aa r X i v : . [ m a t h . N T ] S e p Weak arithmetic equivalence.
Guillermo Mantilla-Soler
Abstract
Inspired by the invariant of a number field given by its zeta function, we definethe notion of weak arithmetic equivalence , and show that under certain ramifica-tion hypotheses, this equivalence determines the local root numbers of the numberfield. This is analogous to a result of Rohrlich on the local root numbers of a ra-tional elliptic curve. Additionally, we prove that for tame non-totally real numberfields, the integral trace form is invariant under arithmetic equivalence.
One of the most fundamental arithmetical invariants of a number field is its Dedekindzeta function. It is well known that pairs of number fields with the same zeta function,
Arithmetically equivalent number fields , share many arithmetic invariants. Among themthe discriminant, unit group, signature, the product of class number times regulator andsome others (see [7, III, § § L -function of asemistable elliptic curve determines its local root numbers (see Theorem 2.24).1 .1 Motivation and results To understand the motivation behind our definitions, it’s important to review Perlis’result on the rational trace form. Let K be a number field of degree n , and let s n be thesymmetric group in n symbols. The Dedekind zeta function ζ K is the L -function of anArtin representation ρ K of G Q , namely the representation obtained by composing thepermutation representation π K : G Q → s n and the natural inclusion j : s n → GL n ( C ).Since we are interested in the equivalence class of the representation ρ K , we think of itas an element in H ( Q , GL n ( C )), and the same for π K . The natural inclusion ι : s n → O n ( Q )induces a map of pointed sets ι ∗ : H ( Q , s n ) → H ( Q , O n ) . Since H ( Q , O n ) classifies isometry classes of non-degenerate rational quadratic formsof dimension n , there exists a quadratic form corresponding to ι ∗ ( π K ). Perlis’ realiza-tion(see [15, Lemma 1.b]) is that such a form is precisely the rational trace form i.e.,the rational quadratic form associated to the bilinear pairing K × K → Q ( x, y ) tr K/ Q ( xy ) . The above result can be interpreted as a relation between the rational trace form ofthe field K and the representation ρ K . Presumably such a relation led Perlis to: Theorem 1.1 (Perlis) . Let K and L be arithmetically equivalent number fields. Then, K and L have isometric rational trace forms. The main ideas behind Perlis’ proof of the above are the following: Using formulas ofSerre for the local Hasse invariants of the trace form Perlis shows that for every prime p the local p -Hasse invariant of the trace form ι ∗ ( π K ) can be written in terms of the p -localStiefel-Whitney class of the representation ρ K . Moreover, due to a formula of Deligne,such numbers can be written in terms of local root numbers of the representations ρ K and det( ρ K ) (see § ρ K at complex conjugation, it can beseen that the signature of ι ∗ ( π K ) is determined by the representation ρ K . It follows,thanks to the Hasse principle, that the isometry class of the rational trace ι ∗ ( π K ) is com-pletely determined by the representation ρ K . On the other hand, by the Chebotarev’sdensity theorem, the representation ρ K is completely determined by ζ K . In particular,two number fields with the same Dedekind zeta function share their rational trace.2 .1.1 Main results Recall that the integral trace form over K is the integral quadratic form, denoted by q K , that is obtained by restricting the rational trace form to the maximal order O K .From an arithmetic point of view, the integral trace form is a better invariant thanthe rational trace form. One can see, for example, that the rational trace form doesnot even determine the discriminant of the number field: any Z / Z -extension of Q hasrational trace isometric to h , , i (for a more general situation see [3, Corollary I.6.5]).On the other hand, the integral trace can characterize the field in some non-trivial cases(see [9]). It is natural then to wonder whether or not arithmetical equivalence impliesequality between integral traces. An immediate observation that one can make fromPerlis’ work is that to ensure an isometry between the rational traces of two numberfields, it is not necessary to have equality between their Dedekind zeta functions butonly local information at finitely many places. With this observation in mind, we setcourse to find out if knowing the local root numbers is sufficient to determine the integraltrace form. Explicitly we prove: Theorem (cf. Theorem 2.9) . Let
K, L be two non-totally real tamely ramified numberfields of the same discriminant and signature. Then, the integral trace forms of K and L are isometric if and only if the p -local root numbers of ρ K and ρ L coincide for everyodd prime p that divides disc( K ) . Since the Dedekind zeta function ζ K determines the discriminant and the signatureof the field K , see [16], Theorem 2.9 gives a two fold generalization of [15, Corollary 1]:1. On one hand, the conclusion of having isometric integral traces is stronger thanhaving isometric rational traces. Example 1.2.
