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Explicit Rational Functions on Fermat Curves and a Theorem of Greenberg

Published online by Cambridge University Press:  04 December 2007

Pavlos Tzermias
Pavlos Tzermias
Affiliation:
Department of Mathematics, P.O. Box 210089, 617 N. Santa Rita, The University of Arizona, Tucson, AZ 85721-0089, U.S.A. E-mail: tzermias@math.arizona.edu
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Abstract

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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.

Keywords

Fermat curvesrational functionsGreenberg's theorem

Information

Type
Research Article
Information
Compositio Mathematica , Volume 122 , Issue 3 , July 2000 , pp. 337 - 345
Copyright
© 2000 Kluwer Academic Publishers