The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension

Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over $\mathbf{Q}(\zeta_{\ell})$, the cyclotomic field of $\ell$-th roots of unity. Suppose that $A(\mathbf{Q}(\zeta_{\ell}))[\ell] \neq 0$. We show that there exists a number field $L$ and a $\mathbf{Z}_p$ extension $L_{\infty}/L$ where the $\ell$-primary fine Selmer group of $A$ grows arbitrarily quickly. This is a fine Selmer group analogue of a theorem of Washington which says that there are certain (non-cyclotomic) $\mathbf{Z}_p$-extensions where the $\ell$-part of the class group can grow arbitrarily quickly. We also prove this for a wide class of non-commutative $p$-adic Lie extensions. Finally, we include several examples to illustrate this theorem.


Introduction
We begin with a fundamental theorem of Iwasawa, which serves as the starting point of Iwasawa theory.Let K be a number field and let K ∞ /K be a Z p -extension: a Galois extension with Galois group isomorphic to the additive group Z p of p-adic integers.For such an extension K ∞ /K, there exists a unique sequence of fields such that each K n /K is a cyclic extension of degree p n .Iwasawa [Iwa59] proved the following now famous theorem about the growth of class numbers in such towers.
Theorem (Iwasawa).Let K be a number field and let K ∞ /K be a Z p extension with layers K n .Suppose that p e n is the exact power of p dividing the class number of K n .Then there exist integers µ, λ, ν such that e n = µp n + λn + ν for all sufficiently large values of n.
A large part of classical Iwasawa theory is devoted to studying the invariants µ and λ in the above formula.In a beautiful paper, Iwasawa showed [Iwa73, Theorem 1] that there are Z p -extensions for which the µ-invariant can be arbitrarily large.
Theorem 1.1 (Iwasawa).Let N ≥ 1.There exists a number field L and a Z p -extension L ∞ /L such that µ ≥ N .Now that we have discussed the p-part of the class group in a Z p -extension, we now discuss the ℓ-part of the class group in a Z p -extension, where ℓ = p are distinct primes.The fundamental theorem in this area is due to Washington [Was78]: Theorem 1.2 (Washington).Let ℓ = p be distinct primes Let K be an abelian extension of the field Q of rational numbers.Let K cyc /K be the cyclotomic Z p extension of K and let ℓ en be the exact power of ℓ dividing the class number of K n .Then e n is bounded as n → ∞.
Based on this, one might reasonably guess that the ℓ-part of the class group is bounded in an arbitrary Z p -extension.But this turns out to be false, as proven in [Was75,Theorem 6].
Theorem 1.3 (Washington).Let N ≥ 1.There exists a number field L and a Z p -extension L ∞ /L such that e n ≥ N p n , where ℓ en is the exact power of ℓ dividing the class number of L n .
The purpose of this article is to discuss analogues of Theorems 1.1 and 1.3 for fine Selmer groups of elliptic curves.Let E be an elliptic curve over a number field F .The Mordell-Weil Theorem says the group E(F ) of rational points is a finitely generated abelian group.This arithmetic of this group is essentially controlled by the Selmer group of E/F .In [Maz72], Mazur introduced the Iwasawa theory of Selmer groups in Z p -extensions of F .The notion of the Fine Selmer group was formally introduced by Coates and Sujatha in [CS05], even though it had been studied by Rubin [Rub00] and Perrin-Riou [PR93,PR95] under various guises in the late 80's and early 90's.In [CS05], Coates and Sujatha showed that these fine Selmer groups have stronger finiteness properties than classical Selmer groups.
In [Kun20, Theorem 4.2], Kundu proved an analogue of Theorem 1.1 for fine Selmer groups.Kundu proved that if A is an abelian variety and N ≥ 1 is an integer, then there exists a number field L and a Z p -extension L ∞ /L such that the µ-invariant of the fine Selmer group of A over L ∞ is at least N .In other words, µ-invariants of fine Selmer groups can be arbitrarily large.And in [KL23, Theorem B], Kundu and Lei proved a Fine Selmer group analogue of Theorem 1.2: in the cyclotomic Z p -extension, the ℓ-part of the Fine Selmer group of A stabilizes.
