Tours de corps de classes de Hilbert pour les corps globaux et ...

UNIVERSITÉ DE LA MÉDITERRANÉE
AIX-MARSEILLE II
Faculté des Sciences de Luminy
THÈSE
pour obtenir le grade de
DOCTEUR DE L’UNIVERSITÉ DE LA MÉDITERRANÉE
Discipline : Mathématiques
présentée et soutenue publiquement
par
Alexandre TEMKINE
le 15 Décembre 2000
Titre
Tours de corps de classes de Hilbert
pour les corps globaux et applications
Directeur de Thèse : Gilles LACHAUD
JURY
M. Lachaud Gilles Directeur de Recherches, I.M.L.
M. Perret Marc Maître de conférences, Université de Lyon
M. Quebbemann Heinz-Georg Professeur, Université d’ Oldenburg Rapporteur
M. Schoof René Professeur, Université de Rome Rapporteur
M. Tsfasman Mikhaïl Directeur de Recherches, I.P.P.I.
Professeur Associé, Université de la Méditerranée
M. Vlădut¸ Serge Directeur de Recherches, I.M.L.

Remerciements
Ma reconnaissance va tout d’abord à mon directeur de thèse, Gilles Lachaud,
dont l’étendue des connaissances et la disponibilité m’ont permis de
mener à bien ce travail.
Je remercie René Schoof et Heinz-Georg Quebbemann d’avoir accepté la
lourde tâche de rapporteur. Merci à Mikhaïl Tsfasman pour m’avoir invité
durant un mois à l’Université indépendante de Moscou et merci à Marc Perret
pour son soutien moral tout au long de cette thèse, quand le besoin s’en faisait
sentir. Je les remercie tous deux, ainsi que Serge Vlădut¸ pour avoir accepté
de faire partie du jury.
Parmi les contacts scientifiques et humains qui ont marqué mon parcours
mathématique avant et pendant cette thèse, je veux citer Philippe Espéret,
Yves Laszlo et Christian Maire, et leur exprimer ma vive reconnaissance.
Merci aussi à Mireille Car et à Jean-Pierre Serre, à l’ensemble des membres
de l’équipe ATI et plus généralement à tous les membres de l’Institut de
Mathématiques de Luminy.
Enfin, hors de l’univers mathématique, mes soutiens ont été innombrables
et tout aussi importants. Je citerai seulement, pêle-mêle, Béa, plusieurs David,
Irène, Jérôme, Corinne, mes parents Jacques et Nadine, mes frères et
soeur Julie, Pierre, Stéphane et Marc, Cédric, Tristan, Christophe, plusieurs
Sophie, Cyril, Dorothée, Manu, Sofien,... .

Table des matières
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
Hilbert class field towers of function fields over finite fields . . . . . . 2
and lower bounds for A(q)
Hilbert class field towers of function fields over finite fields . . . . .21
and lower bounds for A(q), II
Asymptotically good families of unimodular lattices . . . . . . . . . . . . .31
On the splitting rate of places in infinite unramified towers . . . . 41
of number fields

Introduction
Cette thèse est le réunion de quatre articles, chacun d’eux faisant l’objet
d’un chapitre. Au centre de ces quatre articles se trouve la notion de tour de
corps de classes de Hilbert. Etant donné un corps global, c’est-à-dire soit un
corps de nombres, soit un corps de fonctions à une variable sur un corps fini,
et un ensemble fini de places contenant les places archimédiennes dans le cas
des corps de nombres, on peut lui associer par la théorie du corps de classes
son corps de classes de Hilbert, c’est-à-dire son extension abélienne maximale
non ramifiée où les places données sont totalement décomposées. Par
itération, on définit alors une suite croissante de corps globaux, appelée tour
de corps de classes de Hilbert. Sous une condition donnée par le Théorème
de Golod-Shafarevich, cette suite est strictement croissante.
Ce résultat connu depuis le milieu des années soixante a plusieurs applications
: dans le cas des corps de fonctions, il permet d’obtenir des bornes
inférieures de A(q), quantité qui mesure le nombre asymptotique maximal de
points rationnels d’une courbe algébrique projective lisse absolument irréductible
définie sur un corps fini. De telles bornes ont été données notament par
Serre, Perret, Xing et Niederreiter et Xing. L’objet des deux premiers articles
est de donner des améliorations de ces bornes. Le premier article doit paraître
au Journal of Number Theory, le deuxième n’a pas été soumis.
Dans le troisième article, nous utilisons le Théorème de Golod-Shafarevich
dans le cadre des corps de nombres pour prouver l’existence de familles
asymptotiquement bonnes de réseaux unimodulaires. Il a été soumis au Journal
für die Reine Angewandte Mathematik.
Enfin, l’objet de la quatrième partie est l’étude dans le cadre des corps de
nombres du taux de décomposition des places dans les tours infinies de corps
de classes de Hilbert. Il n’a pas été soumis.
1

HILBERT CLASS FIELD TOWERS OF FUNCTION
FIELDS OVER FINITE FIELDS AND LOWER BOUNDS
FOR A(q)
ALEXANDRE TEMKINE
Abstract. We obtain lower bounds for the asymptotic number
of rational points of smooth algebraic curves over finite fields. To
do this we construct infinite Hilbert class field towers with good
parameters. In this way we improve bounds of Serre, Perret, Xing,
and Niederreiter and Xing.
1. introduction
In what follows, Fq is a finite field with q elements, where q is a prime
power. Let K be a function field over Fq or equivalently a projective
smooth absolutely irreducible curve defined over Fq so that Fq is the
constant field of K. We write N(K) for the number of rational places
of K, and g(K) for its genus. According to the Weil-Serre bound (see
[11] for example), we have:
(1) N(K) ≤ q + 1 + g(K) 2q 1/2 ,
where ⌊x⌋ denotes the lower part of x, i.e., ⌊x⌋ is the greatest integer
not exceeding the real number x. To study the asymptotic behaviour of
N(K) when g(K) → ∞, we introduce as usual the following quantity:
Definition. For any prime power q let
N(K)
A(q) = lim sup
g(K)→∞ g(K) .
One can easily deduce from (1) that A(q) ≤ 2q 1/2 . Vlădut¸ and
Drinfeld [13] obtained the following improvement:
(2) for any q, A(q) ≤ q 1/2 − 1.
A famous result by Ihara ([4] and other papers) obtained independently
by Tsfasman, Vlădut¸ and Zink [12] in some cases, and using modular
towers proves that it is an equality if q is a square. Hence
A(q) = q 1/2 − 1 , if q is a square.
For the complete story of this result, see Ihara’s Mathematical Review
of [12].
Date: June 24, 1999.
2

HILBERT CLASS FIELD TOWERS 3
For non-square q, the situation is unclear. Zink [14] used degenerate
Shimura surfaces to prove that if p is a prime, then
A(p 3 ) ≥ 2(p2 − 1)
p + 2
The other best results so far have been obtained by means of Hilbert
class field towers by Serre ([9], [10]), Perret [8], and more recently by
Niederreiter and Xing [6]. Serre [10] has shown that
(3) A(q) ≥ c log q,
for an absolute effective constant c > 0.
Niederreiter and Xing [6] proved that if q is odd and m ≥ 3 is an
integer, then
A(q m ) ≥
2q
⌈2(2q + 1) 1/2 ⌉ + 1 ,
and a similar result if q is even. This result is better than Serre’s for
many values of q m , but not for all since their lower bound remains
the same when q is fixed and m large and prime (if m = m ′ m ′′ with
m ′′ ≥ 3, we can write qm = qm′m ′′
and get a better estimate). In [7]
they improved slightly these results and extended Serre’s method.
For small values of q, the best results so far have been obtained by
Niederreiter and Xing [6], who showed
A(2) ≥ 81
317
A(3) ≥ 62
163
= 0.255 . . .
= 0.3803 . . .
A(5) ≥ 2
3 .
In this paper, we give a generalization of (3), which improves it for all
non-prime q, and also improvements for A(3) and A(5). The second
section of this paper is devoted to preliminaries. The main result of
this paper is:
Theorem. There exists an effective constant c > 0 such that for any
q and any n,
A(q n ) ≥ c n 2 log q
log q
log n + log q .
It improves (3) since Serre proved only:
A(q n ) ≥ c n log q.
Its proof can be found in section 3 for the case of odd characteristic
(Theorem 9) and in section 4 for the case of characteristic 2 (Theorem
13). We also obtain improvements for A(3) and A(5):

4 ALEXANDRE TEMKINE
Theorem. We have
and
A(3) ≥ 8
17
= 0.4705 . . . ,
A(5) ≥ 8
= 0.7272 . . . .
11
The same lower bounds for A(3) and A(5) have been obtained simultaneously
and independently by Angles and Maire [1]. Their constructions
are different of ours. Proof for this Theorem can be found
in section 5 (Theorem 15 for A(3) and Theorem 16 for A(5)). The
method used in this section can also be used to obtain lower bounds
for A(7), A(9), . . . , but it should be noticed that this method is asymptotically
bad, that is the lower bounds for A(q) obtained in this way are
bounded as functions in q. This last section is independent of sections
3 and 4.
Finally, we are deeply grateful to Serre for many remarks on preliminary
versions of this manuscript and to Perret for helpful discussions.
2. Background and preliminary results
In this section, q is a power of a prime number p.
2.1. Background on class field towers. For the following results,
we refer to Cassels and Fröhlich [3]. Let K be a function field over Fq
and S be a finite nonempty set of places of K and OS the S-integral ring
of K, i.e., OS consists of all elements of K that have no poles outside
S. Let ClS be the class group of OS. Let r be the gcd of the degrees of
the places in S, and let l be a prime number. The (S, l)-Hilbert class
field HS,l of K is the maximal unramified abelian extension of K with
Galois group of type (l, l, . . . l) in which all places in S split totally.
If l divides r, it contains the constant field extension by F q l, but if r
and l are prime to each other, there is no constant field extension. By
global class field theory, we know that the Galois group of the extension
HS,l/K is isomorphic to ClS/Cl l S .
Now, let’s define the (S, l)-Hilbert class field tower of K. Let K1 be
the (S, l)-Hilbert class field of K and S1 be the set of places of K1 over
those in S. Recursively, we define Ki to be the (Si−1, l)-Hilbert class
field of Ki−1 and Si the set of places of Ki over those in Si−1. Thus,
we get a tower
K = K0 ⊆ K1 ⊆ K2 ⊆ . . .
Golod and Shafarevich have given a condition for the tower to be infinite
(that is, Ki = Ki−1 for all i ≥ 1) in the case of number fields.
Their proof is valid in the case of function fields over finite fields, even
if the tower contains constant field extension. For an abelian group

HILBERT CLASS FIELD TOWERS 5
B, let dlB denote the l-rank of B. According to Roquette in Cassels
and Fröhlich ([3], chapitre IX) the Golod-Shafarevich Condition can be
expressed as:
Theorem 1. Let K be a function field over Fq and S be a finite nonempty
set of places of K. If dlClS ≥ 1 and
2 (dlClS)
(4) |S| ≤ − dlClS + ɛ
4
where ɛ = 0 if l|q − 1 and ɛ = 1 otherwise, then the (S, l)-Hilbert class
field tower of K is infinite.
2.2. Idea of the proof of the main result. In this paper, we follow
very closely Serre [10], adding the following idea: in order to obtain
a lower bound for A(q n ), we consider class field towers of function
fields over Fq and use the Golod-Shafarevich Condition to split places
of degree n (instead of splitting places of degree one of function fields
over Fqn as Serre did). Thereafter, we apply to the infinite class field
tower a constant field extension by Fqn. In this way, each place of
degree n over Fq gives n places of degree 1 over Fqn without putting
more constraints in the Golod-Shafarevich Condition, since the degree
of places does not appear in this statement. Of course, this method
is better than Serre’s only if n > 1. If n = 1 our proof reduces to
Serre’s proof. This explains why we only recover Serre’s result in the
case where q is prime.
2.3. Quadratic extensions in odd characteristic. From now on, q
is assumed to be odd, unless specified otherwise. Let k = Fq(x) be the
rational function field over Fq. If p is odd, a quadratic extension K/k,
K = k(y) = k( f(x)) is defined by an equation
y 2 = f(x) ∈ Fq[x],
where f is a square-free polynomial. We recall from Stichtenoth [11]
the decomposition law of primes in such extensions, and the genus of
K.
Proposition 2. A finite place P = (P (x)) of k:
• ramifies in K if P (x)|f(x),
• is inert in K if gcd(P (x), f(x)) = 1 and f(x) is not a square
modulo P (x),
• splits (totally) in K if gcd(P (x), f(x)) = 1 and f(x) is a square
modulo P (x).
Moreover, the infinite place ( 1)
of k: x
• ramifies in K if deg(f(x)) is odd,
• is inert in K if 2| deg(f(x)) and f is not monic modulo F ∗2
q ,
• splits (totally) in K if 2| deg f(x) and f is monic modulo F ∗2
q .

