Moduli of local shtukas and Harris's conjecture

Abstract

We prove, under a certain assumption of “Hodge–Newton reducibility”, a strong form of a conjecture of Harris on the cohomology of moduli spaces of mixed-characteristic local shtukas for ${GL}_n$. Our strategy is roughly based on a previous strategy developed by Mantovan in the setting of $p$-divisible groups, but the arguments are completely different. In particular, we reinterpret and generalize the Hodge–Newton filtration of a $p$-divisible group in terms of modified vector bundles on the Fargues–Fontaine curve. We also compute the dualizing complex and compactly supported étale cohomology of any positive Banach–Colmez space over any base; this should be of independent interest.