The Lambda Invariants at CM Points
The Lambda Invariants at CM Points
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DOI：
10.1093/imrn/rnz230
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发表时间：
2018-10
影响因子：
1
通讯作者：
Tonghai Yang;Hongbo Yin;Peng Yu
中科院分区：
文献类型：
--
作者：
Tonghai Yang;Hongbo Yin;Peng Yu
在本文中，我们证明$\lambda (z_1) -\lambda (z_2)$、$\lambda (z_1)$和$1-\lambda (z_1)$都是Borcherds在$X(2) \times X(2)$上的产品。然后，我们利用Bruinier、Kudla和Yang的大CM值公式给出$\lambda (\frac{d+\sqrt d}2)$、$1-\lambda (\frac{d+\sqrt d}2)$和$\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$的范数的显式分解公式，其中的条件为$(d_1, d_2)=1$。最后，我们用这些结果证明了$\lambda (\frac{d+\sqrt d}2)$始终是一个代数整数，并且可以很容易地用于在模$2$的射线类域${\mathbb{Q}}(\sqrt{d})$中构造单元。在此过程中，我们还给出了一类独立的局部Whittaker函数的显式公式。
In this paper, we show that $\lambda (z_1) -\lambda (z_2)$, $\lambda (z_1)$, and $1-\lambda (z_1)$ are all Borcherds products on $X(2) \times X(2)$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $\lambda (\frac{d+\sqrt d}2)$, $1-\lambda (\frac{d+\sqrt d}2)$, and $\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$, with the latter under the condition $(d_1, d_2)=1$. Finally, we use these results to show that $\lambda (\frac{d+\sqrt d}2)$ is always an algebraic integer and can be easily used to construct units in the ray class field of ${\mathbb{Q}}(\sqrt{d})$ of modulus $2$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.