# Fermat curve

## From Wikipedia, the free encyclopedia

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

- Baker, Matthew; Gonzalez-Jimenez, Enrique; Gonzalez, Josep; Poonen, Bjorn (2005), "Finiteness results for modular curves of genus at least 2",
*American Journal of Mathematics*,**127**(6): 1325–1387, arXiv:math/0211394, doi:10.1353/ajm.2005.0037, JSTOR 40068023 - Gross, Benedict H.; Rohrlich, David E. (1978), "Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve" (PDF),
*Inventiones Mathematicae*,**44**(3): 201–224, doi:10.1007/BF01403161, archived from the original (PDF) on 2011-07-13 - Klassen, Matthew J.; Debarre, Olivier (1994), "Points of Low Degree on Smooth Plane Curves",
*Journal für die reine und angewandte Mathematik*,**1994**(446): 81–88, doi:10.1515/crll.1994.446.81</ref> - Tzermias, Pavlos (2004), "Low-Degree Points on Hurwitz-Klein Curves",
*Transactions of the American Mathematical Society*,**356**(3): 939–951, doi:10.1090/S0002-9947-03-03454-8, JSTOR 1195002

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