Hyperelliptic function
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In algebraic geometry, a hyperelliptic curve is an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 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 In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
on the curve; these two concepts are identical for elliptic functions, but different for hyperelliptic functions.


Genus of the curve

The degree of the polynomial determines the genus of the curve: a polynomial of degree 2''g'' + 1 or 2''g'' + 2 gives a curve of genus ''g''. When the degree is equal to 2''g'' + 1, the curve is called an
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 cu ...
. Meanwhile, a curve of degree 2''g'' + 2 is termed 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 In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
. 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 point ''at infinity'' in the
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 (namely, parallel lines) that d ...
. This feature is specific to the case ''n'' > 3. Therefore, in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of
birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
, is meant. To be more precise, the equation defines a
quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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 an open cover of the curve by two affine charts: the one already given by y^2 = f(x) and another one given by w^2 = v^f(1/v) . The glueing maps between the two charts are given by (x,y) \mapsto (1/x, y/x^) and (v,w) \mapsto (1/v, w/v^), 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 In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
, the ramification occurring at the roots of ''f'', and also for odd ''n'' at the point at infinity. In this way the cases ''n'' = 2''g'' + 1 and 2''g'' + 2 can be unified, since we might as well use an automorphism of the projective plane to move any ramification point away from infinity.


Using Riemann–Hurwitz formula

Using the
Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramif ...
, the hyperelliptic curve with genus ''g'' is defined by an equation with degree ''n'' = 2''g'' + 2. Suppose ''f'' : ''X'' → P1 is a branched covering with ramification degree ''2'', where ''X'' is a curve with genus ''g'' and P1 is the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. Let ''g''1 = ''g'' and ''g''0 be the genus of P1 ( = 0 ), then the Riemann-Hurwitz formula turns out to be :2-2g_1 =2(2-2g_0)-\sum_(e_s-1) where ''s'' is over all ramified points on ''X''. The number of ramified points is ''n'', so ''n'' = 2''g'' + 2.


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 space dimension check. Counting constants, with ''n'' = 2''g'' + 2, the collection of ''n'' points subject to the action of the automorphisms of the projective line has (2''g'' + 2) − 3 degrees of freedom, which is less than 3''g'' − 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 In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
, though it is harder to exhibit ''general'' non-hyperelliptic curves with simple models. One geometric characterization of hyperelliptic curves is via Weierstrass points. More detailed geometry of non-hyperelliptic curves is read from the theory of
canonical curve In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
s, the canonical mapping being 2-to-1 on hyperelliptic curves but 1-to-1 otherwise for ''g'' > 2.
Trigonal curve In mathematics, the gonality of an algebraic curve ''C'' is defined as the lowest degree of a nonconstant rational map from ''C'' to the projective line. In more algebraic terms, if ''C'' is defined over the field ''K'' and ''K''(''C'') denotes the ...
s 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 the extension is assumed to be separable. Hyperelliptic curves can be used in hyperelliptic curve cryptography for cryptosystems based on the
discrete logarithm problem In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b' ...
. Hyperelliptic curves also appear composing entire connected components of certain strata of the moduli space of Abelian differentials. Hyperellipticity of genus-2 curves was used to prove Gromov's
filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. Definition ...
in the case of fillings of genus =1.


Classification

Hyperelliptic curves of given genus ''g'' have a moduli space, closely related to the ring of invariants of a binary form of degree 2''g''+2.


History

Hyperelliptic functions were first published by Adolph Göpel (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 savants etc., vol. 11, 1851).


See also

*
Bolza surface In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL_2(3) of order 48 ...
*
Superelliptic curve In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form :y^m = f(x), where m \geq 2 is an integer and ''f'' is a polynomial of degree d\geq 3 with coefficients in a field k; more precisely, it is the smooth pro ...


References

* * A user's guide to the local arithmetic of hyperelliptic curves


Notes

{{DEFAULTSORT:Hyperelliptic Curve Algebraic curves