# On the periods of motives with complex multiplication and a conjecture of Gross–Deligne | Annals of Mathematics

Abstract We prove that the existence of an automorphism of finite order on a 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 Γ-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.

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### 数学年刊（Annals of Mathematics）

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