Abelian Varieties over C vs. Abelian Varieties over C_p

Question

Suppose $A$ is a complex abelian variety. Then $A$ is a complex torus $\mathbb C^g/\Lambda$ where $\Lambda$ is a lattice. On the other hand abelian varieties over $\mathbb C_p$ can have good reduction. But $\mathbb C$ and $\mathbb C_p$ are isomorphic... so, shouldn't abelian varieties over $\mathbb C_p$ be of the form $\mathbb C_p^g/\Lambda$ where $\Lambda$ is now a lattice in $\mathbb C_p$? But then abelian varieties over $\mathbb C_p$ with good reduction wouldn't exist...

Answer 1

Probably the reason is that a lattice in $\mathbb C$ (i.e. a discrete subgroup of $\mathbb C$) is not a discrete subgroup of $\mathbb C_p$ (due to the fact that the norms of the two fields are different!)