The Fontaine–Mazur conjecture for irreducible geometric Galois representations of dimension greater than 2

The Fontaine–Mazur conjecture predicts that every irreducible geometric p-adic Galois representation of the absolute Galois group of the rationals — continuous, unramified outside finitely many primes, and de Rham at p — arises from algebraic geometry (is of geometric origin), and, in the Fontaine–Mazur–Langlands conjecture obtained by combining this with Langlands reciprocity, from automorphic forms. It is essentially settled in dimensions at most 2, but for irreducible representations of dimension greater than 2 it remains largely unsolved: only essentially self-dual cases and potential automorphy over suitable extensions are known. As of August 2026 the problem remains open: all recent progress concerns dimension 2, residual special cases, or foundational structure theory, and no result establishes (or refutes) the conjecture for general irreducible geometric representations of dimension n > 2 over the rationals.

Background

A central theme of the Langlands program is a dictionary between Galois representations and automorphic forms. A p-adic Galois representation of the rationals is a continuous homomorphism $\rho:G_{\mathbb{Q}}\to \mathrm{GL}_n(\overline{\mathbb{Q}}_p)$ , where $G_{\mathbb{Q}}$ is the absolute Galois group of $\mathbb{Q}$ and $p$ is a fixed prime. Following Fontaine and Mazur (1995), such a representation is called geometric if it is unramified at all but finitely many primes and is de Rham at the primes above $p$ : 'de Rham' is the condition, satisfied by the p-adic étale cohomology of smooth proper algebraic varieties, that cuts out exactly the representations coming from reasonable p-adic Hodge theory. The evident sources of geometric representations are algebraic varieties themselves: a representation is of geometric origin if it is a subquotient of $H^i(X_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}}_p)\otimes \mathbb{Q}_p(\chi_p^r)$ for some smooth proper variety $X/\mathbb{Q}$ , cohomological degree $i$ , and integer $r$ , where $\chi_p$ is the p-adic cyclotomic character (the Galois action on all p-power roots of unity). Fontaine and Mazur conjectured that these are the only sources: every irreducible geometric representation of $G_{\mathbb{Q}}$ is of geometric origin. Combined with the Tate conjecture on algebraic cycles (the assertion that the Galois-invariant part of the cohomology of a variety over a finite field is spanned by classes of algebraic cycles), this would force such representations to sit in strictly compatible systems of $l$ -adic representations (families, one for each prime $l$ , arising from a single geometric object and satisfying compatible trace relations under comparison isomorphisms), and the Fontaine–Mazur–Langlands expectation, obtained by combining this with Langlands reciprocity, predicts that every irreducible geometric representation arises from automorphic forms. Here an automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ is an irreducible constituent of a representation of the adelic group $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ on functions on $\mathrm{GL}_n(\mathbb{Q})\backslash\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ invariant under the maximal compact subgroup almost everywhere; the best-understood ones for reciprocity are the cuspidal (irreducible, square-integrable modulo the center) representations that are regular algebraic, meaning their infinitesimal character matches that of a cohomological automorphic form with regular highest weight. The Fontaine–Mazur–Langlands conjecture then predicts that every irreducible geometric representation is automorphic: via some isomorphism $\iota:\overline{\mathbb{Q}}_p\cong\mathbb{C}$ , it is isomorphic to the Galois representation $\rho_{\pi,\iota}$ attached to a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ satisfying Clozel's algebraicity condition (the condition, defined by Clozel, under which $\pi$ is predicted to carry an associated compatible system of geometric Galois representations); in Newton's survey this is Conjecture 2.5.1, obtained by combining the Fontaine–Mazur conjecture with Langlands reciprocity. The converse direction — constructing Galois representations attached to automorphic representations — is a separate problem. Newton's 2022 survey states the Fontaine–Mazur conjecture as its Conjecture 2.4.1 and reviews the state of this reciprocity picture. In dimension $n=1$ the conjecture is essentially class field theory (a theorem of Tate). In dimension $n=2$ the odd case — where complex conjugation has determinant $-1$ , the parity forced by geometry — was settled by Kisin (2009), with Emerton's completed-cohomology methods and Pan's treatment of the residually reducible case (where the reduction modulo p is a direct sum) completing the picture, building on Khare and Wintenberger's proof of Serre's modularity conjecture (the conjecture that every odd irreducible representation of $G_{\mathbb{Q}}$ on a finite-dimensional vector space over a finite field arises from a modular form). In higher dimensions the proven territory is the essentially self-dual case, where the representation is isomorphic to a twist of its own contragredient and the problem can be transferred to unitary Shimura varieties (varieties whose points parametrize abelian varieties with extra endomorphisms, providing the algebraic models and complex structures that the $\mathrm{GL}_n$ symmetric spaces lack); there, modularity lifting theorems — the Taylor–Wiles method as extended by Calegari and Geraghty, which propagates automorphy from a residual representation to its p-adic lifts — give automorphy. For general irreducible geometric representations of dimension $n>2$ , the strongest results are potential automorphy theorems (Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, Thorne): automorphy holds only after restriction to a suitable finite extension of $\mathbb{Q}$ , which falls short of the conjecture over $\mathbb{Q}$ itself. In the converse direction, Harris–Lan–Taylor–Thorne and Scholze attached p-adic Galois representations to arbitrary regular algebraic cuspidal automorphic representations, and Scholze showed that systems of Hecke eigenvalues (eigenvalues for the commuting ring of Hecke operators) occurring in completed cohomology — Emerton's inverse limit of the cohomology of the tower of locally symmetric spaces of $\mathrm{GL}_n$ at p-power level — carry Galois representations; Caraiani and Le Hung proved that these representations satisfy the expected oddness parity condition on complex conjugation. Newton's survey also records a p-adic Langlands reciprocity conjecture in this language — every irreducible, continuous, finitely ramified, odd p-adic representation should occur in completed cohomology — known essentially completely only for $n\le 2$ ; and it notes that for non-regular-algebraic automorphic representations, the simplest case being Maass wave forms (non-holomorphic modular forms, expected to be attached to even Artin representations, i.e. finite-image representations on which complex conjugation has determinant $+1$ ), no general construction of Galois representations is known. The Fontaine–Mazur conjecture in dimension greater than 2 is the precise open core, with the Fontaine–Mazur–Langlands conjecture (Conjecture 2.5.1) as its automorphic companion, obtained by combining it with Langlands reciprocity and supported, in Newton's assessment, by stronger evidence; the adjacent strands just described are documented below as also open.

