Journal de Théorie des Nombres de Bordeaux

Kazuomi Sawatani

Takao Watanabe

Kenji Okuda

Kazuomi SAWATANI, Takao WATANABE et Kenji OKUDA
A note on the Hermite–Rankin constant
Tome 22, no 1 (2010), p. 209-217.
<http://jtnb.cedram.org/item?id=JTNB_2010__22_1_209_0>
© Université Bordeaux 1, 2010, tous droits réservés.
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Article mis en ligne dans le cadre du
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Journal de Théorie des Nombres
de Bordeaux 22 (2010), 209-217
A note on the Hermite–Rankin constant
par Kazuomi SAWATANI, Takao WATANABE et Kenji OKUDA
To the memory of Anne-Marie Bergé
Résumé. Nous généralisons l’inégalité de Poor et Yuen au cas des
constantes γn,k d’Hermite-Rankin et γ′n,k de Bergé–Martinet. En
outre, nous donnons les valeurs exactes de certaines constantes
d’Hermite-Rankin et de Bergé–Martinet de petite dimension en
appliquant certaines inégalités démontrées par Bergé et Martinet
aux valeurs explicites de γ′5, γ′7, γ4,2 et γn, n ≤ 8.
Abstract. We generalize Poor and Yuen’s inequality to the
Hermite–Rankin constant γn,k and the Bergé–Martinet constant
γ′n,k. Moreover, we determine explicit values of some low-
dimensional Hermite–Rankin and Bergé–Martinet constants by
applying Rankin’s inequality and some inequalities proven by
Bergé and Martinet to explicit values of γ′5, γ′7, γ4,2 and γn (n 5 8).
1. Introduction
In the recent paper [PY3], Poor and Yuen proved an inequality among
the Hermite constant γn, the Bergé–Martinet constant γ′n and the constant
cn defined from the dyadic trace. They also studied the condition of the
equality (γ′n)2 = n/cn and applied this to determine low dimensional Bergé–
Martinet’s constants γ′5, γ′6 and γ′7.
In the first half of this paper, we generalize Poor and Yuen’s inequality to
the Hermite–Rankin constant γn,k and the Bergé–Martinet constant γ′n,k. In
the second half, we show by using Rankin’s inequality and some inequalities
proven by Bergé and Martinet in [BM] that explicit values of γ′5, γ′7, γ4,2
and γn (n 5 8) lead us to explicit values of γ6,2, γ′6,2, γ8,2, γ′8,2, γ8,3, γ′8,3
and γ8,4 = γ′8,4.
2. A generalization of Poor and Yuen’s inequality
In order to define γn,k and γ′n,k, we start from the definition of type one
functions. Let Vn be the real vector space of n×n real symmetric matrices
Manuscrit reçu le 31 octobre 2008, révisé le 14 janvier 2009.
210 Kazuomi Sawatani, Takao Watanabe, Kenji Okuda
and Pn the open cone of positive definite symmetric matrices in Vn. We
define the inner product 〈, 〉 : Vn × Vn −→ R by
〈s, t〉 = Tr(st)
for s, t ∈ Vn. A function φ : Pn −→ R>0 is called a type one function if φ
satisfies
(i) φ(λs) = λφ(s) for all λ ∈ R>0, s ∈ Pn, and
(ii) φ(s+ t) = φ(s) + φ(t) for all s, t ∈ Pn.
In addition, if φ is class invariant, i.e., φ(tgsg) = φ(s) holds for all s ∈ Pn
and g ∈ GLn(Z), then φ is called a type one class function. A type one
function is continuous on Pn ([PY, Proposition 2.2]).
Let φ be a type one function. The dual function φ̂ : Pn −→ R>0 of φ is
defined to be
φ̂(s) = inf
t∈Pn
〈s, t〉
φ(t)
.
This φ̂ is also a type one function, and the dual of φ̂ equals φ. If φ is class
invariant, then so is φ̂. A typical example of type one class functions is
given by s 7→
√
n(det s)1/n, which is self-dual.
In order to give another example of type one class functions, we fix a
positive integer k with 1 5 k 5 n − 1. Let Pn be the closure of Pn in
Vn, i.e., Pn is the closed cone of positive semi-definite matrices. Define the
function mk : Pn −→ R=0 by
mk(s) = inf
x1,··· ,xk∈Zn
x1∧···∧xk 6=0
det((txisxj)i,j)
for s ∈ Pn. It is obvious that mk(s) > 0 if s ∈ Pn, and mk(s) = 0 otherwise.
Lemma 2.1. The function m′k(s) = mk(s)1/k in s ∈ Pn is a type one class
function.
Proof. The class invariance and the condition (i) for m′k are trivial. We
show the condition (ii). Since the function x 7→ (detx)1/k in x ∈ Pk is a
type one function, we have
m′k(s+ t) = inf
x1,··· ,xk∈Zn
x1∧···∧xk 6=0
det((txi(s+ t)xj)i,j)1/k
= inf
x1,··· ,xk∈Zn
x1∧···∧xk 6=0
{
det((txisxj)i,j)1/k + det((txitxj)i,j)1/k
}
= inf
x1,··· ,xk∈Zn
x1∧···∧xk 6=0
det((txisxj)i,j)1/k + inf
x1,··· ,xk∈Zn
x1∧···∧xk 6=0
det((txitxj)i,j)1/k
= m′k(s) +m′k(t) .
This shows that m′k is a type one function. 
A note on the Hermite–Rankin constant 211
If k = 1, then m(s) = m′1(s) is none other than the minimum of the
quadratic form x 7→ txsx on the set Zn\{0} of non-zero lattice points. The
dual of m is the dyadic trace, i.e.,
w(s) = m̂(s) = inf
t∈Pn
〈s, t〉
m(t)
,
which plays an important role in Poor – Yuen’s theory.
By using mk, the Hermite–Rankin constant γn,k and the Bergé–Martinet
constant γ′n,k are defined as follows:
γn,k = sup
s∈Pn
mk(s)
(det s)k/n
and γ′n,k = sup
s∈Pn
√
mk(s)mk(s−1) .
A generalization of w(s)/m(s) is given by
cn,k(s) = inf
t∈Pn
〈s, t〉k
mk(s)mk(t)
.
We set
cn,k = inf
s∈Pn
cn,k(s) = inf
(s,t)∈Pn×Pn
〈s, t〉k
mk(s)mk(t)
.
Lemma 2.2. The infimum
inf
(s,t)∈Pn×Pn
〈s, t〉k
mk(s)mk(t)
is attained at some (s0, t0) ∈ Pn × Pn.
Proof. Since mk(s) 5 γn,k(det s)k/n holds for all s ∈ Pn, the function
s 7→ mk(s) is continuous on Pn. Then, by [PY, Lemma 3.6], the infimum
wk(s) = inf
t∈Pn
〈s, t〉
mk(t)1/k
is attained at some ts ∈ Pn. Since wk is the dual of m′k, wk is also a type
one class function, and hence wk(s)/mk(s)1/k depends only on the similar
GLn(Z)-equivalent class R>0 · [s] of s. Therefore, we have
c
1/k
n,k = infs∈Pn
wk(s)
mk(s)1/k
= inf
[s]∈Pn/GLn(Z)
mk(s)=1
wk(s) .
For a constant c > cn,k, we consider the non-empty set
Ωc = {[s] ∈ Pn/GLn(Z) : mk(s) = 1 and wk(s) 5 c1/k} .
From the inequality of the arithmetic and the geometric means, it follow
that
(2.1) n(det s)1/n(det t)1/n 5 〈s, t〉
212 Kazuomi Sawatani, Takao Watanabe, Kenji Okuda
for all (s, t) ∈ Pn × Pn (cf. [BC, Theorem 1]). By using (2.1) and the
definition of γn,k, we have
n · (det s)
1/n
γ
1/k
n,k
5
〈s, ts〉
mk(ts)1/k
= wk(s) .
This implies that Ωc is a subset of
Ω′c = {[s] ∈ Pn/GLn(Z) : mk(s) = 1 and det s 5 n−ncn/k(γn,k)n/k} .
As a consequence, c1/kn,k is represented by
c
1/k
n,k = inf[s]∈Ω′c
wk(s) .
Since Ω′c is compact in Pn/GLn(Z) by [C, Proposition 2.2] and wk is con-
tinuous, this infimum is attained at some [s0] ∈ Ω′c. 
In the case of k = 1, Poor and Yuen proved the inequality:
(2.2) 1
(γn,1)2
5
cn,1
n
5
1
(γ′n,1)2
.
We note that Barnes and Cohn ([BC]) also proved the first inequality
of (2.2).
Theorem 2.1. For 1 5 k 5 n− 1, we have
1
(γn,k)2
5
cn,k
nk
5
1
(γ′n,k)2
.
The equality (γn,k)−2 = cn,k/nk holds if and only if γn,k = γ′n,k.
Proof. By the inequality (2.1), we have
nk · inf
s∈Pn
(det s)k/n
mk(s)
· inf
t∈Pn
(det t)k/n
mk(t)
5 cn,k ,
and hence
nk
(
1
γn,k
)2
5 cn,k .
By the definition of cn,k, the inequality
cn,k 5
〈s, s−1〉k
mk(s)mk(s−1)
= n
k
mk(s)mk(s−1)
holds for all s ∈ Pn. Therefore,
mk(s)mk(s−1) 5
nk
cn,k
.
A note on the Hermite–Rankin constant 213
This gives
(γ′n,k)2 5
nk
cn,k
.
It is known that the equality
n(det s)1/n(det t)1/n = 〈s, t〉
holds if and only if t is similar to s−1. By Lemma 2.2, there exists (s0, t0) ∈
Pn × Pn such that
cn,k =
〈s0, t0〉k
mk(s0)mk(t0)
.
In general, we have
nk
(γn,k)2
5
nk(det s0)k/n(det t0)k/n
mk(s0)mk(t0)
5
〈s0, t0〉k
mk(s0)mk(t0)
= cn,k .
Thus, if (γn,k)−2 = cn,k/nk holds, then t0 must be similar to s−10 and γn,k
is attained at both s0 and s−10 . This implies γn,k = γ′n,k. 
3. Explicit values of some Hermite–Rankin constants
In [BM], Bergé and Martinet proved several inequalities involving γn,k
and γ′n,k, and determined the values γ′2,1 = 2/
√
3, γ′3,1 =
√
3/2 and γ′4,1 =√
2. In the inequality (2.2), Poor and Yuen proved the equality (γ′n,1)2 =
n/cn,1 holds for n 5 8 and n = 24 ([PY3, Theorem 2.4]). By using this,
they determined the following explicit values of Bergé–Martinet constants:
γ′5,1 =
√
2, γ′6,1 =
√
8/3, γ′7,1 =
√
3 .
These explicit values give the following:
Theorem 3.1. γ8,3 = γ8,4 = γ′8,4 = 4.
Proof. If n is even, then the equality γn,n/2 = γ′n,n/2 holds in general ([M,
Corollary 2.8.8]). We recall Rankin’s inequality:
(3.1) γn,k 5 γh,k(γn,h)k/h
holds for 1 5 k < h < n. In the case of n = 8, h = 4, k = 1, we have(
γ8,1
γ4,1
)4
5 γ8,4 .
Since γ8,1 = 2 and γ4,1 =
√
2, this gives 4 5 γ8,4. On the other hand, the
following inequality is known for 1 5 k 5 n/2 ([M, Theorem 2.8.7]):
(3.2) γ′n,2k 5 (γ′n−k,k)2 .
214 Kazuomi Sawatani, Takao Watanabe, Kenji Okuda
We use (3.2) twice. Namely, if we put n = 8, k = 2 and n = 6, k = 1, then
(3.3) 4 5 γ8,4 = γ′8,4 5 (γ′6,2)2 5 (γ′5,1)4 = 4 .
By applying (3.1) to the cases n = 8, h = 4, k = 3 and n = 8, h = 3, k = 1,
we obtain
γ8,3 5 γ4,3(γ8,4)3/4 = 41/4 · 43/4 = 4
and
4 =
(
γ8,1
γ3,1
)3
5 γ8,3 .

