Fontaine–Mazur conjecture

Fontaine–Mazur conjecture
Name	Fontaine–Mazur conjecture
Field	Number theory; Galois representation
Proposer	Jean-Marc Fontaine; Barry Mazur
Year	1990s
Status	Open problem

Fontaine–Mazur conjecture. The Fontaine–Mazur conjecture predicts which p-adic Galois representations of absolute Galois groups of number fields arise from geometry and from automorphic forms, relating arithmetic of algebraic varietys, modular forms, and motivic cohomology. It connects ideas originating with Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, and developed in the milieu including Andrew Wiles, Richard Taylor, and Gerd Faltings, framing a bridge between arithmetic geometry, Langlands program, and Iwasawa theory.

Statement of the conjecture

The conjecture asserts that continuous, irreducible, odd, potentially semistable at primes above a rational prime p, p-adic representations of the absolute Galois group of a number field that are unramified outside finitely many places should come from geometry; more precisely they should appear as subquotients of étale cohomology of projective smooth varietys over number fields or correspond to automorphic representations via the Langlands correspondence. This formulation involves hypotheses named after Jean-Marc Fontaine such as potentially semistable, crystalline, and de Rham conditions, together with parity constraints familiar from Serre's conjecture and the notion of being geometric in the sense of Grothendieck and Deligne. The predicted source from geometry is often described via motives as envisaged by Alexander Grothendieck and formalized in conjectures of Pierre Deligne and Yuri Manin.

Historical background and motivation

Motivated by earlier observations of reciprocity and modularity, the conjecture synthesizes threads from Kronecker Weber theorem, Class field theory, and the reciprocity heuristics culminating in the Langlands program proposed by Robert Langlands. Early patterns came from the study of Tate modules of elliptic curves as in work by John Tate and the modularity of elliptic curves proved by Andrew Wiles and Richard Taylor, which relied on deformation theory of Galois representations developed by Barry Mazur. Fontaine and Mazur formalized the geometricity conditions building on Fontaine's p-adic Hodge theory, which itself stems from interactions among Jean-Pierre Serre, Lucien Szpiro, and developments including Grothendieck's dessins d'enfants and conjectural motivic frameworks advanced by Pierre Deligne and Alexander Grothendieck.

Known cases and partial results

Special cases have been established through modularity lifting theorems proved by Andrew Wiles, Richard Taylor, Christophe Breuil, Fred Diamond, Brian Conrad, and C. Khare in the proof of Fermat's Last Theorem and modularity of elliptic curves over Q. Results proving potential automorphy for certain n-dimensional representations were obtained by Michael Harris, Richard Taylor, Taylor–Wiles method contributors, and extended via Laurent Clozel, Brylinski-type ideas and the work of Peter Scholze employing perfectoid spaces. Fontaine–Mazur is known for two-dimensional odd representations over Q under hypotheses combining Serre's conjecture and modularity lifting, while higher-dimensional and general number field cases remain largely open. Complementary partial results appear in Iwasawa theory contexts studied by Ken Ribet and in p-adic Langlands aspects developed by Colmez and Emerton.

Relations to other conjectures and theories

The conjecture is tightly interwoven with the Langlands correspondence, the notion of motives articulated by Grothendieck and Pierre Deligne, and with Tate conjecture and Hodge conjecture perspectives on realizations of motives. It complements Serre's conjecture on mod p Galois representations and interacts with the Birch and Swinnerton-Dyer conjecture via the arithmetic of elliptic curves, while techniques link to Bloch–Kato conjecture predictions for special values of L-functions as formulated by Florian Sprung and Kazuya Kato. The conjecture also relates to p-adic Hodge theory constructs introduced by Jean-Marc Fontaine such as Bcris and BdR, and to the categorical program of Geometric Langlands studied by Edward Frenkel.

Methods and techniques in proofs and approaches

Approaches exploit deformation theory of Galois representations initiated by Barry Mazur and modularity lifting theorems using the Taylor–Wiles method refined by Fred Diamond, Brian Conrad, and Christopher Skinner. p-adic Hodge theory tools developed by Jean-Marc Fontaine and advanced by Gerd Faltings and Kazuya Kato provide the local conditions (crystalline, semistable, de Rham) used to single out geometric representations. Techniques from automorphy lifting via potential automorphy pioneered by Michael Harris and Richard Taylor employ trace formula inputs developed by James Arthur and Langlands-type functoriality heuristics, while recent progress uses perfectoid space methods of Peter Scholze and cohomological vanishing theorems inspired by Deligne and Grothendieck.

Examples and counterexamples in related settings

Proven instances include Tate modules of elliptic curves over Q and higher-dimensional abelian varieties with complex multiplication as studied by Goro Shimura and Yutaka Taniyama-inspired modularity contexts, while modular forms of weight at least two give two-dimensional geometric representations via Deligne’s construction. Local analogues and non-geometric constructions provide cautions: artfully constructed p-adic representations that fail Fontaine’s local conditions illustrate non-geometric behavior documented in work by Barry Mazur and examples influenced by Serre’s mod p representations. No unconditional counterexample to the original Fontaine–Mazur predictions is known; related negative results appear in settings without finiteness or parity constraints, and pathological representations arise in purely group-theoretic or infinite ramification frameworks studied by Jean-Pierre Serre and John Tate.

Category:Conjectures in number theoryCategory:Galois representations