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 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
1
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