\indent i will discuss the problem of efficiently constructing curves of genus 1 and 2 over finite fields with a prescribed number n of points. in both cases, there are algorithms that, at least heuristically and in practice, run in time polynomial in log n. they are of complex analytic nature, using cm-techniques. time permitting, i will also explain why these techniques are provably insufficient to efficiently construct genus-2 jacobians with a presecribed number of points.