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]
| N-1 | |
| \sum | |
| r=0 |
\phi\left(x+
| 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 ... → Xn+1 → Xn → ... 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:
w(m,n)\sumy\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 Xn. We can define an integral of a step function against φ as
\intfd\phi=
| \sum | |
| x\inXn |
f(x)\phin(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 denote the fractional part normalised to 0 < ≤ 1.
The multiplication theorem for the Hurwitz zeta function
\zeta(s,a)=
| infty | |
| \sum | |
| n=0 |
(n+a)-s
gives a distribution relation
| q-1 | |
| \sum | |
| p=0 |
\zeta(s,a+p/q)=qs\zeta(s,qa) .
Hence for given s, the map
t\mapsto\zeta(s,\{t\})
Recall that the Bernoulli polynomials Bn are defined by
Bn(x)=
| n | |
| \sum | |
| k=0 |
{n\choosen-k}bkxn-k ,
for n ≥ 0, where bk are the Bernoulli numbers, with generating function
| text | |
| et-1 |
=
| infty | |
| \sum | |
| n=0 |
Bn(x)
| tn | |
| n! |
.
They satisfy the distribution relation
Bk(x)=nk-1
| n-1 | |
| \sum | |
| a=0 |
| b | ||||
|
Thus the map
\phin:
| 1 | |
| n |
Z/Z → Q
defined by
\phin:x\mapstonk-1Bk(\langlex\rangle)
is a distribution.[4]
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]
\prodpgb=ga .
One considers the distributions on Z with values in some abelian group V and seek the "universal" or most general distribution possible.
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
gN(r)=
| 1 | |
| |G(N)| |
\sumah\left({\left\langle{
| ra | |
| N |
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.
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-dimensionalp-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.
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
(Tlf)\left(
| a | |
| b\right) |
=f\left(
| la | |
| b |
\right)+
| l-1 | ||
| \sum | f\left({ | |
| k=0 |
| a+kb | |
| lb |
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 = π1+π2 = λp and
ak+2=λpak+1-pak ,
so that
ak=
| k | |
| \pi | |
| 1 |
+
| k | |
| \pi | |
| 2 |
.
. Serge Lang . Cyclotomic Fields I and II . second combined . . . 121 . 3-540-96671-4 . 0704.11038 . 1990 .