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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

$\sum _{r=0}^{N-1}\phi \left(x+{\frac {r}{N}}\right)=\phi (Nx)\ .$

We shall call these ordinary distributions. They also occur in p-adic 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:

$w(m,n)\sum _{y\mapsto x}\phi (y)=\phi (x)$

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

$\int f\,d\phi =\sum _{x\in X_{n}}f(x)\phi _{n}(x)\ .$

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.

Examples

Hurwitz zeta function

The multiplication theorem for the Hurwitz zeta function

$\zeta (s,a)=\sum _{n=0}^{\infty }(n+a)^{-s}$

gives a distribution relation

$\sum _{p=0}^{q-1}\zeta (s,a+p/q)=q^{s}\,\zeta (s,qa)\ .$

Hence for given s, the map $t\mapsto \zeta (s,\{t\})$ is a distribution on Q/Z.

Bernoulli distribution

Recall that the Bernoulli polynomials B_n are defined by

$B_{n}(x)=\sum _{k=0}^{n}{n \choose n-k}b_{k}x^{n-k}\ ,$

for n ≥ 0, where b_k are the Bernoulli numbers, with generating function

${\frac {te^{xt}}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}(x){\frac {t^{n}}{n!}}\ .$

They satisfy the distribution relation

$B_{k}(x)=n^{k-1}\sum _{a=0}^{n-1}b_{k}\left({\frac {x+a}{n}}\right)\ .$

Thus the map

$\phi _{n}:{\frac {1}{n}}\mathbb {Z} /\mathbb {Z} \rightarrow \mathbb {Q}$

defined by

$\phi _{n}:x\mapsto n^{k-1}B_{k}(\langle x\rangle )$

is a distribution.

Cyclotomic units

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

$\prod _{pb=a}g_{b}=g_{a}\ .$

Universal distribution

We consider 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

$g_{N}(r)={\frac {1}{|G(N)|}}\sum _{a\in G(N)}h\left({\left\langle {\frac {ra}{N}}\right\rangle }\right)\sigma _{a}^{-1}\ .$

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.

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 Q_p, 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 Z_D, the limit of the system Z/pⁿD. 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 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 T_l by

$(T_{l}f)\left({\frac {a}{b}}\right)=f\left({\frac {la}{b}}\right)+\sum _{k=0}^{l-1}f\left({\frac {a+kb}{lb}}\right)-\sum _{k=0}^{l-1}f\left({\frac {k}{l}}\right)\ .$

Let f be an eigenfunction for T_p with eigenvalue λ_p in D. The quadratic equation X² − λ_pX + p = 0 has roots π₁, π₂ with π₁ a unit and π₂ divisible by p. Define a sequence a₀ = 2, a₁ = π₁+π₂ = λ_p and

$a_{k+2}=\lambda _{p}a_{k+1}-pa_{k}\ ,$

so that

$a_{k}=\pi _{1}^{k}+\pi _{2}^{k}\ .$

References

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