This is probably a very trivial question, but I haven't been able to find a rigorous explanation anywhere so far or at least haven't understood it.
Assume we have an elliptic curve $E$ over $\mathbb{F}_p$ where $p$ is a prime. Then we often talk about $\text{End}(E)$, the ring of endomorphisms of the curve $E$. The thing I'm confused about is the following. The claim generally is that $\text{End}(E)\cong \mathbb{Z}+\delta\mathbb{Z}$ where $\delta$ is a complex root of the discriminant of a complex quadratic order. So far so good.
Now the thing that confuses me is what is the $\text{End}(E)$ the endomorphism ring of? It can't be the endomorphism ring of the curve $E(\mathbb{F}_p)$ since that is a finite group and it's endomorphism ring must thus also be finite (by a combinatorial argument). My guess is it's the endomorphism ring of $E(\bar{\mathbb{F}_p})$ i.e. $E$ interpreted over the algebraic closure. But then there are places which specifically deal with $\text{End}_{\bar{\mathbb{F}_p}}(E)$ distinguishing it from $\text{End}(E)$.
My other guess was that it might be the endomorphisms of $E(\bar{\mathbb{{F}_p}})$ which are defined over $\mathbb{F}_p$ as morphisms (i.e. (slightly restricted) rational functions if I understand it correctly). The second interpretation is the one I'm leaning towards the most but it would be nice to get confirmation or explanation what I'm missing. Thank you.=
Edit After the discussion with Ferra I've determined that at least part of my issue is in not correctly understanding the abuse of notation used when we say an elliptic curve $E$ (defined over $\mathbb{F}_p$). Without a given base field I'm not sure what this is supposed to mean. In other words, when we write $\text{End}(E)$ which $E$ is meant? Are we in some way thinking of the homogenous polynomial defining $E$? Or are we thinking of the projective variety over the field $E(\overline{\mathbb{F}_p})$? Or are we thinking of the group $(E(\mathbb{F}_p),+,0)$ or $(E(\overline{\mathbb{F}_p}),+,0)$? Or are we thinking of both the group structure and the projective variety structure? I would think the last though I'm still not sure if we have the closure field or the base field. I'm also not sure whether there are group endomorphisms which are not algebraic curve morphisms.
Your are making some confusion between morphisms of groups and morphisms of curves. A morphism of groups $A,B$ is a map $A\to B$ which respects the operations, a morphism of (algebraic) curves $X,Y$ is a map $X\to Y$ whose components can be written as polynomials.
An isogeny of elliptic curves $E,E'$ over a field $k$ is a morphism of algebraic curves $E\to E'$ which is also a morphism of groups $E(\overline{k})\to E'(\overline{k})$.
The set $\text{End}(E)$ is defined as the set of isogenies $E\to E$. Note that if $E$ is defined over $\mathbb F_p$, we are not requiring all elements of $\text{End}(E)$ to be defined over $\mathbb F_p$, but only over $\overline{\mathbb F_p}$. The set of isogenies $E\to E$ defined over $\mathbb F_p$ is (usually, but not always) denoted by $\text{End}_{\mathbb F_p}(E)$. Whenever $\text{End}(E)\simeq \mathbb Z+\delta\mathbb Z$, the two sets coincide because the Frobenius map, which corresponds to $\delta$ in the latter isomorphism, is defined over $\mathbb F_p$. Note that the Frobenius map, which is the one sending $(x,y)\mapsto (x^p,y^p)$ is both a morphism of curves, because its components are polynomials and the coefficients of $E,E'$ lie in $\mathbb F_p$, and a morphism of groups $E(\overline{\mathbb F_p}) \to E(\overline{\mathbb F_p})$ because the addition map $E\times E\to E$ is defined over $\mathbb F_p$.
Note that there is a map $\text{End(E)}\to \text{End}(E(\overline{\mathbb F_p}))$ which sends every isogeny to the induced map on the group of $\overline{\mathbb F_p}$-rational points of $E$. This map is injective, but far from being surjective, since on one hand isogenies have finite kernel while on the other side $E(\overline{\mathbb F_p})$ is an infinite abelian group, so it has many endomorphisms with infinite kernel.
Analogously, if $E$ is defined over $\mathbb F_p$ then there is a map $\text{End(E)}\to \text{End}(E(\mathbb F_p))$ which is not injective since the LHS is infinite while the RHS is not.