Drinfeld's lemma for $F$-isocrystals, I

TL;DR

This paper proves a factorization property of absolutely irreducible $F$-isocrystals on products of schemes over perfect fields of characteristic $p$, extending Drinfeld's lemma to the $p$-adic setting and its applications.

Contribution

It establishes that such $F$-isocrystals decompose as external products, generalizing Drinfeld's lemma to the $p$-adic context and linking to Langlands program techniques.

Findings

Proves irreducible $F$-isocrystals are external products on scheme products.

Extends Drinfeld's lemma from $ar{Q}_ ext{ell}$-sheaves to $F$-isocrystals.

Sets groundwork for $p$-adic Langlands correspondence applications.

Abstract

We prove that in either the convergent or overconvergent setting, an absolutely irreducible $F$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $p$, further equipped with actions of the partial Frobenius maps, is an external product of $F$-isocrystals over the multiplicands. The corresponding statement for lisse $\bar{\mathbb{Q}}_\ell$-sheaves, for $\ell \neq p$ a prime, is a consequence of Drinfeld's lemma on the fundamental groups of absolute products of schemes in characteristic $p$. The latter plays a key role in V. Lafforgue's approach to the Langlands correspondence for reductive groups with $\ell$-adic coefficients; the $p$-adic analogue will be considered in subsequent work with Daxin Xu.