$\mathrm{\ell }$-Independence, slopes, and weights of $\mathrm{\ell }$-adic cohomology groups of rigid spaces

Let $\mathrm{\ell }$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $K=0$, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on $R\mathrm{\Gamma }(X_C,{\mathbb{Q}}_{\mathrm{\ell }})$ is independent of $\mathrm{\ell }$ and is an integer, extending a result of Mieda to the singular case. We also give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i(X_C,{\mathbb{Q}}_{\mathrm{\ell }})$, proving a conjecture of Lu–Zheng. In the general case, we give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H_c^i(X_C,{\mathbb{Q}}_{\mathrm{\ell }})$, and we show that these eigenvalues are algebraic integers and Weil numbers, thereby extending results of Lu–Zheng and Mieda to the positive characteristic case.