  It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an
abelian variety (with trivial trace) defined over the function field of a curve
is bounded by a quantity which depends on the genus of the base curve and on
bad reduction data. Using a function field version of classical $\ell$-descent
techniques, we derive an arithmetic refinement of this bound, extending
previous work of the second and third authors from elliptic curves to abelian
varieties, and improving on their result in the case of elliptic curves. When
the abelian variety is the Jacobian of a hyperelliptic curve, we produce a more
explicit $2$-descent map. Then we apply this machinery to studying points on
the Jacobians of certain genus $2$ curves over $k(t)$, where $k$ is some
perfect base field of characteristic not $2$.
