M_{2}^{1} to “a p^{k}-power basis”.

Lemma 2.3.3. As a k(2)_{∗}-module,

H^{0}M_{2}^{1} ≅ 𝔽_{p} {(x_{3,r}^{s}/v_{2}^{j})^{p^{k}} | k ≥ 0, r ≥ 0, p ∤ s ∈ ℤ, and p ∤ j ≤ a_{3,r}}
⊕ 𝔽_{p} {1/v_{2}^{i} | i ≥ 1} .

Proof. It is sufficient to prove that any 𝔽_{p}-module base x_{3,u}^{m}/v_{2}^{l} (1 ≤ l ≤ a_{3,u}) displayed in Theorem 2.3.2 can be written as a linear sum

x_{3,u}^{m}/v_{2}^{l} = ∑_{λ} a_{λ}x_{λ}, (2.3.4)

where a_{λ} ∈ 𝔽_{p} and each x_{λ} has a form (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} with p ∤ s ∈ ℤ and p ∤ j ≤ a_{3,r}. We shall show it by induction on mp^{u}. Notice that

x_{3,u}^{m}/v_{2}^{l} = v_{3}^{mp^{u}}/v_{2}^{l} + (elements with v_{3}-exponents less than mp^{u}).

When mp^{u} = 1, the only such a base is v_{3}/v_{2}, so it is clear. Suppose that each base x_{3,b}^{a}/v_{2}^{c} (1 ≤ ap^{b} < mp^{u}) can be expressed as (2.3.4) and l = dp^{e} with p ∤ d. Because l ≤ a_{3,u} < p^{u+1}, we may assume that e ≤ u. Define y = x_{3,u}^{m}/v_{2}^{l} −

(x_{3,u−e}^{m}/v_{2}^{d})^{p^{e}} ∈ H^{0}M_{2}^{1}. Since the maximal v_{3}-exponent in y is less than mp^{u}, y has a form (2.3.4) by inductive assumption, hence so does x_{3,u}^{m}/v_{2}^{l}. □

2.4. The construction of the module B^{0}. Let x(sp^{r}/j; k) and y(mp^{r}) be elements of v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}) satisfying

x(sp^{r}/j; k)/v_{1} = (x_{3,r}^{s}/v_{2}^{j})^{p^{k}}/v_{1}  and  y(mp^{r})/v_{1} = 1/v_{1}v_{2}^{mp^{r}}

in H^{0}M_{1}^{2}, and N(s, r, j; k) (resp. N(m; r)) be the maximal integer such that each x(sp^{r}/j; k)/v_{1}^{i} (resp. y(mp^{r})/v_{1}^{i}) with 1 ≤ i ≤ N(s, r, j; k) (resp. 1 ≤ i ≤ N(m; r)) is a cycle of H^{0}M_{1}^{2}.

Proposition 2.4.1. As a k(1)_{∗}-module,

B^{0} = k(1)_{∗} {x(sp^{r}/j; k)/v_{1}^{N(s,r,j;k)} | k ≥ 0, r ≥ 0, p ∤ s ∈ ℤ and p ∤ j ≤ a_{3,r}}
⊕ k(1)_{∗} {y(mp^{r})/v_{1}^{N(m;r)} | r ≥ 0 and p ∤ m ≥ 1}

is isomorphic to H^{0}M_{1}^{2} if it satisfies the following condition for the coboundary map δ : B^{0} → H^{1}M_{2}^{1} in Lemma 2.2.1:

{δ (x(sp^{r}/j; k)/v_{1}^{N(s,r,j;k)}) } ∪ {δ (y(mp^{r})/v_{1}^{N(m;r)}) } is 𝔽_{p}-linearly independent.

Proof. All exactness of the sequence 0 → H^{0}M_{2}^{1} →^{1/v_{1}} B^{0} →^{v_{1}} B^{0} →^{δ} H^{1}M_{2}^{1} is obvious, but Ker δ ⊂ Im v_{1}. So we need to show only this inclusion. Separate the 𝔽_{p}-basis of B^{0} into two parts

A = {x(sp^{r}/j; k)/v_{1}^{N(s,r,j;k)}} ∪ {y(mp^{r})/v_{1}^{N(m;r)}}
B = {x(sp^{r}/j; k)/v_{1}^{l} | 1 ≤ l < N(s, r, j; k)} ∪ {y(mp^{r})/v_{1}^{l} | 1 ≤ l < N(m; r)} .

Then it is obvious that δ(x_{λ}) ≠ 0 ∈ H^{1}M_{2}^{1} for x_{λ} ∈ A, and that δ(y_{μ}) = 0 ∈ H^{1}M_{2}^{1} for y_{μ} ∈ B. Thus for any element z = ∑_{λ} a_{λ}x_{λ} +∑_{μ} b_{μ}y_{μ} of B^{0} (a_{λ}, b_{μ} ∈ ℤ/p), we have δ(z) = ∑_{λ} a_{λ}δ(x_{λ}). The condition implies that all a_{λ} are zero when δ(z) = 0, and so v_{1} ∑_{μ} b_{μ}y_{μ}/v_{1} = z. This completes the proof. □

We will construct the element x(sp^{r}/j; k) from (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} in Section 5, and the element 1/X_{2,r}^{m} from 1/v_{2}^{mp^{r}} as a candidate for y(mp^{r}) in Section 3.

2.5. 𝔽_{p}-linear independence and Coker δ. When we compute the coboundaries δ : B^{0} → H^{1}M_{2}^{1} of x(sp^{r}/j; k)/v_{1}^{N(s,r,j;k)} and y(mp^{r})/v_{1}^{N(m;r)}, we expect each of these images to have an appropriate form so that we can judge whether the set of δ-images is 𝔽-linearly independent or not in H^{1}M_{2}^{1} for use in Proposition 2.4.1. Though the structure of H^{1}M_{2}^{1} (p > 3) has already been computed by Shimomura ([9] and [11]), we don’t need the whole structure of H^{1}M_{2}^{1} for our purpose. We follow the same method as in the proof of [2, Theorem 6.1 (p. 500)].

Consider the long exact sequence

⋯ ⟶ H^{0}M_{2}^{1} ⟶^{δ} H^{1}M_{3}^{0} ⟶^{1/v_{2}} H^{1}M_{2}^{1} ⟶^{v_{2}} H^{1}M_{2}^{1} ⟶^{δ} · · · .

Since the coboundaries on elements of H^{0}M_{2}^{1} have already been computed in [2, Proposition 5.17], we can obtain generators of Ker (v_{2} | H^{1}M_{2}^{1}). An easy calculation shows that Coker(δ : H^{0}M_{2}^{1} → H^{1}M_{3}^{0}) is isomorphic to the following direct sum as a 𝔽_{p}-vector space:

𝔽_{p} {v_{3}^{sp^{r}}h_{0} | either s ∉ ℕ_{0} and even r ≥ 0, or p ∤ s and odd r ≥ 1}

(2.5.1)

⊕𝔽_{p} {v_{3}^{sp^{r}}h_{1} | either s ∉ ℕ_{0} and odd r ≥ 1, or p ∤ s and even r ≥ 0}
⊕𝔽_{p}{ h_{1} } ⊕ 𝔽_{p} {v_{3}^{tp−1}h_{2} | t ∈ ℤ} ⊕ K(3)_{∗}{ ζ_{3} },

where ℕ_{0} = { s ∈ ℤ | p | s + 1 and p^{2} ∤ s + 1 }, and h_{i} is the cohomology class of

t_{1}^{p^{i}}.

Let S be the set of elements

{A_{k} = v_{3}^{b_{k}}h_{a_{k}}/v_{2}^{c_{k}} | 0 ≤ k ≤ n, 0 ≤ a_{k} ≤ 2, c_{1} ≥ ⋯ ≥ c_{n}, v_{3}^{b_{k}}h_{a_{k}} ∈ Coker δ },

and consider the following condition on S:

(2.5.2) (a_{i}, b_{i}, c_{i}) ≠ (a_{j}, b_{j}, c_{j}) for any two elements with i ≠ j.

For a linear sum A = ∑_{k=1}^{n} λ_{k}A_{k}, assume that A = 0 in H^{1}M_{2}^{1} and c_{1} = c_{2} = ⋯ = c_{m} for some m ≤ n. Because H^{1}M_{2}^{1} is a v_{2}-torsion module, we can consider the multiplication by v_{2}^{c_{1}−1}, so that v_{2}^{c_{1}−1}A = ∑_{k=1}^{m} λ_{k}(v_{3}^{b_{k}}h_{a_{k}}/v_{2}) = 0 in Ker v_{2} (≅ Coker δ). By the condition (2.5.2), the set {v_{3}^{b_{k}}h_{a_{k}}} is linearly independent in Coker δ and thus all λ_{k} (1 ≤ λ_{k} ≤ m) should be zero. Iterating this, we conclude that all coefficients λ_{k} are zero and S is linearly independent in H^{1}M_{2}^{1}.

3. Definitions of some elements

In this section we introduce the elements u_{n,k}, w_{n,k}, X_{3,r} and 1/X_{2,r}. We will use these to define many elements in Section 4.

3.1. Moreira’s element u_{n,k} and Shimomura’s element w_{n,k}.

Definition 3.1.1. As is done in [1, §6(4)] or [8, 2.8], we define the element u_{n,k} ∈ v_{n}^{−1}BP_{∗} by the following recursive formula:

u_{n,0} = v_{n}^{−1}  and  ∑_{i+j=k} v_{n+i}u_{n,j}^{p^{i}} = 0 for k ≥ 1.

Remark 3.1.2. In [1] u_{n,k} is defined only for k ≤ n and denoted as u_{n+k}. By definition

u_{n,1} = − v_{n}^{−p−1}v_{n+1},
u_{n,2} = v_{n}^{−p^{2}−p−1}v_{n+1}^{p+1} − v_{n}^{−p^{2}−1}v_{n+2},
u_{n,3} ≡ − v_{n}^{−p^{3}−p^{2}−p−1}v_{n+1}^{p^{2}+p+1} + v_{n}^{−p^{3}−p^{2}−1}(v_{n+1}^{p^{2}}v_{n+2} + v_{n+1}v_{n+2}^{p})
− v_{n}^{−p^{3}−1}v_{n+3} mod (p),

and so on.

Computing the right unit η_{R} on u_{n,n}, Moreira has shown that the K(n)_{∗}-module base ζ_{n} of H^{1}M_{n}^{0} is homologous to ζ_{n}^{p}:

Proposition 3.1.3 ([1], Theorem 6.2.1.1). η_{R}(u_{n,n})−η_{L}(u_{n,n}) ≡ ζ_{n}−ζ_{n}^{p} mod I_{n}.

We now recall the elements w_{n,k} introduced by Shimomura [8]. Define elements T_{j} (j ≥ 0) by T_{0} = 1 and ∑_{i=0}^{j−1} t_{i}η_{R}(v_{j−i}^{p^{i}}) ≡ ∑_{i=1}^{j} v_{i}T_{j−i}^{p^{i−1}} mod (p) for j ≥ 1, and an element e_{n}(x) ∈ v_{n}^{−1}BP_{∗}BP/I_{n} for x ∈ v_{n}^{−1}BP_{∗} by the congruence η_{R}(x) ≡ e_{n}(x) mod I_{n}.

Definition 3.1.4 (cf. [8], 2.10). Define elements w_{n,k} ∈ v_{n}^{−1}BP_{∗}BP (n ≥ 2) inductively on k by

w_{n,0} = 0 and w_{n,k} = ∑_{j=1}^{k} e_{n}(u_{n,k−j}^{p^{j−1}})T_{j}^{p^{n−2}} for k ≥ 1.

Remark 3.1.5. By definition,

w_{n,1} = v_{n}^{−1}t_{1}^{p^{n−1}},
w_{n,2} ≡ − v_{n}^{−p−1}v_{n+1}t_{1}^{p^{n−1}} + v_{n}^{−p}(t_{2}^{p^{n−1}} − t_{1}^{p^{n}+p^{n−1}}) + v_{n}^{−1}t_{1}^{p^{n−1}+1} mod p, v_{1}^{p^{n−2}}.

In general, as an element of M_{2}^{1} ⊗_{A} Γ

w_{2,k} = (−1)^{k−1} \frac{v_{3}^{(p^{k−1}−1)/(p−1)}t_{1}^{p}}{v_{2}^{(p^{k}−1)/(p−1)}} + (elements killed by v_{2}^{(p^{k}−1)/(p−1)−1}).

We note that w_{2,2} is similar to ζ_{2} = − v_{2}^{−p−1}v_{3}t_{1}^{p} + v_{2}^{−p}(t_{2}^{p} − t_{1}^{p^{2}+p}) + v_{2}^{−1}t_{2}. In fact w_{n,n} (n ≥ 2) is related to ζ_{n} by the following congruence (cf. [8, 4.8]):

ζ_{n} − w_{n,n} ≡ ∑_{1≤i<j≤n} u_{n,n−j}^{p^{j−i−1}} ( ∑_{i≤k≤j} t_{k} c(t_{j−k}^{p^{k}}))^{p^{n−i−1}} mod I_{n}.

In particular we have ζ_{2} − w_{2,2} ≡ v_{2}^{−1}(t_{2} − t_{1}^{p+1}) mod I_{2} for n = 2.

Using the notation w_{n,k}, Shimomura [8] has proved the following proposition, which is a generalization of Moreira’s result:

Proposition 3.1.6 ([8], Proposition 2.2). For n ≥ 2,

η_{R}(u_{n,k}) ≡ ∑_{i+j=k} u_{n,i}t_{j}^{p^{i}} − w_{n,k}^{p} − v_{n−1}w_{n,k+1}η_{R}(v_{n}^{−1}) mod I_{n−1} + (v_{n−1}^{p}).

□

3.2. Shimomura’s element X_{3,r}. The elements x_{n,r} ∈ v_{n}^{−1}BP_{∗} (n ≥ 1 and r ≥ 0) have been defined in [2, (5.11)] to express a basis of H^{0}M_{n−1}^{1} (x_{3,r} was listed in 2.3.1). As is shown in [2, Proposition 5.17], the differentials d_{0} : v_{n}^{−1}BP_{∗} → v_{n}^{−1}BP_{∗} ⊗_{A} Γ mod (p, v_{1}, v_{2}^{1+a_{3,r}}) are given by

d_{0}(x_{3,0}) ≡ v_{2}t_{1}^{p^{2}}  and  d_{0}(x_{3,r}) ≡ v_{2}^{a_{3,r}}x_{3,r−1}^{p−1}t_{1}^{p^{δ(r)}} for r ≥ 1,

where δ(r) = 0 for odd r and 1 for even r. However, we need to calculate d_{0}(x_{3,r}) mod (p, v_{1}, v_{2}^{t}) with t > 1 + a_{3,r} for our computation. So we use the following elements X_{3,r} defined by Shimomura [8] instead of x_{3,r}.

Definition 3.2.1. cf. [8, (3.3)]] Define elements X_{3,r} ∈ v_{3}^{−1}BP_{∗} by

X_{3,0} = v_{3},  X_{3,1} = X_{3,0}^{p} + v_{2}^{p}v_{3}^{p}u_{3,1},  X_{3,2} = X_{3,1}^{p} − v_{2}^{p^{2}+p}X_{3,1}^{p−1}u_{3,2},
X_{3,3} = X_{3,2}^{p} + v_{2}^{1+a_{3,3}}X_{3,2}^{p−1}u_{2,1} + v_{2}^{b(3)}X_{3,1}^{p^{2}−p−1}u_{3,3},
X_{3,4} = X_{3,3}^{p} − v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1}(u_{2,2} − u_{3,3}),
X_{3,r} = X_{3,r−1}^{p} + v_{2}^{1+a_{3,r}}X_{3,r−1}^{p−1}u_{2,1} − v_{2}^{b(r)}X_{3,r−2}^{p^{2}−p−1}(u_{2,2} − u_{3,3}) for odd r ≥ 5,
X_{3,r} = X_{3,r−1}^{p} − v_{2}^{b(r)}X_{3,r−2}^{p^{2}−p−1}(2 u_{2,2} − u_{3,3}) for even r ≥ 6.

with b(0) = 1, b(1) = p + 1 and b(r) = a_{3,r} + a_{3,r−1} + 1 = (p^{2} + p + 1)p^{r−2} for r ≥ 2.

Remark 3.2.2. By definition, it is obvious that X_{3,r} ≡ x_{3,r} mod (v_{2}^{1+a_{3,r}}). In addition, they satisfy the congruences

X_{3,1} ≡ v_{3}^{p} mod v_{2}^{p}, X_{3,2} ≡ X_{3,1}^{p} mod v_{2}^{p^{2}+p},

X_{3,2i+1} ≡ X_{3,2i}^{p} mod v_{2}^{a_{3,2i+1}−p}, X_{3,2i+2} ≡ X_{3,2i+1}^{p} mod v_{2}^{b(2i+2)−p^{2}−p−1}

for i ≥ 1.

Then d_{0}(X_{3,r}) is computed as follows. Let s be an integer with p ∤ s.

