On the orbifold coverings associated to integral, ternary quadratic forms

Abstract

The group of (integral) automorphs of a ternary integral quadratic form f acts properly discontinuously as a group of isometries of the Riemann’s sphere (resp. the hyperbolic plane) if f is definite (resp. indefinite) and the quotient has a natural structure of spherical (resp. hyperbolic) orbifold, denoted by \(Q_{f}\). Then f is a B-covering of the form g if \(Q_{f}\) is an orbifold covering of \(Q_{g}\), induced by $$\begin{aligned} T^{\prime }fT=\rho g \end{aligned}$$ where T is an integral matrix and \(\rho >0\) is an integer. Given an integral ternary quadratic form f a number \(\Pi _{f}\), called the B-invariant of the form f, is defined. It is conjectured that if f is a B-covering of the form g then both forms have the same B-invariant. The purpose of this paper is to reduce this conjecture to the case in which g is a form with square-free determinant. This reduction is based in the following main Theorem. Any definite (resp. indefinite) form f is a B-covering of a, unique up to genus (resp. integral equivalence), form g with square-free determinant such that f and g have the same B-invariant. To prove this, a normal form of any definite (resp. indefinite) integral, ternary quadratic form f is introduced. Some examples and open questions are given.