Explicit Tamagawa numbers for certain algebraic tori over number fields
aa r X i v : . [ m a t h . N T ] S e p EXPLICIT TAMAGAWA NUMBERS FOR CERTAIN ALGEBRAICTORI OVER NUMBER FIELDS
THOMAS R ¨UD
Abstract.
Given a number field extension
K/k with an intermediate field K + fixed by a central element of the corresponding Galois group of prime order p ,we build an algebraic torus over k whose rational points are elements of K × sent to k × via the norm map N K/K + . The goal is to compute the Tamagawanumber of that torus explicitly via Ono’s formula that expresses it as a ratioof cohomological invariants. A fairly complete and detailed description ofthe cohomology of the character lattice of such a torus is given when K/k isGalois. Partial results including the numerator are given when the extensionis not Galois, or more generally when the torus is defined by an ´etale algebra.We also present tools developed in SAGE for this purpose, allowing us tobuild and compute the cohomology and explore the local-global principles forsuch an algebraic torus.Particular attention is given to the case when [ K : K + ] = 2 and K is aCM-field. This case corresponds to tori in GSp n , and most examples will bein that setting. This is motivated by the application to abelian varieties overfinite fields and the Hasse principle for bilinear forms. Introduction
The notion of a Tamagawa number as a geometric invariant of an algebraic groupover a number field is now a fairly well understood topic. The Tamagawa numberis defined as a volume of a certain fundamental domain with respect to a canonicalmeasure. It is known that this volume is closely related to local-global principlesand mass formulae. One of the early big contributions to this subject was a formulaby Ono [Ono63] which computes the Tamagawa number τ ( T ) of an algebraic torus.This formula was refined in [Vos98] into the formula τ ( T ) = | Pic( T ) || X ( T ) | , whereboth invariants involved can be computed algebraically in the cohomology of thecharacter lattice X ⋆ ( T ) of T . The formula can however be hard to evaluate inpractical situations, for a general torus. The Tate-Shafarevich group, whose orderis the the denominator of the formula, is famously hard to compute and dependsheavily on the local structure of the splitting field of the torus. In this article, weevaluate the Tamagawa numbers for a particular class of algebraic tori that arisein situations of the following kind.To a bilinear form over a number field k we can associate an adjoint involution on a k -algebra (see [KMRT98]). In the event that this k -algebra is a field K ,the fixed points under the involution form a subfield K + . One can look at theelements of K × whose image under the norm map N K/K + belongs to the base field k . Those elements in fact form the set of points of an algebraic torus and it turnsout that the Tate-Shafarevich group of this torus determines obstructions to theHasse principle for the bilinear form we started with. Such a construction was made and explained in [Cor97] with K being a CM-field and where an example of a toruswith a nontrivial Tate-Shafarevich group was computed.More generally, the motivation of this article was to give tools and results allow-ing one to compute the Tamagawa numbers of arbitrary maximal tori of the alge-braic group GSp n arising as centralizers of regular semisimple elements. Those toriare always split by an ´etale algebra with an involution fixing an index two subfieldin each summand. While some of our results remain true in a more general setting,we focus mainly on the case where the involution is central in the absolute Galoisgroup of k , in particular we give results for CM-´etale algebras. The motivation forthis comes from [AAG + n splitting over aCM-´etale algebra. The same torus and its Tamagawa number were also objects ofinterest in [GSY19] dealing with class numbers of CM algebraic tori.In order to conjecture some of the results presented in this article, it was veryimportant to be able to compute the cohomology of those tori procedurally. There-fore, many tools in SAGE were implemented to define algebraic tori, includingextensive methods to construct and study character lattices. The classes (in thesense of programming) of algebraic tori and G -lattices are to be part of a futurerelease of SAGE. This makes defining character lattices via induction, morphisms,quotients, resolutions, sums, etc much easier and lets us compute their cohomology,and in many cases, the Tate-Shafarevich group. Most of the results conjectured us-ing those tools have then led to proofs presented in this article, but some examplesof Tate-Shafarevich groups are still only known by those computational methods.It is made clear in this article when an example or a result is only known viacomputation.1.1. Statement of the main results.
Throughout this article, T belongs to aclass of algebraic tori including maximal tori of GSp n . More precisely, T is ob-tained by the following construction: T = Ker G m × Spec( k ) R K/k ( G m ) −→ ( x,y ) x − N K/K + ( y ) R K + /k ( G m ) ! , where K, K + are as in § k with intermediate field fixed by a centralelement of the corresponding Galois group of prime order. In the later parts ofthe paper we allow K to be non-Galois, or even more generally, an ´etale algebrawhich is a direct sum of such fields. We start by reformulating Ono’s formula as τ ( T ) = | ˆ H ( k, X ⋆ ( T )) || X ( X ⋆ ( T )) | and our results are based on a description of the structure ofthe character lattice for this class of tori.It turns out that the transfer map (verlagerung) from the Galois group of thesplitting field of the torus to the subgroup of elements fixing the intermediateextension K + is the key concept allowing us to relate the cohomology of X ⋆ ( T ) tothe one of an auxilliary torus, whose cohomology we compute in § H ( k, X ⋆ ( T )), and prove the following in Theorem 7.11: XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 3
Theorem 1.1.
Let T be a torus associated with an extension K/k as above. As-suming that
K/k is Galois and K + is a subfield of K fixed by a central subgroup of Gal(
K/k ) of prime order p , we have that ˆ H ( k, X ⋆ ( T )) is trivial when the p -Sylowsubgroups of Gal(
K/k ) are cyclic, and is isomorphic to Z /p Z otherwise. Next we focus on the denominator of Ono’s formula for this class of tori. Thisnumber depends on which subgroups of the Galois group arise as decompositiongroups, but given a list of subgroups, we describe an algorithmic way to compute X ( T ). In particular, if we only consider restriction maps to cyclic subgroupswe obtain an invariant X C ( T ) with 1 ≤ | X ( T ) | ≤ | X C ( T ) | . This leads to thefollowing results:
Theorem 1.2.
Assuming that
K/k is Galois, let G = Gal( K/k ) , N = Gal( K + /k ) ,and let G p be a p -Sylow subgroup of G . Then • If ˆ H ( k, X ⋆ ( T )) = 0 (i.e. if G p is cyclic) then τ ( T ) = . (Corollary6.4+Theorem 5.6). • If G is abelian then X C ( T ) = Z /p Z if the p -Sylow subgroup of G has a pre-sentation G p = Z /p i Z × · · · Z /p i n Z with n > , i n > max( i , · · · i n − ) , and N is contained in the summand Z /p i n Z . Otherwise, X C ( T ) = X ( T ) =0 . (Proposition 6.7). • If G p is not cyclic and N is not contained in the commutator subgroup of G , then X C ( T ) = Z /p Z if the p -Sylow subgroup of G ab is cyclic or ofthe form described above. Otherwise X ( T ) = X C ( T ) = 0 . (Proposition6.17). We also give a general description of X C ( T ) and exhibit computations for Galoisgroups up to order 256. Then given elements α ∈ X C ( T ) ⊃ X ( T ), we describewhich subgroups need to appear as decomposition groups need so that α / ∈ X ( T ),which in turn allows us to determine X ( T ). In the last two cases described inthe previous theorem, this description takes a simpler form, which we establish inProposition 6.22.In a final section we give an early approach to extending the results, in particularto non-Galois field extensions. First, we briefly comment on the case where K + is not necessarily fixed by a central element, but any normal cyclic subgroup ofGal( K/k ). We also bring attention to the lack of known lower bound for the Tama-gawa number in question, entertaining the possibility of this best lower bound being0. The major part of this section is dedicated to computing explicitly the numeratorof the Tamagawa number for possibly non-Galois fields. For CM-fields, we give acomplete description in terms of a condition that is easy to check computationally.Namely, in § Theorem 1.3.
Let K/ Q be a CM-field with Galois closure K ♯ and Galois group G = Gal( K ♯ / Q ) . We have ˆ H ( Q , X ⋆ ( T )) ⊂ Z / Z . Moreover, ˆ H ( Q , X ⋆ ( T )) = 0 if and only if there is g ∈ G such that |h g i\ G/N | is odd, where N = Gal( K ♯ /K ) . In particular, we give the following data:
Proposition 1.4.
Let K/ Q be a CM-field. Then • If [ K : Q ] = 4 then τ ( T ) = 1 unless K/ Q is Galois with Galois group ( Z / Z ) , in which case τ ( T ) = 2 . (Proposition 7.3) • If [ K : Q ] = 6 then τ ( T ) = 1 . (Proposition 7.13) THOMAS R¨UD • If [ K : Q ] = 8 then we list the values for ˆ H ( Q , X ⋆ ( T )) and X C ( T ) in [R¨u20] . Lastly, we consider the case of CM-´etale algebras and give some results, withintended applications to the formula in [AAG + K = K ⊕ r for some r ≥ K a CM-field,and K + = ( K +1 ) ⊕ r , if we define T K to be the torus associated with K , then τ ( T ) = 2 r − τ ( T K ). This can lead to a contruction of tori with large Tamagawanumbers.Throughout this paper, we used the LMFDB [LMF20] to create concrete exam-ples of CM-fields and applied the results to compute the corresponding interestingTamagawa numbers. Acknowledgement.
I would like to thank Julia Gordon and Jeff Achter forsuggesting this problem and constantly providing great interest and encouragement.I am also immensely grateful to David Roe, who invited me to write the SAGEcode used in this article during a coding sprint in August 2018 at the Institute forMathematics and its Applications in Minneapolis. Since then, David has helpedme format code for SAGE, and provided continuous support and advice.Thank you to Wen-Wei Li with whom I co-authored the appendix of [AAG + +
19, Appendix A] thatoverlap with this paper. His group computed Tamagawa numbers when [ K : K + ] =2 (in the notations used above) and the field extention K is cyclotomic, as well asa few other cases when K is Galois and Gal( K/k ) = Gal(
K/K + ) × Gal( K + /k ).Those results agree with the more general results of Proposition 6.7 and Proposition6.17. They also obtained the same bounds as Corollary 6.8 and examples of CM-fields with conclusions similar to Examples 6.16 and 6.6. Both contributions weredone independently. 2. Preliminaries
Definition of the tori.
Consider a finite Galois extension
K/k of numberfields with intermediate field K + such that [ K : k ] = pn , and [ K : K + ] = p forsome prime number p and some integer n . Throughout this article we will let G = Gal( K/k ), N = h ι i = Gal( K/K + ), and H = Gal( K + /k ). Furthermore, weassume N to be central in G , which is automatically true when p = 2. The notation K + comes from the main goal of this article, which is to give results related to themain theorem of [AAG + k = Q , and K is a Galois CM-field with maximal totally real subfield K + , in particular p = 2. We have an exactsequence(2.1) 1 → N → G → H → . Consider the algebraic torus of rank ( p − n + 1 defined by T = Ker G m × Spec( k ) R K/k ( G m ) −→ ( x,y ) x − N K/K + ( y ) R K + /k ( G m ) ! . XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 5
In particular, T ( k ) = n x ∈ K × : Q p − ℓ =0 ι ℓ ( x ) ∈ k × o . Example . Consider the case where p = 2 and K is a CM-field over k = Q ,with maximal totally real subfield K + . We get T ⊂ GSp n . These tori arise in[AAG +
19] as centralizers of the Frobenius element corresponding to a principallypolarized ordinary abelian variety. Also, in this case, ι is the Rosati involution.For any G -module M , we let ˆ H i ( G, M ) = ˆ H i ( K/k, M ) denote its i -th TateCohomology group, and for all i ∈ Z we define X i ( M ) = Ker ˆ H i ( G, M ) → Y ν ˆ H i ( G v , M ) ! , where ν ranges over primes of K and G v is the corresponding decomposition group(see [PR94]).The goal of this article is to give a computation for the Tamagawa number τ ( T ) = τ k ( T ). We will not recall the definition of Tamagawa numbers, which canbe found in the introduction of [Vos95]. To compute the latter we will focus on theformula given by the following theorem: Theorem 2.2 ([Ono63], p.68) . Let T be an algebraic torus over a number field k and splitting over a Galois extension L . Then τ ( T ) = | ˆ H ( L/K, X ⋆ ( T ) || X ( T ( K )) | . Using Tate-Nakayama duality (see [PR94]) one can rewrite the latter equalitywith X ( T ( K )) = X ( X ⋆ ( T )).We now define an auxilliary torus of rank ( p − n : T = Ker R K/ Q ( G m ) −→ N K/K + R K + / Q ( G m ) ! = R K + / Q R (1) K/K + ( G m ) . Here T ( k ) = n x ∈ K × : Q p − ℓ =0 ι ℓ ( x ) = 1 o . Example . In the same setting as Example 2.1 we have T = T ∩ Sp n . Notethat Sp n is the derived subgroup of GSp n .The two tori sit in the exact sequence(2.2) 1 → T → T → G m → . Character lattices.
