On p 2-Ranks in the Class Field Tower Problem

Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the p2-rank of the class group as a quantity of relevance in the p-class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups have a 2-rank of 5 must have an infinite 2-class field tower. p2-rangs et p-tours de Hilbert Résumé Les récents progrès sur le problème de la 2-tour de Hilbert des corps de nombres portent sur l’infinitude – en particulier pour les corps quadratiques – quand le groupe des classes a un grand 4-rang. Généralisant à tout nombre premier p, nous utilisons les inégalités de type Golod-Safarevic afin d’analyser la contribution du p2-rang du groupe des classes à l’étude de la p-tour de Hilbert. Nous apportons également des résultats partiels en direction de l’infinitude de le 2-tour de Hilbert des corps quadratiques réels lorsque que le 2-rang du groupe des classes vaut 5.


Introduction
The p-class field tower problem for a number field K is the question of whether the maximal unramified p-extension K/K is an infinite extension, or equivalently, whether G = Gal( K/K) is infinite.The first positive answer to the class field tower problem, demonstrating number fields for which K/K is infinite, came from the landmark paper of Golod and Safarevic [2].This is done via what is now known as (one of many forms of) the Golod-Shafarevich inequality, dictating that if G is finite, then for all t ∈ (0, 1) we have the polynomial inequality Here, d = d p Cl(K) denotes the p-rank of the class group of K, and the various r k are invariants of G defined from the Magnus embedding of G into the ring of formal power series over F p in d non-commuting variables.We will not need the precise definitions of these r k , but note that while difficult to compute, they were computable enough to permit the first demonstrations of infinite class field towers.For example, from this it can be deduced that a quadratic field whose class group satisfies d 2 Cl(K) ≥ 6 has an infinite class field tower.A variant of this result due to Schoof [4] gives the same inequality but with the r k replaced by certain cohomologically-defined invariants r k .These new invariants, while still difficult to compute, greatly expanded the collection of number fields for which we could affirmatively answer the class field tower problem.
The central tool of the current paper is a third version of the inequality, due to Maire [3].Here, the invariants r k or r k are replaced by yet another set, m k , which thanks to their concrete arithmetic interpretation can be explicitly computed for many examples.Our main result (Theorem 2.2) is an explicit formula for the second such invariant, m 2 , in terms of the arithmetic of K and that of its maximal elementary abelian unramified p-extension L. This calculation affords us several significant corollaries, two of which we mention here.The first is a correction of a claim in [3] about the effect of the p 2 -rank of the class group on the class field tower problem.The second is an application to 2-class field towers over real quadratic fields: We recall that the original argument of Golod and Shafarevich proves that such towers are infinite when d 2 Cl(K) ≥ 6.The article [1] asks if this requirement can be relaxed to d 2 Cl(K) ≥ 5. Corollary 4.2 gives significant partial progress toward an affirmative answer to this question, showing that it is true under any of a wide range of additional arithmetic hypotheses.Finally, while explicit relationships between the sequences m k , r k , and r k are in general hard to come by, Section 4.3 demonstrates examples where they are provably not all equal.

Notation and Setup
For a group A, we denote its p-rank by and define its p 2 -rank by d p 2 A = d p (A p ).Let K be a number field and p a prime.We denote by K (1) the Hilbert p-class field of K, the maximal unramified abelian p-extension of K, and define the p-class field tower over K recursively by K (n) = (K (n−1) ) (1) for n ≥ 2. We let K denote the top of the tower: K = ∪K (n) .It is easily verified that K/K is Galois, and we put G = Gal( K/K) for the remainder of the paper.Let L i = KG i denote the fixed field of K corresponding to the i-th lower central subgroup , the maximal unramified p-extension of K whose Galois group is elementary abelian.Let E K and A K = Cl p (K) respectively denote the unit group and p-class group of K, and we define the generator and relation rank of G respectively by

