Asymptotic growth patterns for class field towers

Let $p$ be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a $\mathbb{Z}_p$-tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain $\mathbb{Z}_p$-extensions in which a given prime above $p$ is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for $p$-primary class numbers of the number fields in a $\mathbb{Z}_p$-tower.

1. Introduction 1.1.Motivation from Iwasawa theory.Let L be a number field and p be an odd prime number.Choose an algebraic closure L of L. Let H p (L) be the maximal abelian unramified p-extension of L, and let A p (L) denote the Galois group Gal(H p (L)/L).By class field theory, A p (L) is naturally isomorphic to the p-primary part of the class group of L. Setting h p (L) := #A p (L), we refer to h p (L) as the p-class number of L. We set Z p to denote the ring of p-adic integers.A Z p -extension of a number field L is an infinite Galois extension L ∞ /L, such that Gal(L ∞ /L) is isomorphic to Z p (as a topological group).Given a Z p -extension L ∞ /L, and n ∈ Z ≥0 , we set L n /L to be the extension such that L n ⊆ L ∞ and [L n : L] = p n .The field L n is the n-th layer, and we obtain a tower of number fields . Let e n ∈ Z ≥0 be such that p en = h p (L n ).In his seminal work, Iwasawa [Iwa73] showed that there are constants µ, λ ∈ Z ≥0 and ν ∈ Z, such that for all large enough values of n, one has that e n = p n µ + nλ + ν.
Thus, Iwasawa's results show that there are interesting growth patterns for p-class numbers in certain infinite towers of number fields.These results motivate the study of asymptotic growth properties of p Hilbert class field towers along Z p -extensions.Throughout, p will be a fixed odd prime number, and let L ∞ /L be a Z p -extension.Set F(L n ) to denote the maximal unramified pro-p extension of L n , and set G n := Gal(F(L n )/L n ).We identify the abelianization of G n with The field F(L n ) contains a p Hilbert class field tower over L n .In greater detail, for j ∈ Z ≥0 , define H (j) p (L n ) inductively as follows H (0) p (L n ) := L n , H (1)  p (L n ) := H p (L n ), H (j)  p (L n ) := H p H (j−1) p (L n ) for j ≥ 2.
In this way, we obtain a tower of p-extensions of L n whose union is equal to F(L n ).The field F(L n ) is infinite if and only if H Iwasawa studied asymptotic growth patterns for the degrees [H p (L n ) : L n ] considered along the tower that is the second column (from the left margin) in the above diagram.In the spirit of Iwasawa theory, we consider natural growth questions for the pro-p groups G n as n → ∞.We define the exponential growth number ρ(G) of a finitely generated pro-p group G, which is a natural invariant associated with its Hilbert series.In this context, there is a natural analogy with the Hilbert series of an algebraic variety.Let Ω(G) denote the mod-p Iwasawa algebra associated to G, defined as the inverse limit where U ranges over all finite index normal subgroups of G. Let I G be the augmentation ideal of Ω(G), and for [DdSMS99, section 12.3, p. 307]).On the other hand, if a group G has infinite rank, the numbers c n (G) grow at an exponential rate (cf.Proposition 12.17 of loc.cit.).In practice, the Galois groups that arise from infinite p Hilbert class field towers may not be p-adic analytic groups.For many of the Galois groups constructed in this article, the numbers c n (G) increase at an exponential rate.The radius of convergence of H(G; t) is given by R and in this case, H(G, t) is an entire function.The constant ρ(G) measures the exponential growth of subgroups in the Zassenhaus filtration of G.It is clear that if ρ(G) > 1, then G is not an analytic pro-p group.It is shown, in various contexts that pro-p Hilbert class field towers are infinite via an application of the Golod-Shafarevich-Vinberg criterion (cf.[HMR20, Theorem 1.2]).This strategy is developed and employed in the following works: [VK78, Sch86, KL87, JM11, HMR20, HMR21].The class of pro-p groups for which it can be shown that ρ(G) > 1, via the Golod-Shafarevich-Vinberg criterion have special properties, and are known as Golod-Shafarevich groups (cf.Definition 2.4).We refer to [Ers12] for an introduction to the theory of Golod-Shafarevich groups.
