Duke Mathematical Journal

Odd degree number fields with odd class number

Wei Ho, Arul Shankar, and Ila Varma

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For every odd integer n3, we prove that there exist infinitely many number fields of degree n and associated Galois group Sn whose class number is odd. To do so, we study the class groups of families of number fields of degree n whose rings of integers arise as the coordinate rings of the subschemes of P1 cut out by integral binary n-ic forms. By obtaining upper bounds on the mean number of 2-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to 1 as n tends to ) of such fields have trivial 2-torsion subgroups in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen and Lenstra, Cohen and Martinet, Malle, and Dummit and Voight. Additionally, for any order Of of degree n arising from an integral binary n-ic form f, we compare the sizes of Cl2(Of), the 2-torsion subgroup of ideal classes in Of, and of I2(Of), the 2-torsion subgroup of ideals in Of. For the family of orders arising from integral binary n-ic forms and contained in fields with fixed signature (r1,r2), we prove that the mean value of the difference |Cl2(Of)|21r1r2|I2(Of)| is equal to 1, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of |Cl2(Of)|21r1r2|I2(Of)| remains 1 for certain families obtained by imposing local splitting and maximality conditions.

Article information

Duke Math. J., Volume 167, Number 5 (2018), 995-1047.

Received: 5 December 2016
Revised: 11 August 2017
First available in Project Euclid: 3 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R45: Density theorems

number fields class groups narrow class groups Cohen–Lenstra heuristics arithmetic statistics binary n-ic forms


Ho, Wei; Shankar, Arul; Varma, Ila. Odd degree number fields with odd class number. Duke Math. J. 167 (2018), no. 5, 995--1047. doi:10.1215/00127094-2017-0050. https://projecteuclid.org/euclid.dmj/1520046167

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  • [1] O. Beckwith, Indivisibility of class numbers of imaginary quadratic fields, Res. Math. Sci. 4 (2017), no. 20.
  • [2] M. Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), 1031–1063.
  • [3] M. Bhargava, The density of discriminants of quintic rings and fields, Ann. of Math. (2) 172 (2010), 1559–1591.
  • [4] M. Bhargava, Most hyperelliptic curves over $\mathbb{Q}$ have no rational points, preprint, arXiv:1308.0395v1 [math.NT].
  • [5] M. Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, preprint, arXiv:1402.0031v1 [math.NT].
  • [6] M. Bhargava and B. H. Gross, “The average size of the $2$-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point” in Automorphic Representations and $L$-Functions, Tata Inst. Fund. Res. Stud. Math. 22, Tata Inst. Fund. Res., Mumbai, 2013, 23–91.
  • [7] M. Bhargava, B. H. Gross, and X. Wang, A positive proportion of locally soluble hyperelliptic curves over $\mathbb{Q}$ have no point over any odd degree extension, with an appendix by T. Dokchitser and V. Dokchitser, J. Amer. Math. Soc. 30 (2017), 451–493.
  • [8] M. Bhargava and A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2) 181 (2015), 191–242.
  • [9] M. Bhargava and A. Shankar, The average size of the $5$-Selmer group of elliptic curves is $6$, and the average rank is less than $1$, preprint, arXiv:1312.7859v1 [math.NT].
  • [10] M. Bhargava, A. Shankar, and X. Wang, Squarefree values of polynomial discriminants, II, in preparation.
  • [11] M. Bhargava and I. Varma, On the mean number of $2$-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields, Duke Math. J. 164 (2015), 1911–1933.
  • [12] M. Bhargava and I. Varma, The mean number of $3$-torsion elements in the class groups and ideal groups of quadratic orders, Proc. Lond. Math. Soc. (3) 112 (2016), 235–266.
  • [13] M. Bhargava and A. Yang, On the number of integral binary $n$-ic forms having bounded Julia invariant, preprint, arXiv:1312.7339v1 [math.NT].
  • [14] B. J. Birch and J. R. Merriman, Finiteness theorems for binary forms with given discriminant, Proc. Lond. Math. Soc. (3) 24 (1972), 385–394.
  • [15] J. H. Bruinier, Nonvanishing modulo $\ell$ of Fourier coefficients of half-integral weight modular forms, Duke Math. J. 98 (1999), 595–611.
  • [16] H. Cohen and H. W. Lenstra, Jr., “Heuristics on class groups of number fields” in Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. 1068, Springer, Berlin, 1984, 33–62.
  • [17] H. Cohen and J. Martinet, Class groups of number fields: Numerical heuristics, Math. Comp. 48 (1987), 123–137.
  • [18] H. Davenport, On a principle of Lipschitz, J. Lond. Math. Soc. (2) 26 (1951), 179–183.
  • [19] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields, II, Proc. R. Soc. Lond. Ser. A 322 (1971), 405–420.
  • [20] D. S. Dummit and J. Voight, The $2$-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units, preprint, arXiv:1702.00092v1 [math.NT].
  • [21] É. Fouvry and J. Klüners, On the $4$-rank of class groups of quadratic number fields, Invent. Math. 167 (2007), 455–513.
  • [22] C. F. Gauss, Disquisitiones arithmeticae, Yale Univ. Press, New Haven, 1966.
  • [23] P. Hartung, Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by $3$, J. Number Theory 6 (1974), 276–278.
  • [24] K. Horie, A note on basic Iwasawa $\lambda$-invariants of imaginary quadratic fields, Invent. Math. 88 (1987), 31–38.
  • [25] K. Horie, Trace formulae and imaginary quadratic fields, Math. Ann. 288 (1990), 605–612.
  • [26] N. Jochnowitz, Congruences between modular forms and implications for the Hecke algebra, Ph.D. dissertation, Harvard University, Cambridge, Mass., 1976.
  • [27] G. Julia, Étude sur les formes binaires non quadratiques à indéterminées réelles, ou complexes, Mémoires de l’Académie des Sciences de l’Institut de France 55 (1917), 1–296.
  • [28] G. Malle, On the distribution of class groups of number fields, Experiment. Math. 19 (2010), 465–474.
  • [29] J. Nakagawa, Binary forms and orders of algebraic number fields, Invent. Math. 97 (1989), 219–235.
  • [30] J. Nakagawa and K. Horie, Elliptic curves with no rational points, Proc. Amer. Math. Soc. 104 (1988), 20–24.
  • [31] K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms mod $\ell$, Ann. of Math. (2) 147 (1998), 453–470.
  • [32] A. Shankar and J. Tsimerman, Counting $S_{5}$-fields with a power saving error term, Forum Math. Sigma 2 (2014), art. ID e13.
  • [33] C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Publ. Math. Inst. Hautes Études Sci. 89 (1999), 5–126.
  • [34] M. Stoll and J. E. Cremona, On the reduction theory of binary forms, J. Reine Angew. Math. 565 (2003), 79–99.
  • [35] V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397–419.
  • [36] X. Wang, Pencils of quadrics and Jacobians of hyperelliptic curves, Ph.D. dissertation, Harvard University, Cambridge, Mass., 2013.
  • [37] A. Wiles, On class groups of imaginary quadratic fields, J. Lond. Math. Soc. (2) 92 (2015), 411–426.
  • [38] M. M. Wood, Rings and ideals parameterized by binary $n$-ic forms, J. Lond. Math. Soc. (2) 83 (2011), 208–231.
  • [39] M. M. Wood, Parametrization of ideal classes in rings associated to binary forms, J. Reine Angew. Math. 689 (2014), 169–199.