# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 131, 2014, pp. 89–149**

**DOI: 10.4171/RSMUP/131-7**

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

Tomoyuki Abe^{[1]}(1) The University of Tokyo, Kashiwa, Chiba, Japan

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.

*Keywords: *$p$-adic cohomology, arithmetic $\mathcal D$-module, rigid cohomology

Abe Tomoyuki: Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\mathcal D$-modules. *Rend. Sem. Mat. Univ. Padova* 131 (2014), 89-149. doi: 10.4171/RSMUP/131-7