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| 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'''
| | Building and Engineering Specialists Anton Rave from Lively, has hobbies including snooker, property developers in singapore and crossword puzzles. Keeps a trip site and has plenty to write about after going to Cistercian Abbey of Fontenay.<br><br>Feel free to visit my web site ... [http://www.venontab.org/groups/new-launch-property-singapore/ Http://Www.Venontab.Org] |
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| :<math>X^n + Y^n = Z^n.\ </math>
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| Therefore in terms of the [[Euclidean plane|affine plane]] its equation is
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| :<math>x^n + y^n = 1.\ </math>
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| 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'' ≥ 3) there are no nontrivial integer solutions to the Fermat equation; therefore, the Fermat curve has no nontrivial rational points.
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| The Fermat curve is [[non-singular]] and has [[genus (mathematics)|genus]]
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| :<math>(n - 1)(n - 2)/2.\ </math>
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| 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]].
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| ==Fermat varieties==
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| Fermat-style equations in more variables define as [[projective varieties]] the '''Fermat varieties'''.
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| ==Related studies==
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| *{{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 |doi=10.1007/BF01403161}}.
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| [[Category:Algebraic curves]]
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| [[Category:Diophantine geometry]]
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Latest revision as of 04:38, 21 August 2014
Building and Engineering Specialists Anton Rave from Lively, has hobbies including snooker, property developers in singapore and crossword puzzles. Keeps a trip site and has plenty to write about after going to Cistercian Abbey of Fontenay.
Feel free to visit my web site ... Http://Www.Venontab.Org