Corollary 3.1. γ′6,2 = 2, γ6,2 = 32/3.
Proof. By (3.3), γ′6,2 = 2 is trivial. Rankin’s inequality (3.1) gives for n =
6, h = 2, k = 1 (
γ6,1
γ2,1
)2
5 γ6,2 .
Since, γ6,1 = (64/3)1/6 and γ2,1 =
√
4/3, we have 32/3 5 γ6,2. On the other
hand, the inequality
(3.4) (γn,k)n 5 (γn−k,k)n−k(γ′n,k)2k
is known for 1 5 k 5 n/2 ([M, Theorem 2.8.7]). By putting n = 6, k = 2,
we obtain
(γ6,2)6 5 (γ4,2 · γ′6,2)4 = 34
because of γ4,2 = 3/2. 
Corollary 3.2. γ8,2 = γ′8,2 = 3.
Proof. Rankin’s inequality gives for n = 8, h = 2, k = 1
3 =
(
γ8,1
γ2,1
)2
5 γ8,2 .
By (3.2), we have
γ′8,2 5 (γ′7,1)2 = 3 .
Then, by (3.4),
3 5 γ8,2 5 (γ6,2)3/4(γ′8,2)1/2 5 3 .

We show that all γ8,2, γ8,3 and γ8,4 are attained on the E8 lattice and
γ6,2 is attained on the E6 lattice. For a full lattice Λ in the Euclidean space
Rn, det Λ denotes the determinant of Λ, i.e., det Λ = (det g)2 if Λ = gZn
A note on the Hermite–Rankin constant 215
with g ∈ GLn(R). The Hermite–Rankin invariant γn,k(Λ) and the Bergé–
Martinet invarinat γ′n,k(Λ) of Λ are defined as
γn,k(Λ) = inf
Λ′
γ(Λ,Λ′) , where γ(Λ,Λ′) = det Λ
′
(det Λ)k/n
and Λ′ runs over all sublattices in Λ of rank k, and then
γ′n,k(Λ) =
√
γn,k(Λ)γn,k(Λ∗) ,
where Λ∗ denotes the dual lattice of Λ. In terms of lattices, γn,k and γ′n,k
are given by
γn,k = sup
Λ
γn,k(Λ) , γ′n,k = sup
Λ
γ′n,k(Λ) ,
where Λ runs over all full lattices in Rn.
Proposition 3.1. One has γ8,2 = γ8,2(E8) = γ(E8, A2), γ8,3 = γ8,3(E8) =
γ(E8, A3), γ8,4 = γ8,4(E8) = γ(E8, D4) and γ6,2 = γ6,2(E6) = γ(E6, A2).
Proof. From the proof of Rankin’s inequality, it follows that
(3.5) γn,k(Λ) 5 γh,k · (γn,h(Λ))k/h
holds for any lattice Λ of rank n and 1 5 k < h < n (cf. [M, Proof of
Theorem 2.8.6]). For Λ = E8 and n = 8, h = 2, k = 1, (3.5) gives
2 = γ8,1(E8) 5
2√
3
· (γ8,2(E8))1/2.
This implies
3 5 γ8,2(E8) 5 γ8,2 = 3 .
From the table [M, Table 4.10.13], it follows γ(E8, A2) = 3. By a similar
fashion, we have γ8,3 = γ8,3(E8), γ8,4 = γ8,4(E8) and γ6,2 = γ6,2(E6). 
We note that Coulangeon computed γ8,2(E8) = 3, γ7,2(E7) = 3/22/7 and
γ6,2(E6) = 32/3 by another method ([C, Théorème 5.1.1]).
Corollary 3.3. γ′8,3 = 4.
Proof. Since E8 is self-dual, the Bergé–Martinet invariant γ′8,3(E8) coincides
with γ8,3(E8), i.e., γ′8,3(E8) = 4. Then, by [M, Proposition 2.8.4],
4 = γ′8,3(E8) 5 γ′8,3 5 γ8,3 = 4 .

By Theorem 3.1, Corollaries 3.2, 3.3 and the duality relation ([M, Propo-
sition 2.8.5]), Hermite–Rankin and Bergé–Martinet constants of dimension
8 are completely determined.
216 Kazuomi Sawatani, Takao Watanabe, Kenji Okuda
Proposition 3.2. One has
(3.6) 4√
3
5 γ6,3 5
√
6 and 211/7 5 γ7,3 5 24/7 · 32/3 .
These lower bounds are attained on E6 and E7, i.e., we have
γ6,3(E6) = γ(E6, A3) =
4√
3
, γ7,3(E7) = γ(E7, A3) = 211/7
and moreover
γ′7,2(E7) =
√
6 , γ′7,3(E7) = 2
√
2 .
Proof. From Rankin’s inequality (3.1) for n = 8, h = 6, k = 3, it follows
4/
√
3 5 γ6,3. All other inequalities of (3.6) are also obtained by Rankin’s
inequality. Let Λ′ be a sublattice of E6 of rank 3 which attains γ6,3(E6).
Since γ(E8, E6) = 3, we have
γ(E8,Λ′) = det Λ′ = γ6,3(E6)γ(E8, E6)1/2 .
Therefore,
4 = γ8,3(E8) 5 γ(E8,Λ′) = γ6,3(E6)
√
3 .
We know γ(E6, A3) = 4/
√
3 by [M, Table 4.10.13], and hence
4√
3
5 γ6,3(E6) 5 γ(E6, A3) =
4√
3
.
Smilarly, we obtain γ7,3(E7) = 211/7, γ7,4(E7) = γ(E7, D4) = 210/7, and
γ7,5(E7) = γ(E7, D5) = 29/7. Then γ′7,2(E7) =
√
γ7,2(E7)γ7,5(E7) =
√
6
and γ′7,3(E7) =
√
γ7,3(E7)γ7,4(E7) = 2
√
2 follows from the duality relation
γn,k(Λ∗) = γn,n−k(Λ) ([M, Proposition 2.8.5]). 
The same argument as in the proof of Propositon 3.2 yields γ6,4(E6) =
γ6,2(E∗6) = 4/32/3. This implies γ′6,2(E6) = 2 = γ′6,2, namely γ′6,2 is attained
on E6. Since E8 is self-dual, all γ′8,2, γ′8,3, γ′8,4 are attained on E8.
Mayer [Ma, Théorème 3.59] extended Theorem 2.1 to generalized Her-
mite constants over an algebraic number field and applied it to deter-
mine the values γ′3,1(Q(
√
−1)) = γ′3,1(Q(
√
−3)) = 2, γ′4,1(Q(
√
−1)) = 4,
γ′4,1(Q(
√
−3)) = 3 and γ4,2(Q(
√
−1)) = γ4,2(Q(
√
−3)) = 4 ([Ma, Proposi-
tions 3.66, 3.67]).
References
[BC] E. S. Barnes and M. J. Cohn, On the inner product of positive quadratic forms.
J. London Math. Soc. (2) 12 (1975), 32–36.
[BM] A.-M. Bergé and J. Martinet, Sur un problème de dualité lié aux sphères en géométrie
des nombres. J. Number Theory 32 (1989), 14–42.
[C] R. Coulangeon, Réseaux k-extrêmes. Proc. London Math. Soc. 73 (1996), 555–574.
[M] J. Martinet, Perfect Lattices in Euclidean Spaces. Springer-Verlag, 2003.
A note on the Hermite–Rankin constant 217
[Ma] B. Mayer, Constantes d’Hermite et théorie de Voronoï . Thése, Université Bordeaux 1,
2008.
[PY] C. Poor and D. S. Yuen, Linear dependence among Siegel modular forms. Math. Ann.
318 (2000), 205–234.
[PY2] C. Poor and D. S. Yuen, The extreme core. Abh. Math. Sem. Univ. Hamburg 75 (2005),
1–25.
[PY3] C. Poor and D. S. Yuen, The Bergé–Martinet constant and slopes of Siegel cusp forms.
Bull. London Math. Soc. 38 (2006), 913–924.
[R] R. A. Rankin, On positive definite quadratic forms. J. London Math. Soc. 28 (1953),
309–314.
Kazuomi Sawatani
Takao Watanabe
Kenji Okuda
Department of Mathematics
Graduate School of Science
Osaka University
Toyonaka 1-1, Osaka, Japan
E-mail: twatanabe@math.sci.osaka-u.ac.jp


