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...

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...

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

In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way...

} 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...

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...

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...

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

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...

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...

modular curve X0(N) given by τ → –1/Nτ. It is named after Robert Fricke. The Fricke involution also acts on other objects associated with the modular...

supersingular elliptic curves as follows. For a prime number p {\displaystyle p} , the following are equivalent: The modular curve X 0 + ( p ) = X 0 ( p...

X(5) → X(1) = P1, where X(N) is a modular curve of level N. This cover is ramified at 12 points. The modular curve X(5) has genus 0 and is isomorphic...

(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...

In mathematics, Felix Klein's j-invariant or j function is a modular function of weight zero for the special linear group SL ⁡ ( 2 , Z ) {\displaystyle...

Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography...

modular forms and elliptic curves. In the 1950s post-World War II period of mathematics, there was renewed interest in the theory of modular curves due...

ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of the Hecke algebra of Hecke operators...

of two cusps of a modular curve has finite order in the Jacobian variety. Drinfeld, V. G. (1973), "Two theorems on modular curves", Akademija Nauk SSSR...

(see Hartshorne1977, 4.23.6). The modular curve X0(11) has j-invariant −21211−5313, and is isomorphic to the curve y2 + y = x3 − x2 − 10x − 20. The primes...

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

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...

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...