A sharp upper bound for the 2 -torsion of class groups of multiquadratic fields
aa r X i v : . [ m a t h . N T ] S e p A sharp upper bound for the -torsion of classgroups of multiquadratic fields Peter Koymans ∗ and Carlo Pagano † Max Planck Institute for Mathematics, BonnSeptember 18, 2020
Abstract
Let K be a multiquadratic extension of Q and let Cl + ( K ) be its narrow classgroup. Recently, the authors [7] gave a bound for | Cl + ( K )[2] | only in terms of thedegree of K and the number of ramifying primes. In the present work we show thatthis bound is sharp in a wide number of cases. Furthermore, we extend this to rayclass groups. The class group is one of the most fundamental invariants of a number field K . Providingnon-trivial upper bounds for the l -torsion of class groups in terms of the discriminant∆ K/ Q of a general number field K has been an active area of research with connectionsto elliptic curves and diophantine approximation [1, 2, 3, 6, 9, 10, 11, 14].For extensions K/ Q of degree a power of a prime l much more is known. For instancefor l = 2 and K/ Q a quadratic extension, Gauss [4] showed thatdim F Cl + ( K )[2] = ω (∆ K/ Q ) − . Here Cl + ( K ) denotes the narrow class group of the field K and ω ( a ) denotes the numberof prime factors of a non-zero integer a . Recently, the authors [7] generalized Gauss’result to multiquadratic fields. More specifically, we obtained the following result, whichis Theorem 1.1 of [7]. Call a vector ( a , . . . , a n ) ∈ Z n ≥ acceptable if the a i are squarefree,pairwise coprime and only have prime factors congruent to 1 modulo 4. Theorem 1.1.
Let n be a positive integer and let ( a , . . . , a n ) ∈ Z n ≥ be acceptable. Thenwe have dim F Cl + ( Q ( √ a , . . . , √ a n ))[2] ≤ ω ( a · . . . · a n ) · n − − n + 1 . A similar upper bound has subsequently been established by Kl¨uners and Wang in[5, Theorem 2.1] for extensions K/ Q of degree a power of l . However, when specializedto the multiquadratic fields considered above, their bound is in the worst case scenariotwice as large as the one in Theorem 1.1. This work is devoted to showing that thebound in Theorem 1.1 is sharp for every n ∈ Z ≥ .An acceptable vector ( a , . . . , a n ) is said to be maximal in case the inequality ofTheorem 1.1 is an equality. Among other things, we have given a recursive character-ization of maximal vectors (see [7, Theorem 1.2]), which we reproduce now. Write π S for the projection on the coordinates in S , write H +2 ( K ) for the maximal multiquadraticunramified (at all finite places) extension of K and write [ n ] := { , . . . , n } . ∗ Vivatsgasse 7, 53111 Bonn, Germany, [email protected] † Vivatsgasse 7, 53111 Bonn, Germany, [email protected] heorem 1.2. Let n be a positive integer and let ( a , . . . , a n ) be an acceptable vector.Then the following are equivalent. ( a ) The vector ( a , . . . , a n ) is maximal, i.e. dim F Cl + ( Q ( √ a , . . . , √ a n ))[2] = ω ( a · . . . · a n ) · n − − n + 1 . ( b ) For every j ∈ [ n ] , the vector π [ n ] −{ j } ( a , . . . , a n ) is maximal and every prime divisor p of a j splits completely in H +2 ( Q ( {√ a m } m ∈ [ n ] −{ j } )) . ( c ) For every j ∈ [ n ] , the vector π [ n ] −{ j } ( a , . . . , a n ) is maximal and for every primedivisor p of a j , one (or equivalently any) prime above p in the field Q ( {√ a m } m ∈ [ n ] −{ j } ) belongs to + ( Q ( {√ a m } m ∈ [ n ] −{ j } )) . In particular Theorem 1.2 recovers the equality of Gauss’ theorem for n = 1 as aspecial case. It is then natural to ask whether for every positive integer n one can findmaximal vectors of dimension n . As the reader can sense from the characterizationgiven in Theorem 1.2, it is not at all obvious how to do this. A naive inductive approachbased on the Chebotarev Density Theorem runs into severe difficulties, since one needsto simultaneously guarantee splitting of a prime p in a field K q depending on q and of q in a field K p depending on p .To circumvent this problem, we use combinatorial ideas from [12], which we explainhere from first principles in order to make the present work self-contained (see Section2). Our main theorem shows that one can find maximal vectors ( a , . . . , a n ) for every n . Moreover, for any fixed n , we show that Theorem 1.1 is sharp for a wide number ofchoices of ( ω ( a ) , . . . , ω ( a n )). More precisely, we establish the following. Theorem 1.3. ( a ) Take n ∈ Z > and take ( k , . . . , k n ) ∈ Z ≥ × (2 · Z ≥ ) n − . Then thereare infinitely many acceptable vectors ( a , . . . , a n ) with ω ( a i ) = k i for each i ∈ [ n ] and dim F Cl + ( Q ( √ a , . . . , √ a n ))[2] = ω ( a · . . . · a n ) · n − − n + 1 . ( b ) Take ( k , k , k ) ∈ Z ≥ . Then there are infinitely many acceptable vectors ( a , a , a ) with ω ( a i ) = k i for each i ∈ { , , } and dim F Cl + ( Q ( √ a , √ a , √ a ))[2] = ω ( a a a ) · − . We speculate that the condition ( k , . . . , k n ) ∈ Z ≥ × (2 · Z ≥ ) n − can also be removedfor n >
3, but this seems to be out of reach with the techniques employed in this work.We next turn our attention to ray class groups. First of all, let us notice that the2-torsion of the ordinary class group of a number field K can not be larger than the2-torsion of the narrow class group of K . Hence the upper bound in Theorem 1.1 is alsoan upper bound for | Cl( Q ( √ a , . . . , √ a n ))[2] | . Less obvious is whether also this boundis sharp.Similarly, fix an integer c , which we take in this paper to be a squarefree product ofprimes congruent to 1 modulo 4 (see the end of this introduction for some motivation onthis assumption). Then one obtains from Theorem 1.1 and the ray class group sequencedim F Cl( Q ( √ a , . . . , √ a n ) , c )[2] ≤ ω ( a · . . . · a n ) · n − − n + 1 + 2 n · ω ( c ) , where the bound can be reached only if all the prime divisors of c split completely in Q ( √ a , . . . , √ a n ). It is, once more, not obvious whether this bound is sharp. Our nexttheorem settles these questions. Theorem 1.4.
