On \(p\)-adic monodromy (Q1191418)

Let \(f:X\to S\) be a smooth projective morphism of smooth schemes over an open part of \(\text{Spec} \mathbb{Z}\) with \(S=\text{Spec} A\) affine and \(H^ m(X,{\mathcal O}_ X)\) a free \(A\)-module for every \(m\). Section 2 of the paper recalls the construction and some properties of the sheaf \({\mathcal W}{\mathcal O}_ X\) of generalized Witt vectors on \(X\) and gives the important consequences of the above freeness hypothesis for the cohomology groups \(H^ m(X,{\mathcal W}{\mathcal O}_ X)\). There is a specialization homomorphism from \(H^ m(X,{\mathcal W}{\mathcal O}_ X)\) onto the cohomology group \(H^ m(X_ s,{\mathcal W}{\mathcal O}_{X_ s})\) of the sheaf of \(p\)-typical Witt vectors on the fiber of \(f\) at a closed point \(s\in S\) of characteristic \(p\). It is compatible with the Frobenius endomorphism \(F_ p\) on the cohomology groups. Thus one can overview all \(H^ m(X_ s,{\mathcal W}{\mathcal O}_{X_ s})\) \((s\in S)\) from \(H^ m(X,{\mathcal W}{\mathcal O}_ X)\). Choosing a basis for \(H^ m(X,{\mathcal O}_ X)\) and a lifting thereof in \(H^ m(X,{\mathcal W}{\mathcal O}_ X)\) one can associate with the Frobenius operator \(F_ p\) a Hasse-Witt matrix \(B_ p\) with entries in \(A\). Section 3 shows that via étale extension and \(p\)-adic completion of the open part of \(S\) where the matrix \(B_ p\) is invertible one can construct a ring \(R^{\text{ét}}\) such that \(H^ m(X\otimes R^{\text{ét}},{\mathcal W}{\mathcal O}_{X\otimes R^{\text{ét}}})\) has a \({\mathcal W}(R^{\text{ét}})\)-basis consisting of vectors fixed by \(F_ p\). The \(\mathbb{Z}_ p\)-span \(\Lambda\) of this new basis and the matrix, with entries in \(R^{\text{ét}}\), relating it to the original basis of \(H^ m(X,{\mathcal O}_ X)\) should be regarded as \(p\)-adic period lattice and \(p\)-adic period matrix. The main theoretical results of the paper express the action of Frobenius \(F_ p\) on \(H^ m(X_ s,{\mathcal W}{\mathcal O}_{X_ s})\) for every \(s\in S\) where \(B_ p\) is invertible, and the monodromy action of the fundamental group \(\pi_ 1((R/pR)^{\text{ét}})\) on \(\Lambda\) in terms of the \(p\)-adic period matrix. They also relate the \(p\)-adic periods to solutions of the Picard- Fuchs differential equations (= Gauss-Manin connection) on \(\mathbb{H}^ m_{DR}(X/S)\). These results have been known in various formulations since the pioneering work of Dwork [cf. \textit{N. Katz} ``Travaux de Dwork'', Sem. Bourbaki 1971/72, exp. 409, Lect. Notes Math. 317, 167-200 (1973; Zbl 0259.14007)]. The approach via generalized Witt vector cohomology is new and, as demonstrated by examples of hypergeometric curves, makes explicit computations of \(p\)-adic monodromy possible.