May 25, 2026
In the present paper, we show that, for an odd prime number \(p\) and a nontrivial finite Galois extension \(k\) of \(\mathbb{Q}_{p}\), the \(p\)-adic representation of the absolute Galois group of \(k\) determined by a Lubin-Tate formal group over the ring of integers of \(k\) is not Aut-intrinsically Hodge-Tate [in the sense of Hoshi]. This settles the odd-degree cases left open in the previous works of Hoshi and the author and, together with the known even-degree case, completes the picture for finite Galois extensions of \(\mathbb{Q}_{p}\) in the case where \(p\) is odd. This exhibits a sharp contrast, from the viewpoint of anabelian geometry, between the \(p\)-adic cyclotomic character and other \(p\)-adic Lubin-Tate characters.
Let \(p\) be a prime number, \(k\) a finite extension of \(\mathbb{Q}_{p}\), and \(\overline{k}\) an algebraic closure of \(k\). We write \(G_{k}\stackrel{\mathrm{def}}{=} \mathop{\mathrm{Gal}}(\overline{k}/k)\) for the absolute Galois group of \(k\) determined by the algebraic closure \(\overline{k}\). In anabelian geometry, it is natural to discuss conditions for a continuous automorphism of \(G_{k}\) to be induced by a field automorphism of \(k\) [cf., e.g., [1], [2]]. The following theorem studies such conditions from the perspective of \(p\)-adic Hodge theory:
Theorem 1 ([1], Corollary 3.4). Let \(\alpha \colon G_{k} \stackrel{\sim}{\longrightarrow} G_{k}\) be a continuous automorphism of \(G_{k}\). Then the following conditions are equivalent:
The automorphism \(\alpha\) is induced by a field automorphism of \(k\).
For every finite-dimensional continuous representation \(\rho \colon G_{k} \to \mathop{\mathrm{GL}}_{n}(\mathbb{Q}_{p})\) of \(G_{k}\) that is Hodge-Tate, the composite \(G_{k} \stackrel{\alpha}{\to} G_{k} \stackrel{\rho}{\to} \mathop{\mathrm{GL}}_{n}(\mathbb{Q}_{p})\) is Hodge-Tate.
The automorphism \(\alpha\) is HT-qLT-type [cf. [1], Definition 1.3, \(\rm(ii)\)].
It follows from this theorem and [1], Definition 1.3, \(\rm(ii)\), that Lubin-Tate characters play an important role in the study of the geometricity [i.e., the condition to be induced by a field automorphism] of continuous automorphisms of \(G_{k}\).
Moreover, motivated by this theorem, Hoshi defined the following notion for continuous \(p\)-adic representations.
Definition 1 ([3], Definition 1.3). Let \(V\) be a \(\mathbb{Q}_{p}\)-vector space of finite dimension and \(\rho \colon G_{k} \to \mathop{\mathrm{Aut}}_{\mathbb{Q}_{p}}(V)\) a continuous representation. Then we shall say that \(\rho\) is Aut-intrinsically Hodge-Tate if, for an arbitrary continuous automorphism \(\alpha\) of \(G_{k}\), the composite \(\rho \circ \alpha \colon G_{k} \to \mathop{\mathrm{Aut}}_{\mathbb{Q}_{p}}(V)\) is Hodge-Tate.
Let \(\pi \in \mathcal{O}_k\) be a uniformizer of the ring of integers \(\mathcal{O}_k\) of \(k\). In the remainder of the present introduction, we write \(\rho_{k,\pi} \colon G_{k} \to \mathop{\mathrm{Aut}}_{\mathbb{Q}_{p}}(k_{+})\) for the continuous \(p\)-adic representation obtained by forming the composite \[\begin{align} G_{k} \stackrel{\chi_{k,\pi}}{\to} \mathcal{O}_{k}^{\times} \hookrightarrow \mathrm{Aut}_{\mathbb{Q}_{p}}(k_{+}), \end{align}\] where the first arrow is the Lubin-Tate character \(\chi_{k,\pi} \colon G_{k} \to \mathcal{O}_{k}^{\times}\) [i.e., the continuous character determined by a Lubin-Tate formal group law over \(\mathcal{O}_{k}\) associated to \(\pi\)], and the second arrow is the natural inclusion.
It is natural to study which representations are Aut-intrinsically Hodge-Tate. The following theorem is a typical example motivated by this question:
Theorem 2 ([3], Theorem 3.3; [4], Theorem 1.9; [5], Theorem 4.4). Let \(\rho \colon G_k \to \mathop{\mathrm{Aut}}_{\mathbb{Q}_{p}}(V)\) be a continuous \(p\)-adic representation. Then the following assertions hold:
Suppose either that \(\rho\) is one-dimensional or that \(\rho\) is two-dimensional and reducible. Then \(\rho\) is Hodge-Tate if and only if \(\rho\) is Aut-intrinsically Hodge-Tate.
