Duke Mathematical Journal

Odd degree number fields with odd class number

Wei Ho, Arul Shankar, and Ila Varma

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1520046167

Digital Object Identifier
doi:10.1215/00127094-2017-0050

Mathematical Reviews number (MathSciNet)
MR3782066

Zentralblatt MATH identifier
06870399

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

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

Citation

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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