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This extends previous investigations for the case of abelian groups. In particular, the result confirms a conjecture by the second author on this function for arbitrary groups in the nilpotent case. We further prove compatibility of the conjecture with taking wreath products with the cyclic group of order 2 and give examples in degree up to 8. ","lang":"eng"}],"issue":"572","publication":"Journal für die reine und angewandte Mathematik (Crelles Journal)","doi":"10.1515/crll.2004.050","language":[{"iso":"eng"}],"intvolume":"      2004","publication_status":"published","date_updated":"2023-03-06T09:11:16Z","author":[{"id":"21202","full_name":"Klüners, Jürgen","first_name":"Jürgen","last_name":"Klüners"},{"first_name":"G.","last_name":"Malle","full_name":"Malle, G."}],"publication_identifier":{"issn":["0075-4102","1435-5345"]},"title":"Counting nilpotent Galois extensions","year":"2006","external_id":{"arxiv":["math/0112318"]},"citation":{"ieee":"J. Klüners and G. 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Malle. “Counting Nilpotent Galois Extensions.” <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i>, vol. 2004, no. 572, Walter de Gruyter GmbH, 2006, pp. 1–26, doi:<a href=\"https://doi.org/10.1515/crll.2004.050\">10.1515/crll.2004.050</a>.","bibtex":"@article{Klüners_Malle_2006, title={Counting nilpotent Galois extensions}, volume={2004}, DOI={<a href=\"https://doi.org/10.1515/crll.2004.050\">10.1515/crll.2004.050</a>}, number={572}, journal={Journal für die reine und angewandte Mathematik (Crelles Journal)}, publisher={Walter de Gruyter GmbH}, author={Klüners, Jürgen and Malle, G.}, year={2006}, pages={1–26} }","ama":"Klüners J, Malle G. Counting nilpotent Galois extensions. <i>Journal für die reine und angewandte Mathematik (Crelles Journal)</i>. 2006;2004(572):1-26. doi:<a href=\"https://doi.org/10.1515/crll.2004.050\">10.1515/crll.2004.050</a>"},"volume":2004,"user_id":"93826","_id":"34895","publisher":"Walter de Gruyter GmbH","page":"1-26","status":"public"},{"_id":"34891","publisher":"Cellule MathDoc/CEDRAM","page":"607-615","volume":18,"user_id":"93826","status":"public","external_id":{"arxiv":["math/0512260 "]},"citation":{"chicago":"Klüners, Jürgen. “Asymptotics of Number Fields and the Cohen–Lenstra Heuristics.” <i>Journal de Théorie Des Nombres de Bordeaux</i> 18, no. 3 (2006): 607–15. <a href=\"https://doi.org/10.5802/jtnb.561\">https://doi.org/10.5802/jtnb.561</a>.","short":"J. Klüners, Journal de Théorie Des Nombres de Bordeaux 18 (2006) 607–615.","apa":"Klüners, J. (2006). Asymptotics of number fields and the Cohen–Lenstra heuristics. <i>Journal de Théorie Des Nombres de Bordeaux</i>, <i>18</i>(3), 607–615. <a href=\"https://doi.org/10.5802/jtnb.561\">https://doi.org/10.5802/jtnb.561</a>","ieee":"J. Klüners, “Asymptotics of number fields and the Cohen–Lenstra heuristics,” <i>Journal de Théorie des Nombres de Bordeaux</i>, vol. 18, no. 3, pp. 607–615, 2006, doi: <a href=\"https://doi.org/10.5802/jtnb.561\">10.5802/jtnb.561</a>.","ama":"Klüners J. Asymptotics of number fields and the Cohen–Lenstra heuristics. <i>Journal de Théorie des Nombres de Bordeaux</i>. 