Two lectures of upper bounds. Reduction gave $\rho \leq \rho(X_p^{al})$, and the obstruction sharpened this upper bound on the rank.
The rank alone does not determine the Picard lattice: it says nothing about the intersection form, nothing about the Galois action, and exhibits no curve.
Today: produce the classes. A lower bound is a curve you can write down, and enough curves with their intersections is the lattice itself.