Consider the group algebras Z [ G ] and Z [ N ] with respec-tive augmentation ideals J G and J N . By definition N = Gal( K/K + ) = h ι i .As G -modules, we have X ⋆ ( T ) = X ⋆ ( R K + / Q R (1) K/K + ( G m )) = Z [ G ] /L , where L = { a ∈ Z [ G ] : ιa = a } = Z [ G ](1 + ι + · · · + ι p − ) = Ind GN ( J N ).The injection T ⊂ T yields a surjection X ⋆ ( T ) → X ⋆ ( T ), and we get thedescription X ⋆ ( T ) = Z /L where L = L ∩ J G .We recover the corresponding exact sequences(2.3) 0 → Z → X ⋆ ( T ) → X ⋆ ( T ) → , and(2.4) 0 → L → L → Z → . THOMAS R¨UD
For the sake of simplicity we will now write Λ and Λ to denote X ⋆ ( T ) and X ⋆ ( T ) respectively.2.3. Computation of the Tamagawa number for the auxilliary torus.
Thecohomology of T = R K + / Q R (1) K/K + ( G m ) and its character lattice Λ are very easyto compute, and will be useful for the rest of this article. Proposition 2.4.
We have ˆ H i ( G, Λ ) = 0 if i is even, and ˆ H i ( G, Λ ) = N if i isodd.Proof. As a consequence of Shapiro’s Lemma, we haveˆ H i ( K/k, T ( K )) = ˆ H i ( K/K + , R (1) K/K + ( G m )( K )) . Similarly, we get ˆ H i ( G, Λ ) = ˆ H i ( N, X ⋆ ( R (1) K/K + ( G m )). Note that we can write X ⋆ ( R (1) K/K + ( G m )) = Z [ N ] / Z where we identify Z with its diagonal embedding in Z [ N ]. Taking the cohomology of the short exact sequence0 → Z → Z [ N ] → Z [ N ] / Z → , the middle term being cohomologically trivial, we get ˆ H i ( N, Z [ N ] / Z ) = ˆ H i +1 ( N, Z ).Since N is cyclic, its cohomology is 2-periodic, hence ˆ H ( N, Z [ N ] / Z ) = ˆ H ( N, Z ) =ˆ H ( N, Z ) = N as desired. (cid:3) Corollary 2.5.
We have τ ( T ) = p .Proof. We use Propositions 2.4 and 2.2. The numerator is | ˆ H ( G, Λ ) | = p , andthe denominator X (Λ ) is a subgroup of ˆ H ( G, Λ ) = 0, hence τ ( T ) = p . (cid:3) Note that since N is cyclic, there will be primes in K inert over K + , and there-fore, any torus over K + splitting over K will have a trivial denominator in theformula of theorem 2.2, since N itself will appear as the decomposition group of aninert prime.3. Tools for generic computations on algebraic tori in SAGE
Local-global principles, and more generally, cohomological invariants of algebraictori are notoriously hard to compute directly outside of examples fitting in some niceshort exact sequences such as norm-one tori with splitting field having particularlynice decomposition groups. In [Ono63], the author proves that the Tamagawanumber of R (1) K/ Q ( G m ) where K = Q ( √ , √ , √ , √ . This specificextension is chosen because all its decomposition groups are cyclic, and it is abelian,which lets the author use Lyndon’s formula (see [Lyn48, p. 287]) to compute itscohomology groups.More recently, in [HY17], the authors have used GAP to compute cohomology oftori over local fields. In this paper, tori are studied through their character latticeswith action of the Galois group of their minimal splitting field. The latter is seenas a finite subgroup of GL n ( Z ). This forces the user to input the action of thegroup as matrices, and also does not allow for someone to consider Galois groupwith possibly trivial action on the character lattice.To ease the study of such objects, I implemented the classes of algebraic tori and G -lattices (to be seen as lattices of characters of tori) in SAGE. Those methods areto be added in a future release of SAGE and are presently available on my personal XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 7 webpage with documentation. Here we include a brief description of some of thenew SAGE methods with examples of their uses.Examples such as Ono’s can be recreated directly very easily. sage: L. = NumberField([x^2-5, x^2 -29 , x^2 -109 , x^2 -281]) sage: K = L. absolute_field(’e’) sage: from sage. schemes. group_schemes.tori ....: import NormOneRestrictionOfScalars sage: T = NormOneRestrictionOfScalars (K) sage: T. Tamagawa_number() Moreover, the tools implemented for G -lattices provide many ways to create alattice. We now show the construction of Λ and Λ for G = Q , and N = Z ( G ). sage: G = QuaternionGroup() sage: N = G.center () sage: Gm = GLattice(1); Gm Ambient lattice of rank 1 with an action by a group of order 1 sage: IL = Gm. induced_lattice(G) sage: Ld = IL. fixed_sublattice(N) sage: L = Ld. zero_sum_sublattice () sage: Lambda_d = IL. quotient_lattice(Ld); Lambda_d Ambient lattice of rank 4 with a faithful action by a group of order 8 sage: Lambda = IL. quotient_lattice(L); Lambda Ambient lattice of rank 5 with a faithful action by a group of order 8
One can freely use direct sums, pullbacks, restrictions, duals, the norm map,and many more functions to create lattices. Then many methods have been imple-mented with cohomological uses, such as dimension-shifting, (co)flabby resolutions,restrictions. In particular, we have implemented a method to compute the Tate-Shafarevich groups X i for i = 1 ,
2. For i = 1 the program computes the restrictionon ˆ H directly, whereas for i = 2, since all cyclic subgroups of the Galois groupappear as decomposition groups, we can use a flabby resolution of the lattice, toreduce the computation to the case i = 1. More explicitly, given a group G actingon a lattice Λ, we follow constructions made in [HY17] to compute a resolution0 → Λ → P → L → , where P is permutation , i.e. can be writte P = L ℓi =1 Z [ G/H i ] for normal subgroups H , · · · , H ℓ , and L is flabby , meaning ˆ H − ( H, L ) = 0 for all subgroups H ⊂ G .Then by diagram chasing (see [Lor05, Lemma 2.9.1, Proposition 2.9.2]), we get X (Λ) = X ( L ). Moreover, if every decomposition group is cyclic, then X (Λ) =ˆ H ( G, L ).If the group associated to the lattice is the Galois group of a number field ex-tension, then the algorithm will build every decomposition group, otherwise it willassume that every decomposition group is cyclic and the user can input the list ofdesired non-cyclic decomposition groups.Continuing our example with G = Q . sage: for i in range(-5, 6): ....: print("H^"+ str(i)+": Lambda:" ....: ,Lambda . Tate_Cohomology(i),", Lambda_d:", ....: Lambda_d. Tate_Cohomology(i)) ....: ....: H^ -5: Lambda: [] , Lambda_d: [2] H^ -4: Lambda: [4] , Lambda_d: [] H^ -3: Lambda: [2] , Lambda_d: [2] H^ -2: Lambda: [2, 2] , Lambda_d: [] H^ -1: Lambda: [2] , Lambda_d: [2] H^0: Lambda : [4] , Lambda_d: [] H^1: Lambda : [2] , Lambda_d: [2] H^2: Lambda : [2, 2] , Lambda_d: [] H^3: Lambda : [] , Lambda_d: [2] H^4: Lambda : [4] , Lambda_d: [] H^5: Lambda : [2] , Lambda_d: [2]
THOMAS R¨UD
This confirms the computations of the cohomology of Λ done in the previoussection, as well as computing the numerator of τ ( T ), which is | ˆ H ( G, Λ) | = 2.For the denominator, note that the subgroups of Q are either cyclic or Q itself.Therefore, if Q appears as a decomposition group then X ( T ) = 0 and τ ( T ) = 2,otherwise we have sage: Sha = Lambda . Tate_Shafarevich_lattice (2); Sha [2, 2] and so τ ( T ) = .Those methods have been used to compute the Tamagawa numbers of tori forevery field extension up to degree 16 and helped greatly with conjecturing theresults proved in the rest of the article. We note that for some of these cases, noother method of finding the Tamagawa number is known.4. Transfer map as a counting function
Definition and properties.
Let tr = tr GN : G → N denote the usual transfermap as defined in citerotman. We will need some algebraic manipulations on thecharacter lattice of T to build cocycles explicitely. We will use a counting functionsdirectly related to tr, therefore this section will be mostly proving results alreadyknown about tr in our setting, and no prior knowledge of transfer maps.We start with a finite group G and a central subgroup N fitting in the shortexact sequence 1 → N → G → H = G/N → . We let G p , G ab denote a p -Sylow subgroup, and the abelianization of G respec-tively. Write | G p | = p r for some r ∈ N .For g ∈ G we will write g = gN ∈ H . For each coset h ∈ H , let ˆ h ∈ G be arepresentative.Let g ∈ G and h ∈ H . Since h = b hN , we have g b h ∈ c ghN = { ι i c gh : 1 ≤ i ≤ p } .Define the map ψ : G × H → Z /p Z by g b h = ι ψ ( g,h ) c gh . We want to study the map ϕ : G → Z /p Z , g P h ∈ H ψ ( g, h ). It is clear from the definition of the transfermap that we have tr = g ι ϕ ( g ) , but we will show directly that it is well-definedand independent on the choice of representatives. We can immediately check fromthe definition that ˘ ϕ (1) = ˆ ϕ (1) = 0 and ˘ ϕ ( ι ) = ˆ ϕ ( ι ) = | H | , regardless of the choiceof representatives. Lemma 4.1. ϕ does not depend on the choice of representatives.Proof. Fix h ∈ H . Consider another choice of representative map “1˘” such that˘ h = ι ˆ h and ˘ h = ˆ h for all h ∈ H \{ h } . The general case is obtained by repeatingthis process since h = ˆ hN , every representative differ by a power of ι . Write b ψ, ( ψ, ˆ ϕ, ˘ ϕ the corresponding maps. Let g ∈ G \{ , ι } . If h / ∈ { g − h , h } then g ˘ h = g ˆ h = i b ψ ( g,h ) c gh = i \ ψ ( g,h ) ( gh , hence b ψ ( g, h ) = ( ψ ( g, h ). Now observe that g ( h = ιg c h = ιι ˆ ψ ( g,h ) d gh = ι ˆ ψ ( g,h )+1 ( gh , and g ( g − h = g \ g − h = ι ˆ ψ ( g,g − h ) c h = ι ˆ ψ ( g,g − h ) ι − ( h = ι ˆ ψ ( g,g − h ) − ( h . Therefore, ˘ ψ ( g, h ) = ˆ ψ ( g, h ) + 1 and ˘ ψ ( g, g − h ) = ˆ ψ ( g, g − h ) −
1, hence theycompensate each other and ˆ ϕ ( g ) = ˘ ϕ ( g ). (cid:3) XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 9
Remark . For p = 2, the function ϕ counts the parity of the number of repre-sentatives of elements of H which are not sent to another representative under themultiplication by ι . Proposition 4.3. ϕ is a group homomorphism.Proof. For the previous claim, notice that we have ψ ( g g , h ) = ψ ( g , g h )+ ψ ( g , h )for all g , g ∈ G and h ∈ H . Summing this relation over H yields the desiredresult. (cid:3) Corollary 4.4. ϕ factors through ( G ab ) p . If p || H | (equivalently r > ) then N ⊂ Ker( ϕ ) .Proof. ϕ is a group homomorphism and Z /p Z is abelian, so the morphism factorsthrough G ab . Also, by virtue of ϕ being a homomorphism, if g has order coprimeto p then its image is 0.For all h ∈ H we have ψ ( ι, H ) = 1 by definition. So ϕ ( ι ) = | H | hence ϕ ( N ) = 0when p || H | . (cid:3) Lemma 4.5. If G = G p is cyclic, then ϕ is surjective.Proof. Write G = h ε i , with ε p r − = ι . We make the choice d ε i N = ε j where0 ≤ j ≤ p r − −
1. Clearly ε d ε i N = \ ε i +1 N if 0 ≤ i ≤ p r − −
1, and ε \ ε p r − N = ι , so ϕ ( ε ) = 1. (cid:3) Lemma 4.6.
