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=0N−1ϕ(x+rN)=ϕ(Nx) .
Such distributions are called ordinary distributions. They also occur in p-adic integration theory in Iwasawa theory.
Let ... → Xn+1 → Xn → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let X be their projective limit. We give each Xn the discrete topology, so that X is compact. Let φ = (φn) be a family of functions on Xn taking values in an abelian group V and compatible with the projective system:
w(m,n)∑y↦xϕ(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 Xn. We can define an integral of a step function against φ as
∫fdϕ=∑x∈Xnf(x)ϕn(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.
Contents
Examples
Hurwitz zeta function
The multiplication theorem for the Hurwitz zeta function
ζ(s,a)=∑n=0∞(n+a)−s
gives a distribution relation
∑p=0q−1ζ(s,a+p/q)=qsζ(s,qa) .
Hence for given s, the map t↦ζ(s,{t}) is a distribution on Q/Z.
Bernoulli distribution
Recall that the Bernoulli polynomials Bn are defined by
Bn(x)=∑k=0n(nn−k)bkxn−k ,
for n ≥ 0, where bk are the Bernoulli numbers, with generating function
textet−1=∑n=0∞Bn(x)tnn! .
They satisfy the distribution relation
Bk(x)=nk−1∑a=0n−1bk(x+an) .
Thus the map
ϕn:1nℤ/ℤ→ℚ
defined by
ϕn:x↦nk−1Bk(⟨x⟩)
is a distribution.
Cyclotomic units
The cyclotomic units satisfy distribution relations. Let a be an element of Q/Z prime to p and let ga denote exp(2πia)−1. Then for a≠ 0 we have
∏pb=agb=ga .
Universal distribution
One considers the distributions on Z with values in some abelian group V and seek the "universal" or most general distribution possible.
Stickelberger distributions
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
gN(r)=1|G(N)|∑a∈G(N)h(⟨raN⟩)σa−1 .
The group algebras form a projective system with limit X. Then the functions gN form a distribution on Q/Z with values in X, the Stickelberger distribution associated with h.
p-adic measures
Consider the special case when the value group V of a distribution φ on X takes values in a local field K, finite over Qp, or more generally, in a finite-dimensional
p-adic 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 D-submodule of W with K⊗L = W. Up to scaling a measure may be taken to have values in L.
Hecke operators and measures
Let D be a fixed integer prime to p and consider ZD, the limit of the system Z/pnD. Consider any eigenfunction of the Hecke operator Tp with eigenvalue λp prime to p. We describe a procedure for deriving a measure of ZD.
Fix an integer N prime to p and to D. Let F be the D-module of all functions on rational numbers with denominator coprime to N. For any prime l not dividing N we define the Hecke operator Tl by
(Tlf)(ab)=f(lab)+∑k=0l−1f(a+kblb)−∑k=0l−1f(kl) .
Let f be an eigenfunction for Tp with eigenvalue λp in D. The quadratic equation X2 − λpX + p = 0 has roots π1, π2 with π1 a unit and π2 divisible by p. Define a sequence a0 = 2, a1 = π1+π2 = λp and
ak+2=λpak+1−pak ,
so that
ak=π1k+π2k .