## Hiroshima Mathematical Journal

### On integral quadratic forms having commensurable groups of automorphisms

José María Montesinos-Amilibia

#### 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

#### Article information

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

Dates
First available in Project Euclid: 7 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1389102581

Digital Object Identifier
doi:10.32917/hmj/1389102581

Mathematical Reviews number (MathSciNet)
MR3161323

Zentralblatt MATH identifier
1307.44008

#### Citation

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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