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?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
2
|
||||
|
|
5
|
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. |
||
|
|
