Journal de Théorie des Nombres de Bordeaux

Claire Frechette

Mathilde Gerbelli-Gauthier

Alia Hamieh

Naomi Tanabe

Large Sums of Fourier Coefficients of Cusp Forms

Bhawesh Mishra

Paolo Santonastaso

Prime Power Residues and Blocking Sets

Sourabhashis Das

Wentang Kuo

Yu-Ru Liu

On the distribution of the number of distinct generators of $h$-free and $h$-full elements in an abelian monoid

Jonas Iskander

Hari R. Iyer

Jonas ISKANDER et Hari R. IYER
On the average size of 3-torsion in class groups of C2 ≀ H-extensions
Tome 37, no 3 (2025), p. 1031-1039.
https://doi.org/10.5802/jtnb.1350
© Les auteurs, 2025.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 4.0 FRANCE.
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Journal de Théorie des Nombres
de Bordeaux 37 (2025), 1031–1039
On the average size of 3-torsion in class groups of
C2 ≀ H-extensions
par Jonas ISKANDER et Hari R. IYER
Résumé. Les heuristiques de Cohen, Lenstra, et Martinet conduisent à conjec-
turer que la taille moyenne de la p-torsion dans les groupes de classes des
G-extensions d’un corps de nombres est finie. Dans un article de 2021, Lemke
Oliver, Wang et Wood ont démontré cette conjecture dans le cas p = 3 pour
les groupes de permutations G de la forme C2 ≀ H, pour une large famille de
groupes de permutations H, incluant la plupart des groupes nilpotents. Cepen-
dant, leur théorème ne s’applique pas à certains groupes nilpotents d’intérêt,
tels que H = C5. Nous étendons leurs résultats afin de montrer que la taille
moyenne de la 3-torsion dans les groupes de classes des C2 ≀ H-extensions est
finie pour tout groupe nilpotent H.
Abstract. The Cohen–Lenstra–Martinet heuristics lead one to conjecture
that the average size of the p-torsion in class groups of G-extensions of a
number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved
this conjecture in the case of p = 3 for permutation groups G of the form
C2 ≀ H for a broad family of permutation groups H, including most nilpotent
groups. However, their theorem does not apply for some nilpotent groups of
interest, such as H = C5. We extend their results to prove that the average
size of 3-torsion in class groups of C2 ≀ H-extensions is finite for any nilpotent
group H.
1. Introduction
For an extension of number fields K/k with degree n and Galois closure
K̃, let Gal(K/k) denote the Galois group of K̃/k acting as a transitive
permutation group on the n-element set of k-linear embeddings K ↪→ K̃.
Given a transitive permutation group G ⊆ Sn, a G-extension of k is a degree
n extension K/k in Q̄ equipped with an isomorphism Gal(K/k) ∼−→ G as
permutation groups; given a real number X > 1, we denote the set of all
G-extensions K with |Disc(K)| ≤ X by Ek(G, X). Heuristics of Cohen,
Lenstra, and Martinet [3, 4] suggest that the sizes hp(K) of the p-torsion
subgroups of the class groups Cl(K) satisfy the following conjecture.
Manuscrit reçu le 2 septembre 2024, révisé le 19 juillet 2025, accepté le 19 septembre 2025.
2020 Mathematics Subject Classification. 11R29.
Mots-clefs. Class groups, arithmetic statistics.
This work was funded by the Harvard College Research Program.
1032 Jonas Iskander, Hari R. Iyer
Conjecture 1.1. Let k be a number field, G a transitive permutation group,
and p a prime such that p ∤ |G|. Then there exists a constant ck,G,p > 0
such that
(1.1) lim
X→∞
1
|Ek(G, X)|
∑
K∈Ek(G,X)
hp(K) = ck,G,p.
As explained in [11, p. 588], the result [12, Theorem 6.1] shows that the
above is a consequence of the Cohen–Lenstra–Martinet heuristics. Daven-