Let g ∈ G . If ι ∈ h g i then ϕ ( g ) = |h g i\ G | .Proof. Note that ι ∈ h g i therefore h g i\ G ∼ = h g i\ H = F i h g i h i . Consequently h g i\ G = F i h g i ˆ h i . Left multiplication by g preserves each right coset, so bythe same computation as in Lemma 4.5 we have P h ∈h g i h i ψ ( g, h ) = 1, hence ϕ ( g ) = |h g i\ H | = |h g i\ G | . (cid:3) Corollary 4.7. If G p is cyclic, then ϕ is surjective.Proof. Let g be a generator of G p . By Lemma 4.6 ϕ ( g ) = | G p \ H | = | G p \ G | , whichis coprime to p , hence ϕ is surjective. (cid:3) Proposition 4.8. ϕ is the zero map if and only if G p is noncyclic.Proof. We have already shown that ϕ is surjective if G p is cyclic. Now assume that G p is not cyclic. Let g ∈ G . We want to show ϕ ( g ) = 0. • Case 1.
Assume ι ∈ h g i . By Lemma 4.6 we have ϕ ( g ) = |h g i\ G | , whichis still divisible by p since h g i is cyclic and therefore cannot be a p -Sylowsubgroup. So ϕ ( g ) = 0. • Case 2.
Assume ι / ∈ h g i . This means that the sets g i N are all distinct setsfor 0 ≤ i ≤ ℓ where ℓ is the order of g . Decompose h g i H = F i h g i h i . ByLemma 4.1 we are free to pick representatives, so we pick d g i h j = g i ˆ h j forsome representative ˆ h j of h j . Multiplication by g preserves all the cosets,and g d g i h j = gg i ˆ h j = g i +1 ˆ h j = [ g i h j . Therefore ψ ( g, h g i h i ) = 0 for all h i hence ϕ ( g ) = 0. (cid:3) Corollary 4.9.
Let G be a finite group with central subgroup N of order p . If G p iscyclic, then G p has a unique subgroup of order | G p \ G | . In particular this subgroupis normal and G p is its complement.Proof. The idea is to take the kernel of ϕ , and repeat the process to the kernel, andso on, until we get to a group whose p -Sylow is just N , and take its complement.We will proceed by induction on r . Recall that | G p | = p r . If r = 1 then G p = N ∼ = G/ Ker( ϕ ), and Ker( ϕ ) has therefore a complement by Schur-ZassenhausTheorem, so we can write G = Ker( ϕ ) ⋊ N . Assume now that G p is cyclic and r >
1. Let m = | G p \ G | . We have a surjective group homomorphism ϕ : G → Z /p Z ,let M denote its kernel. We know that | M | = | G | /p = p r − m , so we can use ourinduction hypothesis to conclude that M contains a unique normal subgroup of size m , call it C . By uniqueness, C is a characteristic subgroup of K (stable under anyautomorphism), and M is normal in G , therefore C is normal in G . (cid:3) Remark . The assumption that N is central is necessary. For example, thedihedral group D has a normal subgroup of order 3, but no normal group oforder 2.4.2. Application to explicit computation of the first cohomology groups.
In this section we apply the results of the previous section to compute ˆ H directly.In § λ ∈ Z [ G ], let [ λ ] denote its image in the quotient Λ = Z /L , and { λ } denotethe 1-cocycle defined by { λ } g = g [ λ ] − [ λ ]. Lemma 4.11.
For all g ∈ G we have P p − ℓ =0 { ι ℓ g } = 0 .Proof. Such a coboundary is immediately values in L by definition of L as sublatticeof Z [ G ] N of zero-sum vectors. (cid:3) For each h ∈ H fix a choice of preimage ˆ h ∈ G , and for the sake of convenience,we choose ˆ1 = 1.Note that the cohomology of the exact sequence (2.3) gives0 = ˆ H ( G, Z ) → ˆ H ( G, Λ) → ˆ H ( G, Λ ) = N. The equality on the right was proved in Proposition 2.4. Therefore, ˆ H ( G, Λ)embeds as a subgroup of N ∼ = Z /p Z , hence one only needs to find one nontrivial1-cocycle to determine ˆ H ( G, Λ) = N . Theorem 4.12.
Define the coboundary b = { P p − i =0 i P h ∈ H ι i ˆ h } . If G p is not cyclicthen b = pa where a is a nontrivial -cocycle, in particular ˆ H ( G, Λ) = N .Proof. Recall that for all g ∈ G we have g b h = ι ψ ( g,h ) c gh . For g ∈ G and 0 ≤ k ≤ p − I kg = { h ∈ H : ψ ( g, h ) = k } . It is clear that F ≤ k ≤ p − I kg = H . Inparticular, we have that p divides P p − k =0 k | I kg | = P h ∈ H ψ ( g, h ) for all g ∈ G . Alsonote that in Λ, for all g ∈ G we have [ g P p − i =0 ι i ] = [ P p − i =0 ι i ]. XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 11
We only want to know the result modulo p , so the following computation will bedone in Λ /p Λ. b g = g p − X i =0 i X h ∈ H [ ι i ˆ h ] ! − p − X i =0 i X h ∈ H [ ι i ˆ h ]= p − X i =0 i p − X k =0 X h ∈ I kg [ ι i + k c gh ] − p − X i =0 i p − X k =0 X h ∈ I kg [ ι i c gh ]= p − X k =0 X h ∈ I kg p − X i =0 i [ ι i + k c gh ] − p − X k =0 X h ∈ I kg p − X i =0 i [ ι i c gh ]= p − X k =0 X h ∈ I kg p − X i =0 i [ ι i + k c gh ] − p − X i =0 i [ ι i c gh ] ! . We are working modulo p and ι has order p so we have that P p − i =0 i [ ι i + k c gh ] = P p − i =0 ( i − k )[ ι i − k c gh ]. Therefore, b g = p − X k =0 X h ∈ I kg p − X i =0 ( i − k )[ ι i c gh ] − p − X i =0 i [ ι i c gh ] ! = p − X k =0 X h ∈ I kg p − X i =0 − k [ ι i c gh ] = p − X k =0 − k X h ∈ I kg p − X i =0 [ ι i c gh ]= p − X k =0 − k | I kg | p − X i =0 [ ι i ] ! = − p − X i =0 [ ι i ] ! p − X k =0 k | I kg | | {z } = ϕ ( g )=0 = 0 . We have proved that each coordinate of b g is a multiple of p for all g ∈ G , hence a = bp is a 1-cocycle of G with coefficients in Λ. We are left to show that a is not acoboundary. Since P p − ℓ =0 { ι ℓ } = 0 by Lemma 4.11, we can generate all coboundarieswith {{ ι ℓ ˆ h } : ℓ ∈ Z /p Z ℓ = 1 h ∈ H } .We now mimic the computation above to compute b ι ℓ . b ι ℓ = p − X i =0 X h ∈ H i [ ι i + ℓ ˆ h ] − p − X i =0 X h ∈ H i [ ι i ˆ h ] = X h ∈ H p − X i =0 i [ ι i + ℓ ˆ h ] − p − X i =0 i [ ι i ˆ h ] ! = X h ∈ H p + ℓ − X i = ℓ ( i − ℓ )[ ι i ˆ h ] − p − X i =0 i [ ι i ˆ h ] ! = X h ∈ H − ℓ − X i =0 i [ ι i ˆ h ] − p − X i = ℓ ℓ [ ι i ˆ h ] + ℓ − X i =0 ( i + p − ℓ )[ ι i ˆ h ] ! = − X h ∈ H p − X i = ℓ ℓ [ ι i ˆ h ] + ℓ − X i =0 ( ℓ − p )[ ι i ˆ h ] ! = X h ∈ H ℓ − X i =0 p [ ι i ˆ h ] ! − ℓ | H | p − X i =0 [ ι i ] . Therefore, a ι ℓ = P h ∈ H (cid:16)P ℓ − i =0 [ ι i ˆ h ] (cid:17) − ℓ | H | p P p − i =0 [ ι i ]. In particular, using thatfor all h ∈ H we have [ ι ˆ h ] = [ ι ] + P i =1 ([ ι i ] − [ ι i h ]) = P p − i =0 [ ι i ] − P i =1 [ ι i ˆ h ]. a ι = X h ∈ H [ ι ˆ h ] − ℓ | H | p p − X i =0 [ ι i ] = X h ∈ H X i =0 [ ι i ˆ h ] − | H | ( ℓp − p − X i =0 [ ι i ] . Fix ˆ h = 1. If 2 ≤ ℓ ≤ p − { ι ℓ ˆ h } ι = [ ι ℓ +1 ˆ h ] − [ ι ℓ ˆ h ]. Now for ℓ = 0 we have { ˆ h } ι = [ ι ˆ h ] − [ˆ h ] = p − X i =0 [ ι i ] − h ] − p − X i =2 [ ι i ˆ h ] . The [ ι ℓ ˆ h ]-coefficient of a ι for ℓ = 1 is always 1. The goal is to now use that factfor each h ∈ H . Because of this, if a is a sum of coboundaries, it must containssummands spanned by {{ ι ℓ ˆ h } : ℓ = 1 } . However, each of those coboundariesevaluated at ι are of the form { ι ℓ +1 ˆ h − ι ℓ ˆ h } , hence summing them will simplify bya telescopic argument, and there is only one possibility to get the desired describedcoefficients for a ι .Assuming a is a coboundary, then assume k { ˆ h } is a summand for some k ∈ N .On order to have a coefficient 1 at [ˆ h ] the only possibility is to add (2 k + 1) { ι p − ˆ h } ,but the evaluation of the latter at ι has [ ι p − ˆ h ]-coefficient − (2 k + 1). Since thatcoefficient must also be 1, and the only other generating coboundary having nonzero[ ι p − ˆ h ]-coordinate is { ι p − ˆ h } , it means that 3 k +2 { ι p − ˆ h } must also be a summand.We can repeat the process for each power of ι , and we determine that for each h the cocycle a must have a summand of the form P p − i =0 (( i + 1) k + i ) { ι p − i ˆ h } . UsingLemma 4.11, we know that P p − i =0 { ι i ˆ h } = 0, hence this summand can be written as p − X i =0 (( i + 1) k + i ) { ι p − i ˆ h } = p − X i =0 (( i + 1) k + i ) { ι p − i ˆ h } − p − X i =0 { ι i ˆ h } = p − X i =0 ( ik + i ) { ι p − i ˆ h } = ( k + 1) p − X i =0 i { ι i ˆ h } XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 13
However in the basis we picked for Λ, looking modulo p , we have that p − X i =0 i { ι i ˆ h } ι = p − X i =0 i [ ι p − i +1 ˆ h ] − p − X i =0 i [ ι p − i ˆ h ] (mod p ) ≡ p − X i =0 ( i + 1)[ ι p − i ˆ h ] − p − X i =0 i [ ι p − i ˆ h ] ≡ p − X i =0 [ ι p − i ˆ h ] ≡ p − X i =0 [ ι i ] (mod p )Therefore, in our basis for Λ, modulo p , our coboundary ( k + 1) P p − i =0 i { ι i ˆ h } doesnot have any [ ι ℓ ˆ h ]-coordinate, for ℓ = 1, hence a cannot be a sum of those generatingcoboundaries, which yields H ( G, Λ) = 0. (cid:3) Lemma 4.13. If G is cyclic, then for all i we have ˆ H i ( G, Λ ) = (cid:26) if i is odd G/N if i is even .Proof. Since G is cyclic, its cohomology is 2-periodic hence we only need to solve itfor i = 0 ,
1. Using the definition Λ = Z [ G ] /L we have ˆ H i ( G, Λ) = ˆ H i +1 ( G, L ). Theaction of G on L factors through G/N , and L is isomorphic to the augmentationideal of Z [ G/N ], so we have the short exact sequence0 → L → Ind GN ( Z ) = Z [ G/N ] → Z → . Since it is an augmentation ideal, L has no G -fixed point, so ˆ H ( G, L ) = 0. Takingthe cohomology of the sequence above, using Shapiro Lemma, and Hilbert 90, weget 0 → ˆ H ( N, Z ) | {z } = N → ˆ H ( G, Z ) | {z } = G → ˆ H ( G, L ) → ˆ H ( N, Z ) | {z } =0 . We used the cyclicity of G to write ˆ H ( G, Z ) = Z / | G | Z = G . This finishes theproof. (cid:3) Proposition 4.14. If G p is cyclic, then ˆ H ( G, Λ) = 0 .Proof.