6 ALEXANDRE TEMKINE
Hence, the genus g(K) of K satisfies
deg(f(x)) = 2g(K) + 1 or deg(f(x)) = 2g(K) + 2,
according to the parity of deg(f(x)).
2.4. The graph Ωm,n and the subgraph ˜ Ωm,n. We write Ii for the
set of monic irreducible polynomials of degree i of Fq[x], and k still
denotes the rational function field over Fq. Let Ωm,n be the bipartite
graph whose set of vertex is Im ∪ In and whose set of edges Em,n with
Em,n ⊆ Im × In is defined by the following rule:
(D(x), P (x)) ∈ Em,n ⇔ P (x) splits in k( D(x))
⇔ D(x) is a square modulo P (x).
For any vertex P (resp. D), we write δ(P ) (resp. δ(D)) for the number
of vertices to which P (resp. D) is related in the graph Ωm,n (that is,
its degree in the graph, not to be confused, of course, with its degree
as a polynomial).
We write νq(n) for the cardinality of In. We have the inequalities:
Lemma 3. For qn ≥ 16, and for q odd or even, we have
qn 2n ≤ νq(n) ≤ qn
n .
Proof. From [5] (p.126) it follows:
Hence
and
1
n
since x − 2 √ x ≥ x
2
Lemma 4. If
q n − q
q − 1 (qn/2 − 1)
≤ νq(n) ≤ 1
n (qn − q).
1
n (qn − 2q n/2 ) ≤ νq(n) ≤ qn
n ,
1 q
2
n
n ≤ νq(n) ≤ qn
n ,
for x ≥ 16.
(5) q n/2 ≥ 6(m + 1),
then
(6) |Em,n| ≥ qn
3n νq(m).
Proof. For this proof, we use a result by Car [2], that is a lower bound
for the number of P (x) ∈ In such that D(x) ∈ Im is a nonzero square
modulo P (x). Since δ(D) is precisely this number, according to [2]
(Théorème III.6), we have:
∀ D(x) ∈ Im, δ(D) ≥ 1
n
n q
− (m + 1)qn/2 .
2

Hence
HILBERT CLASS FIELD TOWERS 7
|Em,n| ≥
≥ 1
n
D∈Im
δ(D)
n q
− (m + 1)qn/2 νq(m)
2
≥ qn
3n νq(m)
because of (5).
Now, let Ĩn =
P ∈ In / δ(P ) ≥ νq(m)
10
and we denote by ˜ Ωm,n the
bipartite induced subgraph of Ωm,n whose set of vertex is Im ∪ Ĩn
and
whose set of edges is Em,n ∩ .
Im × Ĩn
Lemma 5. With the preceding notations, if
(7) q n/2 ≥ 6(m + 1) and q n ≥ 16,
then
| Ĩn| ≥ qn
4n .
Proof. Using Lemmas 3 and 4, we find:
qn 3n νq(m) ≤ |Em,n| ≤
δ(P ) +
δ(P )
P ∈ Ĩn
P ∈ Ĩn
qn 3n νq(m) ≤ |Em,n| ≤ νq(m)
νq(n) + (10 − 1)|
10
Ĩn|
q
n
3n νq(m) ≤ νq(m)
q
n q
+ 9|Ĩn|
10 n n
1 1
−
n 3 10
≤ 9
10 |Ĩn|
7 q
27
n
n
≤ |Ĩn|
and the result follows.
We now state a theorem that we found in Serre [10]. For convenience,
we recall the proof:
Theorem 6. Let R, S be two finite sets and E ⊆ R × S. Let δ ≥ 1 be
such that every s ∈ S is E-related to at least δ points of R. Let a, b ≥ 1
be integers such that
|R| δ
(8) b ≤ |S| .
a a
Then there exist A ⊆ R, B ⊆ S with |A| = a, |B| = b and A × B ⊆ E.

8 ALEXANDRE TEMKINE
Proof. Let X = {(A, s) / |A| = a, s ∈ S, and A × {s} ⊆ E} and consider
the second projection φ :
φ : X −→ S
(A, s) ↦−→ s
Let R(s) be the subset of R made of the elements r such that (r, s) ∈ E.
We thus have:
|φ −1 (s)| = number of subsets of R(s) with a elements
R(s)
=
a
δ
≥ since R(s) ≥ δ.
a
Then |X| ≥ |S| δ
. a
Now, consider the first projection ψ :
ψ : X −→ set of subsets of R with a elements
(A, s) ↦−→ A
Hence some fiber of ψ has at least |X|
( |R|
a ) elements.
But
|X|
≥ |S| δ
a
≥ b,
|R|
a
|R|
a
so we can choose A whose fiber has more than b elements, and B in
the fiber with |B| = b and we get the result.
3. The construction of the Hilbert class field tower
In this section q is a power of a prime number p, that is assumed to be
odd, and k = Fq(x) is the rational function field over Fq .
3.1. The towers. Let A ⊆ Im and B ⊆ Ĩn. We set a = |A| and
b = |B|. The set of edges Em,n is defined as in section 2. For any
D ∈ A, let kD be the quadratic extension of k defined by y2 = D(x)
and kD = k(y). Let K be the compositum of all the kD when D ∈ A
and finally, let k0 = k(y) be defined by y2 =
D∈A D(x), so that we
have a diagram of field extensions as in figure 1.
Proposition 7. In the preceding notation assume that the following
conditions are satisfied:
(i) A × B ⊆ Em,n,
(ii) a is odd,
(iii) 2b ≤ (a−1)2
(iv) a − 1 ≥ 2.
then
4
− (a − 1),

HILBERT CLASS FIELD TOWERS 9
(i) the extension K/k0 is unramified and Galois with Galois group
(Z/2Z) a−1 ,
(ii) all the places over the places in B split totally in K/k,
(iii) the ( ˜ B, 2)-Hilbert class field tower of k0 is infinite, where ˜ B is
the set of the 2b places of k0 over the places in B.
Moreover, if these conditions are satisfied, we have
✟
✟
✟
✟
✟
✟
✟
✟
✟
✟
✟
✟
✟
A(q n ) ≥ 4nb
ma − 3 .
kD1 = k( √ D1) · · · kDi = k(√ Di) k0 = k( ( D))
❍
❍ ❅
❍
❍
❅
❍
❍
❅
❍ ❅
❍
❍
❍
❅
❍
❍
❅
❍❅
k = Fq(x)
K
Figure 1. Field extensions.
❅
❅ ❅
❅
❅ ❅❅
Proof. By Proposition 2, a finite place D1 is ramified in kD if and only
if D = D1. Hence K is the compositum of a independent quadratic extensions
of k, and K/k is Galois with the Galois group (Z/2Z) a . Since
k0 is a quadratic subextension of K/k, the result on the Galois group
follows.
By Proposition 2, the only possibly ramified places in K/k (and hence
in K/k0) are the places in A and the infinite place ( 1)
of k. However
x
the places in A are ramified in k/k0, hence unramified in K/k0. By
condition (ii), the place(s) over the infinite place ( 1)
of k are unramified
x
in K/k0.
By condition (i), the places in B split totally in K/k.
Hence, K is a subfield of the ( ˜ B, 2)-Hilbert class field of k0.

10 ALEXANDRE TEMKINE
Since the function x2 − x is increasing for x ≥ 2, conditions (iii) and
4
(iv) imply
− d2ClB˜ 4
in the notation of Theorem 1.
Consequently, Theorem 1 shows that the ( ˜ B, 2)-Hilbert class field tower
of k0 is infinite:
k0 ⊆ k1 ⊆ k2 ⊆ . . .
2b ≤ (d2Cl ˜ B ) 2
This tower may contain constant field extensions. However, the constant
field of all these fields is a subfield of Fqn. We apply a (possibly
trivial) constant field extension by the field Fqn to all these function
fields, and we obtain an infinite tower:
k0Fqn ⊆ k1Fqn ⊆ k2Fqn ⊆ . . .
in which the 2nb places of k0Fqn are totally split. These are rational
places. If g denotes the genus of k0 (or equivalently of k0Fqn ), using
the Riemann-Hurwitz genus formula, we see that g − 1 is multiplied in
each extension ki+1Fqn/kiFqn by the degree of the extension, as well as
the number of places over the places in ˜ B. Hence, the ratio of these
two quantities is constant in the tower, and we have:
Since g satisfies
A(q n ) ≥ 2nb
g − 1 .
deg D(x) = 2g + 1 or 2g + 2,
D∈A
according to the parity, we have
2g + 1 ≤ ma
and
ma − 3
g − 1 ≤
2
and the result follows.
3.2. Transcription of the results. Let C1 and C2 be constants to
be choosen (C1 will be large and C2 will be small) and let
log n + log(log q)
m = C1
,
log q
a = ⌈C2n log q⌉ ,
2 a
b = ,
8
where ⌈x⌉ denotes the upper part of x, that is the smallest integer
greater or equal to x, and ⌊x⌋ denotes the lower part of x, that is the

HILBERT CLASS FIELD TOWERS 11
greatest integer not exceeding x.
We now need a technical but elementary lemma.
Lemma 8. For C2 small enough and C1 and qn large enough, we have
νq(m)
(9) b ≤
a
qn
4n
νq(m)
10 .
a
Proof. We must be careful since we use the Stirling formula with functions
depending on two parameters q and n. This formula and other
expressions are considered for q n large, that is for q large or n large,
but not necessarily both. If y is a function depending on q and on n,
we write y = O(1) if there exists M such that
|y| ≤ M for every q ≥ 2 and every n ≥ 1.
For example, if x is a function in q and n that is large for q n large
(x → +∞, when q n → +∞), then the Stirling formula gives
log(x!) = (x + 1
) log(x) − x + O(1).
2
Claim 1. We have
q
a → +∞ and
m
20m → +∞ when qn → +∞.
Proof. The first assertion is trivial. For the second assertion, remark
that m ≥ 1, since q ≥ 3, and let be given B > 0 great enough. There
exists D > 0 such that
If q > B, then
If q ≤ B, then
hence
q n > D ⇒ q > B or n > B B
C 1 .
q m
m
≥ q > B.
n > B B
C 1 ,
C1 log n
> B,
log B
m > B,
and
qm 3m
≥ ≥ m ≥ B,
m m
so that the claim is proved.
Claim 2. If C2 is small enough and C1 and qn are large enough, then
1
m q 2
(10) a ≤
.
20m

12 ALEXANDRE TEMKINE
Proof. What follows is valid for almost all qn . Under this condition,
Claim 1 shows that we have:
qm 40m ≤
m q
,
20m
and ⌈C2n log q⌉ ≤ 2C2n log q.
Hence (10) is satisfied as soon as
(2C2n log q) 2 ≤ qm
40m .
Taking logarithms, we see that it suffices to show that
O(1) + 2 log C2 + 2 log n + 2 log (log q) ≤ m log q − log m,
or
O(1) + 2 log C2 + 2 log n + 2 log (log q) ≤
− log
C1
log n + log(log q)
log(q)
C1
log n + log(log q)
log(q)
log q
Thus, (10) is implied by
log n + log(log q)
O(1) + 2 log C2 + log C1 + 2 log n + 2 log (log q) + log
log(q)
log n + log(log q)
≤ C1
log(q)
.
+ 1
log q,
or equivalently by
log n + log(log q)
O(1) + 2 log C2 + log C1 + 2 log n + 2 log (log q) + log
+ 1
log(q)
≤ C1 (log n + log(log q)) ,
which is surely true for almost all q n , for C1 large enough and for C2
small enough.
Claim 3. If y ≤ x 1
2 , then
(11) log
x!
(x − y)!
= y log x + O(1).
Proof. This is a straightforward application of the Stirling formula.
We return now to the proof of Lemma 8.
What follows is valid for almost all qn . Then (9) is satisfied if
a2
νq(m)
≤
8 a
qn
4n
νq(m)
10 ,
a
which, in turn, using Lemma 3, is implied by
a2 qm
m ≤
8 a
qn
qm 4n a
20m
.

HILBERT CLASS FIELD TOWERS 13
Taking logarithms, we see that we have to show that
qm
! m
O(1) + log n + 2 log a + log qm
− a !
m
≤ n log q + log
q m
20m
qm 20m
!
.
− a !
Using Claim 2 and Claim 3 it suffices to show
m q
O(1) + log n + 2 log a + a log
m
m q
≤ n log q + a log
,
20m
or
Since
O(1) + log n + 2 log a + a log
log
q m
m
qm 20m
≤ log
≤ M
q m
q m
m
qm 20m
m
qm 20m
+ 1
− 1
for a large enough constant M, we only need
≤ n log q.
= O(1)
O(1) + log n + 2 log a + Ma ≤ n log q,
that is, by definition of a,
O(1) + log n + 2 log ⌈C2n log q⌉ + M ⌈C2n log q⌉ ≤ n log q,
and this is surely true for almost all q n if one chooses C2 small enough.
We now state the main result of this section.
Theorem 9. There exists an effective constant c > 0 such that for q
odd,
A(q n ) ≥ c n 2 log(q)
log(q)
log(n) + log(q) .
Proof. Since A(q) > 0 for all q (see Lemma 5.5 in [7]), the result needs
only to be proved for almost all qn (that is, for all except finitely many).
For instance, we can assume that qn ≥ 16, so that by definition of m,
the conditions (7) of Lemma 5 are trivially satisfied for allmost all qn .
We consider the graph ˜ Ωm,n. Using Theorem 6, and Lemmas 5 and 8,
we get A ⊆ Im, B ⊆ Ĩn, with
2 a
|A| = a = ⌈C2n log q⌉ and |B| = b = , and A × B ⊆ Em,n.
8

