Galois representations

Galois representations
Name	Galois representations
Field	Number theory; Algebraic geometry; Representation theory
Introduced by	Évariste Galois; Emil Artin
Related	Langlands program; Modular forms; Étale cohomology

Galois representations are homomorphisms from a Galois group associated to a field extension to a matrix group that encode arithmetic and geometric information. They connect the study of Évariste Galois and Emil Artin with modern tools such as Alexander Grothendieck's Étale cohomology, Andrew Wiles's proof of the Taniyama–Shimura–Weil conjecture, and the broad framework of the Langlands program. These representations serve as a bridge between algebraic number theory, Algebraic geometry, and Representation theory.

Introduction

Galois representations arise by letting a Galois group, typically the absolute Galois group of a number field or a local field, act on algebraic objects producing linear actions into groups like GL_n over fields such as Q_l, F_l, or C. Key historical milestones include ideas from Évariste Galois, the reciprocity laws developed by Richard Dedekind and Emil Artin, and later structural advances by Serre, Tate, and Grothendieck that linked these representations to cohomology theories and automorphic forms.

Definitions and basic examples

A prototypical example is the Galois action on the l-adic Tate module of an elliptic curve E over a number field, giving a continuous representation of the absolute Galois group into GL_2(Q_l). Other standard sources include the action on the l-adic cohomology of a smooth projective variety, motives constructed by Pierre Deligne, and Artin representations arising from finite Galois extensions studied by Emil Artin and Richard Brauer. Classical examples connect to Cyclotomic fields and the Kronecker–Weber theorem via the cyclotomic character, and to the representations attached to modular forms as developed by Jean-Pierre Serre and Nicholas Katz.

Properties and classifications

Important properties include continuity, ramification behavior at primes, Hodge–Tate decomposition at places above l, and purity constraints from the Weil conjectures proved by Pierre Deligne. Classification often treats representations by dimension, coefficient field (such as Q_l or F_l), and local behavior: unramified, crystalline, semistable, or de Rham in the sense of Jean-Marc Fontaine. Local–global principles relate local representations at completions like Q_p to global structures for number fields such as Q or imaginary quadratic fields studied by Carl Friedrich Gauss and Ernst Kummer.

Galois representations in arithmetic geometry

Galois representations appear naturally in the study of rational points on varieties, via the comparison isomorphisms of Grothendieck and Alexander Grothendieck's formulation of the étale fundamental group. The l-adic representations attached to the cohomology of varieties play a central role in proofs of the Weil conjectures by Pierre Deligne and in the study of motives envisioned by Grothendieck and developed by Yves André and Uwe Jannsen. In the context of elliptic curves and abelian varieties, representations link the arithmetic of Andrew Wiles's modularity results to the arithmetic of Gerd Faltings and Faltings's theorem on rational points.

Modularity and reciprocity conjectures

Conjectural correspondences between Galois representations and automorphic forms lie at the heart of the Langlands program proposed by Robert Langlands. Landmark achievements include the modularity theorem proved by Andrew Wiles, Richard Taylor, and collaborators relating 2-dimensional representations to Modular forms, and reciprocity results for Artin representations treated by Herbrand and Hecke in special cases. Extensions include work on potential automorphy by Michael Harris, Richard Taylor, and Nicholas Katz tying representations to Automorphic representations for groups such as GL_n and unitary groups.

Deformation theory and deformation rings

Deformation theory of Galois representations, initiated by Barry Mazur, studies lifts of residual representations over coefficient rings and organizes them into universal deformation rings. Techniques developed by Mazur, Richard Taylor, Andrew Wiles, and Fujwara underpin modularity lifting theorems and the use of congruences between automorphic forms from work of Ken Ribet and Jean-Pierre Serre. Deformation conditions—unramified, flat, crystalline—are encoded in local deformation rings, and patching methods introduced by Taylor–Wiles systems and refined by Calegari–Geraghty yield profound consequences for the Langlands reciprocity.

Applications and major results

Major applications include the proof of the Taniyama–Shimura–Weil conjecture (modularity theorem) by Andrew Wiles and Richard Taylor, which implied Fermat's Last Theorem conjectured by Pierre de Fermat. Results on Sato–Tate distribution for elliptic curves by Richard Taylor and collaborators rely on potential automorphy of Galois representations. The Fontaine–Mazur conjecture formulated by Jean-Marc Fontaine and Barry Mazur frames which p-adic representations arise from geometry, guiding research by Kisin, Emerton, and Colmez. Artin reciprocity and class field theory developed by Emil Artin remain foundational, while contemporary progress in potential automorphy, local-global compatibility, and p-adic Hodge theory continues through work by Peter Scholze, Mark Kisin, Gerd Faltings, and Kazuya Kato.

Category:Algebraic number theory