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.
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.
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@article{CRMATH_2010__348_7-8_363_0, author = {Avni, Nir and Klopsch, Benjamin and Onn, Uri and Voll, Christopher}, 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}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2010.02.019/} }
TY - JOUR AU - Avni, Nir AU - Klopsch, Benjamin AU - Onn, Uri AU - Voll, Christopher 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 UR - http://archive.numdam.org/articles/10.1016/j.crma.2010.02.019/ DO - 10.1016/j.crma.2010.02.019 LA - en ID - CRMATH_2010__348_7-8_363_0 ER -
%0 Journal Article %A Avni, Nir %A Klopsch, Benjamin %A Onn, Uri %A Voll, Christopher %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 %U http://archive.numdam.org/articles/10.1016/j.crma.2010.02.019/ %R 10.1016/j.crma.2010.02.019 %G en %F CRMATH_2010__348_7-8_363_0
Avni, Nir; Klopsch, Benjamin; Onn, Uri; Voll, Christopher. On representation zeta functions of groups and a conjecture of Larsen–Lubotzky. Comptes Rendus. Mathématique, Tome 348 (2010) no. 7-8, pp. 363-367. doi : 10.1016/j.crma.2010.02.019. http://archive.numdam.org/articles/10.1016/j.crma.2010.02.019/
[1] 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] Kirillov theory for compact p-adic groups, Pacific J. Math., Volume 73 (1977), pp. 365-381
[5] Zeta functions of representations of compact p-adic analytic groups, J. Amer. Math. Soc., Volume 19 (2006), pp. 91-118
[6] Representation growth of linear groups, J. Eur. Math. Soc. (JEMS), Volume 10 (2008), pp. 351-390
[7] Polynomial representation growth and the congruence subgroup problem, Israel J. Math., Volume 144 (2004), pp. 293-316
[8] 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
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