Given a number field L, its root discriminant is defined to be D

1/n
L , where D L is its absolute discriminant, and n = [L : Q].It is an old question as to whether there exists an infinite tower of number fields, unramified away from a finite set of primes, for which the root discriminant is bounded.Since the root discriminant is constant in unramified extensions, the above question is related to the constants of Martinet [Mar78] and Odlyzko's bounds [Odl90].In the number field extensions L/L n such that L ⊂ F(L n ), the root discriminant remains bounded.Hajir and Maire [HM01] construct tamely ramified Golod-Shafarevich Galois groups and are able to improve upon Martinet's constants.These root discriminant bounds are further refined by Hajir, Maire and Ramakrishna in [HMR21].One is thus interested in constructing infinite unramified Galois pro-p groups G, such that ρ(G) is large.In this paper, we study the following question.
Question 1.1.Let L be a number field and p be a prime number.Given a Z p -extension L ∞ /L, what can one say about the growth of ρ(G n ), as n → ∞? 1.2.Main results.We consider a variant of the above question, for certain natural Z pextensions in which one of the primes above p is infinitely split.For k ≥ 0, let F [k] (L n ) be the maximal pro-p extension of L n for which • all primes v ∤ p are unramified, • all decomposition groups at primes v|p are abelian, • for all primes v | p, every element of the decomposition group of v has order dividing p k .
Note that For k ≥ 1, there is finite ramification in the extension n /L n is a finite extension, and the proof of [HMR20, Proposition 1.5]).In particular, for all number fields contained in Let L be a number field extension of Q and let p be an odd prime.Let r 1 (resp.r 2 ) be the number of real embeddings (resp.complex embeddings) of L. Let p = p 1 , p 2 , . . ., p g be the primes of L that lie above p, and assume that g ≥ 2. Set e i := e(p i /p) (resp.f i := f (p i /p)) to denote the ramification index (resp.inertial degree) of p i over p.Note that We set U to denote the product g i=2 U i , and Ē to denote the p-adic closure of the image of the group of global units and assume that m ≥ 1.It follows from a standard application of global class field theory that there exists a Z p -extension L ∞ /L in which p is completely split.Note that when L is totally imaginary, the condition m ≥ 1 is automatically satisfied when Theorem 1.2.Assume that the following conditions are satisfied (1) p is odd and there are g > 1 primes of L that lie above p, (2) L contains µ p , the p-th roots of unity, (3) Then, there exists a Z p -extension L ∞ /L in which p 1 is totally split.Moreover, there exists a constant C > 0 (independent of n and k) and n 0 ∈ Z ≥0 , such that for all n ≥ n 0 and k ≥ 1, we have that Let µ(L ∞ /L) denote the Iwasawa µ-invariant for the Z p -extension L ∞ /L.Let T 1 be the number of primes of L that lie above p, that are infinitely decomposed in L ∞ .The constant C can be chosen to be any value such that Remark 1.3.
• We observe that the hypotheses imply that • Note that the condition (2) implies that L is totally imaginary and (3) implies that m ≥ 1.
We prove Theorem 1.2 by adapting the strategy of Hajir, Maire and Ramakrishna [HMR20].The following result illustrates Theorem 1.2.
Corollary 1.4.Let p ≥ 3 and ℓ ≥ 11 be distinct primes and set L := Q(µ pℓ ).Let p be any prime of L that lies above p.Assume that there are g = (ℓ − 1) primes of L that lie above p, i.e., ℓ ≡ 1 mod p.Then, there exists a Z p -extension L ∞ /L in which p is totally split.Let T 1 be the number of primes of L that lie above p, that are infinitely decomposed in L ∞ .For any constant and all large enough values of n.
We also define another notion measuring the size m(G) of a Golod-Shafarevich group G (cf. Definition 2.6).We prove an analogous result for the growth of the numbers 1.3.Organization.Including the introduction, the manuscript consists of three sections.In section 2, we introduce preliminary notions.In greater detail, we develop the Golod-Shafarevich theory of pro-p groups.We prove an explicit criterion (cf.Proposition 2.4) which gives an explicit lower bound for the exponential growth number ρ(G) for a Golod-Shafarevich group G.In section 3, we apply the results from section 2 to prove Theorem 1.2, Corollary 1.4 and Theorem 3.4.1.4.Acknowledgment.This project was started when the third author was at the Centre de recherches mathematiques, Montreal.At the time, the third author's research was supported by the CRM-Simons fellowship.We are very thankful to Katharina Müller and Ravi Ramakrishna for numerous helpful comments.We thank the anonymous referee for the excellent report.