A note on the Hermite–Rankin constant

Frédéric Chapoton

Frédéric CHAPOTON
Fractions de Bernoulli-Carlitz et opérateurs q-Zeta
Tome 22, no 3 (2010), p. 575-581.
<http://jtnb.cedram.org/item?id=JTNB_2010__22_3_575_0>
© Université Bordeaux 1, 2010, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nombres
de Bordeaux » (http://jtnb.cedram.org/), implique l’accord avec
les conditions générales d’utilisation (http://jtnb.cedram.org/
legal/). Toute reproduction en tout ou partie cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 22 (2010), 575-581
Fractions de Bernoulli-Carlitz et
opérateurs q-Zeta
par Frédéric CHAPOTON
Résumé. On introduit une déformation des séries de Dirichlet
d’une variable complexe s, sous la forme d’un opérateur pour
chaque nombre complexe s, agissant sur les séries formelles sans
terme constant en une variable q. On montre que les fractions de
Bernoulli-Carlitz sont les images de certains polynômes en q par
les opérateurs associés à la fonction ζ de Riemann aux entiers
négatifs.
Abstract. Bernoulli-Carlitz fractions and q-Zeta operators
We introduce a deformation of Dirichlet series of one complex
variable s, which is given for each complex number s by an op-
erator acting on formal power series without constant term in
the variable q. We prove that the Bernoulli-Carlitz fractions are
obtained as the image of some simple polynomials in q by the
operators corresponding to the Riemann ζ function at negative
integers.
1. Opérateurs associés aux séries de Dirichlet
A chaque suite (an)n≥1 de nombres complexes, on associe la série de
Dirichlet formelle
(1.1) ζ(a, s) =
∑
n≥1
an
ns
,
où s est un nombre complexe. Le produit de deux telles séries correspond
à la convolution des suites de nombres complexes :
(1.2) ζ(a, s)ζ(b, s) = ζ(a ∗ b, s),
où (a ∗ b)n =
∑
d|n ad bn/d.
Considérons maintenant l’anneau C[[q]] des séries formelles à coefficients
complexes en une variable q. Pour n ≥ 1, on définit le nombre quantique
(1.3) [n] = q
n − 1
q − 1
Manuscrit reçu le 16 septembre 2009.
576 Frédéric Chapoton
et l’opérateur de Frobenius formel
(1.4) Fn(f)(q) = f(qn),
agissant sur C[[q]] et par restriction sur le sous espace qC[[q]] des séries
formelles sans terme constant.
A chaque suite (an)n≥1, on peut alors associer l’opérateur
(1.5) ζq(a, s) =
∑
n≥1
anFn
[n]s
,
qui agit sur qC[[q]]. Cet opérateur est bien défini pour tout nombre com-
plexe s.
Lemme 1.1. Pour tous m,n entiers non nuls, on a la relation
(1.6) Fm
[m]s
Fn
[n]s
= Fmn
[mn]s
.
En particulier, ces deux opérateurs commutent.
Par conséquent,
Proposition 1.1. L’application a 7→ ζq(a, s) est un morphisme de l’al-
gèbre des suites de nombres complexes pour la convolution dans l’algèbre
des opérateurs linéaires sur qC[[q]].
En particulier, si la suite a est multiplicative, l’opérateur ζq(a, s) admet
un produit eulérien, exactement similaire au produit eulérien de la série de
Dirichlet ζ(a, s).
Par exemple, la fonction ζ de Riemann est la série de Dirichlet associée
à la suite constante égale à 1. On a donc un opérateur ζq défini par
(1.7) ζq(s) =
∑
n≥1
Fn
[n]s
.
Cet opérateur admet le produit eulérien
(1.8) ζq(s) =
∏
p
(
1− Fp
[p]s
)−1
,
où le produit porte sur l’ensemble des nombres premiers.
Remarque. Un opérateur Zeta proche de l’opérateur ζq(0) a été introduit
dans [7]. Il utilise plutôt la variable τ telle que q = exp(2iπτ).
Opérateurs q-Zeta 577
2. Fractions de Bernoulli-Carlitz
Introduisons maintenant les fractions de Bernoulli-Carlitz, qui vont jouer
le rôle des nombres de Bernoulli. Ces fractions ont été définies par Carlitz
[1, 2], puis étudiées par Koblitz et Satoh [6, 8]. Elles sont aussi apparues
plus récemment dans [3], avec une motivation algébrique.
On peut les définir par la récurrence suivante : on pose β0 = 1 et
(2.1) q(qβ + 1)n − βn =
{
1 si n = 1,
0 si n > 1.
Dans cette expression, par convention, on pose βi = βi pour tout entier i,
après avoir développé la puissance du binôme.
On obtient ainsi une suite de fractions rationnelles βn dans le corps Q(q).
On peut montrer (voir [1]) que ces fractions n’ont pas de pôle en q = 1 et
que la valeur de βn en q = 1 est le nombre de Bernoulli Bn.
Les premières fractions sont
β0 = 1, β1 = −
1
Φ2
, β2 =
q
Φ2Φ3
,(2.2)
β3 =
q(1− q)
Φ2Φ3Φ4
, β4 =
q(q4 − q3 − 2q2 − q + 1)
Φ2Φ3Φ4Φ5
,(2.3)
où les Φn sont les polynômes cyclotomiques.
Soit B(t) la série génératrice des βn :
(2.4) B(t) =
∑
n≥0
βn
tn
n!
.
La récurrence (2.1) se traduit en une équation fonctionnelle pour B :
(2.5) B(t) = qetB(qt) + 1− q − t.
On vérifie (en utilisant la relation 1 + q[n] = [n + 1]) que la solution de
(2.5) est donnée par
(2.6) B(t) =
∑
n≥0
(1− q)qne[n]t − tq2ne[n]t.
Par dérivations successives par rapport à t, on obtient la formule explicite
(2.7) βn =
∑
k≥1
qk[k]n−1 − (n+ 1)
∑
k≥1
q2k[k]n−1,
valable pour n ≥ 2.
En utilisant l’opérateur ζq(1− n), on obtient donc le résultat suivant.
Théorème 2.1. Pour n ≥ 2, on a
(2.8) βn = ζq(1− n)
(
q − (n+ 1)q2
)
.
578 Frédéric Chapoton
L’égalité (2.8) est un q-analogue naturel du résultat d’Euler qui relie
les valeurs prises aux entiers négatifs par la fonction ζ de Riemann et les
nombres de Bernoulli Bn : pour tout n ≥ 2, on a
(2.9) Bn = ζ(1− n)(−n).
On peut espérer déduire le résultat classique d’Euler par un passage à
la limite q = 1 en un sens approprié. Ceci nécessite sans doute un travail
analytique non trivial, qui reste à faire.
3. Opérateur q-Zeta de Hurwitz
Comme dans le cas classique, les résultats précédents ont des analogues
pour l’opérateur q-Zeta de Hurwitz.
Soit x ∈ Q>0. On note encore Fn+x l’opérateur de Frobenius formel
(3.1) Fn+x(f)(q) = f(qn+x),
qui agit sur les séries de Puiseux sans terme constant en la variable q.
On définit alors l’opérateur Zeta de Hurwitz
(3.2) ζq(s, x) =
∑
n≥0
Fn+x
[n+ x]s
.
Proposition 3.1. Pour tout N ≥ 1 et tout s dans C, on a la relation de
distribution
(3.3)
∑
0≤j<N
FN
[N ]s
ζq(s,
x+ j
N
) = ζq(s, x).
La preuve est immédiate, en utilisant la définition (3.2) et une version
adaptée du Lemme 1.1.
Par ailleurs, on introduit (d’après Carlitz) des q-polynômes de Bernoulli
par la formule symbolique
(3.4) βn(x) = (qxβ + [x])n.
Ce sont des polynômes en la variable qx à coefficients dans Q(q).
La série génératrice des q-polynômes de Bernoulli est alors
(3.5) e[x]tB(qxt).
La série génératrice des βn(x) est donc explicitement donnée par
(3.6) e[x]t
∑
n≥0
(1− q)qne[n]qxt − tq2n+xe[n]qxt.
soit encore (par la relation [x] + qx[n] = [n+ x])
(3.7)
∑
n≥0
(1− q)qne[n+x]t − tq2n+xe[n+x]t.
Opérateurs q-Zeta 579
Par dérivations successives par rapport à t, on obtient la formule explicite
(3.8) qxβn(x) =
∑
k≥0
qk+x[k + x]n−1 − (n+ 1)
∑
k≥0
q2k+2x[k + x]n−1,
valable pour n ≥ 0.
En exprimant ceci avec l’opérateur q-Zeta de Hurwitz, on trouve l’égalité
suivante.
Proposition 3.2. Pour n ≥ 0, on a la relation
(3.9) qxβn(x) = ζq(1− n, x)
(
q − (n+ 1)q2
)
.
4. Opérateurs q-L de Dirichlet
Comme dans le cas classique, on peut généraliser les résultats précédents
aux opérateurs q-L associés aux caractères de Dirichlet, en utilisant l’opé-
rateur q-Zeta de Hurwitz.
Soit χ un caractère de Dirichlet de conducteurN . On définit un opérateur
(4.1) Lq(χ, s) =
∑
n≥1
χ(n)Fn
[n]s
.
On suppose maintenant que χ n’est pas un caractère trivial, pour sim-
plifier.
Proposition 4.1. On a
(4.2) Lq(χ, s) =
∑
0≤j<N
χ(j) FN
[N ]s
ζq(s, j/N).
La preuve est similaire à celle de la relation de distribution.
On introduit alors les analogues des fractions de Bernoulli-Carlitz pour
le caractère χ :
(4.3) βχ,n =
∑
0≤j<N
χ(j) FN
[N ]1−n
(
qj/Nβn(j/N)
)
.
On déduit des propositions 3.2 et 4.1 l’égalité suivante.
Proposition 4.2. Pour n ≥ 0, on a
(4.4) βχ,n = Lq(χ, 1− n)
(
q − (n+ 1)q2
)
.
5. Calcul des valeurs aux entiers négatifs
Soit χ un caractère de Dirichlet et r un entier strictement positif. Comme
χ est périodique, la somme fχ(r) =
∑
n≥1 χ(n)qnr est une fraction ration-
nelle. C’est aussi la valeur Lq(χ, 0)qr. Par exemple, pour le caractère trivial,
on obtient qr/(1− qr).
580 Frédéric Chapoton
Si f est une fonction de la variable r, on note ∆ l’opérateur suivant (qui
est une forme de q-différence) :
(5.1) ∆(f) = f(r + 1)− f(r)
q − 1
.
L’opérateur ∆ agit sur l’ensemble des fractions en q et qr qui, en tant
que fonction de q pour tout r entier fixé, ont des pôles seulement sur le
cercle unité et sont nulles en zéro.
L’opérateur ∆ vérifie
(5.2) ∆(qnr) = [n]qnr.
On a donc, pour tout i ≤ 0 et tout entier r ≥ 1, la valeur
(5.3) Lq(χ, i)qr = ∆−ifχ.
Il en résulte donc que toutes ces valeurs sont des fractions rationnelles
en q, nulles en zéro et avec pôles sur le cercle unité.
Proposition 5.1. Soit P un polynôme en q sans terme constant. Si i ≤ 0,
alors Lq(χ, i)P est une fraction rationnelle en q.
6. Remarques et spéculations
Parmi les différentes fonctions q-Zeta qui sont apparues dans la littéra-
ture [4, 9, 5], certaines sont obtenues par application de l’opérateur q-Zeta
à diverses puissances de q. Il paraît plus naturel d’étudier l’opérateur lui-
même plutôt qu’un choix arbitraire de ses valeurs.
On peut imaginer que les valeurs spéciales des fonctions L correspondent
à l’évaluation des opérateurs q-L en des arguments particuliers. Il faudrait
écrire la valeur spéciale comme un quotient et trouver des q-analogues du
numérateur et du dénominateur.
Vu son produit eulérien, on peut aussi imaginer l’existence d’une équa-
tion fonctionnelle pour l’opérateur ζq(s). Il s’agirait de compléter le produit
eulérien en introduisant un opérateur γ(s) correspondant au facteur archi-
médien. Cet opérateur devrait commuter avec tous les opérateurs Fn[n]s , pour
que le produit eulérien complété reste totalement commutatif.
On observe numériquement la propriété suivante des fractions de
Bernoulli-Carlitz. Leur numérateur a quelques zéros réels positifs, beau-
coup de zéros sur le cercle unité et quelque paires de zéros complexes. Ceci
semble aussi vrai pour les fractions similaires associées aux caractères de
Dirichlet.
Opérateurs q-Zeta 581
Bibliographie
1. Leonard Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987–1000.
MR MR0027288 (10,283g)
2. , q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954), 332–350.
MR MR0060538 (15,686a)
3. Frédéric Chapoton, A rooted-trees q-series lifting a one-parameter family of Lie idempotents,
Algebra Number Theory 3 (2009), no. 6, 611–636. MR 2579388
4. Masanobu Kaneko, Nobushige Kurokawa, and Masato Wakayama, A variation of Euler’s
approach to values of the Riemann zeta function, Kyushu J. Math. 57 (2003), no. 1, 175–192.
MR MR2069738 (2005e :11102)
5. Taekyun Kim, q-Riemann zeta function, Int. J. Math. Math. Sci. (2004), no. 9-12, 599–605.
MR MR2048800 (2005d :11167)
6. Neal Koblitz, On Carlitz’s q-Bernoulli numbers, J. Number Theory 14 (1982), no. 3, 332–339.
MR MR660378 (83k :12017)
7. Ralf Meyer, A spectral interpretation for the zeros of the Riemann zeta function, Mathe-
matisches Institut, Georg-August-Universität Göttingen : Seminars Winter Term 2004/2005,
Universitätsdrucke Göttingen, Göttingen, 2005, pp. 117–137. MR MR2206883 (2006k :11166)
8. Junya Satoh, q-analogue of Riemann’s ζ-function and q-Euler numbers, J. Number Theory
31 (1989), no. 3, 346–362. MR MR993908 (90d :11132)
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Frédéric Chapoton
Institut Camille Jordan
Université Claude Bernard Lyon 1
Bâtiment Braconnier
21 Avenue Claude Bernard
69622 Villeurbanne Cedex, France
E-mail: chapoton@math.univ-lyon1.fr
URL: http://math.univ-lyon1.fr/~chapoton/