Suppose that \(p\) is an odd prime number and that \(k/\mathbb{Q}_{p}\) is a finite Galois extension of even degree. Let \(\pi \in \mathcal{O}_{k}\) be a uniformizer of \(\mathcal{O}_{k}\). If \(\rho\) is isomorphic to the continuous \(p\)-adic representation \(\rho_{k,\pi}\) of \(G_{k}\), then \(\rho\) is not Aut-intrinsically Hodge-Tate.
Suppose that \(p=2\) and that \(k\) contains a primitive \(4\)-th root of unity. Suppose, moreover, that \(k/\mathbb{Q}_p\) is an abelian extension. Let \(\pi \in \mathcal{O}_{k}\) be a uniformizer of \(\mathcal{O}_{k}\). If \(\rho\) is isomorphic to the continuous \(p\)-adic representation \(\rho_{k,\pi}\) of \(G_{k}\), then \(\rho\) is not Aut-intrinsically Hodge-Tate.
Remark 1. We give some remarks on Theorem B.
In the proof of Theorem B, (1), it is essential that the \(p\)-adic cyclotomic character can be reconstructed from \(G_{k}\) in a group-theoretic manner. This is a significant difference between the \(p\)-adic cyclotomic character and other Lubin-Tate characters.
In [3], [4], and [5], Theorem B, (2) is not stated explicitly. However, each of the authors essentially established Theorem B, (2), in [3], [4], and [5]. Theorem B, (2) was first established in [3] for the case where \(k/\mathbb{Q}_{p}\) is an abelian extension of even degree. In [3], by making use of the assumption that the extension is of even degree and abelian, the argument was reduced to the case where \(k/\mathbb{Q}_{p}\) is a quadratic extension.
In [3], [4], and [5], each of the authors constructed an explicit continuous automorphism \(\varphi\) of \(G_{k}\) such that \(\rho_{k,\pi} \circ \varphi\) is not Hodge-Tate in terms of generators and relations established by Jannsen-Wingberg [cf. [6], Theorem 7.5.14].
In the present paper, we show the following theorem, which is a generalization of Theorem B, (2), above:
Theorem 3. Suppose that \(p\) is an odd prime number and that \(k/\mathbb{Q}_{p}\) is a nontrivial finite Galois extension. Let \(\pi \in \mathcal{O}_{k}\) be a uniformizer of \(\mathcal{O}_{k}\). Then the continuous \(p\)-adic representation \(\rho_{k,\pi} \colon G_k \to \mathop{\mathrm{Aut}}_{\mathbb{Q}_{p}}(k_{+})\) is not Aut-intrinsically Hodge-Tate.
Remark 2. We give some remarks on Theorem C.
Here, we note that the novelty of Theorem C lies in the odd-degree cases. Thus, in the proof of Theorem C, we assume that \([k \colon \mathbb{Q}_{p}]\) is odd. However, the proof of Theorem C in the present paper can also be applied to the case in which \(k/\mathbb{Q}_{p}\) is a finite Galois extension of even degree. From that point of view, the proof of Theorem C in the present paper is more uniform than the proofs of Theorem B, (2), in [3] and [4].
In the proof of Theorem C, we do not give an explicit continuous automorphism \(\varphi\) of \(G_{k}\) such that \(\rho_{k,\pi} \circ \varphi\) is not Hodge-Tate. Thus, in the case where \(k/\mathbb{Q}_{p}\) is of even degree, these two methods have both advantages and disadvantages.
At the end of Introduction, we describe the outline of the proof of Theorem C. Let \(\alpha\) be a continuous automorphism of \(G_{k}\). We write \(\alpha_{+}\) for the automorphism of the \(\mathbb{Q}_{p}\)-vector space \(k_{+}\) [obtained by taking the underlying \(\mathbb{Q}_{p}\)-vector space of \(k\)] induced by mono-anabelian reconstruction algorithms [cf. [7], Proposition 3.10, \(\mathrm{(vi)}\); [7], Proposition 3.11, \(\mathrm{(iv)}\); [8], Lemma 1.2]. Under the above notation and definition, we show that if \(\rho_{k,\pi} \circ \alpha\) is Hodge-Tate, then \(\alpha_{+} \in \mathbb{Q}_{p}[\mathrm{Gal}(k/\mathbb{Q}_{p})]\). Then we obtain Theorem C by combining this observation with the theory of mapping class groups and the theory of \(p\)-adic Lie groups.
. Let \(G\) be a topological group and \(\alpha\) a continuous automorphism of the topological group \(G\). Then we shall write \(G^{\mathop{\mathrm{ab}}}\) for the abelianization of \(G\) [i.e., the quotient of \(G\) by the closure of the commutator subgroup of \(G\)] and \(\alpha^{\mathrm{ab}}\) for the continuous automorphism of the topological group \(G^{\mathrm{ab}}\) induced by \(\alpha\) via the functoriality of abelianization.
. In the present paper, every “ring” is assumed to be unital, associative, and commutative. If \(R\) is a ring, then we shall write \(R_{+}\) for the underlying additive group of \(R\) and \(R^{\times} \subset R\) for the multiplicative group of units of \(R\).