Pro-p groups and their Hilbert series
We review some preliminaries and set up some basic notation in this section.Throughout, p shall be an odd prime.Let G be a finitely generated pro-p group, set Ω(G) to denote the completed group algebra of G over F p .More precisely, Ω(G) is defined to be the inverse limit where U runs over all finite index normal subgroups of G.We refer to Ω(G) as the mod-p Iwasawa algebra of G.Many properties of the group G are captured by algebraic properties of Ω(G).We consider the Hilbert series associated with Ω(G).Given a normal subgroup U of G, there is a natural map ι U : G → F p [G/U ], which sends g ∈ G to the element ḡ • 1.Here, ḡ is the congruence class of g modulo U .The map ι : G → Ω(G) is the inverse limit ι := lim ← −U ι U .The augmentation map α : Ω(G) → F p maps each element of the form ι(g) to 1.We shall, by abuse of notation, simply let g ∈ Ω(G) simply denote the element ι(g).Set I G to denote the augmentation ideal of Ω(G), i.e., the kernel of the augmentation map.For n ≥ 1, Ω(G)/I n G is a finite dimensional F p -vector space.
, and set c 0 (G) := 1.The Hilbert series H(G; t) is a formal power series defined by H(G; t) 1 n , we note that the radius of convergence of H(G; t) is given by Given x ∈ G such that x = 1, the depth of x is defined as follows Definition 2.2.The Zassenhaus filtration is defined as follows as n → ∞.These numbers are related to the growth of groups in the Zassenhaus filtration.There is an explicit relationship between the numbers {a n (G)} and {c n (G)}, established by Mináč, Rogelstad and Tân [MRT16].
Assume that G is a finitely presented pro-p group, and let Here, F = σ 1 , . . ., σ d is a free pro-p group generated by d = d(G) elements.
The subgroup R of F is the normal subgroup ρ 1 , . . ., ρ r Norm generated by r = r(G) elements.Given a choice of minimal presentation of G, we define the associated Golod-Shafarevich polynomials.Let Ω(F ) be the mod-p Iwasawa algebra associated to F , and I F be the augmentation ideal of Ω(F ).By a theorem of Lazard [Laz65], the algebra Ω(F ) is isomorphic to the algebra of power series F p u 1 , . . ., u d .Here u i is identified with the element (σ i − 1).The augmentation ideal I F is generated by u 1 , . . ., u d .Let ω F be the depth function on F , defined by setting F }, for x = 1, and ω F (1) := ∞.The map ϕ induces a surjection Ω(F ) → Ω(G), whose kernel we denote by J. Identify Ω(G) with the quotient Ω(F )/J and I G with I F /J.The depth filtration ω G is related to ω F as follows Proposition 2.2.The depth function ω F : F → Z ≥1 ∪ {∞} and associated filtration {G n } n satisfies the following properties Proof.The above mentioned result is [HMR21, Proposition 1 and 2], also see [Koc02, Section 7.4].
Theorem 2.3 (Vinberg's criterion).Let G be a pro-p group and let H(G; t) be the Hilbert series associated to G. Choose a minimal presentation (2.1) of G and let P (G; t) be the associated Golod-Shafarevich polynomial.Recall that R G is the radius of convergence of H(G; t), and set R ′ G := min{1, R G }.Then, for any value t ∈ (0, R ′ G ), the following inequality is satisfied Proof.This result is well known.We refer to [Ers12, Theorem 2.1] and [Vin65] for a proof of the above statement.
The next result shows that if P (G; t) takes a negative value t 0 in the interval (0, 1), then, ρ(G) > 1.Furthermore, it gives us a lower bound for ρ(G), and can therefore be viewed as a refinement of the Golod-Shafarevich-Vinberg criterion (cf.[HMR20, Theorem 1.2]).
Proof.Let R G be the radius of convergence of the Hilbert series H(G; t) and set R ′ G := min{1, R G }.By definition, the coefficients of H(G; t) are all non-negative.Then, for t ∈ (0, R ′ G ), the series H(G; t) converges absolutely to a positive value.By Vinberg's criterion, P (G; t)H(G; t) ≥ 1 for all t ∈ (0, R ′ G ). Since P (G; t 0 ) < 0, it follows that H(G; t) does not converge absolutely at t = t 0 .In other words, t 0 lies outside the domain of convergence.Therefore, there is a value t 0 ∈ (0, 1) so that P (G; t 0 ) < 0. Proposition 2.4 shows that if G is a Golod-Shafarevich group with P (G; t 0 ) < 0, the order of exponential growth satisfies ρ(G) ≥ t −1 0 > 1.In particular G is infinite since the numbers c n (G) grow at an exponential rate as n → ∞.