14 ALEXANDRE TEMKINE
Let à ⊆ A with |Ã| = α = a or a − 1 so that α is odd, and ˜ B ⊆ B
with
2 (α − 1) α − 1
|B| = β = − .
8 2
We now can apply Proposition 7 to à and ˜ B. Conditions (i), (ii), and
(iii) are satisfied. Condition (iv) is clear for almost all qn . So that we
have
A(q n 4nβ
) ≥
mα
− 3
(a−1) 2
a−1
4n −
≥
ma − 3
≥ C3n a
m
for a small enough constant C3, and for almost all q n . Hence
A(q n ) ≥ C3n
C1
8
≥ C4n 2 log q
C5
log n+log q
log q
2
⌈C2n log q⌉
log n+log(log q)
log q
for some constants C4 and C5 and for almost all q n and the result
follows.
4. The case of characteristic 2
In this section q is assumed to be even, i.e., q = 2 t . This case is
very similar to the case of q odd, but technically slightly different since
quadratic extensions are now expressed as Artin-Schreier extensions.
The only problem it carries is that we do not have any result similar
to Car’s result [2] in case of q even. But this little problem is avoided
by a simple modification in the definition of ˜ Ωm,n.
4.1. Quadratic extensions in characteristic 2. Let k = Fq(x) be
the rational function field over Fq. Since q is even, a quadratic extension
K/k, K = k(y) is defined by an equation
y 2 + y = f(x) ∈ Fq(x),
where f is a rational function not of the form g 2 +g for any g ∈ Fq(x). In
fact, we are going to consider only extensions with f(x) = t
i=1
that is defined by equations
y 2 + y =
t
i=1
1
Di(x) ,
1
Di(x) ,

HILBERT CLASS FIELD TOWERS 15
where Di(x) are mutually distinct monic irreducible polynomials of
Fq[x]. We recall from Stichtenoth [11] the decomposition law of primes
in such extensions, and the genus of K.
Proposition 10. A finite place p = (p(x)) of k (c ∈ F q deg p(x) being a
root of p(x))
• ramifies in K with conductor of exponent 2 if p(x) = Di(x) for
some i,
• is inert in K if gcd(p(x), Di(x)) = 1 and TrF q deg p(x) /F2f(c) =
1,
• splits (totally) in K if gcd(p(x), Di(x)) = 1 and TrF q deg p(x) /F2f(c)
= 0.
Moreover, the infinite place ( 1)
of k splits (totally) in K.
x
Hence, the genus g(K) of K satisfies
t
g = −1 + deg Di(x).
i=1
4.2. The graph Ωm,n and the subgraph ˜ Ωm,n. We write Ii for the
set of monic irreducible polynomial of degree i of Fq[x], k still denotes
the rational function field over Fq, we write kD for the quadratic extension
k(y) of k defined by
y 2 + y = 1
D(x) ,
and kDD ′ (for D = D′ , D and D ′ ∈ Fq[x], where D ′ has nothing to do
with the derivative of D) for the quadratic extension k(y) of k defined
by
y 2 + y = 1 1
+
D(x) D ′ (x) .
Let Ωm,n be the bipartite graph whose set of vertex is Im∪In and whose
set of edges is Em,n ⊆ Im × In, with the following rule:
(D(x), P (x)) ∈ Em,n ⇔ P (x) splits in kD.
As in section 2, for any vertex P (resp. D), we write δ(P ) (resp. δ(D)
for the number of vertices to which P (resp. D) is related in Ωm,n.
Moreover, we define a partition Im = I ′ m ∪ I ′′ m by:
I ′ m =
I ′′ m =
D ∈ Im / δ(D) ≥ νq(n)
3
D ∈ Im / δ(D) < νq(n)
.
3
We list the elements of I ′′ m two by two (except maybe one, if |I ′′ m| is
odd), so that
D1, D ′ 1, D2, D ′ 2, . . . Ds, D ′ s

16 ALEXANDRE TEMKINE
are all the elements of I ′′ m except possibly one. Then, we form the set
I ′′′
m
I ′′′
m = {(D1, D ′ 1), (D2, D ′ 2), . . . , (Ds, D ′ s)}
and the graph Ω ′ m,n whose set of vertex is (I ′ m ∪ I ′′′
set of edges is E ′ m,n ⊆ (I ′ m ∪ I ′′′
m) ∪ In, and whose
m) × In, with the following rules:
(D(x), P (x)) ∈ E ′ m,n ⇔ P (x) splits in kD
((Di(x), D ′ i(x)), P (x)) ∈ E ′ m,n ⇔ P (x) splits in kDD ′.
For any (Di, D ′ i) ∈ I ′′′
m, we write δ(Di, D ′ i) for the number of vertices
P ∈ In to which (Di, D ′ i) is related in Ω ′ m,n.
Lemma 11. If
(12)
and
(13)
then
q n
n
q m
m
≥ 48,
≥ 16,
(14) |E ′ m,n| ≥ 1
64
q n
n
q m
m .
Proof. Since at most one place is ramified in kDi , for (Di, D ′ i) ∈ I ′′′
m, we
have:
and also:
2
|{P (x) ∈ In / P (x) is inert in kDi }| ≥
3 νq(n) − 1,
P (x) ∈ In / P (x) is inert in kD ′
2
≥ i 3 νq(n) − 1.
But P (x) splits in kDiD ′ i
kD ′ i
, so that we have:
as soon as P (x) is inert both in kDi and in
P (x) ∈ In / P (x) splits kDiD ′
1
≥ i 3 νq(n) − 2.
Assuming (12) and using Lemma 3, we obtain:
Hence, we have:
P (x) ∈ In / P (x) splits kDiD ′ i
|E ′ m.n| =
D∈I ′ m
δ(D) +
(Di,D ′ i )∈I′′′
m
≥ νq(n)
3 |I′ m| + qn
8n |I′′′
m|
≥ qn
8n (|I′ m| + |I ′′′
m|) ,
≥ q n
8n .
δ(Di, D ′ i)

HILBERT CLASS FIELD TOWERS 17
by Lemma 3.
But, by definition of I ′ m, I ′′ m, and I ′′′
m, we have:
so that
|I ′ m| + 2|I ′′′
m| + 1 ≥ |Im| ≥ qm
2m
|I ′ m| + |I ′′′
m| ≥ 1
≥ qm
8m
using Lemma 3 and assuming (13).
Finally, we have:
2 (|Im| − 1)
≥ 1
m q
− 1
2 2m
|E ′ m,n| ≥ qn q
8n
m
8m ,
giving the result.
Now, let’s denote by δ ′ (P ) the number of vertices in I ′ m ∪ I ′′′
P ∈ In is related in E ′ m,n, let Ĩn =
P ∈ In / δ ′ (P ) ≥ νq(m)
100
mto which
and de-
note by ˜ Ωm,n the bipartite induced subgraph of Ω ′ m,n whose set of vertex
is (I ′ m ∪ I ′′′
m) ∪ Ĩn and whose set of edges is E ′ m,n ∩
Lemma 12. If
and
then
q n
n
q m
m
≥ 48,
≥ 4,
(I ′ m ∪ I ′′′
m) × Ĩn
(15) | Ĩn| ≥ qn
176n .
Proof. The proof is closely similar to that of Lemma 5. Hence, we do
not give it here.
4.3. The main result in characteristic 2. Now, the situation is
very similar to that of odd characteristic. Hence we can state the main
result:
Theorem 13. Assume that q is even. Then, we have the same result
as in odd characteristic, that is there exists an effective constant c > 0
such that we have
A(q n ) ≥ c n 2 log q
log q
log n + log q .
.

18 ALEXANDRE TEMKINE
Proof. We do not present the proof, since everything is very similar to
the case of odd characteristic: the construction of the tower, as well as
the technical Lemmas (with different constants) or the use of Theorem
6. We only remark that there are some minor modifications in the
proof: for instance, one must change m by m + 1, to be sure that
m ≥ 1.
Finally, the Theorem announced in the introduction is simply the sum
of Theorem 9 and Theorem 13.
5. Lower bounds for A(3) and A(5)
In [6], Niederreiter and Xing established a proposition giving a lower
bound for the l-rank of ClS. In their proof, they make no difference
between ClS (the class group of the the integral ring OS) and the Sdivisor
class group, i.e, the quotient of the group of all divisors of K of
degree 0 with support outside S by its subgroup of principal divisors.
In fact, the first one is an extension of the second by Z/rZ, where r is
the gcd of the degrees of the places in S. For the correctness of their
proposition, as Serre noticed, one must add the hypothesis that r = 1.
However, it does not affect the results established in [6]. We now recall
this proposition:
Proposition 14. Let K/k be a finite abelian extension of function
fields. Let T be a finite nonempty set of places of k and S the set of
places of K lying over those in T . Assume that the gcd of the degrees
of the places in S is 1. Then, for any prime l we have
dlClS ≥
dlGP − (|T | − ɛ) − dlG
P
where G = Gal(K/k) and GP is the inertia group of the place P in
K/k, ɛ = 0 if l|q − 1 and ɛ = 1 otherwise. The sum is extended over
all places P of k.
We use now this proposition to obtain lower bounds on A(3) and A(5).
We have used the tables of irreducible polynomials on F3 and F5 in
Lidl and Niederreiter [5].
Theorem 15. We have
Proof. Let
A(3) ≥ 8
17
= 0.4705 . . . .
f(x) = (x 2 + 1)(x 2 + x + 2)(x 2 + 2x + 2)(x 3 + 2x + 1)
(x 3 + 2x + 2)(x 3 + x 2 + 2)(x 3 + x 2 + x + 2)
(x 3 + x 2 + 2x + 1)(x 3 + 2x 2 + 1)(x 3 + 2x 2 + x + 1)
(x 3 + 2x 2 + 2x + 2)(x 4 + x 2 + x + 1)(x 4 + x 3 + x 2 + 1),

HILBERT CLASS FIELD TOWERS 19
written as product of 13 irreducible polynomials.
One can check that f(0) = f(1) = f(2) = 1 = 12 and deg(f) is even.
So, we consider k = F3(x) and K = k(y) with y2 = f(x), and apply
Proposition 14 to K/k, T = x, x + 1, x + 2, 1
, and l = 2 to get:
x
d2ClS ≥ 13 − 4 − 1 = 8
by Proposition 2.
Hence, Theorem 1 shows that the (S, 2)-Hilbert class field tower of K
is infinite, since |S| = 8 by Proposition 2.
We have deg(f) = 38, so that the genus of K is g(K) = 18, and by the
Riemann-Hurwitz genus formula, just as in the proof of Proposition 7
we get:
A(3) ≥ 8
17 ,
giving the result.
Theorem 16. We have
Proof. Let
A(5) ≥ 8
11
= 0.7272 . . . .
f(x) = (x + 1)(x + 2)(x 2 + 2)(x 2 + 3)(x 2 + x + 1)
(x 2 + x + 2)(x 2 + 2x + 3)(x 2 + 2x + 4)(x 2 + 3x + 3)
(x 2 + 3x + 4)(x 2 + 4x + 1)(x 2 + 4x + 2)(x 4 + x 2 + 2),
written as product of 13 irreducible polynomials.
One can check that f(0) = f(1) = f(2) = 4 = 22 and deg(f) is even.
So, we consider k = F5(x) and K = k(y) with y2 = f(x), and apply
Proposition 14 to K/k, T = x, x + 3, x + 4, 1
, and l = 2 to get:
x
d2ClS ≥ 13 − 4 − 1 = 8
by Proposition 2.
Hence, Theorem 1 shows that the (S, 2)-Hilbert class field tower of K
is infinite, since |S| = 8 by Proposition 2.
We have deg(f) = 26, so that the genus of K is g(K) = 12, and by the
Riemann-Hurwitz genus formula, just as in the proof of Proposition 7
we get:
A(5) ≥ 8
11 ,
giving the result.
References
[1] Angles, B., and Maire, C.: A note on tamely ramified towers of global function
fields, preprint, 1999.
[2] Car, M. : Distribution des polynômes irréductibles dans Fq[T ], Acta Arithmetica
LXXXVIII.2 (1999), 141-153.
[3] Cassels, J.W.S., and Fröhlich, A. (eds): Algebraic Number Theory, Academic
Press, London, 1967.

