Twisted limit formula for torsion and cyclic base change
[Formule de multiplicité limite tordue pour la torsion et changement de base cyclique]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 435-471.

Soit G le groupe des points complexes d’un groupe de Lie semi-simple réel dont le rang fondamental est égal à 1, par exemple G=SL 2 ()×SL 2 () ou SL 3 (). Alors le rang fondamental de G est égal à 2 et, selon la conjecture faite dans [3], les réseaux dans G devraient avoir « peu » — dans le sens très faible de « sous-exponentiel en le co-volume » — de torsion homologique. En utilisant le changement de base, nous exhibons des suites de réseaux le long desquelles la torsion homologique croît exponentiellement avec la racine carrée du volume. Ce comportement est déduit d’un théorème général qui compare les torsions L 2 tordues et non tordues dans la situation générale d’un changement de base. Nous utilisons également une version équivariante précise du « Théorème de Cheeger-Müller » démontrée par le second auteur [23].

Let G be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. G=SL 2 ()×SL 2 () or SL 3 (). Then the fundamental rank of G is 2, and according to the conjecture made in [3], lattices in G should have ‘little’ — in the very weak sense of ‘subexponential in the co-volume’ — torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the square root of the volume. This is deduced from a general theorem that compares twisted and untwisted L 2 -torsions in the general base-change situation. This also makes uses of a precise equivariant ‘Cheeger-Müller Theorem’ proved by the second author [23].

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.47
Classification : 11F75,  11F70,  11F72,  58J52
Mots clés : Torsion homologique, multiplicité limite, changement de base
@article{JEP_2017__4__435_0,
     author = {Bergeron, Nicolas and Lipnowski, Michael},
     title = {Twisted limit formula for torsion and cyclic~base change},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {435--471},
     publisher = {Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.47},
     zbl = {1380.11075},
     mrnumber = {3646025},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jep.47/}
}
TY  - JOUR
AU  - Bergeron, Nicolas
AU  - Lipnowski, Michael
TI  - Twisted limit formula for torsion and cyclic base change
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2017
DA  - 2017///
SP  - 435
EP  - 471
VL  - 4
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.47/
UR  - https://zbmath.org/?q=an%3A1380.11075
UR  - https://www.ams.org/mathscinet-getitem?mr=3646025
UR  - https://doi.org/10.5802/jep.47
DO  - 10.5802/jep.47
LA  - en
ID  - JEP_2017__4__435_0
ER  - 
Bergeron, Nicolas; Lipnowski, Michael. Twisted limit formula for torsion and cyclic base change. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 435-471. doi : 10.5802/jep.47. http://www.numdam.org/articles/10.5802/jep.47/

[1] Abert, M.; Bergeron, N.; Biringer, I.; Gelander, T.; Nikolov, N.; Raimbault, J.; Samet, I. On the growth of L 2 -invariants for sequences of lattices in Lie groups (2012) (arXiv:1210.2961)

[2] Barbasch, D.; Moscovici, H. L 2 -index and the Selberg trace formula, J. Funct. Anal., Volume 53 (1983) no. 2, pp. 151-201 | Article | MR 722507 | Zbl 0537.58039

[3] Bergeron, N.; Venkatesh, A. The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu, Volume 12 (2013) no. 2, pp. 391-447 | Article | MR 3028790 | Zbl 1266.22013

[4] Bismut, J.-M.; Zhang, W. An extension of a theorem by Cheeger and Müller, Astérisque, 205, Société Mathématique de France, Paris, 1992 (With an appendix by François Laudenbach) | Numdam

[5] Bismut, J.-M.; Zhang, W. Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal., Volume 4 (1994) no. 2, pp. 136-212 | Article | MR 1262703 | Zbl 0830.58030

[6] Borel, A.; Labesse, J.-P.; Schwermer, J. On the cuspidal cohomology of S-arithmetic subgroups of reductive groups over number fields, Compositio Math., Volume 102 (1996) no. 1, pp. 1-40 | Numdam | MR 1394519 | Zbl 0853.11044

[7] Bouaziz, A. Formule d’inversion d’intégrales orbitales tordues, Compositio Math., Volume 81 (1992) no. 3, pp. 261-290 | MR 1149170 | Zbl 0748.22007

[8] Bredon, G. E. Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York-London, 1972 | MR 413144 | Zbl 0246.57017

[9] Calegari, F. Blog post: torsion in the cohomology of co-compact arithmetic lattices (http://galoisrepresentations.wordpress.com/2013/02/06/)

[10] Calegari, F.; Emerton, M. Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms, Ann. of Math. (2), Volume 170 (2009) no. 3, pp. 1437-1446 | Article | MR 2600878 | Zbl 1195.22015

