L-series and isomorphisms of number fields

doi:10.1016/j.jnt.2020.05.021

Journal of Number Theory

Volume 221, April 2021, Pages 339-357

https://doi.org/10.1016/j.jnt.2020.05.021 Get rights and content

Abstract

Two number fields with equal Dedekind zeta function are not necessarily isomorphic. However, if the number fields have equal sets of Dirichlet L-series then they are isomorphic. We extend this result by showing that the isomorphisms between the number fields are in bijection with L-series preserving isomorphisms between the character groups.

Introduction

Kronecker [8] started a programme to characterise a number field and its extensions using only information about the prime ideals (“primes” from now on). Gaßmann ([4, pp. 671–672]) showed that number fields with the same Dedekind zeta function (called arithmetically equivalent fields) need not be isomorphic (see also [15] and [16]). Even all local information does not suffice: Komatsu [7] gave an example of two non-isomorphic number fields with isomorphic adele rings.

Fortunately, there has also been success: the Neukirch-Uchida Theorem ([11, Satz 2] and [13, Ch. XII, §2]) states that two number fields with isomorphic absolute Galois groups are necessarily isomorphic. Uchida ([20, Main Thm.], see also [13, Ch. XII, §2, Cor. 12.2.2]) proved later that the link between a number field and its absolute Galois group is even stronger: the automorphisms of a number field are in bijection with the outer automorphisms of the absolute Galois group. As a drawback, the structure of the absolute Galois group is, even for $Q$ , not very well understood.

A reasonable object to consider next is the abelianized absolute Galois group. Although it is well-understood by class field theory, it lacks the capacity to uniquely determine the underlying field: Kubota ([9, §4]) gave a classification of the abelianized absolute Galois groups of number fields, which was used by Onabe [14] to show that there exist non-isomorphic imaginary quadratic fields with isomorphic abelianized absolute Galois groups. Stevenhagen and Angelakis ([1, Thm. 4.1]) gave an explicit form of the abelianized absolute Galois group for many imaginary quadratic fields of low class number and found that most of these groups were isomorphic.

For any number field, there exists a Dirichlet L-series of a well-chosen character of odd prime order that does not occur as a Dirichlet L-series of any other non-isomorphic number field, see [3, Thm. 10.1], and the same holds for a pair of quadratic characters ([18, Thm. 2.2.2]). Therefore, if two number fields share all L-series for characters of a certain prime order, then they are isomorphic. However, these theorems do not provide explicit isomorphisms, hence the question arises whether or not one can link the automorphism group of a number field to its Dirichlet L-series.

The main result of this paper is that when one considers the structure of the Dirichlet character group of the absolute Galois group along with their Dirichlet L-series, then one can not only recover the underlying number field, but also its automorphism group.

Let K and $K^{'}$ be number fields, and denote by ${\overset{ˇ}{G}}_{K} [l]$ and ${\overset{ˇ}{G}}_{K^{'}} [l]$ the l-torsion of the character groups of their absolute Galois groups. Moreover, let ${Iso}_{L -series} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ be the set of isomorphisms

such that χ and

ψ (χ)

have the same L-series for any

χ \in {\overset{ˇ}{G}}_{K} [l]

. We prove the following theorem:

Theorem A

Let K and $K^{'}$ be number fields, and let l be any prime number. There exists a bijection

The case

l = 2

was first proven by Gabriele Dalla Torre in his unpublished PhD thesis. The idea of the construction of the map Θ is similar to the approach taken by Neukirch and Uchida: they construct a bijection of primes that preserves the decomposition groups inside the absolute Galois groups. Given a map

ψ \in {Iso}_{L -series} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])

, we first derive a bijection of primes ϕ that is compatible with ψ. What follows is an application of the following theorem:

Theorem B

Let K and $K^{'}$ be number fields of the same degree and denote the primes of $K^{'}$ by $P_{K^{'}}$ . Let $S$ be a subset of the primes of inertia degree 1 of K. Suppose that for some finite extension $L / K$ , $S$ contains $Spl (L / K)$ except for a Dirichlet density zero set. Furthermore, suppose there exists an isomorphism

with an injective norm-preserving map

ϕ : S \to P_{K^{'}}

such that

χ (p) = ψ (χ) (ϕ (p))

for every prime

p \in S

. Then

K ≃ K^{'}

and there is a unique

such that the bijection of primes induced by

σ_{ψ}

equals ϕ on

S

except for finitely many exceptions.

To complete the proof of Theorem A, we check that the map that sends ψ to $σ_{ψ}$ is a bijection.

The proofs of both theorems rely heavily on the Grunwald-Wang theorem (see [2, Ch. X, Thm. 5]), which allows for the creation of characters with specific values on any finite set of primes. Furthermore, we use the Chebotarev density theorem ([12, Ch. VII, §13, p. 545]) to bound degrees of extensions.

Lastly, we state a corollary that can be seen as an analogue of theorems about different types of equivalence of number fields (such as arithmetical or Kronecker equivalence, [6, Ch. II & III]), see for example the Main Theorem of [17] and [5, Satz 1]. Both theorems guarantee that no density zero set of “exceptional primes” can exist.

Corollary

Let K and $K^{'}$ be number fields, and let $S$ be a set of primes of K of Dirichlet density one. Suppose there is an isomorphism

with an injective norm-preserving map

ϕ : S \to P_{K^{'}}

such that

χ (p) = ψ (χ) (ϕ (p))

for any

χ \in {\overset{ˇ}{G}}_{K} [l]

and

p \in S

. Then ϕ can be uniquely extended to a norm-preserving bijection between all primes such that

χ (p) = ψ (χ) (ϕ (p))

for any

χ \in {\overset{ˇ}{G}}_{K} [l]

and any prime

p

of K.

