# 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* ≥ 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

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

## 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.