Une version relative de la conjecture des périodes de Kontsevich-Zagier

Abstract

Integrating the coefficients of this series on the n-dimensional real cube [0,1] n yields a Laurent seriesWhen F is algebraic we say that Joseph AyoubRésumé. -Nous partons d'une série F = r −∞ f r · r où est l'indéterminée et les coefficients f r = f r (z 1 , · · · , z n ) sont des fonctions holomorphes définies sur un voisinage ouvert du polydisque ferméEn intégrant les coefficients de cette série sur le n-cube réel [0, 1] n , on obtient la série de Laurent [0,1] n F. Lorsque F est algébrique nous dirons que [0,1] n F est une série de périodes. Dans cet article, nous cherchons à déterminer les séries algébriques F telles que [0,1] n F est nulle. En principle, ceci fournit des informations sur les propriétés de transcendance des séries de périodes. Notre résultat principal rappelle la conjecture des périodes de Kontsevich-Zagier sous une forme remaniée. Abstract. -We start with a series F = r −∞ f r · r with indeterminate and where the coefficients f r = f r (z 1 , · · · , z n ) are holomorphic functions defined on an open neighborhood of the closed polydisc D n = {(z 1 , · · · , z n ); |z i | ≤ 1}. Integrating the coefficients of this series on the n-dimensional real cube [0,1] n yields a Laurent series [0,1] n F. When F is algebraic we say that [0,1] n F is a series of periods.

Relative Manin–Mumford for Semi-Abelian Surfaces

“…For results on semi-abelian surfaces close to this Main Lemma, see [6], Propositions 4.a, 4.b and Theorem 2. For a broader perspective on algebraic independence of relative periods, see Ayoub's recent paper [4]. The analogy with the Main Theorem is clear, except perhaps for the last conclusion of Part (B) of the Main Lemma (which forces an isotrivial G ≃ G m × E).…”

Rigid cohomology via the tilting equivalence

“…In the final sections of [6] Ayoub proves a series of results on the relations between the categories of rigid motives RigDM(K, Λ) and some categories of motives defined over the residue field k. The most striking result holds for K ∼ = C((t)) where RigDM(K, Λ) is shown to embed in DM(G m,C , Λ) which is the category of motives over the base G m,C (see [12,Section 11]). This fact allowed Ayoub to define a motivic version of the decomposition Galois group as well as some important non-obvious applications ( [5], [7] and [8]).…”

Effective motives with and without transfers in characteristic p

“…Fix a site (C, τ ). The category of complexes of presheaves Ch(Psh(C)) can be endowed with the projective model structure for which weak equivalences are quasi-isomorphisms (maps inducing isomorphisms of homology presheaves) and fibrations 7 are maps F → F ′ such that F (X) → F ′ (X) is a surjection for all X in C (cfr [ . For any C the category Ch(Psh(C)) is equivalent to the category of presheaves on C with values in Ch(Λ).…”