- Fermat varieties
- Related studies
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 > 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
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
- {{citation|first1=Benedict H. |last1=Gross |first2=David E. |last2=Rohrlich |year=1978 |title=Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve |journal=Inventiones Mathematicae |volume=44 |issue=3 |pages=201–224 |url=http://www.kryakin.com/files/Invent_mat_%282_8%29/44/44_01.pdf |archive-url=https://web.archive.org/web/20110713171905/http://www.kryakin.com/files/Invent_mat_(2_8)/44/44_01.pdf |dead-url=yes |archive-date=2011-07-13 |doi=10.1007/BF01403161 }}.
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