p-adic field - Knowlpedia

A $p$ -adic field is a finite extension $F/\mathbb Q_p$ . It is a nonarchimedean local field of characteristic $0$ .

Its normalized discrete valuation

v_F:F^\times\longrightarrow\mathbb Z

determines the valuation ring, maximal ideal, and residue field:

\mathcal O_F=\{x:v_F(x)\geq0\}, \qquad \mathfrak p_F=\{x:v_F(x)>0\}, \qquad k_F=\mathcal O_F/\mathfrak p_F.

The residue field is finite, of cardinality $q_F=p^f$ . A uniformizer $\varpi_F$ satisfies $v_F(\varpi_F)=1$ and generates $\mathfrak p_F$ .

Topology

The ideals $\mathfrak p_F^n$ form a neighborhood basis of $0$ . The ring $\mathcal O_F$ is compact and open, while $F$ is locally compact and totally disconnected; its additive group is locally profinite. The absolute value is commonly normalized by

|x|_F=q_F^{-v_F(x)}.

Scope

Every nonarchimedean local field of characteristic $0$ is $p$ -adic. Local fields of positive characteristic are finite extensions of $\mathbb F_q((t))$ ; they share much of the smooth representation theory but not every $p$ -adic Hodge-theoretic construction.

Langlands role

Smooth admissible representations of $G(F)$ , the Weil and Weil–Deligne groups of $F$ , hyperspecial subgroups, and local $L$ -parameters form the nonarchimedean local side of the program.

References

Jean-Pierre Serre, Local Fields, Springer, 1979.