On abelian cubic fields with large class number
HTML articles powered by AMS MathViewer
- by Jérémy Dousselin;
- Proc. Amer. Math. Soc. 152 (2024), 3255-3264
- DOI: https://doi.org/10.1090/proc/16827
- Published electronically: June 12, 2024
- HTML | PDF
Abstract:
We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of “close” abelian cubic number fields with class numbers as large as possible. We also give a first step toward an explicit lower bound for such extreme values of class numbers of abelian cubic fields.References
- Stephan Baier and Liangyi Zhao, An improvement for the large sieve for square moduli, J. Number Theory 128 (2008), no. 1, 154–174. MR 2382775, DOI 10.1016/j.jnt.2007.03.004
- Stephan Baier and Matthew P. Young, Mean values with cubic characters, J. Number Theory 130 (2010), no. 4, 879–903. MR 2600408, DOI 10.1016/j.jnt.2009.11.007
- Giacomo Cherubini, Alessandro Fazzari, Andrew Granville, Vítězslav Kala, and Pavlo Yatsyna, Consecutive real quadratic fields with large class numbers, Int. Math. Res. Not. IMRN 14 (2023), 12052–12063. MR 4615224, DOI 10.1093/imrn/rnac176
- William Duke, Number fields with large class group, Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, RI, 2004, pp. 117–126. MR 2076589, DOI 10.1090/crmp/036/08
- A. Granville and K. Soundararajan, The distribution of values of $L(1,\chi _d)$, Geom. Funct. Anal. 13 (2003), no. 5, 992–1028. MR 2024414, DOI 10.1007/s00039-003-0438-3
- Karin Halupczok, Large sieve inequalities with general polynomial moduli, Q. J. Math. 66 (2015), no. 2, 529–545. MR 3356836, DOI 10.1093/qmath/hav011
- Karin Halupczok and Marc Munsch, Large sieve estimate for multivariate polynomial moduli and applications, Monatsh. Math. 197 (2022), no. 3, 463–478. MR 4389130, DOI 10.1007/s00605-021-01641-6
- D. R. Heath-Brown, Kummer’s conjecture for cubic Gauss sums. part A, Israel J. Math. 120 (2000), no. part A, 97–124. MR 1815372, DOI 10.1007/s11856-000-1273-y
- Hans Heilbronn, On the class-number in imaginary quadratic fields, Q. J. Math. 5 (1934), no. 1, 150–160.
- Youness Lamzouri, Extreme values of class numbers of real quadratic fields, Int. Math. Res. Not. IMRN 22 (2015), 11847–11860. MR 3456705, DOI 10.1093/imrn/rnv054
- Robert J. Lemke Oliver, Jesse Thorner, and Asif Zaman, An approximate form of Artin’s holomorphy conjecture and non-vanishing of Artin $L$-functions, Invent. Math. 235 (2024), no. 3, 893–971. MR 4701881, DOI 10.1007/s00222-023-01232-2
- J. E. Littlewood, On the Class-Number of the Corpus $P({\surd }-k)$, Proc. London Math. Soc. (2) 27 (1928), no. 5, 358–372. MR 1575396, DOI 10.1112/plms/s2-27.1.358
- Hugh L. Montgomery and Peter J. Weinberger, Real quadratic fields with large class number, Math. Ann. 225 (1977), no. 2, 173–176. MR 427271, DOI 10.1007/BF01351721
- Marc Munsch, A large sieve inequality for power moduli, Acta Arith. 197 (2021), no. 2, 207–211. MR 4189720, DOI 10.4064/aa191212-1-6
- Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI 10.1090/S0025-5718-1974-0352049-8
- Liangyi Zhao, Large sieve inequality with characters to square moduli, Acta Arith. 112 (2004), no. 3, 297–308. MR 2046185, DOI 10.4064/aa112-3-5
Bibliographic Information
- Jérémy Dousselin
- Affiliation: Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
- ORCID: 0000-0002-7850-3818
- Email: jeremy.dousselin@univ-lorraine.fr
- Received by editor(s): March 16, 2023
- Received by editor(s) in revised form: January 8, 2024, and February 8, 2024
- Published electronically: June 12, 2024
- Communicated by: Amanda Folsom
- © Copyright 2024 by the author
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3255-3264
- MSC (2020): Primary 11R29; Secondary 11R42
- DOI: https://doi.org/10.1090/proc/16827
- MathSciNet review: 4767260