Journal de Théorie des Nombres de Bordeaux

Bill Allombert

A new interpretor for PARI/GP

Jean-François Jaulent

Sebastian Pauli

Michael E. Pohst

Florence Soriano–Gafiuk

Computation of 2-groups of positive classes of exceptional number fields

Tetsuya Taniguchi

Tetsuya TANIGUCHI
Prime factors of class number of cyclotomic fields
Tome 20, no 2 (2008), p. 525-530.
<http://jtnb.cedram.org/item?id=JTNB_2008__20_2_525_0>
© Université Bordeaux 1, 2008, tous droits réservés.
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bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
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org/legal/). Toute reproduction en tout ou partie cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
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Journal de Théorie des Nombres
de Bordeaux 20 (2008), 525-530
Prime factors of class number of cyclotomic fields
par Tetsuya TANIGUCHI
Résumé. Soit p un nombre premier impair, r une racine primitive
modulo p et ri ≡ ri (mod p) avec 1 ≤ ri ≤ p − 1. En 2007, R.
Queme a posé la question : le `-rang (` premier impair 6= p) du
groupe des classes d’idéaux du p-ième corps cyclotomique est-il
égal au degré du plus grand diviseur commun sur le corps fini F` de
x(p−1)/2+1 et du polynôme de Kummer f(x) =
∑p−2
i=0 r−ix
i. Dans
cet article, nous donnons une réponse complète à cette question
en produisant un contre-exemple.
Abstract. Let p be an odd prime, r be a primitive root modulo
p and ri ≡ ri (mod p) with 1 ≤ ri ≤ p − 1. In 2007, R. Queme
raised the question whether the `-rank (` an odd prime 6= p) of
the ideal class group of the p-th cyclotomic field is equal to the
degree of the greatest common divisor over the finite field F` of
x(p−1)/2 + 1 and Kummer’s polynomial f(x) =
∑p−2
i=0 r−ix
i. In
this paper, we shall give the complete answer for this question
enumerating a counter-example.
1. Introduction
Let ζn = e2πi/n (n ≥ 2) be a primitive n-th root of unity and Q(ζn) be
the cyclotomic field defined by ζn. Also, let hn be the ideal class number of
Q(ζn). It is well-known that hn can be written as hn = h−n h+n , where h−n and
h+n are the so-called relative and real class numbers of Q(ζn), respectively.
We put that p is an odd prime, N = p−12 , r is a primitive root modulo
p and ri is the least positive residue of ri (i ∈ Z) modulo p, i.e., ri ≡ ri
(mod p) with 1 ≤ ri ≤ p− 1.
In 1850, Kummer [3] established the following formula for h−p :
h−p =
1
(2p)N−1
∣∣∣∣ N∏
j=1
F (ζ2j−1p−1 )
∣∣∣∣,
where
F (x) = r0 + r−1x + r−2x2 + · · ·+ r−(p−2)xp−2.
Manuscrit reçu le 31 octobre 2007.
526 Tetsuya Taniguchi
This formula was slightly modified by Lehmer [4] and it was shown that
(1.1) h−p =
1
(2p)N−1
∣∣∣∣ N∏
j=1
G(ζ2j−1p−1 )
∣∣∣∣,
where G is the “monic” polynomial defined by
(1.2)
G(x) = xp−2F
(
1
x
)
= r−(p−2) + r−(p−3)x + · · ·+ r−1xp−3 + r0xp−2, r0 = 1.
In his papers , Queme investigated the structure of class group of Q(ζp)
and raised the following question.
Question (Queme, 2007): Let p be an odd prime and ` be an odd prime
with ` 6= p. Is the degree of the greatest common divisor over F` (the finite
field with ` elements) of F (x) and xN +1 equal to the `-rank of the `-Sylow
subgroup of relative class group Cl−Q(ζp) of Q(ζp)?
In the present paper, we shall give an answer to the above Question.
This will be done by inspecting the `-part of the class group of Q(ζp) with
an argument about resultants of certain polynomials and by producing
concretely counter-evidence using computer.
2. Some preliminaries on resultants
In this section, we deal with some basic properties of resultants as pre-
liminaries, which will be needed later in order to discuss a prime-divisibility
of h−p in the next section.
Let L be any field. For f ∈ L[x], we write ∂ f for the degree of f and
c(f) for the leading coefficient of f . Given two non-constant polynomials f
and g in L[x], we denote by RL(f, g) the resultant of f and g, that is, using
roots αi and βj in L (an algebraic closure of L) of f and g, respectively,
RL(f, g) = c(f)∂ gc(g)∂ f
∂ f∏
i=1
∂ g∏
j=1
(αi − βj).
Lemma 2.1 (cf. [5]). Let L be any field and f, f1, f2, g, g1, g2 be non-
constant polynomials in L[x]. Then we have
(i) RL(f1f2, g) = RL(f1, g) RL(f2, g),
(ii) RL(f, g1g2) = RL(f, g1) RL(f, g2),
(iii) if f = sg + t (s, t ∈ L[x], ∂ t < ∂ g), then
RL(f, g) =
{
c(g)∂ f−∂ t RL(t, g) if ∂ t > 0,
c(g)∂ f t∂ g if ∂ t = 0.
Prime factors of class number of cyclotomic fields 527
Let Φn (n ≥ 1) be the n-th cyclotomic polynomial, i.e.,
Φn(x) =
n−1∏
k=1
(k,n)=1
(
x− ζkn
)
.
Concerning the resultant of cyclotomic polynomials, we can state
Lemma 2.2 (cf. [1]). Let ϕ be the Euler totient function. Then we get
(i) if n > 1, then
RQ(Φ1,Φn) =
{
u if n = ua (∃u a prime, a ≥ 1),
1 otherwise,
(ii) if n > m > 1 and gcd(m,n) = 1, then RQ(Φm,Φn) = 1,
(iii) if n > m > 1 and gcd(m,n) > 1, then
RQ(Φm,Φn) =
{
uϕ(m) if m | n and nm = u
a (∃u a prime, a ≥ 1),
1 otherwise.
Lemma 2.3. Define two polynomials S and T in Q[x] by
(2.1) S(x) =
∏
d|N Φd(x)
Φ1(x)
=
xN − 1
x− 1
, T (x) =
∏
d|p−1 Φd(x)∏
d|N Φd(x)
= xN + 1.
Then it follows that RQ (S, T ) = 2N−1.
Proof. Using Lemma 2.1, we know
RQ (S, T ) = RQ
(∏
m
′
