Hyperbolic uniformization of Fermat curves
- 101 Downloads
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  in which each Riemann surface F N (ℂ) was described as a quotient of the complex disk by a Fuchsian group Γ.
KeywordsHyperbolic uniformization Fundamental domains Fermat curves
Unable to display preview. Download preview PDF.
- 1.Alsina. M., Bayer, P.: Quaternion orders, quadratic forms and shimura curves. CRM Monograph Series 22, AMS (2004)Google Scholar
- 2.Bayer, P.: Uniformization of certain Shimura curves. In: T. Crespo and Z. Hajto (eds.), Differential Galois Theory, Banach Center Publications 58, 13–26 (2002)Google Scholar
- 3.Bayer, P., Guàrdia, J.: A la recerca de pi, Butlletí de la Societat Catalana de Matemàtiques 17, 7–19 (2002)Google Scholar
- 4.Farkas, H., Kra, I.: Riemann surfaces. Graduate Texts in Mathematics 71, Springer, New York, NY (1992)Google Scholar
- 5.Ford, L.: Automorphic functions, (Chelsea, 1951)Google Scholar
- 8.Lang, S.: Introduction to algebraic and abelian functions. Graduate Text in Mathematics 89, (Springer, 1982)Google Scholar
- 10.Lehner, J.: Discontinuous groups and automorphic functions. AMS Mathematical Surveys 8, Amer. Math. Society (1964)Google Scholar
- 11.Magnus, W.: Noneuclidean tesselations and their groups, Academic Press, 1974Google Scholar
- 12.Nehari, Z.: Conformal mapping. Dover Publications (1952)Google Scholar
- 14.Sansone, G., Gerretsen, J.: Lectures on the theory of functions of a complex variable. Wolters-Noordhoff Publishing (1969)Google Scholar