We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovar and Scholl. This is achieved with the help of Morel's work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.
On the generic part of the cohomology of Shimura varieties of abelian type
On the étale cohomology of Hilbert modular varieties with torsion coefficients
A remark on a conjecture of Buzzard-Gee and the cohomology of Shimura varieties
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