20 ALEXANDRE TEMKINE
[4] Ihara, Y.: On modular curves over finite fields, Discrete subgroups of Lie
Groups, Proc. Internat. Colloq., Bombay, Oxford Univ. Press, 1973, 161-202.
[5] Lidl, R., and Niederreiter, H.: Introduction to Finite Fields and Their Applications,
second ed., Cambridge University Press, Cambridge, 1997.
[6] Niederreiter, H., and Xing, C.: Towers of Global Function Fields with Asymptotically
Many Rational Places and an Improvement on the Gilbert-Varshamov
Bound, Math. Nachr. 195 (1998), 171-186.
[7] Niederreiter, H., and Xing, C.: Curve sequences with asymptotically many
rational points, Contemporary Math. 245 (1999), 3-14.
[8] Perret, M.: Tours Ramifiées Infinies de Corps de Classes, J. Number Theory
38 (1991), 300-322.
[9] Serre, J.-P.: Sur le Nombre des Points Rationnels d’une Courbe Algébrique sur
un Corps Fini, C.R. Acad. Sci. Paris Série 1 Math. 296 (1983), 397-402.
[10] Serre, J.-P.: Rational Points on Curves over Finite Fields, Lecture Notes,
Harvard University, 1985.
[11] Stichtenoth, H.: Algebraic Function Fields and Codes, Springer, Berlin, 1993.
[12] Tsfasman, M.A., Vlădut¸, S.G., and Zink, T: Modular Curves, Shimura Curves,
and Goppa Codes, Better than Varshamov-Gilbert Bound, Math. Nachr. 109
(1982), 21-28.
[13] Vlădut¸, S.G., and Drinfeld, V.G.: Number of Points of an Algebraic Curve,
Funct. Anal. Appl. 17 (1983), 53-54.
[14] Zink, T.: Degeneration of Shimura surfaces and a problem in coding theory,
in Fundamentals of Computation Theory, Budach L. (ed.), Lecture Notes in
Computer Science, Vol. 199, Springer, Berlin, p. 503-511, 1985.
Équipe “Arithmétique et Théorie de l’Information”
I.M.L., C.N.R.S.
Luminy Case 930, 13288 Marseille Cedex 9 - FRANCE
E-mail address: temkine@iml.univ-mrs.fr

HILBERT CLASS FIELD TOWERS OF FUNCTION
FIELDS OVER FINITE FIELDS AND LOWER BOUNDS
FOR A(q), II.
ALEXANDRE TEMKINE
Abstract. We construct infinite Hilbert class field towers with
many places totally splitting in the tower. In this way we obtain
new lower bounds for the asymptotic number of rational points of
smooth algebraic curves over finite fields, improving or extending
bounds of Niederreiter and Xing and W Li and Maharaj.
1. introduction
In what follows, Fq is a finite field with q elements, where q is a prime
power. Let K be a function field over Fq or equivalently a projective
smooth absolutely irreducible curve defined over Fq so that Fq is the
constant field of K. We write N(K) for the number of rational places
of K, and g(K) for its genus. According to the Weil-Serre bound (see
[11] for example), we have:
(1) N(K) ≤ q + 1 + g(K) 2q 1/2 ,
where ⌊x⌋ denotes the lower part of x, i.e., ⌊x⌋ is the greatest integer
not exceeding the real number x. To study the asymptotic behaviour of
N(K) when g(K) → ∞, we introduce as usual the following quantity:
Definition. For any prime power q let
N(K)
A(q) = lim sup
g(K)→∞ g(K) .
One can easily deduce from (1) that A(q) ≤ 2q 1/2 . Vlădut¸ and
Drinfeld [14] obtained the following improvement:
(2) for any q, A(q) ≤ q 1/2 − 1.
A famous result by Ihara ([2] and other papers) obtained independently
by Tsfasman, Vlădut¸ and Zink [13] in some cases, and using modular
towers proves that it is an equality if q is a square. Hence
A(q) = q 1/2 − 1 , if q is a square.
For non-square q, the situation is unclear. The best results so far have
been obtained by means of Hilbert class field towers by Serre [8] [9],
Date: March 12, 2007.
21

22 ALEXANDRE TEMKINE
Perret [7], and more recently by Niederreiter and Xing [4]. Serre [9]
has shown that
(3) A(q) ≥ c log q,
for an absolute effective constant c. We improved this bound previously
in [12] to:
(4) A(q n ) ≥ cn 2 log q
log q
log n + log q ,
for an absolute effective constant c. Recently, W Li and Maharaj gave
in [15] a simpler proof of (4) which allowed them to compute an explicit
constant for qn large (so that we can take c = 1/8 for qn large and odd).
Niederreiter and Xing in [4] and [5] proved that if q is odd and m ≥ 3
is an integer, then
(5) A(q m 2q
) ≥
⌈2(2q + 1) 1/2⌉ + 1 ,
and a similar result if q is even: if q ≥ 4 is even and m ≥ 3 is an odd
integer, then
(6) A(q m q + 1
) ≥
⌈2(2q + 2) 1/2⌉ + 2 .
They extended their result in [6]. This has been improved by W Li and
Maharaj in [15]. To present their most explicit result let us fix some
notation. Let Bn(q) denote the number of places of degree n of the
rational function field Fq(x). By f(q) = O(g(q)) we mean that there is
a constant M > 0 such that |f(q)| ≤ M|g(q)| for all sufficiently large
q. They proved the following
Theorem. Let q be an odd prime power. Let m be an odd integer at
least 3 and n be a positive integer relatively prime to m. Suppose that
3 + 2(2Bn(q) + 1)
Bm(q) >
1/2
.
m − 2
Then we have
(7) A(q mn ) ≥
√
2(m − 2) √ n/2
nq + O(1).
m − 1
For m < n < 2m the conditions of this theorem are satisfied for all q
sufficiently large and the bound above improves the bound (5).
For q even they give the following similar improvement:
Theorem. Let q be a power of 2. Let m be an odd integer at least 3
and n be a positive integer relatively prime to m. Suppose that
⎢
⎢
⎢6
+ 2 4
Bm(q) > ⎣ ⎥
Bn(q)
⎥
⎦ .
m − 1

HILBERT CLASS FIELD TOWERS 23
Then we have
(8) A(q mn √
2√
n/2
) ≥ nq + O(1).
4
For m < n < 2m the conditions of this theorem are satisfied for all q
sufficiently large and the bound above improves the bound (6).
In this paper we give a generalisation of (7) and of (8)which improves
(5) and (6) not only for m < n < 2m but also for m < n < 4m2 . Our
most explicit results are the two following corollaries:
Corollary 1. Let q be an odd prime power. Let m be an integer at least
3, and n be a positive integer relatively prime to m. Then we have
A(q mn √ n/2 nq
) ≥ √ + O(1).
n 2 2m
Corollary 2. Let q be a power of 2. Let m be an integer at least 3,
and n be a positive integer relatively prime to m. Then we have
A(q mn √ n/2 nq
) ≥
2 √ 2 + O(1).
n
2m
Finally, we are deeply grateful to Niederreiter and W Li and Maharaj
for providing us their preprints on the subject.
2. Background on class field towers
In this section, q is a power of a prime number p.
For the following results, we refer to Cassels and Fröhlich [1]. Let K
be a function field over Fq and S be a finite nonempty set of places
of K and OS the S-integral ring of K, i.e., OS consists of all elements
of K that have no poles outside S. Let ClS be the class group of OS.
Let r be the gcd of the degrees of the places in S, and let l be a prime
number. By an l-extension of fields, we mean a Galois extension whose
Galois group is of exponent l. The (S, l)-Hilbert class field HS,l of K
is the maximal unramified abelian l-extension of K in which all places
in S split totally. If l divides r, it contains the constant field extension
by F q l, but if r and l are prime to each other, there is no constant field
extension. By global class field theory, we know that the Galois group
of the extension HS,l/K is isomorphic to ClS/Cl l S .
Now, let’s define the (S, l)-Hilbert class field tower of K. Let K1 be
the (S, l)-Hilbert class field of K and S1 be the set of places of K1 over
those in S. Recursively, we define Ki to be the (Si−1, l)-Hilbert class
field of Ki−1 and Si the set of places of Ki over those in Si−1. Thus,
we get a tower
K = K0 ⊆ K1 ⊆ K2 ⊆ . . .
Golod and Shafarevich have given a condition for the tower to be infinite
(that is, Ki = Ki−1 for all i ≥ 1 in the case of number field.

24 ALEXANDRE TEMKINE
Their proof is valid in the case of function fields over finite fields, even
if the tower contains constant field extension. For an abelian group
B, let dlB denote the l-rank of B. According to Roquette in Cassels
and Fröhlich ([1], chapitre IX) or to Serre [10], the Golod-Shafarevich
Condition can be expressed as:
Theorem 1. Let K be a function field over Fq and S be a finite nonempty
set of places of K. If dlClS ≥ 1 and
(9) dlClS ≥ 2 + 2 (|S| + ɛ) 1/2
where ɛ = 1 if l|q − 1 and ɛ = 0 otherwise, then the (S, l)-Hilbert class
field tower of K is infinite.
3. The construction of the Hilbert class field tower
Recall that Br(q) denotes the number of places of degree r in Fq(x).
The following estimate of the size of Br(q) was proved in [11] (Corollary
V.2.10).
Proposition 2. For any q and any r we have
|Br(q) − qr
| < 2qr/2
r r .
Thus Proposition 2 implies that Br(q) = q r /r + O(q r/2 ).
Theorem 3. Let q be an odd prime power. Let m be an integer at least
3, and n be a positive integer relatively prime to m. Moreover let r be
the smallest integer such that
(10)
Then we have:
Br(qm
) − Br(q)
≥ 3 + 2 (2Bn(q) + 1)
3
1/2
A(q mn ) ≥
2nBn(q)
r 3 + 2 (2Bn(q) + 1) 1/2
.
− 2
Proof. Let D ∈ Fqm[x] an irreducible monic polynomial of degree r
with D /∈ Fq[x]. There are (Br(qm ) − Br(q)) such polynomials. We
write
D(x) = x r + ar−1x r−1 + · · · + a0.
Let σ denote the Frobenius of Fqn. σ is acting by σ(ai) = a qn
i
coefficients of D(x). This make σ act on D(x) by
σ(D)(x) = x r + σ(ar−1)x r−1 + · · · + σ(a0).
on the
We check easily that σ m (D)(x) = D(x), but σ(D)(x) = D(x) since
D /∈ Fq[x] and Fq m ∩ Fq n = Fq. If D(x) = D ′ (x)D ′′ (x) then σ(D)(x)
= σ(D ′ )(x)σ(D ′′ )(x), so that σ acts on the set of irreducible monic
polynomial of degree r belonging to Fq m[x] and not to Fq[x]. We

HILBERT CLASS FIELD TOWERS 25
want to join them by pairs (D, σ(D)) without any repetition. Each
orbit under σ contains at least two elements. Thus there are at least
1
3 (Br(q m ) − Br(q)) such pairs (the worst case is an orbit with 3 ele-
ments wich provides only one pair). We define N by
N =
3 + 2 (2Bn(q) + 1) 1/2
.
By definition of r we can choose N pairs (Di, σ(Di)) for i = 1, . . . N
where the Di and σ(Di) are all different and irreducible.
Let (Di, σ(Di)) be any of these pairs. Let k = Fqm(x) denote the
rational function field over Fq mand ki = k(yi) with
y 2 i = Di(x)σ(Di)(x).
Let H be the compositum of k1, . . . kN and K = k(y) the extension
defined by:
y 2 N
= Di(x)σ(Di)(x).
i=1
Since the polynomials Di and σ(Di), 1 ≤ i ≤ N, are all different and
irreducible, the extensions ki/k, 1 ≤ i ≤ N are linearly disjoint and we
have
N
Gal(H/k) Gal(ki/k) = (Z/2Z) N .
i=1
Note also that K ⊆ H since we can take y = y1 · · · yN. Thus,
Gal(H/K) (Z/2Z) N−1 ,
and so d2Gal(H/K) = N − 1. For 1 ≤ i ≤ N the places Di and σ(Di)
have ramification index 2 in H/k and also ramification index 2 in H/K,
and since there are no other ramified places in H/k, the extension H/K
is unramified.
Now, let T be the set of places of degree n of Fq(x) which can be identified
with a set of places of degree n of k = Fqm(x) since gcd(m, n) = 1,
T = {P (x) ∈ Fq[x] / P (x) is monic irreducible of degree n} .
Thus |T | = Bn(q). Let P (x) ∈ T and α ∈ Fqn a root for P . For all
1 ≤ i ≤ N we have
Di(x)σ(Di)(x) ≡ Di(α)σ(Di)(α) ≡ Di(α)σ(Di)(σ(α))
≡ (Di(α)) 1+qn
mod P.
Since q n is odd, the last element is a nonzero square in the residue
class field of P and so P splits completely in ki/k by Kummer’s theorem.
Thus, if S is the set of places of K lying over those in T , then
|S| = 2Bn(q) and all the places in S split completely in H/K. By the
definition of N,
d2Gal(H/K) = N − 1 ≥ 2 + 2 (2Bq(n) + 1) 1/2 .

26 ALEXANDRE TEMKINE
By the Golod-Shafarevich Condition 1, it follows that the (S,2)-Hilbert
class field tower of K is infinite:
K = K0 ⊆ K1 ⊆ K2 ⊆ . . .
This tower may contain constant field extensions. However, the constant
field of all these fields is a subfield of Fqmn. We apply a (possibly
trivial) constant field extension by the field Fqmn to all these function
fields, and we obtain an infinite tower:
KFqmn ⊆ K1Fqmn ⊆ K2Fqmn ⊆ . . .
in which the 2nBn(q) places of KFqn lying over those in S are totally
split. These are rational places. If g denotes the genus of K (or equiv-
alently of KFqmn ), using the Riemann-Hurwitz genus formula, we see
that g − 1 is multiplied in each extension Ki+1Fqmn/KiFqmn by the degree
of the extension, as well as the number of places over the places
in S. Hence, the ratio of these two quantities is constant in the tower,
and we have
A(q mn ) ≥ 2nBn(q)
g − 1
By Kummer’s theorem we have
so that we get
g − 1 = r
A(q mn ) ≥
3 + 2 (2Bn(q) + 1) 1/2
− 2
2nBn(q)
r 3 + 2 (2Bn(q) + 1) 1/2
.
− 2
Remark 1. To make this result clear, we need an estimation for r. However,
the condition (10) is satisfied with r =
n for all q sufficiently
2m
large. Namely, using Proposition 2, the left side of (10) is
mr 1 q qr
−
3 r r + O(qmr/2
)
as the right side of (10) is
2 √ 2
n/2 q
√ + O(q
n n/4
) .
Since m do not divide n, mr > n and the inegality (10) is satisfied for
2
q large. Using again the estimate of Proposition 2 for Bn(q), it comes
A(q mn √ n/2 nq
) ≥ √ + O(1).
n 2 2m
We thus can state the following corollary which gives a lower bound
twice less than (7) for m < n < 2m (in which case
n = 1) but still
2m
improves (5) for m < n < 4m2 .