. We shall refer to a field isomorphic to a finite extension of \(\mathbb{Q}_{p}\), for some prime number \(p\), as an MLF. Here, “MLF” is to be understood as an abbreviation for “mixed-characteristic local field”. Let \(k\) be an MLF and \(\overline{k}\) an algebraic closure of \(k\). Then we shall write
\(\mathcal{O}_{k}\) for the ring of integers of \(k\),
\(\mathfrak{m}_{k} \subset \mathcal{O}_{k}\) for the maximal ideal of \(\mathcal{O}_{k}\),
\(\underline{k} \stackrel{\mathrm{def}}{=}\mathcal{O}_{k}/\mathfrak{m}_{k}\) for the residue field of \(\mathcal{O}_{k}\),
\(k^{(d=1)} \subset k\) for the [uniquely determined] minimal MLF contained in \(k\),
\(p_{k}\) for the residue characteristic of \(k\),
\(d_{k}\stackrel{\mathrm{def}}{=}[k \colon k^{(d=1)}]\) for the degree of the finite extension \(k/k^{(d=1)}\),
\(f_{k}\stackrel{\mathrm{def}}{=}[\underline{k} \colon \underline{k}^{(d=1)}]\) for the degree of the finite extension \(\underline{k}/\underline{k}^{(d=1)}\) [where we write \(\underline{k}^{(d=1)}\) for the residue field of the ring of integers of the MLF \(k^{(d=1)}\)],
\(G_{k} \stackrel{\mathrm{def}}{=} \mathop{\mathrm{Gal}}(\overline{k}/k)\) for the absolute Galois group of \(k\) determined by the algebraic closure \(\overline{k}\),
\(I_{k} \subset G_{k}\) for the inertia subgroup of \(G_{k}\),
\(P_{k} \subset I_{k}\) for the wild inertia subgroup of \(G_{k}\),
\(\log_{k} \colon \mathcal{O}_{k}^{\times} \to k_{+}\) for the \(p_{k}\)-adic logarithm,
\(\widehat{k^{\times}}\) for the profinite completion of the multiplicative group \(k^{\times}\) of \(k\), and
\(\mathop{\mathrm{rec}}_{k} \colon \widehat{k^{\times}} \stackrel{\sim}{\longrightarrow} G_{k}^{\mathop{\mathrm{ab}}}\) for the isomorphism induced by the reciprocity homomorphism \(k^{\times} \hookrightarrow G_{k}^{\mathop{\mathrm{ab}}}\) in local class field theory.
. We shall refer to a topological group isomorphic to the absolute Galois group of an MLF as a group of MLF-type.
Let us recall [cf.[7], Definition 3.5; [7], Proposition 3.6; [7], Definition 3.10; [7], Proposition 3.11] that there exist functorial group-theoretic algorithms for constructing, from a group of MLF-type \(G\),
a prime number \(p(G)\),
positive integers \(d(G)\), \(f(G)\),
subgroups \(P(G) \subset I(G) \subset G\) of \(G\),
subgroups \(\mathcal{O}^{\times}(G) \subset k^{\times}(G) \subset G^{\mathop{\mathrm{ab}}}\), whose final inclusion \(k^{\times}(G) \subset G^{\mathop{\mathrm{ab}}}\) we denote by \(\mathop{\mathrm{rec}}_{G}\), and
a topological group \(k_{+}(G)\)
which “correspond” to
the prime number \(p_{k}\),
the positive integers \(d_{k}\), \(f_{k}\),
the subgroups \(P_{k} \subset I_{k} \subset G_{k}\) of \(G_{k}\),
the subgroups \(\mathcal{O}_{k}^{\times} \subset k^{\times} \stackrel{\mathop{\mathrm{rec}}_{k}}{\hookrightarrow} G_{k}^{\mathop{\mathrm{ab}}}\), and
the topological group \(k_{+}\),
respectively.
Moreover, it follows from [7], Proposition 3.11, \(\rm(i)\), \(\rm(iv)\), that we have natural homomorphisms \[\begin{align} \mathop{\mathrm{Aut}}(G) \to \mathop{\mathrm{Aut}}(k^{\times}(G)),\;\;\;\mathop{\mathrm{Aut}}(G) \to \mathop{\mathrm{Aut}}(k_{+}(G)). \end{align}\] Since the automorphism of \(G^{\mathop{\mathrm{ab}}}\) induced by an inner automorphism of \(G\) is trivial, it follows from the constructions of \(k^{\times}(G)\) and \(k_{+}(G)\) that the automorphisms of \(k^{\times}(G)\) and \(k_{+}(G)\) induced by an inner automorphism of \(G\) are trivial. Thus, the above two homomorphisms determine group homomorphisms \[\begin{align} \mathop{\mathrm{Out}}(G) \to \mathop{\mathrm{Aut}}(k^{\times}(G)),\;\;\;\mathop{\mathrm{Out}}(G) \to \mathop{\mathrm{Aut}}(k_{+}(G)). \end{align}\] Let \(\alpha\) be an element of \(\mathop{\mathrm{Out}}(G)\). We write \(\alpha^{\times}\) [respectively, \(\alpha_{+}\)] for the image of \(\alpha\) by this homomorphism \(\mathop{\mathrm{Out}}(G) \to \mathop{\mathrm{Aut}}(k^{\times}(G))\) [respectively, \(\mathop{\mathrm{Out}}(G) \to \mathop{\mathrm{Aut}}(k_{+}(G))\)]. In the present paper, we call \(\alpha^{\times}\) and \(\alpha_{+}\) the automorphisms induced from \(\alpha\) by the mono-anabelian reconstruction algorithms.
