Presentations of Galois groups of maximal extensions with restricted ramification
aa r X i v : . [ m a t h . N T ] M a y PRESENTATIONS OF GALOIS GROUPS OF MAXIMAL EXTENSIONSWITH RESTRICTED RAMIFICATION
YUAN LIU
Abstract.
Motivated by the work of Lubotzky, we use Galois cohomology to study thedifference between the number of generators and the minimal number of relations in a pre-sentation of the Galois group G S ( k ) of the maximal extension of a global field k that isunramified outside a finite set S of places, as k varies among a certain family of extensionsof a fixed global field Q . We prove a generalized version of the global Euler-Poincar´e Char-acteristic, and define a group B S ( k, A ), for each finite simple G S ( k )-module A , to generalizethe work of Koch about the pro- ℓ completion of G S ( k ) to study the whole group G S ( k ). Inthe setting of the nonabelian Cohen-Lenstra heuristics, we prove that the objects studied bythe Liu–Wood–Zureick-Brown conjecture are always achievable by the random group thatis constructed in the definition the probability measure in the conjecture. Introduction
For a global field k and a set S of primes of k , we denote by G S ( k ) the Galois group of themaximal extension of k that is unramified outside S . Determining whether G Ø ( k ) is finitelygenerated and finitely presented is a long-existing open question. It is well known by classfield theory that the abelianization of G Ø ( k ) is finitely presented and, in particular, is finitewhen k is a number field. Golod and Shafarevich [GˇS64] constructed the first infinite ℓ -classtower group of a number field, where the ℓ -class tower group of k is the pro- ℓ completion of G Ø ( k ) for a prime integer ℓ . In the book [Koc70], Koch sharpened the Golod-Shafarevichtheorem and employed the Galois cohomology to give new results concerning the minimalnumbers of generators and relations for the ℓ -class tower group of k . Since then, there havebeen many papers studying infinite class field towers and there are conjectures for the upperbound of the minimal numbers of generators of G S ( Q ) as S varies.In the recent decades, the fast-developing area of arithmetic statistics provides anotherpoint of view towards this question. For an odd prime ℓ , the Cohen–Lenstra heuristics[CL84] give predictions of the distributions of ℓ -primary parts of the class groups Cl( k ) as k varies over quadratic number fields. Later, Friedman and Washington [FW89] formulated ananalogous conjecture for global function fields. Moreover, they showed that the probabilitymeasure used for the conjectural distributions in the Cohen–Lenstra heuristics matches theone defined by the random abelian grouplim n ! ∞ Z ⊕ nℓ /n + u random relations (1.1)where the random relations are taken with respect to the Haar measure, and u is chosen tobe 0 and 1 respectively when k varies among imaginary quadratic fields and real quadraticfields. Ellenberg and Venkatesh [VE10] theoretically explained the random group model(1.1) and the value of u , by viewing Cl( k ) as the cokernel of the map sending the S -units of k to the group of fractional ideals of k generated by S with S running along an ascending equence of finite sets of primes of k . Boston, Bush and Hajir [BBH17, BBH19] extendedthe Cohen–Lenstra heuristics to a nonabelian setting considering the distribution of ℓ -classtower groups (for odd ℓ ). In their work, the probability measure in the heuristics is definedby a random pro- ℓ group generalizing (1.1), and the value of u (which is the differencebetween the numbers of relations and generators of a pro- ℓ group in this setting) is obtainedby applying Koch’s argument. Notably, the moment versions of the function field analogsof the Cohen–Lenstra heuristics and the Boston–Bush–Hajir heuristics are both proven, see[EVW16, BW17].The author and her collaborators Wood and Zureick-Brown, in [LWZB19], considered thedistribution question in a more generalized setting. Let Γ be a finite group and Q the globalfield Q or F q ( t ) for a prime power q . For a Galois extension k/Q with Gal( k/Q ) ≃ Γ, define k to be the maximal unramified extension of k , that is split completely at places of k overinfinity, and of order relatively prime to | µ ( Q ) || Γ | and char Q (if non-zero), where µ ( Q ) isthe group of roots of unity of Q . As k varies over Galois extensions of Q with Gal( k/Q ) ≃ Γthat is split completely at ∞ (hence k is totally real if Q = Q ) , [LWZB19] constructsa random group model to give a conjecture predicting the distribtion of Gal( k /k ), andmoreover, proves the moment version of the conjecture in the function field case. That workapproaches the random group model as follows: 1) It proves that Gal( k /k ) is naturally aΓ-group and has two important properties, namely the admissibility (see Definition 4.1) andthe Property E (see Remark 10.8). 2) It constructs a random Γ-grouplim n ! ∞ F n (Γ) (cid:30) [ r − γ ( r )] r ∈ X,γ ∈ Γ , (1.2)satisfying both properties, where F n (Γ) is the free admissible Γ -group (see Section 4 forits definition), X is a set of n + u random elements of F n (Γ), and the square bracketsrepresent the closed normal Γ-subgroup of F n (Γ) generated by those elements inside. 3)Finally, it shows that the moment proven in the function field case matches the moment ofthe probability defined by (1.2) exactly when u = 1. With these evidences, [LWZB19] givesthe conjecture that the random group (1.2) with u = 1 gives the distribution of Gal( k /k )in both the function field case and the number field case.By [LWZB19, Theorem 5.12], for a finite set C of isomorphism classes of finite Γ-groups,the pro- C completion , with respect to the variety of groups generated by C (see Section 5 fordefinition), of the random group (1.2) defines a probability measure supported on all finitepro- C Γ-groups. Moreover, note that, for each n , the quotient in (1.2) is defined by n ( | Γ | − n + u )( | Γ |−
1) relations. So the Liu–Wood–Zureick-Brown conjecture impliesthat the pro- C completion Gal( k /k ) C of Gal( k /k ) is a finite group with probability 1, andalso suggests a bound, depending only on | Γ | , for the difference between the number ofgenerators and the minimal number of relations in a presentation of Gal( k /k ) C .In this paper, we study the theoretical reason behind u = 1 in the Liu–Wood–Zureick-Brown conjecture. Let G Ø , ∞ ( k ) denote the Galois group of the maximal unramified extensionof k that is totally split at every place above ∞ . If all groups in C are of order prime to | µ ( Q ) || Γ | and char( Q ), then G Ø , ∞ ( k ) C = Gal( k /k ) C and we prove u = 1 on each pro- C level. Theorem 1.1.
Let Γ be a nontrivial finite group and Q be either Q or F q ( t ) for q relativelyprime to | Γ | . Let C be a finite set of isomorphism classes of finite Γ -groups all of whoseorders are prime to | µ ( Q ) || Γ | and char( Q ) (if non-zero). Then for a Galois extension k/Q ith Galois group Γ that is split completely over ∞ , we have the following isomorphism of Γ -groups ( Γ acts on the left-hand side via Γ ≃ Gal( k/Q ) ) G Ø , ∞ ( k ) C ≃ F n (Γ) C (cid:30) [ r − γ ( r )] r ∈ X,γ ∈ Γ (1.3) for some positive integer n and some set X consisting of n + 1 elements of F n (Γ) C . The bulk of this paper is devoted to establishing the techniques for proving Theorem 1.1.Motivated by the work of Lubotzky [Lub01], we first translate the question to understandingthe Galois cohomology groups. In Section 3, we construct the free profinite Γ -group F n (Γ) on n generators , and, for a finitely generated profinite Γ-group G , we study the minimalnumber of relations of a presentation defined by a Γ-equivariant surjection π : F n (Γ) ։ G .The minimal number of relations is closely related to the multiplicities of the irreducible G ⋊ Γ-modules appearing as quotients of ker( π ) (Definition 3.1). In Lemma 3.2, we showthat for such a module A with gcd( | A | , | Γ | ) = 1, the multiplicity of A can be computedby a formula involving dim H ( G ⋊ Γ , A ) − dim H ( G ⋊ Γ , A ). So when restricted to thecategory of profinite Γ-groups whose order is prime to | Γ | , by using these multiplicites, weobtain formulas for the minimal number of relations of the presentation F ′ n (Γ) ։ G ′ , where F ′ n (Γ) and G ′ are the pro- | Γ | ′ completions of F n (Γ) and G respectively (Propositions 3.4 and3.7). In particular, the formulas provide an upper bound for the minimal number of relationsof this presentation using dim H ( G, A ) Γ − dim H ( G, A ) Γ , where Γ acts on the cohomolgygroups by conjugation. These upper bound formulas set up the strategy of the proof ofTheorem 1.1. Building upon it, we explore the multiplicities of admissible presentations F n (Γ) ։ G in Section 4 and the multiplicities of pro- C presentations in Section 5, where weobtain formulas that will be directly applied to the proof of Theorem 1.1. Then in Section 6,we define the height of a group and show in Proposition 6.3 that there is an upper bound forthe heights of pro- C groups (not necessarily finitely generated) when C is a finite set. ThenTheorem 6.4 proves the finiteness of G S ( k ) C when S is a finite set of places of k and C is afinite set of finite groups.Therefore, in order to prove Theorem 1.1, we need to deal with the Galois cohomologygroups. In a more general setting, assuming that Q is an arbitrary global field, that k/Q isa Galois extension with Gal( k/Q ) ≃ Γ, and that S is a finite set of primes of k , we want tounderstand δ k/Q,S ( A ) = dim F ℓ H ( G S ( k ) , A ) Γ − dim F ℓ H ( G S ( k ) , A ) Γ , (1.4)for all prime integers ℓ relatively prime to | Γ | and char( Q ), and for all finite simple Gal( k S /Q )-modules A of exponent ℓ . In (1.4), the set S needs to contain enough primes to ensure that k S /Q is Galois (see the definition of the k/Q -closed sets in Section 2), and the Γ action onthe cohomology groups is defined via the conjugation by Gal( k/Q ). In Section 7, we provea generalized version of the Global Euler-Poincar´e Characteristic formula (Theorem 7.1),from which we can compute δ k/Q,S when S is nonempty and contains the primes above ∞ and ℓ if Q is a number field. The proof basically follows the original proof of the GlobalEuler-Poincar´e Characteristic formula, but taking the Γ actions into account creates manytechnical difficulties.In the work of Koch, when dealing with the case that S does not satisfy the assump-tions in Theorem 7.1, the group B S ( k ) plays an important role in the computation ofdim F ℓ H i ( G S ( k ) , F ℓ ) for i = 1 ,
2, and is defined to be the Pontryagin dual of the Kummer roup V S ( k ) = ker k × /k × ℓ −! Y p ∈ S k × p /k × ℓ p × Y p S k × p /U p k × ℓ p ! , where k p is the completion of k at p and U p is the group of units of k p . In Definition 8.1, wedefine a group B S ( k, A ) in a cohomological way ascoker Y p ∈ S H ( k p , A ) × Y p S ( k p , A ) ! H ( k, A ′ ) ∨ ! , in order to generalize Koch’s work to compute δ k/Q,S ( A ) by replacing the trivial module F ℓ with an arbitrary module A . The definition of B S ( k, A ) agrees with the one of B S ( k ) when A = F ℓ (Proposition 8.3). However, Kock’s argument does not directly apply to B S ( k, A ),because the Hasse principle does not always hold for an arbitrary Galois module A (that is,the Shafarevich group X ( k, A ) might be nontrivial). In Section 8, we modify Koch’s workto overcome this obstacle, and show that most properties of B S ( k ) also hold for B S ( k, A ).In particular, one example, clearly showing that the failure of the Hasse principle makes adifference, is that there is a natural embedding X S ( k, A ) ֒ ! B S ( k, A ) for A = F ℓ but not forarbitrary A (Proposition 8.5 and Remark 8.6). In Section 9, we explicitly compute δ k/Q,S ( A )for all S by applying the results from Sections 7 and 8. Finally in Section 10, we give theproof of Theorem 1.1.The methods we developed in this paper can apply to many other interesting situations.First of all, we can study the presentation of G S ( k ) by letting Γ = 1 (so k = Q ), in which caseour result about Γ-presentations is the main result in [Lub01] of Lubotzky (see Remark 3.3).Building on the theorem of Lubotzky, when k is a function field, Shusterman [Shu18] showedthat G Ø ( k ) admits a finite presentation in which the number of relations is exactly thesame as the number of generators (which is called a balanced presentation). Note that in[Shu18] the finite generation of G Ø ( k ) follows by the Grothendieck’s result of the geometricfundamental group of a smooth projective curve defined over a finite field, but when k is anumber field, the finite generation of G Ø ( k ) is not known. We prove an analogous result inSection 9 regarding the number field case. Theorem 1.2.
Let k be a number field and S a finite set of places of k . If G S ( k ) istopologically generated by n elements, then it admits a finite presentation, in which theminimal number of relations is at most [ k : Q ] + n . Then in Section 11, we apply our methods to the situations that are not consideredin Theorem 1.1. Namely, we study the presentation of the pro- ℓ completion of G Ø , ∞ ( k ),denoted by G Ø , ∞ ( k )( ℓ ), for a Galois Γ-extension k/Q in two exceptional cases: 1) Q is anumber field not containing the ℓ -th roots of unity and we do not make any assumptions onthe ramification of ∞ in k (Section 11.1); and 2) Q is a global field containing the ℓ -th rootsof unity (Section 11.2). When considering the ℓ -parts of class groups, it is known for a longtime that the Cohen–Lenstra heuristics need to be corrected in these two cases.Cohen and Martinet [CM87] provided a modification for the case that Q = Q and k/Q is imaginary. Wang and Wood [WW19] proved some results about the probability mea-sures described in the Cohen–Martinet heuristics. From these works, one can see that thedecomposition subgroup Γ ∞ at ∞ of k/ Q crucially affects the probability measures. In emma 11.1, we explicitly compute the upper bounds of multiplicities in a pro- ℓ admissi-ble Γ-presentation of G Ø ( k )( ℓ ), which shows how the multiplicities are determined by Γ ∞ .Then in Corollary 11.2 and Remark 11.3, we prove that, when k/ Q is an imaginary qua-dratic field, G Ø ( k )( ℓ ) can be achieved by a random group model which defines a probabilitymeasure agreeing with the Boston–Bush–Hajir heuristics.When the base field Q contains the ℓ -th roots of unity, we give upper bounds for multiplic-ities in Lemma 11.4 and Corollary 11.5, which suggests that the distributions of G Ø , ∞ ( k )( ℓ )should be different between the function field case and the number field case (Remark 11.6(2)).This difference is not surprising, as Malle observed in [Mal10] that his conjecture regardingthe class groups of number fields does not easily match the result for function fields. So theupper bounds obtained in Corollary 11.5 supports Malle’s observation. The phenomenon re-lated to the presence of the roots of unity has been numerically computed in [Mal08, Mal10],and the random matrices in this setting and their relation with function field counting hasbeen studied in [KS99, Ach06, Ach08, Gar15, AM15]. A correction for roots of unity, providedwith empirical evidence, is presented in [Woo17].For these two exceptional cases, it raises an interesting question whether we can constructrandom pro- ℓ groups that have the multiplicities obtained in Section 11. There are manyfurther questions we would like to understand. First, the techniques in this paper work forany finite set S of places. So we would like to ask whether the random group models (in theabelian, pro- ℓ and pro- C versions) can also be applied to predict the distributions of G S ( k )as k/Q varies among certain families of Γ-extensions. Secondly, the group B S ( k, A ), whichis the generalization of B S ( k ) that we construct in Section 8, has its own interest, becauseit bounds the Shafarevich group via (see Proposition 8.5) X S ( k, A ) ≤ B S ( k, A ) . (1.5)We emphasis here that understanding when X ( k, A ) = B Ø ( k ) holds can help us de-termine whether our upper bound of multiplicities is sharp or not (see how the inequality(1.5) is used in the proof of Proposition 9.4). Last but not least, the techniques establishedin Sections 3, 4 and 5, which use group cohomology to understand the presentation of aΓ-group, are purely group theoretical and independent of the number theory background, sowe hope that they could have other interesting applications.In this paper, we only study the maximal prime-to- | Γ | quotient of G Ø , ∞ ( k ) for a GaloisΓ-extension k/Q , and one can see that this “prime-to- | Γ | ” requirement is necessary in almostevery crucial step. We would like to ask if the ideas of this paper can be generalized to the | Γ | -part of G Ø , ∞ ( k ) too. 2. Notation and Preliminary
Profinite groups and modules.
