Non-Archimedean Local Field

• Non‑archimedean local fields are complete, locally compact fields with a discrete valuation and finite residue field that satisfy the ultrametric inequality.
• They underpin rigorous analytical frameworks by supporting Haar measure and exhibiting unique tree‑like geometric properties in p‑adic analysis.
• Their classification into characteristic zero and positive characteristic drives advances in number theory, representation theory, and non‑archimedean geometry.

A non‐archimedean local field is a complete, locally compact field equipped with a nontrivial absolute value that satisfies the ultrametric inequality, and whose residue field is finite. These fields play a central role in number theory, representation theory, and p‑adic analysis, with applications ranging from the paper of automorphic forms and local L‑functions to rigid analytic geometry and the theory of D‑modules in non‑archimedean settings.

1. Definition and Basic Properties

A non‑archimedean local field $F$ is characterized by the existence of a discrete valuation $$\operatorname{ord}(\cdot)\colon F^\times \to \mathbb{Z}$$ such that the associated absolute value satisfies

$$|x+y| \leq \max\{|x|,|y|\} \quad\text{for all } x,y\in F.$$

Its valuation ring is

$\mathfrak{o} = \{ x \in F : |x|\leq 1\},$

which is a compact, open subring, and the unique maximal ideal is

$\mathfrak{p} = \{ x \in F : |x| < 1\}.$

The residue field $\mathfrak{o}/\mathfrak{p}$ is finite; common examples include $\mathbb{Q}_p$ with $\mathfrak{o} = \mathbb{Z}_p$ and $\mathfrak{p} = p \mathbb{Z}_p$, or the field of formal Laurent series $\mathbb{F}_q((T))$ with $\mathfrak{o} = \mathbb{F}_q[[T]]$.

2. Ultrametricity and Topological Aspects

The non‑archimedean absolute value on $F$ is defined by

$|x| = q^{-\operatorname{ord}(x)},$

where $q$ is the cardinality of the residue field. The ultrametric (or non‑archimedean) property

$|x+y| \le \max\{|x|, |y|\}$

implies that every triangle in $F$ is isosceles with the two longer sides having equal length—a feature that leads to a tree‑like structure in its analytic and geometric counterparts. As a locally compact field, $F$ supports a Haar measure, which is essential in harmonic analysis and the paper of representations of $p$-adic groups.

3. Classification and Examples

Non‑archimedean local fields are classified into two main types:

• Characteristic zero: Finite extensions of the field of $p$-adic numbers $\mathbb{Q}_p$. These fields have discrete valuation groups isomorphic to $\mathbb{Z}$ and finite residue fields of characteristic $p$.
• Positive characteristic: Fields of formal Laurent series $\mathbb{F}_q((T))$ over a finite field $\mathbb{F}_q$; here the absolute value is defined via the degree of the polynomial part.

Both types satisfy the properties of having a discrete valuation, finite residue field, and being complete with respect to their metric induced by the absolute value.

4. Applications in Representation Theory and Number Theory

Non‑archimedean local fields provide the natural setting for the paper of smooth representations of $p$-adic groups such as $\mathrm{GL}_n(F)$ or its inner forms. Their compact open subgroups, such as the principal congruence subgroups $$K(n)=\{g\in \mathrm{GL}_n(\mathfrak{o}) : g\equiv I\mod \mathfrak{p}^n\}$$, play a central role in defining and analyzing local newforms, matrix coefficient decay, and conductors of representations. Techniques such as the Rankin–Selberg method, Whittaker models, and local character expansions rely on the ultrametric structure and the measure theory inherent in $F$.

Furthermore, in arithmetic geometry the interplay between non‑archimedean metrics, formal models, and rigid analytic spaces over $F$ underpins the paper of local heights, intersection theory on Shimura varieties, and integral formulas analogous to those in real differential geometry (for example, in non‑archimedean integral geometry and probabilistic Schubert calculus).

5. Summary Table of Key Attributes

Property	Description	Example
Absolute Value	Non‑archimedean; satisfies $|x+y|\leq\max\{|x|,|y|\}$	$|p|=q^{-1}$ for $\mathbb{Q}_p$
Valuation Ring	Compact open subring $\mathfrak{o}=\{x\mid |x|\le1\}$	$\mathbb{Z}_p$
Maximal Ideal	$\mathfrak{p} = \{x\mid |x|<1\}$	$p\mathbb{Z}_p$
Residue Field	Finite field $\mathfrak{o}/\mathfrak{p}$	$\mathbb{F}_p$ or $\mathbb{F}_q$
Completeness and Local Compactness	Complete and locally compact with respect to the induced metric	All $p$-adic fields

Non‑archimedean local fields are central objects whose arithmetic and analytic structures form the backbone of modern theories in number theory, representation theory, and non‑archimedean geometry. Their distinctive ultrametric properties lead to unique geometric, combinatorial, and analytic phenomena that continue to inform research in both pure and applied mathematics.