--- _id: '31982' abstract: - lang: eng text: "AbstractWe show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold

$$\\Sigma $$

\r\n \ Σ\r\n with Betti number

$$b_1$$

\r\n \r\n \ b\r\n 1\r\n \ \r\n , the order of vanishing of the Ruelle zeta function at zero equals

$$4-b_1$$

\r\n \r\n \ 4\r\n -\r\n \ \r\n b\r\n \ 1\r\n \r\n \ \r\n , while in the hyperbolic case it is equal to

$$4-2b_1$$

\r\n \r\n \ 4\r\n -\r\n \ 2\r\n \r\n b\r\n \ 1\r\n \r\n \ \r\n . This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle

$$S\\Sigma $$

\r\n \ \r\n S\r\n Σ\r\n \ \r\n with harmonic 1-forms on

$$\\Sigma $$

\r\n \ Σ\r\n ." author: - first_name: Mihajlo full_name: Cekić, Mihajlo last_name: Cekić - first_name: Benjamin full_name: Delarue, Benjamin id: '70575' last_name: Delarue - first_name: Semyon full_name: Dyatlov, Semyon last_name: Dyatlov - first_name: Gabriel P. full_name: Paternain, Gabriel P. last_name: Paternain citation: ama: Cekić M, Delarue B, Dyatlov S, Paternain GP. The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. Inventiones mathematicae. 2022;229(1):303-394. doi:10.1007/s00222-022-01108-x apa: Cekić, M., Delarue, B., Dyatlov, S., & Paternain, G. P. (2022). The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. Inventiones Mathematicae, 229(1), 303–394. https://doi.org/10.1007/s00222-022-01108-x bibtex: '@article{Cekić_Delarue_Dyatlov_Paternain_2022, title={The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds}, volume={229}, DOI={10.1007/s00222-022-01108-x}, number={1}, journal={Inventiones mathematicae}, publisher={Springer Science and Business Media LLC}, author={Cekić, Mihajlo and Delarue, Benjamin and Dyatlov, Semyon and Paternain, Gabriel P.}, year={2022}, pages={303–394} }' chicago: 'Cekić, Mihajlo, Benjamin Delarue, Semyon Dyatlov, and Gabriel P. Paternain. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” Inventiones Mathematicae 229, no. 1 (2022): 303–94. https://doi.org/10.1007/s00222-022-01108-x.' ieee: 'M. Cekić, B. Delarue, S. Dyatlov, and G. P. Paternain, “The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds,” Inventiones mathematicae, vol. 229, no. 1, pp. 303–394, 2022, doi: 10.1007/s00222-022-01108-x.' mla: Cekić, Mihajlo, et al. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” Inventiones Mathematicae, vol. 229, no. 1, Springer Science and Business Media LLC, 2022, pp. 303–94, doi:10.1007/s00222-022-01108-x. short: M. Cekić, B. Delarue, S. Dyatlov, G.P. Paternain, Inventiones Mathematicae 229 (2022) 303–394. date_created: 2022-06-20T08:24:17Z date_updated: 2022-06-21T11:55:15Z department: - _id: '548' doi: 10.1007/s00222-022-01108-x intvolume: ' 229' issue: '1' keyword: - General Mathematics language: - iso: eng page: 303-394 publication: Inventiones mathematicae publication_identifier: issn: - 0020-9910 - 1432-1297 publication_status: published publisher: Springer Science and Business Media LLC status: public title: The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds type: journal_article user_id: '70575' volume: 229 year: '2022' ... --- _id: '34890' abstract: - lang: eng text: 'We prove that the 4-rank of class groups of quadratic number fields behaves as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics. ' author: - first_name: Étienne full_name: Fouvry, Étienne last_name: Fouvry - first_name: Jürgen full_name: Klüners, Jürgen id: '21202' last_name: Klüners citation: ama: Fouvry É, Klüners J. On the 4-rank of class groups of quadratic number fields. Inventiones mathematicae. 2006;167(3):455-513. doi:10.1007/s00222-006-0021-2 apa: Fouvry, É., & Klüners, J. (2006). On the 4-rank of class groups of quadratic number fields. Inventiones Mathematicae, 167(3), 455–513. https://doi.org/10.1007/s00222-006-0021-2 bibtex: '@article{Fouvry_Klüners_2006, title={On the 4-rank of class groups of quadratic number fields}, volume={167}, DOI={10.1007/s00222-006-0021-2}, number={3}, journal={Inventiones mathematicae}, publisher={Springer Science and Business Media LLC}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2006}, pages={455–513} }' chicago: 'Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of Quadratic Number Fields.” Inventiones Mathematicae 167, no. 3 (2006): 455–513. https://doi.org/10.1007/s00222-006-0021-2.' ieee: 'É. Fouvry and J. Klüners, “On the 4-rank of class groups of quadratic number fields,” Inventiones mathematicae, vol. 167, no. 3, pp. 455–513, 2006, doi: 10.1007/s00222-006-0021-2.' mla: Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of Quadratic Number Fields.” Inventiones Mathematicae, vol. 167, no. 3, Springer Science and Business Media LLC, 2006, pp. 455–513, doi:10.1007/s00222-006-0021-2. short: É. Fouvry, J. Klüners, Inventiones Mathematicae 167 (2006) 455–513. date_created: 2022-12-23T09:36:15Z date_updated: 2023-03-06T09:12:30Z department: - _id: '102' doi: 10.1007/s00222-006-0021-2 intvolume: ' 167' issue: '3' keyword: - General Mathematics language: - iso: eng page: 455-513 publication: Inventiones mathematicae publication_identifier: issn: - 0020-9910 - 1432-1297 publication_status: published publisher: Springer Science and Business Media LLC related_material: link: - relation: confirmation url: https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/ranks.pdf status: public title: On the 4-rank of class groups of quadratic number fields type: journal_article user_id: '93826' volume: 167 year: '2006' ...