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.
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For a congruence subgroup , the compact modular curve isAt an ordinary point it inherits a coordinate from ; at an elliptic point of stabilizer order a coordinate is the th power of a local disk coordinate; and at a cusp of width a coordinate is .