Assume G p is cyclic. Using Corollary 4.9, we can write G = M ⋊ G p forsome subgroup M of order coprime to p . The corresponding inflation-restrictionexact sequence is:0 → H ( G p , Λ M ) → H ( G, Λ) → H ( M, Λ) G p → H ( G p , Λ M ) → H ( G, Λ) . Again, H ( G, Λ) ⊂ N so in particular, it is p -torsion, but H ( M, Λ) is | M | -torsion,hence the map H ( G, Λ) → H ( M, Λ) G p is trivial. Therefore, H ( G p , Λ M ) = H ( G, Λ).Look at the short exact sequence of M -modules0 → L → Z [ G ] → Λ → . Taking the group cohomology, we get0 → L M → Z [ G/M ] = Z [ G p ] → Λ M → H ( M, L ) → , which gives us the short exact sequence(4.1) 0 → Z [ G p ] /L ∩ Z [ G p ] → Λ M → H ( M, L ) → . Importantly, since M is normal in G , every term in this sequence has a G/M = G p -module structure. Notably, fixed elements of H i ( M, L ) by G p corresponds to theimage of the restriction map H i ( G, L ) → H i ( M, L ) . Now, let Λ p = Z [ G p ] /L ∩ Z [ G p ]. This is the same construction as for one Λ,but replacing G by G p : in particular we can use 4.13 to compute its cohomology.Taking the Tate-cohomology of (4.1), we getˆ H i − ( G p , H ( M, L )) → ˆ H i ( G p , Λ p ) → ˆ H i ( G p , Λ M ) → ˆ H i ( G p , H ( M, L )) . We have that H ( M, L ) is | M | -torsion, and | M | is coprime to p , hence for all i wehave ˆ H i ( G p , H ( M, L )) = 0. Consequently, ˆ H i ( G p , Λ p ) = ˆ H i ( G p , Λ M ), and usingLemma 4.13 we obtain ˆ H i ( G p , Λ M ) = 0, as desired. (cid:3) Remark . Note that in the proof above we do not necessarily have Λ M = Z [ G ] M /L M . Indeed, the group H ( M, L ) might not be trivial. For example, when G = Z / Z , we have M = Z / Z and H ( M, L ) = M . Remark . The proof of Proposition 4.14 shows more generally that whenever G is p -nilpotent, then ˆ H ( G, Λ) = ˆ H ( G p , Λ p ) (with the notations of the proof).5. Computing the rest of the Cohomology groups
We now give a more abstract application of the transfer map to compute allcohomology groups. The following commutative diagram is just a reformulation ofthe short exact sequences (2.3) and (2.4).0 00
L L Z Z [ G ] Z [ G ] 00 Z Λ Λ
00 0This diagram commutes and has both exact rows and columns. Note that anelement of Z in the top right can be lifted to an element of L , which imbeds into Z [ G ]. It can be sent to the left copy of Z [ G ] and into the quotient Λ, this element ishowever trivial in the quotient Λ so by exactness of the sequence below, it belongsto the image of the injection Z → Λ. This gives us a map Z → Z which is just theidentity.Let S be a subgroup of G . We want to compute ˆ H i ( S, Λ). Since Z [ G ] is coho-mologically trivial, as direct sum of induced S -modules, we get the following exactsequence on cohomology. ˆ H i − ( S, Λ ) ˆ H i ( S, Z ) ˆ H i ( S, Λ) ˆ H i ( S, Λ ) ˆ H i +1 ( S, Z ) ˆ H i +1 ( S, Λ) ˆ H i +1 ( S, Λ )ˆ H i ( S, L ) ˆ H i ( S, Z ) ˆ H i +1 ( S, L ) ˆ H i +1 ( S, L ) ˆ H i +1 ( S, Z ) ˆ H i +2 ( S, L ) ˆ H i +2 ( S, L ) XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 15
Note that we will not prove the commutativity of this diagram because it is notneeded. Each vertical arrow is an isomorphism, so we will compute each elementin the bottow row and this will give us the cohomology groups on the top row.
Lemma 5.1.
For any subgroup S ≤ G we have that L ∼ = Ind NSN ( Z ) [ G : NS ] as an S -module.Proof. Let S ′ = N S and decompose G into the right cosets G = ⊔ i S ′ g i and so Z [ G ] is the Z -span of Z [ S ′ g i ], each of them being an S ′ -module. Each summand isisomorphic to Z [ S ′ ] , hence L ∼ = { a ∈ Z [ S ′ ] : ιa = a } [ G : S ′ ] ∼ = Z [ S ′ /N ] [ G : S ′ ] ∼ = Ind S ′ N ( Z ) [ G : S ′ ] . (cid:3) Corollary 5.2.
Let S ≤ G be a subgroup. For all i ∈ Z we have ˆ H i ( S, L ) = (cid:26) N [ G : S ] if ι ∈ S else , and ˆ H i +1 ( S, L ) = 0 . Proof. If ι / ∈ S then L ∼ = Ind SNN ( Z ) [ G : SN ] = Ind S ( Z ) [ G : S ] p , hence is induced. If ι ∈ S then ˆ H i ( S, L ) = ˆ H i ( N, Z ) [ G : S ] . Since N is cyclic, we only need to computeˆ H ( N, Z ) = N and ˆ H ( N, Z ) = 0. (cid:3) Plug this in the second row of the previous diagram. If we pick i odd then wehave(5.1)0 → ˆ H i ( S, Z ) → ˆ H i +1 ( S, L ) → ˆ H i +1 ( S, L ) ϕ i → ˆ H i +1 ( S, Z ) → ˆ H i +2 ( S, L ) → . Corollary 5.3.
Let S ≤ G be a subgroup such that ι / ∈ S . Then for all i ∈ Z wehave ˆ H i +1 ( S, L ) = ˆ H i ( S, Z ) . Consequently ˆ H i ( S, Λ) = ˆ H i ( S, Z ) .Proof. This is a direct consequence of plugging in the results of Corollary 5.2 in(5.1) (cid:3)
Lemma 5.4.
If the transfer map tr GN : G → N is trivial, then ˆ H ( G, Λ) = N and ˆ H ( G, Λ) = G ab , otherwise ˆ H ( G, Λ) = 0 and ˆ H ( G, Λ) = G ab /N . This result isstill true if N is not assumed to be a central subgroup of G .Proof. The sequence (5.1) with S = G and i = 1 is0 → ˆ H ( G, L ) → N ϕ → G ab → ˆ H ( G, L ) → . We used Hilbert 90 to write ˆ H ( G, Z ) = 0. Using the short exact sequence0 → Z → Q → Q / Z → , G (and N ), we note that the middle term is cohomologicallytrivial (it is uniquely divisible), hence we can writeˆ H ( G, L ) = ˆ H ( G, Z [ G/N ]) = ˆ H ( N, Z ) = ˆ H ( N, Q / Z ) = Hom( N, Q / Z ) = N. Likewise ˆ H ( G, Z ) = ˆ H ( G, Q / Z ) = Hom( G, Q / Z ) = G ab . Classically (see [Bro94,Chapter III, section 9]), the corresponding map ˆ H ( N, Z ) → ˆ H ( G, Z ) is inducedby the transfer map. Therefore, the map ϕ is really a map on the dual groupsHom( N, Q / Z ) → Hom( G, Q / Z ) defined by α α ◦ tr GN . We have ˆ H ( G, Λ) =ˆ H ( G, L ) = Ker( ϕ ). The order of N being prime, we have tr GN is either trivial or surjective. In the former case ˆ H ( G, Λ) = N , else ˆ H ( G, Λ) = 0. Plugging this backinto the equation above, we get the corresponding result for ˆ H ( G, Λ) = ˆ H ( G, L ).The assumption of N being central in G has only been used to compute thetriviality of the transfer map in terms of the structure of a p -Sylow subgroup of G .This section has only used the fact that N is normal so far. Thus, we do not needthe assumption that N is central here. (cid:3) Lemma 5.5.
Let S ≤ G with ι ∈ S . If the transfer map tr SN : S → N is trivial,then ˆ H i ( S, Λ) = ˆ H i ( S, Z ) for i even and ˆ H i ( S, Λ) / ˆ H i ( S, Z ) = N [ G : S ] for i odd.Proof. The map ϕ i of equation (5.1) is the natural map from ˆ H i ( S, Ind SN ( Z )) [ G : S ] =ˆ H i ( N, Z ) to ˆ H i ( S, Z ). This is exactly the corestriction map on cohomology, inducedby the transfer map. We refer again to the section on the transfer map in [Bro94,Chapter III, section 9]. Therefore, if the transfer map is trivial, ϕ i is trivial, hencewe get the two exact sequences:0 → ˆ H i ( S, Z ) → ˆ H i +1 ( S, L ) → ˆ H i +1 ( S, L ) → , and 0 → ˆ H i +1 ( S, Z ) → ˆ H i +2 ( S, L ) → , which yields the desired result by replacing ˆ H i ( S, L ) by ˆ H i − ( S, Λ). (cid:3)
With all this preparation, we can now give results for the cohomology groupsinvolved in the computation of the Tamagawa number.
Theorem 5.6.
