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

arXiv2210.14866MaRDI QIDQ6508119

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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 $$-sheaves, for $elleqp$ 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.

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