Comptes Rendus
no. 7-8
Group Theory
On representation zeta functions of groups and a conjecture of Larsen–Lubotzky
[Sur les fonctions zêta de représentations de groupes et sur une conjecture de Larsen–Lubotzky]

Présenté par : Jean-Pierre Serre

Nir Avni1 ; Benjamin Klopsch2 ; Uri Onn3 ; Christopher Voll4
1 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA
2 Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, United Kingdom
3 Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
4 School of Mathematics, University of Southampton, University Road, Southampton SO17 1BJ, United Kingdom
Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 363-367.

On étudie les fonctions zêta dénombrant les représentations linéaires complexes irréductibles de dimension finie de groupes compacts p-adiques analytiques et de groupes arithmétiques. En utilisant une méthode d'intégration p-adique, on démontre que celles de ces fonctions qui sont associées à certains pro-p-groupes p-adiques analytiques satisfont à des équations fonctionnelles. En admettant une conjecture de Serre sur les réseaux dans les groupes semi-simples de rang supérieur 1, on démontre une conjecture de Larsen et Lubotzky pour les groupes algébriques de type A2 définis sur des corps de nombres.

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.02.019

Nir Avni 1 ; Benjamin Klopsch 2 ; Uri Onn 3 ; Christopher Voll 4

1 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA
2 Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, United Kingdom
3 Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
4 School of Mathematics, University of Southampton, University Road, Southampton SO17 1BJ, United Kingdom 
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     title = {On representation zeta functions of groups and a conjecture of {Larsen{\textendash}Lubotzky}},
     journal = {Comptes Rendus. Math\'ematique},
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Nir Avni; Benjamin Klopsch; Uri Onn; Christopher Voll. On representation zeta functions of groups and a conjecture of Larsen–Lubotzky. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 363-367. doi : 10.1016/j.crma.2010.02.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.02.019/

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