HILBERT CLASS FIELD TOWERS 27
Corollary 1. Let q be an odd prime power. Let m be an integer at least
3, and n be a positive integer relatively prime to m. Then we have
A(q mn √ n/2 nq
) ≥ √ + O(1).
n 2 2m
Remark 2. By a more carefully definition of the extensions involved in
the proof, we can obtain a lower bound twice better than ours, hence
recovering exactly (7). For an idea of the necessary modifications, we
refer to the proof of (7) in [15].
We turn now to the case of characteristic 2.
Theorem 4. Let q be a power of 2. Let m be an integer at least 3,
and n be a positive integer relatively prime to m. Moreover let r be the
smallest integer such that
Then we have:
Br(qm
) − Br(q)
≥ 3 + 2
3
2Bn(q)
A(q mn ) ≥
2nBn(q)
2r 3 + 2
2Bn(q)
.
− 2
Proof. The proof is closely similar to that of theorem 3 by replacing
Kummer extensions by Artin-Schreier extensions. Let D ∈ Fqm[x] an
irreducible monic polynomial of degree r with D /∈ Fq[x]. There are
(Br(qm ) − Br(q)) such polynomials. We write
D(x) = x r + ar−1x r−1 + · · · + a0.
Let σ denote the Frobenius of Fqn. σ is acting by σ(ai) = a qn
i
coefficients of D(x). This make σ act on D(x) by
σ(D)(x) = x r + σ(ar−1)x r−1 + · · · + σ(a0).
on the
We check easily that σm (D)(x) = D(x), but σ(D)(x) = D(x) since
D /∈ Fq[x] and Fqm ∩ Fqn = Fq. If D(x) = D ′ (x)D ′′ (x) then σ(D)(x)
= σ(D ′ )(x)σ(D ′′ )(x), so that σ acts on the set of irreducible monic
polynomial of degree r belonging to Fqm[x] and not to Fq[x]. We
want to join them by pairs (D, σ(D)) without any repetition. Each
orbit under σ contains at least two elements. Thus there are at least
1
3 (Br(qm ) − Br(q)) such pairs (the worst case is an orbit with 3 elements
wich provides only one pair). We define N by
N = 3 + 2
2Bn(q) .
By definition of r we can choose N pairs (Di, σ(Di)) for i = 1, . . . N
where the Di and σ(Di) are all different and irreducible.

28 ALEXANDRE TEMKINE
Let (Di, σ(Di)) be any of these pairs. Let k = Fqm(x) denote the
rational function field over Fqmand ki = k(yi) with
y 2 i + yi = 1
Di(x) +
1
σ(Di)(x) .
Let H be the compositum of the k1, . . . kN and K = k(y) the extension
defined by:
y 2 N
1
+ y =
Di(x) +
1
.
σ(Di)(x)
i=1
Since the polynomials Di and σ(Di), 1 ≤ i ≤ N, are all different and
irreducible, the extensions ki/k, 1 ≤ i ≤ N are linearly disjoint and we
have
N
Gal(H/k) Gal(ki/k) = (Z/2Z) N .
i=1
Note also that K ⊆ H since we can take y = y1 + · · · + yN. Thus,
Gal(H/K) (Z/2Z) N−1 ,
and so d2Gal(H/K) = N − 1. For 1 ≤ i ≤ N the places Di and σ(Di)
have ramification index 2 in H/k and also ramification index 2 in H/K,
and since there are no other ramified places in H/k, the extension H/K
is unramified.
Now, let T be the set of places of degree n of Fq(x) which can be identified
with a set of places of degree n of k = Fqm(x) since gcd(m, n) = 1,
T = {P (x) ∈ Fq[x] / P (x) is monic irreducible of degree n} .
Thus |T | = Bn(q). Let P (x) ∈ T and α ∈ Fqn a root for P . For all
1 ≤ i ≤ N we have
TrF q mn/F q n
1
Di(α) +
1
σ(Di)(α)
=
1
TrFqmn/Fqn Di(α) +
=
1
σ(Di)(σ(α))
1
TrFqmn/Fqn Di(α)
1
+ TrFqmn/Fqn σ
Di(α)
=
1
2 TrFqmn/Fqn Di(α)
= 0,
and so the absolute trace
1
TrFqm Di(α) +
1
= 0
σ(Di)(α)
by the transitivity of the trace. It follows then from Kummer’s theorem
and [3] (Theorem 2.25) that P splits completely in ki/k. Thus, if S is

HILBERT CLASS FIELD TOWERS 29
the set of places of K lying over those in T , then |S| = 2Bn(q) and all
the places in S split completely in H/K. By the definition of N,
d2Gal(H/K) = N − 1 ≥ 2 + 2 2Bq(n).
By the Golod-Shafarevich Condition 1, it follows that the (S,2)-Hilbert
class field tower of K is infinite:
K = K0 ⊆ K1 ⊆ K2 ⊆ . . .
This tower may contain constant field extensions. However, the constant
field of all these fields is a subfield of Fqmn. We apply a (possibly
trivial) constant field extension by the field Fqmn to all these function
fields, and we obtain an infinite tower:
KFqmn ⊆ K1Fqmn ⊆ K2Fqmn ⊆ . . .
in which the 2nBn(q) places of KFqn lying over those in S are totally
split. These are rational places. Using [11] (Proposition III.7.8) the
genus g of K satisfies
g − 1 = 2r
3 + 2
Bn(q) − 2.
Hence, as in the preceeding proof it comes
A(q mn ) ≥
2nBn(q)
2r 3 + 2
Bn(q)
.
− 2
Remark 3. The same remark as for Theorem 3 is valid and leads to the
following corollary which is very similar to (8) for m < n < 2m and
still improves (6) for m < n < 4m 2 .
Corollary 2. Let q be a power of 2. Let m be an integer at least 3,
and n be a positive integer relatively prime to m. Then we have
A(q mn √ n/2 nq
) ≥
2 √ 2 + O(1).
n
2m
References
[1] Cassels, J.W.S., and Fröhlich, A. (eds): Algebraic Number Theory, Academic
Press, London, 1967.
[2] Ihara, Y.: On modular curves over finite fields, Discrete subgroups of Lie
Groups, Proc. Internat. Colloq., Bombay, Oxford Univ. Press, 1973, 161-202.
[3] Lidl, R., and Niederreiter, H.: Introduction to Finite Fields and Their Applications,
second ed., Cambridge University Press, Cambridge, 1997.
[4] Niederreiter, H., and Xing, C.: Towers of Global Function Fields with Asymptotically
Many Rational Places and an Improvement on the Gilbert-Varshamov
Bound, Math. Nachr. 195 (1998), 171-186.
[5] Niederreiter, H., and Xing, C.: Global Function Fields With Many Rational
Places And Their Applications, Contemporary Mathematics 225 (1999), 87-
111.

30 ALEXANDRE TEMKINE
[6] Niederreiter, H., and Xing, C.: Curve sequences with asymptotically many rationalplaces,
to appear in the AMS Summer Research Conference Proceedings
(Seattle, 1997).
[7] Perret, M.: Tours Ramifiées Infinies de Corps de Classes, J. Number Theory
38 (1991), 300-322.
[8] Serre, J.-P.: Sur le Nombre des Points Rationnels d’une Courbe Algébrique sur
un Corps Fini, C.R. Acad. Sci. Paris Série 1 Math. 296 (1983), 397-402.
[9] Serre, J.-P.: Rational Points on Curves over Finite Fields, Lecture Notes,
Harvard University, 1985.
[10] Serre, J.-P.: Existence de tours infinies de corps de classes d’après Golod et
Shafarevich, Colloque CNRS, 142 (1996), 231-238.
[11] Stichtenoth, H.: Algebraic Function Fields and Codes, Springer, Berlin, 1993.
[12] Temkine, A.: Hilbert class field towers of function fields over finite fields and
lower bounds for A(q), preprint (1999), to be published in the J. Number
Theory.
[13] Tsfasman, M.A., Vlădut¸, S.G., and Zink, T: Modular Curves, Shimura Curves,
and Goppa Codes, Better than Varshamov-Gilbert Bound, Math. Nachr. 109
(1982), 21-28.
[14] Vlădut¸, S.G., and Drinfeld, V.G.: Number of Points of an Algebraic Curve, in
Funct. Anal. Appl. 17 (1983), 53-54.
[15] W Li, W.-C., and Maharaj, H.: Coverings of curves with asymptotically many
rational points, preprint (1999).
Équipe “Arithmétique et Théorie de l’Information”
I.M.L., C.N.R.S.
Luminy Case 930, 13288 Marseille Cedex 9 - FRANCE
E-mail address: temkine@iml.univ-mrs.fr

ASYMPTOTICALLY GOOD FAMILIES OF
UNIMODULAR LATTICES
ALEXANDRE TEMKINE
Abstract. In this paper we construct asymptotically good families
of unimodular lattices. To do this, we use infinite unramified
towers of number fields.
1. Background on lattices
1.1. Definitions and parameters of lattices. We consider the classical
problem of packing equal non-overlapping spheres in R t . In this
paper, we are only interested in the case of lattice packing (or simply
lattice) that is into additive subgroups of rank t of R t . Moreover we
focus on asymptotical questions, that is for t → +∞. Our purpose is to
construct asymptotically dense lattices with the additionnal property
to be unimodular. This is achieved in Proposition 10. We are deeply
grateful to G. Lachaud for having introduced us to this question.
We denote a lattice by L. The minimum distance of the lattice is
d(L) = min |u − v|.
u,v∈L,u=v
The packing associated to the lattice L is
P =
B (u, d(L)/2) ,
u∈L
where
B (u, r) = x ∈ R t / |x − u| ≤ r
We recall that the density of L can be defined as
∆(L) = lim sup v(P ∩ B(0, r))/v(B(0, r)),
r→+∞
where v(.) is the standard volume in R t .
It is more convenient for asymptotical problems to use another equivalent
parameter, the relative density exponent δ(L) defined by
δ(L) = 1
t log 2 ∆(L).
Note that the relative density exponent is always negative. The closer
to 0 it is, the more dense is the lattice.
Let e1, . . . et be any basis of L, so that L = Ze1 ⊕ · · · ⊕ Zet. The matrix
whose columns are e1, . . . et is called a generator matrix of the lattice.
Date: March 12, 2007.
31

32 ALEXANDRE TEMKINE
The determinant of the lattice is the absolute value of the determinant
of any generator matrix and is denoted by det L. It is clear (or wellknown)
that these parameters are related by the following formula:
(1) ∆(L) = d(L)t Vt
2 t det L ,
where Vt = π t/2 /Γ(t/2 + 1) is the volume of unit ball in R t .
1.2. Asymptotic problems on lattices density. An elementary
calculation using the Stirling formula leads from (1) to the following
asymptotical formula (for t → +∞):
δ(L) = 1
2 log πe 1
2 +
2t t log (d(L))
2
t
det L
t
+ O(log ).
t
We are interested into the dense lattices so that we introduce
δ(t) = sup δ(L)/ L is a lattice in R t ,
and since we focus on asymptotic problems, we consider
δ = lim sup δ(t).
t→+∞
A family of lattices is a set {Lt} of lattices, Lt ⊂ R t , where t runs over
an infinite subset of N. Let
δ({Lt}) = lim sup δ(Lt).
t→+∞
Following [11] (and its analogy with codes), we call families satisfying
δ({Lt}) > −∞ good families. The first well-known upper bound (this
means possibility bound) for ˜ δ is that of Blichfeldt, exposed in [10]:
δ ≤ −0.5,
which has been much later improved by Kabatianski and Levenshtein
(see [5]) into
δ ≤ −0.599 . . . .
On the other hand, the best known lower bound is the existence result
of Minkowski: there exists an (asymptotically good) family of lattices
{Lt} t≥1 satisfying
δ({Lt}) ≥ −1.
Though this result is famous we could not find a complete proof in the
litterature and thus only refer to [1]. However, there exists other proofs
of the same result, for instance by Rogers in [9].

ASYMPTOTICALLY GOOD FAMILIES OF UNIMODULAR LATTICES 33
2. Background and preliminaries on class field towers
2.1. Background on class field towers. For this background we
refer to [3]. Let K be a number field with signature (r1(K), r2(K)).
CK denotes the ideal class group of K. Let l be a prime number.
Global class field theory allows us to define the l-Hilbert class field of
K as the maximal unramified abelian l-extension of K (notice that for
archimedian places, unramified means that no real place complexifies in
the extension). By class field theory the Galois group of this extension
is isomorphic to CK/C l K .
Once given a number field K = K0 and a prime number l we define
recursively a sequence (Kr)r≥1 where Kr is the l-Hilbert class field of
Kr−1. Such a sequence is called a class field tower over K. Thanks to
Golod-Shafarevich’s Theorem, we have a criterion for a class field tower
to be infinite. For an abelian group A, let dl(A) denote the l-rank of
A. If K contains a primitive l-root of unity we set δl,K = 1; otherwise
we set δl,K = 0. The Golod-Shafarevich Theorem can be expressed as:
Theorem (see Roquette in [3]). With the preceeding notation, if
dl(CK) ≥ 2 + 2 r1(K) + r2(K) + δl,K,
then the l-Hilbert class field tower of K is infinite.
2.2. Infinite unramified towers of number field with specific
properties. For our purpose we need to construct infinite unramified
towers of totally real number fields (Kr)r≥1 such that for any r, the
different Diff Kr is a square, i.e., there exists an ideal Jr of the ring
of integers of Kr such that Diff Kr = J 2
r . First we remark that this
condition needs only to be satisfied for K.
Lemma 1. If Diff K is a square then for any unramified extension
Kr/K, Diff Kr is a square.
Proof. Let ℘ be a place of K and ℘r be a place of Kr over ℘. According
to the formula for the different in a tower (cf. [6], Th. 111) and the fact
that the extension is unramified, the different exponent at ℘ and at ℘r
are equal. Diff K is a square if and only if every different exponent is
even so that the result follows.
To apply the Golod-Shafarevich Theorem we need lower bounds for the
l-rank of the class group of a field K. The classical tool to do this is
the following proposition of genus theory:
Proposition 2 (Martinet [8]). Let l be a prime and K/k be a Galois
extension of degree l. Let (r1, r2) be the signature of K. If r places
ramify in K/k then
dl(CK) ≥ r − (r1 + r2) − δl,K.

