  Our main result introduces a new way to characterize two-dimensional finite
ball quotients by algebraicity of their Bergman kernels. This characterization
is particular to dimension two and fails in higher dimensions, as is
illustrated by a counterexample in dimension three constructed in this paper.
As a corollary of our main theorem, we prove, e.g., that a smoothly bounded
strictly pseudoconvex domain G in $\mathbb{C}^2$ has rational Bergman kernel if
and only if there is a rational biholomorphism from G to the 2-dimensional unit
ball.
