Behavior Of the Ring Class Numbers Of a Real Quadratic Field

Abstract

Let $K$ be a real quadratic field $\mathbb{Q}(\sqrt{n})$ with an integer $n=d{f}^2$, where $d$ is the field discriminant of $K$ and $f\ge 1$. Q. Mushtaq found an interesting phenomenon that any totally negative number ${\kappa }_0$ with ${\kappa }^{\sigma }<0$ and ${\kappa }_0^{\sigma }<0$ belonging to the discriminant $n$, attains an ambiguous number ${\kappa }_m$ with ${\kappa }_m{\kappa }_m^{\sigma }<0$ after finitely many actions ${\kappa }_0^{A_j}$ with $0\leqq j\leqq m$ by modular transformations $A_j\in {\mathrm{S}\mathrm{L}}_2^{+}(\mathbb{Z})$. Here $\sigma$ denotes the embedding of $K$ distinct from the identity. In this paper, we give a new aspect for the process to reach an ambiguous number from a totally negative or totally positive number, by which the gap of the proof of Q. Mushtaq's Theorem is complemented. Next, as an analogue of Gauss' Genus Theory, we prove that the ring class number $h_{+}(d{f}^2)$ coincides with the ambiguous class number belonging to the discriminant $n=d{f}^2$, and its behavior is unbounded when $f$ with suitable prime factors goes to infinity using the ring class number formula.