THE p-ADIC LIMITS OF CLASS NUMBERS IN Zp-TOWERS JUN UEKI AND HYUGA YOSHIZAKI Abstract. This article discusses variants of Webers class number problem in the spirit

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THE p-ADIC LIMITS OF CLASS NUMBERS IN Zp-TOWERS

JUN UEKI AND HYUGA YOSHIZAKI

Abstract. This article discusses variants of Weber’s class number problem in the spirit

of arithmetic topology to connect the results of Sinnott–Kisilevsky and Kionke. Let pbe

a prime number. We ﬁrst prove the p-adic convergence of class numbers in a Zp-extension

of a global ﬁeld and a similar result in a Zp-cover of a compact 3-manifold. Secondly, we

establish an explicit formula for the p-adic limit of the p-power-th cyclic resultants of a

polynomial using roots of unity of orders prime to p, the p-adic logarithm, and the Iwasawa

invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic

curves; we complete the list of the cases with p-adic limits being in Zand ﬁnd the cases

such that the base p-class numbers are small and ν’s are arbitrarily large.

Contents

1. Introduction 2

Acknowledgments 4

2. Global ﬁelds 4

3. 3-manifolds 6

4. Alternative proofs 7

5. Cyclic resultants 9

5.1. Signatures 9

5.2. p-adic convergence 9

5.3. Explicit formula 10

6. Knots 13

6.1. Alexander polynomial and Fox–Weber’s formula 13

6.2. Torus knots 14

6.3. Twist knots 15

6.3.1. Observations for K= 4115

6.3.2. Cases with lim |H1(Xpn)tor| ∈ Z16

6.3.3. Can νbe large with rebeing small? 18

6.4. Livingston’s results 20

7. Algebraic curves 21

7.1. A formula for function ﬁelds 21

7.2. Elliptic curves 22

7.2.1. Observations 22

7.2.2. Cases with lim |Cl0(kElpn)| ∈ Z23

7.2.3. Can νbe large with rebeing small? 25

References 27

2020 Mathematics Subject Classiﬁcation. Primary 57K10, 11R23, 11R29; Secondary 11G20, 57M10,

11S05 .

Key words and phrases. Weber’s class number problem, number ﬁeld, knot, elliptic curve, p-adic torsion,

Iwasawa ν-invariant, Iwasawa theory, arithmetic topology.

arXiv:2210.06182v9 [math.NT] 15 Jan 2025

1. Introduction

Since the dawn of algebraic number theory, the quest for number ﬁelds with trivial class

numbers has been of great interest. In the early 19th century, C. F. Gauss studied quadratic

number ﬁelds and asked whether there exist inﬁnitely many real quadratic ﬁelds with trivial

class numbers, which is still an open question. Also in the 19th century, H.Weber studied the

class numbers of cyclotomic Z/2nZ-extensions of Q(cf. [Web86]). So-called Weber’s class

number problem asks whether the class numbers of cyclotomic Z/pnZ-extensions are trivial

for arbitrary prime number p. After many years of eﬀorts, this problem is expected to be

aﬃrmative, nevertheless the assertion is veriﬁed only when (p, n) is (2,≤6),(3,≤3),(5,≤2),

and when 7 ≤p≤19, n= 1 at this moment (cf. [FKM14, Gra21]).

The second author recently approached Weber’s problem using continued fraction expan-

sions and pointed out that the sequence of the class numbers in the cyclotomic Z2-extension

of Qconverges in the ring of 2-adic integers Z2[Yos23, Theorem 5.3]. W. Sinnott also an-

nounced in 1985 a similar result for a cyclotomic Zp-extension of a CM ﬁeld and for the

“minus” class numbers, and Sang-G. Han established an explicit formula [Han91, Theorem

4] by using an analytic argument. Their results are speciﬁc cases of H. Kisilevsky’s theorem

over any global ﬁeld, that is, a ﬁnite extension of Qor Fp′(x), p′being a prime number:

Theorem A (Theorem 2.1, [Kis97, Corollary 2]).Let kp∞be a Zp-extension of a global ﬁeld

k. Then, the sizes of the class groups C(kpn), those of the non-p-subgroups C(kpn)non-p, and

those of the l-torsion subgroups C(kpn)(l)for each prime number lconverge in Zp.

The growth of p-torsions has been extensively studied in the context of Iwasawa theory.

This theorem deﬁnes a numerical invariant, say, the p-adic class number limn→∞ |C(kpn)non-p|

of a Zp-extension with any p-torsion growth.