Proposition 3.2.3. Mod (p, v_{1}, v_{2}^{b(r)+1}), d_{0}(X_{3,r}^{s}) may be expressed as follows. For small values of r,

d_{0}(X_{3,r}^{s}) = \begin{cases}sv_{2}v_{3}^{s}w_{3,1} ( ≡ sv_{2}v_{3}^{s−1}t_{1}^{p^{2}}), & for r = 0 \\ sv_{2}^{p}v_{3}^{sp−1}t_{1} − s v_{2}^{p+1}v_{3}^{sp−1}w_{3,2}, & for r = 1 \\ sv_{2}^{a_{3,2}}X_{3,1}^{sp−1}t_{1}^{p} − sv_{2}^{b(2)}v_{3}^{sp^{2}−p−1}(u_{2,0}t_{2} − w_{3,3}), & for r = 2. \end{cases}

For r = 3,

d_{0}(X_{3,r}^{s}) = sv_{2}^{a_{3,3}}X_{3,2}^{sp−1}t_{1} − s v_{2}^{b(3)}X_{3,1}^{sp^{2}−p−1}{w_{2,2} + u_{2,0}t_{2} − w_{3,3} + u_{3,0}(t_{1}^{p^{2}}t_{2} − t_{3})}.

For r ≥ 4 and even,

d_{0}(X_{3,r}^{s}) = sv_{2}^{a_{3,r}}X_{3,r−1}^{sp−1}t_{1}^{p} − sv_{2}^{b(r)}X_{3,r−2}^{sp^{2}−p−1} ( ζ_{2} + u_{2,0}t_{2} − ζ_{3} ) .

For r ≥ 5 and odd,

d_{0}(X_{3,r}^{s}) = sv_{2}^{a_{3,r}}X_{3,r−1}^{sp−1}t_{1} − sv_{2}^{b(r)}X_{3,r−2}^{sp^{2}−p−1}( ζ_{2} + w_{2,2} − ζ_{3} ).

Proof. Originally, this is proved in [8, Proposition 3.1]. Notice that it is sufficient to prove the s = 1 case because d_{0}(X_{3,r}^{s}) ≡ sX_{3,r}^{s−1}d_{0}(X_{3,r}).

For r = 0, d_{0}(X_{3,0}) ≡ v_{2}t_{1}^{p^{2}} mod p, v_{1}, v_{2}^{2} by (2.1.2). For r = 1, Proposition 3.1.6 shows that

d_{0}(v_{2}^{p}v_{3}^{p}u_{3,1}) ≡ v_{2}^{p}v_{3}^{p−1}t_{1} − v_{2}^{p}t_{1}^{p^{2}} − v_{2}^{p+1}v_{3}^{p−1}w_{3,2}

mod (p, v_{1}, v_{2}^{p+2}). Summing this with d_{0}(X_{3,0}^{p}) ≡ v_{2}^{p}t_{1}^{p^{3}} gives d_{0}(X_{3,1}). For r = 2, we have

d_{0}(−v_{2}^{p^{2}+p}v_{3}^{p^{2}−p}u_{3,2}) ≡ −v_{2}^{p^{2}+p}v_{3}^{p^{2}−p}(u_{3,1}t_{1}^{p} + u_{3,0}t_{2} − w_{3,2}^{p} − v_{2}v_{3}^{−1}w_{3,3})

mod (p, v_{1}, v_{2}^{p^{2}+p+2}). On the other hand, using the congruence v_{3}^{p^{2}−p} ≡ X_{3,1}^{p−1} + v_{2}^{p}v_{3}^{p^{2}−p}u_{3,1} we have

d_{0}(X_{3,1}^{p}) ≡ v_{2}^{p^{2}}X_{3,1}^{p−1}t_{1}^{p} + v_{2}^{p^{2}+p}v_{3}^{p^{2}−p}(u_{3,1}t_{1}^{p} − w_{3,2}^{p}).

Again, summing the terms gives d_{0}(X_{3,2}).

For r = 3, we have

d_{0}(v_{2}^{p^{3}+p}X_{3,2}^{p−1}u_{2,1}) ≡ v_{2}^{1+a_{3,3}}X_{3,2}^{p−1}(u_{2,0}t_{1} − w_{2,1}^{p})
− v_{2}^{b(3)}X_{3,1}^{p^{2}−p−1}(e_{2}(u_{2,1})t_{1}^{p} + u_{2,0}t_{2} + u_{3,0}t_{1}^{p^{2}}t_{2} − w_{3,3}),
d_{0}(v_{2}^{b(3)}X_{3,1}^{p^{2}−p−1}u_{3,3}) ≡ v_{2}^{b(3)}X_{3,1}^{p^{2}−p−1}(∑_{i=0}^{2} u_{3,i}t_{3−i}^{p^{i}} − w_{3,3}^{p}).

mod (p, v_{1}, v_{2}^{b(3)+1}). On the other hand, by the congruences X_{3,1}^{p^{2}−p} ≡ X_{3,2}^{p−1} − v_{2}^{p^{2}+p}X_{3,1}^{p^{2}−p−1}u_{3,2} and v_{3}^{p^{3}−p^{2}−p} ≡ X_{3,1}^{p^{2}−p−1} + v_{2}^{p}X_{3,1}^{p^{2}−p−1}u_{3,1}, we have

d_{0}(X_{3,2}^{p}) ≡ v_{2}^{p^{3}}X_{3,2}^{p−1}t_{1}^{p^{2}} − v_{2}^{b(3)}X_{3,1}^{p^{2}−p−1}(u_{2,0}^{p}t_{2}^{p} − w_{3,3}^{p} + ∑_{i=1}^{2} u_{3,i}t_{3−i}^{p^{i}}).

Using a congruence for w_{2,2} of Remark 3.1.5, we obtain d_{0}(X_{3,3}).

For r = 4, we have

d_{0}(−v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1}u_{2,2}) ≡

− v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1}(−ζ_{2}^{p} − u_{2,0}^{p}t_{2}^{p} + 2ζ_{2} − w_{3,3} − ∑_{1≤i<j≤3} u_{3,3−j}^{p^{j−i−1}}(t_{i} · c(t_{j−i}^{p^{i}}))^{p^{2−i}})

and d_{0}(v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1}u_{3,3}) ≡ v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1}(ζ_{3} − ζ_{3}^{p}) mod (p, v_{1}, v_{2}^{b(4)+1}). On the other hand, by the congruences X_{3,2}^{p^{2}−p} ≡ X_{3,3}^{p−1}+v_{2}^{p^{3}+p}X_{3,2}^{p^{2}−p−1}u_{2,1} and X_{3,1}^{p^{3}−p^{2}−p} ≡ X_{3,2}^{p^{2}−p−1} − v_{2}^{p^{2}+p}X_{3,1}^{p^{3}−p^{2}−p−1}u_{3,2}, d_{0}(X_{3,3}^{p}) is congruent to

v_{2}^{a_{3,4}}X_{3,3}^{p−1}t_{1}^{p}−v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1}{−u_{2,1}t_{1}^{p}+u_{3,2}t_{1}^{p^{2}} +w_{2,2}^{p}+u_{2,0}^{p}t_{2}^{p}−w_{3,3}^{p}+u_{3,0}^{p}(t_{1}^{p^{3}}t_{2}^{p}−t_{3}^{p})} mod (p, v_{1}, v_{2}^{b(4)+1}). Moreover, noticing the congruences

−w_{3,3}^{p} ≡ ζ_{3} − ζ_{3}^{p} − ∑_{i=0}^{2} u_{3,i}t_{3−i}^{p^{i}},
−(u_{3,1}t_{2}^{p} + u_{3,0}t_{3}) + u_{3,0}^{p}t_{1}^{p^{3}}t_{2}^{p} ≡ − ∑_{1≤i<j≤3} u_{3,3−j}^{p^{j−i−1}}[t_{i+1} · c(t_{j−i−1}^{p^{i+1}})]^{p^{2−i}},
−u_{3,0}^{p}t_{3}^{p} ≡ − ∑_{1≤i<j≤3} u_{3,3−j}^{p^{j−i−1}}[t_{i+2} · c(t_{j−i−2}^{p^{i+2}})]^{p^{2−i}},

we have

d_{0}(X_{3,3}^{p}) ≡ v_{2}^{a_{3,4}}X_{3,3}^{p−1}t_{1}^{p} − v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1} (−u_{2,1}t_{1}^{p} + w_{2,2}^{p} + u_{2,0}^{p}t_{2}^{p})

− v_{2}^{b(4)}X_{3,2}^{p^{2}−p−1}(ζ_{3} − ζ_{3}^{p} − ∑_{1≤i<j≤3} u_{3,3−j}^{p^{j−i−1}}[ ∑_{i+1≤k≤j} t_{k} · c(t_{j−k}^{p^{k}})]^{p^{2−i}}).

We obtain d_{0}(X_{3,4}) in the desired expression by summing the terms and using the congruences

\begin{aligned}−u_{2,1}t_{1}^{p} + w_{2,2}^{p} − ζ_{2}^{p} & ≡ & −ζ_{2} + u_{2,0}t_{2}, \\ ∑_{1≤i<j≤3} u_{3,3−j}^{p^{j−i−1}}( ∑_{i≤k≤j} t_{k} · c(t_{j−k}^{p^{k}})^{p^{2−i}}) & ≡ & ζ_{3} − w_{3,3}. \end{aligned}

The r ≥ 5 cases are proved by induction on r. By Proposition 3.1.3, we have congruences

\begin{aligned}d_{0}(v_{2}^{b(r)}X_{3,r−2}^{p^{2}−p−1}u_{2,2}) & ≡ & v_{2}^{b(r)}X_{3,r−2}^{p^{2}−p−1}(ζ_{2} − ζ_{2}^{p}), \\ d_{0}(v_{2}^{b(r)}X_{3,r−2}^{p^{2}−p−1}u_{3,3}) & ≡ & v_{2}^{b(r)}X_{3,r−2}^{p^{2}−p−1}(ζ_{3} − ζ_{3}^{p}). \end{aligned}

We also have the congruences X_{3,r−2}^{(p−1)p} ≡ X_{3,r−1}^{p−1} and

d_{0}(v_{2}^{1+a_{3,r}}X_{3,r−1}^{p−1}u_{2,1}) ≡ v_{2}^{1+a_{3,r}}X_{3,r−1}^{p−1}(u_{2,0}t_{1} − w_{2,1}^{p}) − v_{2}^{b(r)}X_{3,r−2}^{p^{2}−p−1}t_{1} · e_{2}(u_{2,1})

for odd r, and X_{3,r−2}^{(p−1)p} ≡ X_{3,r−1}^{p−1}+v_{2}^{1+a_{3,r−1}}X_{3,r−2}^{p^{2}−p−1}u_{2,1} for even r. Easy calculation shows the desired results for r ≥ 5. □

Moreover, we can obtain the following result in the same way as the proof of Proposition 3.2.3.

Proposition 3.2.4. Mod (p, v_{1}^{2}, v_{2}^{b(r)−1}), d_{0}(X_{3,r}^{s}) is expressed as follows. For small values of r, we get

sv_{2}^{a_{3,2}}X_{3,1}^{sp−1}t_{1}^{p}, r = 2;

sv_{2}^{a_{3,3}}X_{3,2}^{sp−1}(t_{1} − v_{1}w_{2,2}) − sv_{2}^{b(3)−1}X_{3,1}^{sp^{2}−p−1} {v_{2}ζ_{2} − v_{1}t_{1}^{p}(w_{2,2} + v_{2}^{−1}t_{2})} , r = 3;

sv_{2}^{a_{3,4}}X_{3,3}^{sp−1}t_{1}^{p} − sv_{2}^{b(4)−1}X_{3,2}^{sp^{2}−p−1} {v_{2}ζ_{2} − v_{1}(w_{2,3} + t_{1}^{p}ζ_{2})} , r = 4.

For r ≥ 5 and odd, we get

sv_{2}^{a_{3,r}}X_{3,r−1}^{sp−1}(t_{1} − v_{1}w_{2,2}) − sv_{2}^{b(r)−1}X_{3,r−2}^{sp^{2}−p−1} {2 v_{2}ζ_{2} − v_{1}(w_{2,3} + t_{1}^{p}w_{2,2})} .

For r ≥ 6 and even, we get

s v_{2}^{a_{3,r}}X_{3,r−1}^{sp−1}t_{1}^{p} − s v_{2}^{b(r)−1}X_{3,r−2}^{sp^{2}−p−1} {v_{2}ζ_{2} − 2 v_{1}w_{2,3}} .

3.3. The element 1/X_{2,r}^{m}. Here we modify x_{2,r} into X_{2,r} in the analogous way of modifying x_{3,r} into X_{3,r}, and introduce new elements 1/X_{2,r}^{m}.

Definition 3.3.1. Define integers a_{2,r} by a_{2,0} = 1 and a_{2,r} = p^{r} + p^{r−1} − 1 for r ≥ 1, and elements X_{2,r} ∈ v_{2}^{−1}BP_{∗} by

\begin{matrix}X_{2,0} = v_{2}, X_{2,1} = X_{2,0}^{p} + v_{1}^{p}v_{2}^{p}u_{2,1}, X_{2,2} = X_{2,1}^{p} + v_{1}^{1+a_{2,2}}X_{2,1}^{p−1}(u_{1,1} − u_{2,2}), \\ X_{2,r} = X_{2,r−1}^{p} + v_{1}^{1+a_{2,r}}X_{2,r−1}^{p−1}(2 u_{1,1} − u_{2,2}) for r ≥ 3. \end{matrix}

Then we have:

Proposition 3.3.2. Mod (p, v_{1}^{2+a_{2,r}}),

d_{0}(X_{2,r}^{m}) ≡ \begin{cases}mv_{1}v_{2}^{m−1}t_{1}^{p} + \binom{m}{2}v_{1}^{2}v_{2}^{m−2}t_{1}^{2p} & for r = 0 \\ mv_{1}^{p}v_{2}^{mp−1}(t_{1} − v_{1}w_{2,2}) & for r = 1 \\ v_{1}^{a_{2,r}}v_{2}^{(mp−1)p^{r−1}}(2 t_{1} − v_{1}ζ_{2}) & for r ≥ 2. \end{cases}

Proof. It is sufficient to prove the s = 1 case because d_{0}(X_{2,r}^{m}) ≡ mv_{2}^{(m−1)p^{r}}d_{0}(X_{2,r}). The r = 0 case is given by (2.1.1). For r = 1, we have

d_{0}(v_{1}^{p}v_{2}^{p}u_{2,1}) ≡ v_{1}^{p}{ d_{0}(u_{2,1})v_{2}^{p} + η_{R}(u_{2,1})d_{0}(v_{2}^{p}) }
≡ v_{1}^{p}v_{2}^{p}d_{0}(u_{2,1})
≡ v_{1}^{p}(v_{2}^{p−1}t_{1} − t_{1}^{p^{2}}) − v_{1}^{p+1}v_{2}^{p−1}w_{2,2}

mod p, v_{1}^{p+2}. Summing this with d_{0}(X_{2,0}^{p}) gives d_{0}(X_{2,1}) ≡ v_{1}^{p}v_{2}^{p−1}(t_{1} − v_{1}w_{2,2}). (See also [2, Proposition 5.4].) For r = 2, we have

d_{0}(v_{1}^{1+a_{2,r}}X_{2,1}^{p−1}(u_{1,1} − u_{2,2})) ≡ −v_{1}^{p^{2}}X_{2,1}^{p−1}{ (2v_{1}^{p−1}t_{1} − t_{1}^{p}) + v_{1}^{p}(w_{2,2}^{p} − ζ_{2}) }

mod p, v_{1}^{2+a_{2,2}}. Again, summing this with d_{0}(X_{2,1}^{p}) gives

d_{0}(X_{2,2}) ≡ v_{1}^{a_{2,2}}v_{2}^{p^{2}−p}(2t_{1} − v_{1}ζ_{2}).

For r ≥ 3, we have

d_{0}(v_{1}^{1+a_{2,r}}X_{2,r−1}^{p−1}(2u_{1,1} − u_{2,2})) ≡ v_{1}^{a_{2,r}−p+1}X_{2,r−1}^{p−1}{ 2(v_{1}^{p−1}t_{1} − t_{1}^{p}) + v_{1}^{p}(ζ_{2}^{p} − ζ_{2}) }

mod p, v_{1}^{2+a_{2,r}}. This gives d_{0}(X_{2,r}) inductively on r. □

Shimomura [7] has replaced 1/v_{1}^{mp} with 1/x_{1,1}^{m} in computing H^{0}M_{0}^{2} for p = 2. Analogously, we construct elements 1/X_{2,r}^{m} as a substitute for 1/v_{2}^{mp^{r}}. Notice that our 1/X_{2,r}^{m} should be different from X_{2,r}^{−m} because there is no inverse element of X_{2,r} (r ≥ 1) in v_{2}^{−1}BP_{∗}.

Definition 3.3.3. Define elements 1/X_{2,r}^{m} ∈ v_{2}^{−1}BP_{∗} (r ≥ 0 and p ∤ m ≥ 1) by

1/X_{2,0}^{m} = 1/v_{2}^{m} and 1/X_{2,r}^{m} = 2 (1/X_{2,r−1}^{mp}) − X_{2,r}^{m} (1/X_{2,r−1}^{2mp}) for r ≥ 1.

For example, easy calculation shows the congruence

1/X_{2,1}^{m} ≡ 1/v_{2}^{mp} + m v_{1}^{p}v_{3}/v_{2}^{(m+1)p+1} mod (v_{1}^{2p}).

