Hyperbolic uniformization of Fermat curves

Abstract

The ground-breaking research on the uniformization of curves was conducted at the beginning of the last century. Nevertheless, there are few examples in the literature of algebraic curves for which an explicit uniformization is known. In this article we obtain an explicit uniformization of the Fermat curves F N , for each \(N\geq 4\). The results presented here are based in part on an earlier study of the second author [6] in which each Riemann surface F N () was described as a quotient of the complex disk by a Fuchsian group Γ.

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Correspondence to P. Bayer.

Additional information

This work was partially supported by MCYT BFM2000-0627 and BMF2003-01898.

2000 Mathematics Subject Classification Primary—11F03, 11F06; Secondary—11F30

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Bayer, P., Guàrdia, J. Hyperbolic uniformization of Fermat curves. Ramanujan J 12, 207–223 (2006). https://doi.org/10.1007/s11139-006-0074-2

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Keywords

  • Hyperbolic uniformization
  • Fundamental domains
  • Fermat curves