On the periods of motives with complex multiplication and a conjecture of Gross–Deligne

Abstract

We prove that the existence of an automorphism of finite order on a $\overline{\mathbb{Q}}$-variety $X$ implies the existence of algebraic linear relations between the logarithm of certain periods of $X$ and the logarithm of special values of the $\Gamma$-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne [11, p. 205] (This should not be confused with the conjecture by Deligne relating periods and values of $L$-functions.). Our proof relies on the arithmetic fixed-point formula (equivariant arithmetic Riemann-Roch theorem) proved by K. Köhler and the second author in [13] and the vanishing of the equivariant analytic torsion for the de Rham complex.