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
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 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 curves has been studied in depth.
[edit] Fermat varieties
Fermat-style equations in more variables define as projective varieties the Fermat varieties.
[edit] Related studies
- Gross, Benedict H.; Rohrlich, David E. (1978), "Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve", Inventiones Mathematicae 44: 201–224, doi:, http://modular.ucsd.edu/scans/papers/gross/Gross-Rohrlich-Some_results_on_the_Mordell-Weil_groups_of_the_Jacobian_of_the_Fermat_Curve.pdf.