In this paper, groups are always profinite groups.For a group G , a G -group is a group with a continuous G action. If x , · · · are elements of a G -group H , we write [ x , · · · ] for the closed normal G -subgroup of H topologically generatedby x , · · · . If H is a G -group, then we write H ⋊ G for the semidirect product induced by the G action on H , and its multiplication rule is given by ( h , g )( h , g ) = ( h g ( h ) , g g ) for h , h ∈ H and g , g ∈ G . Morphisms of G -groups are G -equivariant group homomorphisms.We write ≃ G to represent isomorphism of G -groups, write Hom G to represent the set of G -equivariant homomophisms, and define G -subgroup and G -quotient accordingly. We say hat a set of elements of a G -group H G -generates H if H is the smallest closed G -subgroupcontaining this set. We say that H is an irreducible G -group if it is a nontrivial G -group andhas no proper, nontrivial G -subgroups. For a positive integer n , a pro- n ′ group is a groupsuch that every finite quotient has order relatively prime to n . The pro- n ′ completion of G is the inverse limit of all pro- n ′ quotients of G . For a prime ℓ , we denote the pro- ℓ and thepro- ℓ ′ completions of G by G ( ℓ ) and G ( ℓ ′ ) respectively.For a group G and a commutative ring R , we denote by R [ G ] the completed R -group ringof G . We use the following notation of G -modulesMod( G ) = the category of isomorphism classes of finite G -modules,Mod( R [ G ]) = the category of isomorphism classes of finite R [ G ]-modules, andMod n ( G ) = the category of isomorphism classes of finite Z /n Z [ G ]-modules.For a prime integer ℓ and a finite F ℓ [ G ]-module A , we define h G ( A ) to be the F ℓ -dimensionof Hom G ( A, A ). We consider the
Grothendieck group K ′ ( R [ G ]), which is the abelian groupgenerated by the set { [ A ] | A ∈ Mod( R [ G ]) } and the relations[ A ] − [ B ] + [ C ] = 0arising from each exact sequence 0 ! A ! B ! C ! R [ G ]). For A, B ∈ Mod( R [ G ]), the tensor product A ⊗ R B endowed with the diagonal action of G is anelement of Mod( R [ G ]). Then K ′ ( R [ G ]) becomes a ring by linear extensions of the product[ A ][ B ] = [ A ⊗ R B ]. If H is a subgroup of G , then the action of taking induced modules Ind HG defines a map from K ′ ( R [ H ]) to K ′ ( R [ G ]), which we will also denote by Ind HG .Let ℓ denote a prime integer. If H is a pro- ℓ ′ subgroup of G , then it follows by theSchur–Zassenhaus theorem that H ( H, A ) = 0 for any A ∈ Mod ℓ ( G ), and hence taking the H -invariants is an exact functor on Mod ℓ ( G ). Moreover, when G is a pro- ℓ ′ group, Mod ℓ ( G )is the free abelian group generated by the isomorphism classes of finite simple F ℓ [ G ]-modules,and elements [ A ] and [ B ] of K ′ ( F ℓ [ G ]) are equal if and only if A and B are isomorphic as F ℓ [ G ]-modules. For an abelian group A , we let A ∨ denote the Pontryagin dual of A .2.2. Galois groups and Galois cohomology.
For a field k , we write k for a fixed choiceof separable closure of k , and write G k for the absolute Galois group Gal( k/k ). For a finite G k -module A , we denote A ′ = Hom( A, k × ). Let k/Q be a finite Galois extension of globalfields. When v is a prime of the field Q , we define S v ( k ) to be the set of all primes of k lyingabove v . Note that the function field F q ( t ) has an infinite place defined by the valuation | · | ∞ := q deg( · ) , but this ∞ place is nonarchemedean. We define S ∞ ( k ) to be the set of allarchemedean places of k , so it is the empty set if k is a function field. We let G Ø , ∞ ( k ) denotethe Galois group of the maximal unramfied extension of k that is split completely at ∞ . Soif k is a number field, then G Ø , ∞ ( k ) is G Ø ( k ). If k is a funtion field, then G Ø , ∞ ( k ) is thequotient of G Ø ( k ) modulo the decomposition subgroups of k at primes above ∞ .Let S be a set of places of k . We let k S denote the maximal extension of k that isunramified outside S , and denote Gal( k S /k ) by G S ( k ) or just G S when the choice of k isclear. The set S is called k/Q -closed if S v ( k ) either is contained in S or intersects emptilywith S for any prime v of Q . When S is k/Q -closed, it is not hard to check by Galois theorythat k S is Galois over Q , and hence each element of Gal( k/Q ) defines an outer automorphismof G S ( k ). We denote N ( S ) = { n ∈ N | n ∈ O × k,S } , here O × k,S is the ring of S -integers of k . Explicitly, if k is a number field, then N ( S ) consistsof the natural numbers such that ord p ( n ) = 0 for all p S ; and if k is a function field, then N ( S ) is the set of all natural numbers prime to char( k ). For a group G , we defineMod S ( G ) = the category of finite G -modules whose order is in N ( S ) . If particular, if Q is a function field, then Mod S ( G ) consists of modules of order prime tochar( Q ).Let k be a global field, and p a prime of k . The completion of k at p is denoted by k p ,and the absolute Galois group and its inertia subgroup of k p is denoted by G p ( k ) and T p ( k )respectively. Let k/Q be a Galois extension of global fields. For a prime v of Q and a prime p ∈ S v ( k ), the Galois group of k p /Q v , denoted by Gal p ( k/Q ), is the decomposition subgroupof Gal( k/Q ) at p . The subgroups Gal p ( k/Q ) are conjugate to each other in Gal( k/Q ) forall p ∈ S v ( k ), so we write Gal v ( k/Q ) for a representative of this conjugacy class. For agroup G and an A ∈ Mod( G ), we write H i ( G, A ) and b H i ( G, A ) for the group cohomologyand the Tate cohomology respectively. For a field k , we denote H i ( k, A ) := H i ( G k , A )and b H i ( k, A ) := b H i ( G k , A ). Let A be a module in Mod( G Q ). Then the Galois groupGal( k/Q ) acts on H i ( k, A ) by conjugation. The conjugation map commutes with inflations,restrictions, cup products and connecting homomorphisms in a long exact sequence, andhence it is naturally compatible with spectral sequences and duality theorems used in thepaper. For a prime v of Q , we consider the Gal( k/Q ) action on ⊕ p ∈ S v ( k ) H i ( k p , A ) defined bythe action on ⊕ p ∈ S v ( k ) H i ( k p , Res G Q G v ( Q ) A ). In other words, Gal( k/Q ) acts on ⊕ p ∈ S v ( k ) H i ( k p , A )by the permutation action on S v ( k ) and by the Gal p ( k/Q )-conjugation on each summand.We similarly define the Gal( k/Q ) action on the product when the each of the local summandis H i ( T p , A ) and the unramified cohomology group H inr ( k p , A ). In particular, the product ofrestriction maps for v H i ( k, A ) ! M p ∈ S v ( k ) H i ( k p , A )respects the Gal( k/Q ) actions. Moreover, one can check that M p ∈ S v ( k ) H i ( k p , A ) ∼ = Ind Gal q ( k/Q )Gal( k/Q ) H i ( k q , A )as Gal( k/Q )-modules for any q ∈ S v ( k ). The same statement holds for the Tate cohomologygroups. We use the following notation for Shafarevich groups X i ( k, A ) = ker H i ( k, A ) ! Y p all places H i ( k p , A ) ! and X iS ( k, A ) = ker H i ( k, A ) ! Y p ∈ S H i ( k p , A ) ! , for a set S of places of k . . Presentations of finite generated profinite Γ -groups A free profinite Γ -group on n generators , denoted by F n (Γ), is defined to be the freeprofinite group on { x i,γ | i = 1 , · · · , n and γ ∈ Γ } , where σ ∈ Γ acts on F n (Γ) by σ ( x i,γ ) = x i,σγ . In other words, F n (Γ) is the largest Γ-group that can be topologically generated by n elementsunder the Γ action. When the choice of Γ is clear, we will denote F n (Γ) simply by F n . Let G be a finitely generated Γ-group. Then when n is sufficiently large, there exists a short exactsequence 1 ! N ! F n ⋊ Γ π ! G ⋊ Γ ! π is defined by mapping Γ identically to Γ, and { x i, Γ } ni =1 to a set of n elements of G that generates G under the Γ action. Note that (3.1) can be viewed as a presentation ofthe group G that is compatible with Γ actions, and we will call it a Γ -presentation of G .The minimal number of relations in the presentation (3.1), which is one of the objects vastlystudied in this paper, is related to the multiplicities of the irreducible F n ⋊ Γ-quotients of N .We define the multiplicity as follows, and one can find that this quantity is similarly definedin [Lub01, LW18, LWZB19]. Definition 3.1.
Given a short exact sequence ! N ! F ω ! H ! of Γ -groups, welet M be the intersection of all maximal proper F ⋊ Γ -normal subgroups of N , and denote N = N/M and F = F/M . Then N is a direct sum of finite irreducible F ⋊ Γ -groups. For anyfinite irreducible F ⋊ Γ -group A , we define m ( ω, Γ , H, A ) to be the multiplicity of A appearingin N . When ω refers to the surjection F ⋊ Γ ! H ⋊ Γ induced by the Γ -equivariant quotient F ! H , we use the notation m ( ω, Γ , H, A ) instead of m ( ω | F , Γ , H, A ) for convenience sake. Consider the short exact sequence (3.1). As in Definition 3.1, we let M be the intersectionof all maximal proper F n ⋊ Γ-normal subgroups of N , and define R = N/M and F = F n /M .Then we obtain a short exact sequence1 ! R ! F ⋊ Γ ! G ⋊ Γ ! . Note that F ⋊ Γ acts on R by conjugation, and maps the factor A m ( π, Γ ,G,A ) of R to itself.When A is abelian, then the conjugation action on A by elements in R is trivial, so the F ⋊ Γaction on A actually factor through G ⋊ Γ, and hence A is a finite simple G ⋊ Γ-module.
Lemma 3.2.
Using the notation above, if A is a finite simple G ⋊ Γ -module such that gcd( | Γ | , | A | ) = 1 , then we have m ( π, Γ , G, A ) = n dim F ℓ A − ξ ( A ) + dim F ℓ H ( G ⋊ Γ , A ) − dim F ℓ H ( G ⋊ Γ , A )dim F ℓ h G ⋊ Γ ( A ) , where ℓ is the exponent of A and ξ ( A ) := dim F ℓ A Γ /A G ⋊ Γ . Remark 3.3.
When Γ is the trivial group, the lemma is the result in [Lub01]
Proof.
Applying the inflation-restriction exact sequence to (3.1), we obtain0 ! H ( G ⋊ Γ , A N ) ! H ( F n ⋊ Γ , A ) ! H ( N, A ) G ⋊ Γ ! H ( G ⋊ Γ , A N ) ! H ( F n ⋊ Γ , A ) . (3.2) lso by gcd( | A | , | Γ | ) = 1, the Hochschild-Serre spectural sequence E ij = H i (Γ , H j ( F n , A )) = ⇒ H i + j ( F n ⋊ Γ , A ) degenerates, so we have that H ( F n ⋊ Γ , A ) ∼ = H ( F n , A ) Γ , which is trivial because F n as a profinite group is free. Note that N acts trivially on A , so H ( N, A ) G ⋊ Γ = Hom( N, A ) F n ⋊ Γ = Hom F n ⋊ Γ ( N, A ) = Hom G ⋊ Γ ( A m ( π, Γ ,G,A ) , A )because A is a simple F ℓ [ G ⋊ Γ]-module and m ( π, Γ , G, A ) is the maximal integer such that A m ( π, Γ ,G,A ) is an F n ⋊ Γ-equivariant quotient of N . Then it follows thatdim F ℓ H ( N, A ) G ⋊ Γ = m ( π, Γ , G, A ) dim F ℓ Hom G ⋊ Γ ( A, A ) . Thus, by (3.2) it suffices to show that dim F ℓ H ( F n ⋊ Γ , A ) = n dim F ℓ A − ξ ( A ).Elements of H ( F n ⋊ Γ , A ) correspond to the A -conjugacy classes of homomorphic sectionsof A ⋊ ( F n ⋊ Γ) ρ ! F n ⋊ Γ. We write every element of F n ⋊ Γ in the form of ( x, γ ) for x ∈ F n and γ ∈ Γ, and similarly, write elements of A ⋊ ( F n ⋊ Γ) as ( a ; x, γ ) for a ∈ A , x ∈ F n and γ ∈ Γ. Then ρ maps ( a ; x, γ ) to ( x, γ ) for any a , x and γ . Note that a section of ρ is totallydetermined by the images of ( x i, Γ ,
1) and (1 , γ ) for i = 1 , · · · , n and γ ∈ Γ, where x i, Γ ’s arethe Γ-generators of F n defined at the beginning of this section. Since gcd( | A | , | Γ | ) = 1, wehave H (Γ , A ) = 0 by the Schur-Zassenhaus theorem, which implies that the restrictions ofall the sections of ρ to the subgroup Γ are conjugate to each other by A . So we only need tostudy the A -conjugacy classes of sections of ρ which map (1 , γ ) to (1; 1 , γ ) for any γ ∈ Γ, andsuch sections are totally determined by the images of ( x i, Γ ,
1) for i = 1 , · · · , n . Let s and s be two distinct sections of this type. Under the multiplication rule of semidirect product,the conjugation of ( a ; x, γ ) by an element α ∈ A is( α − ; 1 , a ; x, γ )( α ; 1 ,
1) = ( α − · a · ( x, γ )( α ); x, γ )= ( α − · ( x, γ )( α ); 1 , a ; x, γ ) , where the last equality uses that A is abelian. Therefore, because of the assumption that s (1 , γ ) = s (1 , γ ) = (1; 1 , γ ) for any γ ∈ Γ, we see that s and s are A -conjugate if andonly if there exists α ∈ A Γ /A G ⋊ Γ such that s ( x, γ ) = ( α − · ( x, r )( α ); 1 , s ( x, γ ) for any x , γ . So { A -conjugacy classes of sections of ρ } = | A Γ /A G ⋊ Γ | − n Y i =1 ρ − ( x i, Γ , | A Γ /A G ⋊ Γ | − | A | n , which proves that dim F ℓ H ( F n ⋊ Γ , A ) = n dim F ℓ A − dim F ℓ ( A Γ /A G ⋊ Γ ). (cid:3) In this paper, instead of the Γ-presentations in the form of (3.1), we want to study thepresentations of pro- | Γ | ′ completions of Γ-groups. We denote the pro- | Γ | ′ completions of F n (Γ) and G by F ′ n (Γ) and G ′ respectively, and write F ′ n for F ′ n (Γ) when the choice of Γ isclear. Then F ′ n and G ′ naturally obtain Γ actions from F n and G , and we have a short exactsequence 1 ! N ′ ! F ′ n ⋊ Γ π ′ ! G ′ ⋊ Γ ! , (3.3)induced by (3.1), which we will call a | Γ | ′ - Γ -presentation of G ′ . roposition 3.4. Use the notation above. Let A be a finite simple G ′ ⋊ Γ -module, anddenote the exponent of A by ℓ . If ℓ divides | Γ | , then m ( π ′ , Γ , G ′ , A ) = 0 . Otherwise, we have m ( π ′ , Γ , G ′ , A )= n dim F ℓ A − ξ ( A ) + dim F ℓ H ( G ′ ⋊ Γ , A ) − dim F ℓ H ( G ′ ⋊ Γ , A )dim F ℓ h G ′ ⋊ Γ ( A ) , (3.4) ≤ n dim F ℓ A − ξ ( A ) + dim F ℓ H ( G, A ) Γ − dim F ℓ H ( G, A ) Γ dim F ℓ h G ⋊ Γ ( A ) , (3.5) where in (3.5) A is viewed as a G ⋊ Γ -module via the surjection G ⋊ Γ ! G ′ ⋊ Γ . Moreover,the equality in (3.5) holds if H (ker( G ! G ′ ) , F ℓ ) = 0 . Remark 3.5.
We see from (3.4) that the multiplicity m ( π ′ , Γ , G ′ , A ) depends on n, Γ , G and A , but not on the choice of the quotient map π ′ . Proof.
It is clear that if ℓ divides | Γ, then m ( π ′ , Γ , G ′ , A ) = 0. Assume ℓ ∤ | Γ | We considerthe following commutative diagram F n ⋊ Γ G ⋊ Γ F ′ n ⋊ Γ G ′ ⋊ Γ , ρ ̟π ρ G π ′ where each of the vertical maps is taking the | Γ | ′ -completion of the first component insemidirect product. If U is a maximal proper F ′ n ⋊ Γ-normal subgroup of ker π ′ such thatker π ′ /U ≃ G ′ ⋊ Γ A , then its full preimage ρ − ( U ) in F n ⋊ Γ is a maximal proper F n ⋊ Γ-normalsubgroup of ker ̟ with ker ̟/ρ − ( U ) ≃ G ′ ⋊ Γ A . So by definition of multiplicities, we havethat m ( π ′ , Γ , G, A ) ≤ m ( ̟, Γ , G, A ). On the other hand, because of gcd( | A | , | Γ | ) = 1, if V isa maximal proper F n ⋊ Γ-normal subgroup of ker ̟ with ker ̟/V ≃ G ′ ⋊ Γ A , then F n ⋊ Γ ։ ( F n /V ) ⋊ Γ factors through ρ , and hence we showed that m ( π ′ , Γ , G, A ) = m ( ̟, Γ , G, A ).Because ̟ defines a Γ-presentation of G ′ , by Lemma 3.2 we obtain the equality (3.4).Let W denote ker ρ G = ker( G ! G ′ ). Because G ′ is the pro- | Γ | ′ completion of G and ℓ ∤ | Γ | , the pro- ℓ completion of W is trivial. So as W acts trivially on A , we have that H ( W, A ) = Hom(
W, A ) = 0 . Then by considering the inflation-restriction exact sequence of1 ! W ! G ⋊ Γ ! G ′ ⋊ Γ ! , we see that H ( G ′ ⋊ Γ , A ) ∼ = H ( G ⋊ Γ , A ) and H ( G ′ ⋊ Γ , A ) ֒ ! H ( G ⋊ Γ , A )where the latter embedding is an isomorphism if H ( W, A ) = 0. Note that H ( W, A ) = H ( W, F ℓ ) ⊕ dim F ℓ A because W acts trivially on A .Finally, since gcd( | A | , | Γ | ) = 1, we have that H i (Γ , A ) = 0 for any i ≥
1, and hence by theHochschild-Serre spectral sequence of1 ! G ! G ⋊ Γ ! Γ ! H i ( G ⋊ Γ , A ) ∼ = H i ( G, A ) Γ for any i . Therefore, we havedim F ℓ H ( G ′ ⋊ Γ , A ) = dim F ℓ H ( G, A ) Γ and dim F ℓ H ( G ′ ⋊ Γ) ≤ dim F ℓ H ( G, A ) Γ , here the equality holds if H ( W, F ℓ ) = 0, and hence we finish the proof. (cid:3) By Remark 3.5, we can define the multiplicities as follows.
