Hiroshima Mathematical Journal

On integral quadratic forms having commensurable groups of automorphisms

José María Montesinos-Amilibia

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We introduce two notions of equivalence for rational quadratic forms. Two $n$-ary rational quadratic forms are commensurable if they possess commensurable groups of automorphisms up to isometry. Two $n$-ary rational quadratic forms $F$ and $G$ are projectivelly equivalent if there are nonzero rational numbers $r$ and $s$ such that $rF$ and $sG$ are rationally equivalent. It is shown that if $F$\ and $G$\ have Sylvester signature $\{-,+,+,...,+\}$ then $F$\ and $G$\ are commensurable if and only if they are projectivelly equivalent. The main objective of this paper is to obtain a complete system of (computable) numerical invariants of rational $n$-ary quadratic forms up to projective equivalence. These invariants are a variation of Conway's $p$-excesses. Here the cases $n$ odd and $n$ even are surprisingly different. The paper ends with some examples

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Hiroshima Math. J., Volume 43, Number 3 (2013), 371-411.

First available in Project Euclid: 7 January 2014

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Primary: 11E04: Quadratic forms over general fields 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds 57M60: Group actions in low dimensions

Integral quadratic form knot link hyperbolic manifold volume automorph commensurability class


Montesinos-Amilibia, José María. On integral quadratic forms having commensurable groups of automorphisms. Hiroshima Math. J. 43 (2013), no. 3, 371--411. doi:10.32917/hmj/1389102581. https://projecteuclid.org/euclid.hmj/1389102581

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See also

  • See: José María Montesinos-Amilibia. Addendum to "On integral quadratic forms having commensurable groups of automorphisms". Hiroshima Math. J., vol. 44, no. 3 (2014), pp. 341-350.