Although our 1/X_{2,r}^{m} is not equal to (X_{2,r}^{m})^{−1} = ∑_{k=0}^{∞}(1 − X_{2,r}^{m})^{k}, the above definition is justified by the congruences (X_{2,1}^{m} −v_{2}^{mp})^{2} ≡ 0 mod (p, v_{1}^{2p}) and (X_{2,r}^{m} − X_{2,r−1}^{mp})^{2} ≡ 0 mod (p, v_{1}^{2(a_{2,r}−p)}) for r ≥ 2. Hereafter we denote Y (1/X_{2,r}^{m}) by Y/X_{2,r}^{m} and (1/X_{2,r}^{m})/v_{1}^{l} by 1/v_{1}^{l}X_{2,r}^{m} for simplicity.

Proposition 3.3.4. Mod (p, v_{1}^{2+a_{2,r}}),

d_{0}(1/X_{2,r}^{m}) ≡ \begin{cases}− m v_{1}t_{1}^{p}/v_{2}^{m+1} + \binom{m+1}{2}v_{1}^{2}t_{1}^{2p}/v_{2}^{m+2} & for r = 0 \\ − m v_{1}^{p}(t_{1} − v_{1}w_{2,2})/v_{2}^{mp+1} & for r = 1 \\ − m v_{1}^{a_{2,r}}(2 t_{1} − v_{1}ζ_{2})/v_{2}^{(mp+1)p^{r−1}} & for r ≥ 2. \end{cases}

Proof. We prove that the all above differentials have the form − d_{0}(X_{2,r}^{m})/v_{2}^{2mp^{r}}.

The case r = 0 is easy. For r ≥ 1, it is shown by induction on r. In fact, we have

d_{0}(1/X_{2,r}^{m}) = 2 d_{0}(1/X_{2,r−1}^{mp}) − d_{0}(X_{2,r}^{m})/X_{2,r−1}^{2mp} − η_{R}(X_{2,r}^{m})d_{0}(1/X_{2,r−1}^{2mp})
≡ 2 d_{0}(1/X_{2,r−1}^{m})^{p} − d_{0}(X_{2,r}^{m})/v_{2}^{2mp^{r}} − v_{2}^{mp^{r}}d_{0}(1/X_{2,r−1}^{2m})^{p}
≡ −2 d_{0}(X_{2,r−1}^{m})^{p}/v_{2}^{2mp^{r}} − d_{0}(X_{2,r}^{m})/v_{2}^{2mp^{r}} + d_{0}(X_{2,r−1}^{2m})^{p}/v_{2}^{3mp^{r}}

≡ −d_{0}(X_{2,r}^{m})/v_{2}^{2mp^{r}} + { d_{0}(X_{2,r−1}^{2m}) − 2 v_{2}^{mp^{r−1}}d_{0}(X_{2,r−1}^{m}) }^{p}/v_{2}^{3mp^{r}}

mod p, v_{1}^{2+a_{2,r}}. By the congruence d_{0}(X_{2,r}^{m}) ≡ mv_{2}^{(m−1)p^{r}}d_{0}(X_{2,r}), the second term is trivial. □

Then we directly obtain the v_{1}-divisibilities of 1/v_{1}X_{2,r}^{m} (= 1/v_{1}v_{2}^{mp^{r}}) in H^{0}M_{1}^{2}.

Corollary 3.3.5. 1/v_{1}X_{2,r}^{m} can be divided by v_{1}^{a_{2,r}−1} in H^{0}M_{1}^{2}, and the image of 1/v_{1}^{a_{2,r}}X_{2,r}^{m} under the coboundary map δ : H^{0}M_{1}^{2} → H^{1}M_{2}^{1} is

δ(1/v_{1}^{a_{2,r}}X_{2,r}^{m}) = \begin{cases}− m h_{1}/v_{2}^{m+1} & for r = 0 \\ − m h_{0}/v_{2}^{mp+1} & for r = 1 \\ − 2m h_{0}/v_{2}^{(mp+1)p^{r−1}} & for r ≥ 2. \end{cases}

□

This corollary asserts that N(m; r) = a_{2,r} when we choose 1/X_{2,r}^{m} to be y(mp^{r}) in Proposition 2.4.1.

Using 2.5, we easily see that { δ(1/v_{1}^{a_{2,r}}X_{2,r}^{m}) } is linearly independent.

4. Preliminary calculations

In Section 5 we shall construct elements x(sp^{r}/j; k) ∈ v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}) and compute the differentials on them. For the sake of this, we define some elements and compute differentials in Subsection 4.1. Based on these results, we describe some inductive methods of constructing the elements x(sp^{r}/j; k) in Subsection 4.2. This is the hardest part in this paper.

Hereafter we assume that p > 3, and that all elements are either in v_{3}^{−1}BP_{∗}/(p) or in v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}). We shall compute the differential d_{0} for positive integers e_{1} and e_{2} in two ways:

d_{0} : v_{3}^{−1}BP_{∗}/(p) ⟶ v_{3}^{−1}BP_{∗}/(p) ⊗_{A} Γ  mod v_{1}^{e_{1}}, v_{2}^{e_{2}},
d_{0} : v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}) ⟶ v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}) ⊗_{A} Γ mod v_{1}^{e_{1}}.

4.1. Some lemmas.

Lemma 4.1.1. Assume that an element X satisfies

d_{0}(X) ≡ v_{1}^{e_{1}}X_{3,r}^{s}t_{1}^{p}/v_{2}^{e_{2}+1} mod v_{1}^{e_{1}+1},

where r ≥ 1, e_{1} ≥ 1 and 1 ≤ e_{2} ≤ a_{3,r} − 2. Then Y = X^{p} − v_{1}^{pe_{1}}v_{3}X_{3,r}^{sp}/v_{2}^{(e_{2}+1)p+1} satisfies

d_{0}(Y) ≡ v_{1}^{pe_{1}}X_{3,r}^{sp}(t_{1} − v_{1}w_{2,2})/v_{2}^{pe_{2}+1} mod v_{1}^{pe_{1}+2}.

The Chromatic E_{1}-term H^{0}M_{1}^{2} for p > 3 35

Proof. We observe that

d_{0}(v_{3}X_{3,r}^{sp}/v_{2}^{(e_{2}+1)p+1}) = d_{0}(1/v_{2}^{(e_{2}+1)p+1}) v_{3}X_{3,r}^{sp} + η_{R}(1/v_{2}^{(e_{2}+1)p+1}) d_{0}(v_{3}X_{3,r}^{sp})
≡ − v_{1}v_{3}X_{3,r}^{sp}t_{1}^{p}/v_{2}^{(e_{2}+1)p+2}
+ (1/v_{2}^{(e_{2}+1)p+1} − v_{1}t_{1}^{p}/v_{2}^{(e_{2}+1)p+2}) d_{0}(v_{3}X_{3,r}^{sp})

mod p, v_{1}^{2}. Since (e_{2} + 1)p + 2 < pa_{3,r}, it is sufficient to compute d_{0}(v_{3}X_{3,r}^{sp}) mod v_{1}^{2}, v_{2}^{pa_{3,r}}. Using (2.1.2) and Proposition 3.2.3, we have

d_{0}(v_{3}X_{3,r}^{sp}) = d_{0}(v_{3}) X_{3,r}^{sp} + η_{R}(v_{3}) d_{0}(X_{3,r}^{sp}) ≡ d_{0}(v_{3}) X_{3,r}^{sp}
≡ (v_{2}t_{1}^{p^{2}} − v_{2}^{p}t_{1} + v_{1}t_{2}^{p})X_{3,r}^{sp}.

Observing the congruence w_{2,2} ≡ − v_{3}t_{1}^{p}/v_{2}^{p+1} + (t_{2}^{p} − t_{1}^{p^{2}+p})/v_{2}^{p} + t_{1}^{p+1}/v_{2} (Remark 3.1.5), we obtain

d_{0}(v_{3}X_{3,r}^{sp}/v_{2}^{(e_{2}+1)p+1}) ≡ X_{3,r}^{sp}t_{1}^{p^{2}}/v_{2}^{(e_{2}+1)p} − X_{3,r}^{sp}t_{1}/v_{2}^{pe_{2}+1} + v_{1}X_{3,r}^{sp}w_{2,2}/v_{2}^{pe_{2}+1}.

Now the result follows easy calculation. □

Given a positive integer j = cp − b ≤ a_{3,r} (0 ≤ b < p), it is convenient to work with the element v_{2}^{b}X_{3,r−1}^{sp}/X_{2,1}^{c} instead of X_{3,r}^{s}/v_{2}^{j}. Define the integer a by a ≡ c (p) with 0 ≤ a < p. Lemma 4.1.2. Assume that c ≤ p for r = 2, that c ≤ a_{3,r−1} − 2 for odd r ≥ 3, or that c ≤ a_{3,r−1} − p − 1 for even r ≥ 4. Then d_{0}(v_{2}^{b}X_{3,r−1}^{sp}/X_{2,1}^{c}) is expressed as

\frac{X_{3,r−1}^{sp}}{v_{2}^{j+b}} \{∑_{i=1}^{b}E(b, i)\}

− \frac{v_{1}^{p}X_{3,r−1}^{sp}}{v_{2}^{j+1}} \{(a + b) t_{1} − v_{1}[a w_{2,2} + ab \frac{v_{3}t_{1}^{p}}{v_{2}^{p+1}} − b(a + b − 1) \frac{t_{1}^{p+1}}{v_{2}} ]\}

mod v_{1}^{p+2}, where E(m, n) = \binom{m}{n}v_{2}^{m−n}(v_{1}t_{1}^{p})^{n}. In particular,

d_{0}(X_{3,r−1}^{sp}/X_{2,1}^{c}) ≡ − a v_{1}^{p}X_{3,r−1}^{sp}(t_{1} − v_{1}w_{2,2})/v_{2}^{j+1} mod (v_{1}^{p+2}). Proof. Notice that

d_{0}(v_{2}^{b}X_{3,r−1}^{sp}/X_{2,1}^{c}) = d_{0}(v_{2}^{b}) X_{3,r−1}^{sp}/X_{2,1}^{c} + η_{R}(v_{2}^{b}) d_{0}(X_{3,r−1}^{sp}/X_{2,1}^{c}), and that η_{R}(v_{2}^{b}) is given in (2.1.1). By the assumption on c, we have

d_{0}(X_{3,r−1}^{s}/v_{2}^{c}) ≡ − a v_{1}X_{3,r−1}^{s}t_{1}^{p}/v_{2}^{c+1} mod (v_{1}^{2}).

Then, Lemma 4.1.1 gives d_{0}(X_{3,r−1}^{sp}/X_{2,1}^{c}) mod (v_{1}^{p+2}). □

Similarly to the r ≥ 2 case, we may consider the element X_{3,1}^{s}/X_{2,1} instead of X_{3,1}^{s}/v_{2}^{p} for r = 1. Lemma 4.1.3.

d_{0} (X_{3,1}^{s}/X_{2,1}) ≡ (s − 1) v_{1}^{p}X_{3,1}^{s}t_{1}/v_{2}^{p+1} + \binom{s}{2}v_{1}^{p}v_{3}^{sp−1}t_{1}^{2}/v_{2} mod (v_{1}^{p+1}).

Proof. Notice that d_{0}(X_{3,1}^{s}/X_{2,1}) = d_{0} (1/X_{2,1}) X_{3,1}^{s} + η_{R} (1/X_{2,1}) d_{0}(X_{3,1}^{s}), and that d_{0} (1/X_{2,1}) X_{3,1}^{s} ≡ − v_{1}^{p}X_{3,1}^{s}t_{1}/v_{2}^{p+1} mod (v_{1}^{p+1}) by Proposition 3.3.4. On the other hand, we have

η_{R} (1/X_{2,1}) ≡ 1/v_{2}^{p} + v_{1}^{p}(v_{3} − v_{2}^{p}t_{1})/v_{2}^{2p+1}  mod v_{1}^{p+1},
d_{0}(X_{3,1}^{s}) ≡ s v_{1}^{p}X_{3,1}^{s−1} {t_{2}^{p^{2}} − t_{1}^{p^{2}}η_{R}(v_{3}^{−1}v_{4})}  mod v_{1}^{p+1}, v_{2}^{p},
≡ s v_{2}^{p}X_{3,1}^{s−1} \{t_{1}^{p^{3}} − d_{0}(v_{3}^{−1}v_{4})\} + \binom{s}{2}v_{2}^{2p}v_{3}^{sp−2}t_{1}^{2}  mod v_{1}, v_{2}^{2p+1}.

Using these congruences, (2.1.2) and (2.1.3), we obtain

η_{R} (1/X_{2,1}) d_{0} (X_{3,1}^{s}) ≡ s v_{1}^{p}X_{3,1}^{s}t_{1}/v_{2}^{p+1} + \binom{s}{2}v_{1}^{p}v_{3}^{sp−1}t_{1}^{2}/v_{2}  mod v_{1}^{p+1}.

Collecting terms gives the result. □

For an integer n with p | n + 1, we denote the integer (n + 1)/p by n′, and the set of integers {s ∈ ℤ | s = s′p − 1 with p ∤ s′} by ℕ_{0}. Then we introduce two elements X_{0}(v_{2}^{j}, X_{3,r}^{s}) and X(v_{2}^{j}, X_{3,r}^{s}) for s ∈ ℕ_{0}, each of which is congruent to X_{3,r}^{s}/v_{2}^{j} mod (v_{1}).

Definition 4.1.4. When s = s′p − 1 ∈ ℕ_{0}, we define X_{0}(v_{2}^{j}, X_{3,r}^{s}) by

X_{0}(v_{2}^{j}, X_{3,r}^{s}) = X_{3,r}^{s}/v_{2}^{j} + j v_{1}X_{3,r+1}^{s′}/s′v_{2}^{j+1+a_{3,r+1}}.

Then we have: Lemma 4.1.5. For odd r and p ∤ j ≤ a_{3,r}, d_{0}(X_{0}(v_{2}^{j}, X_{3,r}^{s})) is expressed as

(1 − j) v_{1}v_{2}^{a_{3,r}−j}v_{3}^{(sp−1)p^{r−1}}ζ_{2} mod v_{1}^{2}

for r ≥ 3 and a_{3,r} − p ≤ j ≤ a_{3,r}, and is expressed as

− j(j + 1) v_{1}^{2}X_{3,r+1}^{s′}t_{1}^{p}/s′v_{2}^{j+2+a_{3,r+1}} − \binom{j + 1}{2}v_{1}^{2}X_{3,r}^{s}t_{1}^{2p}/v_{2}^{j+2} + ε(r, j) mod v_{1}^{3}

for r = 1 and j ≤ p − 2; or for r ≥ 3 and either j = a_{3,r} − p + 2 or j ≤ a_{3,r} − p − 1, where ε(1, j) = 0 and ε(r, j) = v_{1}^{2}v_{2}^{a_{3,r}−j−1}X_{3,r−1}^{sp−1} { a(r)j w_{2,3} + b(r, j) t_{1}^{p}ζ_{2} } for r ≥ 3 with integers a(3) = 1, a(r) = 2 for r ≥ 5 and b(r, j) = j^{2} + (2 − a(r)) j − 1. Proof. Here we prove only for r ≥ 3 case (the r = 1 case is easier). We compute the differential on each term of X_{0}(v_{2}^{j}, X_{3,r}^{s}). For the first term, notice that d_{0}(X_{3,r}^{s}/v_{2}^{j}) = d_{0}(1/v_{2}^{j})X_{3,r}^{s} + η_{R}(1/v_{2}^{j})d_{0}(X_{3,r}^{s}). Easy calculation shows that

d_{0}(X_{3,r}^{s}) ≡ −v_{2}^{a_{3,r}}X_{3,r−1}^{sp−1}(t_{1} − v_{1}w_{2,2} + v_{1}^{2}v_{2}^{−1}t_{1}^{p}w_{2,2}) mod v_{1}^{3}, v_{2}^{2+a_{3,r}}.

Using this and Proposition 3.3.4, we see that d_{0}(X_{3,r}^{s}/v_{2}^{j}) is congruent to

−j v_{1}X_{3,r}^{s}t_{1}^{p} {2 v_{2} − (j + 1) v_{1}t_{1}^{p}} /2v_{2}^{j+2} + v_{1}v_{2}^{a_{3,r}−j−1}X_{3,r−1}^{sp−1}ζ_{2} {v_{2} − (j + 1) v_{1}t_{1}^{p}} mod (v_{1}^{3}). On the other hand, using Propositions 3.2.4 and 3.3.4, we observe that d_{0}(j v_{1}X_{3,r+1}^{s′}/s′v_{2}^{j+1+a_{3,r+1}}) is congruent to

− j(j + 1)v_{1}^{2}X_{3,r+1}^{s′}t_{1}^{p}/s′v_{2}^{j+2+a_{3,r+1}} + j v_{1}X_{3,r}^{s}t_{1}^{p} { v_{2} − (j + 1) v_{1}t_{1}^{p} } /v_{2}^{j+2}

+ v_{1}v_{2}^{a_{3,r}−j−1}X_{3,r−1}^{sp−1} {−j v_{2}ζ_{2} + a(r)j v_{1}w_{2,3} + j( j + 3 − a(r) ) v_{1}t_{1}^{p}ζ_{2}}

mod (v_{1}^{3}). Collecting two terms gives the result for r ≥ 3. □

Definition 4.1.6. For s = s′p − 1 ∈ ℕ_{0} and j ≤ a_{3,r}, we define X(v_{2}^{j}, X_{3,r}^{s}) by

X(v_{2}^{j}, X_{3,r}^{s}) = v_{2}^{b}X_{3,r−1}^{sp}/X_{2,1}^{c} − b v_{1}v_{2}^{b−1}X_{3,r}^{s′p}/s′X_{2,1}^{c+a_{3,r}} + b v_{1}^{p+1}X_{3,r−1}^{sp}u_{2,2}/v_{2}^{j+1}
− (a(r) − 1)b v_{1}^{p+1}v_{2}^{a_{3,r}−j−p}X_{3,r−2}^{(sp−1)p}u_{2,3},

where j = cp − b and 0 ≤ b < p. Moreover, we define DX_{[m,n]}(v_{1}^{e_{1}}, v_{2}^{e_{2}+1}, X_{3,r}^{s}) ∈ v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}) ⊗_{A} Γ for 0 ≤ m < p, 0 ≤ n < p, s = s′p − 1 ∈ ℕ_{0}, and positive integers e_{1} and e_{2} by

DX_{[m,n]}(v_{1}^{e_{1}}, v_{2}^{e_{2}+1}, X_{3,r}^{s}) = −(m + n) v_{1}^{e_{1}−1}X_{3,r}^{s}t_{1}/v_{2}^{e_{2}+1}
+ n(m + n − 2)v_{1}^{e_{1}}X_{3,r+1}^{s′}t_{1}/s′v_{2}^{e_{2}+2+a_{3,r+1}}
+ v_{1}^{e_{1}}X_{3,r}^{s}/v_{2}^{e_{2}+1} · {nζ_{2} + (m − n)w_{2,2}}

Then we obtain the following lemma.

