In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function.
The original examples of distributions occur, unnamed, as functions φ on Q/Z satisfying
∑
r
=
0
N
−
1
ϕ
(
x
+
r
N
)
=
ϕ
(
N
x
)
.
We shall call these ordinary distributions. They also occur in padic integration theory in Iwasawa theory.
Let ... → X_{n+1} → X_{n} → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let X be their projective limit. We give each X_{n} the discrete topology, so that X is compact. Let φ = (φ_{n}) be a family of functions on X_{n} taking values in an abelian group V and compatible with the projective system:
w
(
m
,
n
)
∑
y
↦
x
ϕ
(
y
)
=
ϕ
(
x
)
for some weight function w. The family φ is then a distribution on the projective system X.
A function f on X is "locally constant", or a "step function" if it factors through some X_{n}. We can define an integral of a step function against φ as
∫
f
d
ϕ
=
∑
x
∈
X
n
f
(
x
)
ϕ
n
(
x
)
.
The definition extends to more general projective systems, such as those indexed by the positive integers ordered by divisibility. As an important special case consider the projective system Z/nZ indexed by positive integers ordered by divisibility. We identify this with the system (1/n)Z/Z with limit Q/Z.
For x in R we let ⟨x⟩ denote the fractional part of x normalised to 0 ≤ ⟨x⟩ < 1, and let {x} denote the fractional part normalised to 0 < {x} ≤ 1.
The multiplication theorem for the Hurwitz zeta function
ζ
(
s
,
a
)
=
∑
n
=
0
∞
(
n
+
a
)
−
s
gives a distribution relation
∑
p
=
0
q
−
1
ζ
(
s
,
a
+
p
/
q
)
=
q
s
ζ
(
s
,
q
a
)
.
Hence for given s, the map
t
↦
ζ
(
s
,
{
t
}
)
is a distribution on Q/Z.
Recall that the Bernoulli polynomials B_{n} are defined by
B
n
(
x
)
=
∑
k
=
0
n
(
n
n
−
k
)
b
k
x
n
−
k
,
for n ≥ 0, where b_{k} are the Bernoulli numbers, with generating function
t
e
x
t
e
t
−
1
=
∑
n
=
0
∞
B
n
(
x
)
t
n
n
!
.
They satisfy the distribution relation
B
k
(
x
)
=
n
k
−
1
∑
a
=
0
n
−
1
b
k
(
x
+
a
n
)
.
Thus the map
ϕ
n
:
1
n
Z
/
Z
→
Q
defined by
ϕ
n
:
x
↦
n
k
−
1
B
k
(
⟨
x
⟩
)
is a distribution.
The cyclotomic units satisfy distribution relations. Let a be an element of Q/Z prime to p and let g_{a} denote exp(2πia)−1. Then for a≠ 0 we have
∏
p
b
=
a
g
b
=
g
a
.
One considers the distributions on Z with values in some abelian group V and seek the "universal" or most general distribution possible.
Let h be an ordinary distribution on Q/Z taking values in a field F. Let G(N) denote the multiplicative group of Z/NZ, and for any function f on G(N) we extend f to a function on Z/NZ by taking f to be zero off G(N). Define an element of the group algebra F[G(N)] by
g
N
(
r
)
=
1

G
(
N
)

∑
a
∈
G
(
N
)
h
(
⟨
r
a
N
⟩
)
σ
a
−
1
.
The group algebras form a projective system with limit X. Then the functions g_{N} form a distribution on Q/Z with values in X, the Stickelberger distribution associated with h.
Consider the special case when the value group V of a distribution φ on X takes values in a local field K, finite over Q_{p}, or more generally, in a finitedimensional padic Banach space W over K, with valuation ·. We call φ a measure if φ is bounded on compact open subsets of X. Let D be the ring of integers of K and L a lattice in W, that is, a free Dsubmodule of W with K⊗L = W. Up to scaling a measure may be taken to have values in L.
Let D be a fixed integer prime to p and consider Z_{D}, the limit of the system Z/p^{n}D. Consider any eigenfunction of the Hecke operator T^{p} with eigenvalue λ_{p} prime to p. We describe a procedure for deriving a measure of Z_{D}.
Fix an integer N prime to p and to D. Let F be the Dmodule of all functions on rational numbers with denominator coprime to N. For any prime l not dividing N we define the Hecke operator T_{l} by
(
T
l
f
)
(
a
b
)
=
f
(
l
a
b
)
+
∑
k
=
0
l
−
1
f
(
a
+
k
b
l
b
)
−
∑
k
=
0
l
−
1
f
(
k
l
)
.
Let f be an eigenfunction for T_{p} with eigenvalue λ_{p} in D. The quadratic equation X^{2} − λ_{p}X + p = 0 has roots π_{1}, π_{2} with π_{1} a unit and π_{2} divisible by p. Define a sequence a_{0} = 2, a_{1} = π_{1}+π_{2} = λ_{p} and
a
k
+
2
=
λ
p
a
k
+
1
−
p
a
k
,
so that
a
k
=
π
1
k
+
π
2
k
.