Editing Hyperelliptic curve

{{Short description|Algebraic curve}}
[[File:Example of a hyperelliptic curve.svg|right|thumb|Fig. 1: The graph of the hyperelliptic curve <math>C : y^2 = f(x)</math> where
<math display="block">f(x) = x^5 - 2x^4 - 7x^3 + 8x^2 + 12x = x (x + 1) (x - 3) (x + 2) (x - 2). </math>
]]
In [[algebraic geometry]], a '''hyperelliptic curve''' is an [[algebraic curve]] of [[Genus (mathematics)|genus]] ''g'' > 1, given by an equation of the form
<math display="block">y^2 + h(x)y = f(x)</math>
where ''f''(''x'') is a [[polynomial]] 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 an algebraic variety|function field]] of such a curve, or of the [[Jacobian variety]] on the curve; these two concepts are identical for [[elliptic function]]s, but different for hyperelliptic functions.

==Genus==

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]]. 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 special cases are not called "hyperelliptic". In the case ''g'' = 1 (if one chooses a distinguished point), such a curve is called an [[elliptic curve]].

==Formulation and choice of model==

While this model is the simplest way to describe hyperelliptic curves, such an equation will have a [[Mathematical singularity|singular point]] ''at infinity'' in the [[projective plane]]. 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]], 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 an open cover of the curve by two affine charts: the one already given by
<math display="block">y^2 = f(x) </math>
and another one given by
<math display="block">w^2 = v^{2g+2}f(1/v) .</math>

The glueing maps between the two charts are given by
<math display="block">(x,y) \mapsto (1/x, y/x^{g+1})</math>
and
<math display="block">(v,w) \mapsto (1/v, w/v^{g+1}),</math>
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 (mathematics)|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]], the hyperelliptic curve with genus ''g'' is defined by an equation with degree ''n'' = 2''g'' + 2. Suppose  ''f'' : ''X'' → P<sup>1</sup> is a branched covering with ramification degree ''2'', where ''X'' is a curve with genus ''g'' and P<sup>1</sup> is the [[Riemann sphere]]. Let ''g''<sub>1</sub> = ''g'' and ''g''<sub>0</sub> be the genus of P<sup>1</sup> ( = 0 ), then the Riemann-Hurwitz formula turns out to be
:<math>2-2g_1 =2(2-2g_0)-\sum_{s \in X}(e_s-1)</math>
where ''s'' is over all ramified points on ''X''. The number of ramified points is ''n'', and at each ramified point ''s'' we have ''e<sub>s</sub>'' = 2, so the formula becomes
:<math>2-2\times g =2(2-2\times0)-n\times(2-1)</math>
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) &minus; 3 degrees of freedom, which is less than 3''g'' &minus; 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]],{{clarify|What does the reference to abelian varieties mean?|date=December 2012}}  though it is harder to exhibit ''general'' non-hyperelliptic curves with simple models.<ref>{{cite journal
 | last = Poor | first = Cris
 | doi = 10.1090/S0002-9939-96-03312-6
 | issue = 7
 | journal = Proceedings of the American Mathematical Society
 | mr = 1327038
 | pages = 1987–1991
 | title = Schottky's form and the hyperelliptic locus
 | volume = 124
 | year = 1996| doi-access = free
 }}</ref> 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 curve]]s, the [[canonical bundle#Canonical maps|canonical mapping]] being 2-to-1 on hyperelliptic curves but 1-to-1 otherwise for ''g'' > 2. [[Trigonal curve]]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 [[cryptosystem]]s based on the [[discrete logarithm problem]].

Hyperelliptic curves also appear composing entire connected components of certain strata of the moduli space of Abelian differentials.<ref>{{cite journal |arxiv=math.GT/0201292 | doi=10.1007/s00222-003-0303-x | volume=153 | title=Connected components of the moduli spaces of Abelian differentials with prescribed singularities | year=2003 | journal=Inventiones Mathematicae | pages=631–678 | last1 = Kontsevich | first1 = Maxim | last2 = Zorich | first2 = Anton| issue=3 | bibcode=2003InMat.153..631K | s2cid=14716447 }}</ref>

Hyperellipticity of genus-2 curves was used to prove [[Mikhail Leonidovich Gromov|Gromov]]'s [[filling area conjecture]] 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.{{specify|date=August 2019}}

==History==

Hyperelliptic functions were first published{{citation needed|date=August 2019}} by [[Adolph Göpel]] (1812-1847) in his last paper ''Abelsche Transcendenten erster Ordnung'' (Abelian transcendents of first order) (in [[Crelle's Journal|Journal für die 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]]
* [[Superelliptic curve]]

==References==
*{{Springer|id=Hyper-elliptic_curve|title=Hyper-elliptic curve}}
*[[arxiv:2007.01749|A user's guide to the local arithmetic of hyperelliptic curves]]

==Notes==
{{Reflist}}{{Algebraic curves navbox}}

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[[Category:Algebraic curves]]