Let \(k\) be an MLF, \(\overline{k}\) an algebraic closure of \(k\), and \(\alpha\) an element of \(\mathop{\mathrm{Out}}(G_{k})\). By abuse of notation, we shall denote by \(\alpha_{+} \colon k_{+} \stackrel{\sim}{\longrightarrow} k_{+}\), \(\alpha^{\times} \colon k^{\times} \stackrel{\sim}{\longrightarrow} k^{\times}\) the respective images of \(\alpha_{+} \colon k_{+}(G_{k}) \stackrel{\sim}{\longrightarrow} k_{+}(G_{k})\), \(\alpha^{\times} \colon k^{\times}(G_{k}) \stackrel{\sim}{\longrightarrow} k^{\times}(G_{k})\) by the isomorphisms \(\mathop{\mathrm{Aut}}(k_{+}(G_{k})) \stackrel{\sim}{\longrightarrow} \mathop{\mathrm{Aut}}(k_{+})\), \(\mathop{\mathrm{Aut}}(k^{\times}(G_{k})) \stackrel{\sim}{\longrightarrow} \mathop{\mathrm{Aut}}(k^{\times})\) induced by the isomorphisms \(k_{+} \stackrel{\sim}{\longrightarrow} k_{+}(G_{k})\), \(k^{\times} \stackrel{\sim}{\longrightarrow} k^{\times}(G_{k})\) of [7], Proposition 3.11, \(\rm(i)\), \(\rm(iv)\).
Let \(k\) be an MLF, \(\overline{k}\) an algebraic closure of \(k\), and \(G\) a group of MLF-type. We begin by giving an overview of the remainder of the present paper. First, we recall various notions introduced in [4] for the convenience of the reader. Next, we review the classification of abelian Hodge-Tate representations and prove a key lemma. Finally, we prove the main theorem of the present paper by combining this lemma with certain “mapping class group and \(p\)-adic Lie group techniques” developed in [4].
Theorem 1. Suppose that \(p(G)\) is an odd prime number. Then there exist \(\sigma\), \(\tau\), \(x_{0}, \ldots, x_{d(G)}\in G\), positive integers \(s\), \(t\), an element \(x_{0}^{\prime} \in \overline{\langle\tau, x_{0} \rangle}\), and an element \(x_{1}^{\prime} \in \overline{\langle \sigma,\tau,x_{1}\rangle}\), where “\(\overline{\langle S \rangle}\)” denotes the closed subgroup of \(G\) topologically generated by “\(S\)”, such that the following conditions hold:
The profinite group \(G\) is presented as the profinite group topologically generated by \(\sigma\), \(\tau\), \(x_{0}, \ldots, x_{d(G)}\in G\) and subject to the relations described in the conditions (2), (3), and (4) below.
The closed normal subgroup \(P(G)\) of \(G\) is pro-\(p(G)\) and topologically normally generated by \(x_{0},\ldots,x_{d(G)}\).
The elements \(\sigma\), \(\tau\) satisfy the relation \(\sigma \tau \sigma^{-1}=\tau^{p(G)^{f(G)}}\).
In addition, the generators satisfy one further relation:
for even \(d(G)\), \[\begin{align} \sigma x_{0} \sigma^{-1}=(x_{0}^{\prime})^{t} x_{1}^{p(G)^s}[x_{1},x_{2}][x_{3},x_{4}]\cdots[x_{d(G)-1},x_{d(G)}]; \end{align}\]
for odd \(d(G)\), \[\begin{align} \sigma x_{0} \sigma^{-1}=(x_{0}^{\prime})^{t} x_{1}^{p(G)^s}[x_{1},x_{1}^{\prime}][x_{2},x_{3}]\cdots[x_{d(G)-1},x_{d(G)}]. \end{align}\]
In the remainder of the present paper, we apply the notational conventions introduced in the statement of Theorem 1 in each of the situations in which \(p(G)\) is assumed to be odd. Moreover, if \(p(G)\) is odd, then, for each \(i=1,2,\ldots,d(G)\), write \(y_{i} \in k_{+}(G)\) for the image of \(x_{i}\) in \(k_{+}(G)\) by the composite \(P(G) \hookrightarrow I(G) \to \mathcal{O}^{\times}(G) \to k_{+}(G)\) [cf.conditions (1), (2) of Theorem 1; [7], Definition 3.10, \(\rm(i)\), \(\rm(ii)\), \(\rm(v)\)].
