Prove the Gross–Deligne period formula

Prove that for a Betti–de Rham structure with complex multiplication by an abelian field $F\subset\mathbb{Q}(\zeta_d)$, each rank-one period $P(H_\sigma)$ is, up to an algebraic factor, the gamma-value product specified by the Gross–Deligne conjecture.

Background

The paper invokes the Gross–Deligne conjecture to describe the periods of CM motives and to interpret the periods arising from reducible hypergeometric motives as gamma quotients. The conjecture is stated for a CM Betti–de Rham structure whose coefficient field is an abelian extension of Q\mathbb{Q}. Its validity would make the transcendental quantities in the paper’s evaluation formulas precisely computable in terms of gamma values.

References

The Gross--Deligne conjecture says that $P(H_\sigma)$ has a precise form as a gamma quotient.

— The Arithmetic of Reducible Rank 2 Hypergeometric Motives  (2609.28065 - Rosen, 23 Sep 2026) in Section 3, subsection “Gross--Deligne Periods,” Conjecture \ref{grossdeligne-l-values}