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We will discuss $\infty$-categorical perverse $p$-adic differential equationsover stacks. On one hand, we are going to study some $p$-adic analogous resultsof the Drinfeld's original lemma about the \'etale fundamental groups in the\'etale setting, in the context of $F$-isocrystals closely after Kedlaya andKedlaya-Xu. We expect similar things could also be considered for diamondsafter Scholze, in the context of Kedlaya-Liu's work namely the derived categoryof pseudocoherent Frobenius sheaves, which will induce some categorical form ofDrinfeld's lemma for diamonds motivated by work of Carter-Kedlaya-Z\'abr\'adiand Pal-Z\'abr\'adi. On the other hand, we are going to establish the$\infty$-categorical theory of arithmetic $D$-modules after Abe andGaitsgory-Lurie, which will allow one to construct the rigid Gross $G$-motives.And we are expecting to apply the whole machinery to revisit Weil's conjectureparallel to and after Gaitsgory-Lurie.

$\infty$-Categorical Perverse $p$-adic Differential Equations over Stacks