Explicit Rational Functions on Fermat Curves and a Theorem of Greenberg

This paper is concerned with the arithmetic of curves of the form v^p=u^s(1-u), where p is a prime with p 5 and s is an integer such that 1 s p-2. The Jacobians of these curves admit complex multiplication by a primitive p-th root of unity . We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-)² - and (1-)³-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.