On integral quadratic forms having commensurable groups of automorphisms

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 numbe...

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Bibliographic Details
Published in:Hiroshima mathematical journal 2013-11, Vol.43 (3), p.371-411
Main Author: Montesinos-Amilibia, José María
Format: Article
Language:English
Subjects:
11E04
11E20
57M25
57M50
57M60
automorph
commensurability class
hyperbolic manifold
Integral quadratic form
knot
link
volume
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Summary: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
ISSN:0018-2079
DOI:10.32917/hmj/1389102581