 Research
 Open Access
Characterization of global fields by Dirichlet Lseries
 Gunther Cornelissen^{1},
 Bart de Smit^{2},
 Xin Li^{3},
 Matilde Marcolli^{4, 5, 6} and
 Harry Smit^{1}Email authorView ORCID ID profile
 Received: 24 April 2018
 Accepted: 10 November 2018
 Published: 26 November 2018
Abstract
We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves Lseries.
Keywords
 Class field theory
 Lseries
 Arithmetic equivalence
Mathematics Subject Classification
 11R37
 11R42
 11R56
 14H30
1 Introduction
As was discovered by Gaßmann in 1926 [10], number fields are not uniquely determined up to isomorphism by their zeta functions. A theorem of Tate [20] from 1966 implies the same for global function fields. At the other end of the spectrum, results of Neukirch and Uchida [15, 21, 22] around 1970 state that the absolute Galois group does uniquely determine a global field. The better understood abelianized Galois group again does not determine the field up to isomorphism at all, as follows from the description of its character group by Kubota in 1957 ([13], compare [1, 17]). Funakura ([9, § I], using [8, Thm. 5]) has shown in 1980 that there exists a number field k (of degree 255! over \(\mathbb {Q}\)) and two nonisomorphic abelian extensions \(\mathbb {K}/k\) and \(\mathbb {L}/k\) of degree 4 with a bijection between all Artin Lseries of \(\mathbb {K}/k\) and \(\mathbb {L}/k\) (or, equivalently, between the Lseries of the four occurring characters).
In this paper, we prove that two global fields \(\mathbb {K}\) and \(\mathbb {L}\) are isomorphic if and only if there exists an isomorphism of groups of Dirichlet characters \(\check{\psi }: \ \check{G}^{\mathrm {ab}}_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,\check{G}^{\mathrm {ab}}_{\mathbb {L}}\) that preserves Lseries: \(L_{\mathbb {K}}(\chi ) = L_{\mathbb {L}}(\check{\psi }(\chi ))\) for all \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\). A more detailed series of equivalences can be found in the Main Theorem 3.1 below. To connect this theorem to the discussion in the previous paragraph, observe that the existence of \(\check{\psi }\) without equality of Lseries is the same as \(\mathbb {K}\) and \(\mathbb {L}\) having abelianized Galois groups that are isomorphic as topological groups, and that for the trivial character \(\chi _{\text {triv}}\), we have \(L_{\mathbb {K}}(\chi _{\text {triv}})=\zeta _{\mathbb {K}}\), so that preserving Lseries at \(\chi _{\text {triv}}\) is the same as \(\mathbb {K}\) and \(\mathbb {L}\) having the same zeta function. Contrary to the result of Funakura, our characters do not factor over a fixed Galois group.
In global function fields we explicitly construct the isomorphism of function fields via a map of kernels of reciprocity maps (much akin to the final step in Uchida’s proof [22]). For number fields, we do not need the full hypothesis: we can prove the stronger result that for every number field, there exists a character of any chosen order \(>2\) for which the Lseries does not equal any other Dirichlet Lseries of any other field (see Theorem 10.1). The method here is not via the kernel of the reciprocity map, but rather via representation theory.
We briefly indicate the relation of our work to previous results. The main result was first stated in a 2010 preprint by two of the current authors, who discovered it through methods from mathematical physics ([4], now split into two parts [5] and [6]). For function fields, the main result was proven by one of the authors in [3], using dynamical systems, referring, however, to [4] for some auxiliary results, of which we present the first published proofs in the current paper. Sections 9 and 10 contain an independent proof for the number field case that was found by the second author in 2011. The current paper not only presents full and simplified proofs, but also uses only classical methods from number theory (class field theory, Chebotarev, GrunwaldWang, and inverse Galois theory). In [19], reconstruction of field extensions of a fixed rational function field from relative Lseries is treated from the point of view of the method in Sects. 9 and 10 of this paper.
After some preliminaries in the first section, we state the main result in Sect. 3. The next few sections outline the proof of the various equivalences in the main theorem, using basic class field theory and work of Uchida and Hoshi. In the final sections, we deal with number fields, and use representation theory to prove some stronger results.
2 Preliminaries
In this section, we set notation and introduce the main object of study.
A monoid is a semigroup with identity element. If R is a ring, we let \(R^*\) denote its group of invertible elements.
