Return to Question

Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.

Consider the $p$-power torsion points and adjoin them with $K$.

Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-torsion points of $E_i,~ i=1,2$, where $n$ some positive integers.

Denote the extensions $K_n=K(E_1[p^n])$ and $K_n'=K(E_2[p^n])$.

Consider the corresponding Galois extensions $G_1=\text{Gal}(K_n/K)$ and $G_2=\text{Gal}(K_n'/K)$.

More generally, If further assume $\Lambda=\bigcup_{p,n} E_1[p^n]$ and $\Lambda'=\bigcup_{p,n} E_2[p^n]$, then consider $G_1=\text{Gal}(K(\Lambda)/K)$ and $G_2=\text{Gal}(K(\Lambda')/K)$.

My question-

If $n \to \infty$, does the Galois groups $G_1$ and $G_2$ are isomorphic under some suitable assumptions ?

Does $G_1$ and $G_2$ are map-able by a good map under some suitable assumptions when $n \to \infty$?

Any intuitive idea, discussions are appreciated. Thanks

Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.

Consider the $p$-power torsion points and adjoin them with $K$.

Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-torsion points of $E_i,~ i=1,2$, where $n$ some positive integers.

Denote the extensions $K_n=K(E_1[p^n])$ and $K_n'=K(E_2[p^n])$.

Case I: Consider the corresponding Galois extensions $G_1=\text{Gal}(K_n/K)$ and $G_2=\text{Gal}(K_n'/K)$.

Case II More generally, If further assume $\Lambda=\bigcup_{p,n} E_1[p^n]$ and $\Lambda'=\bigcup_{p,n} E_2[p^n]$, then consider $G_1=\text{Gal}(K(\Lambda)/K)$ and $G_2=\text{Gal}(K(\Lambda')/K)$.

My question-

For both above two cases,

If $n \to \infty$, does the Galois groups $G_1$ and $G_2$ are isomorphic under some suitable assumptions ?

Does $G_1$ and $G_2$ are map-able by a good map under some suitable assumptions when $n \to \infty$?

Any intuitive idea, discussions are appreciated. Thanks

Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.

Consider the $p$-power torsion points and adjoin them with $K$.

Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-torsion points of $E_i,~ i=1,2$, where $n$ some positive integers.

Denote the extensions $K_n=K(E_1[p^n])$ and $K_n'=K(E_2[p^n])$.

Consider the corresponding Galois extensions $G_1=\text{Gal}(K_n/K)$ and $G_2=\text{Gal}(K_n'/K)$.

My question-

If $n \to \infty$, does the Galois groups $G_1$ and $G_2$ are isomorphic under some suitable assumptions ?

Does $G_1$ and $G_2$ are map-able by a good map under some suitable assumptions when $n \to \infty$?

Edit: If further assume $\Lambda=\bigcup_{p,n} E_1[p^n]$ and $\Lambda'=\bigcup_{p,n} E_2[p^n]$, then consider $G_1=\text{Gal}(K(\Lambda)/K)$ and $G_2=\text{Gal}(K(\Lambda')/K)$.

Then if I ask the same above question.

Any intuitive idea, discussions are appreciated. Thanks

Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.

Consider the $p$-power torsion points and adjoin them with $K$.

Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-torsion points of $E_i,~ i=1,2$, where $n$ some positive integers.

Denote the extensions $K_n=K(E_1[p^n])$ and $K_n'=K(E_2[p^n])$.

Consider the corresponding Galois extensions $G_1=\text{Gal}(K_n/K)$ and $G_2=\text{Gal}(K_n'/K)$.

More generally, If further assume $\Lambda=\bigcup_{p,n} E_1[p^n]$ and $\Lambda'=\bigcup_{p,n} E_2[p^n]$, then consider $G_1=\text{Gal}(K(\Lambda)/K)$ and $G_2=\text{Gal}(K(\Lambda')/K)$.

My question-

If $n \to \infty$, does the Galois groups $G_1$ and $G_2$ are isomorphic under some suitable assumptions ?

Does $G_1$ and $G_2$ are map-able by a good map under some suitable assumptions when $n \to \infty$?

Any intuitive idea, discussions are appreciated. Thanks

Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \subseteq \mathbb{Q}_p$.

Consider the $p$-power torsion points and adjoin them with $K$.

Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-torsion points of $E_i,~ i=1,2$, where $n$ some positive integers.

Denote the extensions $K_n=K(E_1[p^n])$ and $K_n'=K(E_2[p^n])$.

Consider the corresponding Galois extensions $G_1=\text{Gal}(K_n/K)$ and $G_2=\text{Gal}(K_n'/K)$.

My question-

If $n \to \infty$, does the Galois groups $G_1$ and $G_2$ are isomorphic under some suitable assumptions ?

Does $G_1$ and $G_2$ are map-able by a good map under some suitable assumptions when $n \to \infty$?

Any intuitive idea, discussions are appreciated. Thanks

Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.

Consider the $p$-power torsion points and adjoin them with $K$.

Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-torsion points of $E_i,~ i=1,2$, where $n$ some positive integers.

Denote the extensions $K_n=K(E_1[p^n])$ and $K_n'=K(E_2[p^n])$.

Consider the corresponding Galois extensions $G_1=\text{Gal}(K_n/K)$ and $G_2=\text{Gal}(K_n'/K)$.

My question-

If $n \to \infty$, does the Galois groups $G_1$ and $G_2$ are isomorphic under some suitable assumptions ?

Does $G_1$ and $G_2$ are map-able by a good map under some suitable assumptions when $n \to \infty$?

Edit: If further assume $\Lambda=\bigcup_{p,n} E_1[p^n]$ and $\Lambda'=\bigcup_{p,n} E_2[p^n]$, then consider $G_1=\text{Gal}(K(\Lambda)/K)$ and $G_2=\text{Gal}(K(\Lambda')/K)$.

Then if I ask the same above question.

Any intuitive idea, discussions are appreciated. Thanks