Comptes Rendus
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]
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
@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/

[1] N. Avni Arithmetic groups have rational representation growth, 2008 | arXiv

[2] N. Avni, B. Klopsch, U. Onn, C. Voll, Representation zeta functions of compact p-adic analytic groups and arithmetic groups, preprint

[3] N. Avni, B. Klopsch, U. Onn, C. Voll, Representation zeta functions for SL3, preprint

[4] R.E. Howe Kirillov theory for compact p-adic groups, Pacific J. Math., Volume 73 (1977), pp. 365-381

[5] A. Jaikin-Zapirain Zeta functions of representations of compact p-adic analytic groups, J. Amer. Math. Soc., Volume 19 (2006), pp. 91-118

[6] M. Larsen; A. Lubotzky Representation growth of linear groups, J. Eur. Math. Soc. (JEMS), Volume 10 (2008), pp. 351-390

[7] A. Lubotzky; B. Martin Polynomial representation growth and the congruence subgroup problem, Israel J. Math., Volume 144 (2004), pp. 293-316

[8] W. Veys; W.A. Zúñiga-Galindo Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra, Trans. Amer. Math. Soc., Volume 360 (2008), pp. 2205-2227

[9] C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math., in press

Cité par Sources :

Commentaires - Politique