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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.
##### MSC:
 11S20 Galois theory 11S31 Class field theory; $$p$$-adic formal groups
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