Graphs of elliptic curves y2 = x3 − x and y2 = x3 − x + 1. If we consider these as curves over the rationals, then the modularity theorem asserts that they can be parametrized by a modular curve.
A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.
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The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Shimura-Weil conjecture or modularity conjecture for ellipticcurves) states...
complex upper-half plane). The points of a modularcurve parametrize isomorphism classes of ellipticcurves, together with some additional structure depending...
mathematics, an ellipticcurve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An ellipticcurve is defined over...
with its derivative can be used to parameterize ellipticcurves and they generate the field of elliptic functions with respect to a given period lattice...
bundle on the moduli stack of ellipticcurves. A modular function is a function that is invariant with respect to the modular group, but without the condition...
Ellipticcurve scalar multiplication is the operation of successively adding a point along an ellipticcurve to itself repeatedly. It is used in elliptic...
In mathematics, ellipticcurve primality testing techniques, or ellipticcurve primality proving (ECPP), are among the quickest and most widely used methods...
In mathematics, the conductor of an ellipticcurve over the field of rational numbers (or more generally a local or global field) is an integral ideal...
classical modularcurve 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...
connection between the modular group and ellipticcurves. Each point z {\displaystyle z} in the upper half-plane gives an ellipticcurve, namely the quotient...
In mathematics, the moduli stack of ellipticcurves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M ell {\displaystyle {\mathcal {M}}_{\textrm...
the moduli problem, which are the points of the modularcurve not corresponding to honest ellipticcurves but degenerate cases, may be difficult to read...
properties of elliptic functions 30 years earlier but never published anything on the subject. Elliptic integral EllipticcurveModular group Theta function...
An important aspect in the study of ellipticcurves is devising effective ways of counting points on the curve. There have been several approaches to do...
mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to ellipticcurves and modular forms. Historically...
branch points of a ramified double cover of the projective line by the ellipticcurve C/⟨1,τ⟩{\displaystyle \mathbb {C} /\langle 1,\tau \rangle }, where the...
the theory of ellipticcurves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with...
semistable ellipticcurve may be described more concretely as an ellipticcurve that has bad reduction only of multiplicative type. Suppose E is an elliptic curve...
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...
In mathematics, a Frey curve or Frey–Hellegouarch curve is the ellipticcurve y 2 = x ( x − α ) ( x + β ) {\displaystyle y^{2}=x(x-\alpha )(x+\beta )}...
mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that...
Taniyama–Shimura conjecture (now known as the modularity theorem) relating ellipticcurves to modular forms. This connection would ultimately lead to...
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