Continued fractions, special values of the double sine function, and Stark units over real quadratic fields
✍ Scribed by Brett A. Tangedal
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 266 KB
- Volume
- 124
- Category
- Article
- ISSN
- 0022-314X
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✦ Synopsis
Let F be a real quadratic field and m an integral ideal of F. Two Stark units, ε m,1 and ε m,2 , are conjectured to exist corresponding to the two different embeddings of F into R. We define new ray class invariants U
(1) m (C + ) and U
(2) m (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 ε m,1 and ε m,2 , assuming they exist, can be expressed simultaneously and symmetrically in terms of U ( 1) andU (2) m (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.