Upper bounds on number fields of given degree and bounded discriminant. (English) Zbl 1504.11121

Oliver, Robert J. Lemke; Thorne, Frank

Duke Math. J. 171, No. 15, 3077-3087 (2022).

The authors give an upper bound for the number \(N_n(X)\) of number fields of degree \(n\) with discriminant bounded by \(X\): \(N_n(X) =O(X^{c(\log n)^2})\) for an explicit constant \(c\). The authors improve previous results of many authors. Their strategy follows the ideas of J. S. Ellenberg and A. Venkatesh [Ann. Math. (2) 163, No. 2, 723–741 (2006; Zbl 1099.11068)], and of J.-M. Couveignes [Ann. Math. (2) 192, No. 2, 487–497 (2020; Zbl 1454.11190)].

Reviewer: Claude Levesque (Québec)

MSC:

11R29

Class numbers, class groups, discriminants

Keywords:

arithmetic statistics; number field counting

Citations:

Zbl 1099.11068; Zbl 1454.11190

Software:

PARI/GP

References:

[1]	J. ALEXANDER and A. HIRSCHOWITZ, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995) no. 2, 201-222. · Zbl 0829.14002
[2]	M. BHARGAVA, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031-1063. · Zbl 1159.11045 · doi:10.4007/annals.2005.162.1031
[3]	M. BHARGAVA, The density of discriminants of quintic rings and fields, Ann. of Math. (2) 172 (2010) no. 3, 1559-1591. · Zbl 1220.11139 · doi:10.4007/annals.2010.172.1559
[4]	M. C. BRAMBILLA and G. OTTAVIANI, On the Alexander-Hirschowitz theorem, J. Pure Appl. Algebra 212 (2008), no. 5, 1229-1251. · Zbl 1139.14007 · doi:10.1016/j.jpaa.2007.09.014
[5]	M. BHARGAVA, A. SHANKAR, T. TANIGUCHI, F. THORNE, J. TSIMERMAN, and Y. ZHAO, Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves, J. Amer. Math. Soc. 33 (2020), no. 4, 1087-1099. · Zbl 1456.11101 · doi:10.1090/jams/945
[6]	H. COHEN, F. DIAZ Y DIAZ, and M. OLIVIER, Enumerating quartic dihedral extensions of \[ \mathbb{Q} \], Compositio Math. 133 (2002), no. 1, 65-93. · Zbl 1050.11104 · doi:10.1023/A:1016310902973
[7]	J.-M. COUVEIGNES, Enumerating number fields, Ann. of Math. (2) 192 (2020), no. 2, 487-497. · Zbl 1454.11190 · doi:10.4007/annals.2020.192.2.4
[8]	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. · Zbl 0212.08101 · doi:10.1098/rspa.1971.0075
[9]	E. P. DUMMIT, Counting G-extensions by discriminant, Math. Res. Lett. 25 (2018), no. 4, 1151-1172. · Zbl 1444.11213 · doi:10.4310/MRL.2018.v25.n4.a5
[10]	J. S. ELLENBERG and A. 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. · Zbl 1099.11068 · doi:10.4007/annals.2006.163.723
[11]	R. HARTSHORNE, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York-Heidelberg, 1977. · Zbl 0367.14001
[12]	DAVID MUMFORD, The Red Book of Varieties and Schemes, expanded ed., Lecture Notes in Mathematics 1358, Springer, Berlin, 1999. · Zbl 0945.14001 · doi:10.1007/b62130
[13]	THE PARI GROUP, PARI/GP version 2.8.0, Univeristy of Bordeaux, 2015.
[14]	WOLFGANG M. SCHMIDT, “Number fields of given degree and bounded discriminant” in Columbia University Number Theory Seminar (New York, 1992), Astérisque 228, Soc. Math. France, Paris, 1995, 189-195. · Zbl 0827.11069
[15]	USER 940, Elementary central binomial coefficient estimates, Mathematics Stack Exchange, 2012, https://math.stackexchange.com/q/58562.