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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...

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  • $\begingroup$ If $p$ does not divide $n$, everything in char $p$ is very similar to char zero. Otherwise, it is not and it is very subtle. E.g. $\Phi_p(x,y) \equiv (x^p-y)(x-y^p)$ (Kronecker's congruence). The standard reference for the subject is the book of Katz and Mazur, but is not an easy read. It is not clear to me exactly what you are asking in your last paragraph. $\endgroup$
    – Felipe Voloch
    Commented Nov 17, 2012 at 14:53
  • $\begingroup$ You are using Kroncker's singular plane curve (and cyclic $n$-isogenies); it isn't the classical modular curve, a smooth affine 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 $(E \rightarrow E') \mapsto (j(E),j(E'))$ is finite, so surjective onto its schematic image, which (by $\mathbf{Z}$-flatness!) is $\Phi_n=0$. In particular, the answer to both questions is affirmative. $\endgroup$
    – user28172
    Commented Nov 17, 2012 at 19:04
  • $\begingroup$ Thanks for the comments. @nosr: Do you know a reference for these? $\endgroup$
    – A.E.
    Commented Nov 18, 2012 at 9:55
  • $\begingroup$ @A.E.: The usual reference is the book of Katz and Mazur that Voloch mentioned, around their discussion of $Y_0(n)$ and $X_0(n)$. (I made a minor misstatement above: I should have said "normal finite type" rather than "regular" -- the underlying Artin stack is regular, but the coarse spaces $X_0(n)$ and $Y_0(n)$ are generally only normal.) But as Voloch indicated, understanding that book requires a lot of technical fluency in algebraic geometry. $\endgroup$
    – user28172
    Commented Nov 22, 2012 at 6:04

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