Take n ∈ Z ≥ and take ( k , . . . , k n ) ∈ (2 · Z ≥ ) n . Let c be a squarefreeinteger divisible only by primes congruent to modulo . Then there are infinitely manyacceptable vectors ( a , . . . , a n ) with ω ( a i ) = k i for each i ∈ [ n ] and dim F Cl( Q ( √ a , . . . , √ a n ) , c )[2] = ω ( a · . . . · a n ) · n − − n + 1 + 2 n · ω ( c ) .
2s a corollary of Theorem 1.4 we obtain the following result on unit groups.
Corollary 1.5.
Let n ∈ Z ≥ . Let c be a squarefree integers with all factors congruent to modulo . Then there exist infinitely many acceptable vectors ( a , . . . , a n ) such that allprime divisors of c split completely in Q ( √ a , . . . , √ a n ) and the unit group O ∗ Q ( √ a ,..., √ a n ) reduced modulo c is entirely contained in the group (cid:18) O Q ( √ a ,..., √ a n ) c (cid:19) ∗ . We remark that, in the context of Corollary 1.5, it is no real loss of generality todemand that all the prime divisors of c are 1 modulo 4. Indeed, we are aiming toconstruct multiquadratic extensions splitting completely at all prime divisors of c andwhose unit group consists entirely of squares modulo c . This then in particular appliesto −
1, which is then a square in F l for every l | c so that l ≡ c = 1)is not sharp, whenever one of the a i is divisible by a prime congruent to 3 modulo 4. Acknowledgments
We thank Alexander Smith for several clarifying emails and conversations about hiswork. We are grateful to the anonymous referee of [7] for having encouraged us to writedown a self-contained proof of Theorem 1.3 in the form of this independent work. Theauthors wish to thank the Max Planck Institute for Mathematics in Bonn for its financialsupport, great work conditions and an inspiring atmosphere.
For completeness we include a self-contained proof of [12, Proposition 3.1]; we claim nooriginality in this section.We let X , . . . , X d be arbitrary non-empty finite sets and put X := X × · · · × X d .In our application the sets X i will consist of acceptable integers a i with ω ( a i ) = k i /
2. Acube C is a product set Y × · · · × Y d with Y i ⊆ X i and | Y i | = 2, in our application wecan think of C as an acceptable vector ( a , . . . , a d ) with ω ( a i ) = k i . It is here that wemake essential use that k i is even. As we see in our next section, we need to find cubes C satisfying certain bilinear conditions. The aim of our next definition is to encapsulatethis in an abstract framework.We write X i for the set X i × X i . For S ⊆ [ d ] and i ∈ [ d ], π i denotes the naturalprojection from Q i ∈ S X i × Q i S X i to X i if i ∈ S and to X i if i S , while pr and pr denote the natural projections from X i to its two factors. Definition 2.1.
Let X , . . . , X d be arbitrary non-empty finite sets and put X := X ×· · · × X d . An additive system A on X is given by a tuple ( C S , C acc S , F S , A S ) indexed bysubsets S ⊆ [ d ] satisfying the following properties • C acc S ⊆ C S ⊆ Q i ∈ S X i × Q i S X i are sets, F S : C S → A S is a map and A S is afinite F -vector space; • we have that C acc S := { x ∈ C S : F S ( x ) = 0 } nd for S = ∅ C S := { x ∈ Y i ∈ S X i × Y i S X i : for all j ∈ S and all y ∈ Y i ∈ S −{ j } X i × Y i ∈ [ d ] − ( S −{ j } ) X i satisfying π k ( x ) = π k ( y ) for k ∈ [ d ] − { j } and π j ( y ) ∈ { pr ( π j ( x )) , pr ( π j ( x )) } , we have y ∈ C acc S −{ j } } ; • suppose that x , x , x ∈ C S and suppose that there exists j ∈ S such that π k ( x ) = π k ( x ) = π k ( x ) for all k ∈ [ d ] − { j } and π j ( x ) = ( a, b ) , π j ( x ) = ( b, c ) , π j ( x ) = ( a, c ) for some a, b, c ∈ X j . Then we have F S ( x ) + F S ( x ) = F S ( x ) . (2.1)Note that we do not quite work with cubes in the above definition, but instead withelements of X × X × · · · × X d × X d . The major difference is that we have also includedelements with equal coordinates. This will be very convenient in the proof of our nextcounting result for C acc S . Later, we shall need to remove such elements, but it is not hardto show that they contribute a vanishingly small proportion. Proposition 2.2.