34 ALEXANDRE TEMKINE
where δl,K = 1 if K contains a primitive l-root of unity and δl,K = 0
otherwise.
It is now easy to construct totally real fields with square different having
an infinite Hilbert class field tower. It suffices to consider any cubic
field K, Galois over Q, having at least 7 primes ramified over Q. Indeed,
applying the last proposition with l = 3 we obtain d3CK ≥ 6 and
the Golod-Shafarevich Theorem can be applied. Moreover if there is
no wild ramification, all the nonzero different exponents are equal to 2,
hence the different is a square. However, even with wild ramification,
Hilbert’s different formula shows that the different is a square. Cubic
cyclic fields are extensively described in [4] and provide explicit examples.
We give the best example (the one with lowest root discriminant).
Example 1. Let e = 9.7.13.19.31.37.43. By ([4], Th. 6.4.6 p.333) there
exists integers u, v such that
e = u2 + 27v2 , u ≡ 6 ( mod 9), 3 ∤ v, u ≡ v ( mod 2), v > 0,
4
and the field K = Q(θ) where θ is a root of the polynomial with
coefficients in Z
P (X) = X 3 − e eu
X −
3 27
is a cubic cyclic field of discriminant e2 . Hence, the different of K is
a square and K admits an infinite class field tower. Finally we obtain
the following proposition:
Proposition 3. The field K admits an infinite class field tower of totally
real fields having square different and its root discriminant satisfies
(Disc K) 1/3 ≈ 838262.
3. Number field lattices
Let K be a number field of degree t. If x ∈ K, let x (1) , . . . x (t) denote
the embeddings of K into C. The norm map from K to Q will be
denoted by NK/Q(.), the trace map by TrK/Q(.). In R t the standard
scalar product of x and y is denoted by (x|y) and the norm of x by
x = (x|x). In what follows we assume that K is either totally real,
either totally complex with complex multiplication. In this case, the
complex conjugation commutes with any embedding of K into C. This
allows us to define x for any x ∈ K. Moreover we define the classical
Q-linear embedding L : K → R t (the so-called Hilbert map). If K is
totally real we set
L : K −→ R t ,
x ↦→ x (1) , . . . , x (t)

ASYMPTOTICALLY GOOD FAMILIES OF UNIMODULAR LATTICES 35
and if λ ∈ K we set
T (λ) =
⎛
⎝ λ(1) . . . . . .
. . . . . . . . .
. . . . . . λ (t)
If K is totally complex and that the embeddings of K are ordered in
such a way that x (j+t/2) = x (j) for 1 ≤ j ≤ t/2, we set
⎞
⎠ .
L : K −→ R t ,
√ √ √ √
(1) (1) (t/2) (t/2)
x ↦→ 2 Re x , 2 Im x , . . . , 2 Re x , 2 Im x ,
and if λ ∈ K with λ (j) = ξ (j) + iη (j) , we set
⎛
⎜
T (λ) = ⎜
⎝
ξ (1) −η (1) . . . . . . . . .
η (1) ξ (1) . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . ξ (t) −η (t)
. . . . . . . . . η (t) ξ (t)
⎞
⎟
⎠ .
The map λ ↦→ T (λ) is a representation of K in the algebra of real
square matrix of order t and in both cases we have for λ, x ∈ K
(2) L(λx) = T (λ)L(x).
If OK is the ring of integers of K and J is a fractional ideal of OK, then
L(J ) is a lattice in R t and also an OK-module (via the representation
T ). To calculate the parameters of L(J ) we set
and we state the following lemma:
Lemma 4. If x, y ∈ K, then
c(J ) = min
x∈J NK/Q(x),
(L(x)|L(y)) = TrK/Q(xy).
Proof. If K is totally real the relation is clear. If it is totally complex
(with complex multiplication) then
t/2
t/2
(L(x)|L(y)) = 2(Re x Re y + Im x Im y) =
j=1
Since x (t/2+j) = x (j) = x (j) it comes
(L(x)|L(y)) =
t
j=1
j=1
x (j) y (j) = TrK/Q(xy).
Lemma 5. With the preceeding notation we have
d(L(J )) ≥ √ t (c(J )) 1/t .
(x (j) y (j) + x (j) y (j) ).

36 ALEXANDRE TEMKINE
Proof. If x ∈ J the preceeding lemma gives
L(x) 2 = TrK/Q(xx).
The arithmetic-geometric mean inequality yields
TrK/Q(xx) ≥ tNK/Q(xx) 1/t = tNK/Q(x) 2/t
≥ t (c(J )) 2/t ,
and we take the square roots.
Lemma 6. If Disc K denotes the discriminant of K we have
det L(J ) = NK/Q(J )|Disc K| 1/2 .
Proof. Let (ω1, . . . , ωt) be a basis of the Z-module J . Let us denote
by Disc(ω1, . . . , ωt) the discriminant of that basis, i.e., the Gram
determinant of the basis relatively to the symmetric bilinear form
(x, y) ↦→ TrK/Q(xy), that is, the determinant of the matrix
M = TrK/Q(ωiωj)
1≤i,j≤t .
Hence if we set
and
U =
⎛
we have det M = (det U) 2 and
⎝ ω(1) 1 . . . ω (1)
. . . . . .
t
. . .
ω (t)
1 . . . ω (t)
t
⎞
⎠ ,
w (i) = (ω (i)
1 , . . . , ω (i)
t ) ∈ C t ,
Disc(ω1, . . . , ωt) = (det(w (1) , . . . , w (t) )) 2 .
The discriminant of K is that of the ring OK and according to ([2],
chap. II, p. 139) we have
Disc J = NK/Q(J ) 2 Disc K.
If K is totally real, U is a generator matrix for L(J ) so that
det L(J ) = |Disc(ω1, . . . , ωt)| 1/2 = |Disc J | 1/2 = NK/Q(J )|Disc K| 1/2 .
If K is totally imaginary a generator matrix for L(J ) is
⎛
⎞
and it comes
Ω = √ ⎜
2 ⎜
⎝
Re ω (1)
1 . . . Re ω (1)
t
Im ω (1)
1 . . . Im ω (1)
t
. . . . . . . . .
Re ω (t/2)
1 . . . Re ω (t/2)
t
Im ω (t/2)
1 . . . Im ω (t/2)
t
⎟
⎠

ASYMPTOTICALLY GOOD FAMILIES OF UNIMODULAR LATTICES 37
det Ω = ( √ 2) t det(Re w (1) , Im w (1) , . . . , Re w (t/2) , Im w (t/2) )
= (−2i) t/2 det((Re w (1) + i Im w (1) ), i Im w (1) , . . . )
= (−2i) t/2 det(w (1) , i w(1) − w (1)
2i
, . . . )
= i t/2 det(w (1) , w (1) , . . . , w (t/2) , w (t/2) )
= ɛi t/2 det(w (1) , . . . , w (t/2) , w (t/2+1) , . . . , w (t) )
where ɛ is the signature of the permutation sending (1, . . . , t) on
(1, t/2 + 1, 2, t/2 + 2, . . . , t/2, t).
Hence we have | det Ω| = Disc(ω1, . . . , ωt) 1/2 and finally
det L(J ) = | det Ω| = |Disc J | 1/2 = NK/Q(J )|Disc K| 1/2 .
Proposition 7. For a number field K of degree t, totally real or with
complex multiplication and an ideal J of OK we have
∆(L(J )) ≥ Vt
2 t
tt/2 ,
|Disc K| 1/2
and
log t
δ(L(J )) ≥ δ0(K) + O( )
t
where
δ0(K) = 1
2 log πe 1
2 −
2 2t log2 |Disc K|.
Proof. Using the preceeding lemmas we have
∆(L(J ) ≥ Vt
2 t
tt/2c(J )
,
NK/Q(J )|Disc K| 1/2
and of course c(J ) ≥ NK/Q(J ). According to ([5], chap.1.4, p.9) we
have
1
t log2 hence
Vt 1
=
2t 2 log πe 1
2 −
2t 2t log2(tπ) + O( 1 1
) =
t2 2 log2 δ(L(J )) ≥ 1
2 log 2
πe
2t
≥ δ0(K) + O(
πe
2t
t
+ O(log ),
t
1
−
2t log2 |Disc K| + 1
2 log log t
2 t + O( )
t
log t
).
t
To construct asymptotically good families of lattices we need families
of number fields (Kr)r≥1 with δ0(Kr) bounded. This is the case with
any unramified tower of number fields since in such a tower the root
discriminant (Disc K) 1/t is constant.

38 ALEXANDRE TEMKINE
Remark 1. The calculation of the parameters of these lattices is due
to Lytsin and Tsfasman [7] for the case J = OK. For the case of an
arbitrary ideal (or of any complete module of K) it is due to Alquié [1].
In [11], Tsfasman presents several constructions of lattices and codes
characterized by the following data encoded by letters: he uses either
number (N ) or function (F ) fields, either additive (A) or multiplicative
(M ) structure, obtains lattices (L) or codes (C ) and the construction
depends on a divisor (D) or not. In the last paragraph he asks for
the non-studied possibilities and for their possible advantages. The
embedding L(J ) we described thereover, encoded by letters (NALD)
is not studied in [11] and presents the following advantage compared
with the embedding of L(OK) which is encoded by letters (NAL): as
soon as J is not principal, the density of L(J ) is strictly better than
that of L(OK). Indeed, if J is not principal then
c(J ) > NK/Q(J )
and the proof of Proposition 7 shows that the inequality for ∆ is
strict. Moreover the inequality for d(L(J )) (Lemma 5) may also be
non-optimal as it is an equality in the case where J = OK and K is totally
real or totally complex (see [11]). However it is not clear whether
it could also provide an improvement for the asymptotical density of
a family of number field lattices. It would come either from an improvement
of Lemma 5 or from a family (Kr, Jr)r≥1 consisting in an
(unramified) tower of number fields together with ideals such that the
ratio c(Jr)/NKr/Q(Jr) growths exponentially with the degree. We do
not know whether this is possible or not.
4. unimodular number field lattices
The dual of a lattice L in R t is defined by
L ∗ = {x ∈ R t /(x|y) ∈ Z for all y ∈ L}.
A lattice L is unimodular if L = L ∗ .
We still consider a number field K of degree t that is assumed to be
either totally real or with complex multiplication. If M is a complete
module of K (for instance an ideal), the complementary module of M
is
M ′ = {x ∈ K/TrK/Q(xy) ∈ Z for all y ∈ M}.
These two notions of duality are related by the following proposition:
Proposition 8. If M denotes the conjugate module of M, we have
L(M) ∗ = L(M ′ ).
Proof. Let (ω1, . . . , ωt) be a basis of M and (α1, . . . , αt) its dual basis
defined by the relationships
TrK/Q(αiωj) = δi,j.

ASYMPTOTICALLY GOOD FAMILIES OF UNIMODULAR LATTICES 39
(α1, . . . , αt) is a basis for M ′ so that L(α1), . . . , L(αt) is a basis of the
lattice L(M ′ ) and a basis of Rt . We have
t
t
(L(ωj)|y) = λi(L(ωj)|L(αi)) =
i=1
for y =
i=1
t
λiL(αi)
i=1
λiTrK/Q(αiωj) = λj
because of Lemma 4. Hence any y ∈ Rt can be written
t
y = (L(ωj)|y)L(αi),
i=1
which establishes the result.
For a number field K as before, we denote the different of the extension
K/Q by Diff K. The next proposition gives a condition on Diff K to
obtain a unimodular lattice.
Proposition 9. Assume K is totally real and that Diff K = J 2 is a
square. Then the lattice L(J −1 ) is unimodular.
Proof. If x ∈ J −1 , then for any y ∈ J −1 , xy ∈ J −2 . Since J −2 =
Diff −1 K = O ′ K , it follows that TrK/Q(xy) ∈ Z. This implies x ∈
(J −1 ) ′ .
Conversely if x ∈ (J −1 ) ′ , then for any y ∈ J −1 and z ∈ OK, we have
TrK/Q(xyz) ∈ Z
so that xy ∈ O ′ K = J −2 . Since it is true for any y ∈ J −1 it implies
x ∈ J −1 . Hence we have (J −1 ) ′ = J −1 and since K is totally real, the
result follows from Proposition 8.
Proposition 10. There exists asymptotically good families of number
field unimodular lattices. For instance there exists a family {Lt} of
unimodular lattices with
δ({Lt}) ≥ −8.791 . . . .
Proof. Let K be a totally real field with infinite class field tower (Kr)r≥1
such that Diff K is a square. Such field do exist thanks to subsection
2.2. Every Kr is totally real and thanks to the formula for the different
in a tower, their different is always a square, Diff Kr = J 2
r . The lattices
L(J −1
r ) are unimodular and they form an asymptotically good family
as follows from Proposition 7 and the fact that the root discriminant
(Disc K) 1/t remains unchanged in the tower. If we consider the field
of the example 3 of subsection 2.2 and use Proposition 7 we obtain an
estimate for the asymptotic density of the family {Lt}:
δ({Lt}) ≥ −8.791 . . . .