Fractions de Bernoulli-Carlitz et opérateurs $q$-Zeta

Pete L. Clark

The period-index problem in WC-groups IV: a local transition theorem

Nicola Mazzari

Nicola MAZZARI
Cohomologial dimension of Laumon 1-motives up to isogenies
Tome 22, no 3 (2010), p. 719-726.
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Journal de Théorie des Nombres
de Bordeaux 22 (2010), 719-726
Cohomologial dimension of
Laumon 1-motives up to isogenies
par Nicola MAZZARI
Résumé. Nous prouvons que la dimension cohomologique de la
catégorie des 1-motifs de Laumon à isogénie près sur un corps
de caractéristique nulle est ≤ 1. En conséquence, cela implique le
même résultat pour la catégorie des structures de Hodge formelles
de niveau ≤ 1 (sur Q).
Abstract. We prove that the category of Laumon 1-motives up
to isogenies over a field of characteristic zero is of cohomological
dimension ≤ 1. As a consequence this implies the same result for
the category of formal Hodge structures of level ≤ 1 (over Q).
1. Introduction
In [6] P. Deligne defined a 1-motive over a field k as Gal(ksep|k)-equivar-
iant morphism [u : X → G(ksep)] where X is a free Gal(ksep|k)-module
and G is a semi-abelian algebraic group over k. They form a category that
we shall denote byM1,k orM1.
Deligne’s definition was motivated by Hodge theory. In fact the category
of 1-motives over the complex numbers is equivalent, via the so called Hodge
realization functor, to the category MHS1 of mixed Hodge structures of level
≤ 1. It is known the the category MHS1 is of cohomological dimension 1
(see [5]) and the same holds forM1,C.
F. Orgogozo proved more generally that for any field k, the category
M1,k ⊗Q is of cohomological dimension ≤ 1 (see [14, Prop. 3.2.4]).
Over a field of characteristic 0 it is possible to define the categoryMa1,k
of Laumon 1-motives generalizing that of Deligne 1-motives (See [11]). In
[3] L. Barbieri-Viale generalized the Hodge realization functor to Laumon
1-motives. He defined the category FHS1 of formal Hodge structures of level
≤ 1 containing MHS1 and proved that FHS1 is equivalent to the category
of Laumon 1-motives over C (compatibly with the Hodge realization).
In this paper we prove that the category of Laumon 1-motives up to
isogenies is of cohomological dimension 1.
Manuscrit reçu le 13 mai 2009.
Classification math.. 14C99, 14L15.
720 Nicola Mazzari
Acknowledgments. The author would like to thank L. Barbieri-Viale for
pointing his attention to this subject and for helpful discussions. The author
also thanks A. Bertapelle for many useful comments and suggestions.
2. Laumon 1-motives
Let k be a (fixed) field of characteristic zero. Let Schk be the category of
schemes over k and Affk be the full sub-category of affine schemes. Accord-
ing to [1, Exp. IV §6.3] the fppf topology on Schk is the one generated by:
the families of jointly surjective open immersions in Schk; the finite families
of jointly surjective, flat, of finite presentation and quasi-finite morphisms
in Affk.
Let Abk be the category of abelian sheaves on Affk w.r.t. the fppf topol-
ogy. We will consider both the category of commutative group schemes and
that of formal group schemes (over k) as full sub-categories of Abk. We
denote by k̄ the algebraic closure of k.
Definition. A Laumon 1-motive over k (or an effective free 1-motive over
k, cf. [2, 1.4.1]) is the data of
(1) a (commutative) formal group F over k, such that LieF is a finite
dimensional k-vector space and F (k̄) = lim[k′:k]<∞ F (k′) is a finitely
generated and torsion-free Gal(k̄/k)-module;
(2) a connected commutative algebraic group scheme G over k;
(3) a morphism u : F → G in the category Abk.
Note that we can consider a Laumon 1-motive (over k)M = [u : F → G]
as a complex of sheaves in Abk concentrated in degree 0, 1.
It is known that any formal k-group F splits canonically as product
F o×F ét where F o is the identity component of F and is a connected formal
k-group, and F ét = F /F o is étale. Moreover, F ét admits a maximal sub-
group scheme F tor , étale and finite, such that the quotient F ét/F tor = F fr
is constant of the type Zr over k̄. One says that F is torsion-free if F tor = 0.
By a theorem of Chevalley any connected algebraic group scheme G is
the extension of an abelian variety A by a linear k-group scheme L that is
product of its maximal sub-torus T with a vector k-group scheme V . See
[7] for more details on algebraic and formal groups.
Definition. A morphism of Laumon 1-motives is a commutative square in
the category Abk. We denote by Ma1 = Ma1,k the category of Laumon k-
1-motives, i.e. the full sub-category of Cb(Abk) whose objects are Laumon
1-motives. We denote by M1 the full sub-category of Ma1 whose objects
are Deligne 1-motives (over k) [6, §10.1.2].
Proposition 2.1. The category Ma1 of Laumon 1-motives (over k) is an
additive category with kernels and co-kernels. In particular let (f, g) be a
Cohomological dimension of Laumon 1-motives 721
morphism from M = [u : F → G] to M ′ = [u′ : F ′ → G′] (i.e. u′f = gu),
then
(2.1) Ker(f, g) = [u∗Ker(g)o → Ker(g)o]
and
(2.2) Coker(f, g) = [Coker(f)fr → Coker(g)]
Proof. See [11, Prop. 5.1.3]. 
Remark. The category of connected algebraic groups is fully embedded
in Ma1 and it is not abelian. So the category of Laumon 1-motives is not
abelian too. In fact consider a surjective morphism of connected algebraic
groups g : G → G′. Then Ker(g) is not necessarily connected and the
canonical map (in the category of connected algebraic groups)
Coim(g) = G/Ker(g)o → Im(g) = G′
is not an isomorphism in general.
According to [14] we define the category Ma1 ⊗ Q of Laumon 1-motives
up to isogenies: the objects are the same of Ma1; the Hom groups are
HomMa1(M,M
′)⊗Z Q.
Remark. Note that a morphism (f, g) : M → M ′ is an isogeny (i.e. an
isomorphism inMa1 ⊗Q ) if and only if f is injective with finite co-kernel
and g is surjective with finite kernel.
Proposition 2.2. The category of Laumon 1-motives up to isogenies is
abelian.
Proof. By constructionMa1⊗Q is an additive category. Let (f, g) :M →M ′
be a morphism of Laumon 1-motives. We know that the group π0(Ker(g)) =
Ker(g)/Ker(g)o is a finite group scheme, hence there exists an integer n
such that the following diagram commutes in Abk
Ker(f)
n·u

0
&&MM
MMM
MMM
MM
Ker(g)o // Ker(g) // π0(Ker(g))
Then n ·u factors through Ker(g)o and it is easy to check that Ker((f, g)) =
[(u∗Ker(g)o)→ Ker(g)o] is isogenous to [Ker(f)→ Ker(g)0].
It follows that Coim(f, g) is isogenous to [(F /Ker(f))fr → G/Ker(g)].
As G/Ker(g)o → G/Ker(g) is an isogeny we get that the canonical map
Coim(f, g)→ Im(f, g) is an isogeny too.
This is enough to prove that the categoryMa1 ⊗Q is abelian. 
722 Nicola Mazzari
Remark. One can also define the category tMa1 of 1-motives with torsion
over k (See [2, Def. 1.4.4]). We note that using the same arguments as in
[4, C.7.3] it is easy to show that there is an equivalence of categories
Ma1 ⊗Q
∼−→ tMa1 ⊗Q .
2.1. Weights. A Deligne 1-motive is endowed with an increasing filtra-
tion (of sub-1-motives) called the weight filtration ([6, §10.1.4]) defined as
follows
Wi =WiM :=

