p-adic distribution
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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
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- {{Citation | last1=Koblitz | first1=Neal | author1-link=Neal Koblitz | title=p-adic Numbers, p-adic Analysis, and Zeta-Functions | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics, vol. 58 | isbn=978-0-387-96017-3 | mr=754003 | year=1984}}
- {{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | last2=Swinnerton-Dyer | first2=P. | title=Arithmetic of Weil curves | doi=10.1007/BF01389997 | mr=0354674 | year=1974 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=25 | pages=1–61| bibcode=1974InMat..25....1M }}
- {{Citation | last1=Washington | first1=Lawrence C. | title=Cyclotomic fields | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-0-387-94762-4 | year=1997}}