We give new arguments that improve the known upper bounds on the maximal number $N_q\left(g\right)$ of rational points of a curve of genus $g$ over a finite field ${𝔽}_q$, for a number of pairs $(q,g)$. Given a pair $(q,g)$ and an integer $N$, we determine the possible zeta functions of genus-$g$ curves over ${𝔽}_q$ with $N$ points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-$g$ curve over ${𝔽}_q$ with $N$ points must have a low-degree map to another curve over ${𝔽}_q$, and often this is enough to...