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]

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

We shall call these 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

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

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

1. Kubert & Lang (1981) p.1
2. Lang (1990) p.53
3. Mazur & Swinnerton-Dyer (1972) p. 36
4. Lang (1990) p.36
5. Lang (1990) p.157
6. 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.