Math Conjectures

Fontaine–Mazur Conjecture

OPENLandmarkConjectureProposed 1995 · Standard version

Canonical statement

Let pp be a prime, put GQ=Gal⁡(Q‾/Q)G_{\mathbb Q}=\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q), and let
ρ:GQ⟶GLn(Q‾p) \rho:G_{\mathbb Q}\longrightarrow \mathrm{GL}_n(\overline{\mathbb Q}_p)
be a continuous irreducible representation, unramified outside finitely many primes and potentially semistable at pp, meaning semistable after restriction to the Galois group of a finite extension of Qp\mathbb Q_p. Then there exist a smooth projective variety X/QX/\mathbb Q, integers i,ji,j, and a finite extension E/QpE/\mathbb Q_p such that ρ\rho, after scalar extension to EE, is a subquotient of Heˊti(XQ‾,E)(j)H^i_{\mathrm{\acute et}}(X_{\overline{\mathbb Q}},E)(j).
View source LaTeX
Let \(p\) be a prime, put \(G_{\mathbb Q}=\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)\), and let
\[
  \rho:G_{\mathbb Q}\longrightarrow
    \mathrm{GL}_n(\overline{\mathbb Q}_p)
\] be a continuous irreducible representation, unramified outside finitely many primes and potentially semistable at \(p\), meaning semistable after restriction to the Galois group of a finite extension of \(\mathbb Q_p\). Then there exist a smooth projective variety \(X/\mathbb Q\), integers \(i,j\), and a finite extension \(E/\mathbb Q_p\) such that \(\rho\), after scalar extension to \(E\), is a subquotient of \(H^i_{\mathrm{\acute et}}(X_{\overline{\mathbb Q}},E)(j)\).
NT-GALOIS-031:[FontaineMazur1995][Kisin2009FontaineMazur][Emerton2011FontaineMazur]
Number theoryAlgebraic geometry

Notes

The Fontaine–Mazur conjecture, formulated by Fontaine and Mazur in 1995 [FontaineMazur1995], proposes an intrinsic characterization of the Galois representations arising from algebraic geometry. It predicts that a continuous irreducible representation ρ:GQ→GLn(Q‾p)\rho:G_{\mathbb Q}\to\mathrm{GL}_n(\overline{\mathbb Q}_p) which is unramified outside finitely many primes and potentially semistable at pp must occur, after a Tate twist, as a subquotient of the étale cohomology of some smooth projective variety over Q\mathbb Q.

In dimension two the conjecture is now largely a theorem over Q\mathbb Q: Kisin proved it for GL2\mathrm{GL}_2 under mild hypotheses [Kisin2009FontaineMazur], and Emerton gave a proof via local-global compatibility in the pp-adic Langlands programme [Emerton2011FontaineMazur]. Both arguments proceed through modularity, showing the representation comes from a modular form and hence from geometry. In higher dimension, modularity and potential-automorphy theorems establish broad classes of regular cases, those with distinct Hodge–Tate weights.

For arbitrary dimension and arbitrary Hodge–Tate weights, in particular for irregular representations where current automorphic technology does not apply, the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.