Modular curve

Summary

A modular curve is the quotient of the upper half-plane by a congruence subgroup of $\mathrm{SL}_2(\Z)$, compactified by adding finitely many cusps. The result is a compact Riemann surface and an algebraic curve defined over a number field. Modular curves parametrize elliptic curves with extra level structure, and their holomorphic differentials correspond to cusp modular forms of weight $2$. They are central in the proof of Fermat's Last Theorem through the modularity of elliptic curves.