[{"issue":"1","publication_status":"published","publication_identifier":{"issn":["0020-9910","1432-1297"]},"citation":{"mla":"Cekić, Mihajlo, et al. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” <i>Inventiones Mathematicae</i>, vol. 229, no. 1, Springer Science and Business Media LLC, 2022, pp. 303–94, doi:<a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>.","bibtex":"@article{Cekić_Delarue_Dyatlov_Paternain_2022, title={The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds}, volume={229}, DOI={<a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>}, 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} }","short":"M. Cekić, B. Delarue, S. Dyatlov, G.P. Paternain, Inventiones Mathematicae 229 (2022) 303–394.","apa":"Cekić, M., Delarue, B., Dyatlov, S., &#38; Paternain, G. P. (2022). The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. <i>Inventiones Mathematicae</i>, <i>229</i>(1), 303–394. <a href=\"https://doi.org/10.1007/s00222-022-01108-x\">https://doi.org/10.1007/s00222-022-01108-x</a>","ama":"Cekić M, Delarue B, Dyatlov S, Paternain GP. The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. <i>Inventiones mathematicae</i>. 2022;229(1):303-394. doi:<a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>","chicago":"Cekić, Mihajlo, Benjamin Delarue, Semyon Dyatlov, and Gabriel P. Paternain. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” <i>Inventiones Mathematicae</i> 229, no. 1 (2022): 303–94. <a href=\"https://doi.org/10.1007/s00222-022-01108-x\">https://doi.org/10.1007/s00222-022-01108-x</a>.","ieee":"M. Cekić, B. Delarue, S. Dyatlov, and G. P. Paternain, “The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds,” <i>Inventiones mathematicae</i>, vol. 229, no. 1, pp. 303–394, 2022, doi: <a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>."},"page":"303-394","intvolume":"       229","year":"2022","author":[{"full_name":"Cekić, Mihajlo","last_name":"Cekić","first_name":"Mihajlo"},{"first_name":"Benjamin","last_name":"Delarue","full_name":"Delarue, Benjamin","id":"70575"},{"first_name":"Semyon","last_name":"Dyatlov","full_name":"Dyatlov, Semyon"},{"full_name":"Paternain, Gabriel P.","last_name":"Paternain","first_name":"Gabriel P."}],"date_created":"2022-06-20T08:24:17Z","volume":229,"date_updated":"2022-06-21T11:55:15Z","publisher":"Springer Science and Business Media LLC","doi":"10.1007/s00222-022-01108-x","title":"The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds","type":"journal_article","publication":"Inventiones mathematicae","status":"public","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title><jats:p>We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Sigma $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>Σ</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula> with Betti number <jats:inline-formula><jats:alternatives><jats:tex-math>$$b_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msub>\r\n                    <mml:mi>b</mml:mi>\r\n                    <mml:mn>1</mml:mn>\r\n                  </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, the order of vanishing of the Ruelle zeta function at zero equals <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-b_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mn>4</mml:mn>\r\n                    <mml:mo>-</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mi>b</mml:mi>\r\n                      <mml:mn>1</mml:mn>\r\n                    </mml:msub>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, while in the hyperbolic case it is equal to <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-2b_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mn>4</mml:mn>\r\n                    <mml:mo>-</mml:mo>\r\n                    <mml:mn>2</mml:mn>\r\n                    <mml:msub>\r\n                      <mml:mi>b</mml:mi>\r\n                      <mml:mn>1</mml:mn>\r\n                    </mml:msub>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>. 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$S\\Sigma $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>S</mml:mi>\r\n                    <mml:mi>Σ</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> with harmonic 1-forms on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Sigma $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>Σ</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>"}],"user_id":"70575","department":[{"_id":"548"}],"_id":"31982","language":[{"iso":"eng"}],"keyword":["General Mathematics"]},{"language":[{"iso":"eng"}],"keyword":["General Mathematics"],"publication":"Inventiones mathematicae","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. "}],"date_created":"2022-12-23T09:36:15Z","publisher":"Springer Science and Business Media LLC","title":"On the 4-rank of class groups of quadratic number fields","issue":"3","year":"2006","user_id":"93826","department":[{"_id":"102"}],"_id":"34890","type":"journal_article","status":"public","author":[{"first_name":"Étienne","full_name":"Fouvry, Étienne","last_name":"Fouvry"},{"first_name":"Jürgen","full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners"}],"volume":167,"date_updated":"2023-03-06T09:12:30Z","doi":"10.1007/s00222-006-0021-2","related_material":{"link":[{"url":"https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/ranks.pdf","relation":"confirmation"}]},"publication_status":"published","publication_identifier":{"issn":["0020-9910","1432-1297"]},"citation":{"ama":"Fouvry É, Klüners J. On the 4-rank of class groups of quadratic number fields. <i>Inventiones mathematicae</i>. 2006;167(3):455-513. doi:<a href=\"https://doi.org/10.1007/s00222-006-0021-2\">10.1007/s00222-006-0021-2</a>","ieee":"É. Fouvry and J. Klüners, “On the 4-rank of class groups of quadratic number fields,” <i>Inventiones mathematicae</i>, vol. 167, no. 3, pp. 455–513, 2006, doi: <a href=\"https://doi.org/10.1007/s00222-006-0021-2\">10.1007/s00222-006-0021-2</a>.","chicago":"Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of Quadratic Number Fields.” <i>Inventiones Mathematicae</i> 167, no. 3 (2006): 455–513. <a href=\"https://doi.org/10.1007/s00222-006-0021-2\">https://doi.org/10.1007/s00222-006-0021-2</a>.","apa":"Fouvry, É., &#38; Klüners, J. (2006). On the 4-rank of class groups of quadratic number fields. <i>Inventiones Mathematicae</i>, <i>167</i>(3), 455–513. <a href=\"https://doi.org/10.1007/s00222-006-0021-2\">https://doi.org/10.1007/s00222-006-0021-2</a>","short":"É. Fouvry, J. Klüners, Inventiones Mathematicae 167 (2006) 455–513.","bibtex":"@article{Fouvry_Klüners_2006, title={On the 4-rank of class groups of quadratic number fields}, volume={167}, DOI={<a href=\"https://doi.org/10.1007/s00222-006-0021-2\">10.1007/s00222-006-0021-2</a>}, number={3}, journal={Inventiones mathematicae}, publisher={Springer Science and Business Media LLC}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2006}, pages={455–513} }","mla":"Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of Quadratic Number Fields.” <i>Inventiones Mathematicae</i>, vol. 167, no. 3, Springer Science and Business Media LLC, 2006, pp. 455–513, doi:<a href=\"https://doi.org/10.1007/s00222-006-0021-2\">10.1007/s00222-006-0021-2</a>."},"page":"455-513","intvolume":"       167"}]