Lemma 4.1.7. Assume that r ≥ 3 is odd and c ≤ a_{3,r−1}−2. Then d_{0}(X(v_{2}^{j}, X_{3,r}^{s})) is expressed as

DX_{[a,b]}(v_{1}^{p+1},v_{2}^{j+1}, X_{3,r}^{s}) + A(v_{2}^{j}, X_{3,r}^{s})

+ a(r)b v_{1}v_{2}^{a_{3,r}−j−p−1}v_{3}^{(sp−1)p^{r−1}}(v_{2}^{p+2−b}ζ_{2}^{p} · η_{R}(v_{2}^{b−1}) + v_{1}^{p}θ_{1}(v_{2}^{j}))

mod v_{1}^{p+2}. Here b and c are the integers in 4.1.6, a ≡ c (p) with 0 ≤ a < p, and the element A(v_{2}^{j}, X_{3,r}^{s}) is defined to be 0 for 0 ≤ b ≤ 1 and to be

\frac{X_{3,r−1}^{sp}}{v_{2}^{j+b}} \{∑_{i=2}^{b}(1 − i)E(b, i)\} − b\frac{v_{1}X_{3,r}^{s′p}}{s′v_{2}^{j+b+a_{3,r+1}}} \{∑_{i=1}^{b−1} E(b − 1, i)\}

for b ≥ 2. Here E(m, n) is as in Lemma 4.1.2 and

θ_{1}(v_{2}^{j}) = (a − 1)(v_{3} − v_{2}^{p}t_{1})ζ_{2}^{p} + {(a + b − 2) v_{2}^{p}t_{1} − v_{2}t_{1}^{p^{2}}}u_{2,2}.

In particular, v_{2}^{p^{2}−1}θ_{1}(v_{2}^{a_{3,r}−p^{2}−p}) = − v_{3}^{p+1}(t_{1} + v_{2}w_{3,2})/v_{2}^{2}.

Proof. The differential on the first term of X(v_{2}^{j}, X_{3,r}^{s}) has already been computed in Lemma 4.1.2. For the second term, notice that −bv_{1}v_{2}^{b−1}X_{3,r}^{s′p}/s′X_{2,1}^{c+a_{3,r}} is congruent to

−bv_{1}X_{3,r}^{s′p}/s′v_{2}^{j+1+a_{3,r+1}} − b(a − 1)v_{1}^{p+1}v_{3}X_{3,r}^{s′p}/s′v_{2}^{j+p+2+a_{3,r+1}}

mod (v_{1}^{p+2}). Using Propositions 3.2.3, 3.2.4 and 3.3.4, we obtain

d_{0}(−bv_{1}v_{2}^{b−1}X_{3,r}^{s′p}/s′X_{2,1}^{c+a_{3,r}})
≡ −bv_{1}X_{3,r}^{s′p}/s′v_{2}^{j+b+a_{3,r+1}}{(v_{2} + v_{1}t_{1}^{p})^{b−1} − v_{2}^{b−1}}
− bv_{1}X_{3,r−1}^{sp}t_{1}^{p}(v_{2} + v_{1}t_{1}^{p})^{b−1}/v_{2}^{j+b}
+ a(r)bv_{1}v_{2}^{a_{3,r}−j−b+1}X_{3,r−2}^{(sp−1)p}ζ_{2}^{p} · η_{R}(v_{2}^{b−1})
+ (a + b − 2)bv_{1}^{p+1}X_{3,r}^{s′p}t_{1}/s′v_{2}^{j+2+a_{3,r+1}}
+ bv_{1}^{p+1}X_{3,r−1}^{sp}/v_{2}^{j+1}

· {w_{2,2}^{p} + (b − 1)t_{1}^{p+1}/v_{2} + (a − 1)(t_{1}^{p^{2}+p} − t_{2}^{p})/v_{2}^{p} + (a − 1)w_{2,2}}

+ bv_{1}^{p+1}v_{2}^{a_{3,r}−j−p−1}X_{3,r−2}^{(sp−1)p}
· {−v_{2}t_{1}^{p^{2}}ζ_{2}^{p} + a(r)(a − 1)ζ_{2}^{p}(v_{3} − v_{2}^{p}t_{1}) − ( a(r) − 1 )v_{2}w_{2,3}^{p}}

mod (v_{1}^{p+2}). For the third term, notice that

d_{0}(bv_{1}^{p+1}X_{3,r−1}^{sp}u_{2,2}/v_{2}^{j+1}) ≡ bv_{1}^{p+1}d_{0}(X_{3,r−1}^{sp}u_{2,2})/v_{2}^{j+1}

mod (v_{1}^{p+2}). Propositions 3.1.3 and 3.2.3 give d_{0}(X_{3,r−1}^{sp}u_{2,2}), and we obtain

d_{0}(bv_{1}^{p+1}X_{3,r−1}^{sp}u_{2,2}/v_{2}^{j+1})

≡ bv_{1}^{p+1}X_{3,r−1}^{sp}(ζ_{2} − ζ_{2}^{p})/v_{2}^{j+1} − bv_{1}^{p+1}v_{2}^{a_{3,r}−j−p}X_{3,r−2}^{(sp−1)p}t_{1}^{p^{2}}(u_{2,2} − ζ_{2}^{p})

mod (v_{1}^{p+2}). For the last term, we may assume that r ≥ 5 because a(3) − 1 = 0. Then it follows that d_{0}(v_{2}^{a_{3,r}−j−p}X_{3,r−2}^{(sp−1)p}u_{2,3}) ≡ v_{2}^{a_{3,r}−j−p}X_{3,r−2}^{(sp−1)p}(u_{2,2}t_{1}^{p^{2}} − w_{2,3}^{p}) mod v_{1} by Propositions 3.1.6 and 3.2.3, and hence

d_{0}(− (a(r) − 1)bv_{1}^{p+1}v_{2}^{a_{3,r}−j−p}X_{3,r−2}^{(sp−1)p}u_{2,3})

≡ − (a(r) − 1)v_{1}^{p+1}v_{2}^{a_{3,r}−j−p}X_{3,r−2}^{(sp−1)p}(u_{2,2}t_{1}^{p^{2}} − w_{2,3}^{p}).

Collecting four terms gives

d_{0}(X(v_{2}^{j}, X_{3,r}^{s}))
≡ X_{3,r−1}^{sp} {(v_{2} + v_{1}t_{1}^{p})^{b} − v_{2}^{b} − bv_{1}t_{1}^{p}(v_{2} + v_{1}t_{1}^{p})^{b−1}} /v_{2}^{j+b}
− bv_{1}X_{3,r}^{s′p} {(v_{2} + v_{1}t_{1}^{p})^{b−1} − v_{2}^{b−1}} /s′v_{2}^{j+b+a_{3,r+1}}
+ a(r)bv_{1}v_{2}^{a_{3,r}−j−b+1}X_{3,r−2}^{(sp−1)p}ζ_{2}^{p} · η_{R}(v_{2}^{b−1})
+ b(a + b − 2)v_{1}^{p+1}X_{3,r}^{s′p}t_{1}/s′v_{2}^{j+2+a_{3,r+1}}
− (a + b)v_{1}^{p}X_{3,r−1}^{sp}t_{1}/v_{2}^{j+1} + v_{1}^{p+1}X_{3,r−1}^{sp}/v_{2}^{j+1} · {bζ_{2} + (a − b)w_{2,2}}

+ a(r)bv_{1}^{p+1}v_{2}^{a_{3,r}−j−p−1}X_{3,r−2}^{(sp−1)p} {(a − 1)ζ_{2}^{p}(v_{3} − v_{2}^{p}t_{1}) − v_{2}t_{1}^{p^{2}}u_{2,2}} .

Then, apply the equation

(X + Y)^{b} − X^{b} − bY (X + Y)^{b−1} = ∑_{i=2}^{b}(1 − i)\binom{b}{i}X^{b−i}Y^{i}

to the first term, and the congruence

X_{3,r}^{s′p} ≡ X_{3,r+1}^{s′} + a(r)s′v_{2}^{b(r+1)}X_{3,r−1}^{sp−1}u_{2,2} mod v_{2}^{b(r+1)} for odd r ≥ 3

to the fourth term. Modifying these terms completes the proof. □

We also set X(v_{2}^{j}, X_{3,1}^{s}) to v_{2}^{b}X_{3,1}^{s}/X_{2,1}−bv_{1}v_{2}^{b−1}X_{3,2}^{s′}/s′X_{2,1}^{p+1} for j = p−b. Then we have:

Lemma 4.1.8. d_{0}(X(v_{2}^{p−1}, X_{3,1}^{s})) is expressed as −3v_{1}^{p}X_{3,1}^{s}t_{1}/v_{2}^{p} mod v_{1}^{p+1}, while d_{0}(X(v_{2}^{p−2}, X_{3,1}^{s})) is expressed as

− 2v_{1}^{2}X_{3,2}^{s′}(t_{1}^{p} − 2v_{1}^{p−1}t_{1})/s′v_{2}^{p^{2}+p}
− v_{1}^{2}X_{3,1}^{s}/v_{2}^{p−1} · {t_{1}^{2p}/v_{2} + 4v_{1}^{p−2}t_{1} − v_{1}^{p−1}(ζ_{2} + t_{2}/v_{2} + C_{1})}
+ v_{1}^{p+1}Z/v_{2}^{p−2}

mod v_{1}^{p+2} for some Z ∈ v_{3}^{−1}BP_{∗}BP, where

C_{1} = (u_{3,0}t_{1}^{p^{2}+1} − w_{3,2})t_{1}^{p} + (u_{3,1} − w_{3,1}^{p})t_{2}^{p} + u_{3,0}^{p}t_{3}^{p}.

Proof. Notice that 1/X_{2,1}^{m} ≡ 1/v_{2}^{mp} + mv_{1}^{p}v_{3}/v_{2}^{(m+1)p+1} mod (v_{1}^{2p}). So we may compute d_{0} on

X_{3,1}^{s}/v_{2}^{p−b} + v_{1}^{p}v_{3}X_{3,1}^{s}/v_{2}^{2p+1−b} − b v_{1}X_{3,2}^{s′}/s′v_{2}^{p^{2}+p+1−b} − b v_{1}^{p+1}v_{3}X_{3,2}^{s′}/s′v_{2}^{p^{2}+2p+2−b}

instead of d_{0}(X(v_{2}^{p−b}, X_{3,1}^{s})). For b = 1 case, we consider the differential mod (v_{1}^{p+1}), so that the last term can be omitted. Because X_{3,1}^{s} ≡ v_{3}^{sp} + v_{2}^{p}v_{3}^{(s−1)p−1}v_{4}

mod v_{2}^{2p} and X_{3,2}^{s′} ≡ X_{3,1}^{s′p} mod v_{2}^{p^{2}+p}, we may compute d_{0} on

v_{2}(v_{3}^{s}/v_{2})^{p} + v_{1}^{p}v_{3}^{sp+1}/v_{2}^{2p} + v_{1}^{p}v_{3}^{(s−1)p}v_{4}/v_{2}^{p} − v_{1}(X_{3,1}^{s′}/s′v_{2}^{p+1})^{p}.

Then, we obtain the desired result using (2.1.1) – (2.1.3), Proposition 3.2.3 and Proposition 3.3.4.

Similar but harder calculation shows the b = 2 case. □

Next we define X_{+}(v_{2}^{jp}, X_{3,r}^{sp}) to modify the differential on X_{0}(v_{2}^{j}, X_{3,r}^{s})^{p}. Let

\begin{aligned}A_{0} & = & −\binom{j+1}{2}v_{1}^{2p}v_{3}^{2}X_{3,r}^{sp}/v_{2}^{(j+2)p+2} + j(j + 1)v_{1}^{2p}v_{3}X_{3,r+1}^{s′p}/s′v_{2}^{(j+2+a_{3,r+1})p+1}, \\ A_{1} & = & v_{1}^{2p}v_{2}^{(a_{3,r}−j−1)p−1}X_{3,r−1}^{(sp−1)p}\{a(r)jv_{2}u_{2,3} − (j + 1)v_{3}u_{2,2}\}, \\ A_{2} & = & v_{1}^{2p−1}v_{3}^{(sp−1)p+2}/v_{2}. \end{aligned}

Using these elements, we define X_{+}(v_{2}^{jp}, X_{3,r}^{sp}) as follows. For r = 1,

X_{+}(v_{2}^{jp}, X_{3,r}^{sp}) = \begin{cases}A_{0} & for j ≤ p − 3 \\ A_{0} + A_{2} & for j = p − 2. \end{cases}

If r ≥ 3 is odd and either j = a_{3,r} − p + 2 or j ≤ a_{3,r} − p − 1, then

X_{+}(v_{2}^{jp}, X_{3,r}^{sp}) = A_{0} + A_{1}.

Then we obtain the following lemma.

Lemma 4.1.9. Mod (v_{1}^{2p+1}),

d_{0}(X_{0}(v_{2}^{j}, X_{3,r}^{s})^{p} + X_{+}(v_{2}^{jp}, X_{3,r}^{sp}))

≡ −j(j + 1)\frac{v_{1}^{2p}X_{3,r+2}^{s′}t_{1}}{s′v_{2}^{jp+2+a_{3,r+2}}} − \binom{j + 1}{2}\frac{v_{1}^{2p}X_{3,r}^{sp}t_{1}^{2}}{v_{2}^{jp+2}}

− (j − 1)v_{1}^{2p}v_{2}^{(a_{3,r}−j−1)p−1}X_{3,r−1}^{(sp−1)p}θ_{2}(v_{2}^{j}),

where θ_{2}(v_{2}^{j}) = 0 for r = 1 and

θ_{2}(v_{2}^{j}) = (j + 1)(v_{3} − v_{2}^{p}t_{1})ζ_{2}^{p} + { (j + 1)v_{2}^{p}t_{1} − v_{2}t_{1}^{p^{2}} }u_{2,2}

for odd r ≥ 3. In particular, v_{2}^{p^{2}−1}θ_{2}(v_{2}^{a_{3,r}−p−1}) = −v_{3}^{p+1}(t_{1} + v_{2}w_{3,2})/v_{2}^{2}.

Proof. Here we prove only for r ≥ 3 case. Other cases are similarly proved. Notice that d_{0}(A_{0}) is congruent to

−\binom{j + 1}{2}v_{1}^{2p}d_{0}(v_{3}^{2}X_{3,r}^{sp})/v_{2}^{(j+2)p+2} + j(j + 1)v_{1}^{2p}d_{0}(v_{3}X_{3,r+1}^{s′p})/s′v_{2}^{(j+2+a_{3,r+1})p+1}

40 Hirofumi Nakai mod (v_{1}^{2p+1}), so that it is sufficient to compute d_{0}(v_{3}^{2}X_{3,r}^{sp}) mod (v_{1}, v_{2}^{(j+2)p+2}) and d_{0}(v_{3}X_{3,r+1}^{s′p}) mod (v_{1}, v_{2}^{(j+2+a_{3,r+1})p+1}). Easy computations show that

d_{0}(v_{3}^{2}X_{3,r}^{sp}) ≡ (v_{2}^{2}t_{1}^{2p^{2}} + v_{2}^{2p}t_{1}^{2} + 2 v_{2}v_{3}t_{1}^{p^{2}} − 2 v_{2}^{p}v_{3}t_{1} − 2 v_{2}^{p+1}t_{1}^{p^{2}+1})X_{3,r}^{sp},
d_{0}(v_{3}X_{3,r+1}^{s′p}) ≡ (v_{2}t_{1}^{p^{2}} − v_{2}^{p}t_{1})X_{3,r+1}^{s′p} + s′v_{2}^{pa_{3,r+1}}X_{3,r}^{sp}t_{1}^{p^{2}}(v_{3} + v_{2}t_{1}^{p^{2}} − v_{2}^{p}t_{1})
−s′v_{2}^{b(r+2)}X_{3,r−1}^{(sp−1)p}ζ_{2}^{p}(v_{3} + v_{2}t_{1}^{p^{2}} − v_{2}^{p}t_{1}).

