Distribution (number theory)

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[1]

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

Such distributions are called ordinary distributions.[2] They also occur in p-adic integration theory in Iwasawa theory.[3]

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.[4]

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[5]

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

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

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.[6] 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

Kubert & Lang (1981) p.1
Lang (1990) p.53
Mazur & Swinnerton-Dyer (1972) p. 36
Lang (1990) p.36
Lang (1990) p.157
Mazur & Swinnerton-Dyer (1974) p.37

Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. 244. Springer-Verlag. ISBN 0-387-90517-0. Zbl 0492.12002.
Lang, Serge (1990). Cyclotomic Fields I and II. Graduate Texts in Mathematics. 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.

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[1] Kubert & Lang (1981) p.1

[2] Lang (1990) p.53

[MSD36-3] Mazur & Swinnerton-Dyer (1972) p. 36

[4] Lang (1990) p.36

[5] Lang (1990) p.157

[MSD37-6] Mazur & Swinnerton-Dyer (1974) p.37