Counting rational points on elliptic and hyperelliptic curves over function fields

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We deduce an upper bound on the number of $S$-integral points, where $S$ is a finite set of places. As a primary application, over small finite fields we bound the $3$-torsion of Jacobians of hyperelliptic curves and the $2$-torsion of Jacobians of trigonal curves. In this setting, these bounds improve on both the trivial geometric bound and the naive inequality coming from the Weil bound, as well as recent upper bounds on $2$-torsion in the work of Bhargava et al.