Given a global field \(\mathbb {K}\), we use the word prime to denote a nonzero prime ideal if \(\mathbb {K}\) is a number field, and to denote an irreducible effective divisor if \(\mathbb {K}\) is a global function field. Let \(\mathscr {P}_{\mathbb {K}}\) be the set of primes of \(\mathbb {K}\). If \(\mathfrak {p}\in \mathscr {P}_{\mathbb {K}}\), let \(v_{\mathfrak {p}}\) be the normalized (additive) valuation corresponding to \(\mathfrak {p}\), \(\mathbb {K}_{\mathfrak {p}}\) the local field at \(\mathfrak {p}\), and \(\mathscr {O}_{\mathfrak {p}}\) its ring of integers. Let \(\mathbb {A}_{\mathbb {K},f}\) be the finite adele ring of \(\mathbb {K}\), \(\widehat{\mathscr {O}}_{\mathbb {K}}\) its ring of finite integral adeles and \(\mathbb {A}_{\mathbb {K},f}^*\) the group of ideles (invertible finite adeles), all with their usual topology. Note that in the function field case, infinite places do not exist, so that in that case, \(\mathbb {A}_{\mathbb {K},f} = \mathbb {A}_{\mathbb {K}}\) is the adele ring, \(\widehat{\mathscr {O}}_{\mathbb {K}}\) is the ring of integral adeles and \(\mathbb {A}_{\mathbb {K},f}^* = \mathbb {A}_{\mathbb {K}}^*\) is the full group of ideles. If \(\mathbb {K}\) is a number field, we denote by \(\mathscr {O}_{\mathbb {K}}\) its ring of integers. If \(\mathbb {K}\) is a global function field, we denote by q the cardinality of the constant field.
Let \(I_{\mathbb {K}}\) be the multiplicative monoid of nonzero integral ideals/effective divisors of our global field \(\mathbb {K}\), so \(I_{\mathbb {K}}\) is generated by \(\mathscr {P}_{\mathbb {K}}\). We extend the valuation to ideals: if \(\mathfrak {m}\in I_{\mathbb {K}}\) and \(\mathfrak {p}\in \mathscr {P}_{\mathbb {K}}\), we define \(v_{\mathfrak {p}}(\mathfrak {m}) \in \mathbb {Z}_{\ge 0}\) by requiring \(\mathfrak {m}= \prod _{\mathfrak {p}\in \mathscr {P}_{\mathbb {K}}} \mathfrak {p}^{v_{\mathfrak {p}}(\mathfrak {m})}\). Let N be the norm function on the monoid \(I_{\mathbb {K}}\): it is the multiplicative function defined on primes \(\mathfrak {p}\) by \(N(\mathfrak {p}):=\# \mathscr {O}_{\mathbb {K}}/\mathfrak {p}\) if \(\mathbb {K}\) is a number field and \(N(\mathfrak {p}) = q^{\deg (\mathfrak {p})}\) if \(\mathbb {K}\) is a function field. Given two global fields \(\mathbb {K}\) and \(\mathbb {L}\), we call a monoid homomorphism \(\varphi : \ I_{\mathbb {K}} \rightarrow I_{\mathbb {L}}\) normpreserving if \(N(\varphi (\mathfrak {m})) = N(\mathfrak {m})\) for all \(\mathfrak {m}\in I_{\mathbb {K}}\).
Let \(G^{\mathrm {ab}}_{\mathbb {K}}\) be the Galois group of a maximal abelian extension \(\mathbb {K}^{\mathrm {ab}}\) of \(\mathbb {K}\), a profinite topological group. There is an Artin reciprocity map \(\mathbb {A}_{\mathbb {K}}^* \rightarrow G^{\mathrm {ab}}_{\mathbb {K}}\). In the number field case, we embed \(\mathbb {A}_{\mathbb {K},f}^*\) into the group of ideles \(\mathbb {A}_{\mathbb {K}}^*\) via \(\mathbb {A}_{\mathbb {K},f}^* \ni x \mapsto (1,x) \in \mathbb {A}_{\mathbb {K}}^*\), restrict the Artin reciprocity map to \(\mathbb {A}_{\mathbb {K},f}^*\) and call this restriction \(\mathrm{rec}_{\mathbb {K}}\). In the function field case, \(\mathrm{rec}_{\mathbb {K}}\) is just the (full) Artin reciprocity map.
For any \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\), the kernel \(\ker \chi \) is an open subgroup of \(G^{\mathrm {ab}}_{\mathbb {K}}\). Therefore, the fixed field of \(\chi \), denoted \(\mathbb {K}_{\chi }\), is a finite abelian extension of \(\mathbb {K}\). Even more: as the extension is finite, there is an n such that \(\chi ^n\) is the trivial character. It follows that \(\text {im} \,\chi \) is a subgroup of the \(n^{\text {th}}\) roots of unity, hence cyclic. As we have an isomorphism \(\text {im} \,\chi \xrightarrow {{\scriptstyle \sim }}\,\mathrm {Gal}(\mathbb {K}_{\chi }/\mathbb {K})\), we obtain that \(\mathbb {K}_{\chi } / \mathbb {K}\) is a finite cyclic extension. Conversely, for any finite cyclic extension \(\mathbb {K}'\) of \(\mathbb {K}\) there exists a character \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\) such that \(\ker \chi = \mathrm {Gal}(\mathbb {K}^{\text {ab}}/\mathbb {K}')\).
The following two lemmas are easy, but we include a proof in the terminology of this paper for lack of a suitable reference.
Lemma 2.1
Let \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\). The primes in \(U(\chi )\) are exactly the primes that are unramified in \(\mathbb {K}_{\chi }\).
