Faltings extension and Hodge-Tate filtration for abelian varieties over $p$-adic local fields with imperfect residue fields
Tongmu He
TL;DR
The paper addresses extending the Hodge-Tate filtration to abelian varieties over $p$-adic local fields with imperfect residue fields by constructing a canonical Faltings extension of $\mathcal{O}_K$ over $\mathbb{Z}_p$ and developing a Fontaine-style injection in this setting. It then builds a $C$-$G_K$-module Hodge-Tate filtration for any abelian variety over $K$, providing an explicit link between global differential forms $H^0(X,\Omega_{X/K}^1)$ and $p$-adic Galois representations through the extension $E=\mathrm{Hom}_{\mathbb{Z}_p}(T_p(X),C)$ and its $G_K$-invariants. A key outcome is a precise splitting criterion: the HT-filtration splits if and only if the image of Fontaine’s injection factors through $C(1)$, with the splitting being unique in that case. The approach relies on foundational $p$-adic Galois cohomology and Hyodo–Fontaine constructions, and avoids deeper theories such as Faltings’ almost purity, while aligning with broader results on HT filtrations in the imperfect residue setting.
Abstract
Let $K$ be a complete discrete valuation field of characteristic $0$ with not necessarily perfect residue field of characteristic $p>0$. We define a Faltings extension of $\mathcal{O}_K$ over $\mathbb{Z}_p$, and we construct a Hodge-Tate filtration for abelian varieties over $K$ by generalizing Fontaine's construction in 1981, where he treated the perfect residue field case.