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 m ( 1 ) ( C + ) and U m ( 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 ε m , 1 and ε m , 2 , assuming they exist, can be expressed simultaneously and symmetrically in terms of U m ( 1 ) ( C + ) and U m ( 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.