p‑adic Galois representation

p‑adic Galois representation
Name	p‑adic Galois representation
Domain	Number theory
Related	Galois group; Local field; Fontaine theory; Hodge–Tate decomposition

p‑adic Galois representation

Introduction

A p‑adic Galois representation is a continuous homomorphism from a Galois group of a field to a GL_n of a p‑adic vector space that plays a central role in modern Andrew Wiles‑era arithmetic, linking Galois group symmetries with elliptic curve and modular form arithmetic, and informing conjectures of Jean-Pierre Serre, Pierre Deligne, Alexander Grothendieck and John Tate. These representations connect the work of Jean-Marc Fontaine and the conjectures of Barry Mazur, relate to the modularity theorem proved by Richard Taylor and Andrew Wiles, and underpin comparisons between \'etale cohomology of varieties studied by Alexander Grothendieck and p‑adic Hodge theoretic objects developed by Jean-Marc Fontaine and Gerd Faltings.

Definitions and basic properties

A p‑adic representation of a Galois group G_K of a local or global field K is a continuous homomorphism rho: G_K -> GL_n(E) where E is a finite extension of the field of p‑adic numbers used in the work of Kurt Hensel, and GL_n(E) appears in the linear algebra context of David Hilbert and Emmy Noether; continuity refers to the profinite topology on G_K studied since Évariste Galois and the p‑adic topology of Kurt Hensel. Such representations enjoy notions of semisimplicity and irreducibility familiar from the representation theory of Ferdinand Georg Frobenius and Issai Schur, and admit invariants such as the Hodge–Tate weights defined by John Tate and the monodromy operator that feature in the work of Jean-Pierre Serre and Pierre Deligne. Local–global compatibility phenomena link representations of decomposition groups at primes studied by Alexander Grothendieck to global Galois representations featured in the proofs by Richard Taylor and Michael Harris.

Examples and classes (crystalline, de Rham, Hodge–Tate, semistable)

Important classes include crystalline, de Rham, Hodge–Tate and semistable representations introduced by Jean-Marc Fontaine and refined by Gerd Faltings, Jean-Pierre Serre and Christophe Breuil. The Tate module of an elliptic curve over a p‑adic field yields a Hodge–Tate representation used in the work of John Tate and Gerd Faltings, while the p‑adic \'etale cohomology of a smooth proper variety as studied by Alexander Grothendieck and Pierre Deligne gives de Rham representations associated to results of Jean-Marc Fontaine and Faltings. Crystalline representations arise in Fontaine’s classification and play roles in the proofs by Christopher Skinner and Andrew Wiles of modularity lifting theorems, whereas semistable representations appear in the study of degenerations as in the work of Kazuya Kato and Luc Illusie.

p‑adic Hodge theory and comparison theorems

p‑adic Hodge theory, developed by Jean-Marc Fontaine, Gerd Faltings, Kazuya Kato and Christopher Breuil, provides comparison isomorphisms between p‑adic \'etale cohomology and de Rham or crystalline cohomologies investigated by Alexander Grothendieck and Jean-Pierre Serre, with key tools such as the period rings B_crys, B_dR and B_st introduced by Jean-Marc Fontaine. The work of Gerd Faltings on the Hodge–Tate decomposition and the subsequent refinements by Jean-Marc Fontaine and Christopher Breuil underpin results applied by Richard Taylor and Michael Harris in automorphy lifting, while the monodromy‑weight conjecture of Pierre Deligne and contributions by Kazuya Kato control the behavior of monodromy operators in the semistable case.

Deformation theory and moduli of p‑adic representations

Deformation theory for p‑adic representations was pioneered by Barry Mazur and later developed by Andrew Wiles, Richard Taylor, Mark Kisin and Christophe Breuil to construct and study universal deformation rings and framed deformation spaces related to modularity lifting and the Taniyama–Shimura–Weil conjecture proved by Andrew Wiles and Richard Taylor. Moduli of p‑adic representations intersect the geometric representation theory of Pierre Deligne and the arithmetic geometry of Alexander Grothendieck through rigid analytic approaches of John Tate and Kiran Kedlaya, and are central to the construction of eigenvarieties studied by Kevin Buzzard and Matthew Emerton.

Applications in number theory and arithmetic geometry

p‑adic Galois representations are instrumental in proofs of major results such as the modularity theorem of Andrew Wiles and Richard Taylor, the proofs of cases of the Fontaine–Mazur conjecture advanced by Frank Calegari and Toby Gee, and the reciprocity laws anticipated by Bernard Dwork and formalized by Grothendieck and Pierre Deligne. They connect the arithmetic of elliptic curve L‑functions considered by Atle Selberg and Ernest Hecke to automorphic forms studied by Robert Langlands and Harris–Taylor, and enable explicit arithmetic applications in the work of Brian Conrad and Fred Diamond on level lowering and congruences between modular forms.

Recent developments and open problems

Recent advances involve the p‑adic Langlands program initiated by Colmez and expanded by Laurent Berger, Matthew Emerton and Pierre Colmez, progress on integral p‑adic Hodge theory by Mark Kisin and Bhargav Bhatt, and advances in moduli approaches by Ana Caraiani and Peter Scholze. Open problems include instances of the Fontaine–Mazur conjecture highlighted by Jean-Pierre Serre and Pierre Deligne, refinements of the p‑adic Langlands correspondence influenced by Robert Langlands and Peter Scholze, and structural questions on the geometry of deformation rings addressed by Barry Mazur and Mark Kisin.

Category:Number theory