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

Let F be a real quadratic field and m an integral ideal of F. Two Stark units, εm,1 and εm,2, are conjectured to exist corresponding to the two different embeddings of F into R. We define new ray class invariants and 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,1 and εm,2, assuming they exist, can be expressed simultaneously and symmetrically in terms of and , 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.