Φm,
∏
n
′′
Φn
)
=
∏
m
′ ∏
n
′′
RQ (Φm,Φn) ,
where the products
∏′
m and
∏′′
n are taken over all positive m and n such
that m | N for m ≥ 2 and n | p−1 with n - N , respectively. Hence, for each
fixed m there exists only one single n = 2em such that RQ(Φm,Φn) 6= 1,
where e = ord2(p − 1) − ord2 m. Here note that if gcd(m,n) = 1, then
RQ(Φm,Φn) = 1. Also, it is clear that if m | n and nm = u
a (∃u a prime,
a ≥ 1), then u = 2.
Using Lemma 2.2 and Möbius’ inversion formula, we can deduce
RQ (S, T ) =
∏
m
′ ∏
n
′′
2ϕ(m) =
∏
m
′
2ϕ(m) = 2
∑′
m
ϕ(m) = 2N−1,
where the sum
∑′
m runs over all m ≥ 2 such that m | N . 
528 Tetsuya Taniguchi
3. Divisibility properties of h−p
In this section, we will discuss the `-divisibility properties (` a prime, ` 6=
p), and as a consequence, we are able to answer to the Question introduced
in Section 1.
Let S, T be the polynomials as in (2.1) and put G0 = G/S for the
polynomial G in (1.2). Here we comment that G0 is a polynomial with
integer coefficients. Indeed,
(x− 1)G0
≡ r−(p−2) + r−(p−2)+(p−1)/2−1 − r0 − r−(p−2)+(p−1)/2
+
(p−3)/2∑
i=1
(
r−(p−2)+i−1 − r−(p−2)+i + r−(p−3)/2+i−1 − r−(p−3)/2+i
)
xi
≡ 0 (mod (x− 1)S),
hence G0 = G/S ∈ Z[x]. We shall state
Proposition 3.1. We get
(i) (2p)N−1h−p = |RQ(G, T )|,
(ii) pN−1h−p = |RQ(G0, T )|.
Proof. Let αi ∈ C, i = 1, . . . , p−2, be the roots of G. From (1.1), we obtain
h−p =
1
(2p)N−1
∣∣∣∣∣∣
N∏
j=1
p−2∏
i=1
(αi − ζ2j−1p−1 )
∣∣∣∣∣∣ = 1(2p)N−1 |RQ(G, T )| ,
as desired in (i). For (ii), we deduce from Lemmas 2.1 and 2.1 that, since
G = SG0,
(2p)N−1h−p = |RQ (G, T )| = |RQ (S, T ) RQ (G0, T )| = 2N−1 |RQ (G0, T )| .
This gives at once pN−1h−p = |RQ(G0, T )| and we are done. 
Here, we should remark that above (i) has been mentioned in Lehmer
[4].
Proposition 3.2. For a prime ` with ` 6= p, we have
(3.1) ` | h−p ⇔ ∂ gcd(G0, T ) ≥ 1 in F` .
Proof. From Proposition 3.1, we can see that
` | h−p ⇔ ` | RQ(G0, T ) ⇔ RF`(G0, T ) = 0,
which yields the proposition. 
We now quote the following one from [8, Lemma 16.15] for stating our
main Proposition 3.4 given below.
Prime factors of class number of cyclotomic fields 529
Proposition 3.3. Let ` be a prime and let L/K be an extension of number
fields of degree prime to `. Let AL and AK be the `-parts of the class groups
of L and K. Then, the natural map AK → AL is injective and
AL ' AK ⊕ (AL/AK).
The next proposition answers to the Question by Queme stated in Section
1.
Proposition 3.4. There exists a pair (p, `) of distinct primes p and ` such
that
rank` Cl−Q(ζp) > ∂ gcd(G, T ) in F`,
where rank` Cl−Q(ζp) means the `-rank of the `-Sylow subgroup of Cl
−
Q(ζp).
Proof. We will produce counter-evidence for the Question by finding a con-
crete pair (p, `) of primes p and ` with p 6= `. In fact, choosing particularly
(p, `) = (3299, 3), we know that
(3.2) ClQ(√−p) ' Z /`2 Z×Z /` Z .
The degree of Q(ζp) over Q(
√
−p) is equal to N = 1649 = 17 × 97, which
is prime to `. From Proposition 3.3, we obtain AQ(ζp) ⊇ Z /`2 Z×Z /` Z,
and hence rank` ClQ(ζp) ≥ 2. Further, since the class-group of Q(
√
−p) is in
the minus eigenspace with respect to complex conjugation acting on ideal
classes and so is its isomorphic image in the class group of Q(ζp), we may
conclude that rank` Cl−Q(ζp) ≥ 2.
On the one hand, we can know that, for the above pair (p, `),
(3.3) gcd(G0, T ) = x + 1 over F`,
and hereby gcd(G, T ) = x + 1 over F`. This implies the assertion immedi-
ately. 
Addendum. We verified the facts (3.2) and (3.3) by computer with aid
of the softwares “GP/PARI CALCULATOR Version 2.3.3 (released)” and
“Mathematica 6.0.1.0 for Linux x86 (32bit)”, respectively. For details, con-
sult with our on-line resources [7]. It should be noted that by several authors
(see e.g. [2, 6]) the isomorphism (3.2) has been also confirmed by computer
for the same pair (p, `) as indicated above.
Acknowledgments. The author thanks professor T.Agoh for his generous
help.
References
[1] Tom M. Apostol. Resultants of cyclotomic polynomials. Proc. Amer. Math. Soc., 24:457–462,
1970.
[2] H. Kisilevsky. Olga Taussky-Todd’s work in class field theory. Pacific J. Math., (Special
Issue):219–224, 1997.
530 Tetsuya Taniguchi
[3] Eduard Ernst Kummer. Bestimmung der Anzahl nicht äquivalenter Classen für die aus λ
ten Wurzeln der Einheit gebildeten complexen Zahlen und die idealen Factoren derselben. J.
Reine Angew. Math., 40:43–116, 1850.
[4] D. H. Lehmer. Prime factors of cyclotomic class numbers. Math. Comp., 31:599–607, 1977.
[5] M. Pohst and H. Zassenhaus. Algorithmic algebraic number theory, volume 30 of Encyclopedia
of Mathematics and its Applications. Cambridge Univ. Press, Cambridge, 1997.
[6] René Schoof. Minus class groups of the fields of the lth roots of unity. Math. Comp., 67:1225–
1245, 1998.
[7] Tetsuya Taniguchi. Program codes of “Prime factors of class number of cyclotomic fields”.
http://www.ma.noda.tus.ac.jp/g/tt/jtnb2008/.
[8] Lawrence C. Washington. Introduction to cyclotomic fields. Springer-Verlag, New York, 2nd
ed., 1997.
Tetsuya Taniguchi
Department of Mathematics,
Tokyo University of Science,
Noda, Chiba 278-8510, Japan
E-mail : taniguti_tetuya@ma.noda.tus.ac.jp
URL: http://www.ma.noda.tus.ac.jp/g/tt/


