Inmathematics, ap-adic zeta function, or more generally ap-adicL-function, is a function analogous to theRiemann zeta function, or more generalL-functions, but whosedomain andtarget arep-adic (wherep is aprime number). For example, the domain could be thep-adic integersZ_p, aprofinitep-group, or ap-adic family ofGalois representations, and the image could be thep-adic numbersQ_p or itsalgebraic closure.

The source of ap-adicL-function tends to be one of two types. The first source—from whichTomio Kubota andHeinrich-Wolfgang Leopoldt gave the first construction of ap-adicL-function (Kubota & Leopoldt 1964)—is via thep-adic interpolation ofspecial values ofL-functions. For example, Kubota–Leopoldt usedKummer's congruences forBernoulli numbers to construct ap-adicL-function, thep-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-adicL-functions arising in this fashion are typically referred to asanalyticp-adicL-functions. The other major source ofp-adicL-functions—first discovered byKenkichi Iwasawa—is from the arithmetic ofcyclotomic fields, or more generally, certainGalois modules overtowers of cyclotomic fields or even more general towers. Ap-adicL-function arising in this way is typically called anarithmeticp-adicL-function as it encodes arithmetic data of the Galois module involved. Themain conjecture of Iwasawa theory (now a theorem due toBarry Mazur andAndrew Wiles) is the statement that the Kubota–Leopoldtp-adicL-function and an arithmetic analogue constructed by Iwasawa theory are essentially the same. In more general situations where both analytic and arithmeticp-adicL-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 ofL-functions contain arithmetic information.

The DirichletL-function is given by the analytic continuation of

${\displaystyle L(s,\chi )=\sum _n\frac{\chi (n)}{n^s}=\prod _{p\text{ prime}}\frac{1}{1-\chi (p)p^{-s}}}$ 

The DirichletL-function at negative integers is given by

${\displaystyle L(1-n,\chi )=-\frac{B_{n,\chi }}{n}}$ 

whereB_n,χ is ageneralized Bernoulli number defined by

${\displaystyle {\displaystyle \sum _{n=0}^{\mathrm{\infty }}{B}_{n,\chi }\frac{t^n}{n!}=\sum _{a=1}^f\frac{\chi (a)t{e}^{at}}{e^{ft}-1}}}$ 

for χ a Dirichlet character with conductorf.

The Kubota–Leopoldtp-adicL-functionL_p(s, χ) interpolates the DirichletL-function with the Euler factor atp removed.More precisely,L_p(s, χ) is the unique continuous function of thep-adic numbers such that

${\displaystyle {\displaystyle L_p(1-n,\chi )=(1-\chi (p)p^{n-1})L(1-n,\chi )}}$ 

for positive integersn divisible byp − 1. The right hand side is just the usual DirichletL-function, except that the Euler factor atp is removed, otherwise it would not bep-adically continuous. The continuity of the right hand side is closely related to theKummer congruences.

Whenn is not divisible byp − 1 this does not usually hold; instead

${\displaystyle {\displaystyle L_p(1-n,\chi )=(1-\chi {\omega }^{-n}(p)p^{n-1})L(1-n,\chi {\omega }^{-n})}}$ 

for positive integersn. Here χ is twisted by a power of theTeichmüller character ω.

p-adicL-functions can also be thought of asp-adic measures (orp-adic distributions) onp-profinite Galois groups. The translation between this point of view and the original point of view of Kubota–Leopoldt (asQ_p-valued functions onZ_p) is via theMazur–Mellin transform (andclass field theory).

Deligne & Ribet (1980), building upon previous work ofSerre (1973), constructed analyticp-adicL-functions for totally real fields. Independently,Barsky (1978) andCassou-Noguès (1979) did the same, but their approaches followed Takuro Shintani's approach to the study of theL-values.

• Barsky, Daniel (1978),"Fonctions zeta p-adiques d'une classe de rayon des corps de nombres totalement réels", inAmice, Y.; Barsky, D.; Robba, P. (eds.),Groupe d'Etude d'Analyse Ultramétrique (5e année: 1977/78), vol. 16, Paris: Secrétariat Math.,ISBN 978-2-85926-266-2,MR 0525346
• Cassou-Noguès, Pierrette (1979), "Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques",Inventiones Mathematicae,51 (1):29–59,Bibcode:1979InMat..51...29C,doi:10.1007/BF01389911,ISSN 0020-9910,MR 0524276
• Coates, John (1989),"On p-adic L-functions",Astérisque (177):33–59,ISSN 0303-1179,MR 1040567
• Colmez, Pierre (2004),Fontaine's rings and p-adic L-functions(PDF)
• Deligne, Pierre; Ribet, Kenneth A. (1980), "Values of abelian L-functions at negative integers over totally real fields",Inventiones Mathematicae,59 (3):227–286,Bibcode:1980InMat..59..227D,doi:10.1007/BF01453237,ISSN 0020-9910,MR 0579702
• Iwasawa, Kenkichi (1969), "On p-adic L-functions",Annals of Mathematics, Second Series,89 (1), Annals of Mathematics:198–205,doi:10.2307/1970817,ISSN 0003-486X,JSTOR 1970817,MR 0269627
• Iwasawa, Kenkichi (1972),Lectures on p-adic L-functions,Princeton University Press,ISBN 978-0-691-08112-0,MR 0360526
• Katz, Nicholas M. (1975), "p-adic L-functions via moduli of elliptic curves",Algebraic geometry, Proc. Sympos. Pure Math., vol. 29, Providence, R.I.:American Mathematical Society, pp. 479–506,MR 0432649
• Koblitz, Neal (1984),p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58, Berlin, New York:Springer-Verlag,ISBN 978-0-387-96017-3,MR 0754003
• Kubota, Tomio;Leopoldt, Heinrich-Wolfgang (1964),"Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen",Journal für die reine und angewandte Mathematik, 214/215:328–339,doi:10.1515/crll.1964.214-215.328,ISSN 0075-4102,MR 0163900
• Serre, Jean-Pierre (1973), "Formes modulaires et fonctions zêta p-adiques", in Kuyk, Willem;Serre, Jean-Pierre (eds.),Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Lecture Notes in Math, vol. 350, Berlin, New York:Springer-Verlag, pp. 191–268,doi:10.1007/978-3-540-37802-0_4,ISBN 978-3-540-06483-1,MR 0404145