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

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