Let X , . . . , X d be arbitrary non-empty finite sets and put X := X ×· · · × X d . Let A be an additive system on X such that | A S | ≤ a for all S ⊆ [ d ] and write δ for the density of C acc ∅ in X . Then we have that | C acc[ d ] | Q i ∈ [ d ] | X i | ≥ δ d · a − d . Proof.
We proceed by induction on d with the case d = 0 being trivial. Fix an element x ∈ Y i ∈ [ d − X i . Let V ( x ) be the subset of a ∈ X d such that ( x, a ) ∈ C acc[ d − and let W ( x ) be the subsetof ( a, b ) ∈ X d such that ( x, ( a, b )) ∈ C acc[ d ] . By definition of an additive system, we seethat W ( x ) naturally injects in V ( x ) × V ( x ). From now on we shall identify W ( x ) withits image in V ( x ) × V ( x ). We claim that W ( x ) defines an equivalence relation on V ( x ).If we apply equation (2.1) with a = b = c , we conclude that for all S ⊆ [ d − y ∈ Q i ∈ S X i × Q i ∈ [ d − − S X i and all a ∈ X d F S ∪{ d } ( y, ( a, a )) = 0 . From this, it follows quickly that W ( x ) is reflexive. Applying equation (2.1) with a = c ,we then get F S ∪{ d } ( y, ( a, b )) + F S ∪{ d } ( y, ( b, a )) = F S ∪{ d } ( y, ( a, a )) = 0 , so that W ( x ) is symmetric. Finally, equation (2.1) with a , b and c arbitrary implies thetransitivity of W ( x ), which establishes the claim.4ur next step is to estimate the number of equivalence classes. To do so, take( x, a ) , ( x, b ) ∈ V ( x ) and { d } ⊆ S ⊆ [ d ]. Then we write ( x, a ) ∼ S ( x, b ) if ( x, a ) ∼ S ′ ( x, b )for all { d } ⊆ S ′ ( S and F S ( y, ( a, b ))) = 0for all y ∈ Q i ∈ S −{ d } X i × Q i ∈ [ d − − S X i satisfying π i ( y ) = π i ( x ) for i ∈ S − { d } and π i ( y ) ∈ { pr ( π i ( x )) , pr ( π i ( x )) } for i ∈ [ d − − S . Note that the equivalence relation ∼ [ d ] is precisely W ( x ).To upper bound the number of equivalence classes, take a collection of points ( x, a i ) ∈ V ( x ) such that ( x, a i ) ∼ S ( x, a j ) for all strict subsets S of [ d ]. Suppose that among thiscollection there are R equivalence classes for ∼ [ d ] , with representatives ( x, b ) , . . . , ( x, b R ).Then we see that the map ( x, b i ) F [ d ] ( x, ( b , b i ))is injective and hence we conclude that R ≤ a . If we proceed in this way, we see thatthe total number of equivalence classes for ∼ [ d ] is bounded by Y { d }⊆ S ⊆ [ d ] a d −| S | ≤ d − Y i =0 a ( d − i ) i = a d − , since for a given S , there are 2 d −| S | choices for y . Define δ ( x ) = | V ( x ) || X d | × Q i ∈ [ d − | X i | . Then it follows that the density of V ( x ) × V ( x ) in Q i ∈ [ d ] X i is δ ( x ) . Then Cauchy’sinequality and our bound for the total number of equivalence classes implies that | W ( x ) | Q i ∈ [ d ] | X i | ≥ δ ( x ) a d − . So far we have proven that | C acc[ d ] | Q i ∈ [ d ] | X i | = P x ∈ Q i ∈ [ d − X i | W ( x ) | Q i ∈ [ d ] | X i | ≥ X x ∈ Q i ∈ [ d − X i δ ( x ) a d − . Another application of Cauchy’s inequality shows that X x ∈ Q i ∈ [ d − X i δ ( x ) a d − ≥ (cid:16)P x ∈ Q i ∈ [ d − X i δ ( x ) (cid:17) a d − · Q i ∈ [ d − | X i | . The average of δ ( x ) over all choices of x equals the density of C acc[ d − in X d × Q i ∈ [ d − X i .The induction hypothesis yields (cid:16)P x ∈ Q i ∈ [ d − X i δ ( x ) (cid:17) a d − · Q i ∈ [ d − | X i | ≥ ( δ d − · a − d − ) a d − = δ d · a − d as desired. In this section we prove Theorem 1.3. The work is divided in two parts. In Subsection3.1 we extract from [7] the basic results that will be needed in the proof, we proveProposition 3.5 and we recall a version of R´edei reciprocity, Theorem 3.7, that will beused later. With these tools in hand, we give the proof of Theorem 1.3 in Subsection3.2. 5 .1 Preparations
The shape of Theorem 1.2 presents a striking resemblance with Definition 2.1. To makethe analogy more stringent one would like to turn the splitting conditions in part ( b ) ofTheorem 1.2 into an additive system : this is precisely the route we are going to follow. Todo so we recall a refinement of Theorem 1.2, which will invoke the language of expansionmaps . We now recall the definition from [7, Section 3.3]. If A is a set, we write F A forthe free F -vector space on A . Definition 3.1.
