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Nearly unimodular quadratic forms. (English) Zbl 0842.11012

A \(\mathbb Z\)-lattice \(L\) is called nearly unimodular if it has a Gram matrix of the form \[ \left( \begin{smallmatrix} a_1 &1 \\ 1 &a_2 & \ddots\\ & \ddots &\ddots \\ &&&a_{n-1} &1\\ &&&1 &a_n \end{smallmatrix} \right)=: [a_1, \dots, a_n ]=: A \] (then we denote \(L\cong A)\). The main result is the following classification theorem: Let \(A= [a_1, \dots, a_n ]\) and \(B= [b_1, \dots, b_n ]\). Suppose \(L\) and \(M\) are positive definite nearly unimodular \(\mathbb Z\)-lattices with \(L\cong A\) and \(M\cong B\). Then \(L\) is isometric to \(M\) if and only if \(B= A\) or \(B= [a_n, \dots, a_1]\).

MSC:

11E12 Quadratic forms over global rings and fields
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