Topic Smooth Affine Group Schemes over Valuation Rings
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Smooth Affine Group Schemes over Valuation Rings

A smooth affine group scheme over a valuation ring \(\mathcal{O}\) is an affine group scheme \(\mathcal{G} / \mathcal{O}\) that is smooth as a morphism of schemes. These give the integral models of algebraic groups, especially relevant for reductive groups over local fields.

Setup

Let \(\mathcal{O}\) be a valuation ring (or more generally, a Dedekind domain) with field of fractions \(F\). A smooth affine \(\mathcal{O}\)-group scheme is an affine group scheme \(\mathcal{G} \to \mathrm{Spec}\, \mathcal{O}\) such that the structure morphism is smooth.

Equivalently: the special fibre \(\bar{\mathcal{G}}\) at the maximal ideal is smooth (with the appropriate finite type / flatness conditions automatically holding).

Generic and special fibres

For a smooth affine \(\mathcal{O}\)-group scheme \(\mathcal{G}\):

  • Generic fibre \(\mathcal{G}_F = \mathcal{G} \otimes_\mathcal{O} F\): affine group scheme over \(F\); typically reductive.
  • Special fibre \(\bar{\mathcal{G}} = \mathcal{G} \otimes_\mathcal{O} \mathbb{F}\) (over the residue field \(\mathbb{F}\)): connected smooth affine group scheme.

Both fibres have the same dimension (smoothness ensures equidimension).

Examples

\(\mathrm{GL}_n / \mathbb{Z}_p\)

\(\mathcal{G} = \mathrm{GL}_n\) as a \(\mathbb{Z}_p\)-group scheme; generic fibre = \(\mathrm{GL}_n(\mathbb{Q}_p)\); special fibre = \(\mathrm{GL}_n(\mathbb{F}_p)\). Smooth and reductive on both sides ("good reduction").

Néron model of an elliptic curve

For an elliptic curve \(E / \mathbb{Q}_p\): the Néron model \(\mathcal{E} / \mathbb{Z}_p\) is a smooth commutative group scheme whose generic fibre is \(E\); the special fibre is the identity component of the reduction (could be elliptic, multiplicative, or additive depending on reduction type).

Parahoric group schemes

For a connected reductive \(G / F\) and a facet \(\mathcal{F}\) of the Bruhat-Tits building: the parahoric group scheme \(\mathcal{G}_\mathcal{F}\) is a specific smooth affine \(\mathcal{O}_F\)-group scheme with generic fibre \(G\). See parahoric group schemes.

Bruhat-Tits "smooth models"

Every connected reductive \(G/F\) admits a smooth affine \(\mathcal{O}_F\)-model; in fact, a family of them indexed by facets of \(\mathcal{B}(G, F)\). The hyperspecial models give the "best" integral structure (when \(G\) is unramified).

Properties

For a smooth affine \(\mathcal{O}\)-group scheme \(\mathcal{G}\):

  • \(\mathcal{O}\)-points: \(\mathcal{G}(\mathcal{O})\) is a topological group (locally compact for \(\mathcal{O}\) compact), compact when \(\mathcal{O}\) is a complete DVR.
  • Smoothness: \(\mathcal{G}(\mathcal{O}) \twoheadrightarrow \bar{\mathcal{G}}(\mathbb{F})\) is surjective (Hensel's lemma + smoothness).
  • Pro-unipotent radical: the kernel \(\ker(\mathcal{G}(\mathcal{O}) \to \bar{\mathcal{G}}(\mathbb{F}))\) is a pro-\(p\) group.

Hensel's lemma and lifting

Theorem (Hensel). For a smooth \(\mathcal{O}\)-scheme \(X\): every \(\mathbb{F}\)-point lifts to an \(\mathcal{O}\)-point. Hence \(X(\mathcal{O}) \twoheadrightarrow X(\mathbb{F})\) is surjective.

For \(\mathcal{G}\) a smooth affine \(\mathcal{O}\)-group scheme: the reduction map \(\mathcal{G}(\mathcal{O}) \twoheadrightarrow \bar{\mathcal{G}}(\mathbb{F})\) is a surjective group homomorphism.

Néron-Ogg-Shafarevich

For an abelian variety \(A / F\) (\(F\) local), \(A\) has good reduction (= extends to a smooth abelian \(\mathcal{O}_F\)-scheme) iff the Galois representation on \(T_\ell(A)\) is unramified for some (equivalently, every) prime \(\ell \ne p\).

This criterion extends to reductive groups: "good reduction" of \(G\) corresponds to \(G\) being unramified.

Construction of integral models

For \(G\) reductive over \(F\):

  1. Conrad-Gabber-Prasad: develops the general theory of smooth affine integral models.
  2. Bruhat-Tits: constructs parahoric models indexed by the building.
  3. Yu: a uniform/alternative construction of Bruhat-Tits parahoric models.

These are all smooth affine \(\mathcal{O}_F\)-group schemes specialising to various reductive / non-reductive integral models.

Role in Langlands

  • Local-global compatibility: smooth integral models at each place are essential for global automorphic theory.
  • Reduction of Shimura varieties: smooth integral models of the reductive group give the smooth integral models of the Shimura variety.
  • Hecke algebras: parahoric / hyperspecial Hecke algebras are defined via smooth integral models.

References

  • Bosch–Lütkebohmert–Raynaud, Néron Models (Ergebnisse 1990).
  • Bruhat–Tits, Groupes réductifs sur un corps local I, II.
  • Conrad–Gabber–Prasad, Pseudo-reductive Groups (2010, 2015).
  • Conrad, Reductive group schemes (Notes, 2014).
  • Yu, Bruhat-Tits theory: a new approach (preprint, 2002).

Prerequisite Topics

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