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**On the Hodge-Newton filtration for \(p\)-divisible groups with additional structures.**
*(English)*
Zbl 1304.14057

The major result of the subject paper shows that a \(p\)-divisible group with additional structures over a complete valuation ring of rank \(1\) with mixed characteristic admits a Hodge-Newton filtration assuming the Newton polygon and Hodge polygon of its special fiber has a nontrivial contact point that is also a break point for the Newton polygon.

More specifically, let \(K/\mathbb{Q}_p\) be a complete discrete valuation field with perfect residue field \(k\). In the PEL symplectic case, a \(p\)-divisible group with additional structures over \(O_K\) is a triple \((H, \iota, \lambda)\) where \(H\) is a \(p\)-divisible group over \(O_K\), \(\iota : O_K \to \text{End}(H)\) is a homomorphism of algebras, \(\lambda : (H, \iota) \to (H^D, \iota^D)\) is a polarization. The following (HN) condition is assumed for all the major results of the paper: the Newton polygon \(\text{Newt}(H_k, \iota, \lambda)\) and the Hodge polygon \(\text{Hdg}(H_k, \iota, \lambda)\) possess a contact point \(x\) outside outside their extremal points, which is also a break point for the Newton polygon. Denote \(\hat{x}\) the symmetric point of \(x\). The main result of the paper is the following:

Theorem. Using the above notation, if \(x\) lies before \(\bar{x}\), then there are unique \(p\)-divisible subgroups with additional structures \((H_1, \iota) \subset (H_2, \iota) \subset (H, \iota)\) over \(O_K\) such that \(\lambda\) induces isomorphisms \((H_1, \iota) \cong ((H/H_2)^D, \iota')\) and \((H_2, \iota) \cong ((H/H_1)^D,\iota')\). The induced filtration \((H_{1k},\iota) \subset (H_{2k},\iota) \subset (H_k,\iota)\) over \(k\) is split. The Newton polygons (resp. Harder-Narasimhan polygons, Hodge polygons) of \((H_1, \iota)\), \((H_2/H_1, \iota)\), and \((H/H_2,\iota)\) are the parts of the Newton polygon (resp. Harder-Narasimhan polygon, Hodge polygon) of \((H, \iota,\lambda)\) up to \(x\), between \(x\) and \(\hat{x}\), and from \(\hat{x}\) on respectively.

The main theorem has several applications. The first application is that the monodromy representations associated to the local systems defined by Tate modules of \(p\)-divisible groups factor through the parabolic subgroups. The second one is to confirm a conjecture of Harris (Conjecture 5.2 in [M. Harris, Prog. Math. 201, 407–427 (2001; Zbl 1025.11038)]) under the assumption of the main theorem.

More specifically, let \(K/\mathbb{Q}_p\) be a complete discrete valuation field with perfect residue field \(k\). In the PEL symplectic case, a \(p\)-divisible group with additional structures over \(O_K\) is a triple \((H, \iota, \lambda)\) where \(H\) is a \(p\)-divisible group over \(O_K\), \(\iota : O_K \to \text{End}(H)\) is a homomorphism of algebras, \(\lambda : (H, \iota) \to (H^D, \iota^D)\) is a polarization. The following (HN) condition is assumed for all the major results of the paper: the Newton polygon \(\text{Newt}(H_k, \iota, \lambda)\) and the Hodge polygon \(\text{Hdg}(H_k, \iota, \lambda)\) possess a contact point \(x\) outside outside their extremal points, which is also a break point for the Newton polygon. Denote \(\hat{x}\) the symmetric point of \(x\). The main result of the paper is the following:

Theorem. Using the above notation, if \(x\) lies before \(\bar{x}\), then there are unique \(p\)-divisible subgroups with additional structures \((H_1, \iota) \subset (H_2, \iota) \subset (H, \iota)\) over \(O_K\) such that \(\lambda\) induces isomorphisms \((H_1, \iota) \cong ((H/H_2)^D, \iota')\) and \((H_2, \iota) \cong ((H/H_1)^D,\iota')\). The induced filtration \((H_{1k},\iota) \subset (H_{2k},\iota) \subset (H_k,\iota)\) over \(k\) is split. The Newton polygons (resp. Harder-Narasimhan polygons, Hodge polygons) of \((H_1, \iota)\), \((H_2/H_1, \iota)\), and \((H/H_2,\iota)\) are the parts of the Newton polygon (resp. Harder-Narasimhan polygon, Hodge polygon) of \((H, \iota,\lambda)\) up to \(x\), between \(x\) and \(\hat{x}\), and from \(\hat{x}\) on respectively.

The main theorem has several applications. The first application is that the monodromy representations associated to the local systems defined by Tate modules of \(p\)-divisible groups factor through the parabolic subgroups. The second one is to confirm a conjecture of Harris (Conjecture 5.2 in [M. Harris, Prog. Math. 201, 407–427 (2001; Zbl 1025.11038)]) under the assumption of the main theorem.

Reviewer: Xiao Xiao (Utica)

### MSC:

14L05 | Formal groups, \(p\)-divisible groups |