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

Abstract

Let $\ell$ 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 $\mathop{\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on $R\Gamma(X_C,\mathbb{Q}_{\ell})$ is independent of $\ell$ 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 $H^i(X_C,\mathbb{Q}_{\ell})$, 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 $H^i_c(X_C,\mathbb{Q}_{\ell})$, 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.