Statement of the Main Theorem
We now turn to the construction of the invariants m i , which form the foundation for the third variant of the Golod-Shafarevich inequality mentioned in the introduction.For each i ≥ 1, define Then when G is finite, i.e., K has a finite p-class field tower, there is an isomorphism ([3], Proposition 4.2) giving a concrete arithmetic interpretation to the relation rank.We then filter ∆ 1 by the images of the higher ∆ i , setting: for i ≥ 2, where the arrows are the obvious norm maps.Finally, we define the invariants m i by m i = dim Fp M i and note that by the filtration, i≥2 m i = r.We can now state the main result of [3]: Theorem 2.1 can be used to prove class field towers infinite using the same type of argument as used in the original Golod-Shafarevich examples: For example, we can re-derive the famous inequality r > d 2  4 for finite towers from the standard argument that the inequality . The goal of the current article is to use the invariants m i to prove certain class field towers infinite, analogously to how refinements of the Golod-Shafarevich theorem (e.g., the refinements of Koch-Venkov [6] and Schoof [4]) provided new examples of infinite class field towers.In particular, whereas Koch-Venkov and Schoof used symmetry arguments to prove that the even invariants vanish (r 2k = r 2k = 0 for k ≥ 1), we will compute the early m-invariants (in this article, specifically m 2 ) in terms of the arithmetic of K and L.
The algebra behind the subsequent numerical results is then rather straight-forward, as in (2.1): Since t k ≤ t 2 for all k ≥ 2 and all t ∈ (0, 1), stronger results come from Theorem 2.1 if we can bound m 2 from above.The article's main result follows this path, culminating in an explicit computation of m 2 in terms of the arithmetic of K and L.

Theorem 2.2. With L/K as above, we have
Here and elsewhere, we abuse notation in using the symbol N = N L/K for all of the obvious norm maps from L to K (e.g., norms of units, ideals, ideal classes,...).This calculation corrects an earlier attempt to bound m 2 from above, namely, the claim given in [3] that we have As a consequence of the theorem, we can see that this inequality does not in general hold without further assumptions, and demonstrate other bounds to replace it.For example, we can deduce the following: We will give the explicit counter-example of 2) in the case that this restricted norm map is not surjective in Example 4.1.

Proof of the main theorem
We begin with the following easily-verified diagram of fields and their Galois groups.In particular, we note that since L/K is the maximal elementary abelian sub-extension of K (1) /K, the Galois group Gal(L/K) is the maximal elementary abelian quotient of A K , isomorphic to (Z/pZ) d .K K (1)  L L (1)  with Galois groups: We begin by identifying the kernel and image of the norm map on ideal classes.
Proof.This follows from the fact that under Artin reciprocity, the norm map on ideal class groups corresponds to the restriction map on Galois groups.That is, we have the following commutative diagram: Now, via the Artin map, we have ker(N ) ∼ = ker(res) ∼ = Gal(L (1) /K (1) ) and

Lemma 3.2. Given a commutative diagram of vectors spaces with exact rows as below,
where ∂ : ker(f 3 ) → cok(f 1 ) denotes the connecting homomorphism from the snake lemma.
Proof.From the snake lemma we have an exact sequence where ∂ is the standard connecting homomorphism.From this we obtain the following short exact sequence: Taking an alternating sum of dimensions of this sequence gives the result.
Next, for any subfield F ⊂ K, recall/define and ) and consider the following commutative diagram: Here the two rows are the well-known exact sequences stemming from the morphism φ L defined by φ where Q is an ideal of L chosen so that Q p = (x), and the two rows are connected by the appropriate norm maps N = N L/K .In anticipation of applying Lemma 3.2, we note that the connecting map ∂ can be made explicit in our context as follows: For Then by commutativity of the rightmost square, N ([x]) = [N (x)] ∈ ∆ K is in the kernel of φ K , so by exactness ∂(c) := [N (x)] ∈ E K .It is now easy to characterize the subgroup of elements of cok(N : . This is the last ingredient needed to apply Lemma 3.2 to the commutative diagram above, which yields the following formula for the key dimension of interest: This calculation reduces the proof of the main theorem to combining a collection of established identities: . Finally, we note that we have The theorem and Lemma 3.1 combine to explain the appearance of the p 2 -rank in the p-class field tower problem.Namely, since and so the dimension of this norm group is bounded above by the p 2 -rank of A K .In cases where this norm map on the p-torsion ideal classes is surjective, we can thus get a lot of mileage from Golod-Shafarevich type arguments by using a large p 2 -rank to demonstrate a small value of m 2 .Unfortunately, this norm map is not always surjective, as we shall see in the next section.

