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Fermat curve

Algebraic curve

Fermat curve
Summary

Algebraic curve

The Fermat cubic surface <math>X^3+Y^3=Z^3</math>

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 2) 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.

algebraic-curves diophantine-geometry
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