Let
K, L be two Galois cubic fields with different discriminant(take for instance the two cubic fields of discriminant 49 and 81 respectively). Aspointed out before we have that q K ⊗ Q ∼ = q L ⊗ Q ∼ = h , , i , but clearly q K = q L .
2. On the other hand, the hypothesis of having the same local root numbers is weakerthan that of having the same Dedekind zeta functions.
Example 1.3.
Take any two non isomorphic tame Galois cubic fields of the samediscriminant (take for instance the two cubic fields of discriminant 8281 = 7 · ).Since their integral traces are isomorphic (see [9, Theorem 3.1]) it follows fromTheorem 2.9 that they have the same root numbers at every prime. However, byLemma 2.18, they do not have the same zeta function.We must however impose some necessary ramification restrictions so that the analogyis still valid in the integral case (see Remark 2.11).3 heorem (cf Theorem 2.10) . Let
K, L be two non-totally real tamely ramified arithmeti-cally equivalent number fields. Then, the integral trace forms q K and q L are isometric. An interpretation of Theorem 2.9 is that in order to know the integral trace, youonly need local information from the Dedekind zeta function at the “bad” places. Sincethe zeta function is a product of local L -functions, it is natural to wonder how thoselocal factors, at the ramified places, influence the behavior of the integral trace. Inspiredby this, we define the notion of weak arithmetic equivalence and show that indeed theintegral trace is determined by local L -functions for a large family of number fields. Definition (cf Definition 2.12) . Let
K, L be two number fields. We say that K and L are weakly arithmetically equivalent if and only if • A prime p ramifies in K if and only if p ramifies in L , • L p ( s, ρ K ) = L p ( s, ρ L ) for p ∈ { p : p | disc( K ) } ∪ {∞} . Theorem (cf. Theorem 2.19) . Let
K, L be two weakly arithmetically equivalent numberfields which are tame and non-totally real. Suppose that any of the following is satisfied:(a) K and L have degree at most (b) K has fundamental discriminant. (c) K and L are Galois over Q .Then, the integral trace forms of K and L are isometric.Remark . See Question 2.21 and the remark after it for further thoughts on Theorem2.19.
We start by recalling briefly how the Dedekind zeta function of a number field can beseen as the Artin L -function of a representation of the absolute Galois group G Q . See[14] and [18] for details and unexplained terminology. Recall that a discriminant is called fundamental if it is the discriminant of a quadratic field. .1.1 Dedekind zeta and Artin representations Let L be a number field with Galois closure ˜ L . Let G ( L ) := Gal( ˜ L/ Q ) and H ( L ) :=Gal( ˜ L/L ). By composing the natural map G Q → G ( L ) with the natural action G ( L ) → Sym( G ( L ) /H ( L )), one gets a permutation representation π L ∈ H ( G Q , S deg( L ) ) i.e., π L = Inf G Q G ( L ) (Ind G ( L ) H ( L ) G Q on the embeddingsof L into ˜ L . The usual inclusions S deg( L ) ֒ → O deg( L ) ( C ) ֒ → GL deg( L ) ( C ) , together with π L , yield an Artin representation ρ L ∈ H ( G Q , GL deg( L ) ( C )) . By theinduction property of Artin L -functions, the Dedekind zeta function ζ L ( s ) of L is nothingother than the Artin L -function L ( s, ρ L ) associated to ρ L . The function L ( s, ρ L ) isdefined as a product of local functions L p ( s, ρ L ) for each finite prime p , where the