Let G be a profinite group and let A ⊆ Hom( G, F ) be a finite, linearlyindependent set with | A | ≥ and χ ∈ A . An expansion map for G with support A andpointer χ is a continuous group homomorphism ψ : G → F [ F A −{ χ } ] ⋊ F A −{ χ } satisfying the following two properties • for every χ ∈ A − { χ } , we have π χ ◦ ψ = χ , where π χ is the projection on thecoordinate of χ in F A −{ χ } ; • we have e χ ◦ ψ = χ , where e χ is the unique non-trivial character of F [ F A −{ χ } ] ⋊F A −{ χ } that sends the subgroup { } ⋊ F A −{ χ } to . If ψ is an expansion map for G Q , we define its field of definition to be L ( ψ ) := Q ker( ψ ) .Denote by χ a the character corresponding to Q ( √ a ). Theorem 3.2.
Let n be a positive integer and let ( a , . . . , a n ) be an acceptable vector.Then the following are equivalent. ( a ) The vector ( a , . . . , a n ) is maximal, i.e. dim F Cl + ( Q ( √ a , . . . , √ a n ))[2] = ω ( a · . . . · a n ) · n − − n + 1 . ( b ) For every T ( [ n ] , every j ∈ [ n ] − T and every prime p | a j , there exists an expansionmap ψ T,p : Gal( H +2 ( Q ( {√ a h : h ∈ T } ∪ {√ p } )) / Q ) → F [ F T ] ⋊ F T with support { χ a h } h ∈ T ∪ { χ p } and pointer χ p .Furthermore, in case one of the two equivalent statements ( a ) , ( b ) holds, then the setof expansion maps described in ( b ) when restricted to the group Gal( H +2 ( Q ( √ a , . . . , √ a n )) / Q ( √ a , . . . , √ a n )) gives a generating set for Cl + ( Q ( √ a , . . . , √ a n )) ∨ [2] .Proof. This follows from [7, Theorem 3.20] and [7, Proposition 4.1].We shall need further understanding of expansion maps, and to this end we recallsome more material from [7, Section 3.3]. Let e i be the i -th basis vector of F T , whichwe can naturally view as an element of F [ F T ]. There is a ring isomorphism F [ F T ] ∼ = F [ t , . . . , t n ] / ( t , . . . , t n )by sending t i to 1 · id + 1 · e i . Under this isomorphism, the action of e i ∈ F T becomesmultiplication by 1 + t i . If ψ T,p is an expansion map, then projection on the monomials t S := Q i ∈ S t i gives a system of 1-cochains ϕ S ( ψ T,p ) : Gal( H +2 ( Q ( {√ a h : h ∈ T } ∪ {√ p } )) / Q ) → F S ⊆ T . These 1-cochains satisfy the recursive equation ϕ S ( στ ) − ϕ S ( σ ) − ϕ S ( τ ) = X ∅6 = U ⊆ S χ U ( σ ) ϕ S − U ( τ ) (3.1)with ϕ ∅ = χ p and χ U = Q i ∈ U χ a i , where the product is taken in F . Reversely, a systemof 1-cochains satisfying equation (3.1) naturally gives rise to an expansion map. Next,a vector ( ψ T −{ i } ,p ) i ∈ T of expansion maps with supports { χ a j } j ∈ T −{ i } ∪ { χ p } and pointer χ p for each i ∈ T , iscalled a commutative vector in case for every i, j ∈ Tϕ T −{ i,j } ( ψ T −{ i } ,p ) = ϕ T −{ i,j } ( ψ T −{ j } ,p ) . Note that Theorem 1.2 implies that a maximal vector ( a , . . . , a n ) must be stronglyquadratically consistent , i.e. we have ( pq ) = 1 for every distinct i, j ∈ [ n ] and every twoprimes p | a i , q | a j . Theorem 3.3.