[X → G] i ≥ 0
[0→ G] i = −1
[0→ T ] i = −2
[0→ 0] i ≤ −3
; grWi M =

[X → 0] i = 0
[0→ A] i = −1
[0→ T ] i = −2
[0→ 0] otherwise
.
According to [4, C.11.1] we extend the weight filtration to Laumon 1-
motives.
Definition. Let M = [u : F → G] be a Laumon 1-motive. The weight
filtration of M is
W−3 = 0 ⊂W−2 = [0→ L] ⊂W−1 = [0→ G] ⊂W0 =M .
Remark.
(1) The morphisms of Laumon 1-motives are compatible w.r.t. the
weight filtration. Also the weight filtration extends to a filtration
on the objects ofMa1 ⊗Q.
(2) Let Modfk be the category of finite dimensional k-vector spaces.
The full sub-category ofMa1 ⊗Q of Laumon 1-motives of weight 0
is equivalent to the category Modfk×RepQ(Gal(k̄/k)) via the functor
F 7→ (Lie(F ),F (k)⊗Q).
3. Cohomological dimension
3.1. Extensions. Let A be any abelian category (we don’t suppose it has
enough injective objects), then we can define its derived category D(A)
and the group of n-fold extension classes ExtnA(A,B) := HomD(A)(A,B[n]),
A,B ∈ A. As usual we identify this group with the group of classes of
Yoneda extensions, i.e. the set of exact sequences
0→ B → E1 → · · · → En → A→ 0
modulo congruences (See [10] or [9]).
We will use the two following well-known facts about extensions and
filtrations.
(1) Let W−2 ⊂ W−1 ⊂ W0 = W be a filtration of W ∈ A. We have the
following exact sequences
γ : 0→W−2 →W−1 →W0/W−2 →W0/W−1 → 0
Cohomological dimension of Laumon 1-motives 723
γ1 : 0→W−2 →W−1 →W−1/W−2 → 0
γ2 : 0→W−1/W−2 →W0/W−2 →W0/W−1 → 0
and γ = γ1 · γ2 ∈ Ext2A(W0/W−1,W−2) by definition. This class is
trivial, i.e. γ = 0 in Ext2A(W0/W−1,W−2). (See [14, Lemma 3.2.5],
or [9, p. 184])
(2) Assume that the objects of A are filtered (by a separated and ex-
haustive filtration W ) and that the morphisms in A are compatible
w.r.t. this filtration. If ExtnA(grWi A, grWj B) = 0 for any i, j, then
ExtnA(A,B) = 0. In fact assume for instance that B has a 3 steps
filtration 0 ⊂ W−2 ⊂ W−1 ⊂ W0 = B: then we have the canonical
exact sequences
0→W−1M ′ →M ′ → grW0 M ′ → 0
0→W−2M ′ →W−1M ′ → grW−1M ′ → 0
By applying Hom(A,−) we get two long exact sequences
· · ·Ext2(A,W−1B′)→ Ext2(A,B)→ Ext2(M, grW0 B) · · ·
· · ·Ext2(A, grW−2B)→ Ext2(A,W−1B)→ Ext2(A, grW−1B) · · ·
from this follows that we can reduce to prove Ext2(A, grWi B) = 0.
This process can be easily adapted to the general case.
Now we can give a sketch of the proof of the main theorem: one first
checks that Ext1Q(M,M ′) = 0 ifM,M ′ are pure of weights w,w′ and w ≤ w′
(a 1-motive is pure if it is isomorphic to one of its graded pieces w.r.t. the
weight filtration). By point (2) above this formally reduces the problem to
checking that if M,M ′,M ′′ are pure respectively of weights 0,−1,−2, then
the Yoneda product of two classes (γ1, γ2) ∈ Ext1Q(M ′,M ′′)×Ext1Q(M,M ′)
is 0. Of course we may assume γ1 and γ2 integral. Then the point is that
γ1 and γ2 glue into a 1-motive and we can conclude by (1) above.
3.2. Main result. From now on we call 1-motive a Laumon 1-motive
(over k) and ExtiQ(M,M ′) is the group of classes of i-fold extensions in
Ma1 ⊗Q.
Theorem 3.1. The category Ma1 ⊗ Q (and in particular FHS1 ⊗ Q) is of
cohomological dimension 1.
Proof. By the general facts on extensions (§3.1 (2)) we can restrict to con-
sider only pure motives M = grWw M and M ′ = grWw′M ′ of weight w and
w′, respectively.
(Equal weights) If w = w′ we can show that Ext1Q(M,M ′) = 0. Let 0 →
M ′ → E → M → 0 be an exact sequence in Ma1 ⊗ Q, then also E is
pure of weight w. We have to consider 3 cases: first note the category
Modfk ×RepQ(Gal(k̄/k)) is semi-simple by Maschke’s lemma [15, p. 47] and
724 Nicola Mazzari
so the claim holds for the weight zero case by point (2) of the remark in
§2.1: also the category of abelian varieties up to isogenies (i.e. w = −1) is
semi-simple by [13, p. 173]; the third case (weight −2) can be reduced to
the first by Cartier duality (See [11, §5]) or proved explicitly.
(Different weights) Fix a 2-fold extension γ ∈ Ext2Q(M,M ′) represented by
0→M ′ → E1 → E2 →M → 0
and let E = Ker(E2 → M). Then we can write γ = γ1 · γ2, where
γ2 ∈ Ext1Q(M,E), γ1 ∈ Ext1Q(E,M ′). Using the canonical exact sequence
induced by weights and the first part of the proof it is easy to reduce to
the case E = gr−1E, i.e. E is an abelian variety.
If w < w′ then γ1 is an extension of an abelian variety E by M ′ which is
a formal group or an abelian variety. Then γ1 = 0 (if M ′ is a formal group
we refer to [2, Lemma A.4.5]).
It remains to study what happens if w > w′. If w or w′ is equal to −1
there is nothing to prove because E is an abelian variety too. So the only
case left is when w = 0 and w′ = −2, i.e. M = F [1], M ′ = L[0]. We want
to reduce to the situation considered in § 3.1 (1). Thus we have to show
that there exists a 1-motive N such that γ1 ∈ Ext1Q(E,L) is represented by
0 → W−2N → W−1N → gr−1N → 0; γ2 ∈ Ext1Q(F [1], E) is represented
by 0→ gr−1N →W0N/W−2 → gr0N → 0.
We claim that γ1 and γ2 can be represented by extensions in the category
Laumon-1-motives. In fact let
γ1 : 0→ L
f⊗n−1−→ G g⊗m
−1
−→ E → 0
be an extension in the category of 1-motives modulo isogenies: f, g are
morphism of algebraic groups, n,m ∈ Z. Then consider the push-forward by
n−1 and the pull-back by m−1, we get the following commutative diagram
with exact rows inMa,fr1 ⊗Q
0 // L
n−1

f/n // G
id

g/m // E
id

// 0
0 // L
f // G
g/m // E // 0
0 // L
id
OO
f // G
id
OO
g // E
m−1
OO
// 0
The exactness of the last row is equivalent to the following: Ker f is finite;
let (Ker g)0 be the connected component of Ker g, then Imf → (Ker g)0 is
surjective with finite kernel K; g is surjective. So after replacing L, E with
isogenous groups we have an exact sequence inMa,fr1
0→ L→ G→ E → 0
Cohomological dimension of Laumon 1-motives 725
Explicitly
0 // L

f // G
id

g // E
id

// 0
0 // L/Ker f

// G

g // E

// 0
0 // Imf/K // G
g // E // 0
0 // Imf/K
id
OO
// G
OO
// G/(Ker g)0
OO
// 0
With similar arguments we can prove that γ2 is represented by an ex-
tension in the categoryMa,fr1
0→ E → N → F [1]→ 0
with N = [u : F → E].
To conclude we have to prove that there is lifting u′ : F → G. First
suppose F = F ét: consider the long exact sequence
HomAbk(F ,G)→ HomAbk(F , E)
∂→ Ext1Abk(F ,L) .
We can consider a (Galois) extension k′/k of finite degree d trivializing F .
By [12, Theorem 3.9] we get the vanishing of Ext1Abk′ (F k′ ,Lk′). Recall that
the multiplication by d on F can be written as the composition
F
can.→ Πk′/kF k′
tr→ F
where Πk′/k is Weil restriction functor and tr is the trace map. This im-
plies that Ext1Abk(F ,L) is torsion, hence ∂u = 0 and the lift exists (up to
isogeny).
In case F = F o is a connected formal group we have a commutative
diagram in Abk
F
 



u

Ĝ
@
@@
@@
@@
π̂ // Ê
?
??
??
??
?
G
π // E
where ?̂ is the formal completion at the origin of ? = G, E. The formal
completion is an exact functor so π̂ is an epimorphism. The category of
726 Nicola Mazzari
connected formal groups is equivalent to Modfk , thus we can choose a section
σ of π̂. Then we can easily construct a (non canonical) lifting of u. 
References
[1] Schémas en groupes. I: Propriétés générales des schémas en groupes. Séminaire de
Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et
A. Grothendieck. Lecture Notes in Mathematics, Vol. 151. Springer-Verlag, Berlin, 1970.
[2] L. Barbieri-Viale and A. Bertapelle, Sharp de Rham realization. Advances in Mathe-
matics Vol. 222 Issue 4, 2009.
[3] Luca Barbieri-Viale, Formal Hodge theory. Math. Res. Lett. 14 (3) (2007), 385–394.
[4] L. Barbieri-Viale and B. Kahn, On the derived category of 1-motives, I.
arXiv:0706.1498v1, 2007.
[5] A. A. Bĕılinson, Notes on absolute Hodge cohomology. In Applications of algebraic K-
theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), volume 55
of Contemp. Math., pages 35–68. Amer. Math. Soc., Providence, RI, 1986.
[6] Pierre Deligne, Théorie de Hodge III. Inst. Hautes Études Sci. Publ. Math. 44 (1974),
5–77.
[7] Michel Demazure, Lectures on p-divisible groups. Lecture Notes in Mathematics, Vol. 302.
Springer-Verlag, Berlin, 1972.
[8] Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nit-
sure, and Angelo Vistoli, Fundamental algebraic geometry. Volume 123 of Mathe-
matical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005.
Grothendieck’s FGA explained.
[9] Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra. Springer Mono-
graphs in Mathematics. Springer-Verlag, Berlin, second edition, 2003.
[10] Birger Iversen, Cohomology of sheaves. Universitext. Springer-Verlag, Berlin, 1986.
[11] Gerard Laumon, Transformation de Fourier généralisée. arXiv:alg-geom/9603004v1,
1996.
[12] J. S. Milne, Étale Cohomology. Princeton University Press, Princeton Mathematical Series,
Vol. 33, 1980.
[13] David Mumford, Abelian varieties. Tata Institute of Fundamental Research Studies in
Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay,
1970.
[14] Fabrice Orgogozo, Isomotifs de dimension inférieure ou égale à un. Manuscripta Math.
115 (3) (2004), 339–360.
[15] Jean-Pierre Serre, Linear representations of finite groups. Springer-Verlag, New York,
1977. Translated from the second French edition by Leonard L. Scott, Graduate Texts in
Mathematics, Vol. 42.
Nicola Mazzari
Università degli Studi di Padova
Via Trieste, 63
35100 Padova, Italy
E-mail: mazzari@math.unipd.it
URL: http://www.math.unipd.it/~mazzari/index.html


