Log-growth filtration and Frobenius slope filtration of \(F\)-isocrystals at the generic and special points
The goal of this paper is to study the relation between the Frobenius slope filtration and the log-growth filtration on a \(\varphi\)-\(\nabla\)-module. The authors prove that, under certain conditions, the two filtrations agree at both ``the generic point'' and ``the special point''. From this, they deduce the truth of a conjecture of Dwork in the special case in which the generic log-growth polygon lies above the special log-growth polygon. This follows as a corollary of the corresponding statement for Frobenius slope polygons. To be more precise, let \(K\) be a complete discrete valuation field of mixed characteristic, equipped with an action \(\varphi\) lifting the Frobenius action on the residue field. Let \(K\lfloor x \rfloor_0\) denote the ring of formal power series with bounded coefficients in \(K\) and \(\mathcal E\) the \(p\)-adic completion of the fraction field of \(K\lfloor x \rfloor_0\). For \(\lambda \in \mathbb R_{\geq 0}\), let \(K \lfloor x\rfloor_\lambda\) denote the ring of power series of log-growth \(\lambda\). Let \(\mathcal A_K(0,1^-)\) denote the open unit disc. For a \(\varphi\)-\(\nabla\)-module \(M\) over \(K\lfloor x\rfloor_0\), let \(\mathrm{Sol}_\lambda(M)\) denote the \(K\)-vector space of solutions \(\mathrm{Hom}_{K\lfloor x\rfloor_0[\nabla]}(M, K\lfloor x\rfloor_\lambda)\). On one hand, one may define a decreasing filtration \(V^\bullet(M)\) on the set of horizontal sections \(V(M)\) of \(M\) over \(\mathcal A_K(0,1^-)\), by taking the orthogonal space of \(\mathrm{Sol}_\bullet(M)\) under the natural pairing. The special log-growth polygon is formed by taking the ranks of the successive quotients of the decreasing filtration \(V^\bullet(M)\). On the other hand, the \(\varphi\)-action on \(V(M)\) induces an increasing slope filtration \(S_\bullet(V(M))\). If \(M'\) is a \(\varphi\)-\(\nabla\)-module over \(\mathcal E\), then pulling it back to the open unit disc around a generic point \(t\) of radius \(1\) defines a \(\varphi\)-\(\nabla\)-module over \(\mathcal E\lfloor X -t\rfloor_0\). One may then invoke the construction above to define a log-growth filtration \(M^\lambda\) and a \(\varphi\) slope filtration \(S_\lambda(M)\); they are called the generic ones. We say a \(\varphi\)-\(\nabla\)-module is pure of bounded quotient (PBQ for short), if the maximal bounded solution quotient of each of \(N_i\) has pure \(\varphi\)-slope. The authors carefully study the possible extensions of two \(\varphi\)-\(\nabla\)-modules over \(\mathcal E\) of pure Frobenius slopes. They prove that \(M\) is bounded (i.e. \(\mathrm{Sol}_0(M)\) is of maximum dimension) if and only if it is a direct sum of pure \(\varphi\)-\(\nabla\)-modules. As a corollary, every \(\varphi\)-\(\nabla\)-module \(M\) admits a filtration such that the associated subquotients are PBQ \(\varphi\)-\(\nabla\)-module. It was proved in an earlier paper of the two authors [J. Inst. Math. Jussieu 8, No. 3, 465--505 (2009; Zbl 1176.12006)] that, for \(M\) a \(\varphi\)-\(\nabla\)-module over \(\mathcal E\) (resp., \(K\lfloor x\rfloor_0\)), the log-growth filtration \(M^\lambda\) (resp., \(V^\lambda(M)\)) is contained in the orthogonal dual of \(S_{\lambda- \lambda_{\mathrm{max}}}(M^\vee)\) (resp., of \(S_{\lambda- \lambda_{{max}}}(V(M^\vee))\)), where \(\lambda_{max}\) is the highest Frobenius slope of \(M\) (resp. \(M \otimes \mathcal E\)). In this paper, the authors study the case of a \(\varphi\)-\(\nabla\)-module \(M\) over \(\mathcal E\), and prove that \(M^\lambda\) equals the orthogonal dual above when \(M\) is PBQ. Then they study the special log-growth filtration, and prove that \(V^\lambda(M)\) is equal to the orthogonal dual of \(S_{\lambda- \lambda_{\mathrm{max}}}(V(M^\vee))\), if \(M \otimes \mathcal E\) is PBQ and \(M\) is horizontal of bounded quotient (HBQ for short), that is, if the maximal bounded quotient of \(M \otimes \mathcal E\) is defined over \(K\lfloor x\rfloor_0\). As a corollary of these two results, they deduce the truth of Dwork's conjecture for a HBQ and PBQ \(\varphi\)-\(\nabla\)-module \(M\) over \(K\lfloor x\rfloor_0\): the generic log-growth polygon lies above the special log-growth polygon, and these two polygons have the same end points. This opens a new approach that might help to prove Dwork's conjecture in general. The authors also point out that their result is different from the version of Dwork's conjecture proved by \textit{Y. André} [Compos. Math. 144, No. 2, 484--494 (2008; Zbl 1161.12002)], where no Frobenius structure was assumed and hence the end points of the generic and special log-growth polygons are not known to coincide. The paper is very well written, mostly self-contained.
scientific article; zbMATH DE number 5884792