Prime factors of class number of cyclotomic fields

Denis Simon

A “class group” obstruction for the equation $Cy^d=F(x,z)$

Aaron Levin

Variations on a theme of Runge: effective determination of integral points on certain varieties

Keisuke Arai

On uniform lower bound of the Galois images associated to elliptic curves

Yahya Ould Hamidoune

Oriol Serra

Gilles Zémor

Yahya Ould HAMIDOUNE, Oriol SERRA et Gilles ZÉMOR
On some subgroup chains related to Kneser’s theorem
Tome 20, no 1 (2008), p. 125-130.
<http://jtnb.cedram.org/item?id=JTNB_2008__20_1_125_0>
© Université Bordeaux 1, 2008, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 20 (2008), 125-130
On some subgroup chains related to Kneser’s
theorem
par Yahya Ould HAMIDOUNE, Oriol SERRA et Gilles ZÉMOR
Résumé. Un résultat récent de Balandraud démontre que pour
toute partie S d’un groupe abélien G, il existe un sous-groupe
H non-trivial tel que l’inégalité |TS| ≤ |T | + |S| − 2 n’a lieu
que si H ⊂ Stab(TS). On remarque que le théorème de Kneser
n’implique que l’inégalité {1} 6= Stab(TS).
Ce renforcement du théorème de Kneser se déduit des proprié-
tés plaisantes d’un certain ensemble partiellement ordonné étudié
par Balandraud. Nous considérons un ensemble partiellement or-
donné analogue pour les groupes non forcément abéliens et à l’aide
d’outils classiques de théorie additive des nombres, généralisons
certains des résultats suscités. En particulier nous obtenons des
démonstrations courtes des résultats de Balandraud dans le cas
abélien.
Abstract. A recent result of Balandraud shows that for every
subset S of an abelian group G there exists a non trivial subgroup
H such that |TS| ≤ |T | + |S| − 2 holds only if H ⊂ Stab(TS).
Notice that Kneser’s Theorem only gives {1} 6= Stab(TS).
This strong form of Kneser’s theorem follows from some nice
properties of a certain poset investigated by Balandraud. We con-
sider an analogous poset for nonabelian groups and, by using
classical tools from Additive Number Theory, extend some of the
above results. In particular we obtain short proofs of Balandraud’s
results in the abelian case.
1. Introduction
In order to avoid switching from multiplicative to additive notation, all
groups will be written multiplicatively.
Kneser’s addition theorem [5] states that if S, T are finite subsets of an
abelian group G then |ST | ≤ |S|+ |T | − 2 holds only if ST is periodic (i.e,
there is a non trivial subgroup H such that HST = ST .) Kneser’s Theorem
is a fundamental tool in Additive number Theory. Proofs of this result may
be found in [4, 5, 6, 7, 8, 9, 11].
Manuscrit reçu le 28 mars 2007.
Supported by the Spanish research Council under project MTM2005-08990 and the Catalan
Research Council under project 2005SGR00253.
126 Yahya Ould Hamidoune, Oriol Serra, Gilles Zémor
In all previously known proofs of Kneser’s Theorem, the subgroup H
depends crucially on both sets S and T . With the goal of breaking this
double dependence in S and T , Balandraud investigated in recent work
[1, 2] the properties of a combinatorial poset that we now present.
Let S be a finite subset containing 1 of a group G and let X ⊂ G.
Following Lee [7], we introduce the following notion (where the reference
to S is implicit): let
X̃ = {x ∈ G : xS = XS}.
A subset X such that X̃ = X is called a full subset. Equivalently, a subset
X is a full subset if for all z 6∈ X we have zS 6⊂ XS.
This notion is rediscovered by Balandraud in [1, 2] where full subsets are
named “cells”, and also by Grynkiewicz [3] where the term “nonextendible
subset” is used. The following lemma is straightforward:
Lemma 1.1 ([1, 2, 7]). Let G be a group and 1 ∈ S ⊂ G be a finite subset.
Then X ⊂ X̃ and X̃S = XS.
We shall use a slightly modified version of Balandraud’s terminology:
A cell (or a full subset) X is called a u-cell if |XS| − |X| = u. A u-cell
with minimal cardinality is called a u-kernel (of S).
Throughout the paper, by a cell we always mean a cell of S.
Balandraud showed that, for a finite set S in an abelian group G, in the
poset of j–cells containing the unity ordered by inclusion with 1 ≤ j ≤
|S| − 2, the set of kernels forms a chain of subgroups. Moreover, if there
exists a u–cell, then there is a unique u–kernel containing the unit element
which is contained in all u–cells containing the unit element.
One of the consequences of this work is a new proof and the following
strengthening of Kneser’s Theorem:
