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

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] 1 . 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.

Periods of Hodge structures and special values of the gamma function

“…Let us emphasize that separating the periods coming from different degrees seems to be out of the reach of the methods used in this paper. Nevertheless, Theorem A implies the original Gross-Deligne conjecture in some interesting cases where H j (X , g ) vanishes in all degrees but one, as already observed by Maillot and Rössler [24,Cor. 4.2].…”

“…This paper pursues a series of works by Weil [37], Gross [19], Shimura [32], Deligne [13], Anderson [2], Colmez [8] and, more recently, Maillot and Rössler [24], aiming to understand the relations between periods of certain motives over Q and values of the gamma function at rational arguments. Recall that the latter is defined by the convergent integral Γ(z) = ∞ 0 t z−1 e −t d t for Re(z) > 0, and extended to a nowhere vanishing meromorphic function on the complex plane, with simple poles at z = 0, −1, −2, .…”

An arithmetic Lefschetz–Riemann–Roch theorem

“…One important reason is that the equivariant ‐genus is closely related to the logarithmic derivative of certain ‐functions. Köhler–Roessler and Maillot–Roessler have shown in [25] and in [29] that the Faltings heights and the periods of C.M. abelian varieties can be expressed as a formula in terms of the special value of logarithmic derivative of ‐functions at 0.…”

CM Periods, CM Regulators, and Hypergeometric Functions, I

“…Concerning the period conjecture, there is a result of Maillot-Roessler [17] using Arakelov theory on the absolute value of the period. Recently, Fresán [13] proved the formula for the alternating product of the determinants for any smooth projective variety with a finite order automorphism by reducing to a result of Saito-Terasoma [23].…”