Let $A_{1,1}$ be the j-line for elliptic curves over $\mathbb{C}$, $A_{1,1}\otimes \mathbb{F}$ is the mod p reduction(here $\mathbb{F}$ is the algebraic closure of a finite field), then can I say the point at infinity of the mod p j-line corresponds to those elliptic curves whose discriminant is divisible by p?
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$\begingroup$ Strictly speaking, you can say whatever you want. :) More seriously, you ask if under some "moduli interpretation" of projective $j$-line, the point at infinity should correspond to non-smooth Weierstrass cubics (not elliptic curves, as those are smooth by definition); note this makes sense in any char. It corresponds to the nodal cubic, not the cuspidal one (over alg. closed field). As for why, one has to first give a real (i.e., not ad hoc) definition of "moduli space" in this context. That is a long story, alas. A hint as to the distinction of nodal vs. cubic is the semistable red. thm. $\endgroup$– BCnrdCommented Sep 14, 2010 at 3:26
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No, you have to be careful with additive reduction. Elliptic curves may have "potential good reduction". In some sense (which takes some work to be made precise) the point at infinity on the mod p j-line correspond to elliptic curves which have multiplicative bad reduction at p together with a nodal cubic. You can parametrize a p-adic neighboorhood of infinity in char zero corresponding to the curves that reduce to infinity mod p using the Tate parameter q. Then for $0 < |q| < 1$ you get Tate curves and for $q=0$ you get the nodal curve. Silverman is, as usual, a good reference.
answered Sep 14, 2010 at 2:16