Let n be a positive integer, and let ( a , . . . , a n ) be an acceptable vector,which is strongly quadratically consistent. Let T ( [ n ] , let j ∈ [ n ] − T and let p be aprime divisor of a j . Then the following are equivalent. ( a ) There exists an expansion map ψ T,p : Gal( H +2 ( Q ( {√ a h : h ∈ T } ∪ {√ p } )) / Q ) → F [ F T ] ⋊ F T with support { χ a h } h ∈ T ∪ { χ p } and pointer χ p . ( b ) There exists a commutative vector of expansion maps ( ψ T −{ i } ,p ) i ∈ T with supports { χ a h } h ∈ T −{ i } ∪ { χ p } and pointer χ p for each i ∈ T , satsfying the followingcondition. For every i ∈ T and every prime divisor q of a i , we have that q splitscompletely in the field of definition of ψ T −{ i } ,p .Proof. This is a special case of [7, Theorem 1.5].In order to prove part (a) of Theorem 1.3, we aim to combine Theorem 3.2 andTheorem 3.3 with Proposition 2.2. An import stepping stone is to guarantee equation(2.1) for the various cochains ϕ S ( ψ T,p ) attached to an expansion map ψ T,p . We nowexplain what this means and how to achieve this.Let n ∈ Z ≥ , let ( k , . . . , k n ) ∈ Z ≥ × (2 · Z ≥ ) n − and let M ∈ Z ≥ . Take Y := Y × · · · × Y n to be a product space, where each Y i is a set of cardinality M consisting of acceptablesquarefree integers. We further require that any two distinct elements in ∪ ni =1 Y i arepairwise coprime and that ω ( z ) = k i for each i ∈ [ n ] − { } and z ∈ Y i , while ω ( z ) = k for z ∈ Y . We call such a Y a (( k , . . . , k n ) , M )-space.Let Y now be a (( k , . . . , k n ) , M )-space. We denote by K ( Y ) the multiquadratic num-ber field obtained by adding all the square roots of the prime divisors of the elements in ∪ ni =1 Y i to Q . Observe that for each prime p ramifying in K ( Y ) / Q , the inertia subgroupsof p in Gal( H +2 ( K ( Y )) / Q ) are cyclic of size 2. For each such prime p we fix a choice ofsuch an inertia element σ p . We will denote this choice by G := { σ p } p | Q ni =1 ( Q y ∈ Yi y ) andrefer to it as a choice of inertia for Y . 7 roposition 3.4. ( a ) Let Y be a (( k , . . . , k n ) , M ) -space together with a choice of inertia G . Let S ( [ n ] and let j ∈ [ n ] − S . Pick a non-trivial divisor d of an element in Y j and pick { a i } i ∈ S with a i a product of elements in Y i for each i ∈ S . Then there existsat most one expansion map ψ ( a i ) i ∈ S ; d ( G ) : Gal( H +2 ( K ( Y )) / Q ) → F [ F { χ ai : i ∈ S and χ ai =0 } ] ⋊ F { χ ai : i ∈ S and χ ai =0 } , with support { χ a i : i ∈ S and χ a i = 0 } ∪ { χ d } and pointer χ d such that ϕ T ( ψ ( a i ) i ∈ S ; d ( G ))( σ ) = 0 for each ∅ 6 = T ⊆ S and each σ ∈ G . ( b ) If ψ ( a i ) i ∈ S ; d ( G ) exists, then it factors through Gal( H +2 ( Q ( {√ a i } i ∈ S , √ d )) / Q ) .Proof. Since Q has no non-trivial unramified extensions, the group Gal( H +2 ( K ( Y )) / Q )is generated by the conjugacy classes of all elements in G . We claim that G already gen-erates Gal( H +2 ( K ( Y )) / Q ). Indeed, if G is any finite group and S ⊆ G , then S generates G if and only if S generates G/ Φ( G ), where Φ( G ) is the Frattini subgroup. Furthermore,for a 2-group we know that the Frattini subgroup Φ( G ) equals G [ G, G ], so that twoconjugate elements have the same image in G/ Φ( G ). This gives part ( a ) immediately,since the requirement ϕ T ( ψ ( a i ) i ∈ S ; d ( G ))( σ ) = 0 for each ∅ 6 = T ⊆ S determines the imageof σ under ψ ( a i ) i ∈ S ; d ( G ).To obtain part ( b ) we start by noticing that L ( ψ ( a i ) i ∈ S ; d ( G )) is an abelian extensionof Q ( {√ a i } i ∈ S , √ d ). We only need to guarantee that it is unramified at all finite places.For this it is enough to notice that for each prime q not dividing a i nor d one has that ψ ( a i ) i ∈ S ; d ( G )( σ q ) = id , precisely thanks to our requirement that ϕ T ( ψ ( a i ) i ∈ S ; d ( G ))( σ q ) = 0 for each ∅ 6 = T ⊆ S . The next proposition gives the sought behavior among expansion maps. For conve-nience we introduce the following notation. Let S ⊆ [ n ] and let U ⊆ [ n ] − S . Let x ∈ Y i ∈ S Y i × Y j ∈ U Y j , then we write c ( x ) := ((pr ( π i ( x ))pr ( π i ( x ))) i ∈ S , ( π j ( x )) j ∈ U )for the vector obtained by multiplying out the double entries of x and leaving unchangedthe single entries of x . Proposition 3.5.
Let Y be a (( k , . . . , k n ) , M ) -space together with a choice of inertia G . Let S ( [ n ] , let j ∈ [ n ] − S and i ∈ S . Pick a non-trivial divisor d of an element in Y j . Let U ⊆ [ n ] − S − { j } . Let x , x , x be three elements of Q i ∈ S Y i × Q u ∈ U Y u suchthat they coincide outside i and such that pr ( π i ( x )) = pr ( π i ( x )) , pr ( π i ( x )) = pr ( π i ( x )) , pr ( π i ( x )) = pr ( π i ( x )) . Suppose ψ c ( x ); d ( G ) and ψ c ( x ); d ( G ) exist. Then the map ψ c ( x ); d ( G ) exists and ϕ T ( ψ c ( x ); d ( G )) = ϕ T ( ψ c ( x ); d ( G )) + ϕ T ( ψ c ( x ); d ( G )) for each ∅ 6 = T ⊆ S . roof. This is now an immediate consequence of Proposion 3.4. Indeed, the maps ϕ T ( ψ ( π i ( c ( x ))) i ∈ S ; d ( G )) + ϕ T ( ψ ( π i ( c ( x ))) i ∈ S ; d ( G ))yield an expansion map from the group Gal( H +2 ( K ( Y )) / Q ) to the group F [ F { χ πi ( c ( x : i ∈ S ∪ U and χ πi ( c ( x =0 } ] ⋊ F { χ πi ( c ( x : i ∈ S ∪ U and χ πi ( c ( x =0 } . Furthermore, the vanishing at all elements of G follows by construction. This gives thedesired conclusion.We next give a more specific version of Theorem 3.3 that encodes the choice of inertiaelements G . We call a (( k , . . . , k n ) , M )-space Y quadratically consistent in case each ofits vectors are strongly quadratically consistent. Theorem 3.6.