Proof
The character \(\chi \) factors through the quotient map \(G^{\mathrm {ab}}_{\mathbb {K}} \twoheadrightarrow \mathrm {Gal}(\mathbb {K}_{\chi }/\mathbb {K})\), giving an injective character \(\overline{\chi }: \mathrm {Gal}(\mathbb {K}_{\chi }/\mathbb {K}) \rightarrow \mathbb {T}\). Under the quotient map, the group \(\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*)\) is mapped surjectively to the inertia group \(I_{\mathfrak {p}}(\mathbb {K}_{\chi }/\mathbb {K})\), hence we have \(\chi (\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*)) = \overline{\chi }(I_{\mathfrak {p}}(\mathbb {K}_{\chi }/\mathbb {K}))\). As \(\overline{\chi }\) is injective, this group is equal to \(\{1\}\) precisely when the inertia group is trivial, i.e. when \(\mathfrak {p}\) is unramified in \(\mathbb {K}_{\chi }\). \(\square \)
Lemma 2.2
For any prime \(\mathfrak {p}\in \mathscr {P}_{\mathbb {K}}\), set \(N_{\mathfrak {p}}:=\bigcap \limits _{\chi : \; \mathfrak {p}\in U(\chi )} \ker \chi \). Then \(\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*) = N_{\mathfrak {p}}\), and the associated fixed field is equal to \(\mathbb {K}^{\mathrm {ur},\mathfrak {p}}\), the maximal abelian extension of \(\mathbb {K}\) unramified at \(\mathfrak {p}\).
Proof
By definition of \(U(\chi )\), for any \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\) with \(\mathfrak {p}\in U(\chi )\) we have \(\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*) \subseteq \ker \chi \), hence \(\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*) \subseteq N_{\mathfrak {p}}.\) As \(\mathscr {O}_{\mathfrak {p}}^*\) is compact and \(\mathrm{rec}_{\mathbb {K}}\) is continuous, \(\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*)\) is compact, and as \(G^{\mathrm {ab}}_{\mathbb {K}}\) is Hausdorff, \(\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*)\) is closed.
3 The main theorem
Theorem 3.1
 (i)There existssuch that

a monoid isomorphism \(\varphi : \ I_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,I_{\mathbb {L}}\),

an isomorphism of topological groups \(\psi : \ G^{\mathrm {ab}}_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,G^{\mathrm {ab}}_{\mathbb {L}}\), and

splits \(s_{\mathbb {K}}: \ I_{\mathbb {K}} \rightarrow \mathbb {A}_{\mathbb {K},f}^* \cap \widehat{\mathscr {O}}_{\mathbb {K}}\), \(s_{\mathbb {L}}: \ I_{\mathbb {L}} \rightarrow \mathbb {A}_{\mathbb {L},f}^* \cap \widehat{\mathscr {O}}_{\mathbb {L}}\)
$$\begin{aligned}&\psi (\mathrm{rec}_{\mathbb {K}}(\mathscr {O}_{\mathfrak {p}}^*)) = \mathrm{rec}_{\mathbb {L}}(\mathscr {O}_{\varphi (\mathfrak {p})}^*) \ \text {for every prime} \ \mathfrak {p}\ \text {of} \ \mathbb {K}, \end{aligned}$$(1)$$\begin{aligned}&\psi (\mathrm{rec}_{\mathbb {K}}(s_{\mathbb {K}}(\mathfrak {m}))) = \mathrm{rec}_{\mathbb {L}}(s_{\mathbb {L}}(\varphi (\mathfrak {m}))) \ \text {for all} \ \mathfrak {m}\in I_{\mathbb {K}}. \end{aligned}$$(2) 
 (ii)There existssuch that for every finite abelian extension \(\mathbb {K}'=\left( \mathbb {K}^{\mathrm {ab}}\right) ^N\) of \(\mathbb {K}\) (N a subgroup in \(G_{\mathbb {K}}^{\mathrm {ab}}\)) with corresponding field extension \(\mathbb {L}' =\left( \mathbb {L}^{\mathrm {ab}}\right) ^{\psi (N)}\) of \(\mathbb {L}\), \(\varphi \) is a bijection between the unramified primes of \(\mathbb {K}'/\mathbb {K}\) and \(\mathbb {L}'/\mathbb {L}\) such that

a normpreserving monoid isomorphism \(\varphi : \ I_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,I_{\mathbb {L}}\), and

an isomorphism of topological groups \(\psi : \ G^{\mathrm {ab}}_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,G^{\mathrm {ab}}_{\mathbb {L}}\)
$$\begin{aligned} \psi ({\text {Frob}}_{\mathfrak {p}}) = {\text {Frob}}_{\varphi (\mathfrak {p})} \text{ in } \mathrm {Gal}(\mathbb {L}'/\mathbb {L}). \end{aligned}$$ 
 (iii)There exists an isomorphism of topological groups \(\psi : \ G^{\mathrm {ab}}_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,G^{\mathrm {ab}}_{\mathbb {L}}\) such thatwhere \(\check{\psi }\) is given by \(\check{\psi }(\chi ) = \chi \circ \psi ^{1}\).$$\begin{aligned} L(\chi ) = L(\check{\psi }(\chi )) \ \text {for all} \ \chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}, \end{aligned}$$(3)
 (iv)
\(\mathbb {K}\) and \(\mathbb {L}\) are isomorphic as fields.
