Math Conjectures

Grothendieck Period Conjecture

OPENLandmarkConjectureProposed c. 1966 · Standard version

Canonical statement

Let MM be a Nori motive over Q\mathbb Q. Let P(M)⊂C\mathcal P(M)\subset\mathbb C be the field generated by all matrix coefficients of the Betti–de Rham comparison isomorphisms for objects in the tensor category generated by MM, and let Gmot(M)G_{\mathrm{mot}}(M) be its motivic Galois group. Then
trdeg⁡QP(M)=dim⁡Gmot(M). \operatorname{trdeg}_{\mathbb Q}\mathcal P(M) =\dim G_{\mathrm{mot}}(M).
View source LaTeX
Let \(M\) be a Nori motive over \(\mathbb Q\). Let \(\mathcal P(M)\subset\mathbb C\) be the field generated by all matrix coefficients of the Betti–de Rham comparison isomorphisms for objects in the tensor category generated by \(M\), and let \(G_{\mathrm{mot}}(M)\) be its motivic Galois group. Then
\[
  \operatorname{trdeg}_{\mathbb Q}\mathcal P(M)
  =\dim G_{\mathrm{mot}}(M).
\]
AG-PERIOD-026:[Grothendieck1966DeRham][BostCharles2016Periods][HuberMullerStach2017Periods][BakkerTsimerman2025Periods]
Algebraic geometryNumber theory

Notes

A period is a complex number obtained by comparing algebraic de Rham cohomology with Betti cohomology, equivalently in classical examples by integrating an algebraic differential form over a topological cycle. Grothendieck proposed that every algebraic relation among the periods of a motive should be forced by its motivic tensors [Grothendieck1966DeRham]. In modern language, this says that the transcendence degree of the period field equals the dimension of the corresponding motivic Galois group.

The conjecture is known only in restricted motivic and transcendence-theoretic settings; Bost and Charles describe both the strength of the prediction and significant special cases [BostCharles2016Periods]. Nori's category gives an unconditional setting in which the motive, comparison maps, and motivic Galois group can all be defined [HuberMullerStach2017Periods]. Recent functional-transcendence theorems establish powerful geometric analogues [BakkerTsimerman2025Periods], but they do not prove the numerical period conjecture over Q\mathbb Q.

The precise remaining equality is trdeg⁡QP(M)=dim⁡Gmot(M)\operatorname{trdeg}_{\mathbb Q}\mathcal P(M)=\dim G_{\mathrm{mot}}(M) for every Nori motive MM. This Nori formulation fixes the version boundary and avoids relying on a conjectural universal category of pure or mixed motives.

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.