Editing Fermat curve (section)

{{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>