localparts are defined by restricting ρ L to a decomposition subgroup G Q p . By looking at theusual Euler product of ζ L ( s ), we see that the local factors are given by L p ( s, ρ L ) = g Y i =1 Ç − ( p − s ) f i å , where g is the number of primes in L lying over p and the f i ’s are the residue degreesof a rational prime p in its decomposition in L . Complete L -function and root numbers. Given an Artin representation ρ , with Artin L -function L ( s, ρ ), its complete L -function Λ( s, ρ ) is defined asΛ( s, ρ ) := A ( ρ ) s/ L ∞ ( s, ρ ) L ( s, ρ ) , where A ( ρ ) is a positive integer divisible only by the finite primes at which ρ ramifies,and L ∞ ( s, ρ ) is a Gamma factor that depends on the value of ρ at complex conjugation.The complete L -function satisfies a functional equationΛ( s, ρ ) = W ( ρ )Λ(1 − s, ρ ∨ ) , where ρ ∨ is the contragradient representation and W ( ρ ) is a complex number called the root number of ρ . Due to a result of Deligne (see [4] and [21, § local root numbers W ( ρ ) = Y p W p ( ρ ) . W p ( ρ ) are complex numbers of norm 1, moreover W p ( ρ ) = 1whenever ρ is unramified at p .In the case of the permutation representation ρ L , A ( ρ L ) is equal to | Disc( L ) | . More-over, since ρ L is an orthogonal representation it is a result of Fr¨ohlich and Queyrut, see[21, § W ( ρ L ) = 1. The local infinite factor of ρ L is given by L ∞ ( s, ρ L ) := Γ r R ( s )Γ r C ( s )where r (resp r ) is the number of real (resp complex) embeddings of L ,Γ R = ( π ) − s/ Γ Å s ã and Γ C = 2(2 π ) − s Γ( s ) , and Γ( s ) is the usual Gamma function. We call the local root numbers W p ( ρ L ) the rootnumbers of the number field L . The connection between root numbers W p ( ρ L ) of ρ L and the trace form tr L/ Q ( x ) wasfirst realized by Perlis by relating the results of Serre on Stiefel-Whitney invariants ofthe representation ρ L and those of Deligne on normalized root numbers. Second Stiefel-Whitney invariant and local root numbers.
Let L be a degree n numberfield of discriminant d . The second Stiefel-Whitney invariant w ( L ) of L , or of ρ L , isa 2-torsion element in the Brauer group of Q defined as follows: Recall the standardpresentation of the symmetric group s n : ¨ t , ..., t n − : t i = 1 for 1 ≤ i ≤ n − , ( t i t i +1 ) = 1 for 1 ≤ i ≤ n − , [ t i , t j ] = 1 for 2 ≤ | i − j | ∂ Let ˜ s n be the ± s n defined by h s , ..., s n − , w : s i = w = 1 for 1 ≤ i ≤ n − , ( s i s i +1 ) = 1 for 1 ≤ i ≤ n − , [ s i , w ] = 1for 1 ≤ i ≤ n − , [ s i , s j ] = w for 2 ≤ | i − j |i where 1 −→ h w i −→ ˜ s n −→ s n −→ s i t i . The extension ˜ s n → s n defines an element ℓ ∈ H ( s n , ± π L : G Q → s n to π ∗ L : H ( s n , ± → H ( G Q , ± w ( L ) = π ∗ L ( ℓ ) . The local p -part w ( L ) p of w ( L ) is the element of the Brauer group of Br ( Q p ) obtained from w ( L ) viarestriction. 6 heorem . Keeping the notation as the above, for all finite pw ( L ) p = h p ( q L )(2 , d ) p where h p ( q L ) and ( · , · ) p denote the local p Hasse-Witt invariant of the trace form andthe Hasse symbol, respectively.Proof.
See [19, Th´eor`eme 1].
Theorem . Keeping the notation as the above, for all finite pw ( L ) p = W p ( ρ L ) W p (det( ρ L )) . Proof.
See [21, § Corollary . h p ( q L ) = (2 , d ) p W p ( ρ L ) W p (det( ρ L )) . The following facts about the integral trace form will be useful in proving our mainresults. We included them here for the reader’s convenience.
