$p$-adic properties of motivic fundamental lines

Olivier Fouquet

doi:10.5802/jep.38

p

-adic properties of motivic fundamental lines
[Propriétés

p

-adiques des droites fondamentales motiviques]

Olivier Fouquet ¹

¹ Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 37-86.

Résumé
Abstract

Nous prouvons la conjecture de compatibilité des droites fondamentales $p$ -adiques avec les spécialisations aux points motiviques pour une large classe de familles $p$ -adiques de représentations galoisiennes (par exemple, les familles provenant de familles $p$ -adiques de représentations automorphes du groupe des unités d’une algèbre de quaternions ou d’un groupe unitaire totalement défini) et en déduisons la compatibilité de la Conjecture Équivariante sur les Nombres de Tamagawa pour ces spécialisations. Néanmoins, nous montrons également que les droites fondamentales ne sont en général pas compatibles avec les spécialisations arbitraires à valeurs dans un anneau intègre de caractéristique zéro. Ceci indique qu’il est nécessaire de modifier la conjecture de [73] en utilisant la cohomologie complétée.

We prove the conjectured compatibility of $p$ -adic fundamental lines with specializations at motivic points for a wide class of $p$ -adic families of $p$ -adic Galois representations (for instance, the families which arise from $p$ -adic families of automorphic representations of the unit group of a quaternion algebra or of a totally definite unitary group) and deduce the compatibility of the Equivariant Tamagawa Number Conjectures for them. However, we also show that fundamental lines are not compatible with arbitrary characteristic zero specializations with values in a domain in general. This points to the need to modify the conjectures of [73] using completed cohomology.

Reçu le : 2016-06-08
Accepté le : 2016-12-14
Publié le : 2017-01-09

MR Zbl

DOI : 10.5802/jep.38

Classification : 11G40, 11F67, 11F70, 11R23, 11F33
Keywords: Iwasawa theory, $p$-adic automorphic forms
Mot clés : Théorie d’Iwasawa, formes automorphes $p$-adiques

Affiliations des auteurs :

Olivier Fouquet ¹

¹ Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Olivier Fouquet. $p$-adic properties of motivic fundamental lines. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 37-86. doi : 10.5802/jep.38. https://jep.centre-mersenne.org/articles/10.5802/jep.38/

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