  Hindry has proposed an analogue of the classical Brauer-Siegel theorem for
abelian varieties over global fields. Roughly speaking, it says that the
product of the regulator of the Mordell-Weil group and the order of the
Tate-Shafarevich group should have size similar to the exponential differential
height. Hindry-Pacheco and Griffon have proved this for certain families of
elliptic curves over function fields using analytic techniques. Our goal in
this work is to prove similar results by more algebraic arguments, namely by a
direct approach to the Tate-Shafarevich group and the regulator. We recover the
results of Hindry-Pacheco and Griffon and extend them to new families,
including families of higher-dimensional abelian varieties.
