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

Abstract

In this paper we investigate the arithmetic aspects of the theory of ${\mathcal{E}}_K^{†}$-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 $(\phi ,\nabla )$-modules over ${\mathcal{E}}_K^{†}$ 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 $H_{\acute{\mathrm{e}}\mathrm{t}}^i(X_{k((t){)}^{\mathrm{sep}}},{\mathbb{Q}}_{\ell })$ for $\ell \ne p$.

29 pages, comments very welcome!

Cited by in corpus (1)

• Rigid cohomology over Laurent series fields II: Finiteness and Poincaré duality for smooth curves