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

$X^n + Y^n = Z^n.\$

Therefore, in terms of the affine plane its equation is

$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

$(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

$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.