Using these and the congruence

X_{3,r+2}^{s′} ≡ X_{3,r+1}^{s′p} − s′v_{2}^{a_{3,r+2}−p}v_{3}X_{3,r}^{sp} − s′v_{2}^{b(r+2)}X_{3,r}^{sp−1}u_{2,2},

we can show that d_{0}(A_{0}) is congruent to

\binom{j + 1}{2}\frac{v_{1}^{2p}X_{3,r}^{sp}(t_{1}^{2p^{2}} − v_{2}^{2p−2}t_{1}^{2})}{v_{2}^{(j+2)p}} + j(j + 1)\frac{v_{1}^{2p}(X_{3,r+1}^{s′p}t_{1}^{p^{2}} − v_{2}^{p−1}X_{3,r+2}^{s′}t_{1})}{s′v_{2}^{(j+2+a_{3,r+1})p}}

− j(j + 1)v_{1}^{2p}v_{2}^{(a_{3,r}−j−1)p−1}X_{3,r−1}^{(sp−1)p}{ζ_{2}^{p}(v_{3} + v_{2}t_{1}^{p^{2}} − v_{2}^{p}t_{1}) + v_{2}^{p}t_{1}u_{2,2}}. On the other hand, d_{0}(A_{1}) is congruent to

jv_{1}^{2p}v_{2}^{(a_{3,r}−j−1)p}d_{0}(a(r)·X_{3,r−1}^{(sp−1)p}u_{2,3})−(j +1)v_{1}^{2p}v_{2}^{(a_{3,r}−j−1)p−1}d_{0}(v_{3}X_{3,r−1}^{(sp−1)p}u_{2,2}). Noticing that (a_{3,r} − j − 1)p ≥ (p − 3)p, we have

d_{0}(X_{3,r−1}^{(sp−1)p}u_{2,3}) ≡ d_{0}(u_{2,3})X_{3,r−1}^{(sp−1)p} + η_{R}(u_{2,3})d_{0}(X_{3,r−1}^{(sp−1)p})

≡ (u_{2,2}t_{1}^{p^{2}} − w_{2,3}^{p})X_{3,r−1}^{(sp−1)p} + (u_{2,3} + u_{2,2}t_{1}^{p^{2}} − w_{2,3}^{p})d_{0}(X_{3,r−1}^{(sp−1)p})

mod (v_{1}). Because d_{0}(X_{3,r−1}^{(sp−1)p}) ≡ −v_{2}^{pa_{3,r−1}}X_{3,r−2}^{(sp^{2}−p−1)p}t_{1}^{p^{2}} mod (v_{1}, v_{2}^{pa_{3,r−1}+1}), the second term remains only for r = 3. Observe that u_{2,3} and w_{2,3}^{p} can be replaced with −v_{2}^{−p^{3}}v_{3}^{p^{2}}u_{2,2} and −v_{2}^{−p^{3}}v_{3}^{p^{2}}ζ_{2}^{p} respectively. So we obtain

d_{0}(a(r)·X_{3,r−1}^{(sp−1)p}u_{2,3})
≡ a(r) · X_{3,r−1}^{(sp−1)p}(u_{2,2}t_{1}^{p^{2}} − w_{2,3}^{p})
+ a(r) · v_{2}^{(a_{3,r−1}−p^{2})p}v_{3}^{p^{2}}X_{3,r−2}^{(sp^{2}−p−1)p}t_{1}^{p^{2}}(u_{2,2} − ζ_{2}^{p})

≡ a(r) · X_{3,r−1}^{(sp−1)p}(u_{2,2}t_{1}^{p^{2}} − w_{2,3}^{p}) + (2 − a(r)) · X_{3,r−1}^{(sp−1)p}t_{1}^{p^{2}}(u_{2,2} − ζ_{2}^{p})

≡ X_{3,r−1}^{(sp−1)p}{ 2 u_{2,2}t_{1}^{p^{2}} − a(r) · w_{2,3}^{p} − (2 − a(r)) · t_{1}^{p^{2}}ζ_{2}^{p} }.

Moreover, it is easy to see that

d_{0}(v_{3}X_{3,r−1}^{(sp−1)p}u_{2,2}) ≡ X_{3,r−1}^{(sp−1)p}{v_{2}t_{1}^{p^{2}}u_{2,2} − v_{2}^{p}t_{1}u_{2,2} − ζ_{2}^{p}(v_{3} + v_{2}t_{1}^{p^{2}} − v_{2}^{p}t_{1})}.

Collecting terms gives d_{0}(X_{+}(v_{2}^{jp}, X_{3,r}^{sp})) as

\binom{j + 1}{2}\frac{v_{1}^{2p}X_{3,r}^{sp}(t_{1}^{2p^{2}} − v_{2}^{2p−2}t_{1}^{2})}{v_{2}^{(j+2)p}} + j(j + 1)\frac{v_{1}^{2p}(X_{3,r+1}^{s′p}t_{1}^{p^{2}} − v_{2}^{p−1}X_{3,r+2}^{s′}t_{1})}{s′v_{2}^{(j+2+a_{3,r+1})p}}

− ε(r, j)^{p} − (j − 1)v_{1}^{2p}v_{2}^{(a_{3,r}−j−1)p−1}X_{3,r−1}^{(sp−1)p}θ_{2}(v_{2}^{j}). This completes the proof for r ≥ 3 case. □

4.2. Some inductive methods of constructing elements x(sp^{r}/j; k). Here we describe some inductive methods of constructing x(k) = x(sp^{r}/j; k). Because (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} = (X_{3,r}^{s}/v_{2}^{j})^{p^{k}} for j ≤ a_{3,r}, we may start with X_{3,r}^{s}/v_{2}^{j} (p ∤ j ≤ a_{3,r}). In general, each x(k) (k ≥ 1) is constructed by adding some appropriate terms to x(k − 1)^{p} so that we can find an element of Coker δ (2.5.1) from the numerator of the leading term of d_{0}(x(k)). There are some patterns for adding terms and constructing x(k) for a large enough k. We first observe the case with small j and s ∉ ℕ_{0}, which is the simplest example of such patterns.

Proposition 4.2.1. Assume that an element W_{1} satisfies

d_{0}(W_{1}) ≡ −jv_{1}^{p}X_{3,r}^{sp}t_{1}/v_{2}^{jp+1} mod v_{1}^{p+1}

for 1 ≤ j ≤ a_{3,r} − 2, then W_{k} (k ≥ 2) defined inductively on k by

\begin{aligned}W_{k} & = & W_{1}^{p} + jv_{1}^{a_{2,2}−p}v_{2}X_{3,r}^{sp^{2}}/X_{2,1}^{jp+1} & for k = 2, \\ & = & W_{k−1}^{p} + 2jv_{1}^{a_{2,k}−p}X_{3,r}^{sp^{k}}/v_{2}^{(jp+1)p^{k−1}−1} & for k ≥ 3 \end{aligned}

satisfies

d_{0}(W_{k}) ≡ −2jv_{1}^{a_{2,k}}X_{3,r}^{sp^{k}}t_{1}/v_{2}^{(jp+1)p^{k−1}} mod v_{1}^{a_{2,k}+1}.

Proof. Using Propositions 3.2.3 and 3.3.4, for k = 2, we have

d_{0} (jv_{1}^{a_{2,2}−p}v_{2}X_{3,r}^{sp^{2}}/X_{2,1}^{jp+1}) ≡ jv_{1}^{a_{2,2}−p+1}X_{3,r}^{sp^{2}}(t_{1}^{p} − 2v_{1}^{p−1}t_{1})/v_{2}^{(jp+1)p},

while for k ≥ 3, we have

d_{0} (2jv_{1}^{a_{2,k}−p}X_{3,r}^{sp^{k}}/v_{2}^{(jp+1)p^{k−1}−1}) ≡ 2jv_{1}^{a_{2,k}−p+1}X_{3,r}^{sp^{k}}(t_{1}^{p} − v_{1}^{p−1}t_{1})/v_{2}^{(jp+1)p^{k−1}}.

The result now follows by an easy calculation. □

We will see that each x(sp^{r}/j; 1) with r ≥ 1, 1 ≤ j ≤ a_{3,r} −2 and s ∉ ℕ_{0} satisfies the condition for W_{1} in the above proposition.

In general, we speculate on the existence of some rules in constructing elements x(k) for large k as the above case. Next we observe another pattern, which occurs in most cases. To state this, we first introduce some elements.

Definition 4.2.2. For e_{1} ≥ 1, e_{2} ≥ 1, s = s′p − 1 ∈ ℕ_{0} and odd r ≥ 3, we define Y_{[a,b]}(v_{1}^{e_{1}p}, v_{2}^{(e_{2}+1)p}, X_{3,r+1}^{s}) as follows: for s′ ∉ ℕ_{0}, it is given by

(a + b)\frac{v_{1}^{(e_{1}−1)p−1}v_{2}X_{3,r}^{sp}}{X_{2,1}^{e_{2}+1}} + a\frac{v_{1}^{e_{1}p}X_{3,r}^{sp}u_{2,2}}{v_{2}^{(e_{2}+1)p}} − (a + b)(b − 2)\frac{v_{1}^{e_{1}p−1}X_{3,r+2}^{s′}}{s′v_{2}^{(e_{2}+1)p+a_{3,r+2}}} ;

for s′ ∈ ℕ_{0}, it is given by

(above terms) + 2a\frac{v_{1}^{e_{1}p}X_{3,r+3}^{s′′}}{s′′v_{2}^{(e_{2}+1)p+b(r+3)}} .

Moreover, we define Z(v_{1}^{e_{1}p^{i}}, v_{2}^{(e_{2}+1)p^{i}}, X_{3,r+i}^{s}) by

−2\frac{v_{1}^{e_{1}p^{i}}v_{3}X_{3,r+i}^{s′p}}{s′v_{2}^{(e_{2}+1)p^{i}+p+1+a_{3,r+i+1}}} − \frac{v_{1}^{e_{1}p^{i}}X_{3,r+i−1}^{sp}u_{2,2}}{v_{2}^{(e_{2}+1)p^{i}}} for s′ ∉ ℕ_{0} and odd r + i,
\frac{v_{1}^{e_{1}p^{i}}X_{3,r+i−1}^{sp}u_{2,2}}{v_{2}^{(e_{2}+1)p^{i}}}  for s′ ∉ ℕ_{0} and even r + i,
−\frac{v_{1}^{e_{1}p^{i}}X_{3,r+i}^{s}}{v_{2}^{(e_{2}+1)p^{i}}} (3u_{2,2} − 2u_{3,3})  fors′ ∈ ℕ_{0},

and DZ(v_{1}^{e_{1}p^{i}}, v_{2}^{(e_{2}+1)p^{i}}, X_{3,r+i}^{s}) ∈ v_{3}^{−1}BP_{∗}/(p, v_{2}^{∞}) ⊗_{A} Γ by

2\frac{v_{1}^{e_{1}p^{i}}X_{3,r+i+1}^{s′}t_{1}}{s′v_{2}^{(e_{2}+1)p^{i}+1+a_{3,r+i+1}}} + \frac{v_{1}^{e_{1}p^{i}}X_{3,r+i}^{s}}{v_{2}^{(e_{2}+1)p^{i}}} (2w_{2,2} − ζ_{2}) for s′ ∉ ℕ_{0} and odd r + i,
2\frac{v_{1}^{e_{1}p^{i}}X_{3,r+i+1}^{s′}t_{1}^{p}}{s′v_{2}^{(e_{2}+1)p^{i}+1+a_{3,r+i+1}}} + \frac{v_{1}^{e_{1}p^{i}}X_{3,r+i}^{s}}{v_{2}^{(e_{2}+1)p^{i}}} (2\frac{t_{2}}{v_{2}} − ζ_{2}) for s′ ∉ ℕ_{0} and even r + i,
−\frac{v_{1}^{e_{1}p^{i}}X_{3,r+i}^{s}}{v_{2}^{(e_{2}+1)p^{i}}} (3ζ_{2} − 2ζ_{3})  fors′ ∈ ℕ_{0}.

Notice that each of these leading terms includes an element of Coker δ. Proposition 4.2.3. Assume that an element x(k) satisfies

d_{0} (x(k)) ≡ DX_{[a,b]}(v_{1}^{e_{1}}, v_{2}^{e_{2}+1}, X_{3,r}^{s}) + v_{1}^{e_{1}}Z_{k}/v_{2}^{e_{2}} mod v_{1}^{e_{1}+1} for some Z_{k} ∈ v_{3}^{−1}BP_{∗}BP, with e_{2} ≡ −b (p) and odd r ≥ 3. Let

x(k + 1) = x(k)^{p} + Y_{[a,b]}(v_{1}^{e_{1}p}, v_{2}^{(e_{2}+1)p}, X_{3,r+1}^{s}).

We compute d_{0}(x(k + 1)). For a ≠ 0, we get

d_{0} (x(k + 1)) ≡ aDZ(v_{1}^{e_{1}p}, v_{2}^{(e_{2}+1)p}, X_{3,r+1}^{s}) + v_{1}^{e_{1}p}Z_{k+1}/v_{2}^{(e_{2}+1)p−1} mod v_{1}^{e_{1}p+1}

for some Z_{k+1} ∈ v_{3}^{−1}BP_{∗}BP. For a = 0, we get

d_{0} (x(k + 1)) ≡ bv_{1}^{e_{1}p+1}X_{3,r+1}^{s}t_{1}^{p}/v_{2}^{(e_{2}+1)p+1} + v_{1}^{e_{1}p}Z′_{k+1}/v_{2}^{(e_{2}+1)p−1} mod v_{1}^{e_{1}p+2}

for some Z′_{k+1} ∈ v_{3}^{−1}BP_{∗}BP. Proof. Here we may ignore elements killed by v_{2}^{(e_{2}+1)p−1}. For the first term of Y_{[a,b]}(v_{1}^{e_{1}p}, v_{2}^{(e_{2}+1)p}, X_{3,r+1}^{s}), notice that (a+b)v_{1}^{(e_{1}−1)p−1}v_{2}X_{3,r}^{sp}/X_{2,1}^{e_{2}+1} is congruent to

(a + b)v_{1}^{(e_{1}−1)p−1}v_{2}X_{3,r}^{sp}{1/v_{2}^{(e_{2}+1)p} − (b − 1)v_{1}^{p}v_{3}/v_{2}^{(e_{2}+2)p+1}}

mod (p, v_{1}^{(e_{1}+1)p−1}). Using (2.1.2), Propositions 3.2.4 and 3.3.4, we have d_{0}((a + b)v_{1}^{(e_{1}−1)p−1}v_{2}X_{3,r}^{sp}/X_{2,1}^{e_{2}+1}) ≡ (a + b)v_{1}^{(e_{1}−1)p}X_{3,r}^{sp}

· \{v_{2}^{p}t_{1}^{p} + \underline{(b − 2)v_{1}^{p−1}v_{2}^{p}t_{1}}_{(A)} + (b − 1)v_{1}^{p}(\underline{t_{1}^{p^{2}+p} − t_{2}^{p}}_{(B)} + \underline{v_{1}v_{2}^{p−1}t_{1}^{p}}_{(C)})\}/v_{2}^{(e_{2}+2)p} mod (v_{1}^{e_{1}p+2}). For the second term, we can easily obtain

d_{0}(av_{1}^{e_{1}p}X_{3,r}^{sp}u_{2,2}/v_{2}^{(e_{2}+1)p}) ≡ \underline{av_{1}^{e_{1}p}X_{3,r}^{sp}(ζ_{2} − ζ_{2}^{p})/v_{2}^{(e_{2}+1)p}}_{(B)}

mod (v_{1}^{e_{1}p+1}) by Proposition 3.1.3. For the third term, using the congruence

X_{3,r+2}^{s′} ≡ X_{3,r+1}^{s′p} − s′v_{2}^{a_{3,r+2}−p}v_{3}X_{3,r}^{sp} mod v_{2}^{b(r+2)−p^{2}−p−1},

(2.1.2), Propositions 3.2.3 and 3.3.4, we have

d_{0}(−(a + b)(b − 2)v_{1}^{e_{1}p−1}X_{3,r+2}^{s′}/s′v_{2}^{(e_{2}+1)p+a_{3,r+2}})

≡ −(a + b)(b − 2)v_{1}^{e_{1}p}X_{3,r+1}^{s′p}t_{1}^{p}/s′v_{2}^{(e_{2}+1)p+1+a_{3,r+2}}

+ (a + b)(b − 2)v_{1}^{e_{1}p−1}X_{3,r}^{sp}\{\underline{−v_{2}^{p}t_{1}}_{(A)}\underline{−v_{1}(t_{1}^{p^{2}+p} − t_{2}^{p})}_{(B)}\underline{−v_{1}^{2}v_{2}^{p−1}t_{1}^{p}}_{(C)}\}/v_{2}^{(e_{2}+2)p}

mod (v_{1}^{e_{1}p+2}). Collecting three terms gives

\begin{aligned}(A) & = & 0, \\ (B) & = & v_{1}^{e_{1}p}X_{3,r}^{sp}/v_{2}^{(e_{2}+1)p}\{(a + b)(t_{1}^{p^{2}+p} − t_{2}^{p})/v_{2}^{p} + a(ζ_{2} − ζ_{2}^{p})\}, \\ (C) & = & (a + b)v_{1}^{e_{1}p+1}X_{3,r}^{sp}t_{1}^{p}/v_{2}^{(e_{2}+1)p+1}. \end{aligned}

Thus d_{0}(Y_{[a,b]}(v_{1}^{e_{1}p}, v_{2}^{(e_{2}+1)p}, X_{3,r+1}^{s})) is congruent to

− (a + b)(b − 2)v_{1}^{e_{1}p}X_{3,r+1}^{s′p}t_{1}^{p}/s′v_{2}^{(e_{2}+1)p+1+a_{3,r+2}} + (a + b)v_{1}^{(e_{1}−1)p}X_{3,r}^{sp}t_{1}^{p}/v_{2}^{(e_{2}+1)p}

+ v_{1}^{e_{1}p}X_{3,r}^{sp}/v_{2}^{(e_{2}+1)p} · {(a + b)(t_{1}^{p^{2}+p} − t_{2}^{p})/v_{2}^{p} + a(ζ_{2} − ζ_{2}^{p})}

mod v_{1}^{e_{1}p+1} for a ≠ 0, and d_{0}(Y_{[0,b]}(v_{1}^{e_{1}p}, v_{2}^{(e_{2}+1)p}, X_{3,r+1}^{s})) is congruent to

− b(b − 2)v_{1}^{e_{1}p}X_{3,r+1}^{s′p}t_{1}^{p}/s′v_{2}^{(e_{2}+1)p+1+a_{3,r+2}}
+ bv_{1}^{(e_{1}−1)p}X_{3,r}^{sp}{v_{2}^{p}t_{1}^{p} + v_{1}^{p}(t_{1}^{p^{2}+p} − t_{2}^{p}) + v_{1}^{p+1}v_{2}^{p−1}t_{1}^{p}}/v_{2}^{(e_{2}+2)p}

mod v_{1}^{e_{1}p+2} for a = 0. Now, recall definitions of ζ_{2} and w_{2,2} and notice the congruences

\begin{aligned}X_{3,r+1}^{s′p} & ≡ & X_{3,r+2}^{s′} + s′v_{2}^{a_{3,r+2}−p}v_{3}X_{3,r+1}^{s} & mod v_{2}^{b(r+2)−p^{2}−p−1}, \\ X_{3,r}^{sp} & ≡ & X_{3,r+1}^{s} & mod v_{2}^{b(r+1)−p^{2}−p−1}. \end{aligned}

Modifying the differentials using the above two congruences completes the proof of this proposition for the case s′ ∉ ℕ_{0}.