Cohomologial dimension of  Laumon 1-motives up to isogenies

John Friedlander

Henryk Iwaniec

Ternary quadratic forms with rational zeros

Toufik Zaimi

Commentaires sur quelques résultats sur les nombres de Pisot

Damian Rössler

Damian RÖSSLER
A note on the ramification of torsion points lying on curves of genus at least
two
Tome 22, no 2 (2010), p. 475-481.
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Journal de Théorie des Nombres
de Bordeaux 22 (2010), 475-481
A note on the ramification of torsion points lying
on curves of genus at least two
par Damian RÖSSLER
Résumé. Soit C une courbe de genre g > 2 définie sur le corps
de fractions K d’un anneau de valuation discret R dont le corps
résiduel est algébriquement clos. On suppose que char(K) = 0 et
que la caractéristique résiduelle p de R n’est pas 2. On suppose
aussi que la jacobienne Jac(C) de C a réduction semi-stable sur
R. On plonge C dans Jac(C) via a un point K-rationnel. Nous
montrons que les coordonnées des points de torsion de Jac(C) qui
se trouvent dans C(K) sont dans l’unique extension modérément
ramifiée du corps engendré par les coordonnées des points de p-
torsion de Jac(C).
Abstract. Let C be a curve of genus g > 2 defined over the
fraction field K of a complete discrete valuation ring R with alge-
braically closed residue field. Suppose that char(K) = 0 and that
the characteristic p of the residue field is not 2. Suppose that the
Jacobian Jac(C) has semi-stable reduction over R. Embed C in
Jac(C) using a K-rational point. We show that the coordinates
of the torsion points lying on C lie in the unique tamely ramified
quadratic extension of the field generated over K by the coordi-
nates of the p-torsion points on Jac(C).
1. Introduction
Let R be a complete discrete valuation ring. Suppose that the residue
field k of R is algebraically closed and of characteristic p > 0. Suppose that
p 6= 2. Let K be the fraction field of R and suppose that char(K) = 0. Let
C be a curve of genus g > 2 defined over K. Let j : C → Jac(C) be the
closed immersion of C into its Jacobian defined by a K-rational point. Let
A := Jac(C). Let A be the Néron model of A over R.
Let L := K(A[p](K̄)) be the extension ofK generated by the coordinates
of the p-torsion points of A(K̄). In particular, L = K if p = 0. Let L′ be
the unique tamely ramified quadratic extension of L.
Finally, let K1 ⊆ K̄ be the field generated over K by the coordinates
of the elements of Tor(A(K̄)) ∩ C(K̄). Here Tor(A(K̄)) is the subgroup of
A(K̄) consisting of elements of finite order.
Manuscrit reçu le 12 mars 2009.
476 Damian Rössler
The aim of this note is to prove the following statement :
Theorem 1.1. Suppose that the connected component of the special fiber
Ak of A is a semi-abelian variety (i.e. A has semi-stable reduction over R).
Then we have
(a) K1 ⊆ L′;
(b) if C is not hyperelliptic then we have K1 ⊆ L.
Corollary 1.2. If p > 3, the inequality
[K1 : K] 6 2 ·#GL2g(Z/3Z) ·#GL2g(Z/pZ)
is verified.
Corollary 1.2 is an immediate consequence of Theorem 1.1 and of Ray-
naud’s criterion of semi-stable reduction. Recall that according to this cri-
terion, A will have semi-stable reduction over K if the l-torsion points of
A are K-rational, for some prime number l such that l 6= p and l > 2 (see
[6, IX]).
For the convenience of the reader, we recall the formula
#GL2g(Z/lZ) = (l2g − 1) · (l2g − l) · · · (l2g − l2g−1),
which is valid for any prime number l.
Theorem 1.1 should be understood as a complement to some results of
Tamagawa (see [10]), Baker-Ribet (see [2]) and Coleman (see [5]).
For instance, with the present notation, suppose that p > 0, that R is
the maximal unramified extension of Qp and that the abelian part of the
connected component of Ak is an ordinary abelian variety. Tamagawa then
proves that K1 is contained in the extension of K generated by the p-th
roots of unity (see [10] or [2, Th. 4.1]). Another example is the following
result of Coleman : if p > max(2g, 5),R is the maximal unramified extension
of Qp and Ak is an abelian variety, then K1 ⊆ K (see [5, Conj. B]).
All these results restrict the size of K1 under various hypotheses on the
special fiber Ak and on the order of absolute ramification of K. The interest
of Theorem 1.1 and its corollary is that they provide a limit for the size of
K1 with no assumption on the absolute ramification of K and little or no
assumptions on the special fiber Ak.
Remark. (1) The avoidance of the prime p = 2 is critical. It appears in
both Lemma 2.4 and Lemma 2.6 and this is exploited at the end of the
proof of Theorem 1.1. It would be interesting to extend the method used
in this note to the case p = 2.
(2) A closer examination of the proof of Theorem 1.1 (b) shows that the
following statement holds. Let x ∈ Tor(A(K)) ∩ C(K). Suppose that C is
hyperelliptic. If x is not a fixed point of the uniquely defined hyperelliptic
involution of C, then the field generated over K by the coordinates of x is
included in L.
A note on the ramification of torsion points 477
Notations. If l is a prime number and G is an abelian group, we write
Torl(G) for the set of elements of Tor(G) whose order is prime to l and
Torl(G) for the set of elements of Tor(G) whose order is a power of l. The
expression Tor0(G) will stand for Tor(G). We shall denote by + the group
law on A(L̄). We shall write divisors on CL̄ in the form
n1P1 ⊕ n2P2 ⊕ · · · ⊕ nrPR
where ni ∈ Z. The symbol ∼ will be used to denote linear equivalence of
divisors.
Ackowledgments. I am grateful to M. Baker and J. Boxall for their feed-
back and for some useful comments.
2. Proof of Theorem 1.1
Let Lt be the maximal tamely ramified extension of L. Let I := Gal(L̄|L),
Iw := Gal(L̄|Lt) and It := Gal(Lt|L). Recall that Iw = 0 if char(k) = 0
and that there is a non-canonical isomorphism It ' ⊕l 6=p, l prime Zl (see [9,
chap. IV]). Furthermore, the group Iw is a pro-p-group if p > 0.
We shall need the following five results.
Theorem 2.1 (monodromy theorem). For any x ∈ Torp(A(L̄)) and any
σ ∈ I, the equation σ2(x)− 2σ(x) + x = 0 is satisfied.
Proof. See [6, IX, 5.12.2] 
Lemma 2.2. The action of Iw on Torp(A(L̄)) is trivial.
Proof. This is a direct consequence of Theorem 2.1. See for instance [2,
Appendix, Lemma A.1]. 
Lemma 2.3. The action of It on Torp(A(Lt)) is trivial.
Proof. We may restrict ourselves to the case where p > 0. Let T ⊆
Torp(A(Lt)) be a finite It-invariant subgroup. We have to show that the
action of It on T is trivial. The action of It on T preserves the order of
elements, hence T is an inner direct sum of Gt-invariants subgroups of
the form (Z/pr)s. Hence we might suppose without loss of generality that
T ' (Z/pr)s for some r, s 6 1. Let Tp be the subgroup of p-torsion elements
of T . The fact that the p-torsion points in A(L̄) are L-rational implies that
the action of Gt on Tp is trivial. Hence the image of Gt in Aut(T ) lies in
the kernel of the natural group map
Aut(T )→ Aut(Tp)
Under the above isomorphism T ' (Z/pr)s, this corresponds to s × s-
matrices of the form Id+pM , whereM is an s×s-matrices with coefficients
in Z/prZ. This last fact is a consequence of the fact that multiplication by
pr−1 induces an It-equivariant isomorphism T/pT → Tp. The calculation
478 Damian Rössler
(Id + pM)pr−1 = Id now shows that the image of It in Aut(T ) is a p-group.
On the other hand, It is a direct sum of pro-l-groups, with l 6= p. The order
of the image of It is thus prime to p. This image is thus trivial. 
Lemma 2.4 (Boxall). Let B be an abelian variety over a field F of char-
acteristic 0. Let l > 2 be a prime number and let L := F (B[l]) be the
extension of F generated by the l-torsion points of B. Let P ∈ Torl(B(L̄))
and suppose that P 6∈ B(L). Then there exists σ ∈ Gal(L̄|L) such that
σ(P )− P ∈ B[l](L̄) \ {0}.
Proof. See [4, Lemme 1] or [8, Prop. 3]. 
Lemma 2.5. Let P ⊕ Q and P ′ ⊕ Q′ be two divisors of degree 2 on CL̄.
Suppose that P ⊕Q and P ′ ⊕Q′ are linearly equivalent.
If C is not hyperelliptic, then the two divisors coincide.
If C is hyperelliptic, then either the two divisors coincide or we have
Q = ι(P ) and Q′ = ι(P ′).
Here ι : C → C is the uniquely defined hyperelliptic involution.
Proof. See [7, IV, Prop. 5.3]. 
Lemma 2.6. Let x ∈ A(L̄) and suppose that x 6= 0. The inequality
(2.1) #
(
C(L̄) ∩ (C(L̄) + x)
)
6 2
is then verified. If C is not hyperelliptic, we even have
(2.2) #
(
C(L̄) ∩ (C(L̄) + x)
)
6 1.
This Lemma is a consequence of [1, Prop. 4]. For the convenience of the
reader, we provide the following proof.
Proof. We shall write O for the K-rational point on C, which is used to
embed C in A.
Let a1, . . . , ar ∈ C(L̄) be pairwise distinct points such that a1 + x, a2 +
x, . . . , ar + x ∈ C(L̄). Let bi := ai + x (i = 1, . . . , r).
Suppose first that r > 1 and that C is not hyperelliptic. We then have a
linear equivalence
(2.3) b1 ⊕ a2 ∼ b2 ⊕ a1
Hence either b1 = b2 or b1 = a1, either of which are ruled out. So we
conclude that if C is not hyperelliptic, then r 6 1. This proves the inequality
(2.2).
Now suppose that C is hyperelliptic and that r > 2. Let ι : C → C be
the corresponding hyperelliptic involution. On top of (2.3), we then have
the further linear equivalence
b2 ⊕ a3 ∼ a2 ⊕ b3
A note on the ramification of torsion points 479
Lemma 2.5 now implies that a2 = ι(b1) and a2 = ι(b3). Hence b1 = b3,
which is impossible. Thus we conclude that r 6 2, if C is hyperelliptic.
This proves the first inequality (2.1). 
We can now start with the proof of Theorem 1.1.
The monodromy theorem 2.1 says that for any x ∈ Torp(A(L̄)) and any
σ ∈ I the equation (σ − Id)2(x) = 0 is satisfied (remember that Torp(·) =
Tor(·) if p = 0). On the other hand, Lemma 2.2 says that Torp(A(L̄)) ⊆
Torp(A(Lt)) and Lemma 2.3 implies that σ(x) = x for any x ∈ Torp(A(Lt))
and any σ ∈ It. Hence the equation
(2.4) (σ − Id)2(x) = 0
is verified for any x ∈ Tor(A(Lt)) and any σ ∈ It. Let x ∈ Tor(A(Lt))∩C(L̄)
and σ ∈ It. The equation (2.4) implies the linear equivalence
(2.5) σ2(x)⊕ x ∼ 2σ(x)
of divisors of degree 2 on CL̄.
First suppose that C is not hyperelliptic; then any two linearly equivalent
divisors of degree 2 on CL̄ coincide (see Lemma 2.5) and the relation (2.5)
thus implies that x = σ(x).1
Summing up, there is an inclusion
Tor(A(Lt)) ∩ C(L̄) ⊆ A(L)
if C is not hyperelliptic.
Now suppose that C is hyperelliptic and let ι : C → C be the uniquely
defined hyperelliptic involution. Lemma 2.5 then implies that either x =
σ(x) as above or that the equations ι(σ(x)) = σ(x) and ι(x) = σ2(x) hold.
Suppose the latter. Since ι is defined over L, the equation ι(σ(x)) = σ(x)
implies that ι(x) = x. This together with the equation ι(x) = σ2(x) implies
that σ2(x) = x.
Now notice that since the group It is abelian, the set (It)2 of squares
of elements is a normal subgroup of It. Let J ′ be the Galois extension of
L defined by (It)2. Since It ' ⊕l 6=p, l prime Zl, we see that [J ′ : L] = 2. In
the last paragraph, we showed that Tor(A(Lt))∩C(L̄) is fixed by (It)2. In
other words, we have shown that Tor(A(Lt)) ∩ C(L̄) ⊆ A(J ′) where J ′ is
the unique tamely ramified quadratic extension of L. In the notation of the
introduction, J ′ = L′.
Summing up, we see that
Tor(A(Lt)) ∩ C(L̄) ⊆ A(L′)
if C is hyperelliptic.
1This calculation is partly the motivation for Ribet’s definition of an "almost rational point"
(see [2, Lemma 2.7]) and is the starting point of Tamagawa’s article [10] (see Prop. 0.2 in that
reference).
480 Damian Rössler
If p = 0, then Lt = K̄ so this completes the proof of Theorem 1.1 in that
case.
Now let x ∈ Tor(A(L̄))∩C(L̄)\C(Lt). Let x = xp + xp be the decompo-
sition of x into its components of prime-to-p torsion and p-primary torsion,
respectively. Lemma 2.3 implies that xp ∈ A(L̄))\A(Lt). Also, using Box-
all’s lemma 2.4 and the fact that L contains the coordinates of the p-torsion
points, we see that there exists σx ∈ Iw such that σx(x)−x = σx(xp)−xp ∈
A[p](L)\{0}. Hence
(2.6) σx(x) ∈
⋃
τ∈A[p](L)\{0}
C(L̄) ∩
(
C(L̄) + τ
)
.
Lemma 2.6 now implies that
(2.7) #(C(L̄) ∩
(
C(L̄) + τ
)
) 6 2
for all τ ∈ A[p](L)\{0}. Notice that if σx(x) ∈ C(L̄) ∩
(
C(L̄) + τ0
)
for a
particular τ0 ∈ A[p](L)\{0}, then we have
{σx(x), σ2x(x), . . . , σpx(x)} ⊆ C(L̄) ∩
(
C(L̄) + τ0
)
(remember that by construction τ0 is fixed by Iw). On the other hand
{σx(x), σ2x(x), . . . , σpx(x)} = {x+ τ0, x+ 2τ0, . . . , x+ pτ0 = x}.
Since p > 2 and τ0 has exact order p in A(K̄), this leads to a contradiction.
Thus we have
Tor(A(L̄)) ∩ C(L̄) ⊆ A(Lt).
Now remember that we have shown above (see the italicized sentences) that
Tor(A(Lt)) ∩ C(L̄) ⊆ A(L) if C is not hyperelliptic and that Tor(A(Lt)) ∩
C(L̄) ⊆ A(L′) if C is hyperelliptic. This concludes the proof of (a) and (b).
References
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[2] Baker, M.; Ribet, K., Galois theory and torsion points on curves. J. Théor. Nombres
Bordeaux 15 (2003), 11–32.
[3] Boxall, J., Autour d’un problème de Coleman. C. R. Acad. Sci. Paris Sér. I Math. 315
(1992), 1063–1066.
[4] Boxall, J., Sous-variétés algébriques de variétés semi-abéliennes sur un corps fini. Number
theory 1992-1993. London Math. Soc. Lecture Note Ser. 215, 69–80. Cambridge Univ. Press,
1995.
[5] Coleman, R., Ramified torsion points on curves. Duke Math. J. 54 (1987), 615–640.
[6] Groupes de monodromie en géométrie algébrique. I. Lecture Notes in Mathematics, Vol.
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A note on the ramification of torsion points 481
[9] Serre, J.-P., Local fields. Graduate Texts in Mathematics 67. Springer-Verlag, 1979. Duke
Math. J. 106 (2001), 281–319.
[10] Tamagawa, A., Ramification of torsion points on curves with ordinary semistable Jacobian
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Damian Rössler
Département de Mathématiques
Bâtiment 425
Faculté des Sciences d’Orsay
Université Paris-Sud
91405 Orsay Cedex, FRANCE
E-mail: damian.rossler@math.u-psud.fr
URL: http://www.math.u-psud.fr/~rossler


