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

:$\sum_^ \phi\left(x + \frac r N\right) = \phi(Nx) \ .$

Such distributions are called ordinary distributions. They also occur in ''p''-adic integration theory in Iwasawa theory.Mazur & Swinnerton-Dyer (1972) p. 36

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:

:$w(m,n) \sum_ \phi(y) = \phi(x)$

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 ''X''_''n''. We can define an integral of a step function against φ as

:$\int f \, d\phi = \sum_ f(x) \phi_n(x) \ .$

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/''n''Z 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.

Examples

Hurwitz zeta function

The multiplication theorem for the Hurwitz zeta function

:$\zeta(s,a) = \sum_^\infty (n+a)^$

gives a distribution relation

:$\sum_^\zeta(s,a+p/q)=q^s\,\zeta(s,qa) \ .$

Hence for given ''s'', the map $t \mapsto \zeta(s,\)$ is a distribution on Q/Z.

Bernoulli distribution

Recall that the ''Bernoulli polynomials'' ''B''_''n'' are defined by

:$B_n(x) = \sum_^n b_k x^ \ ,$

for ''n'' ≥ 0, where ''b''_''k'' are the Bernoulli numbers, with generating function

:$\frac= \sum_^\infty B_n(x) \frac \ .$

They satisfy the ''distribution relation''

:$B_k(x) = n^ \sum_^ b_k\left(\right)\ .$

Thus the map

:$\phi_n : \frac\mathbb/\mathbb \rightarrow \mathbb$

defined by

:$\phi_n : x \mapsto n^ B_k(\langle x \rangle)$

is a distribution.

Cyclotomic units

The cyclotomic units satisfy ''distribution relations''. Let ''a'' be an element of Q/Z prime to ''p'' and let ''g''_''a'' denote exp(2πi''a'')−1. Then for ''a''≠ 0 we haveLang (1990) p.157

:$\prod_ g_b = g_a \ .$

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/''N''Z, and for any function ''f'' on ''G''(''N'') we extend ''f'' to a function on Z/''N''Z by taking ''f'' to be zero off ''G''(''N''). Define an element of the group algebra ''F'' 'G''(''N'')by

:$g_N(r) = \frac \sum_ h\left(\right) \sigma_a^ \ .$

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''.Mazur & Swinnerton-Dyer (1974) p.37 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

:$(T_l f)\left(\frac a b\right) = f\left(\frac\right) + \sum_^ f\left(\right) - \sum_^ f\left(\frac k l \right) \ .$

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 π₁, π₂ with π₁ a unit and π₂ divisible by ''p''. Define a sequence ''a''₀ = 2, ''a''₁ = π₁+π₂ = ''λ''_''p'' and

:$a_ = \lambda_p a_ - p a_k \ ,$

so that

:$a_k = \pi_1^k + \pi_2^k \ .$

References

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* {{cite journal , zbl=0281.14016 , last1=Mazur , first1=B. , author1-link=Barry Mazur , last2=Swinnerton-Dyer , first2=P. , author2-link=Peter Swinnerton-Dyer , title=Arithmetic of Weil curves , journal=Inventiones Mathematicae , volume=25 , pages=1–61 , year=1974 , doi=10.1007/BF01389997

Algebra
Number theory