Definition 3.6.
Let Γ be a finite group, G ′ a finitely generated pro- | Γ | ′ Γ -group. Let A be a finite simple G ′ ⋊ Γ -module and n a sufficiently large integer such that there exists a Γ -equivariant surjection π ′ : F ′ n ! G ′ , we define m ( n, Γ , G ′ , A ) to be m ( π ′ , Γ , G ′ , A ) . Inpartucular, m ( n, Γ , G ′ , A ) can be computed by formula (3.5) . Proposition 3.7.
Use the notation in Proposition prop:d’-cohom. Then the minimal numberof generators of ker π ′ as a closed normal Γ -subgroup of F ′ n is sup ℓ ∤ | Γ | sup A : finite simple F ℓ [ G ′ ⋊ Γ] -modules (cid:24) dim F ℓ H ( G ′ ⋊ Γ , A ) − dim F ℓ H ( G ′ ⋊ Γ , A ) − ξ ( A )dim F ℓ A (cid:25) + n. (3.6) Moreover, this minimal number is ≤ sup ℓ ∤ | Γ | sup A : finite simple F ℓ [ G ′ ⋊ Γ] -modules (cid:24) dim F ℓ H ( G, A ) Γ − dim F ℓ H ( G, A ) Γ − ξ ( A )dim F ℓ A (cid:25) + n (3.7) and the equality holds if H (ker( G ! G ′ ) , F ℓ ) = 0 .Proof. We let M be the intersection of all maximal proper F ′ n ⋊ Γ-normal subgroups of ker π ′ ,and denote R = ker π ′ /M and F = F ′ n /M . Then R is isomorphic to a direct product of finiteirreducible F ⋊ Γ-groups whose orders are coprime to | Γ | . A set of elements of ker π ′ generates R as a closed normal subgroup of F ′ n ⋊ Γ if and only if their images generate R as a normalsubgroup of F ⋊ Γ.By [LW18, Corollaries 5.9, 5.10], if m is a positive integer and A is a finite irreducible F ⋊ Γ-group, then the minimal number of elements of A m that can generate A m as an F ⋊ Γ-group isis ( A is non-abelian l m dim F ℓ h F ⋊ Γ ( A )dim F ℓ A m if A is abelian, where ℓ is the exponent of A. Recall that if A is an abelian simple factor appearing in R , then the F ⋊ Γ action on A factors through G ′ ⋊ Γ, since the conjugation action of R on A is trivial. Therefore, by theargument above and [LW18, Corollary 5.7], the minimal number of generators of R as an F ⋊ Γ-group is sup ℓ ∤ | Γ | sup A : finite simple F ℓ [ G ′ ⋊ Γ]-modules (cid:24) m ( n, Γ , G ′ , A ) dim F ℓ h G ′ ⋊ Γ ( A )dim F ℓ A (cid:25) . Then the proposition follows by Proposition 3.4. (cid:3)
We give a lemma at the end of this section that will be used later.
Lemma 3.8.
Let
E, F and G be Γ -groups such that there exist Γ -equivariant surjections α : E ! F , β : F ! G and a Γ -equivariant section s : F ! E of α . Denote π = β ◦ α . Let A be a finite simple G ⋊ Γ -module. E F G. α π βs We have m ( π, Γ , G, A ) ≤ m ( α, Γ , F, A ) + m ( β, Γ , G, A ) . (2) Moreover, if every Γ -group extension of F by A splits, then m ( π, Γ , G, A ) = m ( α, Γ , F, A )+ m ( β, Γ , G, A ) .Proof. Let S be the set of all maximal proper E ⋊ Γ-normal subgroups U of ker π withker π/U ≃ G ⋊ Γ A . So by definition we have ker π/ ( ∩ U ∈S U ) ≃ A m ( π, Γ ,G,A ) as G ⋊ Γ-modules.Define S = { U ∈ S | ker α ⊂ U } and S = { U ∈ S | ker α U } . One can easily check that there is a natural bijection S ! ( V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V is a maximal proper F ⋊ Γ-normal subgroupof ker β such that ker β/V ≃ A as G ⋊ Γ-modules ) U α ( U ) , and it follows that ker π/ ( ∩ U ∈S U ) ≃ A m ( β, Γ ,G,A ) . Similarly for the set S , there is a bijection S ! ( V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V is a maximal proper E ⋊ Γ-normal subgroupof ker α such that ker α/V ≃ A as F ⋊ Γ-modules ) (3.8) U U ∩ ker αs (ker β ) V − [ V, Let’s justify that (3.8) is a bijection. If U ∈ S , then U ker α is an E ⋊ Γ-normal subgroup ofker π that properly contains U , and therefore we have U ker α = ker π . So U ∩ ker α satisfiesker α/ ( U ∩ ker α ) = ( U ker α ) /U = ker π/U ≃ A as F ⋊ Γ-modules, and hence belongs to thethe right-hand set in (3.8). On the other hand, if V is an element in the right-hand set of(3.8), then we have s (ker β ) V ∈ S becauseker π/ ( s (ker β ) V ) = ( s (ker β ) ker α ) / ( s (ker β ) V )= ker α/ ( s (ker β ) V ∩ ker α )= ker α/V ≃ F ⋊ Γ A, where the equalities above use s (ker β ) ker α = ker π and s (ker β ) ∩ ker α = 1. It’s not hardto check that the composition of the maps in two directions is the identity map. So we seethat ker π/ ( ∩ U ∈S U ) ≃ A m ( α, Γ ,F,A ) . Then ker π/ ( ∩ U ∈S U ) ≃ A m ( π, Γ ,G,A ) is a submodule ofker π/ ( ∩ U ∈S U ) × ker π/ ( ∩ U ∈S U ), which implies (1).If any Γ-group extension of F by A splits, then A m ( α, Γ ,F,A ) ⋊ F is a Γ-quotient of E .Because the Γ-equivariant surjection β factors through an extensions of G by A m ( β, Γ ,G,A ) , wesee that m ( π, Γ , G, A ) ≥ m ( α, Γ , F, A ) + m ( β, Γ , G, A ). So we proved (2). (cid:3) Presentations of finite generated admissible profinite Γ -groups We first recall the definition of the admissible Γ-groups and the free admissible Γ-groupsin [LWZB19].
Definition 4.1.
A profinite Γ -group G is called admissible if it is Γ -generated by elements { g − γ ( g ) | g ∈ G, γ ∈ Γ } and is of order prime to | Γ | . ecall that for each positive integer n , we defined F ′ n to be the pro- | Γ | ′ completion of F n .We let y i,γ to be the image in F ′ n of the generators x i,γ of F n , and therefore F ′ n is the freepro- | Γ | ′ group on { y i,γ | i = 1 , · · · , n and γ ∈ Γ } where σ ∈ Γ acts on F ′ n by σ ( y i,γ ) = y i,σγ .We fix a generating set { γ , · · · , γ d } of the finite group Γ throughout the paper. We denote y i := y i, id Γ and define F n (Γ) to be the closed Γ-subgroup of F ′ n that is generated as a closedΓ-subgroup by the elements { y − i γ j ( y i ) | i = 1 , · · · , n and j = 1 , · · · , d } . We will denote F n (Γ) by F n when the choice of Γ is clear. The following is a list of propertiesof F n (Γ) proven in [LWZB19, Lemmas 3.1, 3.6 and 3.7]:(1) F n is an admissible Γ-group and it does not depend on the choice of the generatingset { γ , · · · , γ d } .(2) There is a Γ-equivariant quotient map ρ n : F ′ n ! F n such that the composition ofthe inclusion F n ⊂ F ′ n with ρ n is the identity map on F n .(3) Define a map of sets for any Γ-group GY : G ! G d g ( g − γ ( g ) , g − γ ( g ) , · · · , g − γ d ( g )) . Then the map Y ( G ) n ! Hom Γ ( F n , G )taking ( Y ( g ) , · · · , Y ( g n )) to the restriction of the map F ′ n ! G with y i g i is abijection.Let G be an admissible Γ-group with a Γ-presentation defined by F n ⋊ Γ π ։ G ⋊ Γ suchthat the reduced map F ′ n ⋊ Γ π ′ ։ G ⋊ Γ satisfies that G is Γ-generated by coordinates of Y ( y i ), i = 1 , · · · , n, (4.1)where y i are the Γ-generators of F ′ n as defined above. In this section, we are interested inthe Γ-presentations of this type, and under the condition (4.1), the restriction of π ′ to theadmissible subgroup F n of F ′ n is surjective. In other words, following by the property (2) of F n listed above, we want to study the | Γ | ′ -Γ-presentation π ′ that factors through the quotientmap ρ n : F ′ n ! F n . We denote π ad = π ′ | F n ⋊ Γ and obtain a short exact sequence1 −! N −! F n ⋊ Γ π ad −! G ⋊ Γ −! , (4.2)and we call it an admissible Γ -presentation of G .Similarly to the previous section, we are interested in the multiplicities of each simplefactors appearing as the quotients of N . Lemma 4.2.
Let G be an admissible Γ -group with an admissible Γ -presentation (4.2) and A a finite simple G ⋊ Γ -module with gcd( | A | , | Γ | ) = 1 . Then we have m ( π ad , Γ , G, A ) = m ( n, Γ , G, A ) − m ( n, Γ , F n , A ) . Proof.
We let ρ n : F ′ n ! F n be the quotient map described in the property (2). Let ̟ be the composition of the following Γ-equivariant surjections and then ̟ defines a | Γ | ′ -Γ-presentation of G . Let ι : F n ! F ′ n be the natural embedding. Then we have the following iagram F ′ n F n G, ρ n ̟ π ad | F n ι Also note that F n is free as a | Γ | ′ -group, so any group extension of F n by A splits. Thereforethe lemma follows immediately from Lemma 3.8. (cid:3) Definition 4.3.
Let G be a Γ -group with an admissible Γ -presentation (4.2) . For a finitesimple G ⋊ Γ -module A with gcd( | A | , | Γ | ) = 1 , we define m ad ( n, Γ , G, A ) to be m ( π ad , Γ , G, A ) .By Lemma 4.2, m ad ( n, Γ , G, A ) = m ( n, Γ , G, A ) − m ( n, Γ , F n , A ) does not depend on thechoice of π ad . Lemma 4.4.
Let A be a finite simple F n ⋊ Γ -module such that gcd( | A | , | Γ | ) = 1 . Then wehave dim F ℓ H ( F n ⋊ Γ , A ) = n dim F ℓ ( A/A Γ ) − ξ ( A ) . Proof.
We use the idea in the proof of Lemma 3.2. Elements of H ( F n ⋊ Γ , A ) correspond tothe A -conjugacy classes of homomorphic sections of A ⋊ ( F n ⋊ Γ) ρ ! F n ⋊ Γ. We use ( g, γ )to represent elements of F n ⋊ Γ, and ( a ; g, γ ) to represent elements of A ⋊ ( F n ⋊ Γ). Again,by the Schur-Zassenhaus theorem, we only need to count the A -conjugacy classes of sectionsof ρ that maps (1; 1 , γ ) to (1 , γ ). In other words, we only need to study the A -conjugacyclasses of Γ-equivariant sections of A ⋊ F n ! F n .By the property (3) of F n , there is a bijection Y ( A ⋊ F n ) n ! Hom Γ ( F n , A ⋊ F n ) taking( Y ( g ) , · · · , Y ( g n )) to the restriction of the map F ′ n ! A ⋊ F n with y i g i . For a Γ-equivariant section s of A ⋊ F n ! F n , the elements s ( y − i γ j ( y i )) in A ⋊ F n must map to y − i γ j ( y j ) ∈ F n for each i = 1 , · · · , n and j = 1 , · · · , d . Therefore, the Γ-equivariant sectionsof A ⋊ F n ! F n are in one-to-one correspondence with elements in Y ( A ⋊ F n ) n which mapto ( Y ( y ) , · · · , Y ( y n )) ∈ Y ( F n ) n under the natural quotient map A ⋊ F n ! F n on eachcomponent.Let’s consider Y ( y i ) and its preimages in Y ( A ⋊ F n ). Note that there is also a naturalembedding Y ( F n ) ֒ ! Y ( A ⋊ F n ) defined by the obvious section of split extension A ⋊ F n ։ F n . So we can fix a g ∈ A ⋊ F n such that Y ( g ) is the image of Y ( y i ) under this embedding,and then Y ( g ) is a preimage of Y ( y i ) under ϕ , where ϕ is the quotient map ( A ⋊ F n ) d ! F dn .The self-bijection ( A ⋊ F n ) d ! ( A ⋊ F n ) d ( a , · · · , a d ) ( ga γ ( g ) − , · · · , ga d γ d ( g ) − )clearly maps Y ( A ⋊ F n ) to itself and ϕ − ( Y ( y i )) to A d . Thus, Y ( A ⋊ F n ) ∩ ϕ − ( Y ( y i )) = Y ( A ⋊ F n ) ∩ A d = Y ( A ) = | A/A Γ | , where the second equality above uses [LWZB19, Lemma 3.5] and the last one uses [LWZB19,Lemma 3.3(2)]. So we’ve shown that there are | A/A Γ | elements in Y ( A ⋊ F n ) mapping to Y ( y i ), and it follows that the number of Γ-equivariant sections of A ⋊ F n ! F n is | A/A Γ | n .Finally, recall that two sections s , s of A ⋊ ( F n ⋊ Γ) ! F n ⋊ Γ are A -conjugate if andonly if s ( g, γ ) = ( α − · ( g, γ )( α ); 1 , s ( g, γ ) for some α ∈ A Γ /A F n ⋊ Γ ,by the computationin the proof of Lemma 3.2. Therefore, we have H ( F n ⋊ Γ , A ) = | A/A Γ | n | A Γ /A F n ⋊ Γ | , and hence weproved the lemma. (cid:3) orollary 4.5. Under the assumptions in Lemma 4.2, we have m ad ( n, Γ , G, A ) = m ( n, Γ , G, A ) − n dim F ℓ A Γ dim F ℓ h G ⋊ Γ ( A ) . Proof.
By Proposition 3.4 and Lemma 4.4, we have m ( n, Γ , F n , A ) = n dim F ℓ A Γ + dim F ℓ H ( F n , A ) Γ dim F ℓ h G ⋊ Γ ( A ) . Because F n is a free | Γ | ′ -group, we see that H ( F n , A ) = 0, and then the corollary followsimmediately by Lemma 4.2. (cid:3) We point out in the next lemma that A Γ is strictly smaller than A when A is a nontrivialmodule. Lemma 4.6. If G is an admissible Γ -group and A is a G ⋊ Γ -group such that Γ acts triviallyon A , then G ⋊ Γ acts trivially on A .Proof. The G ⋊ Γ action on A induces a group homomorphism G ⋊ Γ ! Aut( A ). So itsuffices to show that Γ is not contained in any proper normal subgroup of G ⋊ Γ. Suppose M is a proper normal subgroup containing Γ. Then B := ( G ⋊ Γ) /M is a Γ-quotient of G and Γ acts trivially on B . However, G is admissible, so is generated by elements g − γ ( g ) for g ∈ G and γ ∈ Γ. Then the images of all g − γ ( g ) in the Γ-quotient B generate B but eachof these images is 1, and hence we obtain the contradiction. (cid:3) Presentations of finite generated profinite Γ -groups of level C Let C be a set of isomorphism classes of finite Γ-groups. The variety of Γ -groups generatedby C is defined to be the smallest set C of isomorphism classes of Γ-groups containing C that is closed under taking finite direct products, Γ-quotients and Γ-subgroups. For a givenΓ-group G , we define the pro- C completion of G to be G C = lim − M G/M, where the inverse limit runs over all closed normal Γ-subgroups M of G such that the Γ-group G/M is contained in C . We call a Γ-group G level C if G C = G .We want to emphasis here that we do not require C to be closed under taking groupextensions, and it is different to many works in the literature of studying completions ofgroups. For example, if we let C to be the set containing only the group Z /ℓ Z with thetrivial Γ action, then G C is the maximal quotient of G that is isomorphic to a direct productof Z /ℓ Z on which Γ acts trivially. If we want G C to give us the pro- ℓ completion of G , thenwe need to let C contain all the finite Γ-groups of order a power of ℓ . Lemma 5.1.