Problem Statement

Conjecture (Fontaine–Mazur, as stated in Conjecture 2.4.1 of Newton's survey, following Fontaine–Mazur 1995). Suppose $\rho:G_{\mathbb{Q}}\to \mathrm{GL}_n(\overline{\mathbb{Q}}_p)$ is an irreducible geometric Galois representation, i.e. $\rho$ is continuous, unramified outside a finite set of primes, and de Rham at the primes above $p$ . Must $\rho$ be of geometric origin, that is, a subquotient of $H^i(X_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}}_p)\otimes \mathbb{Q}_p(\chi_p^r)$ for some smooth proper variety $X/\mathbb{Q}$ , some cohomological degree $i$ , and some integer $r$ , where $\chi_p$ is the p-adic cyclotomic character? The problem is to prove or refute this in the range where it remains largely unsolved: irreducible representations of dimension $n>2$ . The Fontaine–Mazur–Langlands conjecture (Conjecture 2.5.1 of Newton's survey), obtained by combining the Fontaine–Mazur conjecture with Langlands reciprocity, is an automorphic consequence of it rather than an equivalent restatement, and Newton reports that the evidence for it surpasses the evidence for the Fontaine–Mazur conjecture itself: is every irreducible geometric representation of dimension $n>2$ automorphic, i.e. isomorphic, via some isomorphism $\iota:\overline{\mathbb{Q}}_p\cong\mathbb{C}$ , to the Galois representation attached to a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ satisfying Clozel's algebraicity condition? Regularity of Hodge–Tate weights (the Hodge–Tate weights at $p$ , i.e. the weights occurring in the $p$ -adic Hodge-theoretic decomposition of $\rho$ at $p$ , being pairwise distinct) and, when $n$ is even, the oddness parity condition on complex conjugation appear in this subject as hypotheses of known theorems — for instance Pan's dimension-2 results — not as restrictions built into either conjecture. A solution must settle Conjecture 2.4.1 as stated for all irreducible geometric representations of dimension $n>2$ over $\mathbb{Q}$ , without restriction to essentially self-dual representations, to representations that become automorphic only over an auxiliary extension, or to any fixed small set of residual representations; since automorphy alone does not deliver the subquotient-of- $H^i(X_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}}_p)$ conclusion, a proof only of the Fontaine–Mazur–Langlands conjecture would not settle Conjecture 2.4.1. The source passage additionally records two adjacent open problems that are not part of this core but remain open: the p-adic Langlands reciprocity conjecture for $n>2$ (every odd irreducible p-adic representation of $G_{\mathbb{Q}}$ , continuous, finitely ramified, appears in completed cohomology) and the construction of Galois representations for non-regular-algebraic automorphic representations, the simplest case being Maass wave forms conjecturally attached to even Artin representations.

References

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