Fermat curve

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

${\displaystyle X^{n}+Y^{n}=Z^{n}.\ }$

Therefore, in terms of the affine plane its equation is

${\displaystyle x^{n}+y^{n}=1.\ }$

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.

The Fermat curve is non-singular and has genus

${\displaystyle (n-1)(n-2)/2.\ }$

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

${\displaystyle n-1.\ }$

Fermat varieties

Fermat-style equations in more variables define as projective varieties the Fermat varieties.

Related studies

• Gross, Benedict H.; Rohrlich, David E. (1978), "Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve" (PDF), Inventiones Mathematicae, 44 (3): 201–224, doi:10.1007/BF01403161.