2006;18(3):607-615. doi:<a href=\"https://doi.org/10.5802/jtnb.561\">10.5802/jtnb.561</a>","bibtex":"@article{Klüners_2006, title={Asymptotics of number fields and the Cohen–Lenstra heuristics}, volume={18}, DOI={<a href=\"https://doi.org/10.5802/jtnb.561\">10.5802/jtnb.561</a>}, number={3}, journal={Journal de Théorie des Nombres de Bordeaux}, publisher={Cellule MathDoc/CEDRAM}, author={Klüners, Jürgen}, year={2006}, pages={607–615} }","mla":"Klüners, Jürgen. “Asymptotics of Number Fields and the Cohen–Lenstra Heuristics.” <i>Journal de Théorie Des Nombres de Bordeaux</i>, vol. 18, no. 3, Cellule MathDoc/CEDRAM, 2006, pp. 607–15, doi:<a href=\"https://doi.org/10.5802/jtnb.561\">10.5802/jtnb.561</a>."},"language":[{"iso":"eng"}],"doi":"10.5802/jtnb.561","publication_identifier":{"issn":["1246-7405"]},"author":[{"full_name":"Klüners, Jürgen","first_name":"Jürgen","last_name":"Klüners","id":"21202"}],"title":"Asymptotics of number fields and the Cohen–Lenstra heuristics","year":"2006","intvolume":"        18","publication_status":"published","date_updated":"2023-03-06T09:12:04Z","date_created":"2022-12-23T09:37:01Z","department":[{"_id":"102"}],"keyword":["Algebra and Number Theory"],"type":"journal_article","publication":"Journal de Théorie des Nombres de Bordeaux","issue":"3","abstract":[{"lang":"eng","text":"We study the asymptotics conjecture of Malle for dihedral groups Dℓ of order 2ℓ, where ℓ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen--Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds. "}]},{"volume":167,"user_id":"93826","_id":"34890","publisher":"Springer Science and Business Media LLC","page":"455-513","status":"public","citation":{"short":"É. Fouvry, J. Klüners, Inventiones Mathematicae 167 (2006) 455–513.","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>.","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>.","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>","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} }","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>","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>."},"doi":"10.1007/s00222-006-0021-2","language":[{"iso":"eng"}],"intvolume":"       167","publication_status":"published","date_updated":"2023-03-06T09:12:30Z","author":[{"full_name":"Fouvry, Étienne","last_name":"Fouvry","first_name":"Étienne"},{"id":"21202","first_name":"Jürgen","last_name":"Klüners","full_name":"Klüners, Jürgen"}],"publication_identifier":{"issn":["0020-9910","1432-1297"]},"title":"On the 4-rank of class groups of quadratic number fields","year":"2006","department":[{"_id":"102"}],"type":"journal_article","keyword":["General Mathematics"],"date_created":"2022-12-23T09:36:15Z","related_material":{"link":[{"relation":"confirmation","url":"https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/ranks.pdf"}]},"abstract":[{"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. ","lang":"eng"}],"publication":"Inventiones mathematicae","issue":"3"},{"date_updated":"2023-03-06T09:13:15Z","publication_status":"published","title":"Cohen–Lenstra Heuristics of Quadratic Number Fields","status":"public","year":"2006","author":[{"last_name":"Fouvry","first_name":"Étienne","full_name":"Fouvry, Étienne"},{"id":"21202","first_name":"Jürgen","last_name":"Klüners","full_name":"Klüners, Jürgen"}],"publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["9783540360759","9783540360766"]},"doi":"10.1007/11792086_4","user_id":"93826","language":[{"iso":"eng"}],"_id":"35958","publisher":"Springer Berlin Heidelberg","related_material":{"link":[{"relation":"confirmation","url":"https://www.researchgate.net/profile/Juergen-Klueners/publication/221451567_Cohen-Lenstra_Heuristics_of_Quadratic_Number_Fields/links/0a85e53298eaed8777000000/Cohen-Lenstra-Heuristics-of-Quadratic-Number-Fields.pdf?origin=publication_detail"}]},"abstract":[{"text":"We