Let
F, G be Γ -groups and ω : F ! G a Γ -equivariant surjection. Let C be a setof isomorphism classes of finite Γ -groups, and ϕ the pro- C completion map F ! F C . Thenwe have the following commutative diagram of Γ -equivariant surjections F GF C G C , ωϕ ω C here ω C is the quotient map modulo F C by ϕ (ker ω ) .Proof. By the set-up, im ω C naturally fit into the right-lower position of this diagram, so it’senough to show that im ω C ≃ G C . First, im ω C is a quotient of G and a quotient of F C , so itis of level C and hence is a quotient of G C . On the other hand, we consider the natural pro- C completion map α : G ! G C , and the composition α ◦ ω : F ! G C . Because G C is of level C , itfollows that ker( α ◦ ω ) ⊇ ker ϕ . Also, because ker ω ⊆ ker( α ◦ ω ), we have that im( α ◦ ω ) = G C is a quotient of F/ (ker ω ker ϕ ) = ( F/ ker ϕ ) / (ker ω/ ker ω ∩ ker ϕ ) = F C / ker ω C = im ω C . Sowe proved that im ω C ≃ G C . (cid:3) Definition 5.2.
For any Γ -equivariant surjection ω : F ! G , we define the pro- C completionof ω to be ω C : F C ! G C in Lemma 5.1. Corollary 5.3.
Under the assumptions in Lemma 5.1, for any finite simple G C ⋊ Γ -module A , we have m ( ω C , Γ , G C , A ) ≤ m ( ω, Γ , G, A ) .Proof. By Lemma 5.1, we have ker ω C = ϕ (ker ω ). If N is a maximal proper F C ⋊ Γ-normalsubgroup of F C such that F C /N ≃ A as G C ⋊ Γ-modules, then its preimage ϕ − ( N ) in F is amaximal proper F ⋊ Γ-normal subgroup of F with F/ϕ − ( N ) ≃ A . So the corollary followsby the definition of the multiplicity. (cid:3) Proposition 5.4.
Let G be an admissible Γ -group, C a set of isomorphism classes of finite Γ -groups and A a finite simple G C ⋊ Γ -module with gcd( | A | , | Γ | ) = 1 . For a fixed positiveinteger n such that there exists an admissible Γ -presentation of G as (4.2) , the multiplicity m ( π C ad , Γ , G C , A ) does not depend on the choice of π ad . So we denote m ( π C ad , Γ , G C , A ) by m C ad ( n, Γ , G, A ) . Then, we have m C ad ( n, Γ , G, A ) ≤ m ad ( n, Γ , G, A ) for any n, G, C and A . Moreover, if m ad ( n, Γ , G, A ) is finite, then the equality holds forsufficiently large C .Proof. Since A is finite, we can find a finite set C ⊂ C of isomorphism classes of finiteΓ-groups such that the map G C ⋊ Γ ! Aut( A ) induced by the G C ⋊ Γ action on A factorsthrough G C ⋊ Γ, and hence A is a simple G C ⋊ Γ-module. Let C ⊂ C ⊂ · · · be an ascendingsequence of finite sets of isomorphism classes of finite Γ-groups with ∪C i = C . For each i ≤ j ,we have that m ( π C i ad , Γ , G C i , A ) ≤ m ( π C j ad , Γ , G C j , A ) ≤ m ( π C ad , Γ , G C , A ) by Corollary 5.3, andhence m ( π C ad , Γ , G C , A ) = lim i ! ∞ m ( π C i ad , Γ , G C i , A ) . Since C i is a finite set of Γ-groups, [LWZB19, Remark 4.9] shows that the multiplicity m ( π C i ad , Γ , G C i , A ) does not depend on the choice of π ad . So we obtained that m ( π C ad , Γ , G C , A )also does not depend on the choice of π ad . The inequality in the proposition follows by m ( π C ad , Γ , G C , A ) ≤ m ( π ad , Γ , G, A ).The last statement in the proposition then automatically follows because m ad ( n, Γ , G, A ) = sup D : set ofΓ-groups m D ad ( n, Γ , G, A ) . (cid:3) . The heights of pro- C groups Definition 6.1.
For a finite group H , we define h ( H ) to be the smallest integer n for whichthere exists a sequence of normal subgroups of H , H ⊳ H ⊳ · · · ⊳ H n = H, such that H i +1 /H i is isomorphic to a direct product of minimal normal subgroups of H/H i .We define the height of H to be b h ( H ) = max { h ( U ) | U is a subquotient of H } . For a profinite group H , the height is defined as b h ( H ) = sup U : finitequotient of H b h ( U ) . Lemma 6.2.
Let G and H be two finite groups. Then b h ( G × H ) ≤ max { b h ( G ) , b h ( H ) } .Proof. It suffices to show that h ( U ) ≤ max { b h ( G ) , b h ( H ) } for any subquotient U of G × H .Each subquotient U of G × H is a quotient of a subgroup V of G × H . Then because h ( U ) ≤ h ( V ), we only need to show that h ( V ) ≤ max { b h ( G ) , b h ( H ) } for any subgroup V ⊂ G × H .We let Proj G and Proj H be the projections mapping G × H to G and H respectively,and denote V G = Proj G ( V ), V H = Proj H ( V ) and V = V / (ker(Proj G ) ker(Proj H )). ThenProj G × Proj H maps V injectively into V G × V H . Let n denote max { b h ( G ) , b h ( H ) } , and thenthere exists a sequence1 ⊳ V G, × V H, ⊳ V G, × V H, ⊳ · · · ⊳ V G,n × V H,n = V G × V H . of normal subgroups of V G × V H of length n , where { V ∗ ,i } for ∗ = G or H is a sequence ofnormal subgroups of V ∗ such that V ∗ ,i +1 /V ∗ ,i is a direct product of minimal normal subgroupsof V ∗ /V ∗ ,i . Assume that A is a minimal normal subgroup of V G contained in V G, . Since V is a subgroup of V G × V H , we have that A ∩ V is normal in V . Then under the surjection V ! V G , A ∩ V maps to a normal subgroup of V G that is contained in A . We see that A ∩ V is either A or 1, because A is minimal normal in V G . In particular, if A ∩ V = A , then it is aminimal normal subgroup of V , because otherwise Proj G maps a minimal normal subgroupof V contained in A ∩ V to a normal subgroup of V G that is properly contained in A . Thus,we showed that V ∩ ( V G, × V H, ) is a direct product of minimal normal subgroups of V .Then by induction on i , we see that { V i := V ∩ ( V G,i × V H,i ) } ni =1 forms a sequence of normalsubgroups of V such that V i +1 /V i is a direct product of minimal normal subgroups of V /V i ,and hence h ( V ) ≤ max { b h ( G ) , b h ( H ) } . (cid:3) Proposition 6.3.
Let Γ be a finite group and C a finite set of isomorphism classes of finite Γ -groups. For any Γ -group G , we have that b h ( G C ) is at most b h C := max { b h ( H ) | H ∈ C} . Proof.
By definition of b h ( G C ), it suffices to prove b h ( G ) ≤ b h C for any G ∈ C . So we justneed to show that the three actions, 1) taking Γ-quotients, 2) taking Γ-subgroups, and 3)taking finite direct products, do not produce groups with larger value of b h . For the first twoactions, it is obvious that if H is a Γ-quotient or a Γ-subgroup of G , then it is a quotient or subgroup of G by forgetting the Γ actions, and hence b h ( H ) ≤ b h ( G ). The last one followsby Lemma 6.2. (cid:3) We finish this section by applying Proposition 6.3 to prove the following number theoreticaltheorem.
Theorem 6.4.
Let k/Q be a Galois global field extension with
Gal( k/Q ) ≃ Γ and S a finite k/Q -closed set of places of k . Let C be a finite set of isomorphism classes of finite Γ -groups.Then G S ( k ) C is a finite group.Proof. By Proposition 6.3, we have that h := b h ( G S ( k ) C ) ≤ b h C is finite. So there exists a sequence of normal subgroups of G S ( k ) C ,1 = H ⊳ H ⊳ · · · ⊳ H h = G S ( k ) C , such that H i +1 /H i is isomorphic to a direct product of minimal normal subgroups of H h /H i .Note that each of the minimal normal subgroups is a (not necessarily finite) direct productof isomorphic finite simple groups. So, for each i , H i +1 /H i as a group is a direct productof finite simple groups. On the other hand, G S ( k ) C is a quotient of G S ( k ), so is the Galoisgroup of an extension of k that is unramified outside S . Therefore, H i +1 /H i is the Galoisgroup of an extension K i /K i +1 of some intermediate global fields between k S and k . Wedenote by S i the set of primes of K i lying above S .For a prime P of K i , the local absolute Galois group G P ( K i ) is finitely generated, sothere are finitely many Galois extensions of K P of a fixed Galois group. Then for a simplegroup E , there exists an integer N E, P ( K i ) for each P ∈ S i , such that any Galois extensionof K i whose Galois group is a subgroup of E has discriminant at most N E, P ( K i ). Let N E,S ( K i ) denote the product Q P ∈ S i N E, P ( K i ). By the Hermite-Minkowski theorem (see[Gos96, Theorem 8.23.5(3)] for the function field version of this theorem), for each finitesimple group E , there are only finitely many extensions of K i that are of Galois group E and of discriminant at most N E,S ( K i ). Therefore, there are finitely many extensions of K i that are of Galois group E and unramified outside S i .Since C is finite, there are only finitely many simple groups that appear as compositionfactors of groups in C (see [LW18, Corollary 6.12]). Now we consider the tower of extensions K i ’s. Note that K h = k and Gal( K h − /K h ) ≃ H h /H h − . By the above argument, weconclude that H h /H h − is a direct product of finite simple groups, that there are finitelymany choices of these finite simple groups, and that for each of them there are finite copiesof this simple group appearing in H h /H h − . So we obtain that H h /H h − is finite, andhence K h − is a finite extension of k . By induction, we see that H i +1 /H i is finite for each i = h − , · · · ,
0, and it follows that G S ( k ) C is finite. (cid:3) A Generalized Version of Global Euler-Poincar´e Characteristic
Throughout this section, we let k/Q be a finite Galois extension of global fields, and S a finite nonempty k/Q -closed set of primes of k such that S ∞ ( k ) ⊆ S . For each A ∈ Mod(Gal( k S /Q )), we define χ k/Q,S ( A ) = H ( G S ( k ) , A ) Gal( k/Q ) H ( G S ( k ) , A ) Gal( k/Q ) H ( G S ( k ) , A ) Gal( k/Q ) , here Gal( k/Q ) acts on H i ( G S ( k ) , A ) by conjugation. We will prove the following theorem. Theorem 7.1.
Use the assumption at the beginning of this section. If A ∈ Mod S (Gal( k S /Q )) has order prime to [ k : Q ] , then we have χ k/Q,S ( A ) = M v ∈ S ∞ ( Q ) b H ( Q v , A ′ ) , M v ∈ S ∞ ( Q ) H ( Q v , A ′ ) Remark 7.2. (1) If k is a function field, then the theorem says that χ k/Q,S ( A ) = 1 since S ∞ ( k ) = Ø.(2) When k = Q , the theorem is exactly the Global Euler-Poincar´e Characteristic For-mula.7.1. Preparation for the proof.Lemma 7.3.
Let G be a profinite group and U an open normal subgroup of G . Let H be anopen subgroup of G and V denote U ∩ H . Then H/V is naturally a subgroup of
G/U , andfor an H -module A we have H i ( U, Ind HG A ) ∼ = Ind H/VG/U H i ( V, A ) as G/U -modules for each i ≥ .Proof. Under the quotient map G ։ G/U , H/V is the image of H , so it is a subgroup of G/U . Then we have Ind HG A = Ind UHG
Ind
HUH A = M σ ∈ G/UH σ (Ind HUH A ) , where we denote by σ (Ind HUH A ) the σU Hσ − -module, whose underlying group is Ind HUH A and the action of τ ∈ σU Hσ − is given by a σ − τ σa . So H i ( U, Ind HG A ) = M σ ∈ G/UH H i ( U, σ (Ind
HUH A ))= M σ ∈ G/UH σ ∗ H i ( U, Ind
HUH A )= Ind H/VG/U H i ( U, Ind
HUH A ) , (7.1)where the second equality follows by U E G and the definition of the conjugation action σ ∗ oncohomology groups, and the last equality is because the quotient map G ! G/U maps a setof representatives of
G/U H to a set of representatives of (
G/U ) / ( H/V ). Since A is an H -module, U H acts on Ind VU A , and moreover, it follows by V = H ∪ U that Ind VU A = Ind HUH A as U H -modules. So we have the isomorphisms of
H/V -modules H i ( U, Ind
HUH A ) = H i ( U, Ind VU A ) ∼ = H i ( V, A ) , (7.2)where the last isomorphism follows by the Shapiro’s lemma. Then we proved the lemma by(7.1) and (7.2). (cid:3) or the rest of this section, we assume S is a nonempty k/Q -closed set of primes of k containing S ∞ and denote G = Gal( k S /Q ) and U = G S ( k ). For each open subgroup H ofGal( k S /Q ) we denote V = U ∩ H and define a map ϕ H,S : Mod( H ) ! K ′ ( Z [ H/V ]) A [ H ( V, A )] − [ H ( V, A )] + [ H ( V, A )] − M P ∈ S ∞ ( K ) b H ( K P , A ′ ) ∨ + M P ∈ S ∞ ( K ) H ( K P , A ′ ) ∨ , where K is the fixed field of V , H/V acts on ⊕ P ∈ S ∞ ( K ) H ( K P , A ′ ) (similarly on Tate coho-mology) by its permutation action on S ∞ ( K ) and by the Gal P ( K/Q ) ∩ H on each summand,and the Pontryagin dual is taking on the classes of K ′ ( Z [ H/V ]).
Lemma 7.4.
Using the notation above, we have the following isomorphisms of
G/U -modulesfor any A ∈ Mod( H ) M p ∈ S ∞ ( k ) H ( k p , Ind HG A ) ≃ Ind
H/VG/U M P ∈ S ∞ ( K ) H ( K P , A ) , (7.3) M p ∈ S ∞ ( k ) b H ( k p , Ind HG A ) ≃ Ind
H/VG/U M P ∈ S ∞ ( K ) b H ( K P , A ) . (7.4) Proof.
It suffices to fix a v ∈ S ∞ ( Q ) and prove (7.3) and (7.4) for places above v . Foreach p ∈ S v ( k ), Ind HG A as a G v ( Q )-module has the following canonical decomposition (see[NSW08, § G G v Ind HG A = M σ ∈G v \ G/H
Ind G v ∩ σHσ − G v σ Res Hσ − G v σ ∩ H A. If v splits completely in k/Q , then Gal v ( k/Q ) = 1 and G p ( k ) = G v ( Q ). So we have thefollowing identities of Gal v ( k/Q )-modules H ( k p , Ind HG A ) = M σ ∈G v \ G/H H ( G v ∩ σHσ − , σ Res Hσ − G v σ ∩ H A )= M σ ∈G v \ G/H σ ∗ H ( σ G v σ − ∩ H, Res Hσ − G v σ ∩ H A ) , (7.5)where the first equality uses the Shapiro’s lemma, and the second follows by definition of theconjugation action on cohomology groups. We let L denote the field fixed by H . For each σ ∈G p \ G/H , the intersection of σ G p σ − and H is G w ( L ) for exactly one w ∈ S v ( L ). Therefore,we have the identity of Gal v ( k/Q )-modules (hence of abelian groups since Gal v ( k/Q ) = 1) H ( k p , Ind HG A ) = M w ∈ S v ( L ) H ( L w , A ) , and hence M p ∈ S v ( k ) H ( k p , Ind HG A ) = Ind G/U M w ∈ S v ( L ) H ( L w , A ) (7.6) ecause the Gal( k/Q )-action on this direct sum is totally determined by its permutationaction on places above v . On the other hand, the assumption that v splits completely in k/Q implies that w splits completely in K for any w ∈ S v ( L ) and then we obtain M P ∈ S v ( K ) H ( K P , A ) = Ind H/V M w ∈ S v ( L ) H ( L w , A ) . (7.7)Thus, (7.6) and (7.7) prove (7.3) in this case. The isomorphism in (7.4) can be proven usingthe exactly same argument.Otherwise, v is ramified in k/Q , so Gal v ( k/Q ) ≃ Z / Z , G p ( k ) = 1 and G P ( K ) = 1 for each p ∈ S v ( k ) and P ∈ S v ( K ). Then (7.4) automatically follows because of b H ( k p , Ind HG A ) = b H ( K P , A ) = 0. Let R be a set of representatives of the right cosets of H in G . Then R naturally acts on S v ( L ) and, moreover, for any w ∈ S v ( L ) and σ , σ ∈ R , σ − σ is containedin G w ( L ) ⊂ Gal(
K/L ) if and only if σ ( w ) = σ ( w ). So we have the following identities ofGal v ( k/Q )-modules H ( k p , Ind HG A ) = M σ ∈ R σA = M w ∈ S v ( L ) w is real A ⊕ M w ∈ S v ( L ) w is imaginary ( A ⊕ τ A ) , where τ denotes the nontrivial element in Gal v ( k/Q ). So we have the following identity ofGal( k/Q )-modules M p ∈ S v ( k ) H ( k p , Ind HG A ) = M w ∈ S v ( L ) w is real Ind
Gal v ( k/Q )Gal( k/Q ) A ⊕ M w ∈ S v ( L ) w is imaginary Ind k/Q ) A. (7.8)Finally, because w ∈ S v ( L ) is imaginary if and only if Gal w ( K/L ) = 1, by applying the sameargument we have Ind
H/VG/U M P ∈ S v ( K ) H ( K P , A )= Ind H/VG/U M w ∈ S v ( L ) w is real Ind
Gal w ( K/L )Gal(
K/L ) A ⊕ M w ∈ S v ( L ) w is imaginary Ind
K/L ) A ∼ = M w ∈ S v ( L ) w is real Ind
Gal v ( k/Q )Gal( k/Q ) A ⊕ M w ∈ S v ( L ) w is imaginary Ind k/Q ) A. (7.9)Thus, (7.3) follows by (7.8) and (7.9). (cid:3) The corollary below immediately follows by Lemmas 7.3 and 7.4.