We will refer to condition (i) as a reciprocity isomorphism, condition (ii) as a finite reciprocity isomorphism, and condition (iii) as an Lisomorphism. We will prove the implication (i) \(\Rightarrow \) (ii) in Proposition 4.2, implication (ii) \(\Rightarrow \) (iii) in Proposition 5.1, and implication (iii) \(\Rightarrow \) (i) in Proposition 6.1. The equivalence with (iv) is proven in 8.1 for function fields and follows from 10.1 for number fields.
4 From reciprocity isomorphism to finite reciprocity isomorphism
Proposition 4.1
Proof

p is uniquely determined by the property \(\left( \mathscr {O}_{\mathfrak {p}}^* / \mathrm{tors}(\mathscr {O}_{\mathfrak {p}}^*) \right) ^p \ne \mathscr {O}_{\mathfrak {p}}^* / \mathrm{tors}(\mathscr {O}_{\mathfrak {p}}^*)\),

\(N(\mathfrak {p}) = p^f = \mathrm{tors}(\mathscr {O}_{\mathfrak {p}}^*) / \mathrm{tors}(\mathscr {O}_{\mathfrak {p}}^*)_p + 1\).
Proof
A prime \(\mathfrak {p}\) of \(\mathbb {K}\) is unramified in \(\mathbb {K}'\) if and only if \(\mathrm{rec}_{\mathbb {K}}(\mathscr {O}^*_\mathfrak {p}) \subseteq N\). Therefore, (1) implies that \(\varphi (\mathfrak {p})\) is unramified in \(\mathbb {L}'\), and the bijection follows by symmetry. Also, \(\mathrm{rec}_{\mathbb {K}}(s_{\mathbb {K}}(\mathfrak {p})) \text{ mod } N\) is independent of the choice of the split \(s_{\mathbb {K}}\), and equal to the Frobenius \(\mathrm {Frob}_\mathfrak {p}\) in \(\mathrm {Gal}(\mathbb {K}'/\mathbb {K})\). Hence from (2) we obtain \(\psi (\mathrm {Frob}_\mathfrak {p}) = \mathrm {Frob}_{\varphi (\mathfrak {p})}\). This proves (ii). \(\square \)
5 From finite reciprocity isomorphism to Lisomorphism
Proposition 5.1
Assume 3.1(ii). Then \(L(\chi ) = L(\check{\psi }(\chi ))\) for all \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\).
Proof
6 From Lisomorphism to reciprocity isomorphism
Proposition 6.1
Assume that there exists an isomorphism of topological groups \(\psi : \ G^{\mathrm {ab}}_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,G^{\mathrm {ab}}_{\mathbb {L}}\) with \(L(\chi ) = L(\check{\psi }(\chi ))\) for all \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\). Then 3.1(i) holds.
Proof
The idea of the proof is to construct a bijection of primes \(\varphi \) with the desired properties via the use of a type of characters that have value 1 on all but one of the primes of a certain norm.
Lemma 6.2
Proof
Definition 6.3

\(u_N(\chi ) = U_N(\chi )\),

\(V_N(\chi ) = \{ \mathfrak {p}\in U_N(\chi ) \mid \chi (\mathfrak {p}) = 1\}\), and

\(v_N(\chi ) = V_N(\chi )\).
Remark 6.4
Let \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\). As \(\chi (\mathfrak {p}) = 1\) for all \(\mathfrak {p}\in U_N(\chi )\), we have \(\mathrm{Re}(\mathscr {X}_N(\chi )) \le u_N(\chi )\). Equality holds precisely when \(\chi (\mathfrak {p}) = 1\) for all \(\mathfrak {p}\in U_N(\chi )\), i.e. \(u_N(\chi ) = v_N(\chi )\).
Lemma 6.5
\(\check{\psi }(\Xi _{\mathbb {K}}^1) = \Xi _{\mathbb {L}}^1\).
Proof
From Lemma 6.2 we obtain \(\chi \in \Xi _{\mathbb {K}}^1 \Longleftrightarrow \mathscr {X}_N(\chi ) = c_N \Longleftrightarrow \mathscr {X}_N(\check{\psi }(\chi )) = c_N \Longleftrightarrow \check{\psi }(\chi ) \in \Xi _{\mathbb {L}}^1\). \(\square \)
Lemma 6.6
For \(k \ge 3\), let \(\mu _k\) denote the kth roots of unity and let \(\zeta = \text {exp}(2\pi i / k)\). Suppose we have \(a_1, \dots , a_n \in \mu _k \cup \{0\}\) such that \(a_1 + \dots + a_n = n  1 + \zeta \). Then there is a j such that \(a_j = \zeta \), and \(a_i = 1\) for all \(i \ne j\).