port and Heilbronn [6] proved Conjecture 1.1 for the case where p = 3,
G = S2, and k = Q in 1971, and Datskovsky and Wright [5] generalized
their results to arbitrary base fields k in 1988. In 2005, Bhargava [2] also
proved (1.1) in the case where p = 2, G = S3, and k = Q.
Recent work of Lemke Oliver, Wang, and Wood [9] further generalizes [5]
by proving Conjecture 1.1 for a much broader class of groups G. Specifically,
they prove the conjecture for p = 3 and arbitrary k when G is a 2-group
containing a transposition, as well as when G = C2 ≀ H for H a transitive
permutation group satisfying |Ek(H, T )| ≪k,H,ϵ T
1
6 +ϵ for every ϵ > 0 and
T ≥ 1.1 In this paper, we further generalize their work by providing an
optimization of an argument from [9] which yields the following result.
Theorem 1.2. Let H ⊆ Sn be a transitive permutation group, and set
G := C2 ≀ H. Suppose that Ek(H, ∞) is nonempty and that there exists a
δ > 0 such that
|Ek(H, T )| ≪k,H T
1
2 −δ
for all T ≥ 1. Then there is a constant ck,G,3 such that
lim
X→∞
1
|Ek(G, X)|
∑
K∈Ek(G,X)
h3(K) = ck,G,3.
Explicitly, we have
(1.2) ck,G,3 =
 ∑
F ∈Ek(H,∞)
h3(F ) Ress=1 ζF (s)
ζF (2) Disc(F )2
·
(
1 + 2
r1(F )
3r1(F )+r2(F )
)
×
 ∑
F ∈Ek(H,∞)
Ress=1 ζF (s)
ζF (2) Disc(F )2
−1 ,
where ζF denotes the Dedekind zeta function for F and r1(F ) and r2(F )
denote the numbers of real and complex embeddings of F , respectively.
1The result of Lemke Oliver et. al. also holds under the weaker condition that∑
F ∈Ek(H,T )
h3(F ) ≪k,H,ϵ X
2
3 +ϵ for all ϵ > 0 and T ≥ 1.
On average 3-torsion 1033
In light of the proof of the weak version of Malle’s conjecture for nilpotent
groups by Alberts [1, Corollary 1.8] (which was also proven in Klüners–
Wang [8, Theorem 1.7] via a different method), we obtain the following
corollary.
Corollary 1.3. The conclusion of Theorem 1.2 holds when G = C2 ≀ H for
any transitive nilpotent permutation group H.
It is possible to specify exactly what transitive nilpotent permutation
groups H covered by this corollary are not covered by [9]. Denote by Hp
any transitive permutation p-group containing a p-cycle. Then, by applying
the classification proved in Corollary 2.4, the new nilpotent groups covered
by our result are groups of the form H5, C2 × H3, C3 × H2, and C5 × H2,
where these are written as direct products of transitive permutation groups.
Regarding non-nilpotent permutation groups, the smallest transitive per-
mutation group H ⊆ Sn for which the conclusion of Theorem 1.2 is not
currently known is H = S3, n = 3. Moreover, if we impose the condition
that 3 ∤ |H|, the smallest such group is H = D5, n = 5. These groups
could be interesting targets for future directions of study. In particular,
the bound on Ek(H, T ) implied by the weak version of Malle’s conjecture
for these groups is not strong enough to apply Theorem 1.2, so a proof of
Corollary 1.3 for H ∈ {S3, D5} would require further innovations.
2. Proof of Results
Throughout the paper, given expressions e1 and e2, we write e1 ≪x1,...,xn
e2 to say that there exists a constant C > 0 depending only on the vari-
ables x1, . . . , xn such that e1 ≤ Ce2 in the domain on which e1 and e2 are
defined. In addition, given an extension of number fields L/K, we write
Disc(L/K) for the norm of the relative discriminant ideal of L/K, and
given any number field L, we write Disc(L) for the absolute value of the
discriminant of L.
The lemma below provides a modified version of Theorem 8.1(2) from [9]
that allows us to prove Conjecture 1.2 for a larger class of groups G.
