Non-commutative Tsfasman–Vlăduţ formula

For a curve $X$ over $\mathbb{F}_q$ let the class number $h_X$ be the order of the finite group of the $\mathbb{F}_q$-points of $\rm{Pic}^0(X)$. In 90's Tsfasman and Vlăduţ proved an asymptotic formula for the growth of the class number $h_{X_i}$ in a sequence of curves $\{X_i\}$ under a restriction that the sequence is asymptotically exact (e.g. given by a tower of curves). I will tell about a natural generalization of their formula in which the class number is replaced by the stacky point-count of $G$-bundles for a given split reductive group $G$. Using a certain inversion formula, we can show that the asymptotic formula does not change if we restrict the count to the semistable locus of $\rm{Bun}_G$. Finally, we also expect that the stacky count can be replaced further by the actual number of semistable $G$-bundles, but for now we can show this only in the case of $G=\rm{GL}_n$ and with some further restrictions on $q$.