Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

Abstract

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) m (C + ) and U (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.

On Shintani’s ray class invariant for totally real number fields

“…For example, in that case, the formula (1.4) was proved by Shintani [5], and the formula (1.5) was essentially obtained by Arakawa [1]. Tangedal [9] and Yamamoto [12] also treated the case of n = 2 by using the theory of continued fractions. We remark that all of them considered only some specific cone decompositions.…”