We recall that the topological group \(k_{+}(G)\) has a natural structure of a \(\mathbb{Q}_{p(G)}\)-vector space of dimension \(d(G)\) [cf.[8], Lemma 1.2] and that, for any continuous automorphism \(\alpha\) of \(G\), the induced automorphism \(\alpha_{+}\) of \(k_{+}(G)\) is an automorphism of \(\mathbb{Q}_{p(G)}\)-vector spaces.
Lemma 2. Suppose that \(d(G)>1\) and \(p(G)\) is odd. Then the \(d(G)\) elements \(y_{1},\ldots,y_{d(G)}\) defined in the discussion following Theorem 1 form a basis of the \(\mathbb{Q}_{p(G)}\)-vector space \(k_{+}(G)\).
Proof. This assertion is none other than [8], Lemma 1.3. ◻
One verifies easily that the isomorphism of topological groups \(k_{+}(G_{k}) \stackrel{\sim}{\longrightarrow} k_{+}\) of [7], Proposition 3.11, \(\rm(iv)\), is also an isomorphism of \(\mathbb{Q}_{p_{k}}\)-vector spaces [here, we have \(p_{k} = p(G_{k})\) — cf.[7], Proposition 3.6]. By abuse of notation, if \(d_{k}>1\) and \(p_{k}\) is odd, then, for each integer \(i\) satisfying \(1 \leq i \leq d_{k}\), we write \(y_{i} \in k_{+}\) for the image of \(y_{i} \in k_{+}(G_{k})\) by the isomorphism \(k_{+}(G_{k}) \stackrel{\sim}{\longrightarrow} k_{+}\) of [7], Proposition 3.11, \(\rm(iv)\). In the remainder of the present paper, if \(d(G)\) [resp. \(d_{k}\)] is greater than one, and \(p(G)\) [resp. \(p_{k}\)] is odd, then we equip \(k_{+}(G)\) [resp. \(k_{+}\)] with this basis, which allows us to identify \(\mathop{\mathrm{Aut}}_{\mathbb{Q}_{p(G)}}(k_{+}(G))\) [resp. \(\mathop{\mathrm{Aut}}_{\mathbb{Q}_{p_{k}}}(k_{+})\)] with \(\mathop{\mathrm{GL}}_{d(G)}(\mathbb{Q}_{p(G)})\) [resp. \(\mathop{\mathrm{GL}}_{d_{k}}(\mathbb{Q}_{p_{k}})\)]. We equip \(\mathop{\mathrm{GL}}_{d(G)}(\mathbb{Q}_{p(G)})\) [resp. \(\mathop{\mathrm{GL}}_{d_{k}}(\mathbb{Q}_{p_{k}})\)] with the natural \(p(G)\)-adic [resp. \(p_{k}\)-adic] Lie group structure.
Next, we review the profinite group structure of \(\mathop{\mathrm{Out}}(G)\). It follows from [6], Theorem 7.4.1, and [9], Proposition 4.4.3, that \(\mathop{\mathrm{Out}}(G)\) has a natural profinite group structure. In the remainder of the present paper, we endow \(\mathop{\mathrm{Out}}(G)\) with this profinite group structure.
Lemma 3. The action \(\mathrm{Out}(G) \curvearrowright k_{+}(G)\) which is defined via the mono-anabelian reconstruction algorithm is continuous. In particular, the induced map \[\begin{align} \Phi \colon \mathop{\mathrm{Out}}(G) \to \mathop{\mathrm{Aut}}_{\mathbb{Q}_{p(G)}}(k_{+}(G)) \stackrel{\sim}{\longrightarrow} \mathrm{GL}_{d(G)}(\mathbb{Q}_{p(G)}) \end{align}\] is continuous and closed.
Proof. This assertion is none other than [4], Lemma 2.13. ◻
Lemma 4. The image of the homomorphism \(\Phi\) of Lemma 3 has a natural \(p(G)\)-adic Lie group structure.
In the remainder of the present paper, we assume that \(p_{k}\) [resp. \(p(G)\)] is an odd prime number and that \(d_{k}\) [resp. \(d(G)\)] is an odd integer greater than one.
Next, we review the subgroup of \(\mathrm{Out}(G)\) that corresponds to a “mapping class group” introduced in the discussion following [4], Lemma 2.15.
Let \(g \stackrel{\mathrm{def}}{=} \frac{d(G) - 1}{2}\), \(S\) a closed orientable surface of genus \(g\;(\geq 1)\), and \(P\) a point on \(S\). We write \(\mathrm{Mod}(S \setminus \{P\})\) for the mapping class group of \(S \setminus \{P\}\).
