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.