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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.

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 )   . {\displaystyle \sum _{r=0}^{N-1}\phi \left(x+{\frac {r}{N}}\right)=\phi (Nx)\ .}

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

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 ) ∑ y ↦ x ϕ ( y ) = ϕ ( x ) {\displaystyle 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 Xn. We can define an integral of a step function against φ as

∫ f d ϕ = ∑ x ∈ X n f ( x ) ϕ n ( x )   . {\displaystyle \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/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

ζ ( s , a ) = ∑ n = 0 ∞ ( n + a ) − s {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }(n+a)^{-s}}

gives a distribution relation

∑ p = 0 q − 1 ζ ( s , a + p / q ) = q s ζ ( s , q a )   . {\displaystyle \sum _{p=0}^{q-1}\zeta (s,a+p/q)=q^{s}\,\zeta (s,qa)\ .}

Hence for given s, the map t ↦ ζ ( s , { t } ) {\displaystyle t\mapsto \zeta (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   , {\displaystyle B_{n}(x)=\sum _{k=0}^{n}{n \choose 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 !   . {\displaystyle {\frac {te^{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 ∑ a = 0 n − 1 b k ( x + a n )   . {\displaystyle B_{k}(x)=n^{k-1}\sum _{a=0}^{n-1}b_{k}\left({\frac {x+a}{n}}\right)\ .}

Thus the map

ϕ n : 1 n Z / Z → Q {\displaystyle \phi _{n}:{\frac {1}{n}}\mathbb {Z} /\mathbb {Z} \rightarrow \mathbb {Q} }

defined by

ϕ n : x ↦ n k − 1 B k ( ⟨ x ⟩ ) {\displaystyle \phi _{n}:x\mapsto n^{k-1}B_{k}(\langle x\rangle )}

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   . {\displaystyle \prod _{pb=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 by

g N ( r ) = 1 | G ( N ) | ∑ a ∈ G ( N ) h ( ⟨ r a N ⟩ ) σ a − 1   . {\displaystyle 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 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 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 ZD, the limit of the system Z/pD. Consider any eigenfunction of the Hecke operator T 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 )   . {\displaystyle (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 Tp with eigenvalue λp in D. The quadratic equation X − 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

a k + 2 = λ p a k + 1 − p a k   , {\displaystyle a_{k+2}=\lambda _{p}a_{k+1}-pa_{k}\ ,}

so that

a k = π 1 k + π 2 k   . {\displaystyle a_{k}=\pi _{1}^{k}+\pi _{2}^{k}\ .}

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
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