Rigid cohomology over Laurent series fields III: Absolute coefficients and arithmetic applications

In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and Frobenius structures, and therefore are naturally $(\varphi,\nabla)$-modules over $\mathcal{E}_K^\dagger$ whenever they are finite dimensional. We also introduce a category of `absolute' coefficients for the theory; the same results are true for cohomology groups with coefficients. We moreover prove a $p$-adic version of the weight monodromy conjecture for smooth (not necessarily proper) curves, and use a construction of Marmora to prove a version of $\ell$-independence for smooth curves over $k(\!(t)\!)$ that includes the case $\ell=p$. This states that after tensoring with $\mathcal{R}_K$, our $p$-adic cohomology groups agree with the $\ell$-adic Galois representations $H^i_{\mathrm{\'{e}t}}(X_{k(\!(t)\!)^\mathrm{sep}},\mathbb{Q}_\ell)$ for $\ell\neq p$.