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...

In number theory, the 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))...

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...

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

rational map with integer coefficients from the classical modular curve X0(N) for some integer N; this is a curve with integer coefficients with an explicit...

Gustav Jakob Jacobi and Niels Henrik Abel in 1827. Bianchi group Classical modular curve Fuchsian group J-invariant Kleinian group Mapping class group Minkowski's...

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

sense that classical modular forms (which are sometimes called elliptic modular forms to emphasize the point) are related to elliptic curves. Jacobi forms...

between the torsion conjecture for elliptic curves over the rationals and the theory of classical modular curves. In the early 1970s, the work of Gérard Ligozat...

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

Rational functions of j {\displaystyle j} are modular, and in fact give all modular functions of weight 0. Classically, the j {\displaystyle j} -invariant was...

Hecke 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...

topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This...

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...

[i]} . The j-line can be seen as giving a coordinatization of the classical modular curve of level 1, X 0 ( 1 ) {\displaystyle X_{0}(1)} , which is isomorphic...

defined by Katz. A classical modular form of weight k can be thought of roughly as a function f from pairs (E,ω) of a complex elliptic curve with a holomorphic...

The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer...

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

moduli space of elliptic curves, given by the j-invariant of an elliptic curve. It is a classical observation that every elliptic curve over C {\displaystyle...

the modularity theorem in 2001. Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo...

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

Lego Modular Buildings (stylized as LEGO Modular Buildings) is a series of Lego building toy set's introduced in 2007, with new sets usually being released...

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...

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

{\mathcal {M}}}_{1,1}} of genus 1 curves with one marked point. This is the stack of elliptic curves. Level 1 modular forms are sections of line bundles...

Shor's algorithm is quantum modular exponentiation, which is by far slower than the quantum Fourier transform and classical pre-/post-processing. There...

objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system...

work of Adolf Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators. Hecke operators can be...

communications. Elliptic-curve Diffie–Hellman key exchange Supersingular isogeny key exchange Forward secrecy Diffie–Hellman problem Modular exponentiation Denial-of-service...

year as the code construction was published, in their paper "Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound". The...