Let Y be a quadratically consistent (( k , . . . , k n ) , M ) -space, together witha choice of inertia G . Let S ( [ n ] and let j ∈ [ n ] − S . Pick a non-trivial divisor d of an element in Y j . Pick furthermore U ⊆ [ n ] − S − { j } . Let a be an element of Q i ∈ S Y i × Q u ∈ U Y u . Then the following are equivalent. ( a ) The map ψ c ( a ); d ( G ) exists. ( b ) For each h ∈ S ∪ U the map ψ π S ∪ U −{ h } ( c ( a )); d ( G ) exists and every prime ramifying in Q ( √ a h ) / Q splits completely in the field of definition of ψ π S ∪ U −{ h } ( c ( a )); d ( G ) Proof.
Proposition 3.4 shows that the vector ( ψ ( π i ( c ( a ))) i ∈ S ∪ U : i = h ; d ( G )) h ∈ S ∪ U is commu-tative. Hence the conclusion follows from Theorem 3.3, provided that we can ensurethat ϕ S ∪ U vanishes on G . But, looking at equation (3.1), we see that we still have thefreedom to twist ϕ S ∪ U by the characters χ p for p ramifying in K ( Y ) / Q .Finally, in order to obtain Theorem 1.3, part ( b ), we recast here (a special caseof) R´edei reciprocity, re-written in the language of expansion maps. Suppose that( a , a , a ) is a strongly quadratically consistent vector. Then there exists an expansionmap ψ a ; a : G Q → F [ F ] ⋊ F such that every prime divisor p of a splits completelyin Q ( √ a , √ a ) / Q .Hence Frob( p ) lands in the central subgroup Gal( L ( ψ a ; a ) / Q ( √ a , √ a )), whichcan be canonically identified with F : here we recall that L ( ψ a ; a ) denotes the field ofdefinition of an expansion map. In what follows Frobenius symbols need to be interpretedas elements of F . Theorem 3.7.
Let ( a , a , a ) be a strongly quadratically consistent vector. Let ψ a ; a :Gal( H +2 ( Q ( √ a , √ a )) / Q ) → F [ F ] ⋊ F and ψ a ; a : Gal( H +2 ( Q ( √ a , √ a )) / Q ) → F [ F ] ⋊ F be expansion maps with supports respectively { χ a , χ a } and { χ a , χ a } andpointers respectively χ a and χ a . Then X p | a Frob L ( ψ a a ) / Q ( p ) = X p | a Frob L ( ψ a a ) / Q ( p ) . Proof.
This is a special case of [8, Theorem 3.3].
Remark 3.8.
Theorem 3.7 has recently been generalized by the authors to more generalexpansion maps, see [8, Theorem 3.3]. It is natural to wonder if this reciprocity lawallows one to generalize the proof of Theorem 1.3 part ( b ) to n > . For every ( k , . . . , k n ) we have been able to construct vectors ( a , . . . , a n ) with ( ω ( a ) , . . . , ω ( a n )) = ( k , . . . , k n ) and | Cl + ( Q ( √ a , . . . , √ a n ))[2] | “large”. However, already for n = 4 , we have not beenable to produce maximal vectors this way. .2 Proof of Theorem 1.3 Let us start with a proposition that immediately yields part ( b ) and will be an importantstep for part ( a ). Proposition 3.9.