A note on the ramification of torsion points lying on curves of genus at least two

Daniel Loughran

Manin’s conjecture for a singular sextic del Pezzo surface

Martin Widmer

Martin WIDMER
Small generators of function fields
Tome 22, no 3 (2010), p. 747-753.
<http://jtnb.cedram.org/item?id=JTNB_2010__22_3_747_0>
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Journal de Théorie des Nombres
de Bordeaux 22 (2010), 747-753
Small generators of function fields
par Martin WIDMER
Résumé. Soit K/k une extension finie d’un corps global, donc K
contient un élément primitif α, c’est à dire K = k(α). Dans cet
article, nous démontrons l’existence d’un élément primitif de pe-
tite hauteur dans le cas d’un corps de fonctions. Notre résultat est
la réponse pour les corps de fonctions à une question de Ruppert
sur les petits générateurs des corps de nombres.
Abstract. Let K/k be a finite extension of a global field. Such
an extension can be generated over k by a single element. The aim
of this article is to prove the existence of a ”small” generator in
the function field case. This answers the function field version of
a question of Ruppert on small generators of number fields.
1. Introduction
Let K be a finite extension of a global field k where global field means
finite extension of either Q or of a rational function field of transcendence
degree one over a finite field. Such an extension is generated by a single
element, and there exists a natural concept of size onK given by the height.
The well-known Theorem of Northcott (originally proved for algebraic num-
bers but easily seen to hold also in positive characteristic) implies that for
each real T there are only finitely many α ∈ K whose height does not exceed
T . In particular there exists a smallest generator. It is therefore natural to
ask for lower and upper bounds for the height of a smallest generator. We
emphasize the situation where d is fixed and K runs over all extensions of
k of degree d.
Several people proved lower bounds for generators; first Mahler [5] for
the ground field k = Q and then Silverman [8] for arbitrary ground fields
(and also higher dimensions) but see also [7], [6] and [4] for simpler results.
For an extension K/k of number fields Silverman’s inequality implies
(1.1) h(1, α) ≥ log |∆K |/(2d(d− 1))− log |∆k|/(2(d− 1))
− [k : Q] log d/(2(d− 1))
for any generator α ofK/k. As shown by examples of Masser (Proposition 1
[6]) and Ruppert [7], this bound is sharp, at least up to an additive constant
Manuscrit reçu le 2 juillet 2009.
748 Martin Widmer
depending only on k and d. A version of Silverman’s bound in the function
field case follows quickly from Castelnuovo’s inequality. For simplicity let us
temporarily assume K and k are finite separable extensions of the rational
function field Fq(t) both with field of constants Fq. We apply Castelnuovo’s
inequality as in [9] III.10.3.Theorem with F = K = k(α), F1 = k and
F2 = Fq(α). Writing gk and gK for the genus of k and K we conclude
[K : Fq(α)] ≥ gK/(d − 1) − dgk/(d − 1) + 1. From (2.1), (2.2) and the
definition of the height in (2.3) we easily deduce h(1, α) ≥ [K : Fq(α)]/d,
and thus
h(1, α) ≥ gK/(d(d− 1))− gk/(d− 1) + 1/d.(1.2)
The discriminant ∆K = qdegDiff(K/Fq(t)) of K/Fq(t) is related to the genus
by the Riemann-Hurwitz formula, more precisely ∆K = q2gK+2([K:Fq(t)]−1).
Thus (1.2) matches with (1.1), at least up to an additive constant depending
only on the degrees of k and K. A similar inequality as in (1.2) was given
by Thunder ([10] Lemma 6).
What about upper bounds for the smallest generator? It seems that
this problem has not been studied much yet. However, at least for number
fields the problem has been proposed explicitly by Ruppert. More precisely
Ruppert ([7] Question 2) addressed the following question.
Question 1 (Ruppert, 1998). Does there exist a constant C = C(d) such
that for each number field K of degree d there exists a generator α of the
extension K/Q with h(1, α) ≤ log |∆K |/(2d) + C?
Ruppert used the non-logarithmic naive height whereas we use the log-
arithmic projective absolute Weil height as defined in [2]. However, it is
easily seen that the question formulated here is equivalent to Ruppert’s
Question 2 in [7]. One can show that there exists always an integral gen-
erator α of K/Q with h(1, α) ≤ log |∆K |/d. For a proof of this simple fact
see [11]. What is more Ruppert showed that Question 1 has an affirmative
answer for K either quadratic or a totally real field of prime degree. In
fact, using Minkowski’s convex body Theorem to construct a Pisot-number
generator, it suffices to assume K has a real embedding and one can drop
the prime degree condition. In this special case the derived small genera-
tor is an algebraic integer. For more details we refer to [11]. On the other
hand, if α is an integral generator of an imaginary quadratic field K then
h(1, α) ≥ log |∆K |/d− log 2. Thus for almost all imaginary quadratic fields
the small generators are not integral elements. Ruppert’s result for d = 2
relies heavily on a distribution result of Duke [3] that does not appear to
have an analogue for higher degrees and is ineffective. As a consequence
Ruppert’s constant C for d = 2 is ineffective.
In this note we introduce a completely different strategy which applies in
the function field case and the number field case. However, it relies on the
Small generators of function fields 749
existence of a certain divisor which is guaranteed under GRH but might
be rather troublesome to establish unconditionally. The aim of this short
note is to answer positively Ruppert’s question in the function field case.
So let k be an algebraic function field with finite constant field k0 and
transcendence degree one over k0. We have the following result.
Theorem 1.1. Let K be a finite field extension of k. There exists an ele-
ment α in K with K = k(α) and a constant C = C(k, [K : k]) depending
solely on k and [K : k] such that
h(1, α) ≤ gK
d(K/k)
+ C
where gK denotes the genus of the function field K with field of constants
K0 and d(K/k) = [K : k]/[K0 : k0].
2. Notation and definitions
Throughout this note we fix an algebraic closure k of k. All fields are
considered to be subfields of k. For any finite extension F of k we write F0
for the field of constants in F ; in other words F0 is the algebraic closure
of k0 in F . When we talk of the field F we implicitly mean the field F
with field of constants F0. We define the geometric degree d(F/k) of the
extension F over k as
d(F/k) = [F : k]
[F0 : k0]
.
Let M(F ) be the set of all places in F . For a place ℘ in M(F ) let O℘ be
the valuation ring of F at ℘; we can identify ℘ with the unique maximal
ideal in O℘. We write F℘ = O℘/℘ for the residue class field and F̂℘ for
the topological completion of F at the place ℘. Write ord℘ for the order
function on F̂℘ normalized to have image in Z∪∞. We extend ord℘ to F̂℘
n
by defining
ord℘(x1, ..., xn) = min1≤i≤n ord℘ xi
with the usual convention ord℘ 0 =∞ > 0. Each non-zero element x of Fn
gives rise to a divisor (x) over F
(x) =
∑
℘∈M(F )
ord℘(x) · ℘.
For a divisor A over F we define the Riemann-Roch space in Fn
Ln(A) = {x ∈ Fn\0; (x) +A ≥ 0} ∪ {0}.
This is a F0 vector space of finite dimension. Denote its dimension over F0
by ln(A).
750 Martin Widmer
