Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants

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 André-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic.