EVENS NORM, TRANSFERS AND CHARACTERISTIC

CLASSES FOR EXTRASPECIAL ppp-GROUPS

Pham Anh Minh

Abstract. Let P be the extraspecial p-group of order p^{2n+1}, of p-rank n+1, and of exponent p if p > 2. Let Z be the center of P and let κ_{n,r} be the characteristic classes of degree 2^{n} −2^{r} (resp. 2(p^{n} −p^{r})) for p = 2 (resp. p > 2), 0 ≤ r ≤ n−1, of a degree p^{n} faithful irreducible representation of P. It is known that, modulo nilradical, the ι^{th}-powers of the κ_{n,r}’s belong to T = Im(H^{∗}(P/Z, 𝔽_{p})/\sqrt{0} →^{Inf} H^{∗}(P, 𝔽_{p})/\sqrt{0}), with ι = 1 if p = 2, ι = p if p > 2. We obtain formulae in H^{∗}(P, 𝔽_{p})/\sqrt{0} relating the κ_{n,r}^{ι}’s to the ones of fewer variables. For p > 2 and for a given sequence r_{0}, …, r_{n−1} of non-negative integers, we also prove that, module nilradical, the element ∏_{r_{i}}κ_{n,i}^{r_{i}}

belongs to T if and only if either r_{0} ≥ 2, or all the r_{i}’s are multiples of p. This gives the determination of the subring of invariants of the symplectic group Sp_{2n}(𝔽_{p}) in T .

1. Introduction

Let p be a prime number. For a given group P, denote by H^{∗}(P) the modp cohomology algebra of P. We are interested in the case where P = P_{n}, the extraspecial p-group of order p^{2n+1}, of p-rank n + 1, and of exponent p if p > 2. It is known (see e.g. [GL, Q]) that, for p > 2 (resp. p = 2), there are exactly n Chern (resp. Stiefel-Whitney) classes κ_{n,r} of degree 2(p^{n} − p^{r}) (resp. 2^{n} − 2^{r}), 0 ≤ r ≤ n − 1, of a degree p^{n} faithful irreducible representation of P; these classes restrict to maximal elementary abelian subgroups of P as Dickson invariants.

Set E = E_{n} = P/Z, with Z the center of P. E is then a vector space of dimension 2n over 𝔽_{p}. Set h^{∗}(P) = H^{∗}(P)/\sqrt{0} (so h^{∗}(P) = H^{∗}(P) for p = 2, by [Q]) and denote by T = T_{n} the subring of h^{∗}(P) equal to the image of the inflation Inf_{P}^{E}, modulo nilradical. For p = 2, it follows from [Q] that all the κ_{n,r}’s belong to T . For p > 2, this fact does not hold, as shown in [GM, G]; however, in [GM], it is also also proved that all p^{th}-powers of the κ_{n,r}’s are, in turn, belonging to T .

For convenience, set ι = 1 for p = 2, and ι = p for p > 2. It follows that, for 0 ≤ r ≤ n − 1, there exits f_{n,r} ∈ H^{∗}(E) such that Inf_{P}^{E}(f_{n,r}) ≑ κ_{n,r}^{ι}. (Here and in what follows, a ≑ b means a = b modulo \sqrt{0}.) Via the inflation map, as elements of h^{∗}(P), f_{n,r} can be identified with κ_{n,r}^{ι}, 0 ≤ r ≤ n − 1. The first aim of this paper is to get an alternating formula expressing f_{n,r} by means of f_{n−1,r−1} and f_{n−1,r}. This work is motivated from the elegant formula, for p > 2 (resp. p = 2), expressing Chern (resp. Stiefel-Whitney) classes of the regular representation r_{A} of the elementary abelian p-group A via such classes of fewer variables. It is known that, if A is of rank m and p > 2 (resp. p = 2), then the 2(p^{m} − p^{r})^{th}-Chern (resp. (2^{m} − 2^{r})^{th}-Stiefel-Whitney) class of r_{A} is the Dickson invariant Q_{m,r} of the same degree with variables in a basis x_{1}, …, x_{m} of βH^{1}(A) (resp. H^{1}(A)), with β the Bockstein homomorphism. These invariants are related by

2000 Mathematics Subject Classification. 20J06.

Key words and phrases. Extraspecial p-groups, Chern classes, Stiefiel-Whitney classes, Evens norm, transfer.

Q_{m,r} = Q_{m−1,r}V_{m}^{p−1} + Q_{m−1,r−1}^{p},
where  V_{m} = ∏_{λ_{i}∈𝔽_{p}}(λ_{1}x_{1} + ⋯ + λ_{m−1}x_{m−1} + x_{m})

= (−1)^{m−1} ∑_{s=0}^{m−1}(−1)^{s}Q_{m−1,s}x_{m}^{p^{s}}

is the Mui invariant.

In so doing, we need to use the Evens norm and transfers from maximal subgroups of P. Some basic properties of the Evens norm, in the relation with modular invariants, are recalled in Section 2. In Section 3, we show how to obtain characteristic classes of P by means of the Evens norm (Theorem 3). Theorem 4 describes the image of such classes via the Evens norm. From this, we obtain formulae related characteristic classes with such classes of fewer variables (Corollary 4).

Let r_{0}, …, r_{n−1} be a sequence of non-negative integers. In Section 4, we prove that, for p > 2, modulo nilradical, the product ∏_{i≥0}κ_{n,i}^{r_{i}} belongs to T if and only if

either r_{0} ≥ 2, or all the r_{i}’s are multiples of p (Theorem 5). This generalizes a result, given in [GL, G], telling that κ_{n,0}^{s} belongs to T provided that either s ≥ 2^{n}, or s ≥ 2 and n ≤ 2. As a consequence, we obtain in the last section the determination of the subring of invariants of the symplectic group in T (Theorem 6).

For convenience, given a subgroup K of a group G, any element of H^{∗}(G) is also considered as an element of H^{∗}(K) via the restriction map Res_{K}^{G}. Also, if K is normal in G, then any element of H^{∗}(G/K) can be considered as an element of H^{∗}(G) via the inflation map Inf_{G}^{G/K}.

2. Evens norm and Mui invariants

Given a polynomial algebra F = 𝔽_{p}[t_{1}, …, t_{k}] and 1 ≤ m ≤ k − 1, define the Mui invariant ([Mu])

\begin{aligned}V_{m+1} & = V_{m+1}(t_{1}, …, t_{m+1}) \\ & = ∏_{λ_{i}∈𝔽_{p}}(λ_{1}t_{1} + ⋯ + λ_{m}t_{m} + t_{m+1}). \end{aligned}  (1)

It follows that (see e.g. [D])

V_{m+1} = (−1)^{m} ∑_{s=0}^{m}(−1)^{s}Q_{m,s}t_{m+1}^{p^{s}}

with Q_{m,s} = Q_{m,s}(t_{1}, …, t_{m}) the Dickson invariants defined inductively as follows (we shall omit the variables, if no confusion can arise) :

Q_{m,m} = 1,
Q_{m,0} = ∏_{λ_{j}∈𝔽_{p} λ_{j} not all equal to 0}(λ_{1}t_{1} + … + λ_{m}t_{m})^{p−1},

Q_{m,s} = Q_{m−1,s}V_{m}^{p−1} + Q_{m−1,s−1}^{p}.

