Complete classification of binary normal regular Hermitian lattices. (English) Zbl 1248.11030

Let \({\mathcal O}\) be the ring of algebraic integers in an imaginary quadratic number field \(K={\mathbb Q}(\sqrt{-m})\), \(m\in {\mathbb Z}\), \(m>0\) and square free. An integral Hermitian lattice \((H,L)\) of rank \(n\) is a finitely generated \({\mathcal O}\)-module \(L\) inside \(K^n\) such that \(KL=K^n\), together with a Hermitian form \(H:L\times L\to {\mathcal O}\). Note that for such lattices, \(H(v,v)\in {\mathbb Z}\) for all \(v\in L\), and we say that \(x\in {\mathbb Z}\) is represented by \((H,L)\) if there exists \(v\in L\) with \(H(v,v)=x\). \((H,L)\) is called normal if the ideal in \({\mathcal O}\) generated by \(\{ H(v,v)\,|\,v\in L\}\) is equal to the ideal generated by \(\{ H(v,w)\,|\,v,w\in L\}\), otherwise it is called subnormal. In this paper, only positive definite integral Hermitian lattices are considered, i.e. lattices for which \(H(v,v)>0\) for all \(v\in L\setminus\{ 0\}\). Such a positive definite lattice is called regular if any \(x\in {\mathbb Z}\) that is represented by all localizations of \(L\) at all places of \(K\) is also represented globally. Regular quadratic or Hermitian lattices and their classification have been studied to quite some extent in the literature. The main result by the authors is the full classification of all positive definite binary normal regular Hermitian lattices over rings of algebraic integers in imaginary quadratic number fields up to similarity. There are altogether \(68\) such lattices, and they are described explicitly in terms of Gram matrices. Those among them that are universal, i.e. that represent all positive integers, are highlighted. Such lattices only exist for \(m\in \{ 1,2,3,5,6,7,10,11,15,19,23,31\}\). The authors also announce new results concerning the case of subnormal regular Hermitian lattices.


11E39 Bilinear and Hermitian forms
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11E41 Class numbers of quadratic and Hermitian forms
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