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Distribution (number theory)

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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.

Contents

The original examples of distributions occur, unnamed, as functions φ on Q/Z satisfying

r = 0 N 1 ϕ ( x + r N ) = ϕ ( N x )   .

We shall call these ordinary distributions. They also occur in p-adic integration theory in Iwasawa theory.

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:

w ( m , n ) y x ϕ ( y ) = ϕ ( 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

f d ϕ = x X n f ( x ) ϕ 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.

Hurwitz zeta function

The multiplication theorem for the Hurwitz zeta function

ζ ( s , a ) = n = 0 ( n + a ) s

gives a distribution relation

p = 0 q 1 ζ ( s , a + p / q ) = q s ζ ( s , q a )   .

Hence for given s, the map t ζ ( s , { t } ) is a distribution on Q/Z.

Bernoulli distribution

Recall that the Bernoulli polynomials Bn are defined by

B n ( x ) = k = 0 n ( n n k ) b k x n k   ,

for n ≥ 0, where bk are the Bernoulli numbers, with generating function

t e x t e t 1 = n = 0 B n ( x ) t n n !   .

They satisfy the distribution relation

B k ( x ) = n k 1 a = 0 n 1 b k ( x + a n )   .

Thus the map

ϕ n : 1 n Z / Z Q

defined by

ϕ n : x n k 1 B k ( x )

is a distribution.

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

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/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 ) = 1 | G ( N ) | a G ( N ) h ( r a N ) σ a 1   .

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. 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

( T l f ) ( a b ) = f ( l a b ) + k = 0 l 1 f ( a + k b l b ) k = 0 l 1 f ( k l )   .

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 = π12λp and

a k + 2 = λ p a k + 1 p a k   ,

so that

a k = π 1 k + π 2 k   .

References

Distribution (number theory) Wikipedia


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