By (1), the Q_{m,s}’s are independent of the choice of generators t_{1}, …, t_{m} of 𝔽_{p}[t_{1}, …, t_{m}]. Hence, if t_{1}, …, t_{m} is a basis of H^{1}(W) (resp. βH^{1}(W)) with W an elementary abelian 2-group (resp. p-group with p > 2) of rank m, we may write

\begin{aligned}Q_{m,s}(t_{1}, …, t_{m}) & = Q_{s}(W) \\ V_{m+1}(t_{1}, …, t_{k}, X) & = V (W, X). \end{aligned}

Mui invariants can be obtained by means of Evens norm map N_{U→W} with U a subgroup of W (see Corollary 1 below). Let us recall that, for every maximal subgroup K of a p-group G, and for ξ ∈ H^{r}(K), we may define the Evens norm map

N_{K→G}(ξ) ∈ H^{pr}(G).

Here are some properties of N_{K→G}. For details of the proof, the reader can refer to [E, M2, Mu].

Proposition 1. Let G, G′ be p-groups and let K be a subgroup of G.

(i) If N is a subgroup of K, then N_{N→G} = N_{K→G}N_{N→K}. (ii) If H is a subgroup of G and G = ∐_{x∈D}KxH, then, for ξ ∈ H^{r}(G),

Res_{H}^{G}N_{K→G}(ξ) = ∏_{x∈D} N_{H∩^{x}K→H}Res_{H∩^{x}K}^{^{x}K}(^{x}ξ).

(iii) If K′ is a subgroup of G′ and f : G′ → G is a homomorphism such that f(K′) ⊂ K and f induces a bijection G′/K′ ≅ G/K of cosets spaces, then, for ξ ∈ H^{r}(K),

N_{K′→G′}(f|_{K′})^{∗}(ξ) = f^{∗}(N_{K→G}(ξ)).

In particular, if N is a normal subgroup of G and N ⊂ K, then, for ξ ∈ H^{r}(K/N),

N_{K→G}Inf_{K}^{K/N}(ξ) = Inf_{G}^{G/N}N_{K/N→G/N}(ξ).

(iv) If ξ, ξ′ ∈ H^{n}(K), then

N_{K→G}(ξ + ξ′) = N_{K→G}(ξ) + N_{K→G}(ξ′)

+ a sum of transfers from proper subgroups of G containing the intersection of the conjugates of K.

Hence, the norm map is in general non-additive, although N_{K→G} ∘ Res_{K}^{G} is.

(v) If ξ ∈ H^{r}(K), ξ′ ∈ H^{s}(K), and [G : K] = n, then

N_{K→G}(ξ · ξ′) = (−1)^{\frac{n(n−1)}{2} rs}N_{K→G}(ξ) · N_{K→G}(ξ′).

(vi) Assume that G = K × E with E = (ℤ/p)^{m}. Consider E as the group of all translations on a vector space S of dimension m over ℤ/p and let W (m) be a E-free acyclic complex with augmentation ϵ : W (m) → ℤ/p.

Let C be a cochain complex of which the cohomology is H^{∗}(K) and set

C^{S} = ⊗_{c∈S}C_{c}, C_{c} = C. Then

N_{K→K×E} = d_{m}^{∗}P_{m},

where H^{r}(C) ⟶^{P_{m}} H_{E}^{p^{m}r}(W (m) ⊗ C^{S}) is the Steenrod power map, and

H_{E}^{p^{m}r}(W (m)⊗C^{S}) ⟶^{d_{m}^{∗}} H^{∗}(E)⊗H^{∗}(C) is induced by the diagonal C → C^{S} and the Kunneth formula. □

In the rest of this section, suppose that W is an elementary abelian p-group of rank n + 1 and U a subgroup of W of index p^{m}.

By Proposition 1 (vi), N_{U→W} = d_{m}^{∗}P_{m}. The first part of the following corollary is then originally due to Mui ([Mu]) and reproved by Okuyama-Sasaki ([OS]); the second one was given in [HM, Proof of Theorem B].

Corollary 1. For p = 2 (resp. p > 2) and for every x ∈ H^{1}(W) (resp. x ∈ βH^{1}(W)),

(i) N_{U→W} (Res_{U}^{W} (x)) = V (W/U, x); (ii) with T the maximal subgroup of W satisfying Res_{T}^{W} (x) = 0,

(−1)^{r}N_{T →W} (Q_{r}(T)) = ∑_{i=r}^{n}(−1)^{i}Q_{i}^{p}(T)x^{p^{i+1}−p^{r+1}}

− [ ∑_{i=r+1}^{n}(−1)^{i}Q_{i}(T)x^{p^{i}−p^{r+1}}][∑_{i=0}^{n}(−1)^{i}Q_{i}(T)x^{p^{i}}]^{p−1}.  □

In the following corollary, G is supposed to be a p-group given by a central extension

0 → ℤ/p → G →^{j} W → 0

and K = j^{−1}(U). Set

H^{ev}(U) = \{ \begin{matrix}H^{∗}(U) & for p = 2, \\ ∑_{n≥0}H^{2n}(U) & for p > 2. \end{matrix}

The following is straightforward from Proposition 1.

Corollary 2. The composition map

H^{ev}(U) →^{Inf_{K}^{U}} H^{∗}(K) →^{N_{K→G}} H^{∗}(G)

is a ring homomorphism. □

In [M2], we proved

Proposition 2. Let ξ ∈ H^{q}(G). Set μ(q) = (h!)(−1)^{hq(p−1)/2} with h = (p − 1)/2 for p > 2. If K = ker u with 0 ≠ u ∈ H^{1}(G), then, by setting v = βu, we have

N_{K→G}Res_{K}^{G}(ξ) = \{ \begin{matrix}∑_{i}Sq^{i}(ξ) · u^{q−i} & for p = 2, \\ μ(q)_{∑_{ϵ=0,1} ; 0≤2i≤q−ϵ}(−1)^{ϵ+i}β^{ϵ}P^{i}(ξ) · v^{(q−2i)h−ϵ}u^{ϵ} & for p > 2, \end{matrix}

with Sq^{i} (resp. P^{i}) the Steenrod operations for p = 2 (resp. p > 2). □

3. Characteristic classes for extraspecial p-groups

Let E = E_{n}, n ≥ 1, be the elementary abelian p-group of rank 2n. Let x_{1}, …, x_{2n} be a basis of H^{1}(E) = Hom(E, 𝔽_{p}) and define

y_{i} = \{ \begin{matrix}x_{i} & for p = 2, \\ βx_{i} & for p > 2, \end{matrix}

1 ≤ i ≤ 2n, with β the Bockstein homomorphism. We have

H^{∗}(E) = \{ \begin{matrix}𝔽_{2}[y_{1}, …, y_{2n}] & if p = 2, \\ Λ[x_{1}, …, x_{2n}] ⊗ 𝔽_{p}[y_{1}, …, y_{2n}] & if p > 2 \end{matrix}

with Λ[s, t, …] (resp. 𝔽_{p}[s, t, …]) the exterior (resp. polynomial) algebra with generators s, t, … over 𝔽_{p}. Let P = P_{n} be the extraspecial p-group given by the central extension

1 → ℤ/p →^{i} P_{n} → E → 1

classified by the cohomology class x_{1}x_{2} + ⋯ + x_{2n−1}x_{2n} ∈ H^{2}(E).

