1.1. CONGRUENCE RELATIONS. 5
be uniquely determined by τ, but we will let στ denote one such lifting. Formula
(1.1.c) becomes λ(X, στ ) = w(X, τ). For an arbitrary σ ∈ IrrF (Δ), we then have
and λ(X, σ) =
d(σ, τ)λ(X, στ ),
assuming that X is a projective Zp[Δ]-module. The above equation for λ(X, σ) is
precisely the congruence relation (1.1.d) which results from the above isomorphism
for σss. Thus, the λ(X, στ )’s determine all of the λ(X, σ)’s.
For example, consider Δ = D2pr , the dihedral group of order 2pr, where p
is an odd prime and r ≥ 0. Then Δ is clearly p-solvable. We have t = 2. The
elements of Irrf(Δ) are τ0 and another 1-dimensional representation τ1. There are
two 1-dimensional representations of Δ over F, σ0 and σ1, whose reductions modulo
m are τ0 and τ1, respectively. Those liftings are unique in this case and are the
irreducible representations which factor through the unique quotient Δ0 of Δ of
order 2. All other representations σ in IrrF (Δ) are of dimension 2 and one has
d(σ, τ) = 1 for both τ’s in Irrf(Δ). For any such σ and any projective module X,
we obtain the congruence relation
λ(X, σ) = λ(X, σ0) + λ(X, σ1) .
However, we should point out that if we use the fact that such a σ is induced from a
1-dimensional representation π of the Sylow p-subgroup Π of Δ, then this relation
is an easy consequence of a congruence relation for the p-group Π. To see this, note
that X is also a projective Zp[Π]-module and so λ(X, π) = λ(X, π0), where π0 is
the trivial character of Π. We have σ
and σ0 ⊕ σ1
using the Frobenius reciprocity law, we have
λ(X, σ) = λ(X, π) = λ(X, π0) = λ(X, σ0) + λ(X, σ1).
A useful general observation is that if Δ contains a normal p-subgroup Π, then
every element τ of Irrf(Δ) must factor through Δ/Π. This is clear since Uτ Π is a
nontrivial subspace of Uτ which is Δ-invariant and hence must coincide with Uτ .
The groups D2pr provides a simple illustration. As another interesting example
(and one of our main guiding examples for this study), suppose that p is an odd
prime and that Δ
for some r ≥ 0. Let Δ0 = PGL2(Z/pZ).
The kernel Π of the obvious homomorphism Δ → Δ0 is a normal p-subgroup of
Δ. Hence the irreducible representations of Δ over a finite field of characteristic p
factor through Δ0. They are easily described and all are defined over Fp. One has
t = p + 1. If p ≥ 5, then Δ is not p-solvable, although it turns out that four of
the τ’s can be lifted to representations in characteristic 0. We will return to this
example in some detail in chapter 7, along with other examples.
Before turning to the arithmetic side of this paper, we make the following
important remark. It will be useful to have a larger class of Zp[Δ]-modules for
which the congruence relations (1.1.d) hold. We consider only finitely-generated
Zp[Δ]-modules. If X is such a module, then one can still define the λ(X, σ)’s for
all σ ∈ IrrF (Δ) since they are determined by V = X ⊗Zp Qp, a representation
space for Δ over Qp. Thus, it would actually be suﬃcient to know that V contains
a Δ-invariant Zp-lattice Y which is projective as a Zp[Δ]-module. If that is so,
we will then say that X is strictly quasi-projective. Equivalently, this means that
there is a Δ-homomorphism X → Y with finite kernel and cokernel. We then have
λ(X, σ) = λ(Y, σ) for all σ ∈ IrrF (Δ) and so by applying formula (1.1.c) to Y ,