An upper bound on the denominator of Eisenstein classes in Bianchi manifolds

Branchereau, Romain

doi:10.5802/aif.3671

An upper bound on the denominator of Eisenstein classes in Bianchi manifolds
[Une borne supérieure sur le dénominateur de classes d’Eisenstein dans les variétés de Bianchi]

Branchereau, Romain ¹

¹ Mathematics Department Mcgill Burnside Hall 805 Sherbrooke Street West Montreal, Quebec H3A 0B9, Canada

Annales de l'Institut Fourier, Online first, 50 p.

Résumé
Abstract

Une conjecture générale de Harder relie le dénominateur de la cohomologie d’Eisenstein de certains espaces localement symétriques à des valeurs spéciales de fonctions $L$ . Dans cet article, nous considérons l’espace localement symétrique $Y_{Γ} = {SL}_{2} (𝒪) ∖ ℍ_{3}$ associé à ${SL}_{2}$ où $K$ est un corps quadratique imaginaire. Berger donne une borne inférieure sur le dénominateur de la cohomologie d’Eisenstein dans certains cas. Nous montrons comment des travaux d’Ito et de Sczech peuvent être utilisés pour donner une borne supérieure en terme de valeurs spéciales d’une fonction $L$ . Quand le nombre de classe de $K$ est égal à un, nous combinons ce résultat avec celui de Berger pour obtenir le dénominateur exact.

A general conjecture of Harder relates the denominator of the Eisenstein cohomology of certain locally symmetric spaces to special values of $L$ -functions. In this paper we consider the locally symmetric space $Y_{Γ} = {SL}_{2} (𝒪) ∖ ℍ_{3}$ associated to ${SL}_{2} (K)$ where $K$ is an imaginary quadratic field. Berger proves a lower bound on the denominator of the Eisenstein cohomology in certain cases. We show how results of Ito and Sczech can be used to prove an upper bound on the denominator in terms of a special value of an $L$ -function. When the class number of $K$ is one, we combine this result with Berger’s result to obtain the exact denominator.

Reçu le : 2023-04-29
Révisé le : 2023-12-06
Accepté le : 2024-02-05
Première publication : 2025-02-10

DOI : 10.5802/aif.3671

Classification : 11F75, 11F67, 11F41
Keywords: Eisenstein cohomology, Bianchi manifolds, Sczech cocycle, $L$-functions
Mots-clés : Cohomologie d’Eisenstein, variété de Bianchi, cocycle de Sczech, fonction $L$

Affiliations des auteurs :

Branchereau, Romain ¹

¹ Mathematics Department Mcgill Burnside Hall 805 Sherbrooke Street West Montreal, Quebec H3A 0B9, Canada

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Branchereau, Romain. An upper bound on the denominator of Eisenstein classes in Bianchi manifolds. Annales de l'Institut Fourier, Online first, 50 p.

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