The purpose of this paper is to prove a fine Selmer group analogue of Theorem 1.3.If ℓ is a prime and A is an abelian variety over a number field F , let R ℓ ∞ (A/F ) denote the ℓ-primary fine Selmer group of A over F .(See Section 5.1 for the precise definition.)If G is an abelian group, the ℓ-rank of G is defined by r ℓ (G) = dim Z/ℓZ G[ℓ].Note that it is possible to have r ℓ (G) = ∞; this is true for example if G contains infinitely many copies of Z/ℓZ.(See [LM16, Section 3] as a reference for this definition.)Here is our main result: Theorem 1.4.Let ℓ = p be distinct primes.Let Q(ζ ℓ ) be the cyclotomic field of ℓ-th roots of unity and let A be an abelian variety over Q(ζ ℓ ).Suppose that A(Q(ζ ℓ ))[ℓ] = 0.For every integer N ≥ 1, there exists a finite extension L/Q(ζ ℓ ) and a Z p -extension L ∞ /L such that for all n ≥ 0, where q = min{ℓ, p}.In particular, More generally, we can also prove this theorem for many non-commutative p-adic Lie-extensions.The statement of our theorem requires a definition: Definition 1.5.A pro-p group Γ is uniform of dimension d if it is topologically finitely generated by d generators and there exists a unique filtration by the p-descending central series of Γ.In other words, we have If F is a number field and p is a prime ideal of F , then the p-class group of F is the quotient of Cl(F ) by the subgroup generated by the ideal class of p.
Assumption 1.Let Γ be a uniform pro-p group with a fixed-point-free automorphism of order m, where m > 2 is a prime different from p. Assume that: (1) there exists a Z/mZ extension of number fields F/F 0 , where F 0 is totally imaginary, (2) the field F contains the p-th roots of unity, (3) there is a unique prime p of F lying over the rational prime p, and (4) the p-part of the p-class group of F is trivial.
Theorem 1.6.Let ℓ = p be distinct primes.Let Q(ζ ℓ ) be the cyclotomic field of ℓ-th roots of unity and let A be an abelian variety over Let Γ be a uniform pro-p group with a fixed-point-free automorphism of order m.If m > 2, assume that Assumption 1 holds.Then for every integer N ≥ 1, there exists a finite extension Let Sel ℓ ∞ (A/F ) denote the ℓ-primary (usual) Selmer group of A over F .(See Section 5.1 for the precise definition.)Since the fine Selmer group is a subgroup of the usual Selmer group, we have the following Corollary 1.7.Retain the notations and assumptions of Theorem 1.6.For every integer N ≥ 1, there exists a finite extension for all n ≥ 0, where q = min{ℓ, p}.In particular, We can give several numerical examples at the end of the paper to show that Assumption 1 often holds.
Strategy.We follow Iwasawa's approach in [Iwa73], which inspired Washington's approach in [Was75, Section VI].We construct a Z d p -extension L ∞ /L where the ℓ-rank of the class group is unbounded.
Theorem 1.8.Let ℓ = p be distinct primes.Let Γ be a uniform pro-p group.If m > 2, then assume Assumption 1.For every integer N ≥ 1, there exists a finite extension L/Q(ζ ℓ ) and a Γ-extension L ∞ /L such that for all n ≥ 0: We then use results of Lim-Murty [LM16] to show that the ℓ-rank of the fine Selmer group is close in size to the ℓ-rank of the class group.Putting these two results together, we conclude that the ℓ-part of the fine Selmer group is unbounded in L ∞ /L, proving Theorem 1.6.
2.1.The case m = 2.By [RZ00, Theorem 2.17], if Γ is a uniform pro-p group with an fixed-pointfree automorphism τ of order m = 2, then Γ ≃ Z d p for some d ≥ 1.To motivate the next proposition, recall that if K is an imaginary quadratic field, then there is a Z p -extension K ∞ /K called the anticyclotomic Z p -extension of K.This extension has the property that infinitely many primes of K split completely in K ∞ .The next proposition generalizes this to Z d p extensions for d ≥ 1. Proposition 2.1.Let K be a CM field such that K/Q is a Z/2dZ-extension.Suppose that there is only prime in K lying over p.Then there is a Z d p -extension K ∞ /K such that infinitely many primes of K split completely in K ∞ .
Proof.This fact is well-known but for lack of a reference, we sketch the proof here.The below construction is reproduced from [Lon12, Section 2].Let K + be the maximal totally real subfield of K.For any integral ideal c ⊆ O K + , let O c = O K + + cO K be the order of conductor c in K.The ring class field K[c]/K of K of conductor c is the Galois extension of K such that there an isomorphism via the Artin map: Suppose that ℓ is a rational prime which is inert in K. Then the ideal class of ℓ is trivial in Cl(K) and hence class field theory gives that ℓ splits completely in any ring class field K[c] of conductor coprime to ℓ. (See [Nek07, Section 2.6.3].)In particular, if ℓ is inert in K/Q and ℓ is coprime to p then ℓ splits completely in K ∞ .By the Chebotarev density theorem, there are infinitely many such primes.

