THOMAE’S FORMULA FOR Z n CURVES By DAVID G. EBIN AND HERSHEL M. FARKAS 1 Introduction A nonsingular Z n curve is one whose equation is given by w n = i=nr−1  i=0 (z − λ i ), where r is a positive integer greater than 1 and λ i = λ j when i = j. This curve (or Riemann surface) is an n-sheeted branched cover of the sphere, branched over the nr points {λ i }. The branching order at each branch point is n − 1, and the genus of the compactified surface (by the Riemann–Hurwitz formula) is g = n−1 2 (nr − 2). We denote the point on the surface above λ i by P i and the points over ∞ by P ∞1 , P ∞2 ,..., P ∞n . In this note, we show how to construct integral divisors  of degree g whose support lies in the branch set and which are non-special. An integral divisor is called non-special if there are no non-zero holomorphic differentials whose di- visors are multiples of it. We use these non-special divisors to build identities satisfied by the theta functions with rational characteristics associated with these surfaces. Thomae’s original work dealt with the case n = 2, hyperelliptic Riemann surfaces. He showed how to associate with each integer characteristic  t ǫ t ǫ ′  =  ǫ 1 ..ǫ g ǫ ′ 1 ..ǫ ′ g  , with the property that the theta function θ  ǫ ǫ ′  (0,) is not zero, a poly- nomial P  ǫ ǫ ′  in nr variables which, when evaluated at λ 0 ,...,λ nr−1 , has the prop- erty that the quotient θ 8  ǫ ǫ ′  (0,) P  ǫ ǫ ′  (λ 0 ,...,λ nr−1 ) is independent of the characteristic. JOURNAL D’ANALYSE MATH ´ EMATIQUE, Vol. 111 (2010) DOI 10.1007/s11854-010-0019-y 289