A natural follow-up question that is not considered in this paper is whether or not Theorem A can be strengthened to include not just isomorphisms, but homomorphisms between number fields.

Section snippets

Preliminaries

In this section we set up notation, introduce the main objects of study, and state some convenient lemmas.

We fix an algebraic closure $\overline{Q}$ of $Q$ throughout the entire paper. We denote number fields by k, K, $K^{'}$ , and N, where usually $N / Q$ is a Galois extension. We use $p$ for the prime ideals of K (that we will call “primes”), $q$ for the primes of $K^{'}$ , and $P$ for the primes of N.

The set of primes of a number field K is denoted $P_{K}$ , and the set of primes lying over a rational prime p is denoted $P_{K, p}$ . Given

Main theorem

The main theorem establishes bijections between the following four sets of isomorphisms:

Definition 3.1

•
$Iso (K, K^{'})$ is the set of field isomorphisms
.
•
${Iso}_{L -series} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ is the set of isomorphisms
for which $L (χ, s) = L (ψ (χ), s)$ holds for all characters $χ \in {\overset{ˇ}{G}}_{K} [l]$ .
•
${Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ is the set of isomorphisms
for which there is a norm-preserving bijection $ϕ : P_{K} \to P_{K^{'}}$ such that $χ (p) = ψ (χ) (ϕ (p))$ for all $χ \in {\overset{ˇ}{G}}_{K} [l]$ and $p \in P_{K}$ .
•
${Iso}_{P}^{δ} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ is the set of isomorphisms
such that there

Isomorphisms between character groups

The aim of this section is to show that any isomorphism $ψ \in {Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ is characterised by its associated bijection of primes.

Lemma 4.1

For any $ψ \in {Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ there is a unique bijection of primes $ϕ : P_{K} \to P_{K^{'}}$ such that $ψ (χ) (ϕ (p)) = χ (p) .$

Proof

Indeed, let $ϕ_{1}$ and $ϕ_{2}$ be bijections $P_{K} \to P_{K^{'}}$ such that $ψ (χ) (ϕ_{1} (p)) = χ (p) = ψ (χ) (ϕ_{2} (p))$ for any $p \in P_{K}$ and $χ \in {\overset{ˇ}{G}}_{K} [l]$ , and suppose for a certain prime $\tilde{p} \in P_{K}$ that $ϕ_{1} (\tilde{p}) \neq ϕ_{2} (\tilde{p})$ . By Lemma 2.1 there exists a $χ^{'} \in {\overset{ˇ}{G}}_{K^{'}} [l]$ such that $χ^{'} (ϕ_{1} (\tilde{p})) = 1$ and $χ^{'} (ϕ_{2} (\tilde{p})) \neq 1$ . Then $χ^{'} (ϕ_{1} (\tilde{p})) = ψ^{-}$

Maps $Θ_{1}$ , $Θ_{2}$ , and $Θ_{3}$

We prove that the three maps $Θ_{1}$ , $Θ_{2}$ , and $Θ_{3}$ of Theorem 3.2 can all be chosen as $ψ \mapsto ψ$ . This establishes a bijection between ${Iso}_{L -series} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ and ${Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ . Each of the following subsections deals with one of the maps.

The map $Θ_{1}$ .

This is a triviality: the conditions imposed on elements of ${Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ are stronger than those imposed on elements of ${Iso}_{P}^{δ} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ . Hence the map $Θ_{1} : ψ \mapsto ψ$ is an injective map ${Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l]) \to {Iso}_{P}^{δ} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ .

The map $Θ_{2}$ .

This

Maps $Θ_{4}$ and $Θ_{5}$

In this section we construct injective maps between $Iso (K, K^{'})$ and ${Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ and between ${Iso}_{P}^{δ} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ and $Iso (K, K^{'})$ . The majority of the section is devoted to the map $Θ_{5}$ : it is constructed by first moving to the Galois closure, finding a suitable automorphism of this Galois closure, and then showing that this restricts to an isomorphism

Elements of $Iso (K, K^{'})$ , ${Iso}_{P} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ , and ${Iso}_{P}^{δ} ({\overset{ˇ}{G}}_{K} [l], {\overset{ˇ}{G}}_{K^{'}} [l])$ come equipped with a bijection of primes, and they are

Acknowledgments

We would like to thank Gabriele Dalla Torre for sharing his proof of Theorem 3.2 for the case $l = 2$ , and Gunther Cornelissen for many helpful discussions and remarks. Many thanks to the referee for their precise and useful comments.

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General SectionL-series and isomorphisms of number fields

Abstract

Introduction

Section snippets

Preliminaries

Main theorem

Isomorphisms between character groups

Maps Θ1, Θ2, and Θ3

Maps Θ4 and Θ5

Acknowledgments

J. Number Theory

J. Number Theory

Imaginary quadratic fields with isomorphic abelian Galois groups

Class Field Theory

Characterization of global fields by Dirichlet L-series

Res. Number Theory

Bemerkungen zur Vorstehenden Arbeit von Hurwitz: Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppen

Math. Z.

Zahlkörper mit gleicher Primzerlegung

J. Reine Angew. Math.

Arithmetical Similarities

On adèle rings of arithmetically equivalent fields

Acta Arith.

Über die Irreducibilität von Gleichungen

Monatsber. K. Preuss. Akad. Wiss. Berlin

General Section
L-series and isomorphisms of number fields

Maps $Θ_{1}$ , $Θ_{2}$ , and $Θ_{3}$

Maps $Θ_{4}$ and $Θ_{5}$