Topic Local Fields
topic admitted

Local Fields

A local field is a non-discrete locally compact topological field of characteristic zero or positive characteristic. Up to isomorphism they fall into the classification:

Local field Description Characteristic
\(\mathbb{R}\) reals \(0\), archimedean
\(\mathbb{C}\) complexes \(0\), archimedean
\(\mathbb{Q}_p\) \(p\)-adic numbers \(0\), non-archimedean
Finite ext. of \(\mathbb{Q}_p\) \(p\)-adic number fields \(0\), non-arch.
\(\mathbb{F}_q((t))\) Laurent series over \(\mathbb{F}_q\) \(p\), non-arch.
Finite ext. of \(\mathbb{F}_q((t))\) function fields of curves \(p\), non-arch.

The non-archimedean case is what enters the Langlands programme for \(p\)-adic groups, buildings, and parahoric subgroups.

Non-archimedean structure

A non-archimedean local field \(F\) carries:

  • a discrete valuation \(v : F^\times \twoheadrightarrow \mathbb{Z}\);
  • a ring of integers \(\mathcal{O} = \{x \in F : v(x) \ge 0\}\), a discrete valuation ring;
  • a maximal ideal \(\mathfrak{m} = \{x : v(x) > 0\}\);
  • a residue field \(k = \mathcal{O}/\mathfrak{m}\), which is finite of size \(q = |k|\);
  • a uniformiser \(\varpi\) generating \(\mathfrak{m}\).

The completion of \(F\) at the valuation is \(F\) itself (the "local" property).

Mixed vs.\ equal characteristic

  • Mixed characteristic: \(\operatorname{char}(F) = 0\), \(\operatorname{char}(k) = p > 0\). Examples: \(\mathbb{Q}_p\), finite extensions.
  • Equal characteristic: \(\operatorname{char}(F) = \operatorname{char}(k) = p > 0\). Examples: \(\mathbb{F}_q((t))\), function fields.

The dichotomy reflects two qualitatively different worlds; for many Langlands-theoretic statements they are treated uniformly, but their detailed structure (in particular ramification, Galois representations) differs.

Galois theory of local fields

For a non-archimedean local field \(F\) with residue field \(k\):

  • \(\operatorname{Gal}(F^{\mathrm{unr}}/F) \cong \operatorname{Gal}(\bar{k}/k) \cong \hat{\mathbb{Z}}\), topologically generated by Frobenius.
  • The full absolute Galois group fits in an extension \(1 \to I_F \to \operatorname{Gal}(\bar{F}/F) \to \hat{\mathbb{Z}} \to 1\) with inertia group \(I_F\).
  • Inertia further filters into wild inertia + tame inertia, giving the ramification filtration.

Why local fields matter for Langlands

  • Local Langlands correspondence: representations of \(G(F)\) for \(G\) a reductive group over \(F\) ↔ certain Galois representations of \(\operatorname{Gal}(\bar{F}/F)\).
  • Local-global principle: global statements (over a number field \(K\)) reduce, in many cases, to compatibility of local statements at each completion.
  • Buildings: the [[node:buildings.bn_pairs_and_tits_systems|Bruhat-Tits building]] of \(G(F)\) is the geometric object that organises the representation theory.
  • Parahoric subgroups: compact open subgroups arising from facets of the Bruhat–Tits building; central to the integral structure theory.

Mathlib formalisation

  • Mathlib.NumberTheory.Padics.* — \(p\)-adic numbers and their extensions.
  • Mathlib.RingTheory.DiscreteValuationRing.* — DVRs and uniformisers.
  • Mathlib.NumberTheory.LocalField — Local field axiomatics (when available).

References

  • Serre, Local Fields (Springer GTM 67, 1979).
  • Neukirch, Algebraic Number Theory (Springer, 1999), ch. II.
  • Cassels–Fröhlich, Algebraic Number Theory (1967).

Prerequisite Topics

Used by