Math Conjectures

Grothendieck–Katz pp-Curvature Conjecture

OPENLandmarkConjectureProposed 1969–1972 · Standard version

Canonical statement

Let KK be a number field, let X/KX/K be a smooth connected variety, and let (E,∇)(E,\nabla) be an algebraic vector bundle with integrable connection on XX. After extending these data over OK[S−1]\mathcal O_K[S^{-1}] for some finite set SS of finite places, reduce at a place p∉S\mathfrak p\notin S of residue characteristic pp. For a local derivation DD on the reduction XpX_{\mathfrak p}, define the pp-curvature by
ψp(D)=∇D p−∇D[p], \psi_{\mathfrak p}(D)= \nabla_D^{\,p}-\nabla_{D^{[p]}},
where D[p]D^{[p]} is the pp-fold iterate of DD as a derivation. If ψp(D)=0\psi_{\mathfrak p}(D)=0 for every local derivation DD and all but finitely many p\mathfrak p, then, after any embedding K↪CK\hookrightarrow\mathbb C, the analytic monodromy of (E,∇)(E,\nabla) on X(C)X(\mathbb C) has finite image; equivalently, the connection becomes trivial after a finite étale cover of XCX_\mathbb C.
View source LaTeX
Let \(K\) be a number field, let \(X/K\) be a smooth connected variety, and let \((E,\nabla)\) be an algebraic vector bundle with integrable connection on \(X\). After extending these data over \(\mathcal O_K[S^{-1}]\) for some finite set \(S\) of finite places, reduce at a place \(\mathfrak p\notin S\) of residue characteristic \(p\). For a local derivation \(D\) on the reduction \(X_{\mathfrak p}\), define the \(p\)-curvature by
\[
  \psi_{\mathfrak p}(D)=
    \nabla_D^{\,p}-\nabla_{D^{[p]}},
\]
where \(D^{[p]}\) is the \(p\)-fold iterate of \(D\) as a derivation. If \(\psi_{\mathfrak p}(D)=0\) for every local derivation \(D\) and all but finitely many \(\mathfrak p\), then, after any embedding \(K\hookrightarrow\mathbb C\), the analytic monodromy of \((E,\nabla)\) on \(X(\mathbb C)\) has finite image; equivalently, the connection becomes trivial after a finite étale cover of \(X_\mathbb C\).
AG-DIFF-021:[Katz1972PCurvature][Shankar2018PCurvature][PatelShankarWhang2021PCurvature][LamLitt2026PCurvature]
Algebraic geometryNumber theory

Notes

The conjecture is known in rank one, for Gauss–Manin connections, and for several rigid, solvable, locally symmetric, and generic-curve cases. The implication from almost-everywhere vanishing pp-curvature to finite monodromy remains open for a general algebraic connection.
Changing the integral model only changes finitely many excluded places. The converse implication—finite monodromy implies vanishing pp-curvature at almost every place—is known and is not the conjectural direction.

References (4)

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.