Fermat curve

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With respect to the quartic Fermat curve x 4 + y 4 = 1 {\displaystyle x^{4}+y^{4}=1} {\displaystyle x^{4}+y^{4}=1}, the hyperbolic lemniscate sine is analogous to the trigonometric tangent function. Unlike slh {\displaystyle \operatorname {slh} } {\displaystyle \operatorname {slh} } and clh {\displaystyle \operatorname {clh} } {\displaystyle \operatorname {clh} }, the functions sin 4 {\displaystyle \sin _{4}} {\displaystyle \sin _{4}} and cos 4 {\displaystyle \cos _{4}} {\displaystyle \cos _{4}} cannot be analytically extended to meromorphic functions in the whole complex plane.[1]

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:

X n + Y n = Z n .   {\displaystyle X^{n}+Y^{n}=Z^{n}.\ } {\displaystyle X^{n}+Y^{n}=Z^{n}.\ }

Therefore, in terms of the affine plane its equation is:

x n + y n = 1.   {\displaystyle x^{n}+y^{n}=1.\ } {\displaystyle x^{n}+y^{n}=1.\ }

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:

( n − 1 ) ( n − 2 ) / 2.   {\displaystyle (n-1)(n-2)/2.\ } {\displaystyle (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 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:

n − 1.   {\displaystyle n-1.\ } {\displaystyle n-1.\ }

Fermat varieties

Fermat-style equations in more variables define as projective varieties the Fermat varieties.

References

  1. Levin (2006) p. 515