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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 (mathematics)|distribution]] in the sense of [[generalised function]].
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| The original examples of distributions occur, unnamed, as functions φ on '''Q'''/'''Z''' satisfying<ref>Kubert & Lang (1981) p.1</ref>
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| :<math> \sum_{r=0}^{N-1} \phi\left(x + \frac r N\right) = \phi(Nx) \ . </math>
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| We shall call these '''ordinary distributions'''.<ref>Lang (1990) p.53</ref> They also occur in ''p''-adic integration theory in [[Iwasawa theory]].<ref name=MSD36>Mazur & Swinnerton-Dyer (1972) p. 36</ref>
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| Let ... → ''X''<sub>''n''+1</sub> → ''X''<sub>''n''</sub> → ... 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''<sub>''n''</sub> the [[discrete topology]], so that ''X'' is [[compact space|compact]]. Let φ = (φ<sub>''n''</sub>) be a family of functions on ''X''<sub>''n''</sub> taking values in an abelian group ''V'' and compatible with the projective system:
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| :<math> w(m,n) \sum_{y \mapsto x} \phi(y) = \phi(x) </math>
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| for some ''weight function'' ''w''. The family φ is then a ''distribution'' on the projective system ''X''.
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| A function ''f'' on ''X'' is "locally constant", or a "step function" if it factors through some ''X''<sub>''n''</sub>. We can define an integral of a step function against φ as
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| :<math> \int f \, d\phi = \sum_{x \in X_n} f(x) \phi_n(x) \ . </math>
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| 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''<nowiki></nowiki>'''Z''' indexed by positive integers ordered by divisibility. We identify this with the system (1/''n'')'''Z'''/'''Z''' with limit '''Q'''/'''Z'''.
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| 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.
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| ==Examples==
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| ===Hurwitz zeta function===
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| The [[multiplication theorem]] for the [[Hurwitz zeta function]]
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| :<math>\zeta(s,a) = \sum_{n=0}^\infty (n+a)^{-s} </math>
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| gives a distribution relation
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| :<math>\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa) \ .</math>
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| Hence for given ''s'', the map <math>t \mapsto \zeta(s,\{t\})</math> is a distribution on '''Q'''/'''Z'''.
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| ===Bernoulli distribution===
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| Recall that the ''[[Bernoulli polynomials]]'' ''B''<sub>''n''</sub> are defined by
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| :<math>B_n(x) = \sum_{k=0}^n {n \choose n-k} b_k x^{n-k} \ ,</math> | |
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| for ''n'' ≥ 0, where ''b''<sub>''k''</sub> are the [[Bernoulli number]]s, with [[generating function]]
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| :<math>\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!} \ .</math>
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| They satisfy the ''distribution relation''
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| :<math> B_k(x) = n^{k-1} \sum_{a=0}^{n-1} b_k\left({\frac{x+a}{n}}\right)\ . </math>
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| Thus the map
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| :<math> \phi_n : \frac{1}{n}\mathbb{Z}/\mathbb{Z} \rightarrow \mathbb{Q} </math> | |
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| defined by
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| :<math> \phi_n : x \mapsto n^{k-1} B_k(\langle x \rangle) </math>
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| is a distribution.<ref>Lang (1990) p.36</ref>
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| ===Cyclotomic units===
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| The [[cyclotomic unit]]s satisfy ''distribution relations''. Let ''a'' be an element of '''Q'''/'''Z''' prime to ''p'' and let ''g''<sub>''a''</sub> denote exp(2πi''a'')−1. Then for ''a''≠ 0 we have<ref>Lang (1990) p.157</ref>
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| :<math> \prod_{p b=a} g_b = g_a \ . </math> | |
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| ==Universal distribution==
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| We consider the distributions on ''Z'' with values in some abelian group ''V'' and seek the "universal" or most general distribution possible.
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| ==Stickelberger distributions==
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| 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''<nowiki></nowiki>'''Z''', and for any function ''f'' on ''G''(''N'') we extend ''f'' to a function on '''Z'''/''N''<nowiki></nowiki>'''Z''' by taking ''f'' to be zero off ''G''(''N''). Define an element of the group algebra ''F''[''G''(''N'')] by
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| :<math> 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} \ . </math>
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| The group algebras form a projective system with limit ''X''. Then the functions ''g''<sub>''N''</sub> form a distribution on '''Q'''/'''Z''' with values in ''X'', the '''Stickelberger distribution''' associated with ''h''.
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| ==p-adic measures==
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| 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'''<sub>''p''</sub>, or more generally, in a finite-dimensional
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| ''p''-adic Banach space ''W'' over ''K'', with valuation |·|. We call φ a '''measure''' if |φ| is bounded on compact open subsets of ''X''.<ref name=MSD37>Mazur & Swinnerton-Dyer (1974) p.37</ref> 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''.
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| ===Hecke operators and measures===
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| Let ''D'' be a fixed integer prime to ''p'' and consider '''Z'''<sub>''D''</sub>, the limit of the system '''Z'''/''p''<sup>''n''</sup>''D''. Consider any [[eigenfunction]] of the [[Hecke operator]] ''T''<sup>''p''</sub> with eigenvalue ''λ''<sub>''p''</sub> prime to ''p''. We describe a procedure for deriving a measure of '''Z'''<sub>''D''</sub>.
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| 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''<sub>''l''</sub> by
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| :<math> (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) \ . </math>
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| Let ''f'' be an eigenfunction for ''T''<sub>''p''</sub> with eigenvalue λ<sub>''p''</sub> in ''D''. The quadratic equation ''X''<sup>2</sup> − λ<sub>''p''</sub>''X'' + ''p'' = 0 has roots π<sub>1</sub>, π<sub>2</sub> with π<sub>1</sub> a unit and π<sub>2</sub> divisible by ''p''. Define a sequence ''a''<sub>0</sub> = 2, ''a''<sub>1</sub> = π<sub>1</sub>+π<sub>2</sub> = ''λ''<sub>''p''</sub> and
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| :<math>a_{k+2} = \lambda_p a_{k+1} - p a_k \ , </math>
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| so that
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| :<math>a_k = \pi_1^k + \pi_2^k \ . </math> | |
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| ==References==
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| {{reflist}}
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| * {{cite book | first1=Daniel S. | last1=Kubert | authorlink1=Daniel Kubert | first2=Serge | last2=Lang | authorlink2=Serge Lang | title=Modular Units | series= Grundlehren der Mathematischen Wissenschaften | volume=244 | publisher=[[Springer-Verlag]] | year=1981 | isbn=0-387-90517-0 | zbl=0492.12002 }}
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| * {{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Cyclotomic Fields I and II | edition=second combined | year=1990 | publisher=[[Springer Verlag]] | series=[[Graduate Texts in Mathematics]] | volume=121 | isbn=3-540-96671-4 | zbl=0704.11038 | year=1990 }}
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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 | url=http://www.springerlink.com/content/l30185r823104886/ | doi=10.1007/BF01389997 }}
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| [[Category:Algebra]]
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| [[Category:Number theory]]
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