The following notation will be used. Set Z = i(ℤ/p), the center of P. For every elementary abelian A of P containing Z, write A/Z = A′; so A = A′ × Z, and A′ is of rank n if A is maximal elementary abelian in P. Fix a generator γ of H^{1}(Z) (resp. βH^{1}(Z)) for p = 2 (resp. p > 2). γ, and also every element of H^{∗}(A′), are then considered as elements of H^{∗}(A) via the inflation maps.

Denote by A the set of maximal elementary abelian subgroups of P. Set h^{∗}(P) = H^{∗}(P)/\sqrt{0}. By [Q], the map induced by the restrictions

h^{∗}(P) →^{Res} ∏_{A∈A} h^{∗}(A)

is injective. Therefore the maps

h^{∗}(E) →^{Inf_{P}^{E}} h^{∗}(P)
and  h^{∗}(E) →^{Res} ∏_{A∈A} h^{∗}(A′)

have the same kernel. Let T = T_{n} be the subring of h^{∗}(P) equal the image of the inflation Inf_{P}^{E}. For elements ξ, η of h^{∗}(E), it follows that Inf_{P}^{E}(ξ) = Inf_{P}^{E}(η) if and only if Res_{A′}^{E}(ξ) = Res_{A′}^{E}(η), for every A ∈ A.

We are now interested in Chern (resp. Stiefel-Whitney) classes, for p > 2 (resp. p = 2), of a degree p^{n} faithful irreducible representation of P. Fix a nontrivial linear character χ of Z. We have then an irreducible character χ̂ of P given by

χ̂(g) = \{ \begin{matrix}p^{n}χ(g) & , if g ∈ Z, \\ 0 & , otherwise. \end{matrix}

Let ρ be a representation of P affording the character χ̂ and set

ζ = ζ_{n} = \{ \begin{matrix}c_{p^{n}}(ρ) & for p > 2, \\ w_{2^{n}}(ρ) & for p = 2, \end{matrix} κ_{n,r} = \{ \begin{matrix}(−1)^{n−r}c_{p^{n}−p^{r}}(ρ) & for p > 2, \\ w_{2^{n}−2^{r}}(ρ) & for p = 2, \end{matrix}

0 ≤ r ≤ n. We have Theorem 1 ([GL, Q]).

(i) In h^{∗}(P),

(ia) 1 + κ_{n,n−1} + ⋯ + κ_{n,0} + ζ_{n} = \{ \begin{matrix}c(ρ) & for p > 2, \\ w(ρ) & for p = 2; \end{matrix}

(ib) the subring of h^{∗}(P) generated of non-nilpotent Chern classes, is gen-

erated by κ_{n,0}, …, κ_{n,n−1}, y_{1}, …, y_{2n}, ζ_{n}.

(ii) For every 0 ≤ i ≤ n − 1 and for every A ∈ A,

Res_{A}^{P}(ζ) = V (A′, γ),
Res_{A}^{P}(κ_{n,i}) = Q_{i}(A′).  □

In [GM], Chern classes of P are also obtained by means of the inflation Inf_{P}^{E} and transfer maps tr_{P}^{K} with K maximal in P. Similar results for the case p = 2 can also be obtained by using the same argument. The result can be stated as follows. Let x be a non-zero element of H^{1}(P) and set H_{x} = ker x. Pick a rank one subgroup U ≠ Z of the center of H_{x}. So H_{x} = P_{n−1} × U. By Künneth formula, we can consider any element of H^{∗}(P_{n−1}) (and of H^{∗}(U)) as an element of H^{∗}(H_{x}). For 0 ≤ r ≤ n − 1, set

χ_{r,x} = \{ \begin{matrix}tr_{P}^{H_{x}}(κ_{n−1,r}ζ_{n−1}^{p−1}) & if n ≥ 2, \\ tr_{P}^{H_{x}}(ζ_{n−1}^{p−1}) & if n = 1. \end{matrix}

For r ≥ 0, define

z_{n}^{(r)} = \{ \begin{matrix}∑_{i=1}^{n}y_{2i−1}y_{2i} & if p = 2 and r = 0, \\ ∑_{i=1}^{n}(y_{2i−1}^{ιp^{r}}y_{2i} − y_{2i−1}y_{2i}^{ιp^{r}}) & otherwise, \end{matrix}

with ι = 1 for p = 2, and ι = p for p > 2. Let A_{0} be an element of A. We have

Theorem 2 ([GM]).

(i) In h^{∗}(P), for 0 ≤ r ≤ n − 1, κ_{n,r} = Q_{r}(P/A_{0}) − ∑_{x∈ℙH^{1}(P/A_{0})}χ_{r,x}.

(ii) There exist f_{n,0}, …, f_{n,n−1} ∈ H^{∗}(E), viewed as elements of H^{∗}(P) via the inflation map, such that

z_{n}^{(n)} + ∑_{i=0}^{n−1}(−1)^{n−i}z_{n}^{(i)}f_{n,i} = 0,

and, for every A ∈ A, Res_{A}^{P}(κ_{n,r}^{ι}) = Res_{A}^{P}(f_{n,r}), 0 ≤ r ≤ n − 1. (iii) There exist h_{i}, 0 ≤ i ≤ n − 1, and a unique η of H^{∗}(E) such that

z_{n}^{(n−1)} = y_{2n}η + ∑_{i=0}^{n−2} h_{i}z_{n}^{(i)},

and, in h^{∗}(P), χ_{n−1,x_{2n}}^{ι} = −Inf(η^{p−1}). Furthermore, for all 0 ≤ r ≤ n − 1 and all ϕ ∈ ℙH^{1}(E), χ_{r,ϕ}^{ι} ∈ Inf_{P}^{E}, as elements of h^{∗}(P). □

EVENS NORM, TRANSFERS AND CHARACTERISTIC CLASSES 7

By [Q], it is known that, for p = 2, all the κ_{n,r}’s and χ_{r,ϕ}’s belong to T . For p > 2, it follows from the above theorem all p^{th}-powers of the κ_{n,r}’s and χ_{r,ϕ}’s belong to T . In fact, by setting

φ = \{ \begin{matrix}y_{2n−1}^{p} − y_{2n−1}y_{2n}^{p−1} & for p > 2, \\ y_{2n−1} & for p = 2, \end{matrix}

we have Corollary 3. In h^{∗}(P),

(i) κ_{n,r}^{ι} = f_{n,r}, 0 ≤ r ≤ n − 1;

(ii) η = (−1)^{n−1}[φf_{n−1,0} + ∑_{i=1}^{n−1}(−1)^{i}(y_{2n−1}^{ιp^{i}} − y_{2n−1}y_{2n}^{ιp^{i}−1})f_{n−1,i}];

(iii) with K = ker x_{2n},

η = \{ \begin{matrix}tr_{P}^{K}(ζ_{n−1}) & for p = 2, \\ (−1)^{n−1}tr_{P}^{K}[ζ_{n−1}^{p−1}∑_{i=0}^{n−1}(−1)^{i}y_{2n−1}^{p^{i}}κ_{n−1,i}] & for p > 2; \end{matrix}

(iv) for 0 ≤ r ≤ n − 1,

χ_{r,x_{2n}}^{ι} = −f_{n−1,r}[φf_{n−1,0} + ∑_{i=1}^{n−1}(−1)^{i}(y_{2n−1}^{ιp^{i}} − y_{2n−1}y_{2n}^{ιp^{i}−1})f_{n−1,i}]^{p−1}.

