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November 2013 On integral quadratic forms having commensurable groups of automorphisms
José María Montesinos-Amilibia
Hiroshima Math. J. 43(3): 371-411 (November 2013). DOI: 10.32917/hmj/1389102581

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

Citation

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José María Montesinos-Amilibia. "On integral quadratic forms having commensurable groups of automorphisms." Hiroshima Math. J. 43 (3) 371 - 411, November 2013. https://doi.org/10.32917/hmj/1389102581

Information

Published: November 2013
First available in Project Euclid: 7 January 2014

zbMATH: 1307.44008
MathSciNet: MR3161323
Digital Object Identifier: 10.32917/hmj/1389102581

Subjects:
Primary: 11E04, 11E20, 57M25, 57M50, 57M60

Rights: Copyright © 2013 Hiroshima University, Mathematics Program

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Vol.43 • No. 3 • November 2013
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