# P-adic L-function > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/P-adic_L-function > Markdown URL: https://mediated.wiki/source/P-adic_L-function.md > Source: https://en.wikipedia.org/wiki/P-adic_L-function > Source revision: 1353715061 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{DISPLAYTITLE:''p''-adic ''L''-function}} In [mathematics](/source/mathematics), a '''''p''-adic zeta function''', or more generally a '''''p''-adic ''L''-function''', is a function analogous to the [Riemann zeta function](/source/Riemann_zeta_function), or more general [''L''-functions](/source/L-function), but whose [domain](/source/domain_of_a_function) and [target](/source/codomain) are ''p-adic'' (where ''p'' is a [prime number](/source/prime_number)). For example, the domain could be the [''p''-adic integers](/source/p-adic_integer) '''Z'''''p'', a [profinite ''p''-group](/source/profinite_group), or a ''p''-adic family of [Galois representation](/source/Galois_representation)s, and the image could be the [''p''-adic numbers](/source/p-adic_number) '''Q'''''p'' or its [algebraic closure](/source/algebraic_closure). The source of a ''p''-adic ''L''-function tends to be one of two types. The first source—from which [Tomio Kubota](/source/Tomio_Kubota) and [Heinrich-Wolfgang Leopoldt](/source/Heinrich-Wolfgang_Leopoldt) gave the first construction of a ''p''-adic ''L''-function {{harv|Kubota|Leopoldt|1964}}—is via the ''p''-adic interpolation of [special values of ''L''-functions](/source/special_values_of_L-functions). For example, Kubota–Leopoldt used [Kummer's congruence](/source/Kummer's_congruence)s for [Bernoulli number](/source/Bernoulli_number)s to construct a ''p''-adic ''L''-function, the '''''p''-adic Riemann zeta function''' ''ζ''''p''(''s''), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). ''p''-adic ''L''-functions arising in this fashion are typically referred to as '''analytic ''p''-adic ''L''-functions'''. The other major source of ''p''-adic ''L''-functions—first discovered by [Kenkichi Iwasawa](/source/Kenkichi_Iwasawa)—is from the arithmetic of [cyclotomic field](/source/cyclotomic_field)s, or more generally, certain [Galois module](/source/Galois_module)s over [towers of cyclotomic fields](/source/tower_of_fields) or even more general towers. A ''p''-adic ''L''-function arising in this way is typically called an '''arithmetic ''p''-adic ''L''-function''' as it encodes arithmetic data of the Galois module involved. The [main conjecture of Iwasawa theory](/source/main_conjecture_of_Iwasawa_theory) (now a theorem due to [Barry Mazur](/source/Barry_Mazur) and [Andrew Wiles](/source/Andrew_Wiles)) is the statement that the Kubota–Leopoldt ''p''-adic ''L''-function and an arithmetic analogue constructed by Iwasawa theory are essentially the same. In more general situations where both analytic and arithmetic ''p''-adic ''L''-functions are constructed (or expected), the statement that they agree is called the main conjecture of Iwasawa theory for that situation. Such conjectures represent formal statements concerning the philosophy that special values of ''L''-functions contain arithmetic information. ==Dirichlet ''L''-functions== The Dirichlet ''L''-function is given by the [analytic continuation](/source/analytic_continuation) of :L(s,\chi) = \sum_n\frac{\chi(n)}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-\chi(p)p^{-s}} The Dirichlet ''L''-function at negative integers is given by :L(1-n, \chi) = -\frac{B_{n,\chi}}{n} where ''B''''n'',''χ'' is a [generalized Bernoulli number](/source/generalized_Bernoulli_number) defined by : \displaystyle \sum_{n=0}^\infty B_{n,\chi}\frac{t^n}{n!