Let N and m be positive integers. Then there exists a product space X := X × · · · × X m , where the X i are disjoint sets of primes congruent to modulo with | X i | = N for each i ∈ [ m ] such that Y i ∈ S X i consists entirely of maximal vectors for every subset S ⊆ [ m ] with ≤ | S | ≤ .Proof. We proceed by induction on m . For m = 1 the statement is trivial. Now supposethat the statement is true for m , so that we have to prove it for m + 1. Pick a productset X × · · · × X m guaranteed by the inductive hypothesis. Consider the set Z of primesthat split completely in H +2 ( K ( X × · · · × X m )) Q ( √− / Q . Thanks to the ChebotarevDensity Theorem, we see that Z is an infinite set.Pick any N -set X m +1 inside Z . Observe that X m +1 is disjoint from each of the X i with i ≤ m , since these are all primes ramifying in H +2 ( K ( X × · · · × X m )) / Q . Next,since X m +1 consists in particular of primes splitting in K ( X × · · · × X m ) / Q , we seethat X i × X i consists entirely of maximal vectors for every distinct i and i . Hencefor each 2-set { i , i } and every point ( p, q ) ∈ X i × X i we have an expansion map ψ p ; q : Gal( H +2 ( Q ( √ p, √ q )) / Q ) → F [ F ] ⋊ F with support { χ p , χ q } and pointer χ q .Thanks to our choice of X m +1 , we have that every x in X m +1 splits completely in L ( ψ p ; q ) whenever i , i ≤ m are distinct. But then Theorem 3.7 yields that p splitscompletely in L ( ψ q ; x ) and q splits completely in L ( ψ p ; x ). Therefore the propositionfollows from Theorem 3.2 and Theorem 3.3. Proof of Theorem 1.3 part ( b ) . By taking m = 3 and N arbitrary large, we see thatProposition 3.9 immediately implies part ( b ) of Theorem 1.3 for ( k , k , k ) = (1 , , Proof of Theorem 1.3 part ( a ) . Take n ∈ Z ≥ and ( k , . . . , k n ) ∈ Z ≥ × (2 Z ≥ ) n − . Fixfurthermore an auxiliary parameter M ∈ Z ≥ . It follows from Proposition 3.5 andProposition 3.9 that we can construct a (( k , . . . , k n ) , M )-space Y := Y × · · · × Y n , equipped with a choice of inertia G such that for any 3-set { i , i , i } ⊆ [ n ], any triple( y i , y i , y i ) ∈ Y i × Y i × Y i and any prime divisor p | y i we have that the map ψ y i ,y i ; p ( G )exists. Fix such a (( k , . . . , k n ) , M )-space Y . Also fix a point y ∈ Y and put e Y := { y } × Y × . . . × Y n . We are going to construct an additive system on e Y for the subsets S of [ n ] − { } . Westart by defining subsets C S ⊆ Y i ∈ S Y i × Y j ∈ [ n ] − S Y j S ⊆ [ n ] − { } . Let us first consider the case that | S | ≤ n −
2. We define C S by the property that a ∈ C S if and only if for each 2-set { i, j } ⊆ [ n ] − S and for everyprime divisor p | π j ( a ), we have that ψ π S ∪{ i } ( c ( a )); p ( G )exists. We next put C [ n ] −{ } to be the set of a in { y } × Q ≤ h ≤ n Y h such that for any j ∈ [ n ] − { } and any prime divisor p of π j ( c ( a )) we have that ψ π [ n ] −{ j } ( c ( a )); p ( G )exists and furthermore ψ π [ n ] −{ ,j } ( c ( a )) , pr k ( π j ( a )); p ( G )exists for all j ∈ [ n ] − { } , k ∈ [2] and p dividing y .We now define the spaces A S . Assume first that | S | ≤ n −
3. We put A S to be thespace of formal F -linear combinations of 5-tuples( x , x , x , x , x ) , where x , x , x ∈ [ n ] − S are pairwise distinct and x ∈ [ k x ] , x ∈ [ k x ]. Instead for | S | ∈ { n − , n − } , we set A S = { } .Let us now define F S : C S → A S . In case | S | > n −
3, we set F S to be the trivial map.Henceforth we assume that | S | ≤ n −
3. Let ( x , x , x , x , x ) be a 5-tuple as above and a ∈ C S . Let p x ( a ) be the x -th prime divisor of π x ( c ( a )), by the natural ordering, andlet p x ( a ) be the x -th prime divisor of π x ( c ( a )). We have that the Frobenius of p x ( a )lands in the center of Gal( L ( ψ π S ∪{ x } ( c ( a )); p x ( a ) ( G )) / Q )thanks to Theorem 3.6 and the definition of C S . Observe that the center of F [ t , . . . , t n ] / ( t , . . . , t n ) ⋊ F n is cyclic of order 2 and generated by t · . . . · t n . Hence to decide whether an element of thecenter is trivial or not one may simply apply the 1-cochain ϕ S ∪{ x } ( ψ π S ∪{ x } ; px a ) ( G ))to the central element. In other words the value ϕ S ∪{ x } ( ψ π S ∪{ x } ( c ( a )); p x ( a ) ( G ))(Frob( p x ( a )))is well-defined and equals 0 if and only if p x ( a ) splits completely in the field of definitionof ψ π S ∪{ x } ( c ( a )); p x ( a ) ( G ). With this preliminary in mind, we define F S ( a ) to be the vectorof A S whose ( x , x , x , x , x )-coordinate equals ϕ S ∪{ x } ( ψ π S ∪{ x } ( c ( a )); p x ( a ) ( G ))(Frob( p x ( a )))for each 5-tuple ( x , x , x , x , x ) as described above. Finally, for each S ⊆ [ n ] − { } ,we put C acc S := F − S (0) . We now establish the following crucial fact.
Proposition 3.10.
