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A note on the geometricity of open homomorphisms between the absolute Galois groups of \(p\)-adic local fields. (English) Zbl 1328.11113
The present paper builds on the previous two papers by S. Mochizuki [Int. J. Math. 8, No. 4, 499–506 (1997; Zbl 0894.11046); J. Math. Sci., Tokyo 19, No. 2, 139–242 (2012; Zbl 1267.14039)]. Let \(k_{\circ}\), \(k_{\bullet}\) be \(p\)-adic local fields (i.e., finite extensions of \(\mathbb{Q}_p\)), \(\bar{k}_{\circ}\), \(\bar{k}_{\bullet}\) their algebraic closures, and \(G_{k_{\circ}}\), \(G_{k_{\bullet}}\) their absolute Galois groups. Let \(\alpha: G_{k_{\circ}} \to G_{k_{\bullet}}\) be an open continuous homomorphism. We say that \(\alpha\) is HT-preserving if for every Hodge-Tate representation \(\phi:G_{k_{\bullet}} \to \mathrm{GL}_n(\mathbb{Q}_p)\), the composite \(G_{k_{\circ}} \mathop{\rightarrow}\limits^{\alpha} G_{k_{\bullet}} \mathop{\rightarrow}\limits^{\phi} \mathrm{GL}_n(\mathbb{Q}_p)\) is also Hodge-Tate. We say that \(\alpha\) is geometric if it arises from an isomorphism of fields \(\bar{k}_{\bullet} \mathop{\rightarrow}\limits^{\sim} \bar{k}_{\circ}\) that determines an embedding \(k_{\bullet} \hookrightarrow k_{\circ}\). The main consequence of the paper is that an open continuous representation \(\alpha: G_{k_{\circ}} \to G_{k_{\bullet}}\) is geometric if and only if it is HT-preserving. The implication “geometric \(\Rightarrow\) HT-preserving” follows from the results in [loc. cit.]. The author proves its reciprocal.
The proof is based on refining some arguments and the observation in [loc. cit.], plus a lemma that ensures the injectivity of open continuous homomorphisms \(\alpha: G_{k_{\circ}} \to G_{k_{\bullet}}\) of HT-qLT-type (actually the lemma is stated for homomorphisms of weakly HT-qLT-type, that is a weaker condition). We say that an open continuous homomorphism \(\alpha:G_{k_{\circ}} \to G_{k_{\bullet}}\) is of HT-qLT-type (“Hodge-Tate-quasi-Lubin-Tate” type) if, for every pair of respective finite extensions \(k'_{\circ}\), \(k'_{\bullet}\) of \(k_{\circ}\), \(k_{\bullet}\) such that \(\alpha(G_{k'_{\circ}}) \subseteq G_{k'_{\bullet}}\), and for every finite Galois extension \(E\) of \(\mathbb{Q}_p\) that admits a pair of embeddings \(\sigma_{\circ}:E \hookrightarrow k'_{\circ}\), \(\sigma_{\bullet}:E \hookrightarrow k'_{\bullet}\), the composite \(G_{k'_{\circ}} \mathop{\longrightarrow}\limits^{\alpha|_{G_{k'_{\circ}}}} G_{k'_{\bullet}} \mathop{\longrightarrow}\limits^{\chi^{\mathrm{LT}}_{\sigma_{\bullet}}} E^{\times}\) is Hodge-Tate, where the character \(\chi^{\mathrm{LT}}_{\sigma_{\bullet}}\) is essentially defined by the Artin map composed with the norm map and an inertia splitting; see the paper for the precise definition.
11S20 Galois theory
11S31 Class field theory; \(p\)-adic formal groups
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