For s′ ∈ ℕ_{0} case, the result is proved using the congruence

d_{0}(2av_{1}^{e_{1}p}X_{3,r+3}^{s′′}/s′′v_{2}^{(e_{2}+1)p+b(r+3)}) ≡ 2av_{1}^{e_{1}p}Y,

where

Y = X_{3,r+2}^{s′}t_{1}^{p}/v_{2}^{(e_{2}+1)p+1+a_{3,r+2}} − X_{3,r+1}^{s}(ζ_{2} − ζ_{3} + t_{2}/v_{2})/v_{2}^{(e_{2}+1)p} + Z/v_{2}^{(e_{2}+1)p−1}

for some Z ∈ v_{3}^{−1}BP_{∗}BP. □

We define Y (v_{1}^{p^{3}+p^{2}−p}, v_{2}^{p^{2}−p}, X_{3,2}^{sp−1}) to be

v_{1}^{p^{3}+p^{2}−2p−1}X_{3,2}^{sp−1}(4v_{2} + v_{1}^{p+1}u_{2,2})/X_{2,1}^{p−1}  for s ∉ ℕ_{0},
(above terms) + 4v_{1}^{p^{3}+p^{2}−p}X_{3,4}^{s′}/s′v_{2}^{p^{3}+p^{2}+a_{3,4}} for s ∈ ℕ_{0}.

44 Hirofumi Nakai

By routine calculation, we have: Lemma 4.2.4. For s ∉ ℕ_{0}, d_{0}(Y (v_{1}^{p^{3}+p^{2}−p}, v_{2}^{p^{2}−p}, X_{3,2}^{sp−1})) is expressed as

(v_{1}^{p^{3}+p^{2}−2p}X_{3,2}^{sp−1}/v_{2}^{p^{2}}){4v_{2}^{p}t_{1}^{p} + v_{1}^{p}(4t_{1}^{p^{2}+p} − 5t_{2}^{p}) + v_{1}^{p}v_{2}^{p}(2ζ_{2} − ζ_{2}^{p} + C_{2} − w_{3,3})}

+ v_{1}^{p^{3}+p^{2}−p}Z/v_{2}^{p^{2}−p−1} mod v_{1}^{p^{3}+p^{2}−p+1}. For s ∈ ℕ_{0}, it is expressed as

(above terms) + 4v_{1}^{p^{3}+p^{2}−p}X_{3,3}^{s}t_{1}^{p}/v_{2}^{p^{3}+p^{2}}

− 4v_{1}^{p^{3}+p^{2}−p}X_{3,2}^{sp−1}(ζ_{2} − ζ_{3} + v_{2}^{−1}t_{2})/v_{2}^{p^{2}−p} + v_{1}^{p^{3}+p^{2}−p}Z′/v_{2}^{p^{2}−p−1} mod v_{1}^{p^{3}+p^{2}−p+1}. Here, Z, Z′ ∈ v_{3}^{−1}BP_{∗}BP, and

C_{2} = ( ∑_{i+j=2} u_{3,i}t_{2−j}^{p^{i}})t_{1}^{p^{2}} + u_{3,0}^{p}t_{1}^{p}(t_{2}^{p^{2}} − t_{1}^{p^{3}+p^{2}}).

Proof. By Proposition 3.3.4,

d_{0}(v_{2}X_{3,2}^{sp−1}/X_{2,1}^{p−1}) = d_{0}(v_{2}X_{3,2}^{sp−1})/X_{2,1}^{p−1} + η_{R}(v_{2}X_{3,2}^{sp−1})d_{0}(1/X_{2,1}^{p−1})
≡ d_{0}(v_{2}X_{3,2}^{sp−1})(1/v_{2}^{p^{2}−p} − v_{1}^{p}v_{3}/v_{2}^{p^{2}+1})
+ η_{R}(v_{2}X_{3,2}^{sp−1}) · v_{1}^{p}(t_{1} − v_{1}w_{2,2})/v_{2}^{p^{2}−p−1}

mod (v_{1}^{p+2}). It is sufficient to compute d_{0}(v_{2}X_{3,2}^{sp−1}) mod (v_{1}^{2}, v_{2}^{p^{2}+1}) and mod (v_{1}^{p+2}, v_{2}^{p^{2}−p}). Notice that

d_{0}(v_{2}X_{3,2}^{sp−1}) = d_{0}(v_{2})X_{3,2}^{sp−1} + η_{R}(v_{2})d_{0}(X_{3,2}^{sp−1}),
≡ d_{0}(v_{2})v_{3}^{(sp−1)p^{2}} + η_{R}(v_{2})d_{0}(v_{3}^{(sp−1)p^{2}}) mod v_{1}^{p+2}, v_{2}^{p^{2}−p}.

Using the results of 3.2, we see that d_{0}(4 v_{1}^{p^{3}+p^{2}−2p−1}v_{2}X_{3,2}^{sp−1}/X_{2,1}^{p−1}) is congruent to

(4 v_{1}^{p^{3}+p^{2}−2p}X_{3,2}^{sp−1}/v_{2}^{p^{2}−p}){t_{1}^{p} + v_{1}^{p}(t_{1}^{p^{2}+p} − t_{2}^{p})/v_{2}^{p}}.

On the other hand, we have

d_{0}(v_{1}^{p^{3}+p^{2}−p}X_{3,2}^{sp−1}u_{2,2}/v_{2}^{p^{2}−p}) ≡ v_{1}^{p^{3}+p^{2}−p}d_{0}(X_{3,2}^{sp−1}u_{2,2})/v_{2}^{p^{2}−p}

mod (v_{1}^{p^{3}+p^{2}−p+1}), and

d_{0}(X_{3,2}^{sp−1}u_{2,2}) = d_{0}(u_{2,2})X_{3,2}^{sp−1} + η_{R}(u_{2,2})d_{0}(X_{3,2}^{sp−1})
≡ (ζ_{2} − ζ_{2}^{p})X_{3,2}^{sp−1} + (u_{2,2} + ζ_{2} − ζ_{2}^{p})d_{0}(X_{3,2}^{sp−1})

mod (v_{1}, v_{2}). According to Proposition 3.2.3, d_{0}(X_{3,2}^{sp−1}) ≡ −v_{2}^{p^{2}}X_{3,1}^{sp^{2}−p−1}t_{1}^{p} + v_{2}^{p^{2}+p+1}v_{3}^{sp^{3}−p^{2}−p−1}(u_{2,0}t_{2} − w_{3,3}) mod (v_{1}, v_{2}^{p^{2}+p+2}). Using this congruence and Definition 3.2.1, we obtain

d_{0} (\frac{v_{1}^{p^{3}+p^{2}−p}X_{3,2}^{sp−1}u_{2,2}}{v_{2}^{p^{2}−p}} ) ≡ \frac{v_{1}^{p^{3}+p^{2}−2p}X_{3,2}^{sp−1}}{v_{2}^{p^{2}−p}} \{(2ζ_{2} − ζ_{2}^{p} − \frac{t_{2}^{p}}{v_{2}^{p}} ) + C_{2} − w_{3,3}\}

+ \frac{v_{1}^{p^{3}+p^{2}−p}Z}{v_{2}^{p^{2}−p−1}}

for some Z ∈ v_{3}^{−1}BP_{∗}BP. Collecting terms gives the result for s ∉ ℕ_{0} case.

For s ∈ ℕ_{0} case, it is easy to see that d_{0}(4 v_{1}^{p^{3}+p^{2}−p}X_{3,4}^{s′}/s′v_{2}^{p^{3}+p^{2}+a_{3,4}}) is congruent to

4 \frac{v_{1}^{p^{3}+p^{2}−p}X_{3,3}^{s}t_{1}^{p}}{v_{2}^{p^{3}+p^{2}}} −4 \frac{v_{1}^{p^{3}+p^{2}−p}X_{3,2}^{sp−1}t_{1}^{p}}{v_{2}^{p^{2}−p}} (ζ_{2}−ζ_{3}+u_{2,0}t_{2})+(elements killed by v_{2}^{p^{2}−p−1}).

□

Proposition 4.2.5. Assume that an element x(2) satisfies

d_{0}(x(2)) ≡ −2s(s − 1)\frac{v_{1}^{p^{2}+p−2}X_{3,1}^{sp−1}t_{1}}{v_{2}^{p−1}} + 2(s − 1)\frac{v_{1}^{p^{2}+p−1}X_{3,2}^{s}t_{1}}{v_{2}^{p^{2}+p}}

+ \binom{s}{2}\frac{v_{1}^{p^{2}+p−1}X_{3,1}^{sp−1}}{v_{2}^{p−1}} (ζ_{2} + \frac{t_{2}}{v_{2}} + C_{1}) + \frac{v_{1}^{p^{2}+p−1}Z}{v_{2}^{p−2}}

mod (v_{1}^{p^{2}+p}) for some Z ∈ v_{3}^{−1}BP_{∗}BP, where C_{1} is as in Lemma 4.1.8. Then x(3) = x(2)^{p} + \binom{s}{2}Y (v_{1}^{p^{3}+p^{2}−p}, v_{2}^{p^{2}−p}, X_{3,2}^{sp−1}) satisfies

d_{0}(x(3)) ≡ s(s − 1)DZ(v_{1}^{p^{3}+p^{2}−p}, v_{2}^{p^{2}−p}, X_{3,2}^{sp−1}) + v_{1}^{p^{3}+p^{2}−p}Z_{3}/v_{2}^{p^{2}−p−1}

mod (v_{1}^{p^{3}+p^{2}−p+1}) for some Z_{3} ∈ v_{3}^{−1}BP_{∗}BP.

Proof. Collect the differentials on x(2)^{p} and \binom{s}{2}Y (v_{1}^{p^{3}+p^{2}−p}, v_{2}^{p^{2}−p}, X_{3,2}^{sp−1}) using the congruences

C_{1}^{p} + C_{2} ≡ w_{3,3} mod I_{3},
X_{3,3}^{s} ≡ X_{3,2}^{sp} − sv_{2}^{p^{3}−1}v_{3}X_{3,2}^{sp−1} mod v_{2}^{p^{3}+p^{2}}.
□

As the next step of Proposition 4.2.3 or 4.2.5, we show the following proposition. Here we state only the result.

Proposition 4.2.6. Assume that an element x(k + 1) satisfies

d_{0}(x(k + 1)) ≡ DZ(v_{1}^{e_{1}p}, v_{2}^{(e_{2}+1)p}, X_{3,r+1}^{s}) + v_{1}^{e_{1}p}Z_{k+1}/v_{2}^{(e_{2}+1)p−1} mod v_{1}^{e_{1}p+1}

for some Z_{k+1} ∈ v_{3}^{−1}BP_{∗}BP, where e_{1} and e_{2} are positive integers and r ≥ 1. Then x(k + i) defined inductively on i ≥ 2 by

x(k + i) = x(k + i − 1)^{p} + Z(v_{1}^{e_{1}p^{i}}, v_{2}^{(e_{2}+1)p^{i}}, X_{3,r+i}^{s})

satisfies

d_{0}(x(k + i)) ≡ DZ(v_{1}^{e_{1}p^{i}}, v_{2}^{(e_{2}+1)p^{i}}, X_{3,r+i}^{s}) + v_{1}^{e_{1}p^{i}}Z_{k+i}/v_{2}^{(e_{2}+1)p^{i}−1} mod v_{1}^{e_{1}p^{i}+1} for some Z_{k+i} ∈ v_{3}^{−1}BP_{∗}BP. □

Proposition 4.2.6 is regarded as the last step in this pattern. We will see in Section 5 that x(k) is constructed in this way in many cases. Actually we have other two inductive ways of constructing x(k), introduced in the following two propositions. Their proofs are relatively easy, so we leave the proofs to the reader.

Proposition 4.2.7. Assume that an element x(k) satisfies

d_{0}(x(k)) ≡ v_{1}^{e_{1}p}v_{2}^{p^{3}−p}v_{3}^{(sp−1)p^{r+i−1}}θ_{i}(v_{2}^{a_{3,r}−p^{3−i}−p^{2−i}})^{p} mod v_{1}^{e_{1}+1}
(= v_{1}^{e_{1}}v_{3}^{e_{3}p}t_{1}^{p}/v_{2}^{2p} + v_{1}^{e_{1}}Z_{k}/v_{2}^{p})

for θ_{i}(v_{2}^{a_{3,r}−p^{3−i}−p^{2−i}}) (i = 1, 2) as in Lemma 4.1.7 or 4.1.9, and some Z_{k} ∈ v_{3}^{−1}BP_{∗}BP. Then x(k + i) defined inductively on i ≥ 1 by

x(k + i) = x(k + i − 1)^{p} − v_{1}^{e_{1}p^{i}}v_{3}^{e_{3}p^{i+1}+1}/v_{2}^{p^{i+2}+2p^{i+1}−pa_{3,i+1}+1} for odd i,
= x(k + i − 1)^{p}  for even i

satisfies

d_{0}(x(k + i)) ≡ v_{1}^{e_{1}p^{i}}v_{3}^{e_{3}p^{i+1}}t_{1}^{p^{δ(i)}}/v_{2}^{p^{i+2}+2p^{i+1}−a_{3,i+2}}
+ v_{1}^{e_{1}p^{i}}Z_{k+i}/v_{2}^{p^{i+1}+p^{i−1}} mod v_{1}^{e_{1}p^{i}+1}

for some Z_{k+i} ∈ v_{3}^{−1}BP_{∗}BP, where δ(i) = 0 for odd i and 1 for even i. □

Proposition 4.2.8. Assume that an element x(k) satisfies

d_{0}(x(k)) ≡ v_{1}^{e_{1}}v_{2}^{e_{2}}v_{3}^{e_{3}p^{r}}w_{2,t} mod v_{1}^{e_{1}+1}

with Max{p^{t−1} − p^{r}, 0} + P (t − 2) + 1 ≤ e_{2}. Then x(k + i) defined inductively on i ≥ 1 by

x(k + i) = x(k + i − 1)^{p} + (−1)^{i−1}v_{1}^{e_{1}p^{i}+P (i−1,1)}v_{2}^{e_{2}p^{i}−P (i−1,1)}v_{3}^{e_{3}p^{r+i}}u_{2,t+i−1}

satisfies

d_{0}(x(k + i)) ≡ (−1)^{i}v_{1}^{e_{1}p^{i}+P (i−1)}v_{2}^{e_{2}p^{i}−P (i−1)}v_{3}^{e_{3}p^{r+i}}w_{2,t+i} mod v_{1}^{e_{1}p^{i}+P (i−1)+1},

where P (i, j) = p^{j}(p^{i−j+1} − 1)/(p − 1) = p^{i} + ⋯ + p^{j} for i ≥ j. We will abbreviate P (i, 0) to P (i). □

Our guide to constructing x(k) is to add suitable elements to x(k − 1)^{p} so that the differential has one of the forms in the assumptions of Propositions 4.2.1, 4.2.6, 4.2.7 and 4.2.8. In fact, we will observe that each case follows one of the above four patterns by k = 5.

5. Proof of the main theorem

In this section we prove our main theorem by defining x(k) (= x(sp^{r}/j; k)) for all cases using the preparatory computations displayed in Section 4. Notice that the smallest integer N with d_{0}(x(k)) ≢ 0 mod p, v_{1}^{N+1} gives the v_{1}-divisibility N(k) (= N(s, r, j; k)) of x(k).

