Geodesic
On the periods of motives with complex multiplication and a conjecture of Gross–Deligne
Vincent Maillot1 ; Damian Roessler2
1 UFR de Mathématiques, Université Paris 7 Denis Diderot, C.N.R.S., 75205 Paris, France
2 UFR de Mathématiques, Université Paris 7 Denis Diderot, C.N.R.S., 75205 Paris , France
Annals of mathematics, Tome 160 (2004) no. 2, pp. 727-754.

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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.
DOI : 10.4007/annals.2004.160.727

Vincent Maillot 1 ; Damian Roessler 2

1 UFR de Mathématiques, Université Paris 7 Denis Diderot, C.N.R.S., 75205 Paris, France
2 UFR de Mathématiques, Université Paris 7 Denis Diderot, C.N.R.S., 75205 Paris , France 
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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. http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.727/

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