A modular curve is a compactified quotient of the upper half-plane by a congruence subgroup of 
. For example, adjoining finitely many cusps to 
gives the modular curve 
. Its noncuspidal points classify
elliptic curves together with the level structure
specified by the subgroup.
Modular Curve
See also
Elliptic Curve, Modular Form, Upper Half-PlaneExplore with Wolfram|Alpha
References
Diamond, F. and Shurman, J. A First Course in Modular Forms. New York: Springer, 2005.Cite this as:
Weisstein, Eric W. "Modular Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModularCurve.html