Timeline for Classical modular curves over characteristic $p$.
Current License: CC BY-SA 3.0
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| Nov 22, 2012 at 6:04 | comment | added | user28172 | @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. | |
| Nov 18, 2012 at 9:55 | comment | added | A.E. | Thanks for the comments. @nosr: Do you know a reference for these? | |
| Nov 17, 2012 at 19:04 | comment | added | user28172 | 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. | |
| Nov 17, 2012 at 14:53 | comment | added | Felipe Voloch | 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. | |
| Nov 17, 2012 at 14:45 | history | asked | A.E. | CC BY-SA 3.0 |