At this point, we introduce a new definition which measures the size of a Golod-Shafarevich group.Let G be a Golod-Shafarevich group and Norm .Then, following [HMR20, p.4, ll.10-17], we get a new minimal presentation as follows.Lift each x i to y i ∈ F , and let R ′ = ρ 1 , . . ., ρ r , y 1 , . . ., y m Norm to be the normal subgroup of F generated by R and the elements y 1 , . . ., y m .Since it is assumed that d(G ′ ) = d(G), this gives a minimal presentation of G ′ .In particular, note that ω F (y i ) ≥ 2 for all i = 1, . . ., m.Following loc.cit., we say that we have cut the group G by the elements y 1 , . . ., y m .Let ϕ ′ : F → G ′ be the composite of ϕ with the quotient map G → G ′ .With respect to the new presentation we find that for t ∈ (0, 1), Definition 2.5.Let G be a Golod-Shafarevich group, and choose a minimal presentation for G such that P (G; t) attains a negative value on (0, 1).We say that a set {y 1 , . . ., y m } ⊂ F is an admissible cutting datum if ω F (y i ) ≥ 2 for all i.
Given an admissible cutting datum {y 1 , . . ., y m }, set x i := ϕ(y i ) and Definition 2.6.Let G be a Golod-Shafarevich group, and choose a minimal presentation for G such that P (G; t) attains a negative value on (0, 1).We define m M (G) to be the smallest value m ∈ Z ≥0 such that there exists an admissible cutting datum {y 1 , . . ., y m+1 }, such that P (G ′ ; t) ≥ 0 for all t ∈ (0, 1).If no such m exists, we set m M (G) := ∞.We set m(G) := min{m M (G) | M}, where the minimum is taken over all minimal presentations M of G.
Remark 2.5.We note that the definition of m(G) does not, to our knowledge, appear in the literature prior to this.
Thus, for any admissible cutting datum {y 1 , . . ., y k }, with k ≤ m(G), the associated quotient G ′ is a Golod-Shafarevich group, and thus in particular is infinite.Like ρ(G), the number m(G) gives one a measure of the size of a Golod-Shafarevich group G.

Growth asymptotics for split prime Z p -extensions
Let L be a number field which satisfies the conditions of Theorem 1.2.In this section, S p (resp.S ∞ ) denote the primes of L that lie above p (resp.∞).Set S to denote the union.Let L/L be a Galois number field extension and let S(L) (resp.S p (L)) be the set of places of L which lie above S (resp.S p ).We shall denote by L S the maximal pro-p algebraic extension of L in which the primes w / ∈ S(L) are unramified.We denote by G S (L) the Galois group Gal(L S /L).Set H S (L) to be the maximal abelian unramified extension of L in which the places v ∈ S(L) are split.The S class group of L is the Galois group Cl S (L) := Gal(H S (L)/L), and is identified with a quotient of the class group Cl(L).
Theorem 3.1.Let L/L be a finite Galois extension.With respect to the above notation, the following relations hold Proof.The above is a special case of [NSW13, Theorem 10.7.3].
We order the set of primes above p such that p i is infinitely decomposed L ∞ for i ∈ [1, T 1 ] and infinitely ramified for i ∈ [T 1 + 1, T 2 ].The primes p i for i ∈ [T 2 + 1, g] are unramified and finitely decomposed in L ∞ .We shall set g i (n), e i (n) and f i (n) to denote g p i (L n /L), e p i (L n /L) and f p i (L n /L) respectively.We find that Before stating the next result, we clarify some standard conventions with regard to our notation.Let f, g, h : Z ≥0 → R ≥0 be non-negative functions.We write f for some absolute constant C > 0.
Corollary 3.2.The following relations hold The result follows therefore as an immediate consequence of Theorem 3.1 and (3.1). For We choose an embedding of or equivalently, a prime v ′ of L n that lies above v.We shall refer to the decomposition group of v ′ |v as the decomposition group at v. The inclusion ι prescribes an inlusion of Gal( Ln,v /L n,v ) into G Ln , and the image of this embedding is identified with the decomposition group at v. Note that the pro-p completion of the decomposition group at v is generated by n v elements.For ease of notation, we set We note that since µ p is contained in L, Let L n be the maximal pro-p extension of L n unramified at all primes v / ∈ S(L n ).Denote by L n the maximal pro-p extension of L n unramified at all primes v / ∈ S(L n ) and such that for all primes v ∈ S p (L n ), the decomposition group is abelian.For k ≥ 0, let F [k] (L n ) be the maximal subfield of L n in which all the elements in the decomposition groups at primes v ∈ S p (L n ) have order dividing p k .It is thus understood that F [0] (L n ) is the maximal unramified pro-p extension of L n in which all primes v ∈ S(L n ) are completely split.We have the following containments We begin with a minimal filtration The group G n is obtained from G n upon cutting by the commutators of the generators of all decomposition groups for primes v ∈ S p (L n ).In greater detail, for each prime v ∈ S p (L n ), let z nv be a set of elements in F mapping to a set of generators of the decomposition group at v. For 1 ≤ i < j ≤ n v , we let z and we obtain a new presentation where, Going modulo the p k -th powers of all generators of decomposition groups, we obtain a presentation Set P n (t) := P( G n , t), P n (t) := P(G n , t), and P n ) for all k ≥ 0. Since we successively cut by elements with depth ≥ 2, we have that Proposition 3.3.For t ∈ (0, 1), P n (t) ≤ 1 − D n t + R n t 2 , where D n , R n are constants satisfying Proof.The group G n is obtained from G n by quotienting by the commutators of each of the decomposition groups at primes v ∈ S p (L n ).At each prime v, we cut by nv 2 elements {z , where we recall that an upper bound for n v is given by (3.2).Furthermore, it follows from (3.4) that ω(z (v) i,j ) ≥ 2. Therefore, we find that for t ∈ (0, 1), where Here, we have invoked the bounds for d(n) and r(n) from Corollary 3.2.
