Fields with small class group in the family $\mathbb{Q}(\sqrt{9m^2+2m})$
Kalyan Chakraborty, Azizul Hoque
TL;DR
This work proves that for the real quadratic fields in the family $k_m=\mathbb{Q}(\sqrt{9m^2+2m})$ with $9m^2+2m$ square-free and $m\equiv 2\pmod{3}$, the class number satisfies $h(9m^2+2m)\ge 4$. The authors leverage partial Dedekind zeta values at $-1$ for certain ideal classes, computed via Lang's formula and generalized Dedekind sums, to show the existence of at least four distinct ideal classes. If $h=4$, they deduce the structure $\mathrm{Cl}(k_m)\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ by parity arguments and the behavior of nonprincipal classes, and they combine these with Siegel's formula to produce a concrete divisor-sum identity linking $\zeta_{k_m}(-1)$ to the class data. Overall, the paper extends results on small class numbers in R-D–type real quadratic fields and provides explicit criteria for the order-four case, enriching the understanding of class group structures in this family.
Abstract
Very recently, Issa and Darrag [Arch. Math. (Basel) 123 (2024), no. 4, 379-383] determined partial Dedekind zeta values for certain ideal classes in the real quadratic fields of the form $\mathbb{Q}(\sqrt{9m^2+2m})$, where $9m^2+2m$ is square-free and $m\equiv 2\pmod 3$ is an odd positive integer. We use these partial Dedekind zeta values to investigate the small class numbers of such fields. More precisely, we prove that the class numbers of the fields in the above mentioned family are at least $4$. Further, we provide a sufficient condition permitting to specify the structure of the class groups of order $4$ in this family of fields.