Proof. Part (i) follows from Theorem 2 (ii), by noting that the restriction map from h^{∗}(P) to ∏_{A∈A}H^{∗}(A) is injective.

We have, by Theorem 2,

z_{n}^{(n−1)} = z_{n−1}^{(n−1)} + (y_{2n−1}^{ιp^{n−1}}y_{2n} − y_{2n−1}y_{2n}^{ιp^{n−1}})

= (y_{2n−1}^{ιp^{n−1}}y_{2n} − y_{2n−1}y_{2n}^{ιp^{n−1}}) + (−1)^{n} ∑_{i=0}^{n−2}(−1)^{i}z_{n−1}^{(i)}f_{n−1,i}

= (y_{2n−1}^{ιp^{n−1}}y_{2n} − y_{2n−1}y_{2n}^{ιp^{n−1}}) + (−1)^{n}[∑_{i=0}^{n−2}(−1)^{i}z_{n}^{(i)}f_{n−1,i}

− y_{2n}φf_{n−1,0} − ∑_{i=1}^{n−2}(−1)^{i}(y_{2n−1}^{ιp^{i}}y_{2n} − y_{2n−1}y_{2n}^{ιp^{i}})f_{n−1,i}]

= (−1)^{n}[∑_{i=0}^{n−2}(−1)^{i}z_{n}^{(i)}f_{n−1,i} − y_{2n}X]

with X = φf_{n−1,0} + ∑_{i=1}^{n−1}(−1)^{i}(y_{2n−1}^{ιp^{i}} − y_{2n−1}y_{2n}^{ιp^{i}−1})f_{n−1,i}. So η = (−1)^{n+1}X. (ii)

and (iv) are proved.

Pick an element A ∈ A. By [GM, Lemma 7.1 and its proof], we have

Res_{A}^{P}(η) = \{ \begin{matrix}−V (B′, y_{2n−1})^{ι} & if A ⊂ K, \\ 0 & otherwise. \end{matrix}

Set

Y = \{ \begin{matrix}tr_{P}^{K}(ζ_{n−1}) & for p = 2, \\ (−1)^{n−1}tr_{P}^{K}[ζ_{n−1}^{p−1}∑_{i=0}^{n−1}(−1)^{i}y_{2n−1}^{p^{i}}κ_{n−1,i}] & for p > 2. \end{matrix}

If A ⊄ K, then Res_{A}^{P}(Y) = 0, by Mackey formula. Suppose that A ⊂ K. Set B = A ∩ P_{n−1}. For p > 2, we have

Res_{A}^{P}(Y) = (−1)^{n−1} ∑_{g∈P/K} ^{g} ∑_{i=0}^{n−1} Res_{A}^{K}((−1)^{i}y_{2n−1}^{p^{i}}κ_{n−1,i}ζ_{n−1}^{p−1})

= (−1)^{n−1} ∑_{g∈P/K} ^{g} ∑_{i=0}^{n−1}(−1)^{i}y_{2n−1}^{p^{i}}Q_{i}(B′)ζ_{n−1}^{p−1}

= (−1)^{n−1} ∑_{i=0}^{n−1}(−1)^{i}y_{2n−1}^{p^{i}}Q_{i}(B′) ∑_{g∈P/K} ^{g}ζ_{n−1,i}^{p−1}

since y_{2n−1} and the Q_{i}(B′)’s are invariant under the action of P/K. Thus

Res_{A}^{P}(Y) = \{ \begin{matrix}Res_{K}^{P}tr_{P}^{K}(ζ_{n−1}) & for p = 2, \\ V (B′, y_{2n−1})Res_{K}^{P}tr_{P}^{K}(ζ_{n−1}^{p−1}) & for p > 2. \end{matrix}

Following [GM, Proposition 4.4], we have

Res_{A}^{P}tr_{P}^{K}(ζ_{n−1}^{p−1}) = −V (B′, y_{2n−1})^{p−1}.

So Res_{A}^{P}(Y) = −V (B′, y_{2n−1})^{ι}.

Since η − Y restricts trivially to every element of A, it follows that η ≑ Y. The corollary is proved. □

We also have

Proposition 3. For 0 ≤ r ≤ n − 1,

κ_{n,r} ≑ − ∑_{x∈ℙH^{1}(P)} χ_{r,x}.

Proof. Let A be an element of A. There exist exactly \frac{p^{n}−1}{p−1} elements of ℙH^{1}(P) of which the kernel contains A. The subset of ℙH^{1}(P) consisting of those elements is nothing but ℙH^{1}(P/A).

Let x be an element of ℙH^{1}(P). It is clear Res_{A}^{P}(χ_{r,x}) = 0 if x ∉ ℙH^{1}(P/A). Hence

Res_{A}^{P}( ∑_{x∈ℙH^{1}(P)} χ_{x,r}) = Res_{A}^{P}( ∑_{x∈ℙH^{1}(P/A)} χ_{r,x}).

Therefore, by [GM, Proof of Theorem 5.2],

\begin{aligned}Res_{A}^{P}( ∑_{x∈ℙH^{1}(P)} χ_{r,x}) & = Res_{A}^{P}(Q_{r}(P/A) − κ_{n,r}) \\ & = −Q_{r}(A′) \end{aligned}

since Res_{A}^{P}(Q_{r}(P/A)) = 0 and Res_{A}^{P}(κ_{n,r}) = Q_{r}(A′). The proposition follows. □

We are now going to obtain characteristic classes of P by using the Evens norm map. We first need

Lemma 1. Fix a generator e of Z. Let H = A ∩ B with A, B ∈ A and let h_{1}, …, h_{k}, e be a basis of H. Then there exist elements g_{1}, …, g_{k} of P satisfying

(i) [g_{i}, g_{j}] = 1, [g_{i}, h_{j}] = \{ \begin{matrix}1 & if i ≠ j, \\ e & if i = j, \end{matrix} , 1 ≤ i, j ≤ k;

(ii) P = ∐_{g∈G}AgB is a double coset decomposition of P with G = ⟨g_{1}, …, g_{k}⟩.

