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 beginning of the theory was a paper of Tomio Kubota and Heinrich Leopoldt, starting from certain congruences applying to special values of Hurwitz zeta-functions. These provided p-adic analogues of the Dirichet 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.[1][2]
References
- Iwasawa, Kenkichi (1972), Lectures on p-adic L-functions, Princeton University Press, ISBN 978-0-691-08112-0, MR0360526
- 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
Notes
- ^ Tomio Kubota and Heinrich Wolfgang Leopoldt, Eine p-adische Theorie der Zetawerte., I, J. Reine Angew. Math., 214/215 (1964), 328–339.
- ^ Nicholas Katz, P-adic L-function via Moduli, pp.479-490 in Algebraic Geometry, Arcata 1974: Proceedings, American Mathematical Society (1974).