Theorem 1.1 (Balandraud). For any non-empty finite subset S of an
abelian group G, there exists a finite subgroup H of G such that for any
finite subset T of G one of the following conditions hold :
• |TS| ≥ |T |+ |S| − 1
• HTS = TS and |TS| ≤ |HS|+ |HT | − |H|
As far as the authors are aware this is a surprising and strong formulation
that was not observed before and does not follow straightforwardly from
the classical forms of Kneser’s Theorem.
The purpose of the present note is to give a short proof that, in the poset
of j–cells that are subgroups ordered by inclusion with 0 ≤ j ≤ |S| − 1, the
set of kernels forms a chain of subgroups. Moreover, each u-kernel of this
poset is unique and contained in all u–cells of this poset. The proof works
for general, not necessarily abelian groups.
Subgroup chains and Kneser’s theorem 127
From this statement Kneser’s theorem allows one to deduce Balandraud’s
results for the abelian case, and in particular Theorem 1.1. Kneser’s The-
orem [5, 6] has several equivalent forms. We use the following one; see e.g
[4, 9]:
Theorem 1.2 (Kneser). Let G be an abelian group and X, Y ⊂ G be finite
subsets such that |XY | ≤ |X|+ |Y | − 2. Then
|XY | = |HX|+ |HY | − |H|,
where H = stab(XY ) = {x : xXY = XY }.
Our main tool is the following Theorem of Olson [10, Theorem 2]. We
give an equivalent formulation here where we use left–cosets instead of
right–cosets.
Theorem 1.3 (Olson [10]). Let X, Y be finite subsets of a group G, and
let H and K be subgroups such that HX = X, KY = Y and KX 6= X,
HY 6= Y . Then
|X \ Y |+ |Y \X| ≥ |H|+ |K| − 2|H ∩K|.
In particular either |X \ Y | ≥ |H| − |H ∩K| or |Y \X| ≥ |K| − |H ∩K|.
We shall use the following lemma.
Lemma 1.2 ([1, 2]). Let G be a group and 1 ∈ S ⊂ G be a finite subset.
Then the intersection of two cells M1,M2 of S is a cell of S.
Proof. Let x /∈ M1 ∩ M2. There is i with x /∈ Mi. Then xS 6⊂ MiS. Hence
xS 6⊂ (M1 ∩M2)S. 
We can now state our main result, namely Theorem 2.1 below.
2. An application of Olson’s Theorem
Balandraud [1, 2] proved that, in the abelian case, the set of kernels
containing the unit element and ordered by inclusion is a chain of subgroups.
In the non abelian case we can prove only that the set of kernels that are
subgroups forms a chain. The abelian case can then be easily recovered, since
Kneser’s Theorem implies (as we shall see below) that a kernel containing
the unit element is a subgroup.
Theorem 2.1. Let S be a finite subset containing 1 of a group G. Let M
be a u–kernel of S which is a subgroup. Let N be a subgroup which is a
v–cell and suppose u, v ≤ |S| − 1.
(i) If either N is a v–kernel or u = v then M ⊂ N or N ⊂ M .
(ii) If N is a v–kernel and v ≤ u then M ⊂ N .
128 Yahya Ould Hamidoune, Oriol Serra, Gilles Zémor
Proof. Suppose that M 6⊂ N and N 6⊂ M . Note that, since M is a cell, if
NMS = MS then NM = M , thus N ⊂ M against our assumption. Hence
we may assume NMS 6= MS and similarly MNS 6= NS. By Theorem 1.3
we have one of the two following cases.
Case 1: |MS| − |(MS) ∩ (NS)| = |(MS) \ (NS)| ≥ |M | − |M ∩ N |. It
follows that |(M∩N)S|−|M∩N | ≤ |(MS)∩(NS)|−|M∩N | ≤ |MS|−|M |.
On the other hand we have u = |MS| − |M | < |S| ≤ |(M ∩ N)S|. Since
|MS| − |M | is a multiple of |M ∩N | we have
u = |MS| − |M | = |(M ∩N)S| − |M ∩N |.
By Lemma 1.2, M∩N is a cell. Since M is a u–kernel, we have M∩N = M,
a contradiction.
Case 2: |NS| − |(NS) ∩ (MS)| = |(NS) \ (MS)| ≥ |N | − |N ∩ M |. It
follows that |(N∩M)S|−|N∩M | ≤ |(NS)∩(MS)|−|N∩M | ≤ |NS|−|N |.
On the other hand we have |NS|−|N | < |S| ≤ |(N∩M)S|. Since |NS|−|N |
is a multiple of |N ∩M | we have
(1) |NS| − |N | = |(N ∩M)S| − |N ∩M |.
Assume first u = v. Then u = |MS|− |M | = |NS|− |N | = |(N ∩M)S|−
|N ∩M |. Since M is a u–kernel, we have M ∩N = M, a contradiction.
Assume that N is a v–kernel. Then (1) implies N ∩ M = N, a contra-
diction. This proves (i).
Assume now that v ≤ u. Suppose M 6⊂ N . By (i) we have N ⊂ M , which
implies in particular that |MS| − |M | is a multiple of N . Therefore, from
u = |MS| − |M | < |S| ≤ |NS| we have u = |MS| − |M | ≤ |NS| − |N | = v
which gives u = v. But then M 6⊂ N and N ⊂ M imply |N | < |M |, and
since N is now a u–cell, this contradicts M being a u–kernel. 
We can now deduce Balandraud’s description for kernels and cells :
Corollary 2.1 (Balandraud [1, 2]). Let G be an abelian group and S ⊂ G
be a finite subset with 1 ∈ S. Let M be a u–kernel of S containing 1 with