Lemma 2.1. Let H ⊆ Sn be a transitive permutation group and set G =
C2 ≀ H. If Ek(H, ∞) is nonempty and there exists δ > 0 such that∑
F ∈Ek(H,T )
h3(F )h2(F )
2
3 ≪k,H T 1−δ
for T ≥ 1, then we have that
lim
X→∞
1
Ek(G, X)
∑
K∈Ek(G,X)
h3(K) = ck,G,3,
where ck,G,3 is the constant given in (1.2).
1034 Jonas Iskander, Hari R. Iyer
Proof. By [9, Theorem 8.1], for any G-extension K/k, there exists a unique
index 2 subfield FK , which is in fact an H-extension of k. Moreover, by [9,
Corollary 3.2], for any number field F , X ≥ 1, and ϵ > 0, we have∑
[K:F ]=2
Disc(K/F )≤X
h3(K/F ) ≪[F :Q],ϵ Disc(F )1+ϵh2(F )2/3X.
Using the fact that h3(K) ≤ h3(K/F )h3(F ) and Disc(K/F ) = Disc(K)Disc(F )2 [10,
Corollary III.2.10], we may then write∑
[K:F ]=2
Disc(K)≤X
h3(K) ≤
∑
[K:F ]=2
Disc(K/F )≤ X
Disc(F )2
h3(F )h3(K/F )
≪[F :Q],ϵ
h3(F )h2(F )2/3X
Disc(F )1−ϵ .
Let X, Y ≥ 1 and 0 < ϵ < δ. We now wish to bound the quantity∑
K∈Ek(G,X),Disc(FK)≥Y
h3(K).
Again by the equality Disc(K) = Disc(FK)2 Disc(K/FK), we know that
if Disc(K) ≤ X, then Disc(FK) ≤ X1/2 and that K/FK is a quadratic
extension of the H-extension FK . Hence, the above sum can be bounded
above by summing h3(K) over all H-extensions F such that Y ≤ Disc(F ) ≤
X1/2 and furthermore summing over all quadratic extensions K of such F
(which certainly includes the sum over all the quadratic extension pairs of
the form K/FK with Disc(K) ≤ X and Disc(FK) ≥ Y ), i.e.∑
K∈Ek(G,X),Disc(FK)≥Y
h3(K) ≤
∑
F ∈Ek(H,X
1
2 )\Ek(H,Y )
∑
[K:F ]=2
Disc(K)≤X
h3(K).
Applying our previous bound for the inner sum yields the upper bound of
∑
K∈Ek(G,X),Disc(FK)≥Y
h3(K) ≪[k:Q],G,ϵ
∑
F ∈Ek(H,X
1
2 )\Ek(H,Y )
h3(F )h2(F )
2
3 X
Disc(F )1−ϵ .
Letting
A(T ) :=
∑
F ∈Ek(H,T )
h3(F )h2(F )
2
3 ≪k,H T 1−δ and ϕ(T ) :=
X
T 1−ϵ
,
On average 3-torsion 1035
we can then use Abel summation to write
∑
F ∈Ek(H,X
1
2 )\Ek(H,Y )
h3(F )h2(F )
2
3 X
Disc(F )1−ϵ
= A(X
1
2 )ϕ(X
1
2 ) − A(Y )ϕ(Y ) −
∫ X 12
Y
A(T )ϕ′(T )dT
≪k,H X
1−δ
2 · X
1+ϵ
2 +
∫ X 12
Y
T 1−δ · X
T 2−ϵ
dT
= X1−
δ
2 +
ϵ
2 +
[
X
(ϵ − δ)T δ−ϵ
]X 12
T =Y
≪ X1−
δ
2 +
ϵ
2 + X
Y δ−ϵ
.
Choosing ϵ = δ/2, we thus obtain∑
K∈Ek(G,X),Disc(FK)≥Y
h3(K) ≪k,G
X
Y δ/2
+ X1−
δ
4 .
This is analogous to the first displayed equation of the proof of [9, Theo-
rem 8.1(1)] with h3(K/FK) replaced by h3(K), which merely introduces a
factor of h3(F ) in the middle expression of the aforementioned displayed
equation.2 Furthermore, the exponent of Y in the denominator here is δ/2
instead of 13n[k:Q] −ϵ as written in [9], and we have an extra error term X
1− δ4
coming from the upper endpoint of the partial summation. However, just as
in the proof of [9, Theorem 8.1(2)], our inequality yields the desired analog
of the “tail estimate” [9, Theorem 5.1], since upon dividing by X (when
averaging over K with bounded discriminant Disc(K) ≤ X) the expression
on the right hand side vanishes in the large X and Y limits. To see why
dividing by X suffices if we wish to average over K in this case, we note
that |Ek(G, X)| is asymptotically linear in X by [7, Theorem 5.8] under
a mild growth condition on |Ek(H, T )|, which is certainly satisfied by our
assumption
∑
F ∈Ek(H,T ) h3(F )h2(F )
2
3 ≪k,H T 1−δ which implies sublinear
growth for |Ek(H, T )|.
Lastly, the rest of proof of [9, Theorem 8.1(2)] establishes a “soft ana-
log” of [9, Theorem 6.5] (i.e. dealing with the sum over G-extensions K such
that FK has small discriminant Disc(FK) ≤ Y ) and applies to G-extensions
for any transitive permutation group H, and is therefore unchanged in our
