Inalgebra andnumber theory, adistribution is a function on a system of finite sets into anabelian group which is analogous to an integral: it is thus the algebraic analogue of adistribution in the sense ofgeneralised function.

The original examples of distributions occur, unnamed, as functions φ onQ/Z satisfying^[1]

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

Such distributions are called ordinary distributions.^[2] They also occur inp-adic integration theory inIwasawa theory.^[3]

Let ... →X_n+1 →X_n → ... be aprojective system of finite sets with surjections, indexed by the natural numbers, and letX be theirprojective limit. We give eachX_n thediscrete topology, so thatX iscompact. Let φ = (φ_n) be a family of functions onX_n taking values in an abelian groupV and compatible with the projective system:

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

for someweight functionw. The family φ is then adistribution on the projective systemX.

A functionf onX is "locally constant", or a "step function" if it factors through someX_n. We can define an integral of a step function against φ as

${\displaystyle \int fd\varphi =\sum _{x\in X_n}f(x){\varphi }_n(x).}$

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 systemZ/n‌Z indexed by positive integers ordered by divisibility. We identify this with the system (1/n)Z/Z with limitQ/Z.

Forx inR we let ⟨x⟩ denote the fractional part ofx normalised to 0 ≤ ⟨x⟩ < 1, and let {x} denote the fractional part normalised to 0 < {x} ≤ 1.

Themultiplication theorem for theHurwitz zeta function

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

gives a distribution relation

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

Hence for givens, the map${\displaystyle t\mapsto \zeta (s,\{t\})}$ is a distribution onQ/Z.

Recall that theBernoulli polynomialsB_n are defined by

${\displaystyle B_n(x)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{n-k})b_k{x}^{n-k},}$

forn ≥ 0, whereb_k are theBernoulli numbers, withgenerating function

${\displaystyle \frac{t{e}^{xt}}{e^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{t^n}{n!}.}$

They satisfy thedistribution relation

${\displaystyle B_k(x)=n^{k-1}\sum _{a=0}^{n-1}{b}_k\left(\frac{x+a}{n}\right).}$

Thus the map

${\displaystyle {\varphi }_n:\frac{1}{n}\mathbb{Z}/\mathbb{Z}\to \mathbb{Q}}$

defined by

${\displaystyle {\varphi }_n:x\mapsto n^{k-1}{B}_k(⟨x⟩)}$

is a distribution.^[4]

Thecyclotomic units satisfydistribution relations. Leta be an element ofQ/Z prime top and letg_a denote exp(2πia)−1. Then fora≠ 0 we have^[5]

${\displaystyle \prod _{pb=a}{g}_b=g_a.}$

One considers the distributions onZ with values in some abelian groupV and seek the "universal" or most general distribution possible.

Leth be an ordinary distribution onQ/Z taking values in a fieldF. LetG(N) denote the multiplicative group ofZ/N‌Z, and for any functionf onG(N) we extendf to a function onZ/N‌Z by takingf to be zero offG(N). Define an element of the group algebraF[G(N)] by

${\displaystyle g_N(r)=\frac{1}{|G(N)|}\sum _{a\in G(N)}h\left(⟨\frac{ra}{N}⟩\right){\sigma }_a^{-1}.}$

The group algebras form a projective system with limitX. Then the functionsg_N form a distribution onQ/Z with values inX, theStickelberger distribution associated withh.

Consider the special case when the value groupV of a distribution φ onX takes values in alocal fieldK, finite overQ_p, or more generally, in a finite-dimensionalp-adic Banach spaceW overK, with valuation |·|. We call φ ameasure if |φ| is bounded on compact open subsets ofX.^[6] LetD be the ring of integers ofK andL a lattice inW, that is, a freeD-submodule ofW withK⊗L =W. Up to scaling a measure may be taken to have values inL.

LetD be a fixed integer prime top and considerZ_D, the limit of the systemZ/pⁿD. Consider anyeigenfunction of theHecke operatorT^p with eigenvalueλ_p prime top. We describe a procedure for deriving a measure ofZ_D.

Fix an integerN prime top and toD. LetF be theD-module of all functions on rational numbers with denominator coprime toN. For any primel not dividingN we define theHecke operatorT_l by

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

Letf be an eigenfunction forT_p with eigenvalue λ_p inD. The quadratic equationX² − λ_pX + p = 0 has roots π₁, π₂ with π₁ a unit and π₂ divisible byp. Define a sequencea₀ = 2,a₁ = π₁+π₂ = λ_p and

${\displaystyle a_{k+2}={\lambda }_p{a}_{k+1}-p{a}_k,}$

so that

${\displaystyle a_k={\pi }_1^k+{\pi }_2^k.}$

• Kubert, Daniel S.;Lang, Serge (1981).Modular Units. Grundlehren der Mathematischen Wissenschaften. Vol. 244.Springer-Verlag.ISBN 0-387-90517-0.Zbl 0492.12002.
• Lang, Serge (1990).Cyclotomic Fields I and II.Graduate Texts in Mathematics. Vol. 121 (second combined ed.).Springer Verlag.ISBN 3-540-96671-4.Zbl 0704.11038.
• Mazur, B.;Swinnerton-Dyer, P. (1974). "Arithmetic of Weil curves".Inventiones Mathematicae.25:1–61.doi:10.1007/BF01389997.Zbl 0281.14016.

• Abstract algebra
• Number theory