establish a link between some heuristic asymptotic formulas (due to Cohen and Lenstra) concerning the moments of the p–part of the class groups of quadratic fields and formulas giving the frequency of the values of the p–rank of these class groups.","lang":"eng"}],"publication":"Lecture Notes in Computer Science","citation":{"chicago":"Fouvry, Étienne, and Jürgen Klüners. “Cohen–Lenstra Heuristics of Quadratic Number Fields.” In <i>Lecture Notes in Computer Science</i>. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. <a href=\"https://doi.org/10.1007/11792086_4\">https://doi.org/10.1007/11792086_4</a>.","short":"É. Fouvry, J. Klüners, in: Lecture Notes in Computer Science, Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.","ieee":"É. Fouvry and J. Klüners, “Cohen–Lenstra Heuristics of Quadratic Number Fields,” in <i>Lecture Notes in Computer Science</i>, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.","apa":"Fouvry, É., &#38; Klüners, J. (2006). Cohen–Lenstra Heuristics of Quadratic Number Fields. In <i>Lecture Notes in Computer Science</i>. Springer Berlin Heidelberg. <a href=\"https://doi.org/10.1007/11792086_4\">https://doi.org/10.1007/11792086_4</a>","bibtex":"@inbook{Fouvry_Klüners_2006, place={Berlin, Heidelberg}, title={Cohen–Lenstra Heuristics of Quadratic Number Fields}, DOI={<a href=\"https://doi.org/10.1007/11792086_4\">10.1007/11792086_4</a>}, booktitle={Lecture Notes in Computer Science}, publisher={Springer Berlin Heidelberg}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2006} }","ama":"Fouvry É, Klüners J. Cohen–Lenstra Heuristics of Quadratic Number Fields. In: <i>Lecture Notes in Computer Science</i>. Springer Berlin Heidelberg; 2006. doi:<a href=\"https://doi.org/10.1007/11792086_4\">10.1007/11792086_4</a>","mla":"Fouvry, Étienne, and Jürgen Klüners. “Cohen–Lenstra Heuristics of Quadratic Number Fields.” <i>Lecture Notes in Computer Science</i>, Springer Berlin Heidelberg, 2006, doi:<a href=\"https://doi.org/10.1007/11792086_4\">10.1007/11792086_4</a>."},"type":"book_chapter","department":[{"_id":"102"}],"place":"Berlin, Heidelberg","date_created":"2023-01-11T09:46:47Z"},{"citation":{"mla":"Klüners, Jürgen. “The Number of S₄-Fields with given Discriminant.” <i>Acta Arithmetica</i>, vol. 122, no. 2, Institute of Mathematics, Polish Academy of Sciences, 2006, pp. 185–94, doi:<a href=\"https://doi.org/10.4064/aa122-2-3\">10.4064/aa122-2-3</a>.","bibtex":"@article{Klüners_2006, title={The number of S₄-fields with given discriminant}, volume={122}, DOI={<a href=\"https://doi.org/10.4064/aa122-2-3\">10.4064/aa122-2-3</a>}, number={2}, journal={Acta Arithmetica}, publisher={Institute of Mathematics, Polish Academy of Sciences}, author={Klüners, Jürgen}, year={2006}, pages={185–194} }","ama":"Klüners J. The number of S₄-fields with given discriminant. <i>Acta Arithmetica</i>. 2006;122(2):185-194. doi:<a href=\"https://doi.org/10.4064/aa122-2-3\">10.4064/aa122-2-3</a>","ieee":"J. Klüners, “The number of S₄-fields with given discriminant,” <i>Acta Arithmetica</i>, vol. 122, no. 2, pp. 185–194, 2006, doi: <a href=\"https://doi.org/10.4064/aa122-2-3\">10.4064/aa122-2-3</a>.","apa":"Klüners, J. (2006). The number of S₄-fields with given discriminant. <i>Acta Arithmetica</i>, <i>122</i>(2), 185–194. <a href=\"https://doi.org/10.4064/aa122-2-3\">https://doi.org/10.4064/aa122-2-3</a>","short":"J. Klüners, Acta Arithmetica 122 (2006) 185–194.","chicago":"Klüners, Jürgen. “The Number of S₄-Fields with given Discriminant.” <i>Acta Arithmetica</i> 122, no. 2 (2006): 185–94. <a href=\"https://doi.org/10.4064/aa122-2-3\">https://doi.org/10.4064/aa122-2-3</a>."