The degree of a place ℘ in M(F ) is defined by degF ℘ = [F℘ : F0]. Let K
be a finite extension of F , and let B be a place inM(K) above ℘. We write
f(B/℘) = [KB : F℘] for the residue degree of B over ℘. Then we have
degKB = [KB : K0] =
[KB : F0]
[K0 : F0]
= [KB : F℘]
[K0 : F0]
[F℘ : F0]
= f(B/℘)
[K0 : F0]
degF ℘.(2.1)
Writing e(B/℘) for the ramification index we also have∑
B|℘
e(B/℘)f(B/℘) = [K : F ],(2.2)
see for example III.1.11.Theorem in [9].
Each divisor A =
∑
℘ a℘℘ over the smaller field F naturally defines a
divisor
A(K) =
∑
℘
∑
B|℘
a℘e(B/℘)B
over the large field K.
As in [10] we define the height h on non-zero x in Kn by
h(x) = −degK(x)
d(K/k)
.(2.3)
Note that the degree of a principal divisor is zero so that the height de-
fines a function on projective space Pn−1(K) over K of dimension n − 1.
This shows also that the height is nonnegative since to evaluate the height
of x we can assume that one coordinate is 1. Moreover, it is absolute in
the following sense. Suppose x ∈ Kn and let D be the divisor given by
D = (x). Let R be a finite extension of K and view x ∈ Rn. Let D(R)
be the divisor over R given by D(R) = (x). By [1] Chap.15, Thm.9 we
have degR(D(R)) = d(R/K) degK(D), and by [1] Chap.15, Thm.2 we have
d(R/k) = d(R/K)d(K/k). Thus h(x) remains unchanged if one views x
in Rn. Therefore the height extends to a projective height on kn. Suppose
x ∈ Ln(A) ⊆ Kn then directly from the definition we see that
h(x) ≤ degK A
d(K/k)
.(2.4)
3. The strategy
Let S be a finite set of places in M(K) such that the following two
properties hold:
(i) for each place p inM(k) there is at most one place in S that lies above p,
(ii)f(B/p) = 1 for all B ∈ S and p ∈M(k) with B | p.
Small generators of function fields 751
A set S with these two properties will be called admissible. Note that for
each field F with k ⊆ F ⊆ K and for all places B,B′ ∈ S, ℘, ℘′ ∈ M(F )
with B | ℘ and B′ | ℘′ we have
B 6= B′ ⇒ ℘ 6= ℘′,(3.1)
f(B′/℘′) = 1.(3.2)
We say the divisor A is admissible if it can be written in the form
A =
∑
B∈S
1 ·B.(3.3)
with an admissible set S.
Lemma 3.1. Suppose A is an admissible divisor, and suppose x = (1, x)
with x /∈ K0 and x ∈ L2(A). Then k(x) = K and h(x) ≤ degK A/d(K/k).
Proof. Suppose k(x) = F ( K, and write (x) =
∑
℘ a℘℘ for the divisor
over F . When we consider (x) as a divisor over K we have (x) =
∑
B aBB
with aB = a℘e(B/℘). Note that none of the coefficients a℘ is positive, and
since x /∈ K0 at least one is negative, say a℘′ . Since (x) lies in L2(A) and A
is an admissible divisor we conclude by (3.1) that there is exactly one place
B′ in M(K) with B′ | ℘′ and by (3.2) that f(B′/℘′) = 1. Together with
(2.2) we deduce that 1 < [K : F ] =
∑
B′|℘′ e(B′/℘′)f(B′/℘′) = e(B′/℘′).
This means that aB′ = a℘′e(B′/℘′) < a℘′ ≤ −1, contradicting the fact
A + (x) ≥ 0. Thus k(x) = K. The remaining statement comes from (2.4).2
Lemma 3.2. Suppose A is an admissible divisor with degK A > gK . Then
there exists α in K with k(α) = K and
h(1, α) ≤ degK A
d(K/k)
.
Proof. We apply the Theorem of Riemann-Roch to the space L1(A) = {x ∈
K\0; (x) + A ≥ 0} ∪ {0} to conclude l1(A) > 1. Therefore we find a α
in L1(A)\K0. Now (x) = (1, α) and (α) have the same pole-divisors, and
since A ≥ 0 we see that (x) lies in L2(A). Applying Lemma 3.1 proves the
lemma. 2
4. Constructing a suitable divisor
In this section we will prove that K admits an admissible divisor of
degree gK + 1 provided gK is large enough.
Lemma 4.1. Let l be a positive integer. The number of places B ∈M(K)
with
degKB = l and
f(B/p) = 1 for p ∈M(k) with B | p
752 Martin Widmer
is
≥ |K0|
l
l
− (2 + 7gK)
|K0|l/2
l
− l[K0 : k0][K : k](|K0|l/2 + (2 + 7gk)|K0|l/4).
Proof. Using Riemann’s hypothesis one can obtain a good lower bound for
the number of places B of fixed degree. For instance V.2.10 Corollary (a)
in [9] tells us that the total number of places B ∈M(K) with degKB = l
is
≥ |K0|
l
l
− (2 + 7gK)
|K0|l/2
l
.
From (2.1) we get degk p = l[K0 : k0]/f(B/p) for p ∈M(k), B | p. Suppose
f = f(B/p) > 1. Applying V.2.10 Corollary (a) again we get the following
upper bound for the number of places p ∈M(k) with degk p = l[K0 : k0]/f
f |k0|l[K0:k0]/f
l[K0 : k0]
+ (2 + 7gk)
f |k0|l[K0:k0]/(2f)
l[K0 : k0]
= f |K0|
l/f
l[K0 : k0]
+ (2 + 7gk)
f |K0|l/(2f)
l[K0 : k0]
≤ |K0|l/2 + (2 + 7gk)|K0|l/4.
Above each place p in M(k) there lie at most [K : k] places B ∈ M(K).
Therefore the number of places B ∈ M(K) satisfying degKB = l and
f(B/p) = f > 1 is
≤ [K : k](|K0|l/2 + (2 + 7gk)|K0|l/4).
Summing over all divisors f of l[K0 : k0] we find that the number of places
B ∈M(K) satisfying degKB = l and f(B/p) > 1 is
≤ l[K0 : k0][K : k](|K0|l/2 + (2 + 7gk)|K0|l/4).
This in turn implies that the number of places B ∈ M(K) satisfying
degKB = l and f(B/p) = 1 is
≥ |K0|
l
l
− (2 + 7gK)
|K0|l/2
l
− l[K0 : k0][K : k](|K0|l/2 + (2 + 7gk)|K0|l/4),
and this proves the lemma. 2
5. Proof of Theorem 1.1
Let d be a positive integer. We can find a constant C1 = C1(k, d) such
that with l = gK + 1
|K0|l
l
− (2 + 7gK)
|K0|l/2
l
− l[K0 : k0][K : k](|K0|l/2 + (2 + 7gk)|K0|l/4) > 0
for all extensions K of k satisfying gK > C1 and [K : k] = d. By virtue of
Lemma 4.1 we conclude that for each suchK there exists a place B ∈M(K)
Small generators of function fields 753
with degKB = gK + 1 and f(B/p) = 1 for p ∈ M(k) with B | p. In
particular there exists an admissible divisor, namely B, with degKB =
gK + 1. From Lemma 3.2 we conclude that if gK > C1 and [K : k] = d then
there exists α ∈ K withK = k(α) and h(1, α) ≤ (gK+1)/d(K/k). There are
only finitely many field extensions K of k of degree d with gK ≤ C1. Hence
there exists a constant C as in Theorem 1.1 depending solely on C1, k, d,
and thus depending solely on k, d, such that the statement of Theorem 1.1
holds for all extensions K of k of degree d.
Acknowledgements
I would like to thank Felipe Voloch for giving me the idea of the proof
of Lemma 4.1 and pointing out the connection with Castelnuovo’s bound
mentioned in the Introduction. Moreover I am grateful to David Masser,
Jeffrey Thunder and Jeffrey Vaaler for many delightful discussions.
References
1. E. Artin, Algebraic numbers and algebraic functions, Gordon and Breach, New York, 1967.
2. E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Cambridge University Press,
2006.
3. W. Duke, Hyperbolic distribution problems and half-integral weight Masss forms, Invent.
Math. 92 (1988), 73–90.
4. J. Ellenberg and A. Venkatesh, Reflection principles and bounds for class group torsion, Int.
Math. Res. Not. no.1, Art. ID rnm002 (2007).
5. K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964),
257–262.
6. D. Roy and J. L. Thunder, A note on Siegel’s lemma over number fields, Monatsh. Math.
120 (1995), 307–318.
7. W. Ruppert, Small generators of number fields, Manuscripta math. 96 (1998), 17–22.
8. J. Silverman, Lower bounds for height functions, Duke Math. J. 51 (1984), 395–403.
9. H. Stichtenoth, Algebraic function fields and codes, Springer, 1993.
10. J. L. Thunder, Siegel’s lemma for function fields, Michigan Math. J. 42 (1995), 147–162.
11. J. D. Vaaler and M. Widmer, On small generators of number fields, in preparation (2010).
Martin Widmer
Institut für Mathematik A
Technische Universität Graz
Steyrergasse 30/II
8010 Graz, Austria
E-mail: widmer@math.tugraz.at


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