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

Journal article

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.

Authors: Maillot, V; Rossler, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Princeton University, Department of Mathematics
• Journal: Annals of Mathematics
• Volume: 160
• Issue: 2
• Pages: 727-754
• Publication date: 2004-09-01
• Acceptance date: 2003-04-14
• DOI: 10.4007/annals.2004.160.727
• EISSN: 1939-8980
• ISSN: 0003-486X
• Language: English