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Olivier Fouquet
$p$-adic properties of motivic fundamental lines
(Propriétés $p$-adiques des droites fondamentales motiviques)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 37-86, doi: 10.5802/jep.38
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Class. Math.: 11G40, 11F67, 11F70, 11R23, 11F33
Mots clés: Théorie d’Iwasawa, formes automorphes $p$-adiques

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.

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