On binary integral quadratic forms having isometric groups of automorphisms

Abstract

The concept of Bianchi equivalence between integral quadratic forms is introduced. Two $n$-ary integral quadratic forms $F$ and $G$ are said to be Bianchi equivalent if there is a real isometry $M$, from $F$ to $\pm G$, sending the group of automorphisms $Aut(F)$ of $F$ isomorphically onto $Aut(G)$. An integer $v>2$ is defined for every binary, indefinite, integral quadratic form $F$ with non-square discriminant such that two of them are Bianchi-equivalent if and only if their $v$-invariants coincide and both are bilateral or unilateral. The number of $\mathbb {Z}$-classes inside such an indefinite Bianchi-class is finite and a practical procedure to construct the $\mathbb {Z}$-classes with the same $v$-invariant is given. It is proved also that an integral quadratic form $F=(a,b,c)$ with $a\ne 0$, $b\ne 0$, $c\ne 0$ is bilateral if and only if the integral quadratic form $F^{\prime }=(a^{2},2b^{2}-ac,c^{2})$ represents $4b^{2}$. The cases of definite forms an indefinite forms with square discriminant are also completely studied. Finally some partial results around the so called Eisenstein problem are given.