We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{ k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the Andr\'e-Oort conjecture for $Y(1)^2_\mathbb{ C}$, which is a theorem of Andr\'e. We use our result to solve the modular support problem for function fields of positive characteristic.
Categories: math.NT math.AG