40 ALEXANDRE TEMKINE
4.1. Other properties of these lattices. These lattices are unimodular
in the sense that det(L(J )) = 1 as any unimodular lattices. In
addition to be unimodular, these lattices have the important property
to be constructive, i.e., they can theorytically be explicitly constructed,
though the calculations are in fact much too long to be practiced (see
[7] for more details). Moreover we notice that these lattices are OKrmodules
(via the representation T ) as shows equation (2).
References
[1] Alquié, D.: Codes, réseaux et empilements de sphères, Thèse de Doctorat,
Université d’Aix-Marseille II, 1997.
[2] Borevich, Z.I., Shafarevich, I.R.: Théorie des Nombres, Gauthier-Villars, Paris,
1967.
[3] Cassels, J.W.S., and Fröhlich, A. (eds): Algebraic Number Theory, Academic
Press, London, 1967.
[4] Cohen, H.: A Course in Computational Algebraic Number Theory, Springer-
Verlag, 1991.
[5] Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 2 nd
edition, Grund. der math. Wiss. 290, Springer, New York Heidelberg, 1993.
[6] Hecke, E.: Lectures on the Theory of Algebraic Numbers, Springer-Verlag,
New York Heidelberg Berlin, 1923.
[7] Litsyn, S.N., Tsfasman, M.A.: Constructive high-dimensional sphere packings,
Duke Math. J., 54 (1987), 147-161.
[8] Martinet, J.: Tours de corps de classe et estimations de discriminants, Invent.
Math. 44 (1978), 65-73.
[9] Rogers, C.A.: Existence Theorem in the Geometry of Numbers, Ann. Math.
II, Ser. 48 (1947), 994-1002.
[10] Siegel, C.L.: Lecture notes on Geometry of Numbers, Springer-Verlag Berlin
Heidelberg, 1989.
[11] Tsfasman, M.A.: Global Fields, Codes and Sphere Packings, Journées
Arithmétiques de Luminy 17-21 juillet 1989. Astérisque 198-199-200 (1991),
373-396.
Équipe “Arithmétique et Théorie de l’Information”
I.M.L., C.N.R.S.
Luminy Case 930, 13288 Marseille Cedex 9 - FRANCE
E-mail address: temkine@iml.univ-mrs.fr

ON THE SPLITTING RATE OF PLACES IN INFINITE
UNRAMIFIED TOWERS OF NUMBER FIELDS
ALEXANDRE TEMKINE
Abstract. Let k (0) ⊂ · · · ⊂ k (n) ⊂ . . . be an infinite unramified
tower of number fields and ℘n a place of k (n) over a place ℘0 in k (0) .
Let fn be the relative inertia degree of ℘n over ℘0 and dn be the
degree of the extension k (n) /k (0) . In this paper we are interested
into comparing the growth of fn to dn. After some general results
on this growth rate, we give for any integer m examples of towers
in which for any place the growth of fn is less than the logarithm
iterated m times of dn.
1. Introduction
Let p be a prime number. If k = k (0) is a number field and S a finite
set of non-archimedean places of k, the (S, p)-Hilbert class field of k is
the maximal extension of k that is abelian elementary of type (p, . . . p)
. By
in which all the places in S are totally split. We denote it by k (1)
S
class field theory the Galois group of this extension is Cl(k,S)/(Cl(k,S)) p
where Cl(k,S) is the quotient of the class group of k by the classes of
the places in S. Let ρS(k), respectively ν(k), denote the p-rank of
Cl(k,S), respectively unit group of k. Throughout all this paper, if K/k
is a field extension and S a set of places in k, S(K) denotes the set
for i ≥ 1 to
. It gives rise to a tower
of places in K over those in S. We define recursively k (i)
S
be the (S(k (i−1)
S
), p)-Hilbert class field of k (i−1)
S
(k (i)
S )i≥0 which we call the (S, p)-class field tower of (k, S). It is said to
be infinite if for any i, k (i)
S
= k(i+1)
S
. If S is empty, we just speak of the
p-class field tower of k. In 1964, Golod and Shafarevich established for
the first time the existence of infinite p-class field towers (for S empty).
Generalized to take into account non-empty sets S, their criterion is as
follows:
Theorem 1 (Golod-Shafarevich). If
ρS(k) ≥ 2 + 2 ν(k) + |S| + 1,
then the (S, p)-class field tower of k is infinite.
In an infinite unramified tower of number fields (k (i) )i≥0 of degree ni
and absolute value of the discriminant Dki
Date: March 12, 2007.
41
, the root discriminant D1/ni
ki

42 ALEXANDRE TEMKINE
remains unchanged. This is one of the interest of these towers since the
Odlyzko-Serre inequalities (cf.[6],[7]) show that it cannot be too small.
More precisely, we have the following theorem:
Theorem 2 (Odlyzko-Serre). Let k be a number field of degree n, of
signature (r1, r2) and with absolute value of the discriminant Dk. If the
generalized Riemann hypothesis (GRH) is valid, then for n → ∞ we
have
π
γ+
Dk ≥ (8πe 2 ) r1 γ 2r2 o(n)
(8πe ) e
where γ is the Euler constant. Without (GRH) we only have
Dk ≥ (4πe γ+1 ) r1 (4πe γ ) 2r2 e o(n) .
Hence it was interesting to find number fields with infinite p-class field
tower and small root discriminant. The best examples (of totally real
or totally complex fields) have been during a long time those of Martinet
[5]. They have been recently improved by Hajir and Maire ([2]
and [3]).
However, as well as the generalization of the Golod-Shafarevich Test
to include finite places, the Odlyzko-Serre bounds have been (very recently)
generalized by Tsfasman and Vlădut¸[9] to take into account the
splitting of finite places. To present some of their results we use their
terminology: (ki)i≥0 is a sequence of distinct number fields of absolute
value of the discriminant Di; we set
gi = log Di
and call it the genus of ki. In any such family (ki) gi → ∞ when i → ∞
since there is only a finite number of number fields with bounded genus.
Let us set
For a prime power q we set
ni = [ki : Q] = r1(ki) + 2r2(ki).
Nq,i = Nq(ki) = |{v ∈ P (ki) : Norm(v) = q}| ,
where P (ki) is the set of non-archimedean places of ki.
We call a family (ki) asymptotically exact if for any prime power q the
limit
and also the limits
φR := lim
i→∞
Nq(ki)
φq := lim
i→∞ gi
r1(ki)
gi
r2(ki)
, φC := lim
i→∞ gi
exist. We can now state the result of Tsfasman and Vlădut¸ that interests
us:

ON THE SPLITTING RATE OF PLACES IN UNRAMIFIED TOWERS 43
Theorem 3 (Tsfasman-Vlădut¸). If (GRH) is valid, then for an asymptotically
exact family of number fields one has
q
φq log q
√ + φR(log 2
q − 1 √ 2π + π γ
+
4 2 ) + φC(log 8π + γ) ≤ 1,
the sum being taken over all prime powers q.
Without (GRH) we only have
2
φq log q
q
∞
m=1
(q m +1) −1 +( γ
2
+ 1
2 +log 2√ π)φR +(γ +log 4π)φC ≤ 1.
They also prove many other important results (in particular a generalization
of the Brauer-Siegel Theorem) but all their results are of an asymptotic
nature. At the end of their paper, they ask several questions.
Among them is the problem to give effective versions of their results
with the remainder terms as good as possible (Problem 8.12, [9]). To
give effective versions of Theorem 3 means to give (upper) bounds depending
on the genus and the degree for the number of places of given
degrees in a number field. Such a result would show that a number
field of given degree and genus cannot have too many places of small
degrees. Since unramified towers of number fields provide asymptotically
good families (cf. Lemma 2.3 [9]) with constant root discriminant
it is interesting to have information about the growth of the degrees of
finite places in such towers which probably cannot be too slow. This
is what we are concerned with in this paper. More precisely, if (ki)i≥0
is a tower of unramified extensions (k0 ⊂ k1 ⊂ . . . ) and ℘i is a place of
ki over a place ℘0 of k0, let fi be the relative inertia degree of ℘i over
℘0 and di = [ki : k0]. Our aim is to compare fi and di when i → ∞
in such a tower. More precisely we are looking for upper bounds for fi
depending on di but not depending on the place ℘i, i.e., we consider
in fact the maximum of all the fi over all places of ki. In section 2
we state general results on these parameters that indicate that it is
reasonnable to think that
fi = O(log di)
for any ℘i ∈ P (ki). However the growth of fi can be much weaker
as we show in section 3 where we give for any integer m examples of
towers such that
fi = O log (m)
di
for any ℘i ∈ P (ki), where log (m) denotes the logarithm iterated m
times.

44 ALEXANDRE TEMKINE
2. General results on the growth of the inertia degree
in unramified towers of number fields
In this section we consider a number field k (0) , its p-class field tower
(k (i) )i≥0 that is assumed to be infinite and a finite place ℘i in ki over
℘0. The relative inertia degree of ℘i over ℘0 is denoted fi and we set
di = [k (i) : k (0) ]. To state our first result we need a Theorem of Maire
[4]:
Theorem 4 (Maire [4]). Let p be a prime number and k a number
field. If the p-part of the ideal class group Clk of k is cyclic, then k has
a finite p-class field tower.
This allows us to prove the following result:
Proposition 5. With the preceeding notation, if k (0) has an infinite
p-class field tower, then
for any place ℘i ∈ P (k (i) ).
fi ≤ di
Proof. The extension k (i) / k (i−1) is elementary of type (p, . . . p). Hence
the relative inertia degree of a place ℘i−1 in this extension which by
class field theory is the order of the Frobenius of ℘i−1 in the Galois
group of the extension is at most p. Since k (i−1) has an infinite p-class
field tower, Theorem 4 shows that the degree of this extension is at
least p 2 . Using the multiplicativity of the relative inertia degree (and
of the degree) we obtain
fi ≤ p i
and di ≥ p 2i
and the result follows.
With some more hypothesis we can do much better. To do this we
need to introduce a conjecture of Fontaine and Mazur.
Conjecture (Fontaine-Mazur). For any number field k, let Γk be the
Galois group of the maximal unramified extension of k with pro-p Galois
group. Then Γk has no infinite p-adic analytic quotient.
This conjecture is related to the question asked by Stark to know
whether the p-class rank of the layers in an infinite p-class field tower
tends to infinity. The connection between Stark’s question and the
Fontaine -Mazur conjecture is extensively studied by Hajir in [1]. In
particular he proves the two following facts which we need:
Theorem 6 (Hajir [1]). Let k be a number field with infinite p-class
field tower. Assume that the Fontaine-Mazur conjecture holds for k or
that k passes the Golod-Shafarevich Test (i.e., ρ(k) ≥ 2+2(ν(k)+1) 1/2
with the notation of the introduction). Then ρ(k (i) ), the p-class rank of
k (i) tends to infinity with i.

ON THE SPLITTING RATE OF PLACES IN UNRAMIFIED TOWERS 45
Practically the hypothesis of this theorem is not too restrictive since
almost all the number fields that are known to have infinite p-class
field tower satisfy the Golod-Shafarevich Test. Even if they do not,
they satisfy a weakened form of it that suffices in fact to conclude that
the p-class rank of the fields in the tower tends to infinity. We refer to
Hajir [1] for more details.
With these facts we can state a stronger result on the growth of the
relative inertia degree.
Proposition 7. If k (0) has an infinite p-class field tower, assume that
the Fontaine-Mazur conjecture holds for k (0) or that k (0) passes the
Golod-Shafarevich Test. Then for any integer N and any place ℘i ∈
P (k (i) ) we have
when i → ∞.
fi = o(d 1
N
i )
Proof. Under these hypotheses, Theorem 6 shows that ρ(k (i) ) → ∞
when i → ∞. As in the proof of Proposition 5 the relative inertia
degree of ℘i−1 in k (i) / k (i−1) is at most p. Summing up from 0 to i − 1
and setting ρi = ρ(k (i) ) we obtain
fi ≤ p i
and di = p ρ0+···+ρi−1 .
Since ρi → ∞ we have i = o(ρ0 + · · · + ρi−1) = (ρ0 + · · · + ρi−1)ɛ(i)
where ɛ(i) → 0 when i → 0. Hence
fi ≤ p (ρ0+···+ρi−1)ɛ(i) = d ɛ(i)
i
and the result follows.
The simplest functions satisfying the conditions of the last Proposition
are the powers of logarithm. For simplicity we propose the following
conjecture:
Conjecture. If k (0) has an infinite p-class field tower, then
fi = O(log di).
3. Examples with slow growth of the inertia degree in
unramified towers of number fields
In this part we give examples of unramified towers of number fields
where the growth of the relative inertia degree regarding to the degree
is much less than that proposed in Conjecture 2. To do this we need
a classical proposition of genus theory. We fix some notation: k is a
number field of signature (r1, r2) and we define δp(k) by
δp = 0 if k contains no primitive p-root of unity and δp = 1 otherwise.
If K / k is a field extension and S a set of places of k then S(K) denotes
the set of places in K over those in S. We can now state the proposition
of genus theory together with the Golod-Shafarevich Test:

