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]

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

Let ... → Xn+1Xn → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let X be their projective limit. We give each Xn the discrete topology, so that X is compact. Let φ = (φn) be a family of functions on Xn 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 Xn. 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.

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 Bn are defined by

for n ≥ 0, where bk 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 ga 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 gN 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 Qp, 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 KL = 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 ZD, the limit of the system Z/pnD. Consider any eigenfunction of the Hecke operator Tp with eigenvalue λp prime to p. We describe a procedure for deriving a measure of ZD.

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 Tl by

Let f be an eigenfunction for Tp with eigenvalue λp in D. The quadratic equation X2  λpX + p = 0 has roots π1, π2 with π1 a unit and π2 divisible by p. Define a sequence a0 = 2, a1 = π12 = λp and

so that

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. 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.