Fermat curve
{{Short description|Algebraic 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 > 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 |title=Finiteness results for modular curves of genus at least 2 |first1=Matthew |last1=Baker |first2=Enrique |last2=Gonzalez-Jimenez |first3=Josep |last3=Gonzalez |first4=Bjorn |last4=Poonen |authorlink4=Bjorn Poonen |journal=American Journal of Mathematics |year=2005 |volume=127 |issue=6 |pages=1325–1387 |doi=10.1353/ajm.2005.0037 |jstor=40068023|arxiv=math/0211394 |s2cid=8578601 }}
- {{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 |url-status=dead |archive-date=2011-07-13 |doi=10.1007/BF01403161 |s2cid=121819622 }}
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- {{citation |title=Low-Degree Points on Hurwitz-Klein Curves |first=Pavlos |last=Tzermias |journal=Transactions of the American Mathematical Society
|volume=356 |issue=3 |year=2004 |pages=939–951 |doi=10.1090/S0002-9947-03-03454-8 |jstor=1195002|doi-access=free }}
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