Proof
Lemma 6.7
\(\check{\psi }(\Xi _{\mathbb {K}}^2) = \Xi _{\mathbb {L}}^2\).
Proof
For any character \(\chi \in \Xi _{\mathbb {K}}^2\) we have \(\chi ^k \in \Xi _{\mathbb {K}}^1\). Thus, by Lemma 6.5, \(\check{\psi }(\chi )^k \in \Xi _{\mathbb {L}}^1\). Hence for any \(\mathfrak {q}\in \mathscr {P}_{\mathbb {L}, N}\) we have that \(\check{\psi }(\chi )(\mathfrak {q}) \in \mu _k \cup \{0\}\). As \(\mathscr {X}_N(\check{\psi }(\chi )) = \mathscr {X}_N(\chi ) = c_N  1 + \zeta \), by the previous lemma there exists a single prime \(\mathfrak {q}_{\check{\psi }(\chi )}\) such that \(\check{\psi }(\chi )(\mathfrak {q}_{\check{\psi }(\chi )}) = \zeta \), while \(\check{\psi }(\chi )(\mathfrak {q}) = 1\) for all \(\mathfrak {q}\ne \mathfrak {q}_{\check{\psi }(\chi )}\). Hence \(\check{\psi }(\Xi _{\mathbb {K}}^2) \subseteq \Xi _{\mathbb {L}}^2\). By symmetry, we have equality. \(\square \)
Lemma 6.8
For every \(\mathfrak {p}' \in \mathscr {P}_{\mathbb {K}, N}\) there exists a character \(\chi \in \Xi _{\mathbb {K}}^2\) such that \(\mathfrak {p}_\chi = \mathfrak {p}'\).
Proof
The GrunwaldWang Theorem [2, Ch. X, Thm. 5] guarantees that there exists a character \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\) such that \(\chi (\mathfrak {p}') = \zeta \) and \(\chi (\mathfrak {p}) = 1\) for all primes \(\mathfrak {p}\ne \mathfrak {p}'\) of norm N, because there exists a character of \(G^{\mathrm {ab}}_{\mathbb {K}_{\mathfrak {p}'}}\) whose fixed field is the unique unramified extension of degree k of \(\mathbb {K}_{\mathfrak {p}'}\). \(\square \)
Lemma 6.9
The map \( \varphi _N: \mathscr {P}_{\mathbb {K}, N} \rightarrow \mathscr {P}_{\mathbb {L}, N} :\mathfrak {p}_\chi \mapsto \mathfrak {q}_{\check{\psi }(\chi )} \) is a welldefined bijection such that for every \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\) and \(\mathfrak {p}\in \mathscr {P}_{\mathbb {K}, N}\) we have \(\chi (\mathfrak {p}) = \check{\psi }(\chi )(\varphi _N(\mathfrak {p}))\).
Proof
This completes the proof of Proposition 6.1. \(\square \)
7 Conditional reconstruction of global fields
Proposition 7.1
Proof
Define the homomorphism \(\widehat{\mathscr {O}}_{\mathbb {K}}^* \times I_{\mathbb {K}} \rightarrow \mathbb {A}_{\mathbb {K},f}^* \cap \widehat{\mathscr {O}}_{\mathbb {K}}\) by \((u,\mathfrak {m}) \mapsto u \cdot s_{\mathbb {K}}(\mathfrak {m})\). It has a complete inverse \(\mathbb {A}_{\mathbb {K},f}^* \cap \widehat{\mathscr {O}}_{\mathbb {K}} \rightarrow \widehat{\mathscr {O}}_{\mathbb {K}}^* \times I_{\mathbb {K}}\) given by \(x \mapsto (x \cdot s_{\mathbb {K}}((x)_{\mathbb {K}})^{1}, (x)_{\mathbb {K}})\). We obtain an isomorphism \(\mathbb {A}_{\mathbb {K},f}^* \cap \widehat{\mathscr {O}}_{\mathbb {K}} \xrightarrow {{\scriptstyle \sim }}\,\widehat{\mathscr {O}}_{\mathbb {K}}^* \times I_{\mathbb {K}}\) (and similarly for \(\mathbb {L}\)).
Corollary 7.2
Proof
The reciprocity map is already defined from \(\mathbb {A}_{\mathbb {K},f}^*\) to \(G^{\mathrm {ab}}_{\mathbb {K}}\), so this follows from (5) by passing to the group of fractions \(\mathbb {A}_{\mathbb {K},f}^*\) of the monoid \(\mathbb {A}_{\mathbb {K},f}^* \cap \widehat{\mathscr {O}}_{\mathbb {K}}\). \(\square \)
We now turn to the reconstruction of isomorphism of fields from the equivalent conditions in the Main Theorem 3.1. For this, we first quote a result about conditions under which an isomorphism of multiplicative groups of fields can be extended to a field isomorphism. Let \(\mathscr {O}_{[\mathfrak {p}]}\) denote the local ring (of \(\mathbb {K}\)) at the prime \(\mathfrak {p}\).