Corollary 7.5.
For any open subgroup H of G and A ∈ Mod( H ) , we have ϕ G,S (Ind HG A ) ≃ Ind
H/VG/U ϕ H,S ( A ) . Lemma 7.6.
The map ϕ G,S is additive on short exact sequences of modules in
Mod S ( G ) . roof. Denote G S ( k ) by G S . Let 0 ! A ! A ! A ! S ( G ). By considering the associated long exact sequence of groupcohomology, we have the following identity of elements in K ′ ( Z [Gal( k/Q )]) X i =0 3 X j =1 ( − i + j +1 [ H i ( G S , A j )] = X i =3 3 X j =1 ( − i + j [ H i ( G S , A j )] + [ δH ( G S , A )] , (7.10)where δ denotes the connecting map H i ! H i +1 (or b H i ! b H i +1 for Tate cohomologygroups) in the long exact sequence. By [NSW08, Theorem (8.6.10)(ii)], for i ≥ j , the restriction map H i ( G S , A j ) ! ⊕ p ∈ S R ( k ) H i ( k p , A j ) is an isomorphism. Note that if p ∈ S R ( k ) then G p ( k ) = Z / Z , and hence we have [ ⊕ p ∈ S R H ( k p , A j )] = [ ⊕ p ∈ S R b H − ( k p , A j )] and[ ⊕ p ∈ S R H ( k p , A j )] = [ ⊕ p ∈ S R b H ( k p , A j )] by [NSW08, Prop. (1.7.1)]. Because b H − ( k p , A j ) and b H ( k p , A j ) are of the same size and with the trivial action of G p , we see that if Gal p ( k/Q ) = 1,then [ b H − ( k p , A j )] = [ b H ( k p , A j )] as elements in K ′ ( Z [Gal p ( k/Q )]) = K ′ ( Z ), and otherwise[ b H − ( k p , A j )] = [ b H ( k p , A j )] = 0. So we have [ ⊕ p ∈ S R b H − ( k p , A j )] = [ ⊕ p ∈ S R b H ( k p , A j )] aselements in K ′ ( Z [ G/U ]).So (7.10) gives X i =0 3 X j =1 ( − i + j +1 (cid:2) H i ( G S , A j ) (cid:3) = (cid:2) δH ( G S , A ) (cid:3) = " M p ∈ S ∞ δH ( k p , A ) = " M p ∈ S ∞ δ b H ( k p , A ) = " M p ∈ S ∞ ker (cid:16) b H ( k p , A ) ! b H ( k p , A ) (cid:17) = " M p ∈ S ∞ coker (cid:16) b H ( k p , A ′ ) ! b H ( k p , A ′ ) (cid:17) ∨ = " M p ∈ S ∞ δ b H ( k p , A ′ ) ∨ (7.11)where the fourth equality and the last one uses the long exact sequence of Tate cohomologygroups, the fifth one uses the local duality theorem [NSW08, Theorem (7.2.17)]. On theother hand, again by [NSW08, Prop. (1.7.1)], the long exact sequence induced by0 ! A ′ ! A ′ ! A ′ ! mplies X j =1 ( − j +1 " M p ∈ S ∞ b H ( k p , A ′ j ) = X j =1 ( − j +1 " M p ∈ S ∞ b H ( k p , A ′ j ) = X j =1 ( − j +1 " M p ∈ S ∞ H ( k p , A ′ j ) + " M p ∈ S ∞ δH ( k p , A ′ ) (7.13)where the last equality follows by the long exact sequence of group cohomology induced by(7.12). Therefore, combining (7.11) and (7.13), we obtain ϕ G,S ( A ) − ϕ G,S ( A ) + ϕ G,S ( A ) = 0 . (cid:3) Lemma 7.7. If ℓ ∈ N ( S ) is a prime, then we have the following identities of elements in K ′ ( F ℓ [Gal( K/Q )]) for any Galois extension K of Q with k ( µ ℓ ) ⊂ K ⊂ k S [ H (Gal( k S /K ) , µ ℓ )] = [ µ ℓ ][ H (Gal( k S /K ) , µ ℓ )] = [ O × K,S /ℓ ] + [Cl S ( K )[ ℓ ]][ H (Gal( k S /K ) , µ ℓ )] = [Cl S ( K ) /ℓ ] − [ F ℓ ] + M P ∈ S \ S ∞ ( K ) F ℓ + M P ∈ S ∞ ( K ) b H ( G P , F ℓ ) , where Cl S ( K ) is the S -class group of K , Cl S ( K )[ ℓ ] is the ℓ -torsion subgroup of Cl S ( K ) ,and O × K,S /ℓ and Cl S ( K ) /ℓ denote the maximal exponent- ℓ quotients of O × K,S and Cl S ( K ) respectively.Proof. The lemma follows directly from the proof of [NSW08, Theorem 8.7.4]. Though theproof of [NSW08, Theorem 8.7.4] only shows these identities when each terms are treated asGrothendieck group elements of Gal(
K/k )-modules, one can check that the ideas there workgenerally for the base field Q instead of k . (cid:3) Proof of Theorem 7.1.
Recall that for any G -module A and v ∈ S ∞ ( Q ), M p ∈ S v ( k ) H ( k p , A ′ ) ∼ = Ind Gal p ( k/Q )Gal( k/Q ) H ( k p , A ′ )as Gal( k/Q )-modules, where p on the right-hand side is an arbitrary place in S v ( k ). So bythe Shapiro’s lemma, we have M p ∈ S v ( k ) H ( k p , A ′ ) Gal( k/Q ) ∼ = H ( k p , A ′ ) Gal p ( k/Q ) = H ( Q v , A ′ ) . (7.14) imilarly, we have M p ∈ S v ( k ) b H ( k p , A ′ ) Gal( k/Q ) ∼ = b H ( k p , A ′ ) Gal p ( k/Q ) = b H ( Q v , A ′ ) , (7.15)where the last equality holds because if Gal p ( k/Q ) = Z / Z , then b H ( k p , A ′ ) = b H ( Q v , A ′ ) =0 because | A ′ | has to be odd as gcd( | A | , [ k : Q ]) = 1. Note that for any M ∈ Mod(Gal( k/Q )),we have M Gal( k/Q ) ≃ ( M ∨ ) Gal( k/Q ) . So we have that the Gal( k/Q )-invariants of M p ∈ S v ( k ) H ( k p , A ′ ) ∨ and M p ∈ S v ( k ) b H ( k p , A ′ ) ∨ are H ( Q v , A ′ ) and b H ( Q v , A ′ ) respectively.We let R denote the ring Q p ∤ [ k : Q ] Z p . Let Θ : K ′ ( R [Gal( k/Q )]) ! Z be the map sendingthe class [ A ] to the size of A Gal( k/Q ) , which is a group homomorphism since taking Gal( k/Q )-invariants is an exact functor in the category of R [Gal( k/Q )]-modules. So we want to showthat Θ ◦ ϕ G,S is the zero map when restricted to modules in Mod S (Gal( k S /Q )) with orderprime to [ k : Q ]. By Lemma 7.6 we just need to showΘ ◦ ϕ G,S ( K ′ ( F ℓ [Gal( E/Q )])) = 0 , (7.16)for any prime integer ℓ ∈ N ( S ) with ℓ ∤ [ k : Q ] and any finite extension E of k that is Galoisover Q . Because the target of the map Θ is free, (7.16) is equivalent to the vanishing ofΘ ◦ ϕ G,S on the torsion-free part of K ′ ( F ℓ [Gal( E/Q )]). Note that, by [NSW08, Lem. (7.3.4)],the Q -linear space K ′ ( F ℓ [Gal( E/Q )]) ⊗ Z Q is generated by classes in the form of Ind C Gal(
E/Q ) A ,where C runs over all cyclic subgroups of Gal( E/Q ) of order prime to ℓ and A runs over classesof K ′ ( F ℓ [ C ]). For such C and A , we denote C the full preimage of C in G = Gal( k S /Q ),and then by Corollary 7.5 and Ind C Gal(
E/Q ) A = Ind CG A , we have that Θ ◦ ϕ G,S (Ind CG A ) = 0 ifand only if Θ ◦ ϕ C,S ( A ) = 0. By setting G to be C , Q to be ( k S ) C and k to be ( k S ) C ∩ U , wefinally reduce the problem to the statement that we will prove below:Θ ◦ ϕ G,S ( A ) = 0 for all A ∈ Mod ℓ ( G ) such that k ( A ) /Q is (7.17)a cyclic extension of Q of order relatively prime to ℓ .We let K = k ( A, µ ℓ ). So under the assumption in (7.17), we have that Gal( K/Q ) is anabelian group of order relatively prime to ℓ , in which case the Hochschild-Serre spectralsequence for the group extension1 ! Gal( k S /K ) ! Gal( k S /k ) ! Gal(
K/k ) ! A degenerates, and then we have for each i ≥ H i (Gal( k S /k ) , A ) ∼ = H i (Gal( k S /K ) , A ) Gal(
K/k ) . (7.18)We first consider the module A = µ ℓ , then K = k ( µ ℓ ) and we let G = Gal( K/Q ). As ℓ ∤ Gal(
K/Q ), in both the number field case (by [NSW08, Cor. (8.7.3)]) and the function eld case (by a standard argument using the divisor group), we have that[ O × K,S /ℓ ] = M P ∈ S ( K ) F ℓ + [ µ ℓ ] − [ F ℓ ]in K ′ ( F ℓ [ G ]). Then since [Cl S ( K )[ ℓ ]] = [Cl S ( K ) /ℓ ] as they are the kernel and the cokernelof the map Cl S ( K ) × ℓ −! Cl S ( K ), by Lemma 7.7 we have X i =0 ( − i [ H i (Gal( k S /K ) , µ ℓ )] = M P ∈ S ∞ ( K ) b H ( K P , F ℓ ) − M P ∈ S ∞ ( K ) H ( K P , F ℓ ) , (7.19)and hence ϕ G,S ( µ ℓ ) = 0 follows easily by (7.18) and by the similar arguments in (7.14) and(7.15). Thus, we have Θ ◦ ϕ G,S ( µ ℓ ) = 0.For a general finite module A ∈ Mod ℓ ( G ), we again denote K = k ( A, µ ℓ ) and G =Gal( K/Q ). We define χ : Mod ℓ ( G ) ! K ′ ( F ℓ [ G ]) M X i =0 ( − i [ H i (Gal( k S /K ) , M )] . Because A and µ ℓ are both trivial Gal( k S /K )-modules, the pairing µ ℓ × Hom( A ′ , F ℓ ) ! Hom( A ′ , µ ℓ ) = A ( ζ , f ) ( x ζ f ( x ) )defines G -isomorphisms via the cup product H i (Gal( k S /K ) , µ ℓ ) ⊗ Z Hom( A ′ , F ℓ ) ∼ −! H i (Gal( k S /K ) , A ) . So we have χ ( A ) = [ A ′∨ ] χ ( µ ℓ ), and hence by (7.19) we have χ ( A ) = [ A ′∨ ] M P ∈ S ∞ ( K ) b H ( K P , F ℓ ) − M P ∈ S ∞ ( K ) H ( K P , F ℓ ) . If Q is a function field, then (7.17) follows immediately after taking the G -invariants on bothsides above.For the rest of the proof we consider the number field case. Let S −∞ ( Q ) be the set ofarchimedean places of Q lying below the imaginary places of K if ℓ = 2, and be the set S ∞ ( Q ) if ℓ is odd. One can easily check that for any module M ∈ Mod ℓ ( G ) (for example, M = A ′ and M = F ℓ ), we have M P ∈ S ∞ ( K ) b H ( K P , M ) − M P ∈ S ∞ ( K ) H ( K P , M ) = X v ∈ S −∞ ( Q ) − h Ind G v G M i , where the group G v is the decomposition subgroup G v ( K/Q ). Also, note that (Ind G v G F ℓ ) ⊗ Z M ∼ = Ind G v G M for any M ∈ Mod ℓ ( G ) and that(Ind G v G M ) G = H ( G, Ind G v G M ) = H ( G v , M ) = M G v . o we haveΘ ◦ ϕ G,ℓ ( A ) = χ ( A ) − M P ∈ S ∞ ( K ) b H ( K P , A ′ ) + M P ∈ S ∞ ( K ) H ( K P , A ′ ) G = Y v ∈ S −∞ ( Q ) (cid:16) − [ A ′∨ ] h Ind G v G F ℓ i + h Ind G v G A ′ i(cid:17) G = Y v ∈ S −∞ ( Q ) (cid:16) − h Ind G v G A ′∨ i + h Ind G v G A ′ i(cid:17) G = Y v ∈ S −∞ ( Q ) A ′ ) G v A ′∨ ) G v which is identically 0 becausedim F ℓ ( A ′∨ ) G v = dim F ℓ Hom( A ′ , F ℓ ) G v = dim F ℓ ( A ′ ) G v = dim F ℓ ( A ′ ) G v , where the first two equalities follow by definition and the last one uses the assumption that ℓ ∤ | G v | . 8. Definition and properties of B S ( k, A )Throughout this section, we assume that k/Q is a finite Galois extension of global fields,and that S is a k/Q -closed set of primes of k (not necessarily nonempty or containing S ∞ ).Let p be a prime of the global field k . We denote G p = G p ( k ) and T p = T p ( k ). Recallthat for a G p -module A of order not divisible by char( k ), the unramified cohomology groupis defined to be H inr ( k p , A ) = im (cid:0) H i ( G p / T p , A ) ! H i ( k p , A ) (cid:1) , where the map is the inflation map. Then we consider the following homomorphism ofcohomology groups Y p ∈ S H ( k p , A ) × Y p S H nr ( k p , A ) ֒ ! Y p H ( k p , A ) ∼ ! Y p H ( k p , A ′ ) ∨ ! H ( k, A ′ ) ∨ . (8.1)The first map is the natural embedding of cohomology groups. The second arrow is anisomorphism because of the local Tate duality theorem [NSW08, Theorems 7.2.6 and 7.2.17].The last map is defined by the Pontryagin dual of the product of restriction map H ( k, A ′ ) ! H ( k p , A ′ ) for each prime p of k . In particular, the composition of the last two maps in (8.1)is the map Y p H ( k p , A ) ! H ( k, A ′ ) ∨ used in the long exact sequence of Poitou-Tate [NSW08, (8.6.10)(i)]. Definition 8.1.
For a global field k , a set S of primes of k , and A ∈ Mod( G k ) of order notdivisible by char( k ) , we define B S ( k, A ) = coker Y p ∈ S H ( k p , A ) × Y p S H nr ( k p , A ) ! H ( k, A ′ ) ∨ ! , here the map is the composition of all maps in (8.1) . Remark 8.2. (1) When A is a finite G Q -module and S is k/Q -closed, the maps in (8.1)are compatible with the conjugation action of Gal( k/Q ) on cohomology groups, so B S ( k, A ) is naturally a Gal( k/Q )-module.(2) Using the language of the Selmer groups, B S ( k, A ) is the Pontryagin dual of theSelmer group of the Galois module A ′ consisting of elements of H ( k, A ′ ) that haveimages inside the subgroup Y p ∈ S × Y p S ker (cid:0) H ( k p , A ′ ) ! H nr ( k p , A ) ∨ (cid:1) ⊂ Y p H ( k p , A ′ ) . under the product of local restriction maps. Proposition 8.3. If A = F ℓ is the trivial G k -module with ℓ = char( k ) , then B S ( k, F ℓ ) is thePontryagin dual of the Kummer group V S ( k, ℓ ) = ker k × /k × ℓ ! Y p ∈ S k × p /k × ℓ p × Y p S k × p /U p k × ℓ p ! . Proof.
By the class field theory, we have H ( k, µ ℓ ) ∼ = k × /k × ℓ , H ( k p , µ ℓ ) ∼ = k × p /k × ℓ p , and H nr ( k p , F ℓ ) ∨ ∼ = k × p /U p k × ℓ p . Then the proposition follows directly from Definition 8.1. (cid:3)
Lemma 8.4.
Let k/Q be a finite Galois extension of global fields, T ⊇ S be k/Q -closed setsof primes of k , and A ∈ Mod(Gal( k S /Q )) be of order not divisible by char( k ) . Then we havethe following exact sequence that is compatible with the conjugation by Gal( k/Q ) H ( G S ( k ) , A ) ֒ ! H ( G T ( k ) , A ) ! Y p ∈ T \ S H ( T p ( k ) , A ) G p ( k ) ! B S ( k, A ) ։ B T ( k, A ) . Proof.