It is said that Gauss’s proof of the quadratic reciprocity law using Gauss sums is based

on his insight into the analogy between knots and prime numbers. In addition, the analogy

between the Alexander–Fox theory for Z-covers and the Iwasawa theory for Zp-extensions

has played an important role since the 1960s (cf. [Maz64, Mor12]). A p-adic reﬁnement of

Alexander–Fox’s theory is of its self-interests, as well as applies to the study of proﬁnite

rigidity (cf. [Uek18, Uek21c, Liu23]). In this view, we establish the Iwasawa-type formula

and an analogue of Theorem 2.1 for unbranched Zp-cover of 3-manifolds.

Theorem B. Let (Xpn→X)nbe a Zp-cover of a compact connected 3-manifold X. Then,

(1) (Theorem 3.2). There exist some λ, µ ∈Z≥0and ν∈Zsuch that for every n≫0, the

size of the p-torsion subgroup satisﬁes

|H1(Xpn)(p)|=pλn+µpm+ν.

(2) (Theorem 3.1). The sizes of the torsion subgroups H1(Xpn)tor, those of the non-ptorsion

subgroups H1(Xpn)non-p, and those of the l-torsion subgroups H1(Xpn)(l)for each prime

number l, of the 1st homology groups converge in Zp.

By S.Kionke’s theorem [Kio20, Theorem 1.1 (ii)] and the Poincar´e duality, the p-adic limit

value limn→∞ |H1(Xpn)non-p|coincides with Kionke’s p-adic torsion. In Sections 2 and 3, we

stick to the homological argument and give proofs to these theorems in a parallel manner.

Afterward, in Section 4, we state a general proposition and discuss alternative proofs.

In several contexts, the size of the n-th layer is calculated by the n-th cyclic resultant

Res(tn−1, f(t)) = Qζn=1 f(ζ) of a certain polynomial 0 ̸=f(t)∈Z[t]. In order to pursue

numerical studies, we establish the following theorems on the p-adic limits of cyclic resultants,

which are detailed versions of [Kis97, Proposition 2]. In the proof, we invoke an elementary

p-adic number theory and the class ﬁeld theory with modulus. Let Cpdenote the p-adic

completion of an algebraic closure of the p-adic numbers Qpand ﬁx an embedding Q→Cp.

Theorem C (Theorem 5.3).Let 0 ̸=f(t)∈Z[t]. Then, the p-power-th cyclic resultants

Res(tpn−1, f(t)) converge in Zp. The limit value is zero if and only if p|f(1). In any

case, if Res(tpn−1, f(t)) ̸= 0 for any n, then the non-p-parts of Res(tpn−1, f(t)) converge

to a non-zero value in Zp. For each prime number l, similar assertions for the l-parts of

Res(tpn−1, f(t)) hold.

Theorem D (Theorem 5.7, a short version).Suppose p∤f(t). Write f(t) = a0Qi(t−αi)

in Q[t] and note that |a0Q|αi|p>1αi|p= 1. Let ξand ζidenote the unique roots of unity

|αi|p= 1. Then

lim

n→∞ Res(tpn−1, f(t)) = (−1)pdegf+#{i||αi|p<1}ξY

i;|αi|p=1

(ζi−1)

holds in Zp. In addition, the non-ppart of Res(tpn−1, f(t)) converges to

(−1)pdegf+#{i||αi|p<1}ξY

i;|αi|p=1,

|αi−1|p=1

(ζi−1)p−νY

i;|αi|p=1,

|αi−1|p<1

log αi

in Zp, where log denotes the p-adic logarithm and ν∈Z∪{∞} is Iwasawa’s invariant deﬁned

by p−ν=Qi;|αi−1|p<1|log αi|p. If all αi’s with |αi−1|p<1 are suﬃciently close to 1, then

pν=|f(1)|−1

pholds.

In the cases of Zp-covers of knots, Fox–Weber’s formula asserts that the cyclic resultants of

the Alexander polynomials coincide with the sizes of torsion subgroups of the 1st homology

groups. We calculate the p-adic limits of |H1(Xpn)tor|for the Zp-covers of torus knots Ta,b

and twist knots J(2,2m) to establish Propositions 6.3, 6.8, 6.10, completing the table of

the cases with the p-adic limits being in Z.Moreover, we give a systematic study of the

Iwasawa ν-invariants and answer the following question (Propositions 6.12, 6.17): Find

Zp-covers (Xepn→Xe)nwith e∈Z>0of twist knots J(2,2n)such that the base p-class

numbers |H1(Xe)(p)|are small and ν’s are arbitrary large. In Subsection 6.4, we discuss

several possible analogues of Weber’s problem for knots; we remark Livingston’s results in

[Liv02] and point out further problems in view of the Sato–Tate conjecture.