In the remainder of the present paper, we regard \(\mathrm{Sp}_{2g}(\mathbb{Z}_{p(G)})\) as a subgroup of \(\mathop{\mathrm{GL}}_{d(G)}(\mathbb{Q}_{p(G)})\) via the injective group homomorphism that is defined by \[A \mapsto \begin{bmatrix} 1 & 0_{1 \times 2g} \\ 0_{2g \times 1} & A \end{bmatrix},\] where \(0_{1 \times 2g}\) [respectively, \(0_{2g \times 1}\)] denotes the \(1 \times 2g\) matrix [respectively, the \(2g \times 1\) matrix] whose entries are all \(0\). Then there exists [cf. the discussion following [4], Lemma 2.15] a map \[\begin{align} \rho \colon \mathop{\mathrm{Mod}}(S \setminus \{P\}) \to \mathop{\mathrm{Out}}(G), \end{align}\] which is not necessarily a homomorphism of groups, such that the following diagram is commutative: \[\begin{tikzcd} \mathop{\mathrm{Out}}(G) \arrow[r, "\Phi"] & \mathop{\mathrm{GL}}_{d(G)}(\mathbb{Q}_{p(G)}) \\ \mathop{\mathrm{Mod}}(S \setminus\{P\}) \arrow[r] \arrow[u, "\rho"] & \mathrm{Sp}_{2g}(\mathbb{Z}) \arrow[u, "\subset"']. \end{tikzcd}\] Here, the lower horizontal arrow is the surjective homomorphism discussed in [4], Theorem 2.17. Write \(D \subset \mathop{\mathrm{Out}}(G)\) for the closed subgroup of \(\mathop{\mathrm{Out}}(G)\) that is topologically generated by the image of \(\rho\).
In what follows, for a \(p(G)\)-adic [resp. \(p_{k}\)-adic] Lie group \(X\), we write \(\dim(X)\) for the dimension of \(X\) as such a Lie group.
Lemma 5. The image of the homomorphism \(\Phi \colon \mathop{\mathrm{Out}}(G) \to \mathrm{GL}_{d(G)}(\mathbb{Q}_{p(G)})\) of Lemma 3 contains \(\mathrm{Sp}_{2g}(\mathbb{Z}_{p(G)})\). In particular, we have \(\dim (\mathrm{Im}(\Phi)) \geq 2g^{2}+g\).
Proof. It follows immediately from Lemma 3, the commutative diagram above, and the fact that the topological closure of \(\mathrm{Sp}_{2g}(\mathbb{Z})\) in \(\mathop{\mathrm{GL}}_{d(G)}(\mathbb{Q}_{p(G)})\) is \(\mathrm{Sp}_{2g}(\mathbb{Z}_{p(G)}) \subset \mathop{\mathrm{GL}}_{d(G)}(\mathbb{Q}_{p(G)})\) that \(\Phi(D) \supset \mathrm{Sp}_{2g}(\mathbb{Z}_{p(G)})\). This completes the proof of the first assertion. The second assertion follows immediately from the first assertion, together with the well-known equality \(\dim(\mathrm{Sp}_{2g}(\mathbb{Z}_{p(G)})) = 2g^{2} + g\). ◻
Next, we review a classification theorem of abelian Hodge-Tate representations.
Definition 6. We shall say that the MLF \(k\) is an absolutely Galois MLF if the extension \(k/k^{(d=1)}\) is a Galois extension.
Definition 7. Suppose that \(k\) is an absolutely Galois MLF. Let \(\pi \in \mathcal{O}_{k}\) be a uniformizer of \(\mathcal{O}_{k}\) and \(\sigma\) an element of \(\mathop{\mathrm{Gal}}(k/k^{(d=1)})\). Then we shall write \[\begin{align} \chi_{\pi,\sigma} \colon G_{k}^{\mathop{\mathrm{ab}}} \stackrel{\mathop{\mathrm{rec}}_{k}^{-1}}{\to} \widehat{k^{\times}} \twoheadrightarrow \mathcal{O}_{k}^{\times} \stackrel{\sigma}{\to} \mathcal{O}_{k}^{\times}, \end{align}\] where the second arrow is the projection determined by \(\pi\).
Theorem 8. Suppose that \(k\) is an absolutely Galois MLF. Let \(\pi \in \mathcal{O}_{k}\) be a uniformizer of \(\mathcal{O}_{k}\) and \(\phi \colon G_{k}^{\mathrm{ab}} \to \mathcal{O}_{k}^{\times}\) a continuous homomorphism. Then the following two conditions are equivalent:
The continuous representation obtained by forming the composite \[\begin{align} G_{k} \twoheadrightarrow G_{k}^{\mathrm{ab}} \stackrel{\phi}{\to} \mathcal{O}_{k}^{\times} \hookrightarrow \mathrm{Aut}_{\mathbb{Q}_{p_{k}}}(k_{+}) \end{align}\] — where the first arrow is the natural surjective continuous homomorphism, and the third arrow is the natural inclusion — is Hodge-Tate.
There exist an integer \(i_{\sigma}\) for each \(\sigma \in \mathrm{Gal}(k/k^{(d=1)})\) and an open subgroup \(J \subset I_{k}\) such that
coincides with
Proof. This assertion is none other than [3], Lemma 1.8. ◻
Definition 9. Let \(\pi \in \mathcal{O}_{k}\) be a uniformizer of \(\mathcal{O}_{k}\). Then we write \(\rho_{k,\pi} \colon G_{k} \to \mathrm{Aut}_{\mathbb{Q}_{p_{k}}}(k_{+})\) for the continuous \(p_{k}\)-adic representation of \(G_{k}\) obtained by forming the composite \[\begin{align} G_{k} \twoheadrightarrow G_{k}^{\mathrm{ab}} \stackrel{\chi_{\pi,\mathop{\mathrm{id}}_{k}}}{\to} \mathcal{O}_{k}^{\times} \hookrightarrow \mathrm{Aut}_{\mathbb{Q}_{p_{k}}}(k_{+}), \end{align}\] where the first arrow is the natural surjective continuous homomorphism, and the third arrow is the natural inclusion.