The genus
The following Jordan decomposition of the local integral trace, for tameextensions, has been obtained by Erez, Morales and Perlis. For details, references andproofs see [12].
Theorem . [12, Theorem 0.1] Let K be a degree n number field and let p be an oddprime which is at worst tamely ramified in K . Then, there exist α, β ∈ Z ∗ p , and aninteger < f ≤ n , such that q K ⊗ Z p ∼ = h , ..., , α i | {z } f M h p i ⊗ h , ..., , β i | {z } n − f . Corollary . Let
K, L be two tamely ramified number fields of the same discriminantand signature. Then, the integral trace forms q K and q L are in the same genus if andonly if h p ( q K ) = h p ( q L ) for every odd prime p . roof. If q K and q L are in the same genus then clearly they have the same local symbolsat every prime. Conversely, let p be an odd prime and suppose that h p ( q K ) = h p ( q L ).Thanks to Theorem 2.4 we can apply [11, Lemma 2.1] to the forms q K ⊗ Z p , q L ⊗ Z p and conclude that q K ⊗ Z p ∼ = q L ⊗ Z p . Since q K ⊗ Z ∼ = q L ⊗ Z (see [11, Proposition 2.7]), and the fields have the samesignature, the result follows. The spinor genus
For details, references and proofs about the spinor genus of theintegral trace see [13].
Theorem . [13, Theorem 2.12] Let K be a number field of degree at least . Then,the genus of integral trace form q K contains only one spinor genus. The main application of the spinor genus is that it gives a way to determine whentwo number fields with ramification at infinity have isometric integral traces.
Proposition . Let
K, L be two non-totally real number fields. Then the forms q K and q L are in the same spinor genus if and only if they are isometric.Proof. Since the discriminant and degree of a number field are invariants of the spinor(resp. isometry) class of its integral trace form, we may assume that both fields havedegree n ≥
3. Since the fields are non-totally real, the forms q K and q L are indefiniteand of dimension at least 3. By Eichler’s Theorem (see [5]) the spinor class and isometryclass coincide for indefinite forms of dimension bigger than 2, hence the result. We now have all we need to give proofs to Theorems 2.9 and 2.10.
Lemma . Let
K, L be number fields of the same discriminant. Then, for all primes p W p (det( ρ K )) = W p (det( ρ L )) . Proof.
Since ρ K is an orthogonal representation, the one dimensional representationdet( ρ K ) → ± δ : Gal( Q ( √ d ) / Q ) → ± , d ∈ Q ∗ / ( Q ∗ ) depends on K . Hence, if σ ∈ G Q , we have that det( ρ K )( σ ) = σ ( √ d ) √ d . On the other hand, a calculation shows that d = disc( K ) (see for example [15, Pg 427,second paragraph]). Since K and L have the same discriminant, the representationsdet( ρ K ) and det( ρ L ) coincide hence so do their root numbers. Theorem . Let
K, L be two tamely ramified number fields of the same discriminantand signature. Then, the integral trace forms q K and q L are in the same spinor genus ifand only if W p ( ρ K ) = W p ( ρ L ) for every odd prime p that divides disc( K ) . In particular,for a tame non-totally real number field, the integral trace form is completely determinedby the local root numbers of the field.Proof. We may assume that the fields have degree at least 3. Thanks to Corollary 2.3and Lemma 2.8 we have that for a prime pW p ( ρ K ) = W p ( ρ L ) if and only if h p ( q K ) = h p ( q L ) . On the other hand, it follows from Theorem 2.6 and Corollary 2.5 that the forms q K and q L are in the same spinor genus if and only if h p ( q K ) = h p ( q L ) for every odd prime p .Since h p ( q K ) = h p ( q L ) = 1 for unramified primes, the result follows. The last assertionin the theorem follows from Proposition 2.7As an immediate consequence of Theorem 2.9, we obtain a generalization of Perlis’result [15, Corollary 1] to the integral trace. Theorem . Let
K, L be two non-totally real tamely ramified arithmetically equivalentnumber fields. Then, the integral trace forms q K and q L are isometric.Proof. Since arithmetically equivalent number fields share discriminants, signatures andlocal root numbers the result follows from Theorem 2.9 and Proposition 2.7.
