Class Numbers of Indefinite Binary Quadratic Forms

Abstract

Let $h(d)$ be the class number of properly equivalent primitive binary quadratic forms $ax^2 + bxy + cy^2$ of discriminant $d = b^2 a - 4ac$. The case of indefinite forms $(d < 0)$ is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let $\alpha (x)$ be an arbitrarily slow monotonically increasing function such that $\alpha (x) \to \infty$. Then

$$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >(\log p)^{\alpha (p)} } \right\} = o(\pi (x)),$$

where $\pi (x) = \# \{ p \leqslant x\}$. 2. Let F be an arbitrary sufficiently large positive constant. Then for $x >x_F$, the relation

$$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >F} \right\} \asymp \frac{{\pi (x)}}{F}$$

holds. 3. The relation

$$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) = 2} \right\} \sim \frac{9}{{19}}A\pi (x)$$

holds, where A is Artin's constant. Hence, for the majority of discriminants of the form $d = 5p^2$, where ${\left( {\frac{{\text{5}}}{p}} \right) = 1}$ , the class numbers are small. This is consistent with the Gauss conjecture concerning the behavior of $h(d)$ for the majority of discriminants $d >0$ in the general case. Bibliography: 22 titles.