p-adic Lie group

p-adic Lie group
Name	p-adic Lie group
Type	Topological group
Field	Number theory; Deligne-style representation theory
Related	p-adic numbers, Lie group, Lie algebra, local fields

p-adic Lie group

A p-adic Lie group is a topological group equipped with the structure of a finite-dimensional analytic manifold over the p-adic numbers Q_p or over a finite extension such as a local field K, combining ideas from Galois-theory, Grothendieck-style arithmetic geometry, and Haar measure-based analysis. These groups play central roles in the work of Tate, Jean-Pierre Serre, Armand Borel, Harish-Chandra, and Bernard Dwork on local-global principles, automorphic forms, and Langlands correspondences.

Definition and basic properties

A p-adic Lie group is a topological group G that is also a finite-dimensional analytic manifold over Q_p or a finite extension K, such that the group operations are analytic maps in the sense of Serre's p-adic analytic functions and Wiles-style deformations. Basic properties mirror those of real Lie groups studied by Sophus Lie and Élie Cartan: G carries a local chart system modelled on K^n, admits a compact open subgroup when G is compact, and supports a unique (up to scaling) Haar measure enabling integration used by Harish-Chandra and Piatetski-Shapiro. Foundational structural results were developed by Serre, Lazard, and Tate.

Examples

Standard examples include matrix groups such as GL_n(Q_p), SL_n(Q_p), and their forms over finite extensions studied by Serre and Borel. Compact analytic examples arise from unit groups of p-adic division algebras linked to Grothendieck's study of Brauer groups and from Lubin–Tate groups connected to Lubin, Tate, and Serre. Pro-p groups that are p-adic analytic appear in the work of Lazard and in Iwasawa developed by Iwasawa and Mazur. Classical arithmetic groups such as SL_2(Z_p) and compact forms related to Shimura and Taniyama contexts also furnish examples used by Taylor and Wiles.

Lie algebra and exponential map

Associated to a p-adic Lie group G is a Lie algebra g over K analogous to the construction by Lie and Cartan in the real case; foundational expositions involve Serre and Lazard. The Lie algebra g captures infinitesimal structure and admits a Campbell–Baker–Hausdorff series convergent in p-adic neighborhoods used by Tate and Grothendieck in deformation contexts. An exponential map exp: g -> G is defined on a neighborhood of 0 in g, and a logarithm map is defined on a neighborhood of the identity in G; these maps are analytic in the sense of Serre and are instrumental in work of Piatetski-Shapiro and Harish-Chandra on orbital integrals.

Structure theory and classification

Structure theory for p-adic Lie groups draws on classification results by Borel, Tits, and Malgrange relating reductive groups over local fields to root data and Bruhat–Tits buildings developed by Bruhat and Tits. The Moy–Prasad filtration, introduced by Moy and Prasad, refines the structure near parahoric subgroups and is central to representation-theoretic classification used by Kottwitz and Piatetski-Shapiro. Classification of compact p-adic analytic groups leverages Lazard's correspondence and connections to Iwasawa studied by Mazur.

Representations and harmonic analysis

Representation theory of p-adic Lie groups is a cornerstone of modern number theory, advanced by Harish-Chandra, Serre, Hervé Jacquet, Robert Langlands, and Pierre Colmez. Smooth admissible representations of reductive p-adic groups such as GL_n(Q_p) were classified in parts by Piatetski-Shapiro and Shahidi and play a critical role in the local Langlands correspondence formulated by Langlands and proved in cases by Harris and Taylor. Harmonic analysis on p-adic groups uses parabolic induction, Hecke algebras developed by Iwahori and Matsumoto, and characters studied by Harish-Chandra and Howe.

Applications and connections

p-adic Lie groups connect to the Langlands via local correspondences proved in many cases by Harris, Taylor, Henniart, and Laumon; to Iwasawa and the work of Iwasawa and Mazur on p-adic L-functions; to Shimura and modularity results of Wiles and Taylor; and to p-adic Hodge theory developed by Fontaine and Colmez. Interactions with Grothendieck-style motives involve contributions from Deligne and Grothendieck.

Topological and analytic aspects

Topologically, p-adic Lie groups are totally disconnected when non-compact, a theme in works by Tate, Serre, and Lazard. Analytic aspects rely on p-adic calculus and rigid analytic geometry introduced by Tate and refined by Berkovich; properties of analytic maps and measures follow tools pioneered by Harish-Chandra and Piatetski-Shapiro. Compact open subgroups, filtration theories of Moy and Prasad, and relations to Bruhat–Tits spaces developed by Bruhat and Tits underpin much of the local harmonic analysis used across arithmetic representation theory.

Category:Lie groups