Remark . In contrast to Perlis’ result on the rational trace, arithmetic equivalencedoes not imply isometry between integral traces. In fact, as the following examplesshow, the ramification conditions imposed on the above theorem are not only sufficientbut also necessary. Example 2.3 of [10] shows that the tameness condition in Theorem2.10 is necessary. On the other hand, if F and L are the number fields defined by thepolynomials p F = x − x − x + 25 x + 755 x + 496 x − x − ,p L = x − x − x − x + 480 x + 793 x + 233 x + 19 , it can be shown, as in the proof of [10, Proposition 2.7], that F and L are non-isomorphicarithmetically equivalent number fields. Furthermore, they are totally real and theircommon discriminant is equal to 5 · · hence they are tamely ramified. A cal-culation in MAGMA shows that their integral traces are not equivalent. This exampleshows that the condition at infinity in Theorem 2.10 is necessary.9 .3 Weak arithmetic equivalence After generalizing Perlis’ work on arithmetic equivalence to the integral trace form, weare ready to go further by using weak arithmetic equivalence . To make statements aboutprime decomposition as general as possible, see for example [11, Remark 2.6], we useConway’s notation ( p = −
1) for the prime at infinity ([1, Chapter 15, § Definition . Let
K, L be two number fields. We say that K and L are weakly arith-metically equivalent if and only if • A prime p ramifies in K if and only if p ramifies in L , • L p ( s, ρ K ) = L p ( s, ρ L ) for p ∈ { p : p | disc( K ) } ∪ {− } . Remark . The second condition above should be interpreted as an equality between L p -factors at every ramified prime. Of course there are fields in which p = − L − ( s, ρ K ) = L − ( s, ρ L ) is equivalent to [ K : Q ] = [ L : Q ].Hence an equivalent statement to Definition 2.12 is that K and L have same degree,same ramified primes and same local p -factors at such a primes.Recall that the decomposition type of a rational prime p in a number field K is thesequence ( f , ..., f g ) consisting of the residue degrees f i of the primes in K lying over p written in increasing order: f ≤ ... ≤ f g . Lemma . Let
K, L be number fields and let S K,L be the set of primes p that areramified in either K or L . Then, K and L are weakly arithmetically equivalent if andonly if K and L have the same degree and for all p ∈ S K,L ∪ {− } , we have that p hasthe same decomposition type in K and L .Proof. This is a simple argument that can be found in the proof vi) ⇒ ii) of [7, III, § g Kp the number of primes in K lying above p . Additionally, we denoteby f Kp the sum of the residue degrees of primes in K above p . Corollary . Let K and L be weakly arithmetically equivalent number fields. Supposethat both fields are tame. Then, they have the same discriminant.Proof. Thanks to Lemma 2.14 we know that [ L : Q ] = [ K : Q ] and that f Kp = f Lp forevery prime p . Since both extensions are tame, we have by [20, III, Proposition 13] thatdisc( K ) = Y p p [ K : Q ] − f Kp = Y p p [ L : Q ] − f Lp = disc( L ) . L is called arithmetically solitary or solitary if one hasthat K is isomorphic to L for any number field K arithmetically equivalent to L . Remark . The notion of weak arithmetic equivalence is quite less restrictive thanthat of arithmetic equivalence. For instance, there exist pairs of non isomorphic weaklyarithmetically equivalent number fields which are either:(1) Galois extensions of Q ,(2) number fields with fundamental discriminant,(3) number fields of degree smaller than 7. Example . The following polynomials, found by using [6], define pairs of non isomor-phic weakly arithmetically equivalent number fields satisfying, respectively, conditions(1), (2) and (3) in the above remark.(1) The polynomials x − x − x + 36743 x + 62118 x − x + 3625 and x − x − x + 48111 x − x − x + 77517 define two non isomor-phic Galois extensions, with Galois group Z / Z , that are weakly arithmeticallyequivalent.(2) The polynomials x − x − x + 52 x + 33 x −