We consider the following commutative diagram X ( k, A ) H ( G S , A ) H ( k, A ) H ( G k S , A ) G S Q p ∈ S H ( k p , A ) × Q p S H nr ( k p , A ) Q p H ( k p , A ) Q p S H ( T p , A ) G p H ( k, A ′ ) ∨ H ( k, A ′ ) ∨ B S ( k, A ) X ( k, A ′ )The exactnesses of the second row and the third row follow from the Hochschild-Serre spectralsequence, and last arrow in the third row is surjective because of the fact that H nr ( G p , A ) = 0 s G p / T p ≃ b Z or 1. The exact sequence of the first column follows from the definitionof B S ( k, A ), and the second column follows from the long exact sequence of Poitou-Tate[NSW08, (8.6.10)]. The right vertical map is injective since G k S is generated by the inertiagroups of primes outside S .We consider the map H ( k, A ) ! Q p S H ( T p , A ) G p in the diagonal of the square diagramon the right. Since H ( G S , A ) is exactly the kernel of this map while X ( k, A ) is containedin this kernel, the top dashed arrow exists and is injective. Then by diagram chasing, wehave an exact sequence X ( k, A ) ֒ ! H ( G S , A ) ! Y p ∈ S H ( k p , A ) × Y p S H nr ( k p , A ) ! H ( k, A ′ ) ∨ ։ B S ( k, A ) . (8.2)We apply the snake lemma to the following diagram Q p ∈ S H ( k p , A ) × Q p S H nr ( k p , A ) H ( k, A ′ ) ∨ Q p ∈ T H ( k p , A ) × Q p T H nr ( k p , A ) H ( k, A ′ ) ∨ Q p ∈ T \ S H ( T p , A ) G p where the horizontal map above is from (8.2), and we obtain the following exact sequence H ( G S , A ) X ( k, A ) ֒ ! H ( G T , A ) X ( k, A ) ! Y p ∈ T \ S H ( T p , A ) G p ! B S ( k, A ) ։ B T ( k, A ) . Note that the inflation map H ( G S , A ) ֒ ! H ( G T , A ) maps the submodule X ( k, A ) to itself,because X ( k, A ) is the kernel of H ( G ∗ , A ) ! Q p H ( k p , A ) for ∗ = S, T . Therefore weproved the exact sequence in the lemma, and it is naturally compatible with the conjugationaction by Gal( k/Q ). (cid:3) Proposition 8.5.
Let k/Q be a finite Galois extension of global fields and S a k/Q -closedset of primes of k . Then for any A ∈ Mod(Gal( k S /Q )) of order not divisible by char( k ) , wehave the following inequality of elements in K ′ (Gal( k/Q ))[ X S ( k, A )] ≤ [ B S ( k, A )] . Proof.
We consider the commutative diagram H ( G S , A ) ֒ ! H ( k, A ) ! H ( k S , A ) G S H ( G S , A ) H ( k, A ) Q p ∈ S H ( k p , A ) Q p H ( k p , A ) α βρ S ρ (8.3) here the first row is the inflation-restriction long exact sequence of 1 ! G k S ! G k ! G S !
1. Then we have an exact sequence H ( G S , A ) ֒ ! H ( k, A ) ! H ( G k S , A ) G S ! X S ( k, A ) ։ β ( X S ( k, A ))following by im α = ker β ⊆ ker ρ ◦ β = ker ρ S = X S ( k, A ). Comparing this exact sequenceto Lemma 8.4 with T = { all primes } , we have H ( k, A ) H ( k S , A ) G S X S ( k, A ) β ( X S ( k, A )) H ( k, A ) Q p S H ( T p , A ) G p B S ( k, A ) B { all primes } ( k, A ) . So by the vertical injection above, we have ker β ֒ ! N := ker( B S ( k, A ) ! B { all primes } ( k, A )).By the diagram in (8.3), we have β (ker ρ S ) ⊆ ker ρ , which means β ( X S ( k, A )) ⊆ X ( k, A ).Also, note that by Definition 8.1 and the Poitou-Tate duality we have B { all primes } ( k, A ) = X ( k, A ′ ) ∨ ∼ = X ( k, A ). Therefore, we obtainker β X S ( k, A ) β ( X S ( k, A )) N B S ( k, A ) B { all primes } ( k, A )Since every map respects the conjugation action by Gal( k/Q ), we have the desired inequality[ X S ( k, A )] ≤ [ B S ( k, A )]. (cid:3) Remark 8.6.
When A = F ℓ is the trivial module, then B { all primes } ( k, F ℓ ) vanishes [NSW08,Proposition 9.1.12(ii)], so there is an embedding X S ( k, F ℓ ) ֒ ! B S ( k, F ℓ ). However, for anarbitrary A , Proposition 8.5 does not give such an embedding. Lemma 8.7.
Let k be a global field and S a set of primes of k containing S ∞ ( k ) . Then forany A ∈ Mod S ( G S ( k )) , we have X S ( k, A ′ ) ∼ = B S ( k, A ) ∨ .Proof. We consider the following commutative diagram Q p H ( k p , A ′ ) Q p ∈ S H ( k p , A ′ ) × Q p S H ( T p , A ′ ) G p Q p H ( k p , A ) ∨ Q p ∈ S H ( k p , A ) ∨ × Q p S H nr ( k p , A ) ∨ , ∼ ∼ where the two vertical arrows are isomorphisms by the Tate local duality theorem and its con-sequence that H ( T p , A ′ ) G p ∼ −! H nr ( k p , A ) ∨ when A is unramified at p and A ) is primeto the characteristic of the residue field of k p (see the proof of [NSW08, Theorem 7.2.15]). hen by definition, we have B S ( k, A ) ∨ = ker H ( k, A ′ ) ! Y p ∈ S H ( k p , A ) ∨ × Y p S H nr ( k p , A ) ∨ ! = ker H ( k, A ′ ) ! Y p ∈ S H ( k p , A ′ ) × Y p S H ( T p , A ′ ) G p ! . So by applying the snake lemma to the following commutative diagram X S ( k, A ′ ) H ( G S , A ′ ) Q p ∈ S H ( k p , A ′ ) B S ( k, A ) ∨ H ( k, A ′ ) Q p ∈ S H ( k p , A ′ ) × Q p S H ( T p , A ′ ) G p H ( k S , A ′ ) G S Q p S H ( T p , A ′ ) G p , we obtain the desired isomorphism X S ( k, A ′ ) ∼ −! B S ( k, A ) ∨ . (cid:3) Corollary 8.8.
For any set S of primes of a global field k and any A ∈ Mod( G S ( k )) oforder not divisible by char( k ) , we have that X ( G S , A ) is finite.Proof. Define T = S ∪ S ∞ ( k ) ∪ S | A | ( k ). By applying Lemma 8.4, we have Y p ∈ T \ S H ( T p , A ) G p ! B S ( k, A ) ։ B T ( k, A ) . (8.4)Since A ∈ Mod T ( G T ), by Lemma 8.7 and [NSW08, Theorem 8.6.4], we have that B T ( k, A ) ∼ = X T ( k, A ′ ) is finite. Also note that H ( k p , A ) is finite [NSW08, Theorem 7.1.8(iv)] and thatthere is a short exact sequence0 ! H nr ( k p , A ) ! H ( k p , A ) ! H ( T p , A ) G p ! . Thus, the direct product Q p ∈ T \ S H ( T p , A ) G p is finite, and hence the corollary follows by(8.4). (cid:3) Determination of δ k/Q,S ( A ) Definition 9.1.
Let k/Q be a finite Galois extension of global fields, S a finite k/Q -closed setof primes of k , ℓ = char( k ) a prime integer not dividing [ k : Q ] , and A ∈ Mod ℓ (Gal( k S /Q )) .We define δ k/Q,S ( A ) = dim F ℓ H ( G S ( k ) , A ) Gal( k/Q ) − dim F ℓ H ( G S ( k ) , A ) Gal( k/Q ) . We will use the notation and assumption in Definition 9.1 throughout this section. Im-mediately following by Theorem 7.1, we have our first case for which δ k/Q,S ( A ) can bedetermined. roposition 9.2. Assume ℓ ∈ N ( S ) and S ⊃ S ∞ ( k ) is nonempty. Then δ k/Q,S ( A ) = log ℓ ( χ k/Q,S ( A )) − dim F ℓ A Gal( k S /Q ) . So in this section, we will consider the cases that are not covered by Proposition 9.2. In § Q is a function field and S = Ø, and obtain a formulafor δ k/Q, Ø ( A ) (Proposition 9.3). Then in § δ k/Q,S ( A ) when k is a number field with S ℓ ( k ) ∪ S ∞ ( k ) S (Proposition 9.4). When k = Q , Theorem 1.2follows by Propositions 9.2 and 9.4. Proof of Theorem 1.2.
We denote G = G S ( k ). Let A be a finite simple G -module and ℓ denote the exponent of A . Since b H ( k p , A ′ ) is naturally a quotient of H ( k p , A ′ ) for each p ∈ S ∞ ( k ), we have log ℓ χ k/k,T ( A ) ≤ T = S ∪ S ℓ ( k ) ∪ S ∞ ( k ). When S ⊃ S ℓ ( k ) ∪ S ∞ ( k ), Proposition 9.2 shows that δ k/k,S ( A ) ≤
0. It follows by definition of ǫ k/k,S ( A ) inProposition 9.4 that ǫ k/k,S ( A ) ≤ [ k : Q ] dim F ℓ A . Also, note that, when S S ℓ ( k ) ∪ S ∞ ( k ), wehave dim F ℓ ( A ′ ) G T ( k ) − dim F ℓ A G S ( k ) ≤ A cannot be µ ℓ if µ ℓ k . So Proposition 9.4shows that δ k/k,S ( A ) ≤ [ k : Q ] dim F ℓ A , and hence the theorem follows by Proposition 3.7. (cid:3) Function field case with S = Ø .Proposition 9.3. Assume k and Q are function fields. Let g = g ( k ) be the (geometric)genus of the curve corresponding to k . Then we have (1) If g = 0 , then δ k/Q, Ø ( A ) = − dim F ℓ A Gal( k Ø /Q ) . (2) If g > , then δ k/Q, Ø ( A ) = dim F ℓ ( A ′ ) Gal( k Ø /Q ) − dim F ℓ A Gal( k Ø /Q ) . Proof.
When g = 0, we have G Ø ( k ) ∼ = b Z by [NSW08, Cor. 10.1.3(i)]. So H ( G Ø , A ) = 0 as b Z has cohomological dimension 1, and H ( G Ø , A ) ∼ = A G Ø by [NSW08, Prop. 1.7.7(i)]. Thenwe see that δ k/Q, Ø ( A ) = − dim F ℓ ( A G Ø ) Gal( k/Q ) = − dim F ℓ ( A G Ø ) Gal( k/Q ) = − dim F ℓ A Gal( k Ø /Q ) , where the second equality uses ℓ ∤ [ k : Q ], which proves (1).For the rest, we assume g >
0. Let κ be the finite field of constants of k and C =Gal( κ/κ ) ∼ = b Z . Then there exists an exact sequence for each jH j ( G Ø ( kκ ) , A ) C ֒ ! H j ( G Ø ( kκ ) , A ) H j ( G Ø ( kκ ) , A ) ։ H j ( G Ø ( kκ ) , A ) C , Frob − (9.1)where Frob is the geometric Frobenious action on the cohomology groups. Note thatGal( kκ/Q ) acts on cohomology groups in (9.1), and1 ! C = Gal( kκ/k ) ! Gal( kκ/Q ) ! Gal( k/Q ) ! k/Q ) acts trivially on the generator Frob of C . Sothe map Frob − kκ/Q ) actions. It follows that H j ( G Ø ( kκ ) , A ) C and H j ( G Ø ( kκ ) , A ) C are in the same class in K ′ ( F ℓ [Gal( kκ/Q )]), and hence they are in thesame class in K ′ ( F ℓ [Gal( k/Q )]), which implies H j ( G Ø ( kκ ) , A ) C ≃ H j ( G Ø ( kκ ) , A ) C (9.2)as Gal( k/Q )-modules as ℓ ∤ [ k : Q ]. Therefore, we have H ( C, H j ( G Ø ( kκ ) , A )) ∼ = H j ( G Ø ( kκ ) , A ) C ≃ H ( C, H j ( G Ø ( kκ ) , A )) s Gal( k/Q )-modules. Then we consider the Hochschild-Serre spectral sequence E ij = H i ( C, H j ( G Ø ( kκ ) , A )) ⇒ H i + j ( G Ø ( k ) , A ) . As C has cohomological dimension 1, E ij = 0 for each i >
1, and hence by [NSW08,Lem. 2.1.3(ii)] we have the following exact sequence for every j ≥ H ( C, H j − ( G Ø ( kκ ) , A )) ֒ ! H j ( G Ø ( k ) , A ) ։ H ( C, H j ( G Ø ( kκ ) , A )) . (9.3)Note that G Ø ( k ) has strict cohomological dimension 3 by [NSW08, Cor. 10.1.3(ii)]. Then as ℓ ∤ [ k : Q ], taking Gal( k/Q )-invariants is exact on (9.3), and by computing the alternatingsum of (9.3) for j = 1 , ,
3, we have X j =1 ( − j dim F ℓ H j ( G Ø ( k ) , A ) Gal( k/Q ) = − dim F ℓ H ( C, H ( G Ø ( kκ ) , A )) Gal( k/Q ) = − dim F ℓ H ( C, H ( G Ø ( kκ ) , A )) Gal( k/Q ) = − dim F ℓ H (Gal( k Ø /Q ) , A ) . Also, [NSW08, Cor. 10.1.3(ii)] shows that G Ø ( k ) is a Poincar´e group of dimension 3 withdualizing module µ , so we have a functorial isomorphism H ( G Ø ( k ) , A ) ∼ = H ( G Ø ( k ) , A ′ ) ∨ .Combining the above computations, we see that δ k/Q, Ø ( A ) = dim F ℓ ( H ( G Ø ( k ) , A ′ ) ∨ ) Gal( k/Q ) − dim F ℓ H (Gal( k Ø /Q ) , A )= dim F ℓ H ( G Ø ( k ) , A ′ ) Gal( k/Q ) − dim F ℓ H (Gal( k Ø /Q ) , A )= dim F ℓ ( A ′ ) Gal( k Ø /Q ) − dim F ℓ A Gal( k Ø /Q ) , where the second equality is because the Gal( k/Q )-invariants of M and M ∨ have the samedimension for any M ∈ Mod ℓ (Gal( k/Q )). (cid:3) Number field case with S ℓ ∪ S ∞ S .Proposition 9.4. Assume k and Q are number fields. Let T = S ∪ S ℓ ( k ) ∪ S ∞ ( k ) . Thenwe have δ k/Q,S ( A ) ≤ log ℓ ( χ k/Q,T ( A )) + dim F ℓ ( A ′ ) Gal( k T /Q ) − dim F ℓ A Gal( k S /Q ) + ǫ k/Q,S ( A ) , where ǫ k/Q,S ( A ) = − P v ∈ I log ℓ k A k v with I = { v ∈ S ℓ ( Q ) such that S v ( k ) S } . In particular, when S = Ø , the equality holds if and only if X ( k, A ) and B Ø ( k, A ) are inthe same class of K ′ ( F ℓ [Gal( k/Q )]) .Proof. First of all, we have inequalities of elements in K ′ ( F ℓ [Gal( k/Q )]),[ H ( G S , A )] ≤ [ X S ( k, A )] + "M p ∈ S H ( k p , A ) ≤ [ B S ( k, A )] + "M p ∈ S H ( k p , A ) , (9.4) k x k v = Rchar( Q v ) − ord v ( x ) where Rchar( Q v ) is the residue characteristic of Q v and ord v is the additivevaluation with value group Z . here the first one uses the definition of X S and the second one uses Proposition 8.5. Thenby applying Lemma 8.4, we have[ B S ( k, A )] − [ H ( G S , A )] = [ B T ( k, A )] − [ H ( G T , A )] + M p ∈ T \ S H ( T p , A ) G p (9.5)Since T contains S ℓ ( k ) ∪ S ∞ ( k ), it follows that [ B T ( k, A )] = [ X T ( k, A )] by Lemma 8.7 andthe Poitou-Tate duality theorem. Also, note that the long exact sequence of Poitou-Tate[NSW08, (8.6.10)] induces an exact sequence X T ( k, A ) ֒ ! H ( G T , A ) ! M p ∈ T H ( k p , A ) ։ H ( G T , A ′ ) ∨ . Therefore we have[ B T ( k, A )] = [ H ( G T , A )] + [ H ( G T , A ′ ) ∨ ] − "M p ∈ T H ( k p , A ) . (9.6)Combining (9.4), (9.5) and (9.6), we have[ H ( G S , A )] − [ H ( G S , A )] ≤ [ H ( G T , A )] − [ H ( G T , A )] + [ H ( G T , A ′ ) ∨ ]+ M p ∈ T \ S H ( T p , A ) G p − M p ∈ T \ S H ( k p , A ) . The dimension of Gal( k/Q )-invariant of the left-hand side above is δ k/Q,S ( A ). On the right-hand side, the dimension of Gal( k/Q )-invariant of [ H ( G T , A )] − [ H ( G T , A )] islog ℓ ( χ k/Q,T ( A )) − dim F ℓ H ( G T , A ) Gal( k/Q ) = log ℓ ( χ k/Q,T ( A )) − dim F ℓ A Gal( k S /Q ) by the definition of χ k/Q,T and the assumption that A is a Gal( k S /Q )-module. Also,dim F ℓ (cid:0) H ( G T , A ′ ) ∨ (cid:1) Gal( k/Q ) = dim F ℓ H ( G T , A ′ ) Gal( k/Q ) = ( A ′ ) Gal( k T /Q ) . So to prove the inequality in the proposition, it suffices to show ǫ k/Q,S ( A ) = dim F ℓ M p ∈ T \ S H ( T p , A ) G p Gal( k/Q ) − dim F ℓ M p ∈ T \ S H ( k p , A ) Gal( k/Q ) (9.7)We first consider v ∈ S ∞ ( Q ) such that S v ( k ) S . Since T p ( k ) = G p ( k ), we know that H ( T p , A ) G p = H ( k p , A ) for each p ∈ S v ( k ). For i = 1 ,
2, we have M p ∈ S v ( k ) H i ( k p , A ) Gal( k/Q ) = (cid:16) Ind
