Publication: On integral quadratic forms having commensurable groups of automorphisms
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2013
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Hiroshima University. Faculty of Science
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Abstract
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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Addendum to ‘‘On integral quadratic forms having commensurable groups of automorphisms’’, disponible en http://projecteuclid.org/euclid.hmj/1419619751