[Torsion analytique, fonctions zêta dynamiques et intégrales orbitales]
L'objet de cette Note est de démontrer une égalité entre la torsion analytique et la valeur en zéro d'une fonction zêta dynamique associée à un fibré vectoriel unitairement plat sur une variété compacte localement symétrique réductive. Nous démontrons aussi une conjecture de Fried.
The purpose of this Note is to prove an identity between the analytic torsion and the value at zero of a dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold, which solves a conjecture of Fried.
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@article{CRMATH_2016__354_4_433_0,
author = {Shen, Shu},
title = {Analytic torsion, dynamical zeta functions and orbital integrals},
journal = {Comptes Rendus. Math\'ematique},
pages = {433--436},
publisher = {Elsevier},
volume = {354},
number = {4},
year = {2016},
doi = {10.1016/j.crma.2016.01.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.01.008/}
}
TY - JOUR AU - Shen, Shu TI - Analytic torsion, dynamical zeta functions and orbital integrals JO - Comptes Rendus. Mathématique PY - 2016 SP - 433 EP - 436 VL - 354 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.01.008/ DO - 10.1016/j.crma.2016.01.008 LA - en ID - CRMATH_2016__354_4_433_0 ER -
%0 Journal Article %A Shen, Shu %T Analytic torsion, dynamical zeta functions and orbital integrals %J Comptes Rendus. Mathématique %D 2016 %P 433-436 %V 354 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.01.008/ %R 10.1016/j.crma.2016.01.008 %G en %F CRMATH_2016__354_4_433_0
Shen, Shu. Analytic torsion, dynamical zeta functions and orbital integrals. Comptes Rendus. Mathématique, Tome 354 (2016) no. 4, pp. 433-436. doi : 10.1016/j.crma.2016.01.008. http://www.numdam.org/articles/10.1016/j.crma.2016.01.008/
[1] The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 379-476
[2] Hypoelliptic Laplacian and Orbital Integrals, Ann. Math. Stud., vol. 177, Princeton University Press, Princeton, NJ, USA, 2011
[3] Equivariant de Rham torsions, Ann. Math., Volume 159 (2004) no. 1, pp. 53-216
[4] An extension of a theorem by Cheeger and Müller, Astérisque, Volume 205 (1992), p. 235
[5] Analytic torsion and the heat equation, Ann. Math., Volume 109 (1979) no. 2, pp. 259-322
[6] Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math., Volume 84 (1986) no. 3, pp. 523-540
[7] Lefschetz formulas for flows, the Lefschetz centennial conference, Mexico City, 1984 (Contemp. Math.), Volume vol. 58, Amer. Math. Soc., Providence, RI, USA (1987), pp. 19-69
[8] Characters, asymptotics and n-homology of Harish–Chandra modules, Acta Math., Volume 151 (1983) no. 1–2, pp. 49-151
[9] Infinite cyclic coverings, Michigan State Univ., E. Lansing, Mich., 1967, Prindle, Weber & Schmidt, Boston, MA, USA (1968), pp. 115-133
[10] R-torsion and zeta functions for locally symmetric manifolds, Invent. Math., Volume 105 (1991) no. 1, pp. 185-216
[11] Intersections of discrete subgroups with Cartan subgroups, J. Indian Math. Soc., Volume 34 (1970) no. 3–4, pp. 203-214
[12] Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., Volume 28 (1978) no. 3, pp. 233-305
[13] Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 721-753
[14] R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210
[15] On the unitary dual of real reductive Lie groups and the modules: the strongly regular case, Duke Math. J., Volume 96 (1999) no. 3, pp. 521-546
[16] S. Shen, Analytic torsion, dynamical zeta functions and orbital integrals, in press.
[17] Unitarizability of certain series of representations, Ann. Math., Volume 120 (1984) no. 1, pp. 141-187
[18] Unitary representations with nonzero cohomology, Compos. Math., Volume 53 (1984) no. 1, pp. 51-90
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