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...

mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over...

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...

complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending...

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

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...

bundle on the moduli stack of elliptic curves. A modular function is a function that is invariant with respect to the modular group, but without the condition...

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

Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at...

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...

In mathematics, the conductor of an elliptic curve over the field of rational numbers (or more generally a local or global field) is an integral ideal...

the moduli problem, which are the points of the modular curve not corresponding to honest elliptic curves but degenerate cases, may be difficult to read...

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...

connection between the modular group and elliptic curves. Each point z {\displaystyle z} in the upper half-plane gives an elliptic curve, namely the quotient...

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...

semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type. Suppose E is an elliptic curve...

In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 {\displaystyle p>0}...

the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with...

In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods...

In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve y 2 = x ( x − α ) ( x + β ) {\displaystyle y^{2}=x(x-\alpha )(x+\beta )}...

Iwasawa–Greenberg main conjectures for a large class of modular forms. As a consequence, for a modular elliptic curve over the rational numbers, they prove that the...

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...

introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and...

Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic...

In mathematics, the moduli stack of elliptic curves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M e l l {\displaystyle {\mathcal {M}}_{\mathrm...

naming conventions. For expressing one argument: α, the modular angle k = sin α, the elliptic modulus or eccentricity m = k2 = sin2 α, the parameter Each...

properties of elliptic functions 30 years earlier but never published anything on the subject. Elliptic integral Elliptic curve Modular group Theta function...

{\displaystyle j} -invariant was studied as a parameterization of elliptic curves over C {\displaystyle \mathbb {C} } , but it also has surprising connections...

mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. Historically...

function. For elliptic curves over the rational numbers, the Hasse–Weil conjecture follows from the modularity theorem: each elliptic curve E over Q {\displaystyle...