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<p>The unit group of the ring of integers of a number field, modulo torsion, is a lattice via the logarithmic Minkowski embedding. We examine the shape of this lattice, which we call the <i>unit shape</i>, within the family of prime degree <i>p</i> number fields whose Galois closure has dihedral Galois group <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(D_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> and a unique real embedding. In the case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(p = 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove that the unit shapes lie on a single hypercycle on the modular surface (in this case, the modular surface is the space of shapes of rank 2 lattices). For general <i>p</i>, we show that the unit shapes are contained in a finite union of translates of periodic torus orbits in the space of shapes.</p>

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Shapes of unit lattices in \(D_p\)-number fields

  • 期刊论文

  • 发表日期：

    2025年11月10日

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  • Robert Harron,
  • Erik Holmes,
  • Sameera Vemulapalli

摘要

The unit group of the ring of integers of a number field, modulo torsion, is a lattice via the logarithmic Minkowski embedding. We examine the shape of this lattice, which we call the unit shape, within the family of prime degree p number fields whose Galois closure has dihedral Galois group \(D_p\) D p and a unique real embedding. In the case \(p = 5\) p = 5 , we prove that the unit shapes lie on a single hypercycle on the modular surface (in this case, the modular surface is the space of shapes of rank 2 lattices). For general p, we show that the unit shapes are contained in a finite union of translates of periodic torus orbits in the space of shapes.