24 and x − x − x − x +37 x + 27 x + 5 define two non isomorphic weakly arithmetically equivalent numberfields with fundamental discriminant equal to 725517561 = 3 ∗ . (3) The polynomials x − x −
15 and x + 10 x − Lemma . Let L be a number field satisfying either (1),(2) or (3) of the above remark.Then, L is solitary.Proof. Items (1) and (3) are part of [7, III, § § Theorem . Let
K, L be two weakly arithmetically equivalent number fields which aretame and non-totally real. Suppose that any of the following is satisfied:(a) K and L have degree at most b) K has fundamental discriminant.(c) K and L are Galois over Q .Then, the integral trace forms q K and q L are isometric.Proof. Part (a) follows from Corollary 2.15 and [11, Theorem 3.3]. Thanks to Lemma2.14, we have that g Kp = g Lp for all ramified prime p , hence (b) follows from [11, Theorem2.15]. If both fields are Galois, then not only every ramified prime p has the samedecomposition type in both fields, but it also has the same ramification index. Thisfollows since both fields have the same degree and discriminant(see Corollary 2.15).Hence, part (c) follows from [11, Proposition 2.14]. Remark . Notice that under the restrictions imposed, Theorem 2.19 gives a positveanswer to the following natural question:
Question . Let
K, L be two weakly arithmetically equivalent number fields which aretame and non-totally real. Are the integral trace forms q K and q L isometric? If we remove the signature condition in Theorem 2.19, we can’t assure the existenceof an isometry between the integral traces. However, by the same argument in the aboveproof, one sees that q K and q L belong to the same spinor genus. In particular, thanksto Theorem 2.9, we have that K and L have the same local root numbers. Hence, wehave: Theorem . Let
K, L be two tame weakly arithmetically equivalent number fields.Suppose that any of the following is satisfied:(a) K and L have degree at most .(b) K has fundamental discriminant.(c) K and L are Galois over Q .Then, K and L have the same local root numbers at every p . Notice that Question 2.21 can be stated in terms of the local behavior of the Artin L -function L ( s, ρ L ) and without any reference to the trace form. Explicitly, thanks toTheorem 2.9, Question 2.21 is equivalent to asking whether or not the equality at allramified primes p between L p ( s, ρ K ) = L p ( s, ρ L ) implies equality between the local rootnumbers of ρ K and ρ L for any pair of number fields K, L that are tame and non-totallyreal. Since the signature condition we imposed on the number fields is only necessaryto get isometry between the integral traces, and not only to get local isometry, we canomit that hypothesis and formulate 2.21 slightly in more general terms:12 uestion . Let
K, L be two tame weakly arithmetically equivalent number fields.Does it follow that K and L have the same local root numbers at every prime p ?Elliptic curves. A natural analog to the Dedekind zeta function L ( s, K ) of a numberfield is the L -function L ( s, E ) of a rational elliptic curve. Using the ℓ -adic Tate module,for some prime ℓ , one sees that L ( s, E ) is the Artin L -function of a Z ℓ -representationof G Q . The notion of arithmetic equivalence in this context is equivalent to the oneof isogeny class, thanks to Falting’s isogeny Theorem. Since this equivalence is quiterestrictive, it seems interesting to see what kind of invariants of an elliptic curve aredetermined by the analog notion of weak arithmetic equivalence. In particular, it isnatural to ask if the analog to Question 2.23 is valid in this context. It turns out thatfor semistable elliptic curves this is the case: Theorem . Let E/ Q , E ′ / Q be two semistable elliptic curves with badramification at the same primes. Suppose that for every bad prime p , the local Hasse-Weil functions of E and E ′ coincide L p ( s, E ) = L p ( s, E ′ ) . Then, for every prime p , E and E ′ have the same local root numbers W p ( E ) = W p ( E ′ ) . Proof.