Let S ≤ G . • If ι / ∈ S then ˆ H ( S, Λ) = 0 and ˆ H ( S, Λ) = S ab . • If ι ∈ S and S p is cyclic, then ˆ H ( S, Λ) = N [ G : S ] − and ˆ H ( S, Λ) = S ab /N . • If ι ∈ S and S p is not cyclic, then ˆ H ( S, Λ) = N [ G : S ] and ˆ H ( S, Λ) = S ab .Proof. By Hilbert 90, we have ˆ H ( S, Z ) = 0. Taking the cohomology of the exactsequence 0 → Z → Q → Q / Z → , since the middle term is uniquely divisible, we have ˆ H ( S, Z ) = ˆ H ( S, Q / Z ) =Hom( S, Q / Z ) = S ab .The exact sequence (5.1) with i = 1 becomes(5.2) 0 → ˆ H ( S, L ) ξ → ˆ H ( S, L ) ϕ → S ab → ˆ H ( S, L ) → . By Corollary 5.3 and Lemma 5.5 we get the first and last cases.If S is cyclic, then ˆ H ( S, L ) = ˆ H ( S, L ) = {{ a i } ∈ N [ G : S ] : Q a i = 1 } = N [ G : S ] − . Therefore, the cokernel of ξ of (5.2) is isomorphic to N . Since we have0 → Coker( ξ ) → S → ˆ H ( S, L ) → , as desired.Now assume S p is cyclic. We can follow the proof of Proposition 4.14 verbatim.Let K S = K ∩ S be the complement of S p in S . We get ˆ H ( S, Λ) = ˆ H ( S p , Λ K S )and look at the cohomology of0 → L → Z [ G ] → Λ → . This time Z [ G ] K S = Z [ G p ] [ K : K S ] . Still following the proof of 4 .
14, the same cohomo-logical sequence yields ˆ H ( S p , Λ K S ) ∼ = ˆ H ( S p , Z [ G p ] [ K : K S ] /L K S ) = ˆ H ( S p , L K S ) = XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 17 ˆ H ( S p , L K S ). The latter corresponds to sum zero elements of ˆ H ( S p , L K S ) = N [ G p : S p ][ K : K S ] = N [ G : S ] . Therefore, ˆ H ( S, Λ) = N [ G : S ] − , and we can then fol-low the same argument as the cyclic case above. (cid:3) Computation of the denominator
Generalities on the denominator.
The goal of this section is to give com-putations of the denominator of the Tamagawa number as stated in Theorem 2.2.
Lemma 6.1.
The group ˆ H ( K/k, T ( K )) is p -torsion, and therefore so is X ( T ( K )) .Proof. We look at the cohomology of the short exact sequence (2.2) and getˆ H ( K/k, T ( K )) → ˆ H ( K/k, T ( K )) → ˆ H ( K/k, G m ( K )) = 0 . Therefore, we have a surjection ˆ H ( K/k, T ( K )) → ˆ H ( K/k, T ( K )), so it sufficesto prove that ˆ H ( K/k, T ( K )) is p -torsion.Now ˆ H ( K/k, T ( K )) = ˆ H ( K/k, R K + / Q R (1) K/K + ( G m )( K )) which in turns equalsˆ H ( K/K + , R (1) K/K + ( G m )( K )) by Shapiro’s Lemma.Now looking at the cohomology of the short exact sequence1 → R (1) K/K + ( G m ) → R K/K + ( G m ) → G m → , we get ˆ H ( K/K + , R (1) K/K + ( G m )( K )) = ˆ H ( K/K + , G m ( K )) = ( K + ) × /N K/K + ( K × ) , which is clearly p -torsion, since N K/K + ( K × ) contains N K/K + (( K + ) ⋆ ) = (( K + ) × ) p . (cid:3) By Tate-Nakayama duality (see [PR94, p.307]), we know that X ( T ( K )) = X ( X ⋆ ( T )). Proposition 6.2.
We have p | G ab [ p ] | ≤ τ ( T ) ≤ p. Proof.
We know that the numerator of Ono’s formula is | ˆ H ( G, Λ) | ≤ p , and thedenominator is a subgroup of ˆ H ( G, Λ) = G ab , and is p -torsion, so it is containedin G ab [ p ]. (cid:3) Proposition 6.3. If G p is cyclic, then X ( T ( K )) = 0 .Proof. By Chebotarev density Theorem, every cyclic subgroup of G appears as thedecomposition group at unramified places. In particular, so does G p . Therefore,we know that X (Λ) ⊂ Ker( ˆ H ( G, Λ) → ˆ H ( G p , Λ)). By restriction-corestriction,we know that the composite mapˆ H ( G, Λ) → ˆ H ( G p , Λ) → ˆ H ( G, Λ)is multiplication by [ G : G p ]. Since the image of X (Λ) through the restrictionmap is trivial, we get that X (Λ) is annihilated by [ G : G p ], which is coprime to p .Therefore, X (Λ) is both p -torsion and [ G : G p ]-torsion, and is hence trivial. (cid:3) Corollary 6.4. If G p is cyclic, then τ ( T ) = = 1 .Proof. Proposition 4.14 tells us that the numerator is 1 whereas Proposition 6.3gives us the triviality of the denominator. (cid:3) If G p is not cyclic, the answer depends on ramification groups and thereforedepends on the particular field K . We will give an explanation of the computationin general, with some bounds and examples.Let C denote the set of cyclic subgroups of G , and let D be the set of subgroupsof G not in C arising as decomposition groups of K/k . In particular, subgroups in D can only occur at some ramified primes. If S is a collection of subgroups of G then we let X i S (Λ) = Ker ˆ H i ( G, Λ) → Y S ∈ S ˆ H i ( S, Λ) ! . Lemma 6.5. If G ∈ S or G p ∈ S , then X S (Λ) = 0 .IProof. The first case is immediate. For the second case, consider the restriction-corestriction maps ˆ H ( G, T ) → ˆ H ( G p , T ) → ˆ H ( G, T ) , the composition of the two maps is multiplication by the index of G p , which iscoprime to p . Since H ( G, T ) is p -torsion, this is an isomorphism, hence the re-striction map ˆ H ( G, T ) → ˆ H ( G p , T ) is an injection. (cid:3) Also, we clearly have X (Λ) = X C ∪ D (Λ) = X C (Λ) ∩ X D (Λ) ⊂ X C (Λ) . We will be now focusing on computing X C (Λ). By Theorem 5.6, assuming G p is not cyclic, we can rewrite this as X C (Λ) = Ker G ab → Y α ∈ G ι/ ∈h α i h α i × Y α ∈ G ι ∈h α i h α i /N . We know that X C ( Z ) = X ( G m ) = 0, in particular, G ab → Q α ∈ G h α i is aninjection.For α ∈ G , the morphism of groups G ab → h α i is not canonical, it is a map onthe Pontryagin’s duals of the groups, it is the map Hom( G, Q / Z ) → Hom( h α i , Q / Z )given by restriction. By virtue of X (Λ) being p -torsion we can restrict the com-putation to elements α of G ab [ p ].We denote the isomorphism G ab → Hom( G, Q / Z ) by g t g . This morphismdepends on the choice of presentation of G ab . We can rewrite Ker( G ab → h α i ) = { g ∈ G ab : t g ( α ) = 0 } . Example . If p = 2 and G is the quaternion group Q , then all proper subgroupsof G are cyclic, and all nontrivial subgroups contain ι . Write G = h α, β i with α = β = ι , βαβ − = α − . Then G ab = ( Z / Z ) . Every element of G ab is 2-torsion, and hence is sent to 2-torsion elements of h α i × h β i , which is the subgroup N × N . However, X C (Λ) = Ker (cid:0) G ab → h α i /N × h β i /N (cid:1) . By our previous point,every element is mapped into N × N , and hence belongs to this kernel. Consequently X (Λ) is trivial if and only if G appears as decomposition group, else X (Λ) =( Z / Z ) .Let K be given by the polynomial x + 68 x + 986 x + 4624 x + 4624. Usingthe LMFDB [LMF20], we know that this is a CM field with Galois group Q ,discriminant 2 · , and both decomposition groups at ramified primes are cyclic, XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 19 isomorphic to Z / Z . Therefore, in this case, τ ( T ) = = . This is the smallestexample of non-integral Tamagawa number for the tori we are interested in. If | G | ≤ G = Q , then τ ( T ) ∈ { , } .6.2. Denominator in the abelian case.
Here G = G ab and G p is not cyclic.We can assume G = G p . Indeed, G will decompose in a direct sum of its Sylowsubgroups, and cyclic subgroups contained in q -Sylow subgroups never contain N when p = q . Therefore, the map G p → Q g ∈ G ℓ h g i is always trivial and does notcontribute to X C (Λ) ⊂ G [ p ] ⊂ G p .Write down G = L ri =1 (cid:0) Z /p i Z (cid:1) n i with n r = 0 and let C = C ⊔ C be thecollection of cyclic subgroups of G , where C is the collection of cyclic groupscontaining N . We can write g ∈ G as g = P ri =1 ~ m i where ~ m i ∈ (cid:0) Z /p i Z (cid:1) n i . Let usdenote the isomorphism G → Hom( G, Q / Z ) via g t g where t P ri =1 ~ m i r X i =1 ~ r i ! = r X i =1 ~ m i · ~ r i p i ∈ Q / Z . Note that in particular t g ( h ) = t h ( g ) for all g, h ∈ G . The goal is to find C = h g i such that ι / ∈ C and t ι ( g ) = 0. Since ι has order p , one can write ι = P ri =1 p i − ~ k i where ~ k i ∈ ( Z /p Z n i ). Let g = P ri =1 ~ m i ∈ G , we have t ι ( g ) = r X i =1 p i − ~ k i · ~ m i p i = 1 p r X i =1 ~ k i · ~ m i . In particular, we can take each ~ m i ∈ ( Z /p Z ) n i . More rigorously, we can use theprojection Z /p i Z → Z /p Z , which yields a projection φ : G → G [ p ] = ( Z /p Z ) P ri =1 n i .We have t ι ( g ) = t ι ( φ ( g )). Define m : G → Z by m r X i =1 ~ m i ! = min ( { i ∈ { , · · · , r } : ~ m i = 0 } ∪ { } ) , it is invariant under the choice of basis. Note that we took G non-cyclic, hence P ri =1 n i ≥ • Case 1. n m ( ι ) ≥
2. Take g with ~ m j = 0 if j = m ( ι ), and ~ m m ( ι ) issome nonzero vector such that ~ k m ( ι ) · ~ m m ( ι ) = 0 in Z /p Z , and ~ m m ( ι ) is notcollinear to ~ k m ( ι ) . This is easy to find, if ~ k m ( ι ) has two nonzero coordinates,take ~ m m ( ι ) with only a 1 at those coordinates, and 0 everywhere else. If ~ k m ( ι ) has only one nonzero coordinate, then take ~ m m ( ι ) to contain two 1’s,one where ~ k m ( ι ) is supported, and one where it’s not. Since the two vectorsare not collinear, we have ι / ∈ h g i as desired. • Case 2. n m ( ι ) = 1 and m ( ι ) < r . Here we repeat the same process asbefore, take g such that ~ m m ( ι ) = (1). If ι ∈ h g i , which happens if ~ k j = 0for j = m ( ι ), then pick ~ m r to be a vector with one 1 and the rest 0. Wecannot have ι ∈ h g i anymore because in this case ~ k r = 0, and so to have ι = g ℓ we would need p r | ℓ , but that would make g ℓ = 0, which is absurd.Again we have found a suitable g . • Case 3. n m ( ι ) = 1 and m ( ι ) = r . In this case, assume we found such a g , then we would need ~ m r = 0. Recall that we can take ~ m r = ( ℓ ) with1 ≤ ℓ ≤ p −
1, and so g has order p r . In particular, p r − g has order p , and since n r = 1, p r − annihilates all smaller factors of G , so m ( p r − g ) = r so p r − g ∈ N and so h g i ∩ N = ∅ . So ι ∈ h g i , and there are no valid choicesfor g .We have proved: Proposition 6.7.
