Fermat curve

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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

Xⁿ + Yⁿ = Zⁿ.

Therefore in terms of the affine plane its equation is

xⁿ + yⁿ = 1,

with integer solutions (not all zero) of the projective equation corresponding to rational number solutions of the affine equation. Proportional integer solutions correspond to the same rational solution. This curve therefore can be used to formulate Fermat's last theorem in a geometric way (hence its name).

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 curves has been studied in depth.

[edit] Related Studies

• Benedict H. Gross and David E. Rohrlich., 1978. “Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve,” Inventiones Mathematicae, volume 44: pp. 201-224: Open Access Copy