[11] Calegari, F.; Emerton, M. Mod-p cohomology growth in p-adic analytic towers of 3-manifolds, Groups Geom. Dyn., Volume 5 (2011) no. 2, pp. 355-366 | Article | MR 2782177 | Zbl 1242.57014

[12] Calegari, F.; Venkatesh, A. A torsion Jacquet–Langlands correspondence (2012) (arXiv:1212.3847)

[13] Clozel, L. Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982) no. 1, pp. 45-115 | Article | Numdam | Zbl 0516.22010

[14] Delorme, P. Théorème de Paley-Wiener invariant tordu pour le changement de base C/R, Compositio Math., Volume 80 (1991) no. 2, pp. 197-228 | MR 1132093 | Zbl 0765.22007

[15] Harish-Chandra Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2), Volume 104 (1976) no. 1, pp. 117-201 | Article | MR 439994 | Zbl 0331.22007

[16] Herb, R. A.; Wolf, J. A. The Plancherel theorem for general semisimple groups, Compositio Math., Volume 57 (1986) no. 3, pp. 271-355 | Numdam | MR 829325 | Zbl 0587.22005

[17] Illman, S. Smooth equivariant triangulations of G-manifolds for G a finite group, Math. Ann., Volume 233 (1978) no. 3, pp. 199-220 | Article | MR 500993 | Zbl 0359.57001

[18] Knudsen, F. F.; Mumford, D. The projectivity of the moduli space of stable curves. I. Preliminaries on ‘det’ and ‘Div’, Math. Scand., Volume 39 (1976) no. 1, pp. 19-55 | Article | MR 437541 | Zbl 0343.14008

[19] Labesse, J.-P. Pseudo-coefficients très cuspidaux et K-théorie, Math. Ann., Volume 291 (1991) no. 4, pp. 607-616 | Article | MR 1135534 | Zbl 0789.22028

[20] Labesse, J.-P.; Waldspurger, J.-L. La formule des traces tordue d’après le Friday Morning Seminar, CRM Monograph Series, 31, American Mathematical Society, Providence, RI, 2013 | Zbl 1272.11070

[21] Langlands, R. P. Base change for GL (2), Annals of Mathematics Studies, 96, Princeton University Press, Princeton, N.J., 1980 | MR 574808

[22] Lipnowski, M. Equivariant torsion and base change, Algebra Number Theory, Volume 9 (2015) no. 10, pp. 2197-2240 | Article | MR 3437760 | Zbl 1377.11066

[23] Lipnowski, M. The equivariant Cheeger–Müller theorem on locally symmetric spaces, J. Inst. Math. Jussieu, Volume 15 (2016) no. 1, pp. 165-202 (See also arXiv:1312.2543v2; version 2 has corrections that we refer to not appearing in the published JIMJ version) | Article | MR 3427597 | Zbl 1335.58021

[24] Lott, J.; Rothenberg, M. Analytic torsion for group actions, J. Differential Geom., Volume 34 (1991) no. 2, pp. 431-481 | Article | MR 1131439 | Zbl 0744.57021

[25] Lück, W. Analytic and topological torsion for manifolds with boundary and symmetry, J. Differential Geom., Volume 37 (1993) no. 2, pp. 263-322 | Article | MR 1205447 | Zbl 0792.53025

[26] Müller, W. Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 721-753 | Article | MR 1189689 | Zbl 0789.58071

[27] Müller, W.; Pfaff, J. On the asymptotics of the Ray-Singer analytic torsion for compact hyperbolic manifolds, Internat. Math. Res. Notices (2013) no. 13, pp. 2945-2983 | Article | MR 3072997 | Zbl 1323.58024

[28] Olbrich, M. L 2 -invariants of locally symmetric spaces, Doc. Math., Volume 7 (2002), pp. 219-237 | MR 1938121 | Zbl 1029.58019

[29] Rohlfs, J. Lefschetz numbers for arithmetic groups, Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989) (Lect. Notes in Math.), Volume 1447, Springer, Berlin, 1990, pp. 303-313 | Article | MR 1082971 | Zbl 0762.11023

[30] Rohlfs, J.; Speh, B. Lefschetz numbers and twisted stabilized orbital integrals, Math. Ann., Volume 296 (1993) no. 2, pp. 191-214 | Article | MR 1219899 | Zbl 0808.11039

[31] Şengün, M. H. On the integral cohomology of Bianchi groups, Experiment. Math., Volume 20 (2011) no. 4, pp. 487-505 | Article | MR 2859903 | Zbl 1269.22007

[32] Serre, J.-P. Cohomologie galoisienne, Lect. Notes in Math., 5, Springer-Verlag, Berlin, 1994 | MR 1324577 | Zbl 0812.12002

Cité par Sources :