p-adic L-function

In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, but taking values in a local field that is a p-adic field rather than the complex numbers. More accurately, the values are allowed to lie in an algebraic closure of a p-adic field, for some given prime number p.

Such functions can sometimes be defined by p-adic interpolation of the values of an ordinary L-function at negative integers. The theory was started by (Kubota & Leopldt 1964), who used Kummer's congruences for Bernoulli numbers, generalized to special values of Hurwitz zeta-functions. These provided p-adic analogues of the Dirichlet L-functions. In subsequent developments, analogues of other types of L-functions were found; connections with module theory were established, setting up another interpretation by means of Iwasawa theory; and the existence of the functions reformulated in terms of a measure theory taking p-adic values.

References

Coates, John (1989), "On p-adic L-functions", Astérisque (177): 33–59, ISSN 0303-1179, MR1040567
Iwasawa, Kenkichi (1969), "On p-adic L-functions", Annals of Mathematics. Second Series, 89: 198–205, ISSN 0003-486X, MR0269627
Iwasawa, Kenkichi (1972), Lectures on p-adic L-functions, Princeton University Press, ISBN 978-0-691-08112-0, MR0360526
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, MR0432649{{citation}}: CS1 maint: extra punctuation (link)
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, MR754003
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, ISSN 0075-4102, MR0163900