Heisenberg群中的计数与等分布
Counting and equidistribution in Heisenberg groups
摘要：
我们大力发展复双曲几何和算术计数或均匀分布的应用程序之间的关系，产生于行动的算术群复双曲空间，特别是在尺寸$2$。证明了厄米型光锥中二次虚数域$K$上整数点的Mertens公式，以及海森堡群中$K$上有理点集的等分布定理.对于$K$上给定的厄米特形式，给出了复射影平面中伽罗瓦共轭正交各向同性的$K$上三次点的计数公式，以及海森堡群中算术链的Cygan直径趋于$0$时的计数和均匀分布结果。
英文摘要：
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to $0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