1 ≤ u ≤ |S| − 2. Then,
(i) M is a subgroup.
(ii) Each u-cell is M–periodic.
(iii) Each v–kernel containing 1 with u < v ≤ |S|−2 is a proper subgroup
of M .
Proof. Let X be a u-cell with u ≤ |S| − 2. By Kneser’s Theorem, the
inequality |XS| − |X| = u ≤ |S| − 2 implies
(2) u = |XS| − |HX| = |HS| − |H|,
where H is the stabilizer of XS. Since X is a cell and HXS = XS, we
have X = HX. Note that, since G is abelian, ({y} ∪ H)S = HS implies
y ∈ Stab(HS) ⊂ Stab(XS), so that y ∈ H. This observation and (2) imply
Subgroup chains and Kneser’s theorem 129
that H is an u–cell. In particular, by taking X = M , the period K of MS
is a u–cell. Since KMS = MS and M is a u–cell, we have K ⊂ KM ⊂ M .
Since M is a u–kernel we have M = K. This proves (i).
Now let H be the stabilizer of XS, where X is a u–cell. As shown in the
preceding paragraph H is also a u–cell. By Theorem 2.1 we have M ⊂ H
and thus MH = H. Since X is a cell and HXS = XS, we have X = HX =
MHX. Hence X ⊂ MX ⊂ MHX = X implies X = MX. This proves (ii).
Finally, by (i), a v–kernel N is a subgroup. By Theorem 2.1 we have
N ⊂ M . 
From Corollary 2.1, one can deduce Theorem 1.1.
Proof of Theorem 1.1. We may assume without loss of generality that
1 ∈ S.
Case 1: There is no m–cell for any 1 ≤ m ≤ |S| − 2.
• either we have |TS| ≥ |S| + |T | − 1 for any non-empty finite T , in
which case the theorem clearly holds with H = {1}.
• or there exists some non-empty finite T such that |TS| ≤ |S| +
|T | − 2. Without loss of generality, we may also suppose 1 ∈ T . Put
X = T̃ . By Lemma 1.1 we have XS = TS. Now X is an m–cell
with m = |XS| − |X| ≤ |TS| − |T | ≤ |T | − 2. But since no such
cell exists for 1 ≤ m ≤ |S| − 2, we have that X = T . Then T
itself must be a cell (a 0-cell) i.e. |TS| = |T |. We therefore have
HT = TH = T = TS = HTS where H is the (necessarily finite)
subgroup generated by S. We have just proved that the theorem
holds in this case with H = 〈S〉.
Case 2: There exists an m–cell with 1 ≤ m ≤ |S| − 2. We may therefore
consider the largest integer u ≤ |S| − 2 for which S admits a u–cell. Let
H be the u–kernel containing 1. Note that u ≤ |S| − 2 implies that H is
different from {1}. Now let T be any finite non-empty subset such that
|TS| − |T | ≤ |S| − 2. We shall prove that HTS = TS.
Put X = T̃ . By Lemma 1.1, XS = TS. Note that we then have |XS| −
|X| ≤ |TS| − |T | ≤ |S| − 2, so that X is a v–cell for some v ≤ u. By
Corollary 2.1 (ii) we have TS = XS = MXS = MTS where M is the
v-kernel containing 1. By part (i) of Corollary 2.1, H is a subgroup of M
so that TS = XS = HTS as well.
Finally, |ST | ≤ |HS|+ |HT | − |H| follows from |ST | being a multiple of
|H|. 
130 Yahya Ould Hamidoune, Oriol Serra, Gilles Zémor
References
[1] E. Balandraud, Une variante de la méthode isopérimétrique de Hamidoune, appliquée au
théorème de Kneser. Annales de l’institut Fourier, to appear.
[2] E. Balandraud, Quelques résultats combinatoires en théorie additive des nombres. Thèse
de doctorat de l’Université de Bordeaux I, May 2006.
[3] D. Grynkiewicz, A step beyond Kemperman’s structure Theorem. Preprint Oct. 2007.
[4] J. H. B. Kemperman, On small sumsets in Abelian groups. Acta Math. 103 (1960), 66–88.
[5] M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen. Math. Zeit. 58
(1953), 459–484.
[6] M. Kneser, Summenmengen in lokalkompakten abelesche Gruppen. Math. Zeit. 66 (1956),
88–110.
[7] R. A. Lee, Proving Kneser’s theorem for finite groups by another e-transform. Proc. Amer.
Math. Soc. 44 (1974), 255–258.
[8] H. B. Mann, Addition Theorems. R.E. Krieger, New York, 1976.
[9] M. B. Nathanson, Additive Number Theory. Inverse problems and the geometry of sumsets.
Grad. Texts in Math. 165, Springer, 1996.
[10] J. E. Olson, On the symmetric difference of two sets in a group. European J. Combin. 7
(1986), 43–54.
[11] T. Tao, V. H. Vu, Additive Combinatorics. Cambridge Studies in Advanced Mathematics
105, Cambridge University Press, 2006.
Yahya Ould Hamidoune
Université Pierre et Marie Curie, Paris 6
Combinatoire et Optimisation - case 189
4 place Jussieu 75252 Paris Cedex 05, France
E-mail : yha@ccr.jussieu.fr
Oriol Serra
Universitat Politècnica de Catalunya
Matemàtica Aplicada IV
Campus Nord - Edif. C3
C. Jordi Girona, 1-3
08034 Barcelona, Spain.
E-mail : oserra@ma4.upc.edu
Gilles Zémor
Institut de Mathématiques de Bordeaux
Université de Bordeaux 1
351 cours de la Libération
33405 Talence, France.
E-mail : zemor@math.u-bordeaux1.fr


