p-adic analysis

In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.

The theory of complex-valued numerical functions on the p-adic numbers is just part of the theory of locally compact groups. The usual meaning taken for p-adic analysis is the theory of p-adic-valued functions on spaces of interest.

Applications of p-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of p-adic functional analysis and spectral theory. In many ways p-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is much simpler. Topological vector spaces over p-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem are different.

See also

• Mahler's theorem, which treats a p-adic analog of Taylor series.
• Hensel's lemma
• Locally compact space
• P-adic quantum mechanics
• Real analysis

References

• A course in p-adic analysis, Alain Robert, Springer, 2000, ISBN 9780387986692
• Ultrametric Calculus: An Introduction to P-Adic Analysis, W. H. Schikhof, Cambridge University Press, 2007, ISBN 978052103287
• P-adic Differential Equations, Kiran S. Kedlaya, Cambridge University Press, 2010, ISBN 9780521768795

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