A parameterized family of quadratic class groups with 3-Sylow subgroups of rank at least three

Abstract

In a recent paper, Kishi and Komatsu have shown that the complex quadratic fields of discriminant

$$\begin{aligned} \Delta _{18} = 4 - 3^{18n+3} \end{aligned}$$

have a class group whose 3-Sylow subgroup is of rank at least three for all integers n. The Kishi and Komatsu paper proves this by looking at the structure of the algebraic fields involved. Our purpose in this paper is to show that the class groups for the order whose discriminant is $\Delta _{18}$ have a 3-SSG of rank at least three, to show this using elementary means that would have been available to Gauss, and to elucidate further the group structure of the subgroup of classes of order 3. More generally, we shall show that $\Delta _{18}$ is actually a special case of a more general sequence of discriminants, all of which can be shown parametrically to have a class group whose 3-Sylow subgroup has rank at least three.

Appendix: Shanks’s algorithm

Shanks’s algorithm to compound two forms

$$\begin{aligned} f_1 = (a_1, b_1, c_1) \end{aligned}$$

and

$$\begin{aligned} f_2 = (a_2, b_2, c_2) \end{aligned}$$

is as follows.

Let $\beta = (b_1 + b_2)/2$, and compute $\mu = \gcd (a_1, \beta )$ and $\nu = \gcd (a_2, \mu )$.

Solve $a_1 x + \beta y = \mu$ for x and y and then

$$\begin{aligned} \mu z/\nu \equiv x \left( \frac{b_1 - b_2}{2} \right) - c_1 y \pmod {a_2/\nu } \end{aligned}$$

for z. Then $f_3$, the form compounded of $f_1$ and $f_2$, is

$$\begin{aligned} (a_1 a_2 / \nu ^2 , b_1 + 2 a_1 z / \nu , *), \end{aligned}$$

where the third coefficient can be computed from the discriminant.