number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex...

A modular elliptic curve is an elliptic curve E that admits a parametrization X0(N) → E by a modular curve. This is not the same as a modular curve that...

The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama–Shimura–Weil conjecture or modularity conjecture for elliptic curves) states...

classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here...

mathematician Sir Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for...

Classical modular curve Fuchsian group J-invariant Kleinian group Mapping class group Minkowski's question-mark function Möbius transformation Modular curve Modular...

)} where ω {\displaystyle \omega } is a canonical line bundle on the modular curve X Γ = Γ ∖ ( H ∪ P 1 ( Q ) ) {\displaystyle X_{\Gamma }=\Gamma \backslash...

rational functions F and G, in the function field of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial of two...

surface Elkies trinomial curves Hyperelliptic curve Classical modular curve Cassini oval Bowditch curve Brachistochrone Butterfly curve (transcendental) Catenary...

asserts that every elliptic curve over Q is a modular curve, which implies that its L-function is the L-function of a modular form whose analytic continuation...