p-adic

English

Etymology

From p +‎ -adic. The letter p follows common number theory usage in representing an arbitrary prime number; the suffix -adic signals that properties of the number p (generally, those due to its being prime, and specifically those of a specified p) are fundamental to the theory and determine the properties of the resulting construction.

Adjective

p-adic (not comparable)

1. (number theory) Of, pertaining to, (ultimately) derived from or defined in the context of p-adic numbers.
   • 1995, Alain Escassut, Analytic Elements in P-adic Analysis, World Scientific, page v:

     The theory of p-adic analytic functions in domains other than simple disks is not very well known yet, although such kind of functions happens to intervene in questions linked to p-adic functional analysis, number theory, and others.

   • 1997, Fernando Q. Gouvêa, p-adic Numbers: An Introduction, Springer, 2nd Edition, page 60,

     Definition 3.3.3 The ring of p-adic integers is the valuation ring

     $\mathbb {Z} _{p}=\{x\in \mathbb {Q} _{p}:|x|_{p}\leq 1\}$.

   • 2007, Nguyen Minh Cheung, Nguyen Van Co, Le Quang Thuan, §12: Harmonic Analysis Over P-adic Field I: Some Equations and Singular Field Operators, N. M. Chuong, Yu V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (editors), Harmonic, Wavelet and P-adic Analysis, World Scientific, page 272,

     The paper is organized as follows:

     2. Preliminaries

     3. A p-adic Cauchy problem

     4. The p-adic Hilbert transform

     5. The boundaries in the p-adic space $L^{q}$

     […]

Usage notes

The synonym ℓ-adic is used in some contexts: for instance, in regard to Galois representations (G-modules, where the Galois group G is a vector space over the field of ℓ-adic numbers), and in algebraic geometry, where ℓ-adic cohomology is one of the "classical" constructions of Weil cohomologies.

Synonyms

• (of, etc., p-adic numbers): ℓ-adic

Derived terms

• p-adic absolute value
• p-adic norm
• p-adic number
• p-adic order
• p-adic ordinal
• p-adic ultrametric

 

Translations

Translations

• Catalan: p-àdic
• Finnish: p-adinen
• French: p-adique
• German: p-adisch
• Italian: p-adico
• Polish: p-adyczny
• Portuguese: p-ádico
• Romanian: p-adic
• Spanish: p-ádico

See also

• ℓ-adic cohomology
• p-adic Riemann zeta function

Further reading

• p-adic number on Wikipedia.Wikipedia
• p-adic L-function on Wikipedia.Wikipedia
• Galois module on Wikipedia.Wikipedia
• Fourier–Deligne transform on Wikipedia.Wikipedia
• Weil cohomology theory on Wikipedia.Wikipedia