We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from -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 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.
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 -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 définis sur des corps de nombres.
Accepted:
Published online:
Nir Avni 1; Benjamin Klopsch 2; Uri Onn 3; Christopher Voll 4
@article{CRMATH_2010__348_7-8_363_0,
author = {Nir Avni and Benjamin Klopsch and Uri Onn and Christopher Voll},
title = {On representation zeta functions of groups and a conjecture of {Larsen{\textendash}Lubotzky}},
journal = {Comptes Rendus. Math\'ematique},
pages = {363--367},
publisher = {Elsevier},
volume = {348},
number = {7-8},
year = {2010},
doi = {10.1016/j.crma.2010.02.019},
language = {en},
}
TY - JOUR AU - Nir Avni AU - Benjamin Klopsch AU - Uri Onn AU - Christopher Voll TI - On representation zeta functions of groups and a conjecture of Larsen–Lubotzky JO - Comptes Rendus. Mathématique PY - 2010 SP - 363 EP - 367 VL - 348 IS - 7-8 PB - Elsevier DO - 10.1016/j.crma.2010.02.019 LA - en ID - CRMATH_2010__348_7-8_363_0 ER -
%0 Journal Article %A Nir Avni %A Benjamin Klopsch %A Uri Onn %A Christopher Voll %T On representation zeta functions of groups and a conjecture of Larsen–Lubotzky %J Comptes Rendus. Mathématique %D 2010 %P 363-367 %V 348 %N 7-8 %I Elsevier %R 10.1016/j.crma.2010.02.019 %G en %F CRMATH_2010__348_7-8_363_0
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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