On some subgroup chains related to Kneser’s theorem

Scott Duke Kominers

Zachary Abel

Scott Duke KOMINERS et Zachary ABEL
Configurations of rank-40r extremal even unimodular lattices (r = 1, 2, 3)
Tome 20, no 2 (2008), p. 365-371.
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Journal de Théorie des Nombres
de Bordeaux 20 (2008), 365-371
Configurations of rank-40r extremal even
unimodular lattices (r = 1, 2, 3)
par Scott Duke KOMINERS et Zachary ABEL
Résumé. Nous montrons que, si L est un réseau unimodulaire
pair extrémal de rang 40r avec r = 1, 2, 3, alors L est engendré
par ses vecteurs de normes 4r et 4r + 2. Notre résultat est une
extension de celui d’Ozeki pour le cas r = 1.
Abstract. We show that if L is an extremal even unimodular
lattice of rank 40r with r = 1, 2, 3, then L is generated by its
vectors of norms 4r and 4r + 2. Our result is an extension of
Ozeki’s result for the case r = 1.
1. Introduction
A lattice of rank n is a free Z-module of rank n equipped with a positive-
definite inner product (·, ·) : L× L→ R. The dual of L, denoted L∗, is the
set
L∗ = {y ∈ L⊗ R : ∀x ∈ L, (x, y) ∈ Z} ,
which itself forms a lattice of the same rank as L. For a lattice vector
x ∈ L, we call (x, x) the norm of x. A lattice L is integral if (x, x′) ∈ Z
for all x, x′ ∈ L, i.e. if and only if L ⊆ L∗. An integral lattice is said to be
unimodular if it is self-dual (L = L∗).
A lattice L is called even if and only if every lattice vector has an even
integer norm, i.e. (x, x) ∈ 2Z for x ∈ L. An even lattice is automatically
integral by the familiar parallelogram identity, 2 (x, x′) = (x + x′, x + x′)−
(x, x)− (x′, x′).
Lattices that are simultaneously even and unimodular are especially rare.
Indeed, such a lattice’s rank must be divisible by 8. Sloane proved that if L
is an even unimodular lattice of rank n then the minimal (nonzero) norm
in L is bounded by
(1.1) min
x∈L
x 6=0
(x, x) ≤ 2bn/24c+ 2
Manuscrit reçu le 5 novembre 2007.
Mots clefs. Even unimodular lattices, extremal lattices, weighted theta series.
366 Scott Duke Kominers, Zachary Abel
(see [2, p. 194, Cor. 21]). An even unimodular lattice of rank n is called
extremal if it attains the bound (1.1).
Ozeki [6, 8] showed that if L is an extremal even unimodular lattice of
rank 32 or 48 then L is generated by its vectors of minimal norm. The first
author [5] showed analogous results for extremal even unimodular lattices of
ranks 56, 72, and 96. In a similar vein, Ozeki [7] showed that if L is extremal
even unimodular of rank 40, then L is generated by its vectors of norms 4
and 6. Here, we extend and slightly simplify Ozeki’s methods, recovering
Ozeki’s rank-40 result and obtaining analogous results for extremal even
unimodular lattices of ranks 80 and 120.
2. Modular forms and theta series
We will use the notationH = {z ∈ C : Im(z) > 0} for the upper half plane
of complex numbers. A modular form of weight k for the group PSL2(Z) is
a holomorphic function f : H → Z which is holomorphic at i∞ and satisfies
f
(
az + b
cz + d
)
= (cz + d)kf(z)
for all
(
a b
c d
)
∈ PSL2(Z). If a modular form f vanishes at z = i∞, it is
called a cusp form.
Let Mk and M0k be the C-vector spaces of modular forms and cusp forms
of weight k respectively. It is known that the Eisenstein series
E4(z) = 1 + 240e2πiz + 2160e4πiz + 6720e6πiz + · · · and
E6(z) = 1− 504e2πiz − 16632e4πiz − 122976e6πiz − · · · ,
which are modular forms of weights 4 and 6 respectively, freely generate
the spaces Mk in the sense that any nonzero modular form can be written
uniquely as a weighted homogeneous polynomial in E4 and E6. This implies
that dim(Mk) = 0 for k odd, negative, or k = 2; that dim(M2k) = 1 and
dim(M02k) = 0 for k = 0, 2 ≤ k ≤ 5 and k = 7; and that multiplication
by the weight-12 modular form ∆ = 12−3(E34 − E26) defines an isomor-
phism Mk−12
∼→M0k . More information on the theory of modular forms for
PSL2(Z) can be found in [9].
