Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\mathcal{D}$-modules

Abstract

The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic $\mathcal{D}$-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of SGA4. As another application, we compare the push-forward as arithmetic $\mathcal{D}$-modules and the rigid cohomologies taking Frobenius into account. These theorems will be used to prove "$p$-adic Weil II" and a product formula for $p$-adic epsilon factors.