Proof. The existence of the g_{j} satisfying (i) follows from [M1]. Assume that agb = a′g′b′ with a, a′ ∈ A, b, b′ ∈ B, g, g′ ∈ G. It follows that [g, h_{i}] = [g′, h_{i}], 1 ≤ i ≤ k, hence g = g′. As |G| = p^{k}, (ii) is obtained. The lemma follows. □

The following notation will be used. Let C be the cyclic group of order p and fix a generator u of H^{1}(C) (resp. H^{2}(C)) for p = 2 (resp. p > 2). Set Γ = P × C. If H is a subgroup of P, by Künneth formula, every element of H^{∗}(H) (resp. H^{∗}(C)) can be considered as an element of H^{∗}(H × C). We have

Lemma 2. Let A, B be elements of A and let v ∈ H^{1}(A × C) (resp. H^{2}(A × C)) for p = 2 (resp. p > 2). Assume that Res_{Z×C}^{A×C}(v) = μγ + λu with μ, λ ∈ 𝔽_{p}, then

Res_{B×C}^{Γ}N_{A×C→Γ}(v) = μV (B′, γ) + λV (B′, u).

Proof. Let P = ∪_{g∈G}BgA be the double coset decomposition of P given in Lemma

1. Set H = B ∩ A. We have

Res_{B×C}^{Γ}N_{A×C→Γ}(v) = ∏_{g∈G} N_{(B∩A^{g})×C→B×C}Res_{(B∩A^{g})×C}^{A^{g}×C}(^{g}v)

= ∏_{g∈G} N_{H×C→B×C}Res_{H×C}^{A×C}(^{g}v)  since A is normal
= N_{H×C→B×C}( ∏_{g∈G} Res_{H×C}^{A×C}(^{g}v))  by Corollary 2

= N_{H×C→B×C}( ∏_{g∈G}(μ^{g}γ + λu)

= N_{H×C→B×C}(V (H′, μγ + λu))  by Lemma 1
= N_{Z×C→B×C}(μχ + λu)  by Corollary 1
= μV (B′, γ) + λV (B′, u)  by Corollary 2.

The lemma follows. □

The following shows that characteristic classes of P can be obtained by means of the Evens norm map.

Theorem 3. Let A be an element of A and let v be an element of H^{1}(A) (resp. βH^{1}(A)) for p = 2 (resp. p > 2) satisfying Res_{Z}^{A}(v) = γ. Set

ζ_{A,v} = N_{A×C→Γ}(v + u) − N_{A×C→Γ}(v).

Then, as elements of h^{∗}(P),

ζ_{A,v} = (−1)^{n} ∑_{s=0}^{n}(−1)^{s}κ_{n,s}u^{p^{s}}.

Proof. For every B ∈ A, by Lemma 2, we have

Res_{B×Z}^{Γ}(ζ_{A,v}) = Res_{B×Z}^{Γ}N_{A×Z→Γ}(v + u) − Res_{B×Z}^{Γ}N_{A×Z→Γ}(v)
= V (B′, γ + u) − V (B′, γ)
= V (B′, u),

since V (B′, X), as a function on X, is additive. By Theorem 1(ii),

Res_{B×Z}^{Γ}(ζ_{A,v}) = Res_{B×Z}^{Γ}[(−1)^{n} ∑_{s=0}^{n} κ_{n,s}u^{p^{s}}].

So ζ_{A,v} ≑ (−1)^{n}∑_{s=0}^{n}(−1)^{s}κ_{n,s}u^{p^{s}}. □

Remark. Write N_{A×C→Γ} = N. It follows from the above theorem and from Corollary 1 that

N (v + u) − N (v) − N (u) ≑ (−1)^{n} ∑_{s=0}^{n}(−1)^{s}[κ_{n,s} − Q_{s}(P/A)]u^{p^{s}}.

According to Proposition 1 (iv), the κ_{n,s}’s can be expressed are sums of transfers from maximal subgroups of P × C. Such formulae are the ones given in Theorem 2.

Let a_{1}, a_{2}, …, a_{2n−1}, a_{2n} be elements of P satisfying x_{i}(a_{j}) = δ_{ij} with δ_{ij} the Kronecker symbol, 1 ≤ i, j ≤ 2n. Suppose now that K is the maximal subgroup of P given by K = ker x_{2n}. So K ≅ P_{n−1} × ⟨a_{2n−1}⟩ ≅ P_{n−1} × ℤ/p. Write y_{2n} = y, N_{K×C→Γ} = N, and, for 0 ≤ r ≤ n − 1, χ_{r,x_{2n}} = χ_{r}. Define

θ_{n−1,r} = Res_{K}^{P}(χ_{r}) ∈ H^{∗}(K),

and θ_{n−1} = u^{p^{n}} + ∑_{r=0}^{n−1}[(−1)^{n−r}u^{p^{r}}(−θ_{n−1,r} + κ_{n−1,r−1}^{p})] ∈ H^{∗}(K × C) (with the

convention that κ_{n−1,−1} = 0). We prove Theorem 4. As elements of h^{∗}(P),

(−1)^{n}N (θ_{n−1}) = ∑_{s=0}^{n}(−1)^{s}κ_{n,s}^{p}u^{p^{s+1}},
(−1)^{r}N (κ_{n−1,r}^{ι}) = ∑_{i=r}^{n−1}(−1)^{i}κ_{n−1,i}^{ιp}y^{ι(p^{i+1}−p^{r+1})}

− [ ∑_{i=r+1}^{n−1}(−1)^{i}κ_{n−1,i}^{ι}y^{ι(p^{i}−p^{r+1})}][∑_{i=0}^{n−1}(−1)^{i}κ_{n−1,i}^{ι}y^{ιp^{i}}]^{p−1},

for 0 ≤ r ≤ n − 2. Proof. For convenience, write κ_{n−1,r} = κ_{r}, 0 ≤ r ≤ n − 1. Let A be an element of A and set

\begin{aligned}X & = N (θ_{n−1}), Y_{r} = N (κ_{n−1,r}^{ι}), \\ Z_{r} & = ∑_{i=r}^{n−1}(−1)^{i}κ_{i}^{ιp}y^{ι(p^{i+1}−p^{r+1})} − [ ∑_{i=r+1}^{n−1}(−1)^{i}κ_{i}^{ι}y^{ι(p^{i}−p^{r+1})}][∑_{i=0}^{n−1}(−1)^{i}κ_{i}^{ι}y^{ιp^{i}}]^{p−1}, \end{aligned}

0 ≤ r ≤ n − 2. Consider the following cases:

- A ⊂ K: by setting B = A ∩ P_{n−1}, we have A = B × ⟨a_{2n−1}⟩. So

Res_{K×C}^{Γ}(X) = ∏_{x∈⟨a_{2n}⟩} ^{x}θ_{n−1}.

As the θ_{n−1,r}’s belong to Im Res_{K}^{P}, they are invariant under the action of a_{2n}. Hence

Res_{A×C}^{Γ}(X) = ∏_{x∈⟨a_{2n}⟩} Res_{A×C}^{K×C}(^{x}θ_{n−1})

= Res_{A×C}^{K×C}(θ_{n−1}^{p})

= u^{p^{n+1}} + [∑_{r=0}^{n−1}[(−1)^{n−r}u^{p^{r}}(Q_{r}(B′)V (B′, y_{2n−1})^{p−1} + Q_{r−1}^{p}(B′))]^{p}

= V (A′, u)^{p}

= Res_{A×C}^{Γ}((−1)^{n} ∑_{s=0}^{n}(−1)^{s}κ_{n,s}^{p}u^{p^{s+1}}).

