Explicit Rational Functions on Fermat Curves and a Theorem of Greenberg

Abstract

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 complexion by a primitive p-th root of unity ζ. We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-ζ)² - and (1-ζ)³-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.