Consequences
Before turning to explicit corollaries of the main theorem, we pause for the brief general remark that the theorem implies that the ultimate goal of bounding m 2 from above can be achieved via two principal routes: • Finding elements of E K which are norms from Λ L (including, in particular, norms from E L ).
This illuminates the perspective that the likelihood of a number field K having an infinite p-class field tower increases directly with the non-triviality of the norm maps on ideal classes and units from L.

A Counter-Example to Inequality (2.2)
All computations were done in SAGE [5].
Then the claim of inequality (2.2) predicts that We will show that, to the contrary, m 2 = 2, but first note that verifying that m 2 ≥ 2 is easier.We simply compute in SAGE that both norm maps [2] are the trivial map.Then the Main Theorem gives We include some auxiliary calculations to show that in fact m 2 = 2 by showing that −1 ∈ N (∆ L ).The genus field L of K is given by L = K( √ 13, √ −3), and A L ∼ = (8, 8, 4).We have SAGE choose a basis {c i } 3 i=1 of A L [2], choose representative ideals I i ∈ c i , and let x i be a generator for I 2 i .We check that for SAGE's particular choice of x 1 , x 2 , and x 3 , we have From this we see that the image of Incidentally, this computation makes clear the oversight which led to the inequality.The claim in the proof of [3,Corollary 3.2] that we can find the described elements σ 1 , . . ., σ 4 is tantamount to the claim that the surjection on 2-torsion.The example above shows how this can be false from a purely group-theoretic perspective: Taking abstract groups G = (8, 8, 4) and H = (4, 4) as in the class groups of the example above, there is no surjection from G to H 2 which restricts to a surjection G [2]  Next, we return to the application from the introduction concerning real quadratic fields.Suppose K is real quadratic with fundamental unit ε, and d = d 2 A K = 5.Then as in the first sentence of the proof of the main theorem, we have =: e and so r ≤ 5 + e.Now by Theorem 2.1, if K has a finite 2-class field tower, then for all t ∈ (0, 1), we have It is trivial to verify that for each possible value of e (note e ≤ 2 for real quadratic fields), this inequality is violated for some t ∈ (0, 1) if m 2 ≤ 7 − e.Using Theorem 2.2 to compute m 2 , this is equivalent to This inequality thus provides a sufficient condition for K to have an infinite 2-class field tower.We continue to develop this expression.In particular, note that since E L ⊂ Λ L , we have It is easy to enumerate the list of possible dimensions of the three F 2vector spaces appearing in ( * ) for which the inequalities in ( * ) and ( * * ) are all satisfied, providing the next corollary.
It is desirable to have a version of this result which does not depend on knowing anything about N K/K (E K ).To that end, we extract from the previous corollary a slightly weaker sufficient condition that is in practice vastly simpler to evaluate.[2]).Even more directly, we note that this condition is satisfied, and hence K has an infinite 2-class field tower, if any of the following hold: • d 2 (N (A L [2])) ≥ 2; • d 2 (N (A L [2])) = 1 and at least one of −1 or ε are norms from Λ L .
Finally, to return to the topic of p 2 -ranks, recall that d 2 N (A L [2]) is bounded above by the 4-rank of A K .This tells us, for example, that if d 4 A K = 0, only the third of the three conditions in the above list will be viable as an argument to show that K has an infinite 2-class field tower via this method.

A Comparison of Invariants
While the computation of m 2 does not improve upon the result of Koch-Venkov for quadratic imaginary number fields and p odd, the simplicity of Theorem 2.2 in this case permits us to demonstrate a distinction between the m-invariants and the two types of r-invariants in this case.Namely, by Koch-Venkov [6] and Schoof [4]

Corollary 4 . 2 .
Suppose K is a real quadratic field with d 2 A K = 5.Then K has an infinite 2-class field tower if either

Corollary 4 . 3 .
Suppose K is a real quadratic field with d 2 A K = 5.Then K has an infinite 2-class field tower if respectively, we have r 2 = 0 and r 2 = 0. Applying Theorem 3, sinceE K = E p K = {±1}, we conclude simply that m 2 = d − d p N (A L [p]).But since N (A L [p]) ⊂ A p K [p],we conclude m 2 = d if, for example, K has a cyclic class group.