The classical modular curve $\Phi_n(X,Y) \in \mathbb{Z}[X,Y]$ for $n \in \mathbb{Z}_{\geq 2}$ relates the $j$-invariants of elliptic curves $E_1$ and $E_2$ defined over $\mathbb{C}$ in the sense that if there is an isogeny $f: E_1 \rightarrow E_2$ of degree $n$ then $\Phi_n(j(E_1), j(E_2))=0$.

My first question is that is there a version of these classical modular curves for all $n$ over characteristic $p$? As I know if $p$ does not divide $n$, then $\Phi_n(X,Y)$ (mod $p$) can be used for finite fields (and for their algebraic extensions), but I don't know if $\Phi_p(X,Y)$ (mod $p$) satisfies this relation.

My second question is that if my first question has a positive answer, i.e. we have classical modular curves for all $n$ which works for algebraic extensions of $\mathbb{F}_p$, does this modular curve gives a similar relation for any elliptic curves defined over some function field, e.g. $\mathbb{F}_p(t)$.

Thanks...

cyclic$n$-isogenies); it isn't the classical modular curve, asmoothaffine curve over $\mathbf{C}$ (or $\mathbf{Q}$ or $\mathbf{Z}[1/n]$). The correct definition, hard to describe explicitly, is a regular flat affine scheme $Y$ over $\mathbf{Z}$. The key is that the morphism $Y \rightarrow \mathbf{A}^2_{\mathbf{Z}}$ induced by $(E \rightarrow E') \mapsto (j(E),j(E'))$ isfinite, so surjective onto its schematic image, which (by $\mathbf{Z}$-flatness!) is $\Phi_n=0$. In particular, the answer to both questions is affirmative. $\endgroup$