Slopes and weights of $\ell$-adic cohomology of rigid spaces

We prove that Frobenius eigenvalues of $\ell$-adic cohomology and $\ell$-adic intersection cohomology of rigid spaces over $p$-adic local fields are algebraic integers and we give bounds for their $p$-adic valuations.

As an application, we deduce bounds for their weights, proving conjectures of Bhatt, Hansen, and Zavyalov.

We also give examples of monodromy-pure perverse sheaves on projective curves with non monodromy-pure cohomology, answering a question of Hansen and Zavyalov.