Torsion points on elliptic curve when base field changes

Question

There are two questions related to elliptic curve

(1)Let E be an algebraic elliptic curve over k where k:algebraic closed field with characteristic zero

Let K be an any extension field of k so we can view E as an elliptic curve over K

Then is it right that $E(k)^{tor}=E(K)^{tor}?$

$E(k)[N]\subset E(K)[N]$ and $|E(k)[N]|=N^2=|E(\overline K)[N]|$ implies that E(k)[N]=E(K)[N]

So we can conclude that $E(k)^{tor}=E(K)^{tor}?$ Is it right?

(2)Let $E_1, E_2$ be algebraic elliptic curves over $\overline{\mathbb{Q}}$

Assume that $E_1, E_2$ are isomorphic over $\mathbb{C}$(View $E_1, E_2$ elliptic curve over $\mathbb{C}$)

Can we show that this morphism is actually defined over $\overline{\mathbb{Q}}$?

It looks like that this statement hold for any algebraic curve $C_1, C_2$

Thank you.

Yes $E(K)=E(k)$ this is due to the division polynomials giving solutions to each $[N]P=O$. Yes the isomorphism over $\Bbb{C}$ must be defined over $\overline{\Bbb{Q}}$. This is because it is defined over a finitely generated $\overline{\Bbb{Q}}$ algebra. If it contains a non algebraic element then it will have infinitely many homomorphisms to $\overline{\Bbb{Q}}$ giving infinitely many solutions to $[N]P = O$ in $C_2$, a contradiction. — reuns, Jan 27 '23 at 10:54

0 Answers0