The theta function ΘL : H → Z associated to a lattice L is defined by
ΘL(z) =
∑
x∈L
eπi(x,x)z;
it is a generating function encoding the norms of L’s vectors. For a ho-
mogeneous harmonic polynomial P ∈ C[x1, . . . , xn], i.e. a homogeneous
polynomial for which
∑n
j=1
∂2P
∂x2j
≡ 0, we define the weighted theta
series ΘL,P by
ΘL,P (z) =
∑
x∈L
P (x)eπi(x,x)z.
Rank-40r extremal even unimodular lattices 367
As shown in [9, 3], if L is an even unimodular lattice of rank n then ΘL is a
modular form of weight n2 , and if in addition P is a homogeneous harmonic
polynomial of degree d, then ΘL,P is a modular form of weight n2 + d.
3. Main result
We denote by Pd,x0(x) the “zonal spherical harmonic polynomial” of
degree d, related to the Gegenbauer polynomial by
(3.1) Pd,x0(x) = Gd((x, x0) , ((x, x) (x0, x0))
1/2),
where Gd(·, ·) is the homogeneous polynomial of degree d such that Gd(t, 1)
is the Gegenbauer polynomial of degree d evaluated at t [1].
We let L be an extremal even unimodular lattice of rank 40r (where
r ∈ {1, 2, 3}), and adopt the notation used by Ozeki in [7]: For an even
unimodular lattice L, we denote by Λ2m(L) the set of vectors in L having
norm 2m. We denote by L2m(L) the sublattice of L generated by Λ2m(L),
and similarly denote by L2m1+2m2(L) the sublattice of L generated by
Λ2m1(L) ∪ Λ2m2(L).
We define a(2k, L) = |Λ2k(L)|. It is clear that the theta series ΘL is given
by ΘL(z) =
∑∞
k=0 a(2k, L)e
2kπiz. We note that
4r = 2b5r/3c+ 2 = min{2k > 0 : a(2k, L) 6= 0}
is the minimal norm of vectors in L and use the notation
Nj(x) = |{y ∈ Λ4r(L) : (x, y) = j}| ,
Mj(x) = |{y ∈ Λ4r+2(L) : (x, y) = j}| .
Using the involution y ←→ −y of Λm(L), we see that we have Nj(x) =
N−j(x) and Mj(x) = M−j(x) for any j ∈ R and x ∈ L⊗ R.
We will show the following configuration result, which directly extends
Ozeki’s [7] result for extremal even unimodular lattices of rank 40:
Theorem 3.1. For r = 1, 2, 3 and L extremal even unimodular of rank
40r, we have L = L4r+(4r+2)(L).
Proof. We partition L into its equivalence classes modulo L4r+(4r+2)(L).
We need only show that any class [x] ∈ L/L4r+(4r+2)(L) is represented by
a vector x0 ∈ [x] with (x0, x0) ≤ 4r + 2.
Now, we suppose there exists some equivalence class [x0] ∈
L/L4r+(4r+2)(L) where x0 6= 0 is a representative of minimal norm with
(x0, x0) = 2t for some t ≥ 2r + 2. We have the inequality
(3.2) |(x0, x)| ≤ 2r for all x ∈ Λ4r(L).
Indeed, if (x0,±x) > 2r, then L contains a vector x∓ x0 with norm
(x∓ x0, x∓ x0) = (x, x)∓ 2(x, x0) + (x0, x0) < (x0, x0),
368 Scott Duke Kominers, Zachary Abel
contradicting the minimality of x0.
Similarly, we have
(3.3) | (x0, x) | ≤ 2r + 1 for all x ∈ Λ4r+2(L).
From (3.2) and (3.3), we have the equations
∑
x∈Λ4r(L)
(x, x0)
2k =
2r∑
j=1
2 · j2k ·Nj(x0),(3.4)
∑
x∈Λ4r+2(L)
(x, x0)
2k =
2r+1∑
j=1
2 · j2k ·Mj(x0),(3.5)
for all k > 0.
We extract from the theta series ΘL of L the coefficients a(4r, L) and
a(4r + 2, L). We observe immediately from (3.4) and (3.5) that∑
x∈Λ4r(L)
(x, x0)
0 = a(4r, L),(3.6)
∑
x∈Λ4r+2(L)
(x, x0)
0 = a(4r + 2, L).(3.7)
Since L is even unimodular of rank 40r, we have ΘL,Pd,x0 ∈M
0
20r+d for any
d > 0. By comparing power-series coefficients, we then observe
ΘL,Pd,x0 ≡ 0 for d ∈ {2, . . . , 4r − 2, 4r + 2},(3.8)
ΘL,P4r,x0 ≡ c1∆
2r for a constant c1,(3.9)
ΘL,P4r+4,x0 ≡ c2E4∆
2r for a constant c2.(3.10)
From (3.8), we obtain the equations
(3.11) ∑
x∈Λ4r(L)
(x, x0)
2d = a(4r, L)
1 · 3 · · · (2d− 1)
40r · (40r + 2) · · · (40r + 2d− 2)
(8r)dtd
and
(3.12)
∑
x∈Λ4r+2(L)
(x, x0)
2d =
a(4r + 2, L)
1 · 3 · · · (2d− 1)
40r · (40r + 2) · · · (40r + 2d− 2)
(8r + 4)dtd,
Rank-40r extremal even unimodular lattices 369
for d ∈ {2, . . . , 4r − 2, 4r + 2}. We obtain from (3.9)
(3.13)
∑
x∈Λ4r+2(L)
P4r,x0(x) = c4r
∑
x∈Λ4r(L)
P4r,x0(x),
where ∆4r = e(4r)πiz + c4re(4r+1)πiz + O(e(4r+2)πiz). Similarly, (3.10) gives
(3.14)
∑
x∈Λ4r+2(L)
P4r+4,x0(x) = c4r+4
∑
x∈Λ4r(L)
P4r+4,x0(x),
where E4∆4r = e(4r)πiz + c4r+4e(4r+1)πiz + O(e(4r+2)πiz).
