Fig. 1: The graph of the hyperelliptic curve where
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form
where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 2 > 4 with n distinct roots, and h(x) is a polynomial of degree < g + 2 (if the characteristic of the ground field is not 2, one can take h(x) = 0).
A hyperelliptic function is an element of the function field of such a curve, or of the Jacobian variety on the curve; these two concepts are identical for elliptic functions, but different for hyperelliptic functions.
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In algebraic geometry, a hyperellipticcurve is an algebraic curve of genus g > 1, given by an equation of the form y 2 + h ( x ) y = f ( x ) {\displaystyle...
There are two types of hyperellipticcurves, a class of algebraic curves: real hyperellipticcurves and imaginary hyperellipticcurves which differ by the...
1, and the numerator is a quadratic. As an example, consider the hyperellipticcurve C : y 2 + y = x 5 , {\displaystyle C:y^{2}+y=x^{5},} which is of...
g − 1. When C is a hyperellipticcurve, the canonical curve is a rational normal curve, and C a double cover of its canonical curve. For example if P is...
The hyperellipticcurves are prototypic examples. An unramified covering then is the occurrence of an empty ramification locus. Morphisms of curves provide...
of Waterloo. He is the creator of hyperellipticcurve cryptography and the independent co-creator of elliptic curve cryptography. Koblitz received his...
generalise the elliptic integrals to all curves over the complex numbers. They include for example the hyperelliptic integrals of type ∫ x k d x Q ( x ) {\displaystyle...
linear systems is used in the classification of algebraic curves. A hyperellipticcurve is a curve C {\displaystyle C} with a degree 2 {\displaystyle 2} morphism...
curves which has both a hyperelliptic locus and a non-hyperelliptic locus. The non-hyperellipticcurves are all given by plane curves of degree 4 (using the...
surface is the Kummer variety of the Jacobian variety of a smooth hyperellipticcurve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution...
hyperellipticcurve to the projective line are all hyperelliptic Weierstrass points and these exhausts all the Weierstrass points on a hyperelliptic curve...
d\geq 5} is a hyperellipticcurve, and the case m = 3 {\displaystyle m=3} and d ≥ 4 {\displaystyle d\geq 4} is an example of a trigonal curve. Some authors...
the solution of Diophantine equations, and when generalized to hyperellipticcurves, the study of the Sato–Tate conjecture. First assume K {\displaystyle...
(an abelian surface): what would now be called the Jacobian of a hyperellipticcurve of genus 2. After Abel and Jacobi, some of the most important contributors...
written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira...
codimension of the curve in P n {\displaystyle \mathbf {P} ^{n}} . The canonical mapping for a hyperellipticcurve has image a rational normal curve, and is 2-to-1...
Neal Koblitz, adjunct professor, creator of elliptic curve cryptography and hyperellipticcurve cryptography Doug Stinson, professor, author of Cryptography:...
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