Remark 10. It follows from [11], \(\rm III\), §A.4, Proposition 4, that \(\rho_{k,\pi}\) is isomorphic to the continuous \(p_{k}\)-adic representation determined by a Lubin-Tate character [i.e., a continuous character determined by a Lubin-Tate formal group over \(\mathcal{O}_{k}\)].
Remark 11. It follows from [11], \(\rm III\), §A.1, Corollary 2, that the Hodge-Tate-ness of continuous \(p_{k}\)-adic representations of \(G_{k}\) is independent of the choice of representatives of an inertial equivalence class [cf., e.g., [1], Definition 1.2, \(\rm(i)\)]. Moreover, one verifies easily that the inertial equivalence class of \(\chi_{\pi, \mathop{\mathrm{id}}_{k}}\) is independent of the choice of a uniformizer of \(\mathcal{O}_{k}\). Thus, the choice of a uniformizer of \(\mathcal{O}_{k}\) is inessential for the discussion that follows.
The following proposition is one of the key ingredients of the present paper:
Proposition 12. Suppose that \(k\) is an absolutely Galois MLF. Let \(\pi \in \mathcal{O}_{k}\) be a uniformizer of \(\mathcal{O}_{k}\) and \(\alpha\) a continuous automorphism of \(G_{k}\). If \(\rho_{k,\pi} \circ \alpha\) is Hodge-Tate, then there exists an integer \(i_{\sigma}\) for each \(\sigma \in \mathrm{Gal}(k/k^{(d=1)})\) such that \[\begin{align} \alpha_{+}=\sum_{\sigma \in \mathrm{Gal}(k/k^{(d=1)})} i_{\sigma} \cdot \sigma. \end{align}\] In particular, if \(\rho_{k,\pi} \circ \alpha\) is Hodge-Tate, then \(\alpha_{+} \in \mathbb{Q}_{p_{k}}[\mathrm{Gal}(k/k^{(d=1)})] \cap \mathrm{Aut}_{\mathbb{Q}_{p_{k}}}(k_{+}) \subset \mathrm{End}_{\mathbb{Q}_{p_{k}}}(k_{+})\).
Proof. Suppose that \(\rho_{k,\pi} \circ \alpha\) is Hodge-Tate. Then it follows from Theorem 8 that there exist an integer \(i_{\sigma}\) for each \(\sigma \in \mathrm{Gal}(k/k^{(d=1)})\) and an open subgroup \(J \subset I_{k}\) such that
coincides with
Thus, it follows from the definition of \(\alpha^{\times}\) that there exist an integer \(i_{\sigma}\) for each \(\sigma \in \mathrm{Gal}(k/k^{(d=1)})\) and an open subgroup \(U \subset \mathcal{O}_{k}^{\times}\) such that
coincides with
where, we write \(\sigma_{i_{\sigma}}\) for the endomorphism of \(\mathcal{O}_{k}^{\times}\) defined by \(x \mapsto \sigma(x)^{i_{\sigma}}\). Let us recall that it follows from the definition of \(\alpha_{+}\) and [7], Proposition 3.11, \(\rm(iv)\), that the following diagram commutes: \[\begin{tikzcd} k_{+} \arrow[r, "\alpha_{+}"] & k_{+} \\ \mathcal{O}_{k}^{\times} \arrow[r, "\alpha^{\times}"'] \arrow[u, "\log_{k}"] & \mathcal{O}_{k}^{\times} \arrow[u, "\log_{k}"']. \end{tikzcd}\] Thus, it follows from [2], Lemma 4.1, together with the commutativity of the diagram above, that for each \(\sigma \in \mathrm{Gal}(k/k^{(d=1)})\) there exists an integer \(i_{\sigma}\) such that \[\begin{align} \alpha_{+}=\sum_{\sigma \in \mathrm{Gal}(k/k^{(d=1)})} i_{\sigma} \cdot \sigma. \end{align}\] This completes the proof of Proposition 12. ◻
With the above preparations, we now prove the main theorem of the present paper.
Theorem 13. Suppose that \(k\) is an absolutely Galois MLF, that \(p_k\) is odd, and that \(d_k\) is odd and greater than one. Then the continuous \(p_{k}\)-adic representation \(\rho_{k,\pi}\) is not Aut-intrinsically Hodge-Tate [cf. Definition in Introduction].
Proof. We shall write \(Z\) for the image of \(\mathbb{Q}_{p_{k}}[\mathrm{Gal}(k/k^{(d=1)})]^{\times}\) in \(\mathrm{GL}_{d_{k}}(\mathbb{Q}_{p_{k}})\) via the isomorphism \(\mathop{\mathrm{Aut}}_{\mathbb{Q}_{p_{k}}}(k_{+}) \stackrel{\sim}{\longrightarrow} \mathrm{GL}_{d_{k}}(\mathbb{Q}_{p_{k}})\) determined by the basis \(y_{1}, \ldots, y_{d_{k}}\) of \(k_{+}\).
