ag.algebraic geometry – What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman’s book? Answer

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ag.algebraic geometry – What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman’s book?

  1. let $E_1$ other $E_2$ be Elliptic curves over the field $K$ other $l\neq\text{char}(K)$ be a prime number. let $T_l(E_i)$ is the Tate module of $E_i$, $i=1.2$. Then the natural map
    $$\mathrm{Hom}_K(E_1,E_2)\otimes\mathbb{Z}_l\longrightarrow\mathrm{Hom}_K(T_l(E_1),T_l(E_2))$$ is an isomorphism if:

i) $K$ is a finite field.

ii) $K$ is a number field.

  1. let $K$ be a number field and $E/K$ be an elliptic curve without complex multiplication. let $$\rho_l:G_{\bar{K}/K}\longrightarrow\mathrm{Aut}(T_l(E))$$ be the $l$-adic representation of $G_{\bar{K}/K}$ associated to $E$. Then:

i) $\rho_l(G_{\bar{K}/K})$ is of finite index in $\mathrm{Aut}(T_l(E))$ for all primes $l\neq\text{char}(K)$.

ii)$\rho_l(G_{\bar{K}/K})=\mathrm{Aut}(T_l(E))$ for all but finitely many primes $l$.

I have recently started studying about Elliptic curves from the book of Silverman (The Arithmetic of Elliptic Curves) and I am a proper beginner to the theory of Elliptic curves so I am looking for the proof of 1 I checked out the cited paper of Tate in the book which further directs me to an article which is written in German so I couldn’t read up anything there. And, for 2 I couldn’t really understand the proof from the source cited in the book.

So, if anyone could explain me the idea behind the proofs of these two theorems or maybe just how to visualize these two theorems geometrically or algebraically I would really appreciate it.

PS I am not looking for a proper proof for either of this theorem.

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