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

TL;DR

This paper develops an $ abla$-categorical framework for $p$-adic differential equations over stacks, extending foundational results and conjectures in arithmetic geometry and motives using advanced categorical and $p$-adic techniques.

Contribution

It introduces an $ abla$-categorical approach to $p$-adic differential equations over stacks, extending Drinfeld's lemma and developing the theory of arithmetic $D$-modules in this context.

Findings

Establishes $ abla$-categorical analogues of Drinfeld's lemma for stacks and diamonds.

Constructs the rigid Gross $G$-motives within the $ abla$-categorical framework.

Provides new tools to approach Weil's conjecture using $ abla$-categorical $p$-adic methods.

Abstract

We will discuss $\infty$-categorical perverse $p$-adic differential equations over stacks. On one hand, we are going to study some $p$-adic analogous results of the Drinfeld's original lemma about the \'etale fundamental groups in the \'etale setting, in the context of $F$-isocrystals closely after Kedlaya and Kedlaya-Xu. We expect similar things could also be considered for diamonds after Scholze, in the context of Kedlaya-Liu's work namely the derived category of pseudocoherent Frobenius sheaves, which will induce some categorical form of Drinfeld's lemma for diamonds motivated by work of Carter-Kedlaya-Z\'abr\'adi and Pal-Z\'abr\'adi. On the other hand, we are going to establish the $\infty$-categorical theory of arithmetic $D$-modules after Abe and Gaitsgory-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 conjecture parallel to and after Gaitsgory-Lurie.