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.
@article{10_4007_annals_2004_160_727,
author = {Vincent Maillot and Damian Roessler},
title = {On the periods of motives with complex multiplication and a conjecture of {Gross{\textendash}Deligne}},
journal = {Annals of mathematics},
pages = {727--754},
year = {2004},
volume = {160},
number = {2},
doi = {10.4007/annals.2004.160.727},
mrnumber = {2123937},
zbl = {1077.14032},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.727/}
}
TY - JOUR AU - Vincent Maillot AU - Damian Roessler TI - On the periods of motives with complex multiplication and a conjecture of Gross–Deligne JO - Annals of mathematics PY - 2004 SP - 727 EP - 754 VL - 160 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.727/ DO - 10.4007/annals.2004.160.727 LA - en ID - 10_4007_annals_2004_160_727 ER -
%0 Journal Article %A Vincent Maillot %A Damian Roessler %T On the periods of motives with complex multiplication and a conjecture of Gross–Deligne %J Annals of mathematics %D 2004 %P 727-754 %V 160 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.727/ %R 10.4007/annals.2004.160.727 %G en %F 10_4007_annals_2004_160_727
Vincent Maillot; Damian Roessler. On the periods of motives with complex multiplication and a conjecture of Gross–Deligne. Annals of mathematics, Tome 160 (2004) no. 2, pp. 727-754. doi: 10.4007/annals.2004.160.727
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