Fermat curve
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates 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 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
This means genus 0 for the case n = 2 and genus 1 only for n = 3. 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 gonalityFermat-style equations in more variables define as projective varieties the Fermat varieties.Related studies