In the cases of constant Zp-extensions of function ﬁelds, the cyclic resultants of the Frobe-

nius polynomials coincide with the sizes of the degree zero divisor class groups. In Section

7, we recollect basic facts of function ﬁelds, state an analogue of Fox–Weber’s formula for

constant extensions of function ﬁelds (Proposition 7.2), and study elliptic curves over ﬁnite

ﬁelds. We point out conditions for the p-adic limit value being 0 and 1 using the notions

of supersingular primes and anomalous primes, as well as complete the list of the cases with

the p-adic limits being in Z(Proposition 7.8, 7.10). We also give a systematic study of

the Iwasawa ν-invariant and answer the following question (Propositions 7.12, 7.13): Find

constant Zp-extensions (kepn/ke)nwith e∈Z>0of the function ﬁeld of elliptic curves over

Flsuch that the base p-class numbers |Cl0(ke)(p)|are small and ν’s are arbitrarily large.

Note that we have intentionally kept our materials to the very basic, such as torus knots,

twist knots, and elliptic curves, to raise questions in a broad scope. Our results in this article

were initially announced by the authors at several conferences in 2021–2022. This article

contains a detailed revisiting of Kisilevsky’s short article [Kis97]. Our numerical study of the

p-adic limits gives explicit examples of Kionke’s p-adic torsions introduced in [Kio20]. Recent

related works are due to A. Gothandaraman [G.23] and M. Ozaki [Oza22] (See Remarks 5.4

and 4.2). In addition, C. Deninger points out that there would exist a common generalization

of our work and his [Den20].

Acknowledgments. The authors would like to express their sincere gratitude to Cristopher

Deninger, Yoshinosuke Hirakawa, Teruhisa Kadokami, Tomokazu Kashio, Hershy Kisilevsky,

Satoshi Kumabe, Moemi Matsumoto, Daichi Matsuzuki, Tomoki Mihara, Yasushi Mizusawa,

Manabu Ozaki, Makoto Sakuma, Shin-ichiro Seki, Jordan Schettler, and the anonymous

referee of the journal for useful information and fruitful conversations. We dedicate this

article to Toshie Takata. The ﬁrst and second authors have been partially supported by

JSPS KAKENHI Grant Number JP19K14538 and 22J10004 respectively.

2. Global fields

A number ﬁeld is a ﬁnite extension of Q. A function ﬁeld is a ﬁnite extension of the

rational function ﬁeld Fp′(x) of one variable over a ﬁnite ﬁeld Fp′,p′being a prime number.

A global ﬁeld is a number ﬁeld or a function ﬁeld. For a global ﬁeld k, let C(k) denote

the ideal class group Cl(k) if kis a number ﬁeld, and the degree-zero divisor class group

Cl0(k) if kis a function ﬁeld. Note that C(k) is always a ﬁnite group. We regard C(k) as

a multiplicative group. For any ﬁnite abelian group Gand a prime number l, let G(l)and

Gnon-pdenote the l-torsion subgroup and non-ptorsion subgroup of Grespectively. The size

of a ﬁnite set Xis written as |X|. A Zp-extension kp∞of a global ﬁeld kis a direct system

(kpn)nof Z/pnZ-extensions or its union Snkpn. The following theorem was initially proved

by Kisilevsky [Kis97, Corollary 2]. We note that although Kisilevsky’s proof is short and

clear, we here give our original proof with a purpose.

Theorem 2.1. Let kp∞be a Zp-extension of a global ﬁeld k. Then, the sizes of the class

groups C(kpn), those of the non-p-subgroups C(kpn)non-p, and those of the l-torsion subgroups

C(kpn)(l)for each prime number lconverge in Zp.

Proof. It is well-known (see Remark 2.2 below) that for any n≫0, the class ﬁeld theory

or it converges to 0 in Zp. Thus, it suﬃces to prove for each prime number l̸=pand n∈Z>0

the congruence formula of relative class numbers

|C(kpn)(l)|/|C(kpn−1)(l)| ≡ 1 mod pn.(2.1)

Deﬁne the relative norm map Nr : C(kpn)→C(kpn−1):[a]7→ Qp−1

i=0 aτi, where τis a generator

of Gal(kpn/kpn−1)∼

=Z/pZ.

The map Nr : C(kpn)(l)→C(kpn−1)(l)on the l-parts is surjective. Indeed, there is a natural

homomorphism ι: C(kpn−1)(l)→C(kpn)(l)and the composition map Nr ◦ι: C(kpn−1)(l)→

C(kpn−1)(l)is given by x7→ xp. Since l̸=p, this map Nr ◦ιis an isomorphism and hence Nr

is surjective.

