CHAPTER 1

Introduction.

Let F be a finite extension of Q. Fix a prime p and let F∞ denote the unique

subfield of F (μp∞ ) such that Γ = Gal(F∞/F ) is isomorphic to Zp, the additive

group of p-adic integers. One refers to F∞ as the cyclotomic Zp-extension of F .

Suppose that K is a finite Galois extension of F such that K ∩ F∞ = F . Let

K∞ = KF∞, the cyclotomic Zp-extension of K. Then K∞ is Galois over F and

G = Gal(K∞/F ) is isomorphic to Δ × Γ, where Δ = Gal(K/F ). Iwasawa theory is

often concerned with a compact Zp-module X which has a natural action of such

a Galois group G. The questions that we will consider in this paper concern the

structure of X just as a Zp[Δ]-module. The structure of X as a module over the

Iwasawa algebra Λ = Zp[[Γ]] will not play a significant role.

Assume that X is a finitely generated, torsion-free Zp-module and hence a free

Zp-module. This turns out to be so in many interesting cases. Let λ(X) denote its

Zp-rank. One can study the action of Δ on X by considering V = X ⊗Zp Qp, a

vector space over Qp of dimension λ(X) and a representation space for the group

Δ. The module X will be a Δ-invariant Zp-lattice in V . If the order of Δ is

not divisible by p, then one sees easily that X is determined up to isomorphism

as a Zp[Δ]-module by V . Furthermore, X will be projective as a Zp[Δ]-module.

However, if p |Δ|, then V can have non-isomorphic Δ-invariant Zp-lattices and it is

possible that none will be projective. If X happens to be a projective Zp[Δ]-module,

then its isomorphism class is again determined by V .

Let IrrF (Δ) denote the set of irreducible representations of Δ (up to isomor-

phism) over a field F. We choose F to be a finite extension of Qp containing all

m-th roots of unity where m ≥ 1 is divisible by the order of all elements of Δ.

Then all σ ∈ IrrF (Δ) are absolutely irreducible. For each such σ, let Wσ denote

the corresponding F-representation space for Δ and let n(σ) = dimF (Wσ). One

can decompose VF = V ⊗Qp F as a direct sum of the Wσ’s, each occurring with

a certain multiplicity. We denote this multiplicity by λ(X, σ). We then have the

obvious formula

(1.0.a) λ(X) = dimF

(

VF

)

=

σ

n(σ)λ(X, σ)

where σ runs over IrrF (Δ). The representation space V is determined by the

λ(X, σ)’s. One simple relationship that they satisfy is that λ(X, σ) = λ(X, σ ) for

σ, σ ∈ IrrF (Δ) if their characters χσ and χσ are conjugate over Qp. We refer to

these equalities as the conjugacy relations.

Our primary objective in this paper is to study another more subtle type of

relationship involving the λ(X, σ)’s which arises when the order of Δ is divisible

by p and X is projective as a Zp[Δ]-module. These new relationships, which we

refer to as congruence relations, owe their existence to the fact that there are

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