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  • Explicit Rational Functions on Fermat Curves and a Theorem of Greenberg

  • Pavlos Tzermias
  • Journal:
    Compositio Mathematica / Volume 122 / Issue 3 / July 2000
    Published online by Cambridge University Press:
    04 December 2007, pp. 337-345
    Print publication:
    July 2000
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  • This paper is concerned with the arithmetic of curves of the form vp=us(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.