Representations of ternary quadratic forms and the class number of imaginary quadratic fields (Q1071046)

Let \(A\) be a definite rational quaternion algebra of class number 1 and \(\Lambda\) a maximal order in \(A\). Let \(m\) be a positive integer not divisible by 4 and write \(m=m_ 0f^ 2\), \(m_ 0\) squarefree. Denote by \(T(m)\) the number of primitive \(\mu\in \Lambda\) with trace 0 and reduced norm \(m\), and let \(h(m)\) denote the order of the ideal class group of proper \(O_ f\)-ideals in \(\mathbb Q(\sqrt{-m})\). Let \(\omega (m)\) denote the number of units in \(O_ f\) and \(| \Lambda^{\times}|\) the order of the unit group, \(\Lambda^{\times}\), of \(\Lambda\). Then the following theorem is proved: Suppose that \(T(m)>0\). Then \[ \omega (m) T(m)/h(m)=| \Lambda^{\times}| \varepsilon_ m\quad\text{if}\quad q| m,\quad\text{and}\quad 2| \Lambda^{\times}| \varepsilon_ m\quad\text{if}\quad q\nmid m \] where \(\varepsilon_ m=1\) if \(m\equiv 1,2 \bmod 4\), \(\varepsilon_ m=2\) if \(m\equiv 7 \bmod 8\), \(\varepsilon_ m=4\) if \(m\equiv 3 \bmod 8\). The proof follows the ideas of B. A. Venkov (1929) in more modern terminology.