单位格的量子化与质量逃逸
Shapes of unit lattices and escape of mass
摘要：
研究了全实立方数域上单位群对数嵌入模曲面上的点集。我们猜想这个集合是稠密的，证明了它的闭包包含可数个显式曲线，并给出了一个证明它有非空内部的策略。这些结果是通过构造显式的阶族（推广所谓的最简立方域）和计算它们的单位群得到的。我们还解决了与这些订单相关的对角群的紧凑轨道的质量逃逸的问题。
英文摘要：
We study the collection of points on the modular surface obtained from the logarithm embeddings of the groups of units in totally real cubic number fields. We conjecture that this set is dense and show that its closure contains countably many explicit curves and give a strategy to show that it has non-empty interior. The results are obtained by constructing explicit families of orders (generalizing the so called simplest cubic fields) and calculating their groups of units. We also address the question of escape of mass for the compact orbits of the diagonal group associated to these orders.
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