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 satisfyingKubert & Lang (1981) p.1

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

Such distributions are called ordinary distributions.Lang (1990) p.53 They also occur in p-adic integration theory in Iwasawa theory.Mazur & Swinnerton-Dyer (1972) p. 36

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/n{{zwnj}}Z 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

: $\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!} \ .$

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.Lang (1990) p.36

=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 haveLang (1990) p.157

: $\prod_{p b=a} g_b = g_a \ .$

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/N{{zwnj}}Z, and for any function f on G(N) we extend f to a function on Z/N{{zwnj}}Z 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.Mazur & Swinnerton-Dyer (1974) p.37 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

: $a_{k+2} = \lambda_p a_{k+1} - p a_k \ ,$

so that

: $a_k = \pi_1^k + \pi_2^k \ .$

References

{{cite book | first1=Daniel S. | last1=Kubert | authorlink1=Daniel Kubert | first2=Serge | last2=Lang | authorlink2=Serge Lang | title=Modular Units | series= Grundlehren der Mathematischen Wissenschaften | volume=244 | publisher=Springer-Verlag | year=1981 | isbn=0-387-90517-0 | zbl=0492.12002 }}
{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Cyclotomic Fields I and II | edition=second combined | publisher=Springer Verlag | series=Graduate Texts in Mathematics | volume=121 | isbn=3-540-96671-4 | zbl=0704.11038 | year=1990 }}
{{cite journal | zbl=0281.14016 | last1=Mazur | first1=B. | author1-link=Barry Mazur | last2=Swinnerton-Dyer | first2=P. | author2-link=Peter Swinnerton-Dyer | title=Arithmetic of Weil curves | journal=Inventiones Mathematicae | volume=25 | pages=1–61 | year=1974 | doi=10.1007/BF01389997 }}

Category:Abstract algebra

Category:Number theory