Also, for 0 ≤ r ≤ n − 2,

Res_{A×C}^{Γ}(Y_{r}) = ∏_{x∈⟨a_{2n}⟩} Res_{A×C}^{K×C}(x^{∗}κ_{n−1,r}^{ι})
= [Res_{A×C}^{K×C}(κ_{n−1,r})^{ι}]^{p}
= Q_{r}^{ιp}(B′)
= (−1)^{r}Res_{A×C}^{Γ}(Z_{r});

12 PHAM ANH MINH

- A ⊄ K: by setting H = K ∩ A, we have

Res_{A×C}^{Γ}(X) = N_{H×C→A×C}Res_{H×C}^{K×C}(X)

= N_{H×C→A×C}(V (H′, u)^{p})

= V (A′, u)^{p}

= Res_{A×C}^{Γ}((−1)^{n} ∑_{s=0}^{n}(−1)^{s}κ_{n,s}^{p}u^{p^{s+1}}),

Res_{A×C}^{Γ}(Y_{r}) = N_{H×C→A×C}Res_{H×C}^{K×C}(Y_{r})

= N_{H×C→A×C}(Q_{r}^{ι}(H/Z)) = (−1)^{r}Res_{A×C}^{Γ}(Z_{r}).

The theorem follows. □

Formulae relating the κ_{n,r}^{ι} to such classes of fewer variables are given by Corollary 4. For 0 ≤ r ≤ n − 1, as elements of h^{∗}(P),

\begin{aligned}κ_{n,r}^{ι} & = κ_{n−1,r−1}^{ιp} + κ_{n−1,r}^{ι}[[∑_{i=0}^{n−1}(−1)^{i}κ_{n−1,i}^{ι}y^{ιp^{i}}]^{p−1} \\ & + [κ_{n−1,0}^{ι}φ + ∑_{i=1}^{n−1}(−1)^{i}κ_{n−1,i}^{ι}(y_{2n−1}^{ιp^{i}} − y_{2n−1}y^{ιp^{i}−1})]^{p−1}]. \end{aligned}

Proof. By Corollary 2, we have

N (θ_{n−1}) = N (u^{p^{n}}) + ∑_{r=0}^{n−1} N [(−1)^{n−r}u^{p^{r}}(−θ_{n−1,r} + κ_{n−1,r−1}^{p})]

= V (y, u)^{p^{n}} + ∑_{r=0}^{n−1}(−1)^{n−r}V (y, u)^{p^{r}}[N (−θ_{n−1,r}) + N (κ_{n−1,r−1}^{p})]

= (u^{p} − uy^{p−1})^{p^{n}} + ∑_{r=0}^{n−1}(−1)^{n−r}(u^{p} − uy^{p−1})^{p^{r}}[N (−θ_{n−1,r}) + N (κ_{n−1,r−1}^{p})]

= u^{p^{n+1}} + ∑_{r=0}^{n−1}(−1)^{n−r}u^{p^{r+1}}[N (−θ_{n−1,r}) + N (κ_{n−1,r−1}^{p})

+ y^{(p−1)p^{r+1}}(N (−θ_{n−1,r+1}) + N (κ_{n−1,r}^{p}))] + (−1)^{n}uy^{p−1}N (−θ_{n−1,0}). By the Frobenius formula, the cup-product of χ_{r} with each of x_{2n}, y_{2n} vanishes.

As the transfer commutes with Steenrod operations, we have, by Proposition 2 and Theorem 4,

u^{p^{n+1}} + ∑_{r=0}^{n−1}(−1)^{n−r}κ_{n,r}^{p}u^{p^{r+1}} = N (θ_{n−1})

= u^{p^{n+1}} + ∑_{r=0}^{n−1}(−1)^{n−r}u^{p^{r+1}}[−χ_{r}^{p} + N (κ_{n−1,r−1}^{p})

+ y^{(p−1)p^{r+1}}(−χ_{r+1}^{p} + N (κ_{n−1,r}^{p}))]

= u^{p^{n+1}} + ∑_{r=0}^{n−1}(−1)^{n−r}u^{p^{r+1}}[−χ_{r}^{p} + N (κ_{n−1,r−1}^{p}) + y^{(p−1)p^{r}+1}N (κ_{n−1,r}^{p})].

Therefore

∑_{r=0}^{n−1}(−1)^{n−r}κ_{n,r}^{ι}u^{ιp^{r}} = ∑_{r=0}^{n−1}(−1)^{n−r}u^{ιp^{r}}[−χ_{r}^{ι} + N (κ_{n−1,r−1}^{ι}) + y^{(ιp−1)p^{r}}N (κ_{n−1,r}^{ι})].

Hence

κ_{n,r}^{ι} = −χ_{r}^{ι} + N (κ_{n−1,r−1}^{ι}) + y^{(p−1)ιp^{r}}N (κ_{n−1,r}^{ι}).

Since

N (κ_{n−1,r−1}^{ι}) + y^{(p−1)ιp^{r}}N (κ_{n−1,r}^{ι}) ≑κ_{n−1,r−1}^{ιp} + κ_{n−1,r}^{ι}[∑_{i=0}^{n−1}(−1)^{i}κ_{n−1,i}^{ι}y^{ιp^{i}}]^{p−1},

by Theorem 4, we obtain

κ_{n,r}^{ι} ≑ −χ_{r}^{ι} + κ_{n−1,r−1}^{ιp} + κ_{n−1,r}^{ι}[∑_{i=0}^{n−1}(−1)^{i}κ_{n−1,i}^{ι}y^{ιp^{i}}]^{p−1}.

The corollary follows from Corollary 3. □

4. The subring 𝔽_{p}[κ_{n,0}, …, κ_{n,n−1}] ∩ T

In this section, p is supposed to be an odd prime. It was proved in [GL, G] that, κ_{n,0}^{s} ∈ T provided that s ≥ 2^{n}, or s ≥ 2 and n ≤ 2. This result can be sharpened as follows. Let ℜ_{n} be the set consisting of sequences R = (r_{0}, r_{1}, …, r_{n−1}) of non-negative integers. For R = (r_{0}, …, r_{n−1}) ∈ ℜ_{n} and for m > 0, set

\begin{aligned}s_{R} & = ∑_{i≥0}r_{i}, \\ κ_{m}^{R} & = \{ \begin{matrix}∏_{i=0}^{m−1}κ_{m,i}^{r_{i}} & if m ≤ n, \\ ∏_{i=0}^{n−1}κ_{m,i}^{r_{i}} & if m > n. \end{matrix} \end{aligned}

The main purpose of this section is to prove the following Theorem 5. Let R = (r_{0}, …, r_{n−1}) be an element of ℜ_{n}. As an element of h^{∗}(P), κ_{n}^{R} belongs to T if and only if one of the following conditions is satisfied:

(R_{1}) r_{0} ≥ 2;

(R_{2}) r_{0} = 0 and all the r_{i}’s, i > 0, are multiples of p.