(Ker Nr)(l)to obtain the assertion. Put G= Gal(kpn/k)∼

=Z/pnZand let σbe a generator of

G. For each [a]∈(Ker Nr)(l),let G[a] denote the G-orbit of [a]. If [a]̸= 1, then |G[a]|=pn.

Indeed, suppose that |G[a]|< pn. Then |G[a]|divides pn−1and we have that [a] = [aσpn−1].

Note that σpn−1generates the group pn−1Z/pnZ∼

=Gal(kpn/kpn−1)< G and put τ=σpn−1.

Since [a]∈(Ker Nr)(l), we have [a]p= [Qp−1

i=0 aτi] = 1. Since l̸=p, we obtain [a] = 1.

Now the G-orbital decomposition yields that Ker Nr(l)≡1 mod pn, hence the claimed

formula (∗). Therefore, both (|C(kpn)(l)|)nand (|C(kpn)non-p|)nare p-adic Cauchy sequences

and converge in the completed ring Zp, and so does (|C(kpn)|)n.□

Remark 2.2. The following well-known argument completes the ﬁrst paragraph of the proof.

(1) For each number ﬁeld k, let e

kdenote the Hilbert class ﬁeld, that is, the maximal

unramiﬁed abelian extension of k. Then the class ﬁeld theory asserts that Cl(k)∼

=Gal(e

k/k).

If k′/k is a ramiﬁed extension of degree p, then we have e

k∩k′=kand that e

kk′/k′is an

unramiﬁed extension of degree |Cl(k)|, and hence |Cl(k)|divides |Cl(k′)|= deg( e

k′/k).

If kp∞/k is a Zp-extension, then the inertia group of a ramiﬁed prime is an open subgroup

of Zp= Gal(kp∞/k), and hence kpn/kpn−1is a ramiﬁed p-extension for any n≫0.

(2) For a function ﬁeld k, let e

kdenote the maximal unramiﬁed abelian extension of k. Let

Fp′denote the algebraic closure of Fp′, so that we have Gal(Fp′/Fp′)∼

Z= lim

←−rZ/rZ. Then

an analogue of the class ﬁeld theory assets that Cl0(k)∼

=Gal(e

k/kFp′).

(i) If k′/k is a constant extension of degree p, then by k′Fp′=kFp′,e

k/kFp′is a subextension

of e

k′/k′Fp′, and hence |Cl(k)|divides |Cl(k′)|.

(ii) If k′/k is a geometric ramiﬁed extension of degree p, then a similar argument to (1)

yields that |Cl(k)|divides |Cl(k′)|.

For a Zp-extension kp∞/k of a function ﬁeld, kpn/kpn−1is a constant extension for all

n∈Z>0and (i) applies, or kpn/kpn−1is a geometric ramiﬁed extension for all n≫0 and (ii)

applies. In the latter case, we always have p=p′.

Remark 2.3. In a view of the analogy between number ﬁelds and function ﬁelds, Iwasawa

pointed out so-called Iwasawa’s class number formula (cf. [Iwa59],[Was97, Section 7.2]),

which asserts that if kp∞is a Zp-extension of a number ﬁeld k, then there exists some

λ, µ, ν ∈Z≥0such that for any n≫0,

|C(kpn)(p)|=pλn+µpn+ν

holds. A similar formula with µ= 0 holds for a constant Zp-extension of a function ﬁeld

[Ros02, Theorem 11.5] and λis related to the genus of an algebraic curve in several senses.

In many literature of number theory, the suﬃx is shifted as k′

n=kpn. Note that λ′n+

µ′pn+ν′=λ(n−1) + µpn−1+νimplies µ=pµ′,λ=λ′,ν=ν′+λ.

Gold–Kisilevsky [GK88] pointed out that in a geometric Zp-extension the p-parts can grow

arbitrarily fast. Even in such a case, Theorem 2.1 persists.

Remark 2.4. Let l̸=pbe a prime number. Washington [Was78] proved that in a cyclotomic

Zp-extension of a number ﬁeld abelian over k, for each prime number l̸=p, the l-part of the

class numbers are bounded, and hence the sequence is constant for n≫0. The assertion on

the l-part in our Theorem 2.1 is a weak generalization of Washington’s one.

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THEp-ADICLIMITSOFCLASSNUMBERSINZp-TOWERSJUNUEKIANDHYUGAYOSHIZAKIAbstract.ThisarticlediscussesvariantsofWeber’sclassnumberprobleminthespiritofarithmetictopologytoconnecttheresultsofSinnott–KisilevskyandKionke.Letpbeaprimenumber.Wefirstprovethep-adicconvergenceofclassnumbersinaZp-extensionofaglobalfie...

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