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

Let image be a real quadratic field and image an integral ideal of image. Two Stark units, image and image, are conjectured to exist corresponding to the two different embeddings of image into image. We define new ray class invariants image and image associated to each class image of the narrow ray class group modulo image and dependent separately on the two different embeddings of image into image. 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 image and image, assuming they exist, can be expressed simultaneously and symmetrically in terms of image and image, thus giving a canonical expression for every existent Stark unit over image as a product of double sine function values. We prove that Stark units do exist as predicted in certain special cases.