Assume G is abelian. If G p = L ri =1 (cid:0) Z /p i Z (cid:1) n i with n r = 0 and C is the collection of cyclic subgroups of G , we have X C (Λ) = 0 unless G p is notcyclic, n r = 1 and N is equal to the p -torsion elements of the summand Z /p r Z , inwhich case X C (Λ) ∼ = N . The last condition corresponds to n r = 1 and m ( ι ) = r , hence N cannot besupported on any summand but Z /p r Z . This case is not common. Corollary 6.8. If G is abelian then τ ( T ) ∈ { , p } .Example . Assume G = ( Z / Z ) . Here r = 1 and n = 2 = 1 so by Proposition6.7 we get X ( T ) = 1. Example . Assume G = Gal( K/k ) ∼ = Z / Z × Z / Z with ι = (0 , H = ( Z / Z ) . As it turns out, H ( H, Λ ) ∼ = H and ι ∈ Ker( G → H ), hence if G does not appear itself as a decomposition group, then X ( T ) = N and τ ( T ) = 1. Example . In the same spirit as previous examples, an immediate application ofthe proposition implies the triviality of X ( T ) when G is abelian and (2.1) splits.6.3. General case.
The computation only depends on the p -torsion points of theabelianization of G , therefore we will assume without loss of generality that G is anon-cyclic p -group.Let G ′ denote the commutator subgroup of G .Let α ∈ G such that α / ∈ G ′ and α p ∈ G ′ (we only care about the p -torsion pointsof G ab ). We fix an isomorphism G/G ′ ∼ = Hom( G, Q / Z ) so that there is g ∈ G suchthat t α ∈ h t g i and t α ( g ) = 0. Indeed, to do so, we can write a presentation of G ab as Q i ( Z /p n i Z ) where t α is only supported on one summand, then we can take thesame morphism as in previous subsection and pick g ∈ G such that t g is a generatorof that summand.If ι / ∈ h g i then ˆ H ( h g i , Λ) = h g i , and the image of α under ˆ H ( G, Λ) → ˆ H ( h g i , Λ) is not trivial, therefore t α / ∈ X C (Λ). If ι ∈ g , then since t α is p -torsion,so is its image under the restriction map G ab → h g i , and h g i is cyclic, hence has aunique subgroup of p -torsion elements, hence t α is sent into N ⊂ h g i . By Theorem5.6 we have ˆ H ( h g i , Λ) = h g i /N , hence t α is sent to 0 via the restriction map. As aside note, this further explains why we only need to cover p -torsion elements, sincethe rest need not be sent into p -torsion elements of h g i .Again, by normality of N and unicity of p -torsion elements in cyclic groups, wehave that ι ∈ h g i if and only if ι ∈ h α i . Also by normality of N in G , if ι belongsto the cyclic group generated by α , it belongs to the cyclic group generated by anyconjugate of α , so the condition ι ∈ h α i only depends on t α and not the choice oflift.Let us restate those observations in a couple lemmas. Lemma 6.12.
Assume G p is non-cyclic. Fix α ∈ G with image t α in G ab ⊃ X (Λ) . If t α ∈ X (Λ) , then α p ∈ G ′ and ι ∈ h α i . XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 21
Lemma 6.13.
Fix α ∈ G satisfying the conditions of Lemma 6.12. We have t α / ∈ X C (Λ) if and only if and there is t g ∈ G ab such that t α ( g ) = 0 and ι / ∈ h g i .Proof. This follows immediately from the discussion above and Theorem 5.6 (cid:3)
Example . We have computed X C (Λ) in Example 6.6. We have seen that forevery g ∈ G we have t g ∈ X C (Λ). We can check that for every g ∈ G we have g ∈ h ι i = G ′ so both conditions hold. Remark . Assume G is abelian, i.e. G ′ = { } . The conditions of Lemma 6.12become: α is p -torsion and ι ∈ h α i . In particular N ⊂ h α i and the two groups havethe same size, hence the condition can be reduced to α ∈ N . We therefore recoverthe observation of the previous subsection that X (Λ) ⊂ N in the abelian case.Therefore, if G is abelian then | X (Λ) | ∈ { , p } . The abelian case already shows that the conditions of Lemma 6.12 are onlynecessary and not sufficient. This is because in the choice of g ∈ G such that t α ( g ) = 0 we have required so far that t α ∈ h t g i . We now give a non-abelianexample. Example . Assume p = 2 and G is the dihedral group D . We write a presen-tation of D = h α, β i with α = β = 1, and βαβ = α . There is only one choiceof N , that is N = Z ( G ) and ι = α . Similarly to Example 6.6, we have G ′ = h ι i and G ab = ( Z / Z ) . We choose a presentation sending α to (1 ,
0) and β to (0 , G, Q / Z ) as in the previous subsection.In this case ι / ∈ h β i so β / ∈ X (Λ). Also, α satisfies the conditions of 6.12, but t α ( αβ ) = 1 /
2, and ι / ∈ h αβ i , hence α / ∈ X (Λ) either. We have just shown that X (Λ) = 0 and if Gal( K/k ) = G then τ ( T ) = 2.Also note that D and Q are the only nonabelian groups of order 8 so we havecomputed all possible Tamagawa numbers for | G | = 8. Only two of those groupshave potentially nontrivial denominator, those groups are Q and Z / Z × Z / Z where N is respectively Z ( Q ) and h (0 , i . Proposition 6.17. If ι / ∈ G ′ then X C (Λ) only depends on G ab , and can be com-puted by Proposition 6.7. In particular, X (Λ) ⊂ N .Proof. In order to have both α p ∈ G ′ and ι ∈ h α i , we need α p = 0, and therefore |h α i| = p = | N | , thus N = h α i and we are left to find an element β ∈ G ab such that t α ( β ) = 0 as in the previous subsection. (cid:3) Remark . Applying Proposition 6.17 comes with one crucial caveat. In theprevious section, we assume G abelian and G p not cyclic. This is due to us knowingthe case when G p is cyclic. However, it is possible to have G not abelian with G p non cyclic, but the p -Sylow of G ab is cyclic (and containing the nontrivial projectionof N ). In that case, the computations done in the previous section remain true andwe obtain X C (Λ) = N (since, with notations of that section, n r = 1 and m ( ι ) = r ).In order for X (Λ) to be trivial in that case, we need a decomposition group whose p -Sylow is not cyclic and whose projection on G ab is onto. In general, we will seein Proposition 6.22, that if X C (Λ) = Z /p Z , then X (Λ) = 0 if and only if thereis a subgroup with non-cyclic p -Sylow subgroups such that the projection on thesummand of the p -Sylow subgroup of G ab containing N is onto. Remark . Proposition 6.17 covers in particular the case where (2.1) splits.
The only difficulty of the computation comes therefore when N ⊂ G ′ . As shownin Example 6.6, we can have | X (Λ) | > | N | .For any α ∈ G let us write I α = { g ∈ G : t α ( g ) = 0 } and ˙ I α = ∩ g ∈ I α h g i . We canrewrite Lemmas 6.12 and 6.13 as: α ∈ X C (Λ) if and only if α p ∈ G ′ and ι ∈ ˙ I α .As we saw, this implies in particular, that ι ∈ h α i . Proposition 6.20.
Assume there is = t α ∈ X C (Λ) . Then t α ∈ X (Λ) if andonly if { D ∈ D : D p is not cyclic, D ∩ I α = ∅ } = ∅ . In other words, t α / ∈ X (Λ) if and only if there is a ramified prime p ∈ K such that the corresponding decom-position group G ( p ) has non-cyclic p -Sylow subgroup and contains some element of I α .Proof. This is simply because if S ∈ { D ∈ D : D p is not cyclic, D ∩ I α = ∅ } thenby Theorem 5.6 we get that the image of α under the restriction map ˆ H ( G, Λ) → ˆ H ( S, Λ) = S ab is nonzero since it is supported on some g ∈ I α ∩ S . (cid:3) Example . Let G = Z / Z × Z / Z and ι = (0 , I ι = { (1 , , (0 , } and therefore ˙ I ι = N . In order to have X (Λ) = 0we would need to have a non-cyclic decomposition group containing (1 ,
1) or (0 , G ,so X (Λ) = 0 if and only if K/k contains an inert prime.This argument together with Proposition 6.7 yields a full description of thetriviality of X (Λ) in the case where G is abelian. Proposition 6.22. If G is Galois and abelian, and X C (Λ) = 0 , then write G p = Q ℓk =1 Z /p i k Z with ℓ > and i ℓ > max( i , · · · , i ℓ − ) with N ⊂ Z /p ℓ Z accordingto Proposition 6.7. Then X (Λ) = 0 if and only if there is a ramified prime for K/k with non-cyclic decomposition group H such that the projection of H on thesummand Z /p i ℓ Z is onto.This method also computes X (Λ) when ι does not belong to the derived subgroup G ′ according to Proposition 6.17.Proof. This is a direct consequence of the fact that, in this case, α ∈ I α if and onlythe projection of h α i on Z /p i ℓ Z is onto. (cid:3) Results for 2-groups.
When p = 2, we used Lemma 6.13 to compute X C (Λ)for G with | G | < G the first 29631 groups of order 256. The resultsare available at [R¨u20]. We can get the following bounds. Proposition 6.23.
Assume | G | ≤ . Then | X C (Λ) | ∈ { , , , } . The onlycase with | X C (Λ) | = 8 happens for G = M (2) . D . In particular, for such Galoisgroups we have ≤ τ ( T ) ≤ . Here M (2) . D is the small group of GAP label (128 , D by M (2) acting via D / Z / Z = ( Z / Z ) .So far over half the groups of order 256 were checked, and no group was foundthat would give[] | X C (Λ) | ≥ XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 23 Extensions of the results
Bounds for the Tamagawa number.
It is not clear whether the boundsin Proposition 6.2 are tight. The upper bound certainly is, and the lower bound istight as we have seen in Example 6.6. Other groups computed in § G ab [ p ] is much smaller than | G | . If G p is abelian, we knowthat the denominator is at most p , and so 1 ≤ τ ( T ) ≤ p .It is not clear in general if we can have N = G ′ and X C (Λ) = G/N . The onlysuch example currently is G = Q . For p = 2 there is no other example of size ≤ p , there might not be universal lower bound for all such tori.In particular, there might be algebraic tori in the general symplectic groups withTamagawa numbers arbitrarily small, but if true this would require tori of verylarge ranks to get | X C (Λ) | > Non-central subgroup.
The main application of this article happens when p = 2, in which case N being normal and central are equivalent. When taking p > p central rather than the more general p normal.This more restricted setting allowed us to reformulate most of results in terms of p -Sylow subgroups of G , whereas for N normal, we would have needed to writeeverything in terms of the transfer mapping. Most results in the article remaintrue, including the bound of Proposition 6.2. However, Proposition 4.8 is not trueanymore, therefore the following computations should be written in terms of thetransfer mapping rather than p -Sylow subgroups in the vein of Lemma 5.4 whichstill holds true. Example . Assume G = D and N is its unique normal subgroup of order 3.Then G is cyclic, but ˆ H ( G, Λ) = Z / Z . This is indeed a direct consequence ofthe triviality of the transfer map in that case. The assertion follows from Lemma5.57.3. Non-Galois extension.
