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
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.
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Dedicado con cariño y agradecimiento a Mayte Lozano Imízcoz en su septuagésimo cumpleaños.
Supported by MEC-MTM 2015-63612P.
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Montesinos-Amilibia, J.M. On the orbifold coverings associated to integral, ternary quadratic forms. RACSAM 112, 717–749 (2018). https://doi.org/10.1007/s13398-017-0472-x
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DOI: https://doi.org/10.1007/s13398-017-0472-x
Keywords
- Integral quadratic form
- Knot
- Link
- Hyperbolic manifold
- Volume
- Automorph
- Commensurability class
- Integral equivalence
- Rational equivalence
- Projective equivalence
- Bianchi equivalence
- Conway’s excesses
- p-adic symbols