Odd degree number fields with odd class number

Abstract

For every odd integer $n\ge 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ 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 ${\mathbb{P}}^1$ 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 $\infty$) of such fields have trivial $2$-torsion subgroup 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-Lenstra-Martinet-Malle and Dummit-Voight. Additionally, for any order ${\mathcal{O}}_f$ of degree $n$ arising from an integral binary $n$-ic form $f$, we compare the sizes of ${\mathrm{Cl}}_2({\mathcal{O}}_f)$, the $2$-torsion subgroup of ideal classes in ${\mathcal{O}}_f$, and ${\mathcal{I}}_2({\mathcal{O}}_f)$, the $2$-torsion subgroup of ideals in ${\mathcal{O}}_f$. For the family of orders arising from integral binary $n$-ic forms and contained in fields with fixed signature $(r_1,r_2)$, we prove that the mean value of the difference $|{\mathrm{Cl}}_2({\mathcal{O}}_f)|-2^{1-r_1-r_2}|{\mathcal{I}}_2({\mathcal{O}}_f)|$ 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 $|{\mathrm{Cl}}_2({\mathcal{O}}_f)|-2^{1-r_1-r_2}|{\mathcal{I}}_2({\mathcal{O}}_f)|$ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.

Final version. 38 pages

References in corpus (8)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• The geometric sieve and the density of squarefree values of invariant polynomials
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• Squarefree values of polynomial discriminants I
• On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
• Rings and ideals parametrized by binary n-ic forms
• Squarefree values of polynomial discriminants II
• On the number of integral binary $n$-ic forms having bounded Julia invariant

Cited by in corpus (10)

• The $2$-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units
• Chabauty without the Mordell-Weil group
• Squarefree values of polynomial discriminants II
• Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensions
• On Malle's conjecture for nilpotent groups, I
• The second moment of the size of the $2$-Selmer group of elliptic curves
• Counting integral points on symmetric varieties with applications to arithmetic statistics
• Monogenicity and 2-torsion in the class group of number fields of odd degree
• Geometry-of-numbers methods in the cusp
• Asymptotics of $p$-torsion subgroup sizes in class groups of monogenized cubic fields