2The implied proof of [9, Theorem 8.1(2)] would also differ from the displayed equation in
this way.
1036 Jonas Iskander, Hari R. Iyer
situation. These modifications allow us to proceed as in the proof of [9, The-
orem 8.1(2)], and in this manner the theorem follows. The explicit formula
for ck,G,3 comes directly from [9, Section 6]. □
Proof of Theorem 1.2. Applying the substitution h6(F ) = h2(F )h3(F ) and
the trivial bound h(F ) ≪ϵ Disc(F )
1
2 +ϵ for any ϵ > 0, we may write∑
F ∈Ek(H,T )
h3(F )h2(F )
2
3 =
∑
F ∈Ek(H,T )
h3(F )
1
3 h6(F )
2
3
≪ϵ
∑
F ∈Ek(H,T )
Disc(F )
1
2 +ϵ.
From here, the bound |Ek(H, T )| ≪k,H T
1
2 −δ yields, for any ϵ > 0,∑
F ∈Ek(H,T )
h3(F )h2(F )
2
3 ≪ϵ,k,H T 1−δ+ϵ.
Choosing ϵ = δ/2 and applying Lemma 2.1 yields the theorem. □
To explore the implications of Theorem 1.2 for nilpotent groups, the
following lemma of [8], which decomposes transitive nilpotent permutation
groups into their p-Sylow subgroups, proves useful.
Lemma 2.2 ([8, Lemma 3.1]). A transitive nilpotent permutation group
G ⊆ Sn is permutation isomorphic to the natural direct product of transitive
permutation p-groups Gp ⊆ Snp ,
G ∼=
∏
p
Gp, with n =
∏
p
np,
where the Gp are isomorphic to the p-Sylow subgroups of G and np is the
maximal p-power dividing n.
Given a transitive permutation group H ⊆ Sn and an element h ∈ H, let
orb(h) denote the number of cycles (including fixed points) in the permu-
tation corresponding to h, and let a(H) := min{n − orb(h) : 1 ̸= h ∈ H}.
Alberts showed that when H is nilpotent, the quantity a(H) controls the
growth rate of Ek(H, T ) as a function of T . More precisely, he proved that
for all ϵ > 0 and T ≥ 1, we have |Ek(H, T )| ≪k,ϵ T
1
a(H) +ϵ [1, Corollary 1.8].
In light of this result, it is natural to ask whether nilpotent groups H achiev-
ing a particular value of a(H) admit a simple classification. The following
lemma is helpful for answering this question.
Lemma 2.3. Let H be a nontrivial transitive nilpotent permutation group,
and use Lemma 2.2 to write H ∼=
∏
p Hp for some transitive permutation
p-groups Hp ⊆ Snp. Then we have
a(H)
n
= min
p|n
(p − 1)mp
np
,
On average 3-torsion 1037
where mp is the smallest positive integer such that Hp contains a product
of mp p-cycles.
Proof. We will prove equality by showing that the left hand side is both less
than or equal to and greater than or equal to the right hand side. To see that
a(H) ≤ minp|n
(
(p−1) ·mp · nnp
)
, let p | n be a prime, and choose an element
hp ∈ Hp which is a product of mp p-cycles. Then letting H ′ :=
∏
q ̸=p Hq,
we see from the definition of a product of two permutation groups that
(hp, 1) ∈ Hp × H ′ ∼= H decomposes as a product of mp · |H ′| = mp · nnp
p-cycles. By the definition of a(H), it follows that a(H) ≤ (p − 1) · mp · nnp .
For the remaining inequality, adopt the notation |g| for the order of a
group element g ∈ G. Choose an element h ∈ H such that n − orb(h) =
a(H), let p be an arbitrary prime dividing |h|, and set h0 := h|h|/p. Then
h0 has order p. Consequently, if we set H ′ :=
∏
q ̸=p Hq as before and let
(hp, h′) ∈ Hp × H ′ be the element corresponding to h0 under the isomor-
phism Hp × H ′ ∼= H, we see that (h′)p = 1, hence that h′ = 1 because
p ∤ |H ′|. It follows in turn that hp has order p and hence is a product of
p-cycles. Letting m ≥ mp denote the number of p-cycles in hp, we conclude
that h0 is a product of m · |H ′| = m · nnp p-cycles, hence that
a(H) = n − orb(h) ≥ n − orb(h0) = (p − 1) · m ·
n
np
≥ (p − 1) · mp ·
n
np
. □
We can apply Lemma 2.3 to produce a classification of transitive nilpo-