The case m > 2.
Here is the main result: Proposition 2.2.Let Γ be a uniform pro-p group.Assume the Assumption 1. Then there exists a Galois extension K/F and a Γ-extension K ∞ /K such that infinitely many primes of K split completely in K ∞ .Now let F be a number field and let F max,p be the maximal pro-p extension of F unramified outside the primes above p.The number field F is called p-rational if Gal(F max,p /F ) is pro-p free.
Lemma 2.3.Let F be a number field with a primitive p-th root of unity.Then F is p-rational if and only if there exists a unique prime p above p and the p-part of the p-class group of F is trivial.
Lemma 2.4.Keep the notations and assumptions from Proposition 2.2.Let n be an integer such that [F 0 : Q]p n ≥ 2d and let K 0 (resp.K) be the n-th layer of the cyclotomic Z p -extension of F 0 (resp.F ). Then there exists an intermediate field K ⊂ K ∞ ⊂ K max, p such that K ∞ is Galois over K 0 with Galois group Γ ⋊ τ .Suppose τ acts fixed-point-freely on Γ.Then every place of K 0 which is inert in K/K 0 splits completely in K ∞ /K.
Proof.This follows from [HM19, Proposition 3.6] and [HM19, Proposition 3.7] if F is p-rational.But Lemma 2.3 gives conditions for F to be p-rational and F satisfies those conditions.Proposition 2.2 now follows from Lemma 2.4, because by the Chebotarev density theorem there are infinitely many primes that are inert in the cyclic extension K/K 0 .

Construction of the Γ-extension L ∞ /L
Fix N ≥ 1.In this section we will construct a Γ-extension L ∞ /L with the properties in Theorem 1.8, i.e: such that r ℓ (Cl(L n )) ≥ N p n for all n ≥ 0.
Proposition 3.1.Let N ≥ 1 be an integer.Let Γ be a uniform pro-p group of dimension d with a fixed-point-free automorphism of order m.If m > 2, assume Assumption 1.Let K ∞ /K denote the Γ-extension from Proposition 2.1 (resp.Proposition 2.2) if m = 2 (resp.m > 2).There exists a finite extension L/K and a Γ-extension L ∞ /L satisfying the following: (1) The extension L ∞ contains K ∞ .Furthermore, for all n ≥ 0, the number of primes ramifying in L n /K n is at least (2) We have [L n : Q] ≥ mℓ(ℓ − 1)dp n for all n ≥ 0.
Proof.We only prove the case m > 2; the proof of m = 2 is identical and left to the reader.By Proposition 2.2, infinitely many primes of K split completely in K ∞ /K.Let t ≥ 1 be an integer (to be chosen later) and primes v 1 , . . ., v t in K that split completely in K ∞ /K.
Proof of Claim.To see this, consider the ideal class for some α ∈ K. Then ord vi (α) ≥ 1 for all i = 1, . . .t.We can ensure that ord vi (α) is exactly one by dividing w through by v i if necessary.This proves the claim.Now let α ∈ K be such that ord vi (α) = 1 for all i = 1, . . ., t.Put L = K(α 1/ℓ , ζ ℓ ).Then L/K(ζ ℓ ) cyclic degree ℓ extension where v 1 , . . ., v t ramify.Put L ∞ = K ∞ L. Then L ∞ /L is a Γ-extension.We summarize this in a diagram: The primes v 1 , . . ., v t ramify in L/K.Furthermore, all the primes of K n lying over v 1 , . . ., v t must ramify in L n as well.Since each v i splits completely, there are tp n such primes of K n .Therefore, the number of primes of L n /K n that ramify is at least tp n .Now set t := N + mdℓ(ℓ − 1).

This proves Property (1).
To prove Property (2), we just count degrees in the above field diagram, noting that [L n : L] = dp n .This completes the proof.4. Growth of class groups in L ∞ /L: the proof of Theorem 1.8 We want to show that the ℓ-part of the class group in the Γ-extension L ∞ /L is unbounded.Our main tool to do this is the so-called ambiguous class number formula.Definition 4.1.Let ℓ be a prime.Let K be a number field and L/K be a cyclic Z/ℓZ-extension.
The subgroup of Cl(L) consisting of strongly ambiguous classes is denoted by Am st (L/K).
The following is given in [Kun20, Proposition 4.5].
