Drinfeld's lemma for $F$-isocrystals, II: Tannakian approach

TL;DR

This paper establishes a Tannakian framework for Drinfeld's lemma concerning isocrystals with Frobenius actions, advancing the transfer of Langlands correspondence techniques from $\, ext{$ ext{l}$}$-adic to $p$-adic contexts.

Contribution

It introduces a Tannakian approach to Drinfeld's lemma for isocrystals, providing a key step towards $p$-adic Langlands correspondence transfer.

Findings

Proves a Tannakian form of Drinfeld's lemma for isocrystals.

Discusses motivic and local variants of the lemma.

Facilitates transfer of Langlands correspondence from $\, ext{$ ext{l}$}$-adic to $p$-adic coefficients.

Abstract

We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from $\ell$-adic to $p$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.