Prove the Period Conjecture in general

Establish the Period Conjecture for mixed motives over fields of algebraic numbers by showing, for every motive M over such a field F, that the F-dimension of its period space P⟨M⟩ equals the Q-dimension of the associated Nori algebra A(M).

Background

The Period Conjecture predicts that all linear relations between period numbers arise from functoriality of motives, leading to a quantitative equality between the dimension of the period space and the endomorphism algebra attached to the motive. The paper proves sharp bounds and equalities in specific hereditary settings (e.g., 1-motives), but emphasizes that in general the conjecture is not resolved.

This statement identifies the overarching open problem, beyond the cases treated in the paper, to prove the full Period Conjecture for general motives.

References

In general the version of the Period Conjecture is wide open, but we get explicit upper bounds.

— Dimension formulas for period spaces via motives and species  (2405.21053 - Huber et al., 2024) in Introduction

It is conjectured that this period map is an isomorphism \S3.2.

— Mishiba's Conjecture on the Coaction of $\infty$-adic Multiple Zeta Values  (2609.30823 - Tsui, 25 Sep 2026) in Section 1, Subsection 1.1, “Classical Background”