Ballini–Lombardo–Verzobio's $L^1$ Katz–Sarnak conjecture for $\mathcal A_g$

Let $g\ge 2$ be an integer. For each prime $l$, let $v_l(t)$ denote the local factor defined from the trace distribution in $\operatorname{GSp}_{2g}(\mathbb{Z}_l)$, with $v_p(t)=1$, and let $\operatorname{ST}_g$ be the Sato–Tate trace density. Ballini–Lombardo–Verzobio's $L^1$ Katz–Sarnak conjecture for $\mathcal A_g$.

$$\sum_{t\in \mathbb{Z}}\left|\frac{\#\mathcal A_g(\mathbb{F}_p,t)}{\# \mathcal A_g(\mathbb{F}_p)}-\frac{1}{\sqrt p} \operatorname{ST}_g(t/\sqrt p)\prod_{l}v_l(t)\right| \longrightarrow 0$$

as $p\rightarrow\infty$. This is the corresponding global trace-distribution prediction for principally polarized abelian varieties, with local factors correcting the naïve Katz–Sarnak model. The conjecture remains open in general, while the paper proves related results for some cases.