In this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant $-31$ and Fourier coefficients of a weight $16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant $-23$ .

Let K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$. For a positive integer N, let KN be the ray class field of K modulo $N {\cal O}_K$. By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for which KNab is the maximal abelian extension of K unramified outside prime ideals dividing $N{\cal O}_K$.

Among the values of a binary quadratic form, there are many twins of fixed distance. This is shown in quantitative form. For quadratic forms of discriminant −4 or 8 a corresponding result is obtained for triplets.

Habib Muzaffar of Carleton University has pointed out to the authors that in their paper $\left[ \text{A} \right]$ only the result

$${{\pi }_{K,d}}\left( x \right)\,+\,{{\pi }_{{{K}^{-1}},d}}\left( x \right)\,=\,\frac{1}{h\left( d \right)}\,\frac{x}{\log x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}x} \right)$$

follows from the prime ideal theorem with remainder for ideal classes, and not the stronger result

$${{\pi }_{K,d}}\left( x \right)\,=\,\frac{1}{2h\left( d \right)}\,\frac{x}{\log \,x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}\,x} \right)$$

stated in Lemma 5.2. This necessitates changes in Sections 5 and 6 of $\left[ \text{A} \right]$. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of $\left[ \text{A} \right]$, each and every occurrence of $o\left( s-1 \right)$ should be replaced by $o\left( 1 \right)$.

Sections 5 and 6 of $\left[ \text{A} \right]$ have been rewritten to incorporate the abovementioned correction and are given below. They should replace the original Sections 5 and 6 of $\left[ \text{A} \right]$.

Let $d$ be the discriminant of an imaginary quadratic field. Let $a$, $b$, $c$ be integers such that

$${{b}^{2}}-4ac=d,a>0,\gcd (a,b,c)=1.$$

The value of $\left| \eta ((b+\sqrt{d})/\left. 2a) \right| \right.$ is determined explicitly, where $\eta \left( z \right)$ is Dedekind’s eta function

$$\eta (z)\,=\,{{e}^{\pi iz/12}}\,\prod\limits_{m=1}^{\infty }{(1-{{e}^{2\pi imz}})\,\,\,(im(z)>0).}$$

Let h(m) denote the class number of the quadratic field Q(√m). In this paper necessary and sufficient conditions for h (m) to be divisible by 16 are determined when m = −p, where p is a prime congruent to 1 modulo 8, and when m = −2p, where p is a prime congruent to ±1 modulo 8.