The rest of the section is devoted to the proof of the theorem. By Corollary 3 (i), κ_{n}^{R} ∈ T if R satisfies (R_{2}). We shall prove

Proposition 4. If R ∈ ℜ_{n} satisfies (R_{1}), then κ_{n}^{R} ∈ T .

By [GL, L], the proposition holds for n = 1. Suppose inductively that it holds for n − 1. Set K = ker x_{2n} = P_{n−1} × ℤ/p and T′ = Im Inf_{K}^{K/Z} + \sqrt{0}. Write

w = Res_{K}^{P}(y_{2n−1}) and ψ = (−1)^{n−1}∑_{j=0}^{n−1}(−1)^{j}κ_{n−1,j}w^{p^{j}}. We have

Res_{K}^{P}(κ_{n,j}) ≑ κ_{n−1,j−1}^{p} + κ_{n−1,j}ψ^{p−1}, 0 ≤ j ≤ n.

So, for every element R ∈ ℜ_{n}, as elements of h^{∗}(K),

Res_{K}^{P}(κ_{n}^{R}) = ∏_{j=0}^{n−1}[κ_{n−1,j−1}^{p} + κ_{n−1,j}ψ^{p−1}]^{r_{j}}

= κ_{n−1,0}^{r_{0}}ψ^{r_{0}(p−1)} ∏_{j=1}^{n−1}[κ_{n−1,j−1}^{p} + κ_{n,j}ψ^{p−1}]^{r_{j}}

= κ_{n−1}^{R}ψ^{(p−1)s_{R}} + ∑_{r_{0}≤t<s_{R}} ρ_{t}ψ^{(p−1)t}  (2)

with ρ_{t} ∈ h^{∗}(P_{n−1}).

Lemma 3. Let S = (s_{0}, …, s_{n−1}) be an element of ℜ_{n} with s_{0} ≥ 1, and let x be a non-zero element of H^{1}(P). Then

κ_{n}^{S}χ_{0,x} ∈ T .

Proof. Without loss of generality, we may assume that x = x_{2n}. So K = ker x. Since s_{0} ≥ 1, by (2), we have

Res_{K}^{P}(κ_{n}^{S}) ≑ ∑_{U∈U} κ_{n−1}^{U}w^{t_{U}} ψ  (3)

with U a subset of

{R = (r_{0}, …, r_{n−2}) ∈ ℜ_{n−1}|r_{0} ≥ 1}.

Let U = (u_{0}, …, u_{n−2}) be an element of U. Since

κ_{n−1}^{U}κ_{n−1,0} = κ_{n−1,0}^{u_{0}+1} ∏_{i=1}^{n−2} κ_{n−1,i}^{u_{i}}

and u_{0} +1 ≥ 2, it follows from the inductive hypothesis that κ_{n−1}^{U}κ_{n−1,0}, and hence κ_{n−1}^{U}κ_{n−1,0}w^{t_{U}}, belong to T′. So, via the inflation map, κ_{n−1}^{U}κ_{n−1,0}w^{t_{U}} belongs to T .

We then have, as elements of h^{∗}(P),

κ_{n}^{S}χ_{0,x} = κ_{n}^{S}tr_{P}^{K}(κ_{n−1,0}ζ_{n−1}^{p−1})
= tr_{P}^{K}(Res_{K}^{P}(κ_{n}^{S}) · κ_{n−1,0}ζ_{n−1}^{p−1}) by Frobenius formula
= ∑_{U∈U} tr_{P}^{K}(κ_{n−1}^{U}κ_{n−1,0}w^{t_{U}} ψζ_{n−1}^{p−1})  by (3)

= ∑_{U∈U} κ_{n−1}^{U}κ_{n−1,0}w^{t_{U}} · tr_{P}^{K}(ψζ_{n−1}^{p−1}),

which implies κ_{n}^{S}χ_{0,x} ∈ T , by Corollary 3 (iii). □

Proof of Proposition 4. Let R = (r_{0}, …, r_{n−1}) be an element of ℜ_{n}. By Corollary 3 (i) , κ_{n}^{R} ∈ T if R satisfies (R_{2}). Suppose that r_{0} ≥ 2. Set S = (r_{0} − 1, r_{1}, …, r_{n−1}). We then have

κ_{n}^{R} = κ_{n}^{S}κ_{n,0}
= − ∑_{x∈ℙH^{1}(P)} κ_{n}^{S}χ_{0,x}  by Proposition 3.

Since r_{0} − 1 ≥ 1, by Lemma 3, κ_{n}^{S}χ_{0,x} ∈ T , for every x ∈ ℙH^{1}(P). So κ_{n}^{R} ∈ T . The proposition is proved. □

Consider ψ, and also the right hand side of (2), as polynomials with variable w and with coefficients in h^{∗}(P_{n−1}). We have

Lemma 4. Let R = (r_{0}, …, r_{n−1}) be an element of ℜ_{n} with s_{R} ≠ 0 mod p. Then, for 0 ≤ i ≤ n − 2,

(i) Res_{K}^{P}(κ_{n}^{R}) ≑ s_{R}(−1)^{i+n}κ_{n−1}^{R}κ_{n−1,i}α^{p^{n−1}[(p−1)s_{R}−1]+p^{i}} + other terms; (ii) κ_{n−1}^{R}κ_{n−1,i} ∈ T if κ_{n}^{R} ∈ T .

Proof. For t < s_{R}, deg(ψ^{(p−1)t}) ≤ p^{n−1}(p − 1)s_{R} − p^{n} + p^{n−1}; hence

deg(ψ^{(p−1)t} < min(p^{n−1}(p − 1)s_{R} − 1, p^{n−1}[(p − 1)s_{R} − 2] + p^{i}).

So (i) follows from (2) and the fact that

ψ^{(p−1)s_{R}} = [∑_{i=0}^{n−1}(−1)^{i}κ_{n−1,i}α^{p^{i}}]^{(p−1)s_{R}}
= −s_{R}(−1)^{i+n−1}κ_{n−1,i}α^{p^{n−1}[(p−1)s_{R}−1]+p^{i}} + other terms.

Write

Res_{K}^{P}(κ_{n}^{R}) = ∑_{i≥0} ϱ_{i}w^{i}

with ϱ_{i} ∈ h^{∗}(P_{n−1}). If κ_{n}^{R} ∈ T , then Res_{K}^{P}(κ_{n}^{R}) belongs to T′, so all the ϱ_{i}’s lie in T′. (ii) is then a direct consequence of (i). □

The proof of the theorem is completed by Proposition 4 and the following Lemma 5. If κ_{n}^{R} ∈ T with R = (r_{0}, …, r_{n−1}) ∈ ℜ_{n}, then R satisfies (R_{1}) or (R_{2}).

Proof. By [L], the lemma holds for n = 1. Assume that it holds for n − 1.

