 Modular curve

For a congruence subgroup $\Gamma$, the compact modular curve isAt an ordinary point it inherits a coordinate from $\mathfrak{h}$; at an elliptic point of stabilizer order $e$ a coordinate is the $e$th power of a local disk coordinate; and at a cusp of width $h$ a coordinate is $e^{2\pi i\tau /h}$.

Modular curves are fundamental objects in number theory and algebraic geometry that arise in the study of modular forms and elliptic curves. They provide a geometric way to understand properties of these mathematical structures. ### Definition A modular curve, often denoted as \( X(N) \) for some integer \( N \), parametrizes isomorphism classes of elliptic curves together with additional level structure.