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: Therefore, in terms of the affine plane its equation is: 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 > 2) there are no nontrivial integer solutions to the Fermat equation; therefore, the Fermat curve has no nontrivial rational points.

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