The -tuple { ( C S , C acc S , F S , A S ) } S ⊆ [ n ] −{ } defined above is an addi-tive system on e Y . Furthermore, | A S | ≤ n · ( P ni =1 k i ) for each S ⊆ [ n ] − { } .Finally, for all a ∈ C [ n ] −{ } we have that the vector ( π i ( c ( a ))) i ∈ [ n ]: χ πi ( c ( a )) =0 is amaximal vector of dimension |{ i ∈ [ n ] : χ π i ( c ( a )) = 0 }| . roof. Equation (2.1) is satisfied thanks to Proposition 3.5. The bound on | A S | followsfrom straightforward counting. The maximality claim is a consequence of Theorem 3.2and Theorem 3.3.We now finish the proof of Theorem 1.3, part ( a ). Due to Proposition 3.10 andProposition 2.2 we deduce that there exists a positive number c ( k ,...,k n ) , depending onlyon the vector ( k , . . . , k n ), such that there are at least c ( k ,...,k n ) · M n − vectors a ∈ { y }× Q ≤ i ≤ n Y i with ( π i ( c ( a ))) i ∈ [ n ]: χ πi ( c ( a )) =0 maximal. On the other hand,no more than ( n − · M n − vectors a in { y } × Q ≤ i ≤ n Y i are such that pr ( π i ( a )) =pr ( π i ( a )) for some i . It follows that there at least c ( k ,...,k n ) · M n − − ( n − · M n − vectors in C [ n ] −{ } with distinct coordinates. Each of them gives a maximal vector c ( a )such that ω ( π i ( c ( a ))) = k i for each i ∈ [ n ]. Precisely 2 n − choices of a will give rise to the same vector when passingto c ( a ). All in all we have obtained at least c ( k ,...,k n ) · M n − − ( n − · M n − n − distinct multiquadratic fields Q ( √ a , . . . , √ a n ) with ( a , . . . , a n ) a maximal vector andwith ω ( a i ) = k i for each i ∈ [ n ]. For M going to infinity this quantity goes to infinity,which gives us the desired conclusion. In this section we give a proof of Theorem 1.4 and Corollary 1.5. We start by demon-strating that Corollary 1.5 is a simple consequence of Theorem 1.4. Denote by K := Q ( √ a , . . . , √ a n ) a field satisfying the conclusion of Theorem 1.4. Recall that we havean exact sequence 0 → ( O K /c ) ∗ O ∗ K → Cl(
K, c ) → Cl( K ) → . To ease the notation, let us denote by A the group ( O K /c ) ∗ O ∗ K . This gives the inequalitydim F Cl(
K, c )[2] ≤ dim F Cl( K )[2] + dim F A [2] ≤ ω ( a · . . . · a n ) · n − − n + 1 + 2 n · ω ( c ) . The second inequality can be an equality only ifdim F Cl( K )[2] = ω ( a · . . . · a n ) · n − − n + 1 , and dim F A [2] = 2 n · ω ( c ) , thanks to Theorem 1.1 (for the first equation) and simple counting (for the secondequation). Therefore we deduce fromdim F Cl(
K, c )[2] = ω ( a · . . . · a n ) · n − − n + 1 + 2 n · ω ( c )12hat dim F Cl( K )[2] = ω ( a · . . . · a n ) · n − − n + 1and dim F A [2] = 2 n · ω ( c ) . Observe that we have a surjection ϕ : ( O K /c ) ∗ ( O K /c ) ∗ → A A .
The above equations imply thatdim F ( O K /c ) ∗ ( O K /c ) ∗ = 2 n · ω ( c ) = dim F A [2] = dim F A A , whence ϕ is an isomorphism. On the other handker( ϕ ) = im (cid:18) red c ( K ) : O ∗ K O ∗ K → ( O K /c ) ∗ ( O K /c ) ∗ (cid:19) , where red c ( K ) is the natural reduction map modulo c . We conclude that the mapred c ( K ) is trivial as desired.It remains to prove Theorem 1.4. To this end we switch to the set-up of the proofof Theorem 1.3 and indicate the necessary modifications. First of all, we recall thatthe choice of y was arbitrary, so we are allowed to take y := c . Now suppose that( c, a , . . . , a n ) is a maximal vector such that all expansion maps have totally real field ofdefinition. Then we claim thatdim F Cl( Q ( √ a , . . . , √ a n ) , c )[2] = ω ( a · . . . · a n ) · n − − n + 1 + 2 n · ω ( c ) . Surely we have thatdim F Cl( Q ( √ a , . . . , √ a n ))[2] = ω ( a · . . . · a n ) · n − − n + 1 . But now observe that the collection of characters { ϕ T ( ψ a ,...,a n ; l ( G )) } l | c prime , T ⊆ [ n ] is linearly independent and generates a subspace ofCl( Q ( √ a , . . . , √ a n ) , c ) ∨ [2]linearly disjoint from Cl( Q ( √ a , . . . , √ a n )) ∨ [2]by ramification considerations. This gives precisely the 2 n · ω ( c ) additional characters inCl( Q ( √ a , . . . , √ a n ) , c ) ∨ [2] and therefore yieldsdim F Cl( Q ( √ a , . . . , √ a n ) , c )[2] = ω ( a · . . . · a n ) · n − − n + 1 + 2 n · ω ( c )as desired.We still need to explain how one ensures that all expansion maps are totally real.First of all, we indicate how Proposition 3.9 can be modified to ensure that all themaps ψ y ; y ( G ) are totally real. In this case we use a more general version [13] of R´edeireciprocity, which includes − A S to encode the splitting condition at infinity, and the maps F S are also extendedaccordingly. With these modifications in mind, one proceeds exactly with the sameargument as in Theorem 1.3. 13 eferences [1] M. Bhargava, A. Shankar, T. Taniguchi, F. Thorne, J. Tsimerman and Y. Zhao.Bounds on 2-torsion in class groups of number fields and integral points on ellipticcurves. J. Amer. Math. Soc. , to appear.[2] J. Ellenberg, L.B. Pierce and M.M. Wood. On l -torsion in class groups of numberfields. Algebra Number Theory l -torsion in classgroups. arXiv preprint: Disquisitiones Arithmeticae . 1801.[5] J. Kl¨uners and J. Wang. l -torsion bounds for the class group of number fields withan l -group as Galois group. arXiv preprint: J. Amer. Math. Soc.
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