Is there a substitution which transforms every Fermat curve into an elliptic curve?
$begingroup$
A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,zin mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two different elliptic curves. To transform the Fermat Curve of degree $3$, the substitutions
$$ a=frac{12z}{x+y},quad b=frac{36(x-y)}{x+y} $$
produce $b^2=a^3-432$, an elliptic curve. Similarly for the Fermat Curve of degree $4$, the substitutions
$$ a=frac{2(y^2+z^2)}{x^2},quad b=frac{4y(y^2+z^2)}{x^3} $$
give $b^2=a^3-4a$. However, the substitutions used are not at all obvious, which leads me to wonder,
Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?
How can we even begin to prove this? I suspect the proof or disproof of this statement will be way above my level; I truly have no clue where to begin. Can someone help out? If this is somehow an open problem, then any links to literature is also appreciated!
Edit: This question has been asked and answered on MO. (Yay!)
number-theory algebraic-geometry elliptic-curves algebraic-curves substitution
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
$endgroup$
add a comment |
$begingroup$
A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,zin mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two different elliptic curves. To transform the Fermat Curve of degree $3$, the substitutions
$$ a=frac{12z}{x+y},quad b=frac{36(x-y)}{x+y} $$
produce $b^2=a^3-432$, an elliptic curve. Similarly for the Fermat Curve of degree $4$, the substitutions
$$ a=frac{2(y^2+z^2)}{x^2},quad b=frac{4y(y^2+z^2)}{x^3} $$
give $b^2=a^3-4a$. However, the substitutions used are not at all obvious, which leads me to wonder,
Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?
How can we even begin to prove this? I suspect the proof or disproof of this statement will be way above my level; I truly have no clue where to begin. Can someone help out? If this is somehow an open problem, then any links to literature is also appreciated!
Edit: This question has been asked and answered on MO. (Yay!)
number-theory algebraic-geometry elliptic-curves algebraic-curves substitution
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
$endgroup$
2
$begingroup$
In the language of algebraic geometry, you're asking whether every Fermat curve has a nonconstant morphism to an elliptic curve.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 1:31
add a comment |
$begingroup$
A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,zin mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two different elliptic curves. To transform the Fermat Curve of degree $3$, the substitutions
$$ a=frac{12z}{x+y},quad b=frac{36(x-y)}{x+y} $$
produce $b^2=a^3-432$, an elliptic curve. Similarly for the Fermat Curve of degree $4$, the substitutions
$$ a=frac{2(y^2+z^2)}{x^2},quad b=frac{4y(y^2+z^2)}{x^3} $$
give $b^2=a^3-4a$. However, the substitutions used are not at all obvious, which leads me to wonder,
Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?
How can we even begin to prove this? I suspect the proof or disproof of this statement will be way above my level; I truly have no clue where to begin. Can someone help out? If this is somehow an open problem, then any links to literature is also appreciated!
Edit: This question has been asked and answered on MO. (Yay!)
number-theory algebraic-geometry elliptic-curves algebraic-curves substitution
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
$endgroup$
A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,zin mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two different elliptic curves. To transform the Fermat Curve of degree $3$, the substitutions
$$ a=frac{12z}{x+y},quad b=frac{36(x-y)}{x+y} $$
produce $b^2=a^3-432$, an elliptic curve. Similarly for the Fermat Curve of degree $4$, the substitutions
$$ a=frac{2(y^2+z^2)}{x^2},quad b=frac{4y(y^2+z^2)}{x^3} $$
give $b^2=a^3-4a$. However, the substitutions used are not at all obvious, which leads me to wonder,
Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?
How can we even begin to prove this? I suspect the proof or disproof of this statement will be way above my level; I truly have no clue where to begin. Can someone help out? If this is somehow an open problem, then any links to literature is also appreciated!
Edit: This question has been asked and answered on MO. (Yay!)
number-theory algebraic-geometry elliptic-curves algebraic-curves substitution
number-theory algebraic-geometry elliptic-curves algebraic-curves substitution
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
edited Dec 21 '18 at 9:50
YiFan
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
asked Dec 17 '18 at 0:12
YiFanYiFan
3,7511527
3,7511527
2
$begingroup$
In the language of algebraic geometry, you're asking whether every Fermat curve has a nonconstant morphism to an elliptic curve.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 1:31
add a comment |
2
$begingroup$
In the language of algebraic geometry, you're asking whether every Fermat curve has a nonconstant morphism to an elliptic curve.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 1:31
2
2
$begingroup$
In the language of algebraic geometry, you're asking whether every Fermat curve has a nonconstant morphism to an elliptic curve.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 1:31
$begingroup$
In the language of algebraic geometry, you're asking whether every Fermat curve has a nonconstant morphism to an elliptic curve.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 1:31
add a comment |
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$begingroup$
In the language of algebraic geometry, you're asking whether every Fermat curve has a nonconstant morphism to an elliptic curve.
$endgroup$
– Eric Wofsey
Dec 17 '18 at 1:31