Hyperelliptic curve
Encyclopedia
In algebraic geometry
, a hyperelliptic curve (over the complex numbers) is an algebraic curve
given by an equation of the form
where f(x) is a polynomial
of degree n > 4 with n distinct roots. A hyperelliptic function is a function from the function field
of such a curve; or possibly on the Jacobian variety
on the curve, these being two concepts that are the same for the elliptic function
case, but different in this case.
of the curve: a polynomial of degree 2g + 1 or 2g + 2 gives a curve of genus g. When the degree is equal to 2g + 1, the curve is called an imaginary hyperelliptic curve
. Meanwhile, the curve that has degree 2g + 2 is mentioned a real hyperelliptic curve
. This statement about genus remains true for g = 0 or 1, but those curves are not called "hyperelliptic". Rather, the case g = 1 (if we choose a distinguished point) is an elliptic curve
. Hence the terminology.
at infinity in the projective plane
. This feature is specific to the case n > 4. Therefore in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model, equivalent in the sense of birational geometry
, is meant.
To be more precise, the equation defines a quadratic extension of C(x), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization (integral closure) process. It turns out that after doing this, there is a cover of the curve with two affine pieces: the one already given by
and another one given by
.
The glueing maps between the two pieces are given by
and
wherever they are defined.
In fact geometric shorthand is assumed, with the curve C being defined as a ramified double cover of the projective line
, the ramification occurring at the roots of f, and also for odd n at the point at infinity. In this way the cases n = 2g + 1 and 2g + 2 can be unified, since we might as well use an automorphism
of the projective line to move any ramification
point away from infinity.
dimension check. Counting constants, with n = 2g + 2, the collection of n points subject to the action of the automorphisms of the projective line has (2g + 2) − 3 degrees of freedom, which is less than 3g − 3, the number of moduli of a curve of genus g, unless g is 2. Much more is known about the hyperelliptic locus in the moduli space of curves or abelian varieties, though it is harder to exhibit general non-hyperelliptic curves with simple models. One geometric characterization of hyperelliptic curves is via Weierstrass point
s. More detailed geometry of non-hyperelliptic curves is read from the theory of canonical curves, the canonical mapping being 2-to-1 on hyperelliptic curves but 1-to-1 otherwise for g > 2. Trigonal curves are those that correspond to taking a cube root, rather than a square root, of a polynomial.
The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if it is assumed to be separable.
Hyperelliptic curves can be used in hyperelliptic curve cryptography
in cryptosystem
s based on the discrete logarithm problem.
(1812-1847) in his last paper Abelsche Transcendenten erster Ordnung (Abelian transcendents of first order) (in Journal für reine und angewandte Mathematik
, vol. 35, 1847). Independently Johann G. Rosenhain worked on that matter and published Umkehrungen ultraelliptischer Integrale erster Gattung (in Mémoires des sa vanta etc., vol. 11, 1851).
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a hyperelliptic curve (over the complex numbers) is an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
given by an equation of the form
where f(x) is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
of degree n > 4 with n distinct roots. A hyperelliptic function is a function from the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
of such a curve; or possibly on the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
on the curve, these being two concepts that are the same for the elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
case, but different in this case.
Genus of the curve
The degree of the polynomial determines the genusGenus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
of the curve: a polynomial of degree 2g + 1 or 2g + 2 gives a curve of genus g. When the degree is equal to 2g + 1, the curve is called an imaginary hyperelliptic curve
Imaginary hyperelliptic curve
A hyperelliptic curve is a particular kind of algebraic curve.There exist hyperelliptic curves of every genus g \geq 1. If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves...
. Meanwhile, the curve that has degree 2g + 2 is mentioned a real hyperelliptic curve
Real hyperelliptic curve
A hyperelliptic curve is a class of algebraic curves. Hyperelliptic curves exist for every genus g \geq 1. The general formula of Hyperelliptic curve over a finite field K is given byC : y^2 + h y = f \in K[x,y]...
. This statement about genus remains true for g = 0 or 1, but those curves are not called "hyperelliptic". Rather, the case g = 1 (if we choose a distinguished point) is an elliptic curve
Elliptic curve
In 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 in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
. Hence the terminology.
Formulation and choice of model
While this model is the simplest way to describe hyperelliptic curves, such an equation will have a singular pointMathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
at infinity in the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
. This feature is specific to the case n > 4. Therefore in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model, equivalent in the sense of birational geometry
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
, is meant.
To be more precise, the equation defines a quadratic extension of C(x), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization (integral closure) process. It turns out that after doing this, there is a cover of the curve with two affine pieces: the one already given by
and another one given by
.The glueing maps between the two pieces are given by
and
wherever they are defined.
In fact geometric shorthand is assumed, with the curve C being defined as a ramified double cover of the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
, the ramification occurring at the roots of f, and also for odd n at the point at infinity. In this way the cases n = 2g + 1 and 2g + 2 can be unified, since we might as well use an automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
of the projective line to move any ramification
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
point away from infinity.
Occurrence and applications
All curves of genus 2 are hyperelliptic, but for genus ≥ 3 the generic curve is not hyperelliptic. This is seen heuristically by a moduli spaceModuli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
dimension check. Counting constants, with n = 2g + 2, the collection of n points subject to the action of the automorphisms of the projective line has (2g + 2) − 3 degrees of freedom, which is less than 3g − 3, the number of moduli of a curve of genus g, unless g is 2. Much more is known about the hyperelliptic locus in the moduli space of curves or abelian varieties, though it is harder to exhibit general non-hyperelliptic curves with simple models. One geometric characterization of hyperelliptic curves is via Weierstrass point
Weierstrass point
In mathematics, a Weierstrass point P on a nonsingular algebraic curve C defined over the complex numbers is a point such that there are extra functions on C, with their poles restricted to P only, than would be predicted by looking at the Riemann–Roch theorem...
s. More detailed geometry of non-hyperelliptic curves is read from the theory of canonical curves, the canonical mapping being 2-to-1 on hyperelliptic curves but 1-to-1 otherwise for g > 2. Trigonal curves are those that correspond to taking a cube root, rather than a square root, of a polynomial.
The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if it is assumed to be separable.
Hyperelliptic curves can be used in hyperelliptic curve cryptography
Hyperelliptic curve cryptography
Hyperelliptic curve cryptography is similar to elliptic curve cryptography insofar as the Jacobian of a hyperelliptic curve is an Abelian group on which to do arithmetic, just as we use the group of points on an elliptic curve in ECC.-Definition:...
in cryptosystem
Cryptosystem
There are two different meanings of the word cryptosystem. One is used by the cryptographic community, while the other is the meaning understood by the public.- General meaning :...
s based on the discrete logarithm problem.
Classification
Hyperelliptic curves of given genus g have a moduli space, closely related to the ring of invariants of a binary form of degree 2g+2.History
Hyperelliptic functions were first published by Adolph GöpelAdolph Göpel
Adolph Göpel was a German mathematician who did the first paper on hyperelliptic functions.- Life and work :...
(1812-1847) in his last paper Abelsche Transcendenten erster Ordnung (Abelian transcendents of first order) (in Journal für reine und angewandte Mathematik
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
, vol. 35, 1847). Independently Johann G. Rosenhain worked on that matter and published Umkehrungen ultraelliptischer Integrale erster Gattung (in Mémoires des sa vanta etc., vol. 11, 1851).