},"external_id":{"arxiv":["math/0411484"]},"status":"public","user_id":"93826","volume":122,"page":"185-194","publisher":"Institute of Mathematics, Polish Academy of Sciences","_id":"34892","abstract":[{"text":"We prove that the number of quartic S4--extensions of the rationals of given discriminant d is $O_\\eps(d^{1/2+\\eps})$ for all $\\eps>0$. For a prime number p we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor p or p² is bounded above by O(p¹/²log(p)²). ","lang":"eng"}],"publication":"Acta Arithmetica","issue":"2","type":"journal_article","keyword":["Algebra and Number Theory"],"department":[{"_id":"102"}],"date_created":"2022-12-23T09:40:25Z","date_updated":"2023-03-06T09:52:41Z","publication_status":"published","intvolume":"       122","year":"2006","title":"The number of S₄-fields with given discriminant","publication_identifier":{"issn":["0065-1036","1730-6264"]},"author":[{"full_name":"Klüners, Jürgen","first_name":"Jürgen","last_name":"Klüners","id":"21202"}],"doi":"10.4064/aa122-2-3","language":[{"iso":"eng"}]},{"extern":"1","citation":{"mla":"Burban, Igor, et al. “Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves.” <i>Global Aspects of Complex Geometry</i>, Springer, 2006, pp. 83–129.","bibtex":"@inbook{Burban_Bodnarchuk_Drozd_Greuel_2006, title={Vector bundles and torsion free sheaves on degenerations of elliptic curves}, booktitle={Global Aspects of Complex Geometry}, publisher={Springer}, author={Burban, Igor and Bodnarchuk, L. and Drozd, Yu. and Greuel, G.-M.}, year={2006}, pages={83–129} }","ama":"Burban I, Bodnarchuk L, Drozd Yu, Greuel G-M. Vector bundles and torsion free sheaves on degenerations of elliptic curves. In: <i>Global Aspects of Complex Geometry</i>. Springer; 2006:83–129.","ieee":"I. Burban, L. Bodnarchuk, Yu. Drozd, and G.-M. Greuel, “Vector bundles and torsion free sheaves on degenerations of elliptic curves,” in <i>Global Aspects of Complex Geometry</i>, Springer, 2006, pp. 83–129.","apa":"Burban, I., Bodnarchuk, L., Drozd, Yu., &#38; Greuel, G.-M. (2006). Vector bundles and torsion free sheaves on degenerations of elliptic curves. In <i>Global Aspects of Complex Geometry</i> (pp. 83–129). Springer.","chicago":"Burban, Igor, L. Bodnarchuk, Yu. Drozd, and G.-M. Greuel. “Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves.” In <i>Global Aspects of Complex Geometry</i>, 83–129. Springer, 2006.","short":"I. Burban, L. Bodnarchuk, Yu. Drozd, G.-M. Greuel, in: Global Aspects of Complex Geometry, Springer, 2006, pp. 83–129."},"publication":"Global Aspects of Complex Geometry","department":[{"_id":"602"}],"type":"book_chapter","date_created":"2023-05-05T22:27:45Z","date_updated":"2023-05-07T01:38:44Z","publication_status":"published","author":[{"id":"72064","full_name":"Burban, Igor","last_name":"Burban","first_name":"Igor"},{"last_name":"Bodnarchuk","first_name":"L.","full_name":"Bodnarchuk, L."},{"full_name":"Drozd, Yu.","last_name":"Drozd","first_name":"Yu."},{"full_name":"Greuel, G.-M.","last_name":"Greuel","first_name":"G.-M."}],"year":"2006","title":"Vector bundles and torsion free sheaves on degenerations of elliptic curves","status":"public","user_id":"49063","language":[{"iso":"eng"}],"_id":"44531","publisher":"Springer","page":"83–129"}]