Abstract
Computations using theta functions with characteristic show that the modular curve X(9) is the complete intersection of two cubics. The holomorphic differentials and Weierstrass gap sequence are also computed.
Similar content being viewed by others
References
R. Brooks, H.M. Farkas, and I. Kra, “Number theory, theta identities, and modular curves,” in Extremal Riemann Surfaces (Quine and Sarnak, eds.), AMS Contemporary Mathematics, Vol. 201, 1997.
H.M. Farkas and Y. Kopeliovich, “New theta constant identities,” Israel J. of Mathematics 82 (1993), 133-140.
H.M. Farkas, Y. Kopeliovich, and I. Kra, “Uniformization of modular curves,” Communications in Analysis and Geometry 4(2), (1996).
A. Hurwitz, “Über Relationen zwischen Klassenanzahlen binären quadratische Formen von negativer Determinante,” Mathematische Annalen 25 (1885), 157-196.
F. Klein, “Über die transformation siebenter ordnung der elliptischen functionen,” Math. Ann. 25 (1885), 157-196.
K. Knopp, Modular Functions in Analytic Function Theory, Markham Publishing Company, Chicago, 1970.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan, Kanô memorial lectures 1, Vol. 11, Iwannami Shoten and Princeton University Press, 1971.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kopeliovich, Y., Quine, J. On the Curve X(9). The Ramanujan Journal 2, 371–378 (1998). https://doi.org/10.1023/A:1009703117263
Issue Date:
DOI: https://doi.org/10.1023/A:1009703117263