TAILIEUCHUNG - Đề tài " On the periods of motives with complex multiplication and a conjecture of GrossDeligne "

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]1 . | Annals of Mathematics On the periods of motives with complex multiplication and a conjecture of Gross-Deligne By Vincent Maillot and Damian Roessler Annals of Mathematics 160 2004 727 754 On the periods of motives with complex multiplication and a conjecture of Gross-Deligne By Vincent Maillot and Damian Roessler 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 r-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 1. Our proof relies on the arithmetic fixed-point formula equivariant arithmetic Riemann-Roch theorem proved by K. Kohler and the second author in 13 and the vanishing of the equivariant analytic torsion for the de Rham complex. 1. Introduction In the following article we shall be concerned with the computation of periods in a very general setting. Recall that a period of an algebraic variety defined by polynomial equations with algebraic coefficients is the integral of an algebraic differential against a rational homology cycle. In his article 16 formule 26 p. 303 Lerch proved see also 3 that the abelian integrals that arise as periods of elliptic curves with complex multiplication . whose rational endomorphism ring is an imaginary quadratic field can be related to special values of the T-function. A special case of his result is the following identity already known to Legendre 15 1-ere partie no. 146 147 p. 209 i dt 2334 r3 1 J0 y 1 k2 sin2 t 8 3 where k sin 12 which is associated to an elliptic curve whose rational endomorphism ring is isomorphic to Q y 3 . The formula of Lerch now known 1This should not be confused with the conjecture by Deligne relating periods and .

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