46 ALEXANDRE TEMKINE
Proposition 8 (Tsfasman-Vlădut¸). Let S = {℘1, . . . , ℘t} and Q =
{Q1, . . . , Qr} be disjoint sets of prime ideals of k, and let t0 be the
number of principal ideals in S. Consider a number field extension
K / k of prime degree p ramified exactly at Q. Let τ be the number
of real places in k becomming complex in K / k. Set s = |S(K)|. We
write ρS(K) for the p-rank of ClK,S(K) instead of ρS(K)(K). Then we
have
ρS(K) ≥ r − (r1 + r2) + τ − δp(k) − s + t0.
Moreover if
r ≥ s − t0 + r1 + r2 + δp + 2 − τ + 2 p(r1 + r2 − τ/2) + δp + s
then K has an infinite unramified (S(K), p)-class field tower (where
S(K) splits completely).
Proof. See Theorem 6.1 and Lemma 6.4 in [9].
In [1] Hajir describes a construction showing that the growth of the
p-rank of the class group in an infinite unramified tower of number
fields can be linear relatively to the degree. Our idea is to iterate his
construction so as to have a slow growth of the inertia degree. It allows
us to state our main result.
Theorem 9. For any integer m there exists a number field k0 and an
infinite unramified tower k0 ⊂ k1 ⊂ k2 ⊂ . . . such that for any places
℘0 in k0 and ℘i in ki over ℘0 we have
fi = o(log (m) di)
where fi is the relative inertia degree of ℘i over ℘0 and di = [ki : k0].
Moreover if we set
2m M := 3 + 2
2m − 1 (2m2 − 1) + 2,
the root discriminant rd of the fields in the tower is bounded by
1
rd ≤ 2M ( 2 +ɛ) ,
for every ɛ > 0 and m large enough (depending on ɛ).
Proof. For simplicity we take p = 2 but the same construction can
be achieved with any prime number. Assume m is fixed. Recall that
if K / k is a field extension and S a set of places of k, S(K) denotes
the set of places of K over those in S. The proof is divided in two steps.
- Step 1. We assume we have a number field k (0)
0
having an infinite
(S(k (0)
0 ), 2)-class field tower with S = S1 ∪ . . . Sm where the sets Si are

ON THE SPLITTING RATE OF PLACES IN UNRAMIFIED TOWERS 47
disjoint sets of prime numbers. Let k (0)
0
infinite (S, 2)-class field tower of k (0)
0 . We set
and
ai =
pj∈Sj
pj
S (i) = Si ∪ · · · ∪ Sm
⊂ · · · ⊂ k (0)
n
⊂ . . . be the
for i = 1 . . . m + 1 (in particular S (1) = S and also S (m+1) is empty).
Moreover we define βi for i = 1, . . . , m by
Si(k
βi :=
(0)
0 )
2i−1 −
r1(k (0)
0 ) + r2(k (0)
0 ) − 2 (i+1) (0)
S (k o ) .
and we assume βi ≥ 1.
We now consider the following diagram of field extensions
k (0)
n
k (0)
0
k (0) √
n a1
k (0) √
0 a1
k (1)
n
k (1)
0
. . . k (m−1) √
n am−1
. . . k (m−1) √
0 am−1
k (m)
n
k (m)
0
where k (i)
n is defined recursively to be the (S (i+1) , 2)-Hilbert class field
of k (i−1)
n
tension of k (0)
0
( √ ai) for i = 1 . . . m. It is clear that k (m)
n is an unramified ex-
√
a1 . . . am−1 . The infinite unramified tower we finally
consider is the tower
k (0)
0 ( √ a1 . . . am−1) ⊂ k (m)
0 ⊂ · · · ⊂ k (m)
n .
Let ℘n be any place of k (m)
n and ℘0 (respectively ℘) denote its restriction
to k (0) √ (0)
0 a1 . . . am−1 (resp. to k 0 ). Let fn denote the rel-
ative inertia degree of ℘n over ℘ and set di = [k (i)
n : k (0)
0 ] (in fact we
should write di(n) but for simplicity we forget about n for the moment).
Since they differ from the same parameters relatively to the extension
k (m)
n / k (0) √
0 a1 . . . am−1 from a factor at most 2 we can compare the
growth of fn and dm(n).
Let us estimate fn. In the extension k (i) √ (i)
n ai+1 / k n the relative
inertia degree is at
most 2. The same thing is true in the extension
since this extension is elementary of type (2, . . . , 2).
k (i+1)
n / k (i) √
n ai+1
Hence if f ′ n denotes the relative inertia degree of the place in the extension
k (0)
n / k (0)
0 of degree d0 we have
fn ≤ 2mf ′ n = o(d0)

48 ALEXANDRE TEMKINE
by Proposition 5.
Now let us compare d0 and dm+1. To do this we fix the following
notation for i = 1, . . . , m:
• s (i+1)
:=
S (i+1)
),
• r2 := r2(k (i−1)
n ),
• r := Si(k (i−1)
n ) ,
• r1 := r1(k (i−1)
n
k (i−1)
n
• ρi := ρS (i+1) (i−1)
(k n ( √ ai)) ,
• d ′ n := [k (0)
n : k (0)
0 ].
( √
,
ai)
These notation fit with those of Proposition 8 that we want to apply to
the extension k (i−1)
n ( √ ai) / k (i−1)
n . Moreover with the notation of this
Proposition we have
• τ = 0,
• δ2 = 0,
• t0 ≥ 0.
We have of course
and
r1 = di−1r1(k (0)
0 ),
r2 = di−1r2(k (0)
0 ),
s (i+1)
≤ 2di−1 S (i+1) (k (0)
0 ) .
The set Si(k (0)
0 ) is totally split in k (i−1)
n
/ k (0)
0
except possibly in the
intermediary extensions k (j)
n ( √ aj+1) / k (j)
n for j = 0 . . . i − 2. Hence we
have
r ≥ di−1
2i−1
Si(k (0)
0 ) .
Using Proposition 8 it comes
ρi ≥ di−1
2i−1
Si(k (0)
0 ) − di−1 r1(k (0)
0 ) + r2(k (0)
0 ) − 1 − 2di−1 S (i+1) (k (0)
≥
0 )
⎛
Si(k
di−1 ⎝
(0)
0 )
2i−1 −
r1(k (0)
0 ) + r2(k (0)
0 ) − 2 S (i+1) (k (0)
⎞
0 ) ⎠
− 1
≥ di−1βi − 1.
Now we have
[k (i)
n : k (i−1)
n
and this implies
] = [k (i)
since βi ≥ 1. It comes
n : k (i−1)
n
[k (i)
n : k (i−1)
n ] = di
di−1
( √ ai)][k (i−1)
n
di ≥ di−12 di−1
( √ ai) : k (i−1)
n ] = 2 ρi+1
,
= 2 ρi+1 ≥ 2 di−1

so that
ON THE SPLITTING RATE OF PLACES IN UNRAMIFIED TOWERS 49
di−1 = O(log di)
for i = 1, . . . , m. Finally we obtain
and
d0 = O(log (m) dm).
fn = o(log (m) dm(n)).
- Step 2. To complete the proof we need to construct a field k and a
set of primes S = S1 ∪ · · · ∪ Sm such that the (S(k), 2)-class field tower
of k is infinite and the previously defined βi are greater than one. We
also need to estimate the root discriminant of the bottom field of the
tower. Our field k will be an imaginary quadratic field. We set for
i = 0, . . . , m − 1:
• sm−i = |Sm−i(k)| = (2 m ) i+1 ,
• s (m−i) = S (m−i) (k) .
Under these conditions βm−i ≥ 1 is implied by
Since
we have
sm−i ≥ 2 m−i (s (m−i+1) + 1).
s (m−i+1) + 1 = sm−i+1 + · · · + sm−1 + sm + 1
= (2 m ) i + · · · + 2 m + 1
= (2m ) i+1 − 1
2 m − 1
≤ (2m ) i+1
2 m − 1 ,
2 m−i (s (m−i+1) + 1) ≤
m(i+1) 2m−i
2
2m − 1
≤ 2 m(i+1)
≤ sm−i,
for i ≥ 1, so that βm−i ≥ 1 for i ≥ 1. Of course we also have trivially
βm ≥ 1.
It now suffices to have a quadratic imaginary field with at least
s = s (1) = 2m
2m − 1 (2m2 − 1)
primes splitting totally in its infinite (S, 2)-class field tower. This is
easely achieved by use of Proposition 8. With the notation of this
Proposition we now have:
• t = s (1) ,
• t0 = s (1) ,
• τ = 1,

50 ALEXANDRE TEMKINE
• r1 = 1,
• r2 = 0,
• δ2 = 1.
Thus we take r := 3 + 2 √ 2 + s (1) and k = Q
−p (1) . . . p (r) where
p (i) is the i th prime number and a set S consisting of a subset of size
s (1) of the set of primes in k over p (r+1) , . . . , p (r+s(1) ) (or even all this
set by taking S1 larger than planed). According to the first step of
the
proof, the bottom field of the tower with large splitting rate is
k p (r+1) . . . p (r+s (1)
) . Its root discriminant (and thus the root dis-
criminant of all the fields in the tower) is
⎛
r+s
rd = 2 ⎝
(1)
p i
⎞
⎠
Using the prime number theorem and the estimation
p ≤ e (1+ɛ)n
p≤n
i=1
for n ≥ n0(ɛ) (see [8], p.20) we obtain the announced upper estimation
for rd.
We end this paper by a few questions.
Questions. In Theorem 9, the estimation of the root discriminant is
going to infinity very fast with m. Is it necessarely so? More precisely,
can we have such examples with more reasonnable root discriminant?
At best, is it possible to bound the root discriminant independently of
m? This question is clearly related to the following one: does there
exist an infinite unramified tower of number fields k0 ⊂ k1 ⊂ k2 ⊂ . . .
such that with the previously defined fi and di we have for every place
1/2
fi = o(log (m) di) for every m?
As explained in the introduction the splitting rate of places in such
a tower probably cannot be too great. However we do not have any
explicit lower bound for fi. As pointed out by Tsfasman in a personnal
communication, the proof above suggests a natural lower bound. Recall
that log ∗ is the function defined by
log ∗ x = m if and only if log (m) x < 1 and log (m−1) x ≥ 1.
Then it is natural to ask if there exists a constant M possibly depending
on k such that with the preceeding notation
fi ≥ M log ∗ di?
,

ON THE SPLITTING RATE OF PLACES IN UNRAMIFIED TOWERS 51
References
[1] Hajir, F.:On the growth of p-class groups in p-class field towers J. Algebra,
1997, 188, 256-271.
[2] Hajir, F., Maire, C.:Tamely ramified towers and discriminant bounds for number
fields, Preprint Université de Bordeaux I, 1999.
[3] Hajir, F., Maire, C.:Tamely ramified towers and discriminant bounds for number
fields II, in preparation.
[4] Maire, C.:Finitude de tours et p-tours T -ramifiées modérées, S-décomposées,
J. de Théorie des Nombres de Bordeaux, 1996, 8, 47-73.
[5] Martinet, J.:Tours de corps de classes et estimations de discriminants, Invent.
Math., 1978, 44, 65-73.
[6] Odlyzko, A.M.: Lower bounds for discriminant of number fields, Acta. Arith.,
1976, 29, 275-297.
[7] Poitou, G.: Sur les petits discriminants, Sém. Delange-Pisot-Poitou,
1976/1977, exp. 6.
[8] Tenenbaum, G.: Introduction a la théorie analytique et probabilistique des
nombres, Cours Spécialisés, Collection SMF, Numéro 1, Soc. Math. de France
(1995).
[9] Tsfasman, M.A., Vlădut¸, S.G.: Asymptotic Properties of Global Fields
and Generalized Brauer-Siegel Theorem, preprint 99-30, Institut de
Mathématiques de Luminy, 1999, to be published in the Duke Math. Journal.
Équipe “Arithmétique et Théorie de l’Information”
I.M.L., C.N.R.S.
Luminy Case 930, 13288 Marseille Cedex 9 - FRANCE
E-mail address: temkine@iml.univ-mrs.fr

RÉSUMÉ : Au centre de cette thèse se trouve la notion de tour de corps
de classes de Hilbert. Etant donné un corps global, c’est-à-dire soit un corps
de nombres, soit un corps de fonctions à une variable sur un corps fini, et
un ensemble fini de places contenant les places archimédiennes dans le cas
des corps de nombres, on peut lui associer par la théorie du corps de classes
son corps de classes de Hilbert, et par itération, une suite croissante pour
l’inclusion de corps globaux. Sous une condition donnée par le Théorème de
Golod-Shafarevich, cette suite est strictement croissante. Ce résultat bien
connu a plusieurs applications : dans le cas des corps de fonctions, ou de
manière équivalente des courbes algébriques sur les corps finis, il nous permet
d’améliorer des minorations de A(q), paramètre qui mesure le nombre
asymptotique maximal de points rationnels d’une telle courbe définie sur Fq.
C’est l’objet des deux premières parties. Dans la troisième partie, nous utilisons
ce théorème dans le cadre des corps de nombres pour prouver l’existence
de familles asymptotiquement bonnes de réseaux unimodulaires. Enfin, l’objet
de la quatrième partie est l’étude dans le cadre des corps de nombres du
taux de décomposition des places dans les tours infinies de corps de classes
de Hilbert.
TITLE :
Hilbert class field towers
for global fields and applications
ABSTRACT : Given a global field, i.e., a number field or a function field in
one variable over a finite field, and a finite set of places including the infinite
ones in the case of number fields, class field theory allows us to define its
Hilbert class field. Iterating this process we obtain an increasing sequence
of global fields. Under a condition given by the Golod-Shafarevich Theorem,
the sequence is strictly increasing. This well-known result has several applications
: in the function field case, it allows us to improve lower bounds for
A(q), a number that measures the asymptotic maximal number of rational
points of an algebraic curve defined over Fq. It is the aim of the first two
parts. In the third part we use this theorem in the setting of number fields
to prove the existence of asymptotically good families of unimodular lattices.
The last part is devoted to the study of the rate of decomposition of places
in infinite Hilbert class field towers of number fields.
DISCIPLINE : Mathématiques
MOTS-CLÉS :
Corps de nombres Corps de fonctions
Corps finis Corps de classes
Points rationnels Réseaux
Institut de Mathématiques de Luminy - UPR 9016

163 Av. de Luminy - case 907
13288 Marseille Cedex 9 - France