Elliptic and Hypergeometric FunctionsFor an elliptic curve E over the complex numbers C we have already observed in (7.3) of the Introduction that the group E(C) is a compact group isomorphic to the product of two circles. This assertion ignores the fact that there is a complex structure on...
Algebraically, the discriminant is nonzero when the right-hand side has three distinct roots. In the classical case of anelliptic curveover thecomplex numbers, the discriminant has a geometric interpretation. If, then the elliptic curve is nonsingular and hascurve genus1, i.e., it is atorus...
An elliptic curve over real numbers may be defined as the set of points (x,y) which satisfy an elliptic curve equation of the form: y2 = x3 + ax + b, where x, y, a and b are real numbers. Each choice of the numbers a and b yields a different elliptic curve. For example, a...
The Weil pairing on elliptic curves over C 来自 Semantic Scholar 喜欢 0 阅读量: 9 作者: S Galbraith 摘要: To help motivate the Weil pairing, we discuss it in the context of elliptic curves over the field of complex numbers. 被引量: 2 年份: 2005 ...
Elliptic curvesroot numbercomplex multiplicationquadratic twistsLetE/Fbe an elliptic curve defined over a number fieldF. Suppose thatEhas complex multiplication overF, i.e.EndF(E)is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a ...
Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and ...
To make Bézout’s theorem rigorous you have to work over an algebraically complete field (e.g. the complex numbers rather than the real numbers), you have to count intersections with multiplicity, and you have to add points at infinity, i.e. you have to work in a projective plane. ...
Existence of curves of genus three on a product of two elliptic curvesLet E be an elliptic curve over the field of complex numbers, and let A be the abelian surface tE*E. It seems interesting to study if A contains a smooth curve of genus g. In the case when g=2, Hayashida and ...
To be precise, suppose is an elliptic curve over with known endomorphism ring (for simplicity let’s take ). Let be an elliptic curve such that there is an isogeny of degree . Suppose we are also given where generate the subgroup of points of order on . The attacker wants to know . ...
Figure 1 shows a picture of an elliptic curve over the real numbers where a is –1 and b is 1. Elliptic curves satisfy some interesting mathematical properties. The curve is symmetric around the x axis, so that if (x,y) is a point on the curve, then (x,–y) is also on the curv...