Gal p ( k/Q )Gal( k/Q ) H i ( k p , A ) (cid:17) Gal( k/Q ) = H i ( k p , A ) Gal p ( k/Q ) = H i ( Q v , A ) , here the second equality uses the Shapiro’s lemma and the last one follows by the assump-tion that ℓ ∤ [ k : Q ]. Therefore, we havedim F ℓ M p ∈ S v ( k ) H ( T p , A ) G p Gal( k/Q ) − dim F ℓ M p ∈ S v ( k ) H ( k p , A ) Gal( k/Q ) = dim F ℓ H ( Q v , A ) − dim F ℓ H ( Q v , A ) , which always equals 0 since Q v is a cyclic group ([NSW08, Proposition 1.7.6]).Finally, we consider v ∈ S ℓ ( Q ) such that S v ( k ) S . Since A is a Gal( k S /Q )-module, foreach v in this case, A is unramified at v , i.e. T p ( k ) acts trivially on A for each p ∈ S v ( k ),and hence the inflation-restriction exact sequence gives0 ! H ( G p / T p , A ) ! H ( k p , A ) ! H ( T p , A ) G p ! . (9.8)Because G p / T p is procyclic, we have that H ( G p / T p , A ) ∼ = A G p / T p ; and by the same argumentfrom (9.1) to (9.2) in the function field case, we have an isomorphism H ( G p / T p , A ) ≃ A G p / T p = A G p that is compatible with the conjugation action by Gal p ( k/Q ). So we see thatdim F ℓ M p ∈ S p ( k ) H ( G p / T p , A ) Gal( k/Q ) = dim F ℓ (cid:16) Ind
Gal p ( k/Q )Gal( k/Q ) H ( G p / T p ( k ) , A ) (cid:17) Gal( k/Q ) = dim F ℓ H ( G p / T p ( k ) , A ) Gal p ( k/Q ) = dim F ℓ A G p ( Q ) . (9.9)Therefore, we computedim F ℓ M p ∈ S v ( k ) H ( T p , A ) G p Gal( k/Q ) − dim F ℓ M p ∈ S v ( k ) H ( k p , A ) Gal( k/Q ) = dim F ℓ M p ∈ S v ( k ) H ( k p , A ) Gal( k/Q ) − dim F ℓ M p ∈ S v ( k ) H ( k p , A ) Gal( k/Q ) − dim F ℓ A G v ( Q ) = dim F ℓ H ( k p , A ) Gal p ( k/Q ) − dim F ℓ H ( k p , A ) Gal p ( k/Q ) − dim F ℓ A G v ( Q ) = dim F ℓ H ( Q v , A ) − dim F ℓ H ( Q v , A ) − dim F ℓ A G v ( Q ) = − log ℓ k A k v . The first equality above uses (9.8) and (9.9), the second one uses the Shapiro’s lemma,the third one uses the assumption that ℓ ∤ [ k : Q ], and the last one uses the Tate’s localEuler-Poincar`e Characteristic formula [NSW08, Theorem 7.3.1]. Then we proved (9.7).When S = Ø, we have X ( k, A ) = H ( G Ø , A ), so the first inequality in (9.4) is anequality, and hence we have the last statement in the proposition. (cid:3) Proof of the main theorem
In this section, we will prove Theorem 1.1. We assume that Γ is a nontrivial finite group, Q = Q or F q ( t ) with gcd( q, | Γ | ) = 1, and let k/Q be a Galois extension with Gal( k/Q ) ≃ Γ.By Theorem 6.4, G Ø ( k ) C is a finite Γ-group when C is finite, so that we can talk about a -presentation F n (Γ) ! G Ø ( k ) C for a sufficiently large n . In § G of G Ø ( k ) such that G C ≃ G Ø ( k ) C as Γ-groups. With the help of thegroup G , we employ the cohomology of G Ø to compute the multiplicities in a pro- C admissibleΓ-presentation of G Ø ( k ) C . In § m C ad ( n, Γ , G Ø ( k ) C , A ), andthen the compute multiplicities m C ad ( n, Γ , G Ø , ∞ ( k ) C , A ) for a finite simple G Ø , ∞ ( k ) C ⋊ Γ-module A . Using these multiplicities, finally in § C admissible Γ-presentation F n (Γ) C ! G Ø , ∞ ( k ) C can be normally generated by elements { r − γ ( r ) } r ∈ X,γ ∈ Γ with X a subset of F n (Γ) of cardinality n + 1.Note that in Theorem 1.1, k/Q is assumed to be split completely at ∞ , and the Γ-groupsin C are of order prime to | µ ( Q ) | , | Γ | and char( Q ). However, in the proof, we do not use theseassumptions until § k/Q is a Galois field extensionwith Gal( k/Q ) ≃ Γ and that C is a finite set of isomorphism classes of finite Γ-groups oforder prime to | Γ | .10.1. Construction of a specific finitely generated quotient of G Ø ( k ) . Because G Ø ( k ) C is finite, when n is sufficiently large, there exists a Γ-equivariant surjection π : F n (Γ) ! G Ø ( k ) C , where F n (Γ) is the free profinite Γ-group defined in Section 3. Then π factorsthrough π C : F n (Γ) C ! G Ø ( k ) C as defined in Definition 5.2. Lemma 10.1.
Use the notation above. If A is a finite simple G Ø ( k ) C ⋊ Γ -module with m ( π C , Γ , G Ø ( k ) C , A ) > , then A ⋊ G Ø ( k ) C ∈ C .Proof. We denote G Ø ( k ) C by G for convenience purposes. If m ( π C , Γ , G , A ) >
0, then thereis a Γ-group extension 1 ! A ! H ̟ ! G ! , such that H is a quotient of F C n , and so H ∈ C . We let E be the fiber product H × G H defined by ̟ , i.e. E = { ( x, y ) ∈ H × H | ̟ ( x ) = ̟ ( y ) } . Note that E is a subgroup of H × H , so is in C . There is a natural diagonal embedding H ֒ ! E mapping x to ( x, x ), and a normal subgroup { ( a, | a ∈ A } of E that is isomorphic to A .From this, we see that E ≃ A ⋊ H , where the H action on A factors through ̟ ( H ) = G . Soby taking the quotient map ̟ on the subgroup H of E , we obtain that A ⋊ G is a quotientof E , and therefore we proved the lemma. (cid:3) Now we fix a finite simple G Ø ( k ) C ⋊ Γ-module A with m ( π C , Γ , G Ø ( k ) C , A ) >
0, andconstruct the desired quotient of G Ø ( k ) for A . We let ϕ denote the quotient map G Ø ( k ) ! G Ø ( k ) C , and again let G denote G Ø ( k ) C . We define G to be the quotient of G Ø ( k ) satisfyingthe following Γ-group extension1 ! A m ( ϕ , Γ ,G ,A ) ! G ̟ ! G ! . (10.1)By definition of the multiplicities, G is well-defined. Since G is a quotient of G Ø ( k ), wehave that G C is exactly G . Then we claim that the extension (10.1) is “completely nonsplit”(that is, if a subgroup of G maps surjectively onto G , then it has to be G itself). Indeed,if it’s not completely nonsplit, then G has a Γ-quotient isomorphic to A ⋊ G , and henceby Lemma 10.1 we have that A ⋊ G ∈ C , which violates G C = G .Similarly, we define G , G , · · · to be the Γ-quotients of G Ø ( k ) inductively via1 ! A m ( ϕ i , Γ ,G i ,A ) ! G i +1 ̟ i ! G i ! , here the map ϕ i is the quotient map G Ø ( k ) ! G i . Using the argument in the previousparagraph, we see that each of these group extensions is completely nonsplit, and G C i = G for each i . Then we take the inverse limit G := lim − i G i . Intuitively, the profinite group G is the maximal extension of G in G Ø ( k ) that can beobtained via group extensions by A . Lemma 10.2. (1)
A subset of G is a generator set if and only if its image in G generates G . (2) The map π : F n (Γ) ! G defined at the beginning of this subsection factors through G . (3) Let ϕ be the natural quotient map G Ø ( k ) ! G . Then Hom G (cid:0) (ker ϕ ) ab , A (cid:1) = 0 .Proof. The group extension ̟ i : G i +1 ! G i is completely nonsplit, so any lift of a generatorset of G i is a generator set of G i +1 . So we have (1) by induction, and then (2) follows.Note that G acts on the abelianization (ker ϕ ) ab of ker ϕ by conjugation. Suppose thatHom G ((ker ϕ ) ab , A ) = 0. Then it means that ϕ factors through a group extension H of G bya kernel A . However, G does not have such a group extension in G Ø ( k ) by definition. So weproved (3). (cid:3) Determination of the multiplicity of A . We continue to use notation and as-sumptions given previously in this section. In particular, we remind the reader that A is afixed finite simple G Ø ( k ) C ⋊ Γ-module where Γ ≃ Gal( k/Q ), and G is defined to be depend-ing on A . The goal of this subsection is to compute the multiplicity of A in an admissibleΓ-presentation of G Ø , ∞ ( k ) C . The Γ-group G plays a very important role in this computation. Lemma 10.3.
Let ℓ be the exponent of A and assume that ℓ = char( Q ) is prime to | Γ | .Then we have dim F ℓ H ( G, A ) Γ − dim F ℓ H ( G, A ) Γ ≤ δ k/Q, Ø ( A ) . Proof.
We consider the Γ-equivariant short exact sequence1 ! M ! G Ø ( k ) ϕ ! G ! . By the inflation-restriction exact sequence, we have0 ! H ( G, A ) ! H ( G Ø ( k ) , A ) ! H ( M, A ) G ! H ( G, A ) ! H ( G Ø ( k ) , A ) , (10.2)which is compatible with the conjugation action by Γ. Since M acts trivially on A , we seethat H ( M, A ) G = Hom G ( M ab , A ) = 0 by Lemma 10.2(3). So by taking the Γ-invariants on(10.2) and computing the dimensions, we have thatdim F ℓ H ( G, A ) Γ − dim F ℓ H ( G, A ) Γ ≤ dim F ℓ H ( G Ø ( k ) , A ) Γ − dim F ℓ H ( G Ø ( k ) , A ) Γ = δ k/Q, Ø ( A ) . (cid:3) Starting from now, we assume that C is a finite set of isomorphism classes of finite Γ-groupsall of whose orders are prime to | Γ | , char Q and | µ ( Q ) | . Let b π denote the Γ-equivariantsurjective map F n (Γ) ! G used in Lemma 10.2(2). Then the pro- C completion of b π is π C : F C n ։ G Ø ( k ) C . If Q = Q , then G Ø ( k ) C is exactly G Ø , ∞ ( k ) C . If Q is a function field, then Ø /k is not split completely at primes over ∞ . Instead, G Ø , ∞ ( k ) is the Γ-quotient of G Ø ( k )obtained via modulo the decomposition subgroup Gal p ( k Ø /k ) of one prime p of k above ∞ (because Γ acts transitively on all the primes of k above ∞ ). Since this decompositionsubgroup Gal p ( k Ø /k ) is isomorphic to b Z and G is a quotient of G Ø ( k ), we can define g n to be an element of G that is the image of one generator of Gal p ( k Ø /k ). In other words,denoting G the quotient G modulo the Γ-closed normal subgroup generated by g n , we havethe following diagram F n G G F n C G Ø ( k ) C G Ø , ∞ ( k ) C , b ππ ̟ η/ [ g n ] π C ̟ C η C (10.3)where the vertical maps are taking pro- C completions. To make the notation consistentbetween the number field and the function field cases, when Q = Q , we let g n = 1 and hence η and η C in (10.3) are both identity maps. First of all, we want to determine m ( b π, Γ , G, A ). Proposition 10.4.
Let ℓ be the exponent of A . Assume ℓ = char( Q ) is relatively prime to | µ ( Q ) || Γ | . If Q = Q , then m ( b π, Γ , G, A ) ≤ ( n + 1) dim F ℓ A − dim F ℓ A Γ h G ⋊ Γ ( A ) . If Q = F q ( t ) and A = µ ℓ , then m ( b π, Γ , G, A ) ≤ n dim F ℓ A − dim F ℓ A Γ h G ⋊ Γ ( A ) . Remark 10.5.
Recall that in Theorem 1.1 we assume that k/Q is split completely at ∞ .In the number field case, the module µ ℓ is not a Gal( k Ø /Q )-module, so A = µ ℓ . In thefunction field case, µ ℓ is a Gal( k Ø /Q )-module but not a Gal( k Ø , ∞ /Q )-module, so we excludethe situation that A = µ ℓ . Proof.
By the assumptions, we can apply Proposition 3.4 to compute the multiplicities.Because ℓ ∤ | Γ | , we have for i = 1 , H i ( G ⋊ Γ , A ) = H i ( G, A ) Γ . Then by Lemma 10.3,we have m ( b π, Γ , G, A ) ≤ n dim F ℓ A − ξ ( A ) + δ k/Q, Ø ( A ) h G ⋊ Γ ( A ) . (10.4)So we just need to compute δ k/Q, Ø ( A ).In the function field case, since k/Q is split completely above ∞ , the genus of k is positive.By Proposition 9.3(2), δ k/Q, Ø ( A ) = dim F ℓ Hom
Gal( k Ø /Q ) ( A, µ ℓ ) − dim F ℓ A Gal( k Ø /Q ) . Recall that A is a simple F ℓ [Gal( k Ø /Q )]-module that is not µ ℓ , so we see that δ k/Q, Ø ( A ) is − A = F ℓ , and is 0 otherwise. So we proved the result in function field case. n the number field case that Q = Q , we need to compute each terms in the formula fromProposition 9.4. Let T = S ℓ ( k ) ∩ S ∞ ( k ). In this case, ℓ is odd as µ ⊂ Q . First, we applyTheorem 7.1 log ℓ χ k/Q,T ( A ) = − dim F ℓ H ( Q ∞ , A ′ ) = − dim F ℓ ( A ′ ) Gal( C / R ) , where the first equality uses b H ( Q ∞ , A ′ ) = 0 because of G ∞ ( Q ) = 2 and [NSW08, Proposi-tion 1.6.2(a)]. Then because A is a simple F ℓ [Gal( k Ø /Q )]-module, it is totally real and hence( A ′ ) Gal( C / R ) = Hom C / R ( A, µ ℓ ) = 0. So we have log ℓ χ k/Q,T ( A ) = 0. Then note that ǫ k/Q, Ø ( A )in the formula in Proposition 9.4 is dim F ℓ A in this case, and we obtain δ k/Q, Ø ( A ) ≤ dim F ℓ Hom
Gal( k T /Q ) ( A, µ ℓ ) − dim F ℓ A Gal( k Ø /Q ) + dim F ℓ A, where the right-hand side is 0 if A = F ℓ and is dim F ℓ A otherwise. So we proved the numberfield case. (cid:3) Lemma 10.6.
Use the assumptions in Proposition 10.4. Consider the function field caseand the diagram (10.3) . When n is sufficiently large, we have m ( ̟, Γ , G , A ) ≤ ( n + 1) dim F ℓ A − dim F ℓ A Γ h G ⋊ Γ ( A ) Proof.
Again, we use x , · · · , x n to denote the generators of F n . We can make n large toassume b π ( x n ) = g n (recall that the multiplicity depends on n but not on the choice of ̟ ).Then we have a commutative diagram F n F n − G G , λ/ [ x n ] b π ̟ φη/ [ g n ] where the top map are defined by modulo the Γ-closed normal subgroup generated by x n . Note that the composition of the top and the right arrows satisfies the conditionsin Lemma 3.8(2), so we have m ( ̟, Γ , G , A ) = m ( λ, Γ , F n − , A ) + m ( φ, Γ , G , A ) . By the statement and the computation of H i ( F n ⋊ Γ , A ) in the proof of Lemma 3.2, we seethat m ( λ, Γ , F n − , A ) = dim F ℓ Ah G ⋊ Γ ( A ) . So it suffices to prove m ( φ, Γ , G , A ) ≤ m ( b π, Γ , G, A ) , (10.5)which will immediately follow after we prove the following embedding { U | max. proper F n − ⋊ Γ-normal subgroup of ker φ s.t. ker φ/U ≃ G ⋊ Γ A } κ ֒ −! { V | max. proper F n ⋊ Γ-normal subgroup of ker b π s.t. ker b π/V ≃ G ⋊ Γ A } mapping U to λ − ( U ) ∩ ker b π .Since ker ̟ = ker b π ker λ , for each U in the first set, we haveker b π (cid:30) λ − ( U ) ∩ ker b π = λ − ( U ) ker b π (cid:30) λ − ( U ) = ker ̟ (cid:30) λ − ( U ) ≃ G ⋊ Γ A, o the map κ is well-defined. Also, if V = κ ( U ), thenker ̟ (cid:30) V ker λ = ker b π ( V ker λ ) (cid:30) V ker λ = ker b π (cid:30) ker b π ∩ ( V ker λ ) . (10.6)Since V ⊂ ker b π and ker b π/V is a simple module, the last quotient is either 1 or isomorphicto A . On the other hand, both of V and ker λ are contained in λ − ( U ), so is V ker λ .Then (10.6) implies that V ker λ = λ − ( U ). So we see that if κ ( U ) = κ ( U ) = V , then λ − ( U ) = λ − ( U ) and hence U = U . So we conclude that κ is injective. (cid:3) Proposition 10.7.