5.1. Definitions of x(0) and the differentials. Now we start with X_{3,r}^{s}/v_{2}^{j} (p ∤ j ≤ a_{3,r} and p ∤ s ∈ ℤ). Using Propositions 3.2.4 and 3.3.4, we can easily calculate d_{0}(X_{3,r}^{s}/v_{2}^{j}) mod v_{1}^{2} :

d_{0}(X_{3,0}^{s}/v_{2}^{j}) ≡ −v_{1}v_{3}^{s}t_{1}^{p}/v_{2}^{2} + sv_{1}v_{3}^{s−1}(t_{2}^{p} − t_{1}^{p^{2}+p})/v_{2},

and for r = 1 or for even r ≥ 2,

d_{0}(X_{3,r}^{s}/v_{2}^{j}) ≡ −jv_{1}X_{3,r}^{s}t_{1}^{p}/v_{2}^{j+1}.

Finally, for odd r ≥ 3,

d_{0}(X_{3,r}^{s}/v_{2}^{j}) ≡ −jv_{1}X_{3,r}^{s}t_{1}^{p}/v_{2}^{j+1} − sv_{1}v_{2}^{a_{3,r}−j}X_{3,r−1}^{sp−1} (jt_{1}^{p+1}/v_{2} + w_{2,2}) .

Moreover, we have

\begin{aligned}d_{0}(X_{3,0}^{s}/v_{2}) & ≡ & v_{1}v_{2}^{p−1}v_{3}^{s−1}w_{2,2} & mod (v_{1}^{2}), \\ & ≡ & v_{1}v_{2}^{p−2}v_{3}^{s−1}ζ_{2}(v_{2} − v_{1}t_{1}^{p}) & mod (v_{1}^{3}) \end{aligned}

for p | (s − 1). Then the coboundary δ : H^{0}M_{1}^{2} → H^{1}M_{2}^{1} on X_{3,r}^{s}/v_{1}v_{2}^{j} is

δ(X_{3,r}^{s}/v_{1}v_{2}^{j}) = −jv_{3}^{sp^{r}}t_{1}^{p}/v_{2}^{j+1} + (elements killed by v_{2}^{j}).

According to 2.5, its numerator belongs to Cokerδ, when s ∉ ℕ_{0} = {s′p−1 | p ∤ s′} and r ≥ 1 is odd or when r ≥ 0 is even by (2.5.1). In this case, we can set x(0) to X_{3,r}^{s}/v_{2}^{j} with the v_{1}-divisibility N(0) = 1. In another case, we need to modify X_{3,r}^{s}/v_{2}^{j} so that its coboundary includes an element of Cokerδ.

5.1.1. For odd r ≥ 3 and s ∈ ℕ_{0}.

(i) For a_{3,r} − p ≤ j ≤ a_{3,r}. We set x(0) to X_{0}(v_{2}^{j}, X_{3,r}^{s}). By Lemma 4.1.5, we have

d_{0} (x(0)) ≡ (1 − j)v_{1}v_{2}^{a_{3,r}−j}v_{3}^{(sp−1)p^{r−1}}ζ_{2} mod v_{1}^{2},

and so N(0) = 1. Notice that this differential is trivial either for j = a_{3,r} − p + 2 ≡ 1 (p) or for j ≤ a_{3,r} − p − 1.

(ii) Either for j = a_{3,r} − p + 2 or for j ≤ a_{3,r} − p − 1. We also set x(0) to X_{0}(v_{2}^{j}, X_{3,r}^{s}). Then, Lemma 4.1.5 shows that

d_{0}(x(0)) ≡ −j(j + 1)v_{1}^{2}X_{3,r+1}^{s′}t_{1}^{p}/s′v_{2}^{j+2+a_{3,r+1}} − \binom{j + 1}{2}v_{1}^{2}X_{3,r}^{s}t_{1}^{2p}/v_{2}^{j+2} + ε(r, j)

mod (v_{1}^{3}), and so N(0) = 2. Note that all elements but ε(r, j) are vanished when p | (j + 1), and ε(r, j) = 0 unless r ≥ 3 is odd and a_{3,r} − p^{2} − p ≤ j ≤ a_{3,r} − 2p, in which case

ε(r, j) = −a(r)v_{1}^{2}v_{2}^{a_{3,r}−j−1}v_{3}^{(sp−1)p^{r−1}}w_{2,3}.

(iii) For p | (j +1), p^{2} ∤ (j +p+1) and j ≤ a_{3,r} −p^{2} −p. Set x(0) = X(v_{2}^{j}, X_{3,r}^{s}).

Notice that we replace X_{0}(v_{2}^{a_{3,r}−p^{2}−p}, X_{3,r}^{s}) with X(v_{2}^{a_{3,r}−p^{2}−p}, X_{3,r}^{s}) for j = a_{3,r} − p^{2} − p although ε(r, a_{3,r} − p^{2} − p) ≠ 0. Then Lemma 4.1.7 shows that

d_{0}(x(0)) ≡ DX_{[j′,1]}(v_{1}^{p+1}, v_{2}^{j+1}, X_{3,r}^{s}) + a(r)v_{1}^{p+1}v_{2}^{a_{3,r}−j−p−1}v_{3}^{(sp−1)p^{r−1}}θ_{1}(v_{2}^{j}) mod v_{1}^{p+2}. Note that d_{0}(x(0)) ≢ 0 mod (v_{1}^{p+1}), and so N(0) = p when p^{2} ∤ (j + p + 1).

(iv) For p^{2} | (j + p + 1) and j ≤ a_{3,r} − p^{2} − 2p. We first define B(j) by

B(j) = X(v_{2}^{j}, X_{3,r}^{s}) for j ≤ a_{3,r} − p^{2} − 3p;
B(a_{3,r} − p^{2} − 2p) = X(v_{2}^{a_{3,r}−p^{2}−2p}, X_{3,r}^{s}) − a(r)v_{1}^{p+1}v_{3}^{(sp−1)p^{r−1}+p+2}/2v_{2}^{2}.

Then, let

x(0) = \begin{cases}B(j) & for s′ ∉ ℕ_{0} \\ B(j) − 2 v_{1}^{p+1}X_{3,r+2}^{s′′}/s′′v_{2}^{j+1+b(r+2)} & for s′ ∈ ℕ_{0}. \end{cases}

Easy calculation shows that d_{0}(x(0)) ≡ −DZ(v_{1}^{p+1}, v_{2}^{j+1}, X_{3,r}^{s}) mod (v_{1}^{p+2}), and so N(0) = p + 1.

5.1.2. For r = 1 and s ∈ ℕ_{0}. For j ≤ p − 2, we set x(0) to X_{0}(v_{2}^{j}, X_{3,1}^{s}). Then Lemma 4.1.5 shows that

d_{0}(x(0)) ≡ −j(j + 1)v_{1}^{2}X_{3,2}^{s′}t_{1}^{p}/s′v_{2}^{j+2+a_{3,2}} − \binom{j + 1}{2}v_{1}^{2}X_{3,1}^{s}t_{1}^{2p}/v_{2}^{j+2}

mod (v_{1}^{3}), and so N(0) = 2. For j = p − 1, we set x(0) to X(v_{2}^{p−1}, X_{3,1}^{s}). By Lemma 4.1.8, we have d_{0}(x(0)) ≡ −3v_{1}^{p}X_{3,1}^{s}t_{1}/v_{2}^{p} mod (v_{1}^{p+1}), and so N(0) = p.

5.2. Definitions of x(k) (k ≥ 1) and the differentials. We complete the proof of our main theorem by defining x(k) and computing the differentials for all cases, based on the computations in k = 0 case.

5.2.1. For r = 0.

(i) For p ∤ (s − 1). We set x(1) to X_{3,1}^{s}/X_{2,1}. Then Lemma 4.1.3 shows that

d_{0}(x(1)) ≡ (s − 1)v_{1}^{p}X_{3,1}^{s}t_{1}/v_{2}^{p+1} + \binom{s}{2}v_{1}^{p}v_{3}^{sp−1}t_{1}^{2}/v_{2}

mod v_{1}^{p+1}, and so N(1) = p. For k = 2, we set

x(2) = x(1)^{p} + \binom{s}{2}v_{1}^{p^{2}−2}X(v_{2}^{p−2}, X_{3,1}^{sp−1}).

Then Lemma 4.1.8 implies

d_{0} (\binom{s}{2}v_{1}^{p^{2}−2}X(v_{2}^{p−2}, X_{3,1}^{sp−1}))

≡ −(s − 1)\frac{v_{1}^{p^{2}}X_{3,2}^{s}}{v_{2}^{p^{2}+p}} (t_{1}^{p} − 2v_{1}^{p−1}t_{1})

− \binom{s}{2}\frac{v_{1}^{p^{2}}X_{3,1}^{sp−1}}{v_{2}^{p−1}} \{\frac{t_{1}^{2p}}{v_{2}} + 4v_{1}^{p−2}t_{1} − v_{1}^{p−1} (ζ_{2} + \frac{t_{2}}{v_{2}} + C_{1})\} + \frac{v_{1}^{p^{2}+p−1}Z_{2}}{v_{2}^{p−2}}

mod v_{1}^{p^{2}+p}, and hence d_{0}(x(2)) is congruent to

− 2s(s − 1)\frac{v_{1}^{p^{2}+p−2}X_{3,1}^{sp−1}t_{1}}{v_{2}^{p−1}} + 2(s − 1)\frac{v_{1}^{p^{2}+p−1}X_{3,2}^{s}t_{1}}{v_{2}^{p^{2}+p}}

+ \binom{s}{2}\frac{v_{1}^{p^{2}+p−1}X_{3,1}^{sp−1}}{v_{2}^{p−1}} (ζ_{2} + \frac{t_{2}}{v_{2}} + C_{1}) + \frac{v_{1}^{p^{2}+p−1}Z_{2}}{v_{2}^{p−2}}

mod v_{1}^{p^{2}+p}. Notice that d_{0}(x(2)) ≢ 0 mod (v_{1}^{p^{2}+p−1}), and so N(2) = p^{2} + p − 2. For k = 3, we set x(3) to x(2)^{p} + \binom{s}{2}Y (v_{1}^{p^{3}+p^{2}−p}, v_{2}^{p^{2}−p}, X_{3,2}^{sp−1}). Then Lemma 4.2.5 shows that

d_{0} (x(3)) ≡ s(s − 1)DZ(v_{1}^{p^{3}+p^{2}−p}, v_{2}^{p^{2}−p}, X_{3,2}^{sp−1}) + v_{1}^{p^{3}+p^{2}−p}Z_{3}/v_{2}^{p^{2}−p−1}

mod (v_{1}^{p^{3}+p^{2}−p+1}) for some Z_{3} ∈ v_{3}^{−1}BP_{∗}BP, and so N(3) = p^{3} + p^{2} − p. Applying Proposition 4.2.6, we can define x(k) inductively on k ≥ 4 and obtain N(k) = p^{k} + p^{k−1} − p^{k−2}. □

Recall that ℕ_{1} = {(ap^{2} − p − 1)p^{r} + 1 | p ∤ a, r ≥ 1 : odd}.

(ii) For p | (s − 1) and s ∉ ℕ_{1}. We have already computed d_{0}(x(0)) ≡ v_{1}v_{2}^{p−1}v_{3}^{s−1}w_{2,2} mod v_{1}^{2}. Thus we can define x(k) inductively on k ≥ 1 using Proposition 4.2.8 and obtain N(k) = P (k). □

(iii) For s ∈ ℕ_{1}. In this case s is expressed as (s_{0}p − 1)p^{r_{0}} + 1 (s_{0} ∈ ℕ_{0} and odd r_{0} ≥ 1). We set x(1) to

x(0)^{p} − 1/a(r_{0} + 2)v_{1}^{p−1}X(v_{2}^{(a_{3,r_{0}+1}−p+2)p}, X_{3,r_{0}+2}^{s_{0}}) − v_{1}^{2p}v_{2}^{p^{2}−2p−1}v_{3}^{(s−1)p+1}u_{2,2}

with a(3) = 1 and a(r_{0} + 2) = 3 for r_{0} ≥ 3. Then the differential on the second term is given by Lemma 4.1.7. For the third term, we have

d_{0}(−v_{1}^{2p}v_{2}^{p^{2}−2p−1}v_{3}^{(s−1)p+1}u_{2,2})
≡ v_{1}^{2p}v_{2}^{p^{2}−2p−1}v_{3}^{(s−1)p}{v_{3}ζ_{2}^{p} + (v_{2}^{p}t_{1} − v_{2}t_{1}^{p^{2}})(u_{2,2} − ζ_{2}^{p})}

mod (v_{1}^{2p+1}). Collecting three terms gives

d_{0}(x(1)) ≡ −1/a(r_{0} + 2)DX_{[2,1]}(v_{1}^{2p}, v_{2}^{(a_{3,r_{0}+1}−p+2)p}, X_{3,r_{0}+2}^{s_{0}})

mod v_{1}^{2p+1}, and so N(1) = 2p − 1. Applying Propositions 4.2.3 and 4.2.6, we can define x(k) inductively on k ≥ 2 and obtain N(k) = 2p^{k}. □

5.2.2. For even r ≥ 2. Define x(1) by

\begin{aligned}x(1) & = & X_{3,r}^{sp}/X_{2,1}^{a_{3,r}−1} + sv_{1}^{p}v_{3}^{(sp−1)p^{r}+2}/2v_{2}^{2} & for j = a_{3,r} − 1, \\ & = & X_{3,r}^{sp}/X_{2,1}^{j} & for j ≤ a_{3,r} − 2. \end{aligned}

Then, in both cases we have

d_{0}(x(1)) ≡ −jv_{1}^{p}X_{3,r+1}^{s}(t_{1} − v_{1}w_{2,2})/v_{2}^{jp+1}

mod (v_{1}^{p+2}). Notice that d_{0}(x(1)) ≢ 0 mod (v_{1}^{p+1}), and so N(1) = p when s ∉ ℕ_{0}. Then we can define x(k) inductively on k ≥ 2 by Proposition 4.2.1 and obtain N(k) = a_{2,k}. When s ∈ ℕ_{0}, notice that the above differential can be rewritten as

d_{0}(x(1)) ≡ DX_{[j,0]}(v_{1}^{p+1}, v_{2}^{jp+1}, X_{3,r+1}^{s}),

and so N(1) = p. Applying Propositions 4.2.3 and 4.2.6, we can define x(k) inductively on k ≥ 2 and obtain N(k) = p^{k} + p^{k−1}. □

5.2.3. For odd r ≥ 1 and s ∉ ℕ_{0}. Define x(1) by

x(1) = X_{3,1}^{sp}/X_{2,1}^{j}  for r = 1,
= X_{3,r}^{sp}/X_{2,1}^{j} − sv_{1}^{p}v_{2}^{(a_{3,r}−j)p}v_{3}^{(sp−1)p^{r}}u_{2,2} for odd r ≥ 3.

For odd r ≥ 3 and j ≤ a_{3,r} − p − 1, d_{0}(X_{3,r}^{sp}/X_{2,1}^{j}) has already been computed in Lemma 4.1.2. We also obtain d_{0}(X_{3,r}^{sp}/X_{2,1}^{j}) for other cases by easy computations. We first give the case r = 1:

d_{0}(X_{3,1}^{sp}/X_{2,1}^{p−1}) = v_{1}^{p}X_{3,1}^{sp}t_{1}/v_{2}^{(p−1)p+1} − sv_{1}^{p}v_{3}^{(sp−1)p+1}t_{1}^{p}/v_{2},

and for j ≤ p − 2,

d_{0}(X_{3,1}^{sp}/X_{2,1}^{j}) = −jv_{1}^{p}X_{3,1}^{sp}t_{1}/v_{2}^{jp+1}.

Now let r ≥ 3 be odd. Then

d_{0}(X_{3,r}^{sp}/X_{2,1}^{a_{3,r}}) = v_{1}^{p}X_{3,r}^{sp}t_{1}/v_{2}^{pa_{3,r}+1} − sv_{1}^{p}v_{3}^{(sp−1)p^{r}}(ζ_{2}^{p} − w_{2,2}),
d_{0}(X_{3,r}^{sp}/X_{2,1}^{a_{3,r}−1}) = 2v_{1}^{p}X_{3,r}^{sp}t_{1}/v_{2}^{(a_{3,r}−1)p+1} − sv_{1}^{p}v_{3}^{(sp−1)p^{r}}(v_{2}^{p}ζ_{2}^{p} + 2v_{3}t_{1}^{p}/v_{2}),

and for j ≤ a_{3,r} − 2, we have

d_{0}(X_{3,r}^{sp}/X_{2,1}^{j}) = −jv_{1}^{p}X_{3,r}^{sp}t_{1}/v_{2}^{jp+1} − sv_{1}^{p}v_{2}^{(a_{3,r}−j)p}v_{3}^{(sp−1)p^{r}}ζ_{2}^{p}.

Additionally, we consider the element −sv_{1}^{p}v_{2}^{(a_{3,r}−j)p}v_{3}^{(sp−1)p^{r}}u_{2,2} for odd r ≥ 3, on which the differential is congruent to

−sv_{1}^{p}v_{2}^{(a_{3,r}−j)p}v_{3}^{(sp−1)p^{r}}(ζ_{2} − ζ_{2}^{p})

mod v_{1}^{p+1}. Collecting terms shows that d_{0}(x(1)) is congruent to

v_{1}^{p}X_{3,r}^{sp}t_{1}/v_{2}^{pa_{3,r}+1} + sv_{1}^{p}v_{3}^{(sp−1)p^{r}}(t_{1}^{p+1} − t_{2})/v_{2} for odd r ≥ 3 and j = a_{3,r},
−jv_{1}^{p}X_{3,r}^{sp}t_{1}/v_{2}^{(a_{3,r}−1)p+1} − sv_{1}^{p}v_{3}^{(sp−1)p^{r}+1}t_{1}^{p}/v_{2} for j = a_{3,r} − 1,
−jv_{1}^{p}X_{3,r}^{sp}t_{1}/v_{2}^{jp+1}  for j ≤ a_{3,r} − 2

mod v_{1}^{p+1}, and so N(1) = p.