Lemma 7.3
 (i)
\( F(1 + \mathfrak {p}\mathscr {O}_{[\mathfrak {p}]}) = 1 + \varphi (\mathfrak {p}) \mathscr {O}_{[\varphi (\mathfrak {p})]}\) (as sets, or, equivalently, as subgroups),
 (ii)
\(v_{\varphi (\mathfrak {p})} \circ F = v_{\mathfrak {p}}.\)
Proof
This follows immediately by results of Uchida for global function fields [22, Lemma 8–11], as explained in the introduction of [12]) and by Hoshi for number fields [12, Thm. D]. \(\square \)
Theorem 7.4
Assume that \(\Psi \) in (7) above satisfies \(\Psi (\mathbb {K}^*) = \mathbb {L}^*\). Then the extension of that isomorphism to a map \(\mathbb {K}\rightarrow \mathbb {L}\) by setting \(0 \mapsto 0\) is an isomorphism of fields.
Proof
8 Reconstruction of global function fields
In function fields, we can immediately apply the results of the previous section:
Proposition 8.1
If one of the equivalent conditions (i)–(iii) of Theorem 3.1 holds for two global function fields \(\mathbb {K}\) and \(\mathbb {L}\), then they are isomorphic as fields.
Proof
From the commutativity of diagram (7) we find that \( \Psi (\ker (\mathrm{rec}_{\mathbb {K}})) = \ker (\mathrm{rec}_{\mathbb {L}}). \) For a global function field \(\mathbb {K}\) we have \(\ker (\mathrm{rec}_{\mathbb {K}}) = \mathbb {K}^*\) and therefore the result follows from Theorem 7.4. \(\square \)
9 The number field case: an auxiliary result
In this section we prove the existence of certain Galois extensions of number fields with prescribed Galois groups; a result that we will use in the next section to prove the reconstruction of number fields from the consideration of specific induced representations.
Proposition 9.1
Remark 9.2
The semidirect product \(C^n \rtimes G\) is also known as the wreath product of C and the group G considered as a permutation group on G / H. For any extension \(\mathbb {L}\) of \(\mathbb {K}\) with Galois group C, the Galois group of the Galois closure of \(\mathbb {L}\) over \(\mathbb {Q}\) is a subgroup of this wreath product. The proposition asserts that the wreath product itself (i.e., the maximal subgroup, which can be viewed as the ‘generic’ case), actually occurs for some \(\mathbb {L}\).
We give a selfcontained proof, but the result also follows from [14, Thm. IV.2.2]; or, for C of order 3 one can use the existence of a generic polynomial for C and apply [7, Prop. 13.8.2].
Proof of Proposition 9.1
Let \(p \ne 2\) be a prime that is totally split in \(\mathbb {N}\) and denote by \(\mathfrak {p}_1, \dots , \mathfrak {p}_n\) the primes in \(\mathbb {K}\) lying above p. There exists a Galois extension \(\widetilde{\mathbb {K}}/\mathbb {K}\) with Galois group C in which the prime \(\mathfrak {p}_1\) is inert, while \(\mathfrak {p}_2, \dots , \mathfrak {p}_n\) are totally split (this follows, e.g., from the GrunwaldWang theorem).
Let X be the set of field homomorphisms from \(\mathbb {K}\) to \(\mathbb {N}\). Since G acts on \(\mathbb {N}\), we get an action of G on X by composition. This action is transitive and the stabilizer of the inclusion map \(\iota \in X\) is H, so X is isomorphic to G / H as a Gset. For each \(\sigma \in X\) we now consider \(\widetilde{\mathbb {N}}_{\sigma } := \widetilde{\mathbb {K}} \otimes _{\mathbb {K}, \sigma } \mathbb {N}\), where \(\mathbb {N}\) is viewed as a \(\mathbb {K}\)algebra through \(\sigma :{\mathbb {K}} \rightarrow \mathbb {N}\). The Caction on \(\widetilde{\mathbb {K}}\) induces a Caction on \(\widetilde{\mathbb {N}}_{\sigma }\) by \(\mathbb {N}\)algebra automorphisms. Setting \(P_\sigma \) to be the set of primes of \(\mathbb {N}\) that contain \(\sigma (\mathfrak {p}_1)\), we see that \(\widetilde{\mathbb {N}}_{\sigma }\) is a Galois extension of \(\mathbb {N}\) with Galois group C for which the primes in \(P_\sigma \) are inert and all other primes of \(\mathbb {N}\) over p are totally split.
