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...

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...

)} 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...

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...

Classical modular curve Erdős lemniscate Hurwitz surface Mandelbrot curve Polynomial lemniscate Sinusoidal spiral Superellipse Bowditch curve Brachistochrone...

associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise...

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

number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by...

field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio...

cryptosystems based on modular exponentiation in Galois fields, such as the RSA cryptosystem and ElGamal cryptosystem. Elliptic curves are applicable for...

notion of supersingular elliptic curves as follows. For a prime number p, the following are equivalent: The modular curve X0+(p) = X0(p) / wp, where wp is...

(PDF), MSRI Publications, 49: 1–10 Chen, Imin (1999), "On Siegel's Modular Curve of Level 5 and the Class Number One Problem", Journal of Number Theory...

this curve has a complicated form, it is natural and conceptually significant in the number theory of elliptic curves. The equation describes a modular curve...

Enrique; Gonzalez, Josep; Poonen, Bjorn (2005), "Finiteness results for modular curves of genus at least 2", American Journal of Mathematics, 127 (6): 1325–1387...

Taniyama–Shimura conjecture (now known as the modularity theorem) relating elliptic curves to modular forms. This connection would ultimately lead to...

varieties Shimura variety Modular curve Elliptic cohomology Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag...

In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined...

pairing Hyperelliptic curve Klein quartic Modular curve Modular equation Modular function Modular group Supersingular primes Fermat curve Bézout's theorem...

function Weil conjectures Modular form modular group Congruence subgroup Hecke operator Cusp form Eisenstein series Modular curve Ramanujan–Petersson conjecture...

curves Hyperelliptic curve Klein quartic Classical modular curve Bolza surface Macbeath surface Polynomial lemniscate Fermat curve Sinusoidal spiral Superellipse...

are modular, and in fact give all modular functions of weight 0. Classically, the j-invariant was studied as a parameterization of elliptic curves over...

face) is the modular curve X(5); this explains the relevance for number theory. More subtly, the (projective) Klein quartic is a Shimura curve (as are the...

operators on Heegner points on the classical modular curve X0(N) as well as on the Drinfeld modular curve XDrin 0(I). These buildings with complex multiplication...

Eichler–Shimura congruence relation between the local L-function of a modular curve and the eigenvalues of Hecke operators. In 1959, Shimura extended the...

Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators. It was...