Combining the equations (3.6), (3.7), (3.11), (3.12), (3.13), and (3.14)
with (3.4) and (3.5), we obtain a system of 4r + 4 homogeneous linear
equations in the 4r + 3 unknowns
N0(x0), . . . , N2r(x0),M0(x0), . . . ,M2r+1(x0).
At this stage, we diverge from our natural generalization of Ozeki’s orig-
inal methods and obtain the (extended) determinants of these inhomoge-
neous linear systems; these determinants must vanish because the system
is overdetermined.
For r = 1, 2, 3, these determinants are respectively
255375874114131196233 · (t − 2) · t · (6t − 13) ·
(
10t2 − 55t + 77
)
,(3.15)
21323275167101161310234418436473 · (t − 4) · t · Q2(t),(3.16)
2244348526713117137176234311137159146111675713731 · (t − 6) · t · Q3(t),(3.17)
where Q2(t) is the irreducible quintic
10768t5 − 242280t4 + 2202310t3 − 10101795t2 + 23361877t− 21771246
and Q3(t) is the irreducible septic
19989882674056909935t7 − 892881426107875310430t6
+17258039601222654151533t5 − 187053310321121904306075t4
+1227398249908229181423784t3 − 4874010945909263810320032t2
+10840974078436271024624064t− 10414527769923133690990080.
In each case, there are no integer solutions t ≥ 2r + 2. However, we had
assumed the existence of an equivalence class
[x0] ∈ L/L4r+(4r+2)(L)
with minimal-norm representative x0 6= 0 having (x0, x0) = 2t for integral
t ≥ 2r + 2; since no such t exists, all equivalence classes must be generated
by vectors having norms 4r and 4r + 2. 
370 Scott Duke Kominers, Zachary Abel
4. Concluding remarks
A quick inspection will show that our results are the only possible imme-
diate extensions of Ozeki’s methods. In the cases r ≥ 4, it is not possible
to extract sufficiently many linear conditions by these exact techniques, as
the dimensions of the relevant spaces of cusp forms grow too large.
However, using different analysis, Elkies [4] has shown a stronger result
than our Theorem 3.1 in the r = 3 case: If L is an extremal even unimodular
lattice of rank 120 then L = L12(L). This result for rank-120 lattices is
analogous to Ozeki’s [6, 8] results in dimensions 32 and 48, and to the first
author’s [5] results in dimensions 56, 72, and 96.
Acknowledgements
The authors are especially grateful to Professor Noam D. Elkies, under
whose direction this research was conducted. They appreciate his instruc-
tion, support, and commentary on drafts of this article. The authors also
thank Denis Auroux, Katherine Paur, and Erin Schlumpf for their transla-
tion assistance and an anonymous referee for especially helpful comments
and advice.
References
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101 (2002), 151–158.
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Springer-Verlag, New York, 1999.
[3] W. Ebeling, Lattices and Codes (2nd edition). Vieweg, Germany, 2002.
[4] N. D. Elkies, On the quotient of an extremal Type II lattice of rank 40, 80, or 120 by the
span of its minimal vectors. Preprint.
[5] S. D. Kominers, Configurations of extremal even unimodular lattices. To appear, Int. J.
Num. Thy. (Preprint arXiv:0706.3082, 21 Jun 2007.)
[6] M. Ozeki, On even unimodular positive definite quadratic lattices of rank 32. Math. Z. 191
(1986), 283–291.
[7] M. Ozeki, On the structure of even unimodular extremal lattices of rank 40. Rocky Mtn. J.
Math. 19 (1989), 847–862.
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(1986), 247–287.
[9] J.-P. Serre, A Course in Arithmetic. Springer-Verlag, New York, 1973.
Rank-40r extremal even unimodular lattices 371
Scott Duke Kominers
Department of Mathematics
Harvard University
c/o 8520 Burning Tree Road
Bethesda, Maryland, 20817, USA
E-mail : kominers@fas.harvard.edu
URL: http://www.scottkom.com/
Zachary Abel
Department of Mathematics
Harvard University
c/o 17134 Earthwind Drive
Dallas, Texas, 75248, USA
E-mail : zabel@fas.harvard.edu
URL: http://people.fas.harvard.edu/~zabel/


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