tent groups H for some small values of a(H). For simplicity, we focus on the
case where H is not a 2-group, as the case of 2-groups is already covered
by Theorem 1.1 of [9].
Corollary 2.4. Let H be a nontrivial transitive nilpotent permutation
group, and set a := a(H). Consider the finite set of triples
Ta := {(p, m, n′) ∈ Z3≥1 : (p − 1) · m · n′ = a, p prime, p ∤ n′}.
Then there exists a triple (p, m, n′) ∈ Ta such that H ∼= Hp × H ′, where:
(i) Hp is a transitive permutation p-group containing a product of m
p-cycles; and
(ii) H ′ is a transitive nilpotent permutation group of order n′.
In particular, suppose H is not a 2-group. Then:
(1) We have a(H) ≥ 2.
(2) If a(H) = 2, then H is a 3-group containing a 3-cycle.
(3) If a(H) = 3, then H ∼= C3 × H2 where H2 is a 2-group containing
a transposition.
1038 Jonas Iskander, Hari R. Iyer
(4) If a(H) = 4, then one of the following holds:
(a) H = C2 × H3 where H3 is a 3-group containing a 3-cycle.
(b) H is a 3-group containing a product of two 3-cycles.
(c) H is a 5-group containing a 5-cycle.
(5) If a(H) = 5, then H = C5 × H2, where H2 is a 2-group containing
a transposition.
Proof. Adopting the notation from Lemma 2.3, we know that there exists
a prime p such that a(H) = (p − 1) · mp · nnp . This in turn allows us to write
H ∼= Hp × H ′, where Hp is a transitive permutation p-group containing a
product of mp p-cycles and H ′ is a transitive permutation group of order
n
np
, proving the first part of the corollary. The rest of the corollary follows
by noting that
T1 = {(2, 1, 1)}, T2 = {(2, 2, 1), (3, 1, 1)},
T3 = {(2, 3, 1), (2, 1, 3)}, T4 = {(2, 4, 1), (3, 1, 2), (3, 2, 1), (5, 1, 1)},
T5 = {(2, 5, 1), (2, 1, 5)}.
The first entry in each set corresponds to a case in which H is a 2-group
and hence is excluded. □
By applying Corollary 2.4 to the cases not covered directly by Theo-
rem 1.2, we obtain Corollary 1.3.
Proof of Corollary 1.3. If H is a 2-group, then G = C2 ≀ H is a transitive
permutation 2-group containing a transposition, so the result follows by [9,
Theorem 1.1] on the average size of 3-torsion in class groups of such ex-
tensions. Suppose that H is not a 2-group, in which case H necessarily
satisfies a(H) ≥ 2. If a(H) ≥ 3, then since H is a transitive nilpotent
permutation group, [1, Corollary 1.8] implies that for all number fields k,
ϵ > 0 and T ≥ 1, we have |Ek(H, T )| ≪k,ϵ T
1
a(H) +ϵ ≤ T
1
3 +ϵ, which satis-
fies the hypothesis of Theorem 1.2 of this paper and therefore implies the
result, i.e. the conclusion of Theorem 1.2. Lastly, if a(H) = 2, then Corol-
lary 2.4(i) implies that H is a 3-group. This case is covered by the remark
after Theorem 8.1 in [9]. □
Acknowledgments. The authors thank Melanie Wood and Michael Kural
for advising this project and giving many helpful suggestions, as well as
Jiuya Wang for providing comments on an earlier draft of the paper.
References
[1] B. Alberts, “The weak form of Malle’s conjecture and solvable groups”, Res. Number
Theory 6 (2020), no. 1, article no. 10 (23 pages).
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162 (2005), no. 2, p. 1031-1063.
On average 3-torsion 1039
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Jonas Iskander
Department of Mathematics, Harvard University
Cambridge, MA 02138, United States
E-mail: jonasiskander@college.harvard.edu
Hari R. Iyer
Department of Mathematics, Princeton University
Princeton, NJ 08540, United States
E-mail: hi9184@princeton.edu


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