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Binary Hermitian forms over a cyclotomic field. (English) Zbl 1211.11046

Let \(F=\mathbb{Q}(\zeta)\), where \(\zeta\) is a primitive \(5\)th root of unity, let \(k=\mathbb{Q}(\sqrt{5})\) its real subfield and let \(\cdot': F \to F, \;x \mapsto x'\) be the (unique) automorphism of the field \(F\) that sends \(\zeta\) to \(\zeta^3\). The paper under review investigates binary Hermitian forms \(\phi: F^2 \to k\) subject to the condition that \(\phi\) and \(\phi'\), defined by \(\phi'(x,y)=\phi(x,y)'\), are positive definite. The sum \(\hat{\phi}:=\phi+\phi'\) can be considered as a (positive definite) quadratic form over \(\mathbb{Q}\). Set \[ m(\phi):= \inf_{v \in \mathcal{O}^2 \setminus \{0\}} \hat{\phi}(v), \] where \(\mathcal{O}\) is the ring of integers of \(F\). Vectors \(v\in \mathcal{O}^2\setminus \{0\}\) with \(\hat{\phi}(v)=m(\phi)\) are called minimal. The form \(\phi\) is called perfect, if no other binary Hermitian form \(\psi\) satisfies \(m(\phi)=m(\psi)\) and has the same set of minimal vectors.
In the paper under review it is shown that all perfect forms \(\phi\) are \(\text{GL}_2(\mathcal{O})\)-conjugate. Moreover the set of minimal vectors \(M(\phi)\) is determined up to \(\text{GL}_2(\mathcal{O})\)-conjugacy for every binary Hermitian form \(\phi\). This is done via the computation of the so-called Voronoï-polyhedron, a polyhedral geometric object in the real coefficient space of binary Hermitian forms, stable under the \(\text{GL}_2(\mathcal{O})\)-action, whose cones and vertices correspond bijectively and \(\text{GL}_2(\mathcal{O})\)-equivariantly to binary Hermitian forms, respectively to minimal vectors modulo torsion units.

MSC:

11E39 Bilinear and Hermitian forms
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Software:

Magma; polymake
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Full Text: DOI arXiv

References:

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