General SectionL-series and isomorphisms of number fields
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 , 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 be number fields, and denote by and the l-torsion of the character groups of their absolute Galois groups. Moreover, let be the set of isomorphisms such that χ and have the same L-series for any . We prove the following theorem:
Theorem A Let K and be number fields, and let l be any prime number. There exists a bijection Theorem B Let K and be number fields of the same degree and denote the primes of by . Let be a subset of the primes of inertia degree 1 of K. Suppose that for some finite extension , contains except for a Dirichlet density zero set. Furthermore, suppose there exists an isomorphism with an injective norm-preserving map such that for every prime . Then and there is a unique such that the bijection of primes induced by equals ϕ on except for finitely many exceptions.
To complete the proof of Theorem A, we check that the map that sends ψ to 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 be number fields, and let be a set of primes of K of Dirichlet density one. Suppose there is an isomorphism with an injective norm-preserving map such that for any and . Then ϕ can be uniquely extended to a norm-preserving bijection between all primes such that for any and any prime 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 of throughout the entire paper. We denote number fields by k, K, , and N, where usually is a Galois extension. We use for the prime ideals of K (that we will call “primes”), for the primes of , and for the primes of N.
The set of primes of a number field K is denoted , and the set of primes lying over a rational prime p is denoted . Given
Main theorem
The main theorem establishes bijections between the following four sets of isomorphisms: Definition 3.1 is the set of field isomorphisms . is the set of isomorphisms for which holds for all characters . is the set of isomorphisms for which there is a norm-preserving bijection such that for all and . is the set of isomorphisms such that there
Isomorphisms between character groups
The aim of this section is to show that any isomorphism is characterised by its associated bijection of primes.
Lemma 4.1 For any there is a unique bijection of primes such that
Proof Indeed, let and be bijections such that for any and , and suppose for a certain prime that . By Lemma 2.1 there exists a such that and . Then
Maps , , and
We prove that the three maps , , and of Theorem 3.2 can all be chosen as . This establishes a bijection between and . Each of the following subsections deals with one of the maps.
The map .
This is a triviality: the conditions imposed on elements of are stronger than those imposed on elements of . Hence the map is an injective map .
The map .
This
Maps and
In this section we construct injective maps between and and between and . The majority of the section is devoted to the map : 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 , , and 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 , and Gunther Cornelissen for many helpful discussions and remarks. Many thanks to the referee for their precise and useful comments.
References (20)
On the equation
J. Number Theory
(1977)On the class numbers of arithmetically equivalent fields
J. Number Theory
(1978)- et al.
Imaginary quadratic fields with isomorphic abelian Galois groups
- et al.
Class Field Theory
(2008) - et al.
Characterization of global fields by Dirichlet L-series
Res. Number Theory
(2019) Bemerkungen zur Vorstehenden Arbeit von Hurwitz: Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppen
Math. Z.
(1926)Zahlkörper mit gleicher Primzerlegung
J. Reine Angew. Math.
(1978)Arithmetical Similarities
(1998)On adèle rings of arithmetically equivalent fields
Acta Arith.
(1984)Über die Irreducibilität von Gleichungen
Monatsber. K. Preuss. Akad. Wiss. Berlin
(1880)