秩2对数单位格的几何与形状
On the Geometry and Shapes of Rank 2 Log Unit Lattices
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中文总结 AI 辅助
本文研究单位秩为2的数域的对数单位格形状，利用伽罗瓦群刻画其形状位置，证明特定六次域的单位形状可唯一确定数域同构类，并给出其对数单位格正交的条件及比例下界。
中文摘要 AI 辅助
每个数域都会典范地产生两个格：其整数环与对数单位格。前者的形状已得到广泛研究，而后者（称为单位形状）的形状却鲜为人知。本文深入分析了单位秩为2的数域的单位形状。首个主要结果在许多情形下，依据数域的伽罗瓦闭包的伽罗瓦群，刻画了单位形状在秩2格形状空间基本域内的位置，并确定这些单位形状何时为超越数。接着，证明了对于全虚的D₆非CM六次域，单位形状可唯一确定该数域的同构类；而在CM情形下该结论不成立。最后，针对D₆非CM虚六次域的某些子族，给出了其对数单位格正交的简单充分条件，并提供了具有正交对数单位格的数域比例的下界。
英文摘要
Every number field canonically gives rise to two lattices: its ring of integers and its log unit lattice. While the shapes of the former have undergone extensive research, far less is known about the shapes of the latter, referred to as unit shapes. This paper presents an in-depth analysis of the unit shapes of number fields with unit rank 2. Our first main result characterizes, in many cases, the location of unit shapes within the fundamental domain of the space of rank 2 lattice shapes in terms of the Galois group of the field's Galois closure, and determines when these unit shapes are transcendental. Next, we establish that the unit shape uniquely determines the field up to isomorphism for totally imaginary $D_6$ non-CM sextic fields; this result fails in the CM case. Finally, for certain subfamilies of $D_6$ non-CM imaginary sextics, we offer a simple sufficient condition for their log unit lattices to be orthogonal and provide lower bounds on the proportion of fields with orthogonal log unit lattice.
