Continued fractions, special values of the double sine function, and Stark units over real quadratic fields☆
Continued fractions, special values of the double sine function, and Stark units over real quadratic fields☆
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连分数、双正弦函数的特殊值、实二次域上的斯塔克单位☆
DOI：
10.1016/j.jnt.2006.09.011
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发表时间：
2007
影响因子：
0.7
通讯作者：
Brett A. Tangedal
中科院分区：
文献类型：
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作者：
Brett A. Tangedal
设F是实二次域，m是F的积分理想，我们猜想存在两个Stark单位，εm，1和εm，2，对应于F到R的两种不同嵌入.我们定义了新的射线类不变量Um(1)(C+)和Um(2)(C+)，它们分别与模m的窄射线类群的每一类C+有关，并且分别依赖于F到R的两种不同的嵌入.我们证明了两个斯塔克单位εm，1和εm，2，假设它们存在，可以同时对称地表示为Um(1)(C+)和Um(2)(C+)，从而给出了F上每个存在的斯塔克单位作为双正弦函数值的乘积的正则表达式。我们证明了在某些特殊情况下，斯塔克单位确实存在。
Let F be a real quadratic field and m an integral ideal of F. Two Stark units, εm,1and εm,2, are conjectured to exist corresponding to the two different embeddings of F into R. We define new ray class invariants Um(1)(C+) and Um(2)(C+) associated to each class C+of the narrow ray class group modulo m and dependent separately on the two different embeddings of F into R. These invariants are defined as a product of special values of the double sine function in a compact and canonical form using a continued fraction approach due to Zagier and Hayes. We prove that both Stark units εm,1and εm,2, assuming they exist, can be expressed simultaneously and symmetrically in terms of Um(1)(C+) and Um(2)(C+), thus giving a canonical expression for every existent Stark unit over F as a product of double sine function values. We prove that Stark units do exist as predicted in certain special cases.