Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-20T00:37:53.515Z Has data issue: false hasContentIssue false
  • English
  • Français

A Diophantine Equation Associated to X0(5)

Published online by Cambridge University Press:  01 February 2010

Imin Chen
Imin Chen
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada, ichen@math.sfu.ca, http://www.math.sfu.ca/~ichen/

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Several classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat equations.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 116 - 121
Copyright
Copyright © London Mathematical Society 2005

References

1.Bosma, W., Cannon, J. and Playoust, C., ‘The MAGMA algebra system, I. The user language’, Computational algebra and number theory (London, 1993), J. Symbolic Comput. 24 (1997) 235–265.CrossRefGoogle Scholar
2.Breuil, C., Conrad, B., Diamond, F. and Taylor, R., Modularity of elliptic curves over Q: wild 3-adic exercises’, J. Amer. Math. Soc. 14 (2001) 843–939.CrossRefGoogle Scholar
3.Chen, I. and Yui, N., ‘Singular values of Thompson series’, Groups, difference sets, and the Monster (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ. 4 (ed. Arasu, K. T., Dillon, J. F., Harada, K., Segal, S. and Solomon, R., Gruyter, De, Berlin, 1996)255–326.Google Scholar
4.Darmon, H., ‘Faltings plus epsilon, wiles plus epsilon, and the generalized Fermat equation’, C. R. Math. Acad. Sci. Soc. R. Can. 19 (1997) 2–14.Google Scholar
5.Kenku, M. A., The modular curves Xo(65) and Xo(91) and rational isogeny’, Math. Proc. Cambridge Philos. Soc. 87 (1980) 15–20.CrossRefGoogle Scholar
6.Mazur, B., ‘Rational isogenies of prime degree’, Invent. Math. 44 (1978) 129–162.CrossRefGoogle Scholar
7.Papadopoulos, I., ‘Sur la classification de Néron des courbes elliptiques en caractéristique résiduelle 2 et 3’, J. Number Theory 44 (1993) 119–152.CrossRefGoogle Scholar
8.Ribet, K., ‘On modular representations of Gal(ℚ | ℚ) arising from modular forms’, Invent. Math. 100 (1990)431–476.CrossRefGoogle Scholar