we compare the absolute and relative de rham–witt complexes considered by hesselholt and madsen (ann. sci. école norm. sup. 37 (2004), 1–43; ann. of math. (2) 158 (2003), 1–113) and by langer and zink (j. inst. math. jussieu 3 (2004), 231–314), which both generalize the classical de rham–witt complex of bloch, deligne, and illusie (ann. sci. école norm. sup. (4) 12 (1979), 501–661) from $\mathbb{f}_p$-schemes to $\mathbb{z}_{(p)}$-schemes. from this comparison, we derive a gauss–manin connection on the crystalline cohomology of x/wn(s) for a smooth family x/s.

We determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension $E/F$ of $p$-adic fields of characteristic zero. We study the case where the Levi component $M\simeq \text{G}{{\text{L}}_{n}}\left( E \right)\times {{U}_{m}}\left( F \right)$, with $n\,\equiv \,m\,\left( \bmod \,2 \right)$. This, along with earlier work, determines the poles of the local Rankin-Selberg product $L$-function $L\left( s,\,{\tau }'\,\times \,\tau \right)$, with ${\tau }'$ an irreducible unitary supercuspidal representation of $\text{G}{{\text{L}}_{n}}\left( E \right)$ and $\tau $ a generic irreducible unitary supercuspidal representation of ${{U}_{m}}\left( F \right)$. The results are interpreted using the theory of twisted endoscopy.