Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

Let $G$ be a semi-simple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve{G}$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$ of genus at least two. Denote by $\operatorname{Bun}_{G}$ the moduli stack of $G$-bundles on $C$ and $\operatorname{LocSys}_{\breve{G}}$ the moduli stack of $\breve{G}$-local systems on $C$. Let $D_{\operatorname{Bun}_{G}}$ be the sheaf of crystalline differential operators on $\operatorname{Bun}_{G}$. In this paper we construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}(\operatorname{LocSys}_{\breve{G}}^{0}))$ of quasi-coherent sheaves on some open subset $\operatorname{LocSys}_{\breve{G}}^{0}\subset \operatorname{LocSys}_{\breve{G}}$ and bounded derived category $D^{b}(D_{\operatorname{Bun}_{G}}^{0}\text{-}\text{mod})$ of modules over some localization $D_{\operatorname{Bun}_{G}}^{0}$ of $D_{\operatorname{Bun}_{G}}$. This generalizes the work of Bezrukavnikov and Braverman in the $\operatorname{GL}_{n}$ case.