This follows immediately from Rohrlich’s formula for local root numbers [17,Proposition 3].In our analogy between rational elliptic curves and number fields, the conductor playsthe role of the discriminant. Henceforth, we can think of semistability for an ellipticcurve as the analog, for a number field, of having square free discriminant. Keeping inmind this analogy we see that Theorem 2.22 part (b) is the number theoretic versionof Rohrlich’s theorem. The following shows, as in the case of elliptic curves, that thehypothesis of having square free (conductor/discriminant) can not be removed fromTheorem 2.22. In particular, the following gives a negative answer to Questions 2.21and 2.23.
Lemma . Let K and L be the number fields defined by x − x + 4 x + 68 x + 152 and x − x − x + 121 respectively. Then, K and L are tame non-totally real weaklyarithmetically equivalent number fields with different root numbers at p = 7 and p = 43 .Proof. The fields K and L have signature (0 ,
2) and discriminant d = (7 · · . Inparticular, K and L are tame. Let S = { , , } . The following table contains, foreach prime p in S , its decomposition type ( f , ..., f g ), and respective ramification indices13 e , ..., e g ), in the fields K and L . p K (1 ,
1) (1 ,
3) (1 ,
1) (2 ,
2) (1 ,
1) (2 , L (1 ,
1) (2 ,
2) (1 ,
1) (1 ,
3) (1 ,
1) (1 , K and L are weakly arithmetically equivalent. Usingthe decomposition given in [12, Theorem 0.1], or by direct computation, we see that q K ⊗ Z ∼ = h , , , i , q L ⊗ Z ∼ = h , , , i and q K ⊗ Z ∼ = h , , , i , q L ⊗ Z ∼ = h , , , i . In particular, h ( q K ) = 1 = − h ( q L ) and h ( q K ) = − = 1 = h ( q L ) . Therefore, arguing as in the first part of the proof of Theorem 2.9, we see that W p ( ρ K ) = W p ( ρ L )for p = 7 , Acknowledgements
In the first place I would like to thank the referee for the various constructive and quitevaluable comments and suggestions on the paper. I thank Lisa (Powers) Larsson for herhelpful comments on a previous version of this paper, and to Val´ery Mah´e for providingme with a reference for Rohrlich’s theorem.
References [1] J.H. Conway, N.J.A. Sloane,
Sphere packings, lattices and groups , Third edition.,Springer-Verlag New York, Inc. (1999).[2] P.E. Conner, N. Yui,
The additive characters of the Witt ring of an algebraicnumber field , Can. J. Math., Vol. XL, No. 3 (1988), 546-588.[3] P.E. Conner, R. Perlis,
A survey of trace forms of algebraic number fields , WorldScientific, Singapore, 1984. 144] P. Deligne,
Les constantes locales de l’´equation fonctionelle de la fonction L d’Artind’une repres´esentation orthogonale , Inv. Math. (1976), 296–316.[5] M. Eichler, Quadratische Formen und Orthogonal Gruppen , Springer-Verlag, 1952[15, 16].[6] J. Jones,
Number Fields Database, http://hobbes.la.asu.edu/NFDB/.[7] N. Klingen,
Arithmetical similarities. Prime decomposition and finite group theory,
Oxford Mathematical Monographs. Oxford Science Publications. The ClarendonPress, Oxford University Press, New York, 1998.[8] T. Kondo,
Algebraic number fields with the discriminant equal to that of a quadraticnumber field,
J. Math. Soc. Japan (1995), 3136.[9] G. Mantilla-Soler, Integral trace forms associated to cubic extensions , Algebra Num-ber Theory, (2010), 681-699.[10] G. Mantilla-Soler,
On number fields with equivalent integral trace forms , Interna-tional Journal of Number Theory, (2012), 1569-1580.[11] G. Mantilla-Soler,
On the Arithmetic determination of the trace , preprint (2013)arXiv:1308.2187.[12] G. Mantilla-Soler,
The genus of the Integral trace form , Authors webpage.[13] G. Mantilla-Soler,
The Spinor genus of the Integral trace form preprint (2013)arXiv:1306.3998.[14] J. Martinet,
Character theory and Artin L-functions, in Algebraic number fields ,Academic Press, New York (1977), 1-87.[15] R. Perlis,