Proof of Theorem 1.2.It follows from Proposition 3.3 that for t ∈ (0, 1), where n is obtained from G n by quotienting by the p k -th powers of the generators of the decomposition groups at the primes v ∈ S p (L n ).The total number of elements we quotient by is (3.9) Let y 1 , . . ., y R ′ n be the elements that generate the decomposition groups at the primes v ∈ S p (L n ).By Proposition 2.2, we have that Therefore, for t ∈ (0, 1), we find that Note that by (3.9), R ′ n = O(p n ).We estimate both sides of the inequality (3.10).Therefore, we obtain the following asymptotic estimates Therefore, for large enough values of n, we have that Q(t n ) < 0. By the estimates in the statement of Proposition 3.3, (3.13) Therefore, in particular, we find that t n ∈ (0, 1) for all large enough values of n.Since P [k] n (t) ≤ Q(t) for all t ∈ (0, 1), it follows that P With respect to notation above, Let C > 0 be a constant for which Then, we find that for all large enough values of n, we find that t −1 n > Cp n .Since for large enough values of n, P n (t n ) < 0 and t n ∈ (0, 1), it follows from Proposition 2.4 that n , and this proves the result.
Proof of Corollary 1.4.We show that the conditions of Theorem 1.2 are satisfied.
Theorem 3.4.Let p be a prime number and L be a number field for which the conditions of Theorem 1.2 are satisfied.Then, there exists a constant c > 0 (independent of n and k) and n 0 ∈ Z ≥0 , such that for all n ≥ n 0 and k ≥ 1, we have that Proof.We set m n := m G [k] n , and assume without loss of generality that m n < ∞.We obtain a lower bound on m n .Choose a minimal presentation n → 1 such that there exists t 0 ∈ (0, 1) for which P n , t 0 ) < 0 for the associated Golod-Shafarevich polynomial.Let {y 1 , . . ., y m+1 } ⊂ F be an admissible cutting datum.Note that by definition, ω F (y i ) ≥ 2 for all i.Setting x i := ϕ(y i ), denote by G ′ the quotient G [k] n / x 1 , . . ., x m+1 Norm .By definition, the admissible cutting datum {y 1 , . . ., y m+1 } can be chosen so that P (G ′ , t) ≥ 0 for all t ∈ (0, 1).From the proof of Theorem 1.2, we have that P [k] n (t) ≤ Q(t), where Q(t) := 1 − D n t + R n t 2 + R ′ n t p k .For t ∈ (0, 1), we find that P (G ′ , t) ≤ P [k]  n (t) + (m + 1)t 2 ≤ Q(t) + (m + 1)t 2 = 1 − D n t + (R n + m + 1)t 2 + R ′ n t p k ≤ 1 − D n t + (R n + m + 1)t 2 + R ′ n t p .In what follows, we set a n := Dn 2(Rn+m+1) .From the estimates (3.13), it follows that a n ∈ (0, 1) for large enough values of n.Therefore, assume that n is large enough so that a n ∈ (0, 1).Then, P (G ′ , a n ) ≥ 0, i.e., ) for all j ≥ 1.These Hilbert class field towers, when viewed along the Z p -extension, are represented by the following diagram Q.
. The quantity ρ(G) measures the order of exponential growth of the quotients I n G /I n+1 G cf. [Laz65, Lemma 3.4 and Theorem 3.5, pp.204-205] for further details.

n
(t n ) < 0 for all large enough values of n, and all values of k ≥ 1.