An analogue of the Brauer–Siegel theorem for abelian varieties in positive characteristic

Consider a family of abelian varieties $A_i$ of fixed dimension defined over the function field of a curve over a finite field. We assume finiteness of the Shafarevich–Tate group of $A_i$. We ask then when does the product of the order of the Shafarevich–Tate group by the regulator of $A_i$ behave asymptotically like the exponential height of the abelian variety. We give examples of families of abelian varieties for which this analogue of the Brauer–Siegel theorem can be proved unconditionally, but also hint at other situations, where the behaviour is different. We also prove interesting inequalities between the degree of the conductor, the height and the number of components of the Néron model of an abelian variety.