(i) For k ≥ 2 and j ≤ a_{3,r} − 2. Using Proposition 4.2.1, we can define x(k) inductively on k so that

d_{0}(x(k)) ≡ −2jv_{1}^{a_{2,k}}X_{3,r}^{sp^{k}}t_{1}/v_{2}^{(jp+1)p^{k−1}}

mod (v_{1}^{1+a_{2,k}}) and obtain N(k) = a_{2,k}. □

(ii) For k ≥ 2 and j = a_{3,r} − 1. Let

x(2) = x(1)^{p} + jv_{1}^{a_{2,2}−p}X_{3,r}^{sp^{2}}/v_{2}^{a_{3,r+2}−p^{2}} + sv_{1}^{p^{2}}v_{2}^{p^{2}}v_{3}^{(sp−1)p^{r+1}}u_{2,2}.

Then easy calculation shows that

d_{0}(x(2)) ≡ −sv_{1}^{p^{2}+1}v_{2}^{p^{2}−1}v_{3}^{(sp−1)p^{r+1}}w_{2,3}

mod (v_{1}^{p^{2}+2}), and so N(2) = p^{2} +1. Applying Proposition 4.2.8, we can define x(k) inductively on k ≥ 3 and obtain N(k) = P (k) − p^{k−1}. □

(iii) For k ≥ 2, odd r ≥ 3 and j = a_{3,r}. Let x(2) = x(1)^{p} − v_{1}^{p^{2}−1}X_{3,r+2}^{s}/v_{2}^{a_{3,r+2}},

and x(3) = x(2)^{p} + sv_{1}^{p^{3}+p}v_{3}^{(sp−1)p^{r+2}+1}/v_{2}^{p+1}. Then easy computations show that

d_{0}(x(2)) ≡ −sv_{1}^{p^{2}+1}v_{3}^{(sp−1)p^{r+1}}t_{1}^{p}/v_{2} mod v_{1}^{p^{2}+2},
d_{0}(x(3)) ≡ −sv_{1}^{p^{3}+p}v_{3}^{(sp−1)p^{r+2}}(t_{1} − v_{1}w_{2,2})/v_{2} mod v_{1}^{p^{3}+p+2}
( ≡ sDX_{[1,0]}(v_{1}^{p^{3}+p+1}, v_{2}, X_{3,r+2}^{sp−1})),

and so N(2) = p^{2} + 1 and N(3) = p^{3} + p. Applying Propositions 4.2.3 and 4.2.6, we can define x(k) inductively on k ≥ 4 and obtain N(k) = p^{k} + p^{k−2} + p^{k−3}. □

5.2.4. For r = 1 and s ∈ ℕ_{0}.

(i) For j = p − 1. First we define x̃ by x̃ = x(0)^{p} + 3v_{1}^{a_{2,2}−p}X_{3,1}^{sp}/v_{2}^{p^{2}−1}.

Observe that d_{0}(x̃) ≡ −3v_{1}^{a_{2,2}}X_{3,1}^{sp}t_{1}/v_{2}^{p^{2}} mod (v_{1}^{1+a_{2,2}}). Then, let x(1) =

x̃+3v_{1}^{a_{2,2}}X_{3,3}^{s′}/s′v_{2}^{p^{2}+a_{3,3}}, x(2) = x^{p}+3v_{1}^{a_{2,3}−p}X_{3,2}^{sp}/v_{2}^{a_{3,3}−p}, and x(3) = x(2)^{p}+

3v_{1}^{a_{2,4}−p}v_{3}^{sp^{4}}/v_{2}^{p^{4}−1} + 3v_{1}^{a_{2,4}}X_{3,5}^{s′}/s′v_{2}^{p^{4}+a_{3,5}}. Easy computations show that

\begin{matrix}d_{0}(x(1)) ≡ −3v_{1}^{a_{2,2}}v_{2}v_{3}^{(sp−1)p}ζ_{2}, d_{0}(x(2)) ≡ −3v_{1}^{a_{2,3}}v_{3}^{sp^{3}}t_{1}/v_{2}^{p^{3}}, \\ d_{0}(x(3)) ≡ 3v_{1}^{a_{2,4}}v_{3}^{(sp−1)p^{3}+p}t_{1}/v_{2}^{p} \end{matrix}

mod v_{1}^{1+a_{2,k+1}}, and so N(k) = a_{2,k+1} (1 ≤ k ≤ 3). For k ≥ 4, we define x(k) inductively on k by x(k) = x(k − 1)^{p} − 3v_{1}^{a_{2,k+1}−p}v_{3}^{(sp−1)p^{k}+p^{k−2}}/v_{2}^{p^{k−2}−1}. Parallel calculation to the proof of Proposition 4.2.1 shows that

d_{0}(x(k)) ≡ 3v_{1}^{a_{2,k+1}}v_{3}^{(sp−1)p^{k}+p^{k−2}}t_{1}/v_{2}^{p^{k−2}}

mod (v_{1}^{1+a_{2,k+1}}), and so we obtain N(k) = a_{2,k+1}. □

(ii) For j ≤ p − 2. We set x(1) to X_{0}(v_{2}^{j}, X_{3,1}^{s})^{p} + X_{+}(v_{2}^{jp}, X_{3,1}^{sp}). As is already computed in Proposition 4.1.9,

d_{0}(x(1)) ≡ −j(j + 1)v_{1}^{2p}X_{3,3}^{s′}t_{1}/s′v_{2}^{jp+2+a_{3,3}} − \binom{j + 1}{2}v_{1}^{2p}X_{3,2}^{s}t_{1}^{2}/v_{2}^{jp+2}

mod (v_{1}^{2p+1}), and so N(1) = 2p. For k ≥ 2, we can construct x(k) as in 5.2.5 (ii) and obtain N(k) = 2p^{k} + p^{k−1} − p^{k−2}. □

5.2.5. For odd r ≥ 3 and s ∈ ℕ_{0}.

(i) For a_{3,r} − p ≤ j ≤ a_{3,r} (j ≠ a_{3,r} − p + 2). We set

x(1) = x(0)^{p} + (1 − j)v_{1}^{p}v_{2}^{(a_{3,r}−j)p}v_{3}^{(sp−1)p^{r}}u_{2,2}.

Easy calculation shows that for a_{3,r} − 1 ≤ j ≤ a_{3,r},

d_{0}(x(1)) ≡ (1 − j)v_{1}^{p}v_{2}^{(a_{3,r}−j)p}v_{3}^{(sp−1)p^{r}}ζ_{2} mod v_{1}^{p+1},

and for a_{3,r} − p ≤ j ≤ a_{3,r} − 2,

d_{0}(x(1)) ≡ (j − 1)v_{1}^{p+1}v_{2}^{(a_{3,r}−j)p−1}v_{3}^{(sp−1)p^{r}}w_{2,3} mod v_{1}^{p+2}.

For j = a_{3,r} and k ≥ 2, we can define x(k) as in 5.2.3 (iii). For j = a_{3,r} − 1, we set x(2) to x(1)^{p} − 3/2v_{1}^{p^{2}−1}X(v_{2}^{(a_{3,r+1}−p+1)p−1}, X_{3,r+2}^{s}). Then d_{0}(x(2)) is congruent to

3v_{1}^{p^{2}+p−1}X_{3,r+2}^{s}(2t_{1} − v_{1}ζ_{2})/2v_{2}^{(a_{3,r+1}−p+1)p} + 3v_{1}^{p^{2}+p}v_{2}^{p^{2}−p}v_{3}^{(sp−1)p^{r+1}}t_{1}^{p^{2}}u_{2,2}

(= −3/2DX_{[1,1]}(v_{1}^{p^{2}+p}, v_{2}^{(a_{3,r+1}−p+1)p}, X_{3,r+2}^{s}) + 3v_{1}^{p^{2}+p}Z_{2}/v_{2}^{2p+1})

mod (v_{1}^{p^{2}+p+1}) for some Z_{2} ∈ v_{3}^{−1}BP_{∗}BP, and so N(2) = p^{2}+p−1. Applying Propositions 4.2.3 and 4.2.6, we can define x(k) inductively on k ≥ 3 and obtain N(k) = p^{k} + p^{k−1}. On the other hand, we can apply Proposition 4.2.8 to a_{3,r} − p ≤ j ≤ a_{3,r} − 2 case and obtain N(k) = P (k). □

(ii) Either for j = a_{3,r} − p + 2, or for j ≤ a_{3,r} − p − 1 and p ∤ (j + 1). For j = a_{3,r} − p − 2, we set

x(1) = X_{0}(v_{2}^{a_{3,r}−p−2}, X_{3,r}^{s})^{p} + X_{+}(v_{2}^{(a_{3,r}−p−2)p}, X_{3,r}^{sp}) + 2v_{1}^{2p}v_{3}^{(sp−1)p^{r}+p+2}/v_{2}^{2}.

Otherwise, we set

x(1) = X_{0}(v_{2}^{j}, X_{3,r}^{s})^{p} + X_{+}(v_{2}^{jp}, X_{3,r}^{sp}).

We now use Proposition 4.1.9, to compute d_{0}(x(1)) mod (v_{1}^{2p+1}). For j = a_{3,r} − p − 1,

d_{0}(x(1)) ≡ −2\frac{v_{1}^{2p}X_{r+2}^{s′}t_{1}}{s′v_{2}^{b(r+2)−p^{2}−p+1}} − \frac{v_{1}^{2p}X_{3,r+1}^{s}t_{1}^{2}}{v_{2}^{a_{3,r+1}−p^{2}−p+2}} + 3v_{1}^{2p}v_{2}^{p^{2}−1}X_{3,r−1}^{(sp−1)p}θ_{2}(v_{2}^{a_{3,r}−p−1}).

Otherwise,

d_{0}(x(1)) ≡ −j(j + 1)\frac{v_{1}^{2p}X_{r+2}^{s′}t_{1}}{s′v_{2}^{jp+2+a_{3,r+2}}} − \binom{j + 1}{2}\frac{v_{1}^{2p}X_{3,r+1}^{s}t_{1}^{2}}{v_{2}^{jp+2}} .

So N(1) = 2p. For k = 2, we set

x(2) = x(1)^{p} − \binom{j + 1}{2}v_{1}^{2p^{2}−2}X(v_{2}^{(jp+2)p−2}, X_{3,r+2}^{s}).

Using Proposition 4.1.7, we now compute d_{0}(x(2)), getting

3v_{1}^{2p^{2}}v_{2}^{p^{3}−p}v_{3}^{(sp−1)p^{r+1}}θ_{2}(v_{2}^{a_{3,r}−p−1})^{p}  mod v_{1}^{2p^{2}+1} for j = a_{3,r} − p − 1,
−\binom{j+1}{2}DX_{[2,2]}(v_{1}^{2p^{2}+p−1}, v_{2}^{(jp+2)p−1}, X_{3,r+2}^{s}) mod v_{1}^{2p^{2}+p} otherwise.

Thus, N(2) = 2p^{2} for j = a_{3,r} − p − 1, and N(2) = 2p^{2} + p − 2 for other cases. In the former case, we can construct x(k) inductively on k ≥ 3 using Proposition 4.2.7 and obtain N(k) = 2p^{k}. In the latter case, Propositions 4.2.3 and 4.2.6 work well, and so we obtain N(k) = 2p^{k} + p^{k−1} − p^{k−2}. □

(iii) For j ≤ a_{3,r} − 2p and p | (j + 1). For a_{3,r} − p^{2} ≤ j ≤ a_{3,r} − 2p, we have already shown that

d_{0}(x(0)) ≡ −a(r)v_{1}^{2}v_{2}^{a_{3,r}−j−1}v_{3}^{(sp−1)p^{r−1}}w_{2,3}

mod (v_{1}^{3}) in 5.1.1 (ii). Then we can define x(k) inductively on k ≥ 1 using Proposition 4.2.8 and obtain N(k) = p^{k} + P (k). For j ≤ a_{3,r} − p^{2} − p, 5.1.1 (iii) and (iv) also show how to compute the congruence class of d_{0}(x(0)) mod p, v_{1}^{p+2}. For p^{2} ∤ (j + p + 1), we get

DX_{[j′,1]}(v_{1}^{p+1}, v_{2}^{j+1}, X_{3,r}^{s}) + a(r)v_{1}^{p+1}v_{2}^{a_{3,r}−j−p−1}v_{3}^{(sp−1)p^{r−1}}θ_{1}(v_{2}^{j}).

For p^{2} | (j + p + 1), we get

−DZ(v_{1}^{p+1}, v_{2}^{j+1}, X_{3,r}^{s}).

When p^{2}|(j + p + 1), we can construct x(k) inductively on k ≥ 1 using Proposition 4.2.6 and obtain N(k) = p^{k+1} + p^{k}. On the other hand, we set x(1) to x(0)^{p} + Y_{[j′,1]}(v_{1}^{p^{2}+p}, v_{2}^{(j+1)p}, X_{3,r+1}^{s}) when p^{2} ∤ (j + p + 1). Then d_{0}(x(1)) is expressed as follows. For p^{2} ∤ (j + 1),

d_{0}(x(1)) ≡ j′DZ(v_{1}^{p^{2}+p}, v_{2}^{(j+1)p}, X_{3,r+1}^{s}) mod v_{1}^{p^{2}+p+1}.

For j = a_{3,r} − p^{2} − p,

d_{0}(x(1)) ≡ a(r)v_{1}^{p^{2}+p}v_{2}^{p^{3}−p}v_{3}^{(sp−1)p^{r+1}}θ_{1}(v_{2}^{a_{3,r}−p^{2}−p})^{p}  mod v_{1}^{p^{2}+p+1}.

For p^{2} | (j + 1) and j ≤ a_{3,r} − 2p^{2} − p,

d_{0}(x(1)) ≡ v_{1}^{p^{2}+p+1}X_{3,r+1}^{s}t_{1}^{p}/v_{2}^{(j+1)p+1}  mod v_{1}^{p^{2}+p+2}.

When p^{2} ∤ (j + 1), we construct x(k) inductively on k ≥ 2 using Proposition 4.2.6 and obtain N(k) = p^{k+1} + p^{k}. When j = a_{3,r} − p^{2} − p, we can construct x(k) inductively on k ≥ 2 as in j = a_{3,r} − p − 1 case and k ≥ 3 (discussed in 5.2.5 (ii)), and obtain N(k) = p^{k+1} + p^{k}. When p^{2} | (j + 1) and j ≤ a_{3,r} − 2p^{2} − p, we set x(2) to x(1)^{p} − v_{1}^{p^{3}+p^{2}+p}v_{3}X_{3,r+1}^{sp}/v_{2}^{(j+1)p^{2}+p+1}. Since

d_{0}(−v_{1}^{p^{3}+p^{2}+p}v_{3}X_{3,r+1}^{sp}/v_{2}^{(j+1)p^{2}+p+1})
≡ −v_{1}^{p^{3}+p^{2}+p}X_{3,r+2}^{s}/v_{2}^{(j+1)p^{2}+p} {(t_{1}^{p^{2}} − v_{2}^{p−1}t_{1}) + v_{1}v_{2}^{p−1}w_{2,2}}

mod (v_{1}^{p^{3}+p^{2}+p+2}), we have

d_{0}(x(2)) ≡ −DX_{[1,0]}(v_{1}^{p^{3}+p^{2}+p+1}, v_{2}^{(j+1)p^{2}+1}, X_{3,r+2}^{s}),

and so N(2) = p^{3} + p^{2} + p. Applying Propositions 4.2.3 and 4.2.6, we can define x(k) inductively on k ≥ 3 and obtain N(k) = p^{k+1} + p^{k} + p^{k−1} + p^{k−2}. Finally we have completed the computations for all cases. □

54 Hirofumi Nakai

Now we can prove our main theorem.

Proof of Theorem. For each 𝔽_{p}-basic element (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} of Lemma 2.3.3, we have already constructed the element x(k) (= x(sp^{r}/j; k)), which satisfies the congruence x(sp^{r}/j; k) ≡ (x_{3,r}^{s}/v_{2}^{j})^{p^{k}} mod (v_{1}), and determined the v_{1}-divisibility N(k) (= N(s, r, j; k)) of x(k) as the smallest integer with d_{0}(x(k)) ≢ 0 mod (v_{1}^{N(k)+1}). We also define y(mp^{r}) by y(mp^{r}) = 1/X_{2,r}^{m} and obtain N(m; r) = a_{2,r} (Corollary 3.3.5). Using these results, let B^{0} be the direct sum of the cyclic modules as in Proposition 2.4.1. The linear independence of the set

{δ(x(sp^{r}/j; k)/v_{1}^{N(s,r,j;k)})} ∪ {δ(1/v_{1}^{a_{2,r}}X_{2,r}^{m})}

is verified by checking with condition (2.5.2). This completes the proof. □

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Department of Mathematics, Faculty of Science, Osaka City University nakai@sci.osaka-cu.ac.jp http://www.sci.osaka-cu.ac.jp/˜ nakai/

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