1
$\begingroup$

I am not entirely sure this question is research-level question, but I have tried stack exchange and have received no response there.

During a conversation with a professor, I was informed of the following statement: $x^4+kx^2y^2+y^4=z^2$ parametrizes all elliptic curves and fixes all the 4-torsion points. Here by fixing all the 4-torsion points I believe he meant that for any $t$, the projection $E\rightarrow \mathbb{P}^1$ sends any 4-torsion point on $E$ to one of $\{0,1,-1,i,-i,\infty\}$. I wonder if this is true, and is there any reference or proof of this.

I have thought about the following verification: by $X=\frac{x^2}{y^2}$ and $Y=\frac{zx}{y^3}$ one gets $X^3+kX^2+X=Y^2$, and solve the 4-torsion condition via explicit computations. But firstly, as professor Elkies mentioned here, this is not an isomorphism but a 2-isogeny, I am not sure if this will affect the verification. And secondly, I think there might be some insights in this other than just computations.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.