The number of ramified primes in number fields of small degree
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- by Robert J. Lemke Oliver and Frank Thorne PDF
- Proc. Amer. Math. Soc. 145 (2017), 3201-3210
Abstract:
In this paper we investigate the distribution of the number of primes which ramify in number fields of degree $d \leq 5$. In analogy with the classical Erdős-Kac theorem, we prove for $S_d$-extensions that the number of such primes is normally distributed with mean and variance $\log \log X$.References
- Karim Belabas, Manjul Bhargava, and Carl Pomerance, Error estimates for the Davenport-Heilbronn theorems, Duke Math. J. 153 (2010), no. 1, 173–210. MR 2641942, DOI 10.1215/00127094-2010-007
- M. Bhargava, A.C. Cojocaru, and F. Thorne, The square sieve and the number of $A_5$-quintic extensions of bounded discriminant, Work in preparation.
- K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213–1237. MR 1415795, DOI 10.1090/S0025-5718-97-00846-6
- K. Belabas and E. Fouvry, Sur le 3-rang des corps quadratiques de discriminant premier ou presque premier, Duke Math. J. 98 (1999), no. 2, 217–268 (French). MR 1695199, DOI 10.1215/S0012-7094-99-09807-1
- Manjul Bhargava, Higher composition laws. III. The parametrization of quartic rings, Ann. of Math. (2) 159 (2004), no. 3, 1329–1360. MR 2113024, DOI 10.4007/annals.2004.159.1329
- Manjul Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031–1063. MR 2183288, DOI 10.4007/annals.2005.162.1031
- Manjul Bhargava, Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Int. Math. Res. Not. IMRN 17 (2007), Art. ID rnm052, 20. MR 2354798, DOI 10.1093/imrn/rnm052
- Manjul Bhargava, The density of discriminants of quintic rings and fields, Ann. of Math. (2) 172 (2010), no. 3, 1559–1591. MR 2745272, DOI 10.4007/annals.2010.172.1559
- Patrick Billingsley, The probability theory of additive arithmetic functions, Ann. Probability 2 (1974), 749–791. MR 466055, DOI 10.1214/aop/1176996547
- Manjul Bhargava, Arul Shankar, and Jacob Tsimerman, On the Davenport-Heilbronn theorems and second order terms, Invent. Math. 193 (2013), no. 2, 439–499. MR 3090184, DOI 10.1007/s00222-012-0433-0
- Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Enumerating quartic dihedral extensions of $\Bbb Q$, Compositio Math. 133 (2002), no. 1, 65–93. MR 1918290, DOI 10.1023/A:1016310902973
- Peter J. Cho and Henry H. Kim, Effective prime ideal theorem and exponents of ideal class groups, Q. J. Math. 65 (2014), no. 4, 1179–1193. MR 3285767, DOI 10.1093/qmath/hau002
- Peter J. Cho and Henry H. Kim, Low lying zeros of Artin $L$-functions, Math. Z. 279 (2015), no. 3-4, 669–688. MR 3318244, DOI 10.1007/s00209-014-1387-2
- Henri Cohen, Enumerating quartic dihedral extensions of $\Bbb Q$ with signatures, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 2, 339–377 (English, with English and French summaries). MR 1990000
- H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 491593, DOI 10.1098/rspa.1971.0075
- J. Ellenberg, L. B. Pierce, and M. Matchett Wood, On $\ell$-torsion in class groups of number fields, ArXiv e-prints, June 2016.
- Jordan S. Ellenberg and Akshay Venkatesh, The number of extensions of a number field with fixed degree and bounded discriminant, Ann. of Math. (2) 163 (2006), no. 2, 723–741. MR 2199231, DOI 10.4007/annals.2006.163.723
- Zev Klagsbrun and Robert J. Lemke Oliver, The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion, Mathematika 62 (2016), no. 1, 67–78. MR 3430377, DOI 10.1112/S0025579315000121
- Gunter Malle, On the distribution of Galois groups. II, Experiment. Math. 13 (2004), no. 2, 129–135. MR 2068887
- Greg Martin and Paul Pollack, The average least character non-residue and further variations on a theme of Erdős, J. Lond. Math. Soc. (2) 87 (2013), no. 1, 22–42. MR 3022705, DOI 10.1112/jlms/jds036
- Arul Shankar and Jacob Tsimerman, Counting $S_5$-fields with a power saving error term, Forum Math. Sigma 2 (2014), Paper No. e13, 8. MR 3264252, DOI 10.1017/fms.2014.10
- Takashi Taniguchi and Frank Thorne, Secondary terms in counting functions for cubic fields, Duke Math. J. 162 (2013), no. 13, 2451–2508. MR 3127806, DOI 10.1215/00127094-2371752
- Takashi Taniguchi and Frank Thorne, An error estimate for counting $S_3$-sextic number fields, Int. J. Number Theory 10 (2014), no. 4, 935–948. MR 3208868, DOI 10.1142/S1793042114500080
- Frank Thorne and Maosheng Xiong, Distribution of zeta zeroes for trigonal curves over a finite field, Preprint, 2014.
- Andrew Yang, Distribution problems associated to zeta functions and invariant theory, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Princeton University. MR 2713725
- Yongqiang Zhao, On sieve methods for varieties over finite fields, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 3187553
Additional Information
- Robert J. Lemke Oliver
- Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
- MR Author ID: 894148
- Email: robert.lemke_oliver@tufts.edu
- Frank Thorne
- Affiliation: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29201
- MR Author ID: 840724
- Email: thorne@math.sc.edu
- Received by editor(s): July 26, 2016
- Published electronically: April 26, 2017
- Additional Notes: The first author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship at Stanford University
The second author was partially supported by the National Science Foundation under Grant No. DMS-1201330. - Communicated by: Ken Ono
- © Copyright 2017 Robert Lemke Oliver and Frank Thorne
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3201-3210
- MSC (2010): Primary 11K65, 11R16, 11R21
- DOI: https://doi.org/10.1090/proc/13467
- MathSciNet review: 3652776