p-adic distribution

In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of p-adic numbers.

Definition

If X is a topological space, a distribution on X with values in an abelian group G is a finitely additive function from the compact open subsets of X to G. Equivalently, if we define the space of test functions to be the locally constant and compactly supported integer-valued functions, then a distribution is an additive map from test functions to G. This is formally similar to the usual definition of distributions, which are continuous linear maps from a space of test functions on a manifold to the real numbers.

p-adic measures

A p-adic measure is a special case of a p-adic distribution, analogous to a measure on a measurable space. A p-adic distribution taking values in a normed space is called a p-adic measure if the values on compact open subsets are bounded.

References

• Colmez, Pierre (2004), Fontaine's rings and p-adic L-functions, http://www.math.jussieu.fr/~colmez/tsinghua.pdf 
• 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 
• Mazur, Barry; Swinnerton-Dyer, P. (1974), "Arithmetic of Weil curves", Inventiones Mathematicae 25: 1–61, doi:10.1007/BF01389997, ISSN 0020-9910, MR0354674 
• Washington, Lawrence C. (1997), Cyclotomic fields (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4