For \(g\in G\) and \(\sigma \in X\) there is a natural field isomorphism \(\widetilde{g}_\sigma :\widetilde{\mathbb {N}}_{\sigma }\rightarrow \widetilde{\mathbb {N}}_{g\sigma }\) given by \(x\otimes y\mapsto x \otimes gy\) that extends the map \(\mathbb {N}\rightarrow \mathbb {N}\) given by \(y\mapsto gy\). Combining these maps for all \(\sigma \in X\) we obtain an automorphism of the tensor product \(\mathbb {M}\) that permutes the factors of the tensor product by the gaction on X. Thus, we have extended the Gaction on \(\mathbb {N}\) to a Gaction on \(\mathbb {M}\). Since each \(\widetilde{g}_\sigma \) is Cequivariant, the subgroup of \(\mathrm{Aut}\,(\mathbb {M})\) generated by G and \(C^n\) is the semidirect product \(C^n \rtimes G\). As the cardinality of this group is the field degree of \(\mathbb {M}\) over \(\mathbb {Q}\) we see that \(\mathbb {M}\) is a Galois extension of \(\mathbb {Q}\) with Galois group \(C^n \rtimes G\), and that \(\mathbb {K}\) is the invariant field of \(C^n\rtimes H\). \(\square \)
10 Characterization of number fields
Using the previous section we prove a stronger version of Proposition 8.1 for number fields.
Theorem 10.1
Let \(\mathbb {K}\) be a number field and let \(k \ge 3\). Then there exists a character \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\) of order k such that every number field \(\mathbb {L}\) for which there is a character \(\chi ' \in \check{G}^{\mathrm {ab}}_{\mathbb {L}}\) with \(L_{\mathbb {L}}(\chi ') = L_{\mathbb {K}}(\chi )\) is isomorphic to \(\mathbb {K}\).
We will use the following basic facts about Artin Lseries of representations of the absolute Galois group \(G_{\mathbb {K}}:=\mathrm {Gal}(\bar{\mathbb {Q}} / \mathbb {K})\) for a number field \(\mathbb {K}\) within a fixed algebraic closure \(\bar{\mathbb {Q}}\) of \(\mathbb {Q}\).
Lemma 10.2
 (a)
For any two representations \(\rho \) and \(\rho '\) of \(G_{\mathbb {Q}}\), \(L_{\mathbb {Q}} (\rho ) = L_{\mathbb {Q}}(\rho ')\) is equivalent to \(\rho \xrightarrow {{\scriptstyle \sim }}\,\rho '\).
 (b)
For \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\), we have \(L_{\mathbb {K}}(\chi ) = L_{\mathbb {Q}}(\mathrm {Ind}^{G_\mathbb {Q}}_{G_\mathbb {K}} (\chi ))\).
 (c)For any two number fields \(\mathbb {K}\) and \(\mathbb {L}\) within \(\bar{\mathbb {Q}}\) and characters \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}\) and \(\chi '\in \check{G}^{\mathrm {ab}}_{\mathbb {L}}\) with \(L_{\mathbb {K}} (\chi ) = L_{\mathbb {L}}(\chi ')\) we have an isomorphism of representations of \(G_\mathbb {Q}\)and the fixed fields \(\mathbb {K}_\chi \) of \(\chi \) and \(\mathbb {L}_{\chi '}\) of \(\chi '\) have the same normal closure over \(\mathbb {Q}\).$$\begin{aligned} \mathrm {Ind}^{G_\mathbb {Q}}_{G_\mathbb {K}} (\chi )\xrightarrow {{\scriptstyle \sim }}\,\mathrm {Ind}^{G_\mathbb {Q}}_{G_\mathbb {L}} (\chi ') \end{aligned}$$
Proof
Fact (a) follows from Chebotarev’s density theorem, comparing Euler factors. The basic fact (b) is due to Artin, see [16, VII.10.4.(iv)]. The isomorphism in (c) follows from (a) and (b). The last statement follows from the fact that the normal closure of \(\mathbb {K}_\chi \) over \(\mathbb {Q}\) is the fixed field of the kernel of the representation \(\mathrm {Ind}^{G_\mathbb {Q}}_{G_\mathbb {K}} (\chi )\). \(\square \)
By a monomial structure of a representation \(\rho \) of a group G we mean a Gset \({\mathscr {L}}\) consisting of 1dimensional subspaces of \(\rho \) that is Gstable (i.e. \(gL\in \mathscr {L}\) for all \(g\in G\) and \(L\in \mathscr {L}\)), and such that as a vector space we have \(\rho =\bigoplus _{L\in \mathscr {L}} L\). By choosing a single nonzero vector of L for each \(L\in \mathscr {L}\) one obtains a basis of \(\rho \) such that for every \(g \in G\) the action of g on \(\rho \) is given by a matrix with exactly one nonzero element in each row and in each column. If H is a subgroup of G and \(\chi \) a linear character of H, then the induced representation \(\rho =\mathrm {Ind}^{{G}}_{{H}}(\chi )\) naturally produces a monomial structure \(\mathscr {L}\) that is isomorphic to G / H as a Gset.