The definition of our tori also makes sense in thesetting of a non-Galois extension
K/k . Let K ♯ denote the corresponding Galoisclosure. We then have the following extension diagram kK + K ♯ KpG N N where G = Gal( K ♯ /k ), N = Gal( K ♯ /K + ), and N = Gal( K ♯ /K ). Note that if p = 2, we have that K/K + is necessarily Galois, and so N is normal in N .We recall the definition of our torus of rank [ K + : k ]( p −
1) + 1:(7.1) T = Ker G m × Spec( k ) R K/k ( G m ) −→ ( x,y ) x − N K/K + ( y ) R K + /k ( G m ) ! . Observe that X ⋆ ( R K + /k G m ) = Z [ G/N ], and X ⋆ ( R K/k G m ) = Z [ G/N ], andtherefore the norm map N K/K + : R K/k G m → R K + /k G m yields the corresponding map on the character lattices Z [ G/N ] → Z [ G/N ] defined by gN X hN ∈ N /N ghN = g · X h ∈ N /N hN . By definition of T as kernel, we get the definition of X ⋆ ( T ) through the dualsequence: 0 → Z [ G/N ] → Z ⊕ Z [ G/N ] → X ⋆ ( T ) → , where the first map is the direct sum of the augmentation map and the norm mapabove. This corresponds with the description made in [Cor97].When K = K ♯ is Galois, then N = { } , and hence we get N = N as in theprevious sections. Remark . We now write a description of X ⋆ ( T ) which is not particularly inter-esting for theoretical purposes, but does simplify its construction in SAGE.Let φ : Z [ G/N ] → Z [ G/N ] be the map defined above. Its image is the span (as G -module) of φ ( N ). One can check that φ | Z [ G/N ] N : Z [ G/N ] N → Z [ G/N ] N isonto and contains φ ( N ). Consequently, the image of φ can be written Z [ G/N ] N ,which we use to denote G · Z [ G/N ] N . If K + /k is Galois then N is normal in G ,hence Z [ G/N ] N = Z [ G/N ] and Im( φ ) = Im( φ | Z [ G/N ] ) = Z [ G/N ] N .Using this description, we can write X ⋆ ( T ) = Z [ G/N ] /J Z [ G/N ] N , where J denotes the augmentation ideal. This explains why we build the N -invariant elements of Z [ G/N ] and then complete the result into a G -submodulein the following examples. Building the map φ and its image is also possible, butmore cumbersome.7.4. Non-Galois extensions of degree 4.
Assume K is a non-Galois field ex-tension of degree 4; in this setting K + /k is quadratic hence Galois. Looking atthe transitive subgroups of S , we observe that the only possibility of having anintermediate field K + /k of degree 2 is to have G be a group of order 4 or D .Therefore since K is non-Galois, we get G = D . Taking N to be any non-normal subgroup of D of degree 2 and N any (normal) subgroup of D of order4 containing N . Then SAGE computations show that | ˆ H ( G, X ⋆ ( T )) | = 2 and | X | ≤ | X C ( X ⋆ ( T )) | = 1. sage: G = DihedralGroup(4) sage: N2 = [h for h in G.subgroups() if h.order () == 2][1] sage: N1 = [h for h in G.subgroups() if h.order () == 4][0] sage: N2.is_normal(G) False sage: N2.is_normal(N1) True sage: Z = GLattice(N2 , 1) sage: IL = Z. induced_lattice(G) sage: SM = IL. fixed_sublattice(N1) sage: SL = SM. complete_submodule (). zero_sum_sublattice() sage: L = IL. quotient_lattice(SL) sage: L. Tate_Cohomology(1) [] sage: L. Tate_Shafarevich_lattice (2) [] Proposition 7.3.
Let
K/k be an extension of degree with intermediate extension K + /K of degree . Define T as in (7.1). If K/k is Galois and
Gal(
K/k ) = ( Z / Z ) then we get τ ( T ) = 2 , else τ ( T ) = 1 . XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 25
Proof.
The only non-Galois case is computed above. The numerator in the case
K/k
Galois follows from Theorem 4.12 for G = ( Z / Z ) and Proposition 4.14 for G = Z / Z . The denominator in the Galois case is always trivial, when G = Z / Z this follows from Proposition 6.3 and when G = ( Z / Z ) it was done in Example6.9. (cid:3) Example . Motivated by the application of the Tamagawa number in the formulaof [AAG + X over F definedby the equation y = x + x + x − x −
1. The Frobenius polynomial of this curveis x − x + x − x + 9 which yields a non-Galois extension of Q . Therefore if T is the centralizer of this Frobenius in GSp , we get τ ( T ) = 1.7.5. The numerator of the Tamagawa number for general non-Galois ex-tensions.
As before, we define the auxilliary torus T = Ker R K/ Q ( G m ) −→ N K/K + R K + /k ( G m ) ! = R K + / Q R (1) K/K + ( G m ) . We have the exact sequence(7.2) 1 → T → T → G m → . In particular, 0 → Z → X ⋆ ( T ) → X ⋆ ( T ) → , and by Hilbert 90, we get | ˆ H ( G, X ⋆ ( T )) | ≤ | ˆ H ( G, X ⋆ ( T )) | .Let Λ = X ⋆ ( T ) and Λ = X ⋆ ( T ). From the same reasoning as for T , we getthe exact sequence 0 → Z [ G/N ] → Z [ G/N ] → Λ → . The corresponding cohomology yields0 = ˆ H ( N , Z ) → ˆ H ( G, Λ ) → ˆ H ( N , Z ) = N ab1 → ˆ H ( N , Z ) = N ab2 . The map N ab1 → N ab2 is just the restriction map on the dual groups ϕ :Hom( N , Q / Z ) → Hom( N , Q / Z ).If K/K + is Galois, then N is normal in N , therefore if α ∈ Ker( ϕ ), then α | N = 0 and hence α is induced from a character of N /N = Z /p Z . In this case,ˆ H ( G, Λ ) = Ker( ϕ ) = ( Z /p Z ) ab = p . In the case K/K + is not Galois, and hence N is not normal, if α ∈ Ker( ϕ ) then Ker( α ) is a normal subgroup of N containing N , but N has prime index, hence α = 0. Therefore we proved: Proposition 7.5.
We have τ ( T ) ≤ | ˆ H ( G, Λ ) | , and ˆ H ( G, Λ ) = Z /p Z if K/K + is Galois, else ˆ H ( G, Λ ) = 0 .Proof. We have shown the second assertion. The first one comes from the formulaof Theorem 2.2 and the cohomology of character lattices of (7.2). (cid:3)
Remark . If K + contains a p -root of unity, then K/K + is always Galois sinceit has degree p , in particular when p = 2 any quadratic extension is Galois so inthis case we always get τ ( T ) ≤
2. Note that from the classical results, if p is thesmallest prime dividing | N | = [ K ♯ : K + ] then N is automatically normal in N .In particular this holds when N is a p -group. The cohomology of the sequence of character lattices associated to (7.2) yields0 → ˆ H ( G, Λ) → ˆ H ( G, Λ ) → ˆ H ( G, Z ) . Now using the fact that ˆ H ( G, Λ ) = Ker(Hom( N , Q / Z ) → Hom( N , Q / Z )), weget: Proposition 7.7.
We have the equality ˆ H ( G, Λ) = Ker (cid:16) ˆ H ( G, Λ ) → ˆ H ( G, Z ) (cid:17) = { α ∈ Hom( N , Q / Z ) : α | N = 0 , α ◦ tr GN = 0 } . The last equality of the Proposition seems a little ambiguous since tr GN is valuedin N ab1 , but note that elements of Hom( N , Q / Z ) factor through N ab1 and hence weonly need to look at pre-transfers.7.6. The case of CM-fields.
Assume k = Q and let K be a CM-field, withmaximal totally real subfield K + . Let ρ denote the involution given by complexconjugation, it is central in G . Since K + is totally real, we also have ρ ∈ N , but K/K + is imaginary so ρ / ∈ N . Lemma 7.8.
We have ˆ H ( G, Λ) ⊂ Z / Z , and ˆ H ( G, Λ) = Z / Z if and only if tr GN ( G ) ⊂ N .Proof. The first assertion follows from the factˆ H ( G, Λ) ⊂ ˆ H ( G, Λ ) = Hom( N /N , Q / Z ) = Z / Z . Here Hom( N /N , Q / Z ) is identified with characters of N trivial on N . Theonly nontrivial element of this group is the character α : N → Q / Z defined by α ( N ) = 0 and α ( ρN ) = 1 / H ( G, Λ) in the previous section, we only needto determine if the image of tr GN is contained in Ker( α ) = N . If this is the casethen ˆ H ( G, Λ) = Z / Z , else ˆ H ( G, Λ) = 0. (cid:3)
Corollary 7.9. If [ K + : Q ] is odd, then ˆ H ( G, Λ) = 0 .Proof.
Assume [ K + : Q ] is odd. Since ρ is central, we have tr GN ( ρ ) = ρ | G/N | = ρ ([ K + : Q ]) = ρ / ∈ N since K is not totally real. (cid:3) Corollary 7.10. If K is a CM-field of degree then ˆ H ( G, Λ) = 1 . We now know the numerators for all extensions of degree 4 and 6.
Theorem 7.11.
Let K/ Q be a CM-field and Λ = X ⋆ ( T ) . We have ˆ H ( G, Λ) ⊂ Z / Z . Moreover, ˆ H ( G, Λ) = 0 if and only if there is g ∈ G such that |h g i\ G/N | is odd, where G = Gal( K ♯ / Q ) and N = Gal( K ♯ /K ) .In particular, if K/ Q is Galois, then we recover the result that ˆ H ( G, Λ) = 0 ifand only if the -Sylow subgroups of Gal( K/ Q ) ∼ = G/N are cyclic.Proof. We will use 7.8 and compute the transfer map tr GN . This proof will besomewhat technical so the reader can refer to a simplified example going throughthe main ideas of the proof in Example 7.12.Decompose G into the left cosets: G = F ni =1 x i N and write X = { x , · · · , x n } .Fix g ∈ G . Let ˜ σ ∈ S X such that gxN = ˜ σ ( x ) N for all x ∈ X . Let ˜ σ = ˜ ζ · · · ˜ ζ ℓ be the decomposition of ˜ σ as product of disjoint cyclic permutations. XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 27
For all 1 ≤ i ≤ ℓ write ˜ ζ i = ( x i · · · x i ri ). Write g ˜ ζ i = x − i g r x i = ( x − i gx i )( x − i gx i ) · · · ( x − i ri gx i ) ∈ N . Then tr GN ( g ) = Q ℓi =1 g ˜ ζ i N ′ where N ′ denotes the commutator subgroup of N .Note that this cycle decomposition is equivalent to a decomposition of G into doublecosets h g i\ G/N via G = F ℓj =1 F r j i =1 g i x j N = F ℓj =1 h g i x j N . In particular, ℓ = |h g i\ G/N | .We have that N = N ⊔ ρN , so G = ( F ni =1 x i N ) ⊔ ( F ni =1 ρx i N ) . Let ρX = { ρx : x ∈ X } , and let σ ∈ S X ⊔ ρX such that gxN = σ ( x ) N for all x ∈ X ⊔ ρX .Since for all x ∈ X we have gxN = σ ( x ) N , then either gxN = ˜ σ ( x ) N or gxN = ρ ˜ σ ( x ) N , hence for all x ∈ X we have σ ( x ) = ˜ σ ( x ) or σ ( x ) = ρ ˜ σ ( x ).Decompose σ = ζ · · · ζ t into disjoint cycles, where t = |h g i\ G/N | . Define theaction of ρ on S X ∪ ρX via ρ · τ ( x ) = τ ( ρx ).Clearly for all 1 ≤ i ≤ t , both ζ i and ρ · ζ i appear in the decomposition of σ sincethe latter is ρ -invariant.Let X ′ ⊂ X denote a set of representatives of h g i\ G/N . For all 1 ≤ j ≤ ℓ wehave the identification ˜ ζ j = ˜ ζ x for some x ∈ X ′ , where ˜ ζ x denotes the action of ˜ σ onthe orbit of x . Likewise for all 1 ≤ i ≤ t we have ζ i = ζ x or ζ i = ζ ρx = ρ · ζ x for some x ∈ X ′ . Let x ∈ X ′ , the cycle ˜ ζ x corresponds to two permutations ζ x and ζ ρx . If ζ x = ζ ρx then ˜ ζ x has the same length as ζ x and ρ is central hence g ˜ ζ x = g ζ x = g ζ ρx ,thus g ˜ ζ x ∈ N . If ζ x = ζ ρx , then the length of ζ x is twice the size of ˜ ζ x , and inparticular g ζ x = (cid:16) g ˜ ζ x (cid:17) and g ζ x ∈ ρN . Note that η := (cid:12)(cid:12)(cid:12)n x ∈ X ′ : g ˜ ζ x / ∈ N o(cid:12)(cid:12)(cid:12) = |{ x ∈ X ′ : ζ x = ρ · ζ x }| = |{ ≤ i ≤ t : ζ i = ρ · ζ i }| . On the one hand tr GN ( g ) = Q ℓi =1 g ˜ ζ x ∈ ρ η N belongs to N if and only if η is even.On the other hand, η is the number of fixed points of the action of the involution ρ on { ζ , · · · , ζ t } , hence η ≡ t (mod 2). We got tr GN ( g ) ∈ N if and only if t is even,and t = |h g i\ G/N | . We conclude that Im(tr GN ) ⊂ N if and only if |h g i\ G/N | iseven for all g ∈ G and then we can use Lemma 7.8.If K/ Q is Galois, then N is normal in G , so h g i\ G/N is even for all g ∈ G ifand only if every cyclic subgroup of Gal( K/ Q ) ∼ = G/N has even index, which istrue if and only if the 2-Sylow subgroups of G/N are not cyclic. (cid:3) Example . Using the notations of the previous theorem, we will give a mini-malist example to explain the reasonning.Assume | G/N | = 3 and we pick the left coset representatives x, y, z ∈ G . Thenwe have G = G i =0 ρ i xN ⊔ ρ i yN ⊔ ρ i zN . Pick g ∈ G with associated permutations ˜ σ ∈ S G/N and σ ∈ S G/N as in thetheorem. We will also assume that ˜ σ = ˜ ζ = ( x y z ), and σ sends x to ρx , and y to ρy . Since ˜ σ ( z ) = x , there are then two possibilities for the image of z under σ . • σ ( z ) = x . We can therefore write σ = ( x ρy z )( ρx y ρz ) = ζ x ζ ρx , and ρ per-mutes both (disjoint) cycles. Equivalently |h g i\ G/N | = 2. By centralityof ρ we have g ˜ ζ = g ζ x = g ζ ρx ∈ N . • σ ( z ) = ρx . Now σ = ( x ρy z ρx y ρz ) = ζ x . This is the case when ζ x isfixed under the action of ρ . Equivalently |h g i\ G/N | = 1. In particular, wehave g ˜ ζ ∈ ρN , and g ζ x = ( g ˜ ζ ) ∈ N .This is the argument we use in the proof, the image of the pre-transfer will belong in ρ i N where i is the number of fixed cycles in the decomposition of σ . The non-fixedcycles come in pairs, so this number has the same parity as |h g i\ G/N | . Examples computed by SAGE..