} = \sum_{a=1}^f\frac{\chi(a)te^{at}}{e^{ft}-1} for ''χ'' a [Dirichlet character](/source/Dirichlet_character) with conductor ''f''. ==Definition using interpolation== The Kubota–Leopoldt ''p''-adic ''L''-function ''L''''p''(''s'', ''χ'') interpolates the Dirichlet ''L''-function with the Euler factor at ''p'' removed. More precisely, ''L''''p''(''s'', ''χ'') is the unique continuous function of the ''p''-adic number ''s'' such that : L_p(1-n, \chi) = \left(1-\chi(p)p^{n-1}\right) L(1-n, \chi) for positive integers ''n'' divisible by ''p'' − 1. The right hand side is just the usual Dirichlet ''L''-function, except that the Euler factor at ''p'' is removed, otherwise it would not be ''p''-adically continuous. The continuity of the right hand side is closely related to the [Kummer congruence](/source/Kummer_congruence)s. When ''n'' is not divisible by ''p'' − 1 this does not usually hold; instead : L_p(1-n, \chi) = \left(1-\chi\omega^{-n}(p)p^{n-1}\right) L\left(1-n, \chi\omega^{-n}\right) for positive integers ''n''. Here ''χ'' is twisted by a power of the [Teichmüller character](/source/Teichm%C3%BCller_character) ''ω''. ==Viewed as a ''p''-adic measure== ''p''-adic ''L''-functions can also be thought of as [''p''-adic measures](/source/p-adic_measure) (or [''p''-adic distributions](/source/p-adic_distribution)) on ''p''-profinite Galois groups. The translation between this point of view and the original point of view of Kubota–Leopoldt (as '''Q'''''p''-valued functions on '''Z'''''p'') is via the [Mazur–Mellin transform](/source/Mazur%E2%80%93Mellin_transform) (and [class field theory](/source/class_field_theory)). ==Totally real fields== {{harvtxt|Deligne|Ribet|1980}}, building upon previous work of {{harvtxt|Serre|1973}}, constructed analytic ''p''-adic ''L''-functions for totally real fields. Independently, {{harvtxt|Barsky|1978}} and {{harvtxt|Cassou-Noguès|1979}} did the same, but their approaches followed [Takuro Shintani](/source/Takuro_Shintani)'s approach to the study of the ''L''-values. ==References== *{{Citation | last1=Barsky | first1=Daniel | editor1-last=Amice | editor1-first=Y. | editor1-link = Yvette Amice | editor2-last=Barsky | editor2-first=D. | editor3-last=Robba | editor3-first=P. | title=Groupe d'Étude d'Analyse Ultramétrique (5e année: 1977/78) | chapter-url=http://www.numdam.org/item?id=GAU_1977-1978__5__A9_0 | publisher=Secrétariat Mathématique | location=Paris | isbn=978-2-85926-266-2 | mr=525346 | year=1978 | volume=16 | chapter=Fonctions zêta ''p''-adiques d'une classe de rayon des corps de nombres totalement réels}} *{{Citation | last1=Cassou-Noguès | first1=Pierrette | title=Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta ''p''-adiques | doi=10.1007/BF01389911 | mr=524276 | year=1979 | journal=[Inventiones Mathematicae](/source/Inventiones_Mathematicae) | issn=0020-9910 | volume=51 | issue=1 | pages=29–59| bibcode=1979InMat..51...29C }} *{{Citation | last1=Coates | first1=John | title=On ''p''-adic ''L''-functions | url=http://www.numdam.org/item?id=SB_1988-1989__31__33_0 | mr=1040567 | year=1989 | journal=Astérisque | issn=0303-1179 | issue=177 | pages=33–59}} *{{Citation | last1=Colmez | first1=Pierre | title=Fontaine's rings and ''p''-adic ''L''-functions | url=http://www.math.jussieu.fr/~colmez/tsinghua.pdf | year=2004 | access-date=2011-05-12 | archive-date=2012-03-11 | archive-url=https://web.archive.org/web/20120311101908/http://www.math.jussieu.fr/~colmez/tsinghua.pdf | url-status=dead }} *{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Ribet | first2=Kenneth A. | title=Values of abelian ''L''-functions at negative integers over totally real fields | doi=10.1007/BF01453237 | mr=579702 | year=1980 | journal=[Inventiones Mathematicae](/source/Inventiones_Mathematicae) | issn=0020-9910 | volume=59 | issue=3 | pages=227–286| bibcode=1980InMat..59..227D }} *{{Citation | last1=Iwasawa | first1=Kenkichi | title=On ''p''-adic ''L''-functions | jstor=1970817 | mr=0269627 | year=1969 | journal=[Annals of Mathematics](/source/Annals_of_Mathematics) |series=Second Series | issn=0003-486X | volume=89 | pages=198–205 | doi=10.2307/1970817 | issue=1 | publisher=Annals of Mathematics}} *{{Citation | last1=Iwasawa | first1=Kenkichi | title=Lectures on ''p''-adic ''L''-functions | publisher=[Princeton University Press](/source/Princeton_University_Press) | isbn=978-0-691-08112-0 | mr=0360526 | year=1972}} *{{Citation | last1=Katz | first1=Nicholas M. |author-link=Nick Katz | title=Algebraic geometry |series=Proc. 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