Editing Fermat curve

{{Short description|Algebraic curve}}
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[[File:FermatCubicSurface.PNG|thumb|The Fermat cubic surface <math>X^3+Y^3=Z^3</math>]]
In [[mathematics]], the '''Fermat curve''' is the [[algebraic curve]] in the [[complex projective plane]] defined in [[homogeneous coordinates]] (''X'':''Y'':''Z'') by the '''Fermat equation:'''
:<math>X^n + Y^n = Z^n.\ </math>

Therefore, in terms of the [[Euclidean plane|affine plane]] its equation is:
:<math>x^n + y^n = 1.\ </math>

An integer solution to the Fermat equation would correspond to a nonzero [[rational number]] solution to the affine equation, and vice versa. But by [[Fermat's Last Theorem]] it is now known that (for ''n''&nbsp;>&nbsp;2) there are no nontrivial integer solutions to the Fermat equation; therefore, the Fermat curve has no nontrivial rational points.

The Fermat curve is [[Algebraic curve#Singularities|non-singular]] and has [[genus (mathematics)|genus]]:

:<math>(n - 1)(n - 2)/2.\ </math>

This means genus 0 for the case ''n'' = 2 (a [[conic]]) and genus 1 only for ''n'' = 3 (an [[elliptic curve]]). The [[Jacobian variety]] of the Fermat curve has been studied in depth. It is isogenous to a product of simple abelian varieties with [[complex multiplication]].

The Fermat curve also has [[gonality]]:

:<math>n-1.\ </math>

==Fermat varieties==
Fermat-style equations in more variables define as [[projective varieties]] the '''Fermat varieties'''.

==Related studies==
*{{citation |title=Finiteness results for modular curves of genus at least 2 |first1=Matthew |last1=Baker |first2=Enrique |last2=Gonzalez-Jimenez |first3=Josep |last3=Gonzalez |first4=Bjorn |last4=Poonen |authorlink4=Bjorn Poonen |journal=[[American Journal of Mathematics]] |year=2005 |volume=127 |issue=6 |pages=1325–1387 |doi=10.1353/ajm.2005.0037 |jstor=40068023|arxiv=math/0211394 |s2cid=8578601 }}
*{{citation|first1=Benedict H. |last1=Gross |first2=David E. |last2=Rohrlich |year=1978 |title=Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve |journal=[[Inventiones Mathematicae]] |volume=44 |issue=3 |pages=201–224 |url=http://www.kryakin.com/files/Invent_mat_%282_8%29/44/44_01.pdf |archive-url=https://web.archive.org/web/20110713171905/http://www.kryakin.com/files/Invent_mat_(2_8)/44/44_01.pdf |url-status=dead |archive-date=2011-07-13 |doi=10.1007/BF01403161 |s2cid=121819622 }}
*{{citation |title=Points of Low Degree on Smooth Plane Curves |first1=Matthew J. |last1=Klassen |first2=Olivier |last2= Debarre |journal=[[Journal für die reine und angewandte Mathematik]] |volume=1994 |issue=446 |year=1994 |pages=81–88 |doi=10.1515/crll.1994.446.81|s2cid=7967465 |arxiv=alg-geom/9210004 }}
*{{citation |title=Low-Degree Points on Hurwitz-Klein Curves |first=Pavlos |last=Tzermias |journal=[[Transactions of the American Mathematical Society]]
|volume=356 |issue=3 |year=2004 |pages=939–951 |doi=10.1090/S0002-9947-03-03454-8 |jstor=1195002|doi-access=free }}
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