Proposition 4.2 (Ambiguous Class Number Formula).Let ℓ be a prime.Let K be a number field and L/K be a cyclic Z/ℓZ-extension with σ a generator of the Galois group Gal(L/K).Then where T is the number of ramified primes in L/K.Proof of Theorem 1.8.Observe that for each n ≥ 1, the extension L n /K n is cyclic of degree ℓ.Applying Proposition 4.2 to L n /K n , we have the number of primes ramifying in L n /K n is at least We have [L n : Q] ≥ mℓ(ℓ − 1)dp n for all n ≥ 0.
By Property (2), we have Combining these, we obtain: for all n ≥ 0. This completes the proof of Theorem 1.8. 5. Application to fine Selmer groups: the proof of Theorems 1.6 5.1.Review of fine Selmer Group.Let F be a number field and p a prime.Let A be an abelian variety over F and let S be a finite set of primes containing S p ∪ S bad ∪ S ∞ .Denote by F S the maximal extension of F unramified outside S.
The usual p ∞ -Selmer group of A is defined by Here v runs through all the primes of F .The fine Selmer group of A is defined by the exact sequence 5.2.Proof of Theorem 1.6.Let F be a number field and S a finite set of places of F .The S-class group of F , denoted Cl S (F ), is the quotient of Cl(F ) by the subgroup generated by the ideal classes of prime ideals in S. The following proposition is proven in [LM16, Lemma 4.3].
Proposition 5.1.Let A be an dimensional Abelian variety over a number field F .Let S be a finite set of primes containing S ℓ ∪ S bad ∪ S ∞ .Suppose that A(F )[ℓ] = 0. Then where dim(A) denotes the dimension of the Abelian variety A.
We will first relate the S-class group of F to the class group of F .
Lemma 5.2.Let L be a number field and let ℓ be a rational prime.Let S be a finite set of places of L containing the primes above ℓ.Let s 0 be the number of finite primes in S. Then for all n: Proof.We have reproduced this proof from [Kun20, Lemma 4.6, Step A].Let s n be the number of finite primes of L n lying over a prime of S. Consider the following short exact sequence for all n [NSW13, Lemma 10.3.12],Z s0 → Cl(L n ) → Cl S (L n ).Taking ℓ-ranks of this sequence, we obtain [LM16, Lemma 3.2]: Proof of Theorem 1.6.Recall that we have two distinct primes ℓ = p and an abelian variety A defined over Q(ζ ℓ ).Let S = S p ∪ S bad ∪ S ∞ .Let s 0 be the number of finite places of S. We can apply Theorem 1.8 (replacing N with N + 2s 0 ) to construct a Γ-extension L ∞ /L such that r ℓ (Cl(L n )) ≥ (N + 2s 0 )p n for all n ≥ 0. Proposition 5.1 tells us that for all n ≥ 0: Example 2. We now look at Z d p extensions for d = 3.As in the previous example, let E = 11a1, ℓ = 5 and p = 3. Set N = 10.We will construct a Z Consider the CM field K = Q(ζ 9 ).Let K ∞ be the Z 3 3 -extension of K given in Proposition 2.1.We have t = N + 2dℓ(ℓ − 1) = 90.We want to find t = 90 primes v 1 , . . ., v t that are inert in K.By the well-known splitting laws for primes in cyclotomic fields, every rational prime which is ≡ 2, 5 modulo 9 is inert in K.Here is a list of 90 such primes: Let α be the product of these primes: Now put L = K(ζ 5 , 5 √ α).And let L ∞ = K ∞ L. Then L ∞ /L is a Z 2 3 -extension.Theorem 1.8 says that r 5 (Cl(L n )) ≥ 10 • 3 nfor all n ≥ 0. And Theorem 1.6 says thatr 5 (R 5 ∞ (E/L n )) ≥ 10 • 3 nfor all n ≥ 0.Example 3. We discuss a non-commutative example of Γ.The nilpotent uniform groups of dimension d = 3 are parametrized, up to isomorphism, by a parameter s ∈ N.They are given by (see [RZ00, Section 7, Theorem 7.4])Γ(s) = x, y, z : [x, z] = [y, z] = 1, [x, y] = z p s .The groups Γ(s) are non-abelian; they fit in the exact sequence1 → Z p → Γ(s) → Z 2 p → 1.If p ≡ 1 modulo 3, the group Γ(1) has an automorphism τ of order 3 which has no fixed points (see [HM19, Proposition 4.1]).Therefore m = 3.Put Γ = Γ(1).Let E = 19a1.Since E(Q)[3] = 0, let ℓ = 3, and p = 7.Let N = 6.We will construct a Γ-extension L ∞ /L such that r 5 (R 5 ∞ (E/L n )) ≥ 6 • 3 n for all n ≥ 0.