We have determined the Tamagawa num-bers of the tori corresponding to CM-fields of degree 4, and we have seen the nu-merator of Ono’s formula is always trivial for CM-fields of degree 6. Moreover,using SAGE, one can check that X C ( T ) is trivial in the latter case. We get thefollowing proposition. Proposition 7.13.
Let K/ Q be a CM -field of degree , then τ ( T ) = 1 . For degree 8 extensions, one can easily compute the numerator and X C ( T ); theresults are available at [R¨u20]. We can notice that there are only 6 different groupsGal( K ♯ / Q ) such that X C ( T ) = 0, in all those cases we have X C ( T ) = Z / Z . Example . Assume K is the CM-field defined by the polynomial x + 8 x +17 x + 9 x + 1. The corresponding Galois group is C : S , of order 192 withaction 8T39 on the roots. We get τ ( T ) = 2. The totally real subfield is defined by x − x − x + 4 x + 3 and has Galois group S . Example . Let f , f , f the three following polynomials: f = x + 14 x + 36 x + 28 x + 4 ,f = x + 28 x + 250 x + 868 x + 961 ,f = x + 14 x + 39 x + 32 x + 8 . Let K , K , K be the corresponding CM-fields with respective tori T , T , and T .All have Galois group Gal( K ♯i / Q ) = C .D with label (32 , τ ( T ) = 2and τ ( T ) = τ ( T ) = 1. Indeed, despite the extensions having the same Galoisgroup, they correspond to a different transitive action on 8 points, respectively8T19, 8T20, 8T21. Here both K +2 and K +3 are Galois, with respective Galoisgroups Z / Z and ( Z / Z ) . The field K +1 is not Galois, the Galois group of itsGalois closure is D . Example . Let K be the non-Galois CM-field defined by f = x − x + 27 x − x + 331 x − x + 1513 x − x + 1801 . The Galois group of its closure is the Semidihedral group QD = Q ⋊ Z / Z .Every decomposition group here is cyclic, hence using the table for extensions ofdegree 8 in [R¨u20] we get that τ ( T ) = . This example is particularly interestingbecause in the Galois case, Proposition 6.3 tells us that if the numerator of Ono’sformula equals 1, then so does the denominator. However in this non-Galois case,we can have trivial ˆ H ( Q , X ⋆ ( T )) but not X ( T ).7.8. Tori split over a CM-´Etale algebra.
Let K = L ri =1 K i be an ´etale algebraover Q with the intermediate ´etale subalgebra K + = L mj =1 K + j . We can againdefine the tori XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 29 T K = Ker G m × Spec( Q ) R K/ Q ( G m ) −→ ( x,y ) x − N K/K + ( y ) R K + / Q ( G m ) ! , and T K = Ker R K/ Q ( G m ) −→ N K/K + R K + / Q ( G m ) ! = R K + / Q R (1) K/K + ( G m ) . Assume that K is a CM-´etale algebra, with complex involution ρ and K + isthe ´etale algebra fixed by ρ . Since each K + i is totally real and ρ is the complexconjugation, we have r = m , and for each 1 ≤ i ≤ r we can define T K i = Ker G m × Spec( Q ) R K i / Q ( G m ) −→ ( x,y ) x − N Ki/K + i ( y ) R K + i / Q ( G m ) ! , and T K i = Ker R K i / Q ( G m ) −→ N Ki/K + i R K + i / Q ( G m ) ! = R K + i / Q R (1) K i /K + i ( G m ) . We have T K = Q ri =1 T K i , and T K ⊂ T K ⊂ Q ri =1 T K i with corresponding exactsequences(7.3) 1 → T K → T K → G m → , and(7.4) 1 → T K → r Y i =1 T K i → G r − m → . The respective exact sequences of character lattices are(7.5) 0 → Z → X ⋆ ( T K ) → X ⋆ ( T K ) → , and(7.6) 1 → Z r − → r M i =1 X ⋆ ( T K i ) → X ⋆ ( T K ) → . Thanks to the previous section we have computedˆ H ( Q , X ⋆ ( T K )) = r M i =1 ˆ H ( Q , X ⋆ ( T K i )) = ( Z / Z ) r and Theorem 7.11 gives us a formula for L ri =1 ˆ H ( Q , X ⋆ ( T K i )) . The cohomology of (7.5) and (7.6) gives us an estimate of the numerator of τ ( T ).We have the inequality r Y i =1 | ˆ H ( Q , X ⋆ ( T K i )) | ≤ | ˆ H ( Q , X ⋆ ( T K )) | ≤ r . Let K ♯ be a Galois extension of Q splitting every T K i , and hence splitting T K too. Let G = Gal( K ♯ / Q ). For each i ∈ { , · · · , r } we let K ♯i denote the Galois closure of K i , and K + i its maximal totally real subfield. We write G i = Gal( K ♯i / Q ), H i = Gal( K ♯ /K ♯i ) and N i = Gal( K ♯i /K i ).Continuing with the cohomology of (7.6), we get0 → ˆ H ( G, X ⋆ ( T K )) → ˆ H ( G, X ⋆ ( T K )) = r M i =1 ˆ H ( G, X ⋆ ( T K i )) | {z } =( Z / Z ) r → ˆ H ( G, Z ) . We have computed the rightmost arrow in the previous section for a single CM-field,this time we have to consider the sum of those mappings. If g ∈ G and 1 ≤ i ≤ r ,we let ν ig = |h gH i i\ ( G/H i ) /N i | |h gH i i\ G i /N i | ∈ Q / Z . Let k = ( k , · · · , k r ) ∈ ( Z / Z ) r , and define ϕ i ∈ Hom( G, Q / Z ) by ϕ k ( g ) := P ri =1 k i ν ig . To sum up the results of Theorem 7.11, we get that
Lemma 7.17. ˆ H ( G, X ⋆ ( T )) = Ker( ϕ ) , where ϕ : (cid:26) ( Z / Z ) r → Hom( G, Q / Z ) k ϕ k . We can recover previous computations of ˆ H ( G, X ⋆ ( T K i )) = Ker( ϕ ◦ η i ) where η i : Z / Z → ( Z / Z ) r is the injection in the i th coordinate. In particular, we againhave ˆ H ( G, X ⋆ ( r Y i =1 T K i )) r M i =1 Ker( ϕ ◦ η i ) ⊂ Ker( ϕ ) = ˆ H ( G, X ⋆ ( T )) . Corollary 7.18.
Assume K = L ri =1 ( K i ) ⊕ j i for some j , · · · , j r ∈ N . Define I = { ≤ i ≤ r : ˆ H ( G, X ⋆ ( T K i )) = 0 } . Then we get M i/ ∈ I ( Z / Z ) j i ⊕ M i ∈ I ( Z / Z ) j i − ≤ ˆ H ( G, X ⋆ ( T )) ≤ r M i =1 ( Z / Z ) j i . Proof.
For each i , let ˜ ϕ i denote the restriction of ϕ to the labels corresponding to K j i i .Let 1 ≤ i ≤ r . If ˆ H ( G, X ⋆ ( T K i )) = Z / Z , then ˜ φ i is identically zero, hence( Z / Z ) j i = Ker( ˜ ϕ i ). If however ˆ H ( G, X ⋆ ( T K i )) = 0, then ˜ ϕ i will send everyzero-sum vector of ( Z / Z ) j i to 0, hence ( Z / Z ) j i − = Ker( ˜ ϕ i ). (cid:3) Corollary 7.19.
Assume K = L ri =1 ( K i ) ⊕ j i for some j , · · · , j r ∈ N such that Gal( K ♯ / Q ) = Q ri =1 Gal( K ♯i / Q ) . We get ˆ H ( G, X ⋆ ( T )) = r M i =1 ( Z / Z ) j i − ˆ H ( G i , X ⋆ ( T K i )) . Proof.
In that case, the triviality of each ϕ i can be tested independently on eachquotient G/H i = G i . Therefore, following the notations of the proof of Corollary7.18, we have Q ri = i Ker( ˜ ϕ i ) = Ker( ϕ ). (cid:3) Example . Let K = ( K ) ⊕ r where K / Q is a Galois CM-field with Galois group G = G . By Corollary 7.19 and Theorem 4.12 we know that ˆ H ( Q , X ⋆ ( T )) =( Z / Z ) r − if the 2-Sylow subgroups of G are cyclic, otherwise ˆ H ( Q , X ⋆ ( T )) = XPLICIT TAMAGAWA NUMBERS OF ALGEBRAIC TORI 31 ( Z / Z ) r . Since K is Galois, we have ˆ H ( G, X ⋆ ( T K )) = ˆ H ( G, X ⋆ ( T )) ⊕ r = 0 byCorollary 5.2. Therefore the cohomology of (7.5) yields0 → ˆ H ( G, X ⋆ ( T K )) → ˆ H ( G, X ⋆ ( T K )) → ˆ H ( G, Z ) → ˆ H ( G, X ⋆ ( T K )) → . The first arrow has trivial cokernel if G is non-cyclic, else it has cokernel Z / Z .Therefore, we get ˆ H ( G, X ⋆ ( T K )) = ˆ H ( G, Z ) if G is not cyclic, else we getˆ H ( G, X ⋆ ( T K )) = ˆ H ( G, Z ) /N . The same logic works for the cohomology of anysubgroup of G . Therefore, we get X ( T K ) = X ( T K ). We can conclude τ ( T K ) = 2 r − τ ( T K ) . The exact same resonning gives us the following proposition.
Proposition 7.21.
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