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]}
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:
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
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.
Contents
Examples
Hurwitz zeta function
The multiplication theorem for the Hurwitz zeta function
gives a distribution relation
Hence for given s, the map is a distribution on Q/Z.
Bernoulli distribution
Recall that the Bernoulli polynomials B_{n} are defined by
for n ≥ 0, where b_{k} are the Bernoulli numbers, with generating function
They satisfy the distribution relation
Thus the map
defined by
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]}
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
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/p^{n}D. 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
Let f be an eigenfunction for T_{p} with eigenvalue λ_{p} in D. The quadratic equation X^{2} − λ_{p}X + p = 0 has roots π_{1}, π_{2} with π_{1} a unit and π_{2} divisible by p. Define a sequence a_{0} = 2, a_{1} = π_{1}+π_{2} = λ_{p} and
so that
References
- 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. Zbl 0281.14016. doi:10.1007/BF01389997.