Let A be a finite simple G Ø , ∞ ( k ) C ⋊ Γ -module of exponent ℓ = char( k ) relatively prime to | µ ( Q ) || Γ | . When n is sufficiently large, there exists an admissible Γ -presentation F n (Γ) ։ G Ø , ∞ ( k ) C , and m C ad ( n, Γ , G Ø , ∞ ( k ) C , A ) ≤ m ad ( n, Γ , G , A ) ≤ ( n + 1)(dim F ℓ A − dim F ℓ A Γ ) h G Ø , ∞ ( k ) C ⋊ Γ ( A ) . Remark 10.8.
The proposition shows that m C ad ( n, Γ , G Ø , ∞ ( k ) C , F ℓ ) = 0. In other words, G Ø , ∞ ( k ) C does not admit any nonsplit central group extension1 ! F ℓ ! e G ⋊ Γ ! G Ø , ∞ ( k ) C ⋊ Γ ! , such that e G is of level C . This is equivalent to the solvability (i.e. the existence of the dashedarrow) of the following embedding problem G Ø , ∞ ( k ) C ⋊ Γ1 F ℓ e H ⋊ Γ H ⋊ Γ 1 α for any nonsplit central group extension in the lower row with e H of level C , and for anysurjection α . In [LWZB19], this solvability is called the Property E of G Ø , ∞ ( k ) and is provenusing the classical techniques of embedding problems. So Proposition 10.7 provides a newproof of the Property E by counting multiplicites. Proof.
By [LWZB19, Proposition 2.2], G Ø , ∞ ( k ) is an admissible Γ-group, so is G , because G is a Γ-quotient of G Ø , ∞ ( k ). Since G Ø , ∞ ( k ) C is finite, when n is large, there exist elements a , · · · , a n of G Ø , ∞ ( k ) C such that { Y ( a i ) } ni =1 forms a generator sets of G Ø , ∞ ( k ) C . Then wechoose a preimage b i ∈ G of a i , and hence { Y ( b i ) } ni =1 generates G by Lemma 10.2(1).Recall that the multiplicity does not depend on the choice of presentation, so we assume ̟ in (10.3) maps y i ∈ F n to b i ∈ G for each i = 1 , · · · , n . Then the restriction ̟ | F n is anadmissible Γ-presentation of G . We have by Corollary 4.5 that m ad ( n, Γ , G , A ) = m ( n, Γ , G , A ) − n dim F ℓ A Γ h G ⋊ Γ ( A ) . Then the desired result follows by Propositions 5.4, 10.4 and 10.6. (cid:3) Existence of the presentation (1.3) . Finally, we will prove that when n is suffi-ciently large, there exists a set X of F C n containing n + 1 elements for which the followingisomorphism, which is (1.3) in Theorem 1.1, holds G Ø , ∞ ( k ) C ≃ F n (Γ) C (cid:30) [ r − γ ( r )] r ∈ X,γ ∈ Γ . In Proposition 10.7, we showed that when n is sufficiently large, there is an admissibleΓ-presentation, denoted by 1 ! N ! F C n ̟ C ad −! G Ø , ∞ ( k ) C ! . Let M be the intersection of all maximal proper F C n ⋊ Γ-normal subgroups of N , and define R = N/M and F = F C n /M . Note that because C is finite, we have that F C n is finite[Neu67, Corollary 15.72]. Then R is a finite direct product Q ti =1 A m i i of finite irreducible F ⋊ Γ-groups A i ’s. Assume A i and A j are not isomorphic as F ⋊ Γ-groups if i = j . When a factor A i is abelian, its multiplicity m i is m C ad ( n, Γ , G Ø , ∞ ( k ) C , A i ) computed in Proposition 10.7.Let X be a subset of F C n . Then the closed F C n ⋊ Γ-normal subgroup generated by { r − γ ( r ) } r ∈ X,γ ∈ Γ is N if and only if the closed F ⋊ Γ-normal subgroup generated by { r − γ ( r ) } r ∈ X,γ ∈ Γ is R , where X is the image of X in R . Recall the properties of F n listed at the beginning of §
4. Because of the property (1), in the definition of Y in (3), we can take the generator set { γ , · · · , γ d } to be the whole group Γ, then { r − γ ( r ) } r ∈ X,γ ∈ Γ = Y ( { r } r ∈ X ) and { r − γ ( r ) } r ∈ X,γ ∈ Γ = Y ( { r } r ∈ X ) . By [LWZB19, Proposition 4.3], for a fixed integer u , the probability that the images underthe map Y of n + u random elements of R generate R as an F ⋊ Γ-normal subgroup isProb([ Y ( { r , · · · , r n + u } )] F ⋊ Γ = R )= Y ≤ i ≤ tA i abelian m i − Y j =0 (1 − h F ⋊ Γ ( A i ) j | Y ( A i ) | − n − u ) Y ≤ i ≤ tA i non-abelian (1 − | Y ( A i ) | − n − u ) m i . This product in the formula is a finite product. By [LWZB19, Lemma 3.3(2)], we have | Y ( A i ) | = | A i | / | A Γ i | for each i . Note that Lemma 4.6 shows that | Y ( A i ) | > A i isnon-abelian, so the product over non-abelian factors in the above formula is always positive.The term for an abelian factor A i is positive if and only if m i ≤ ( n + u ) log ℓ | Y ( A i ) | h G Ø , ∞ ( k ) C ⋊ Γ ( A i ) = ( n + u )(dim F ℓ A i − dim F ℓ A Γ i ) h G Ø , ∞ ( k ) C ⋊ Γ ( A i ) . Therefore, by Proposition 10.7, R can be F ⋊ Γ-normally generated by the Y -values of n + 1elements, and hence we finish the proof.11. Exceptional cases
We will discuss the cases that are not covered by the Liu-Wood-Zureick-Brown conjecture,using the techniques developed in this paper. In this section, the base field Q can be anyglobal field. If Q is a number field, we denote r and r the numbers of the real embeddingsand the complex embeddings of Q respectively.Again, we let Γ be a finite group and k/Q a Galois extension of global fields withGal( k/Q ) ≃ Γ. We assume that ℓ is a prime integer that is not char( Q ) and is prime o | Γ | . Recall that G Ø ( k )( ℓ ) denotes the pro- ℓ completion of G Ø ( k ). So G Ø ( k )( ℓ ) is the Ga-lois group of the maximal unramified pro- ℓ extension of k , which we will denote by k Ø ( ℓ ) /k .Note that G Ø ( k )( ℓ ) is finitely generated, because dim F ℓ H ( k Ø , F ℓ ) is the minimal number ofgenerators of G Ø ( k )( ℓ ) and is finite. So when n is sufficiently large, there is a Γ-presentation π : F ′ n (Γ) ! G Ø ( k )( ℓ ). Moreover, we assume, throughout this section, that the ℓ -primarypart of the class group of Q is trivial. Then G Ø , ∞ ( k )( ℓ ) is admissible by the proof of[LWZB19, Proposition 2.2], and hence we can assume that the presentation π induces anadmissible presentation, i.e. π ad := π | F n is surjective.In this section, we use the assumptions above and study the multiplicities from the pre-sentation π ad in the following two cases:(1) When Q is a number field with µ ℓ Q , and k/Q is not required to be split completelyat S ∞ ( Q ) (see Section 11.1).(2) When Q contains the ℓ -roots of unity µ ℓ (see Section 11.2).We will compare the multiplicities in these two cases with the multiplicities from Theo-rem 1.1, to see why the random group model used in the Liu–Wood–Zureick-Brown conjec-ture cannot be applied to these two exceptional cases.We point out that we study only G Ø ( k )( ℓ ) instead of G Ø ( k ) C for a general C , simplybecause we want to keep the computation easy in this section and there is no previous workdiscussing these two exceptional cases beyond the distribution of ℓ -class tower groups. Onecan generalize the argument in this section to any finite set C .11.1. Other signatures.
Assume Q is a number field with µ ℓ Q (so ℓ is odd), and k is aΓ-extension of Q . For each v ∈ S ∞ ( Q ), we denote Γ v to be the decomposition subgroup at v of the extension k/Q . Lemma 11.1.
For a finite simple F ℓ [Gal( k Ø ( ℓ ) /Q )] -module A , we have m ad ( n, Γ , G Ø ( k )( ℓ ) , A ) ≤ ǫ k/Q, Ø ( A ) − r − if A = F ℓ ǫ k/Q, Ø ( A ) + n − r − r if A = µ ℓ ǫ k/Q, Ø ( A ) + ( n − r ) dim F ℓ A − P v ∈ S R ( Q ) dim F ℓ A/A Γ v − ( n + 1) dim F ℓ A Γ h Gal( k Ø ( ℓ ) /Q ) ( A ) otherwise.Proof. Let T be S ∞ ( k ) ∪ S ℓ ( k ). Since ℓ is odd, b H ( Q v , A ′ ) = 0 for any v ∈ S ∞ ( Q ), and hencewe have log ℓ ( χ k/Q,T ( A )) = − X v ∈ S ∞ ( Q ) dim F ℓ H ( Q v , A ′ )= − X v ∈ S C ( Q ) dim F ℓ A − X v ∈ S R ( Q ) dim F ℓ A/A Γ v . The last equality is because:(1) If v ∈ S C ( Q ), then G v ( Q ) = 1 acts trivially on both µ ℓ and A .(2) If v ∈ S R ( Q ), then G v ( Q ) ≃ Z / Z acts on µ ℓ as taking the inverse. Since the ac-tion of G v ( Q ) on A factors through Γ v , and Γ v acts on A/A Γ v as taking inverse,we have dim F ℓ ( A ′ ) G v ( Q ) = dim F ℓ Hom G v ( Q ) ( A, µ ℓ ) = dim F ℓ Hom G v ( Q ) ( A/A Γ v , µ ℓ ) =dim F ℓ A/A Γ v . y Proposition 9.4, we have δ k/Q,S ( A ) ≤ ǫ k/Q, Ø ( A ) − r − A = F ℓ ǫ k/Q, Ø ( A ) − r − r + 1 if A = µ ℓ ǫ k/Q, Ø ( A ) − r dim F ℓ A − P v ∈ S R ( Q ) dim F ℓ A/A Γ v otherwise,where A can be µ ℓ only if µ ℓ ⊂ k . So the desired result following by Proposition 3.4,Corollary 4.5 and Proposition 5.4. (cid:3) Corollary 11.2.
Let k/ Q be an imaginary quadratic field such that k = Q ( √− , and γ denote the nontrivial element of Γ = Gal( k/Q ) ≃ Z / Z . For an odd prime ℓ , we have thefollowing isomorphism of Γ -groups G Ø ( k )( ℓ ) ≃ F n (Γ)( ℓ ) (cid:30) [ r − γ ( r )] r ∈ X (11.1) for sufficiently large positive integer n and some set X consisting of n elements of F n (Γ)( ℓ ) . Remark 11.3.
If we choose the n elements of set X randomly with respect to the Haarmeasure, then the quotient in (11.1) gives a random group that defines a probability measureon all n -generated pro- ℓ admissible Γ-groups. By taking n ! ∞ , there is a limit probabilitymeasure, which can be computed using formulas in [LWZB19]. The discussion in [LWZB19, § Proof.
When Q = Q and k is imaginary quadratic, we have r = 1, r = 0, ǫ k/ Q , Ø ( A ) =dim F ℓ A , and Γ ∞ = Γ. Let A be a finite simple F ℓ [Gal( k Ø ( ℓ ) / Q )]-module. Also, µ ℓ k forany odd ℓ , since k = Q ( √− A = µ ℓ . By Lemma 11.1 , when n is sufficiently large, wehave m ad ( n, Γ , G Ø ( k )( ℓ ) , A ) ≤ A = F ℓ n (dim F ℓ A − dim F ℓ A Γ ) h Gal( k Ø ( ℓ ) / Q ) A otherwise.Note that Γ ≃ Z / Z implies that the normal subgroup of F n (Γ)( ℓ ) ⋊ Γ generated by Y ( X )is exactly [ r − γ ( r )] r ∈ X . Thus, the corollary follows by [LWZB19, Proposition 4.3]. (cid:3) When Q contains the ℓ -th roots of unity. In this subsection, we assume µ ℓ ⊂ Q .In this case, µ ℓ becomes the trivial Gal( k Ø /Q )-module F ℓ , which makes the multiplicities ina presentation of G Ø ( k )( ℓ ) significantly different from the previous cases. Lemma 11.4.
Assume µ ℓ ⊂ Q . For a finite simple F ℓ [Gal( k Ø ( ℓ ) /Q )] -module A , we have (1) If Q is a function field and the genus of k is not , then δ k/Q, Ø ( A ) = 0 . (2) If Q is a number field, then δ k/Q, Ø ( A ) ≤ ǫ k/Q, Ø ( A ) − r dim F ℓ A .Proof. Because of µ ℓ ⊂ Q , we havedim F ℓ ( A ′ ) Gal( k Ø /Q ) = dim F ℓ ( A ∨ ) Gal( k Ø /Q ) = dim F ℓ A Gal( k Ø /Q ) . (11.2)Then the first statement follows directly by Proposition 9.3(2). For the rest we assumethat Q is a number field and denote T = S ℓ ( k ) ∪ S ∞ ( k ). If ℓ is odd, then the assumption µ ℓ ⊂ Q implies that Q is totally imaginary. Then we can easily see by Theorem 7.1 that og ℓ χ k/Q,T ( A ) = − r dim F ℓ A , and hence the statement for odd ℓ follows by Proposition 9.4and (11.2). If ℓ = 2, then we first want to compute for each v ∈ S ∞ ( Q )dim F ℓ b H ( Q v , A ′ ) − dim F ℓ H ( Q v , A ′ ) . (11.3)For each v ∈ S C ( Q ), we have G v ( Q ) = 1, and hence (11.3) equals to − dim F ℓ A . For each v ∈ S R ( Q ), the assumption ℓ ∤ | Γ | implies that | Γ | is odd. So for each p ∈ S v ( k ), p is real and G p ( k ) acts trivially on A ′ , which implies that b H ( k p , A ′ ) = H ( k p , A ′ ). Then (11.3) equals 0,and we obtain the statement for ℓ = 2 by Proposition 9.4 and (11.2). (cid:3) Then by the same arguments used in §
10, we obtain the following bounds for the multi-plicity of A . Corollary 11.5.
Assume µ ℓ ⊂ Q . When k is a function filed, we assume that the genus of k is positive. Let A be a finite simple F ℓ [Gal( k Ø ( ℓ ) /Q )] -module. Then for a sufficiently large n , we have m ad ( n, Γ , G Ø , ∞ ( k )( ℓ ) , A ) ≤ ( n + 1) dim F ℓ ( A ) − ξ ( A ) − n dim F ℓ A Γ h G Ø , ∞ ( k )( ℓ ) ⋊ Γ ( A ) if Q is a function field ( n − r ) dim F ℓ ( A ) + ǫ k/Q, Ø ( A ) − ξ ( A ) − n dim F ℓ A Γ h G Ø , ∞ ( k )( ℓ ) ⋊ Γ ( A ) if Q is a number field . Remark 11.6. (1) The readers can compare the corollary with Proposition 10.7. When A = F ℓ and Q is Q ( ζ ℓ ) or a funtion field containing µ ℓ , one can check that the upperbound of the multiplicity is positive, which suggests the failure of the Property E of G Ø , ∞ ( k ). Therefore, the random group model used in the Liu–Wood–Zureick-Brownconjection is not expected to work in this exceptional case.(2) If the upper bounds in Corollary 11.5 are sharp, then it also suggests that we shouldnot expect the coincidence of the distributions of G Ø , ∞ ( k )( ℓ ) between the functionfield case and the number field case, since r and ǫ k/Q, Ø ( A ) appear in the upper boundfor number fields. For example, when Q = Q , ℓ = 2 or Q = Q ( ζ ) , ℓ = 3, the upperbound in the corollary equals the one for function fields. However, when Q = Q ( ζ ℓ )with ℓ >
3, the upper bound is( n + ( ℓ − /
2) dim F ℓ ( A ) − ξ ( A ) − n dim F ℓ A Γ h G Ø , ∞ ( k )( ℓ ) ⋊ Γ ( A ) , which is strictly larger than the upper bound for function fields. Acknowledgements.
The author would like to thank Nigel Boston and Melanie MatchettWood for helpful conversations and encouragement which inspire the author to work onthe questions studied in this paper . She also thanks Nigel Boston, Mark Shusterman andPreston Wake for comments on and corrections to an early draft of the paper.
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Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI48104 USA
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