We prove Theorem 13 by contradiction. Suppose that the continuous \(p_k\)-adic representation \(\rho_{k,\pi}\) is Aut-intrinsically Hodge-Tate. Then it follows from Proposition 12 that, for any \(\alpha \in \mathop{\mathrm{Out}}(G_k)\), it holds that \(\alpha_{+}, \alpha_{+}^{-1} \in \mathbb{Q}_{p_k}[\mathop{\mathrm{Gal}}(k/k^{(d=1)})]\). In particular, the image of the continuous group homomorphism \(\Phi\) [cf. Lemma 3] is contained in \(Z\).
We first consider the case where \(g \stackrel{\mathrm{def}}{=} \frac{d_{k} - 1}{2} \geq 2\). In light of the fact that \(Z \subset \mathrm{GL}_{d_{k}}(\mathbb{Q}_{p_{k}})\) is a closed subgroup, we endow \(Z\) with the natural \(p_{k}\)-adic Lie group structure [cf. [10], Theorem 9.6]. Then it follows from [4], Lemma 2.14, (2), (3), that \(\dim (Z) \leq d_{k}=2g+1\). Thus, it follows from the [easily verified] inequality \(2g^{2}+g > 2g+1\) and Lemma 5 that there exists an automorphism \(\alpha\) of \(G_{k}\) such that \(\Phi(\alpha) \notin Z\). This contradicts the above observation that the image of \(\Phi\) is contained in \(Z\). This completes the proof of Theorem 13 in the case where \(g \geq 2\).
Finally, we consider the case where \(g = 1\) [i.e., \(d_{k} = 3\)]. In this case, since [it is immediate that] the Galois group \(\mathop{\mathrm{Gal}}(k/k^{(d=1)})\) is abelian, it follows that the group \(Z\) is abelian. On the other hand, one verifies easily that the group \(\mathrm{Sp}_{2}(\mathbb{Z}_{p_k})\) is not abelian. Thus, it follows from Lemma 5 that the \(\mathrm{Im}(\Phi)\) is not abelian. In particular, there exists an automorphism \(\alpha\) of \(G_{k}\) such that \(\Phi(\alpha) \notin Z\). This contradicts the above observation that the image of \(\Phi\) is contained in \(Z\). This completes the proof of Theorem 13 in the case where \(g = 1\), hence also of Theorem 13. ◻
Remark 14. Let \(p\) be a prime number and \(\overline{\mathbb{Q}_{p}}\) an algebraic closure of \(\mathbb{Q}_{p}\). Then it is well-known that the Lubin-Tate character over \(\mathbb{Z}_{p}\) determined by the uniformizer \(p\) coincides with the \(p\)-adic cyclotomic character. Thus, the continuous \(p\)-adic representations \(G_{\mathbb{Q}_{p}} \stackrel{\mathrm{def}}{=} \mathop{\mathrm{Gal}}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p}) \to \mathbb{Q}_{p}^{\times}\) determined by the Lubin-Tate characters over \(\mathbb{Z}_{p}\) [i.e., with respect to arbitrary choices of uniformizers of \(\mathbb{Z}_{p}\)] are Aut-intrinsically Hodge-Tate [cf. Remark 11; [2], Proposition 1.1]. This reflects a fundamental distinction between \(\mathbb{Q}_p\) and its nontrivial finite Galois extensions from the point of view of anabelian geometry in the case where \(p\) is odd.
Remark 15. It is straightforward to see that a similar proof strategy applied in the proof of Theorem 13 may also be applied in the case where \(k\) is an absolutely Galois MLF with even \(d_k\). We leave the routine details to the interested reader.
Remark 16. In [4], the author of the present paper proved the following assertion:
Suppose that \(p_{k}\) is odd, that \(d_{k}\) is even, and that \(k\) is an absolutely Galois MLF. Let \(\varphi\) be the automorphism of \(G_{k}\) defined by the following equalities [cf.Theorem 1]: \[\begin{align} \varphi(\sigma)=\sigma,\;\varphi(\tau)=\tau,\;\varphi(x_{2})=x_{2}x_{1},\;\varphi(x_{i})=x_{i}\;(i \neq 2). \end{align}\] Then the continuous \(p_{k}\)-adic representation \(\rho_{k,\pi} \circ \varphi\) is not Hodge-Tate.
On the other hand, we cannot obtain an explicit automorphism of \(G_{k}\) that violates the Aut-intrinsic Hodge-Tate-ness of Lubin-Tate characters in the above proof of Theorem 13.
I would like to express my sincere gratitude to Professor Yuichiro Hoshi for his numerous insightful discussions and warm encouragement. I am also grateful to Reiya Tachihara for carefully reading my drafts and providing invaluable comments. I am especially thankful to Professor Yuichiro Hoshi for his detailed feedback on the drafts.