Spherical varieties and norm relations in Iwasawa theory

Loeffler, David

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 1021-1043.

Les familles des classes de cohomologie compatibles pour l’application norme, définies pour les variétés de Shimura et pour d’autres espaces arithmétiques symétriques, jouent un rôle important dans la théorie d’Iwasawa des formes automorphes. Dans cette note, nous développons une approche systématique pour établir la compatibilité dans le cas « vertical » (c’est-à-dire dans le cas où le niveau ne change qu’en un nombre premier fixé $p$), à la fois pour la cohomologie de Betti et la cohomologie étale, en révélant une relation inattendue avec la théorie des variétés sphériques. Cette machinerie peut être utilisée pour construire de nouveaux exemples de telles familles, éventuellement donnant naissance à la fois à de nouvelles constructions des systèmes d’Euler et à de nouvelles fonctions $L$ $p$-adiques : par example, nous obtenons des familles anticyclotomiques de cycles algébriques sur les variétés de Shimura pour les groupes $U\left(n\right)\times U(n+1)$ et $U\left(2n\right)$.

Norm-compatible families of cohomology classes for Shimura varieties, and other arithmetic symmetric spaces, play an important role in Iwasawa theory of automorphic forms. The aim of this note is to give a systematic approach to proving “vertical” norm-compatibility relations for such classes (where the level varies at a fixed prime $p$), treating the case of Betti and étale cohomology at once, and revealing an unexpected relation to the theory of spherical varieties. This machinery can be used to construct many new examples of norm-compatible families, potentially giving rise to new constructions of both Euler systems and $p$-adic $L$-functions: examples include families of algebraic cycles on Shimura varieties for $U\left(n\right)\times U(n+1)$ and $U\left(2n\right)$ over the $p$-adic anticyclotomic tower.

Classification : 11F67, 11R23, 14M17
Mots clés : Euler systems, norm relations, spherical varieties

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