I've been trying to learn about elliptic curves since high school and have taught myself some abstract algebra to understand it better. I have only used online resources and Abstract Algebra by Dummit and Foote, so I apologize if I am overlooking something very obvious.
I read the proof outlined here that $[\mathbb{Q}(j(I):\mathbb{Q}]<h_{k}$ for an ideal $I\subseteq O_{k}$ of the ring of integers of an imaginary quadratic field $K$. I follow the proof of it but am having trouble with the following concept which seems pretty fundamental. When talking about $End(E)$ of an elliptic curve, I had always assumed that this referred to any abelian group homomorphism of $E$ to $E$. I see that the structure of $End(E_{\Lambda})$ is usually determined by looking at the endomorphism ring of $\mathbb{C}/\Lambda$, which I also took just to mean any abelian group homomorphism between $\mathbb{C}/\Lambda$ and itself. The site I referenced mentioned "isogenies (of elliptic curves) and analytic maps fixing 0 (of C modulo a lattice) are the same thing." Either I am overlooking something obvious or misunderstand the difference between an isogony and a homomorphism.
I looked it up, and it appears the only extra condition an isogeny affords is that it is surjective. What I dont understand is what this condition does: I am assuming it makes the connection between $End(E_{\Lambda})$ and $End(\mathbb{C}/{\Lambda})$ "nicer", but why would this be? If I am right about that, it seems pretty ad hoc to me though I know there must be some natural reason for this requirement. How does this lead us to the condition that the corresponding map between $\mathbb{C}/\Lambda$ for both elliptic curves is analytic?
Thank you for any help!
On the $y^2=x^3+ax+b$ algebraic side an isogeny is any homomorphism given by a rational map. In particular $(x,y)\mapsto (\overline{x}, \overline{y})$ (complex conjugaison) on $y^2=x^3+1$ is not an isogeny. On the complex torus side an isogeny is any homomorphism given by a complex analytic map. In particular $z\mapsto \overline{z}$ is not an isogeny on $\Bbb{C/(Z+iZ})$. The main theorem is that the Weierstrass elliptic function gives a correspondence between the two, the map being $z\mapsto (\wp_L(z),\wp_L'(z))$.
Note that the Frobenius $(x,y)\mapsto (x^5,y^5)$ is an isogeny/endomorphism on $y^2=x^3+1$ defined over $\Bbb{F}_5$. This is what makes endomorphism rings more complicated in characteristic $p$.