Suppose that κ_{n}^{R} ∈ T with R = (r_{0}, …, r_{n−1}) and r_{0} < 2. It follows that ζ = Res_{K}^{P}(κ_{n}^{R}) ∈ T′. Conider ζ as a polynomial with variable w and with coefficients in h^{∗}(P_{n−1}). By (2), we have

ζ = κ_{n−1}^{R}ψ^{(p−1)s_{R}} + ∑_{0≤t<s_{R}} ρ_{t}ψ^{(p−1)t}
= κ_{n−1}^{R}w^{p^{n−1}(p−1)s_{R}} + other terms  (3)

which implies κ_{n−1}^{R} ∈ T′. By the induction hypothesis, r_{0} = 0 and r_{1}, …, r_{n−2} are multiples of p. So s_{R} = r_{n−1} mod p. If s_{R} ≠ 0 mod p, it follows from Lemma 4 that

κ_{n−1,1}^{r_{1}} … κ_{n−1,n−3}^{r_{n−3}}κ_{n−1,n−2}^{r_{n−2}+1} = κ_{n−1,n−2}κ_{n−1}^{R} ∈ T′

which contradicts the induction hypothesis, since r_{n−2} = 0 mod p implies r_{n−2}+1 ≠ 0 mod p. So s_{R} = 0 mod p, hence r_{n−1} = 0 mod p. The lemma follows. □

Let x be a non-zero element of H^{1}(P). By Theorem 2 (iii), there exists a unique η_{x} ∈ H^{∗}(E) such that, as elements of H^{∗}(E)/(z_{n}^{(1)}, …, z_{n}^{(n−2)}),

z_{n}^{(n−1)} = η_{x}βx.  (4)

Note that H_{x} = ker x can be identified with P_{n−1} × ℤ/p. Pick a non-zero element v_{x} of βH^{1}(P) satisfying

0 ≠ Res_{H_{x}}^{P}(v_{x}) ∈ ker Res_{P_{n−1}}^{H_{x}}

and define ψ_{x} = (−1)^{n−1}∑_{j=0}^{n−1}(−1)^{j}κ_{n−1,j}v_{x}^{p^{j}}.

Let R be an element of ℜ_{n} satisfying (R_{1}) or (R_{2}). By Theorem 5, κ_{n}^{R} and κ_{n−1}^{R} both belong to T . It is then interesting to find out formulae relating κ_{n}^{R} and κ_{n−1}^{R}. If R satisfies (R_{2}), the formula can be derived from Corollary 4. In the case where R satisfies (R_{1}), the formula follows from

Corollary 5. Let R = (r_{0}, …, r_{n−1}) be an element of ℜ_{n} with r_{0} ≥ 2. Then, as elements of T ,

κ_{n}^{R} = −κ_{n−1}^{r_{0}} ∑_{x∈ℙH^{1}(P)} η_{x}ψ_{x}^{(r_{0}−1)(p−1)−1} ∏_{j=1}^{n−1}[κ_{n−1,j−1}^{p} + κ_{n−1,j}ψ_{x}^{p−1}]^{r_{j}}.

Proof. Set S = (r_{0} − 1, r_{2}, …, r_{n−1}) and U = (r_{0} − 2, r_{1}, …, r_{n−1}). It follows from the proof of Proposition 4 that

κ_{n}^{R} = − ∑_{x∈ℙH^{1}(P)} κ_{n}^{S}χ_{0,x}

= − ∑_{x∈ℙH^{1}(P)} κ_{n}^{S}tr_{P}^{H_{x}}(κ_{n−1,0}ζ_{n−1}^{p−1})

= − ∑_{x∈ℙH^{1}(P)} tr_{P}^{H_{x}}[Res_{H_{x}}^{P}(κ_{n}^{S})κ_{n−1,0}ζ_{n−1}^{p−1}]

= − ∑_{x∈ℙH^{1}(P)} tr_{P}^{H_{x}}[Res_{H_{x}}^{P}(κ_{n}^{U})κ_{n−1,0}^{2}ψ_{x}^{p−1}ζ_{n−1}^{p−1}]

= − ∑_{x∈ℙH^{1}(P)} tr_{P}^{H_{x}}(ψζ_{n−1}^{p−1}κ_{n−1,0}^{r_{0}}ψ_{x}^{(p−1)(r_{0}−1)−1} ∏_{j=1}^{n−1}[κ_{n−1,j−1}^{p} + κ_{n−1,j}ψ_{x}^{p−1}]^{r_{j}}).

Since r_{0} ≥ 2, it follows from Theorem 5 that

ρ_{x} = κ_{n−1,0}^{r_{0}}ψ_{x}^{(p−1)(r_{0}−1)−1} ∏_{j=1}^{n−1}[κ_{n−1,j−1}^{p} + κ_{n−1,j}ψ_{x}^{p−1}]^{r_{j}}

belongs to T , for any x ∈ ℙH^{1}(P). Hence

κ_{n}^{R} = − ∑_{x∈ℙH^{1}(P)} ρ_{x}tr_{P}^{H_{x}}(ψ_{x}ζ_{n−1}^{p−1})
= − ∑_{x∈ℙH^{1}(P)} ρ_{x}η_{x}  by (4)  □

5. Symplectic invariants

Recall that the symplectic group Sp_{2n} = Sp_{2n}(𝔽_{p}) is the group consisting of linear transformations of E which preserve the nondegenerate symplectic form x_{1}x_{2} + ⋯ + x_{2n−1}x_{2n} ∈ H^{2}(E). It is clear that z_{n}^{(0)}, …, z_{n}^{(n−1)} belong to the subring of invariants of Sp_{2n} in 𝔽_{p}[y_{1}, …, y_{2n}]. According to a result of Quillen ([Q]) for p = 2, and of Tezuka-Yagita ([TY]) for p > 2,

T = 𝔽_{p}[y_{1}, …, y_{2n}]/(z_{n}^{(0)}, …, z_{n}^{(n−1)}).

There is then an induced action of Sp_{2n} on T . Set

ℜ′ = {R ∈ ℜ_{n}|R satisfies (R_{1}) or (R_{2})},

and let ℜ′′ be the subset of ℜ_{n} consisting of elements R = (r_{0}, r_{1}, …, r_{n−1}) of ℜ_{n} satisfying the following two conditions:

• 0 ≤ r_{i} ≤ p − 1 for i > 0;

• r_{0} = 3, or r_{0} = 2 and r_{1}, …, r_{n−1} are not all equal to 0. Let T^{Sp_{2n}} be the ring of invariants of Sp_{2n} in T . The following is then straight-

forward from Theorem 5 and [G, Prop. 21].

Theorem 6. T^{Sp_{2n}} is the subring of 𝔽_{p}[κ_{n,1}, …, κ_{n,n−1}] given by:

(i) for p = 2, T^{Sp_{2n}} = 𝔽_{2}[κ_{n,0}, …, κ_{n,n−1}]; (ii) for p > 2,

(iia) as a vector space over 𝔽_{p}, T^{Sp_{2n}} has a basis {κ_{n}^{R}|R ∈ ℜ′}; (iib) as a module over the polynomial algebra 𝔽_{p}[κ_{n,0}^{2}, κ_{n,1}^{p}, …, κ_{n,n−1}^{p}],

T^{Sp_{2n}} is free and is generated by {1, κ_{n}^{R}|R ∈ ℜ′′}. □

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Department of Mathematics, College of Science, University of Hue, Dai hoc Khoa hoc, 77 Nguyen Hue, Hue, Vietnam. E-mail address: paminh@dng.vnn.vn