Proof of Theorem 10.1
If \(L_{\mathbb {L}}(\chi ') = L_{\mathbb {K}}(\chi )\) for two characters \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}, \chi ' \in \check{G}^{\mathrm {ab}}_{\mathbb {L}}\), the lemma implies that \(\mathrm {Ind}^{G_\mathbb {Q}}_{G_\mathbb {K}} (\chi )\) has two monomial structures, one arising from \(\chi \) and one from \(\chi '\). We see that \(\mathbb {K}\) and \(\mathbb {L}\) are isomorphic as number fields if and only if these two monomial structures are isomorphic as \(G_\mathbb {Q}\)sets (note that they are transitive \(G_\mathbb {Q}\)sets). In order to prove Theorem 10.1 it therefore suffices to choose \(\chi \) in such a way that the representation \(\mathrm {Ind}^{G_\mathbb {Q}}_{G_\mathbb {K}} (\chi )\) only has a single monomial structure.
To finish the proof we will show that \(\mathscr {L}\) is the unique monomial structure on the representation \(\rho \). Suppose that \(\mathscr {M}\) is another monomial structure on \(\rho \). The trace of the element \(c=(\zeta , 1, \ldots ,1)\in C^n\) on \(\rho \) is equal to \(n1+\zeta \). On the other hand, c permutes the elements of \(\mathscr {M}\), so the trace of c on \(\rho \) is also equal to the sum of kth roots of unity \(\zeta _M \in \mu _k\) where M ranges over those lines \(M\in \mathscr {M}\) with \(cM=M\), and \(\zeta _M\) is the scalar by which c then acts on M. Since \(k\ge 3\) and \(\mathscr {L}\) and \(\mathscr {M}\) have the same number of elements, Lemma 6.6 implies that \(cM=M\) for all \(M\in \mathscr {M}\). It follows that c acts trivially on the set \(\mathscr {M}\). Since the Gconjugates of c generate \(C^n\) we deduce that \(C^n\) acts trivially on the set \(\mathscr {M}\). Thus, \(\mathscr {M}\) consists of 1dimensional \(C^n\)submodules of \(\rho \). This implies that \(\mathscr {M}\subset \mathscr {L}\), so \(\mathscr {M}=\mathscr {L}\) for cardinality reasons.
Remark 10.3
Not every representation has a unique monomial structure: consider the isometry group of a square, the dihedral group \(D_4\) of order 8, with its standard 2dimensional representation. It has two distinct monomial structures (consisting of the axes and the diagonals) and these are not isomorphic as \(D_4\)sets.
By realizing \(D_4\) as a Galois group over \(\mathbb {Q}\) this gives rise to quadratic fields \(\mathbb {K}\) and and \(\mathbb {L}\) with quadratic characters \(\chi \in \check{G}^{\mathrm {ab}}_{\mathbb {K}}[2]\) and \(\chi '\in \check{G}^{\mathrm {ab}}_{\mathbb {L}}[2]\) that satisfy \(L_{\mathbb {K}} (\chi ) = L_{\mathbb {L}}(\chi ')\) while \(\mathbb {K}\) is not isomorphic to \(\mathbb {L}\). This shows that the method of proof of the theorem fails without the assumption that \(k\ge 3\). Concretely, \(\mathrm {Gal}(\mathbb {Q}(\root 4 \of {2}, i)/\mathbb {Q}) \xrightarrow {{\scriptstyle \sim }}\,D_4\), and we find \(L_{\mathbb {K}}(\chi )=L_{\mathbb {L}}(\chi ')\) for \(\mathbb {K}=\mathbb {Q}(\sqrt{2})\), \(\mathbb {L}=\mathbb {Q}(i\sqrt{2})\) and \(\chi \) and \(\chi '\) uniquely determined by \(\mathbb {K}_{\chi } = \mathbb {Q}(\root 4 \of {2})\) and \(\mathbb {L}_{\chi '}=\mathbb {Q}(i \sqrt{2}, (1+i)\root 4 \of {2})\).
In [18, Thm. 3.2.2] it is shown that \(\mathbb {K}=\mathbb {Q}(\root 8 \of {5})\) provides a counterexample to the statement of the theorem for \(k=2\). On the other hand, in [18, Thm. 2.2.2], a similar method as in our proof is used to show that every number field is characterized uniquely by the Lseries of two suitable quadratic characters.
11 Comparison of different methods of proof
On the other hand, it is not possible to copy the proof of Theorem 10.1 for function fields, as this would force fixing a rational subfield \(\mathbb {F}_q(t)\) inside both \(\mathbb {K}\) and \(\mathbb {L}\) (that plays the role of \(\mathbb {Q}\) in the number field proof), for which there are infinitely many, noncanonical, choices. However, Theorem 10.1 does hold in the relative setting of separable geometric extensions of a fixed rational function field of characteristic not equal to 2, compare [19]. It is unclear to us whether the analogue of Theorem 10.1 holds for a global function field without fixing a rational subfield. It does seem that Lseries of global function fields, as polynomials in \(q^{s}\), contain less arithmetical information than their number field cousins (compare [11]).
Notes
Declarations
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