Comptes Rendus Mathématique

Annemarie Luger

Mitja Nedic

On quasi-Herglotz functions in one variable

Qing-Hu Hou

Hao Pan

Zhi-Wei Sun

Comptes Rendus
Mathématique
Qing-Hu Hou, Hao Pan and Zhi-Wei Sun
A new theorem on quadratic residues modulo primes
Volume 360 (2022), p. 1065-1069
https://doi.org/10.5802/crmath.371
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1065-1069
https://doi.org/10.5802/crmath.371
Number theory / Théorie des nombres
A new theorem on quadratic residues modulo
primes
Qing-HuHoua, Hao Panb and Zhi-Wei Sun∗, c
a School of Mathematics, Tianjin University, Tianjin 300350, People’s Republic of
China
b School of Applied Mathematics, Nanjing University of Finance and Economics,
Nanjing 210046, People’s Republic of China
c Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic
of China
URL: http://maths.nju.edu.cn/~zwsun
E-mails: qh_hou@tju.edu.cn, haopan79@zoho.com, zwsun@nju.edu.cn
Abstract. Let p > 3 be a prime, and let ( ·p ) be the Legendre symbol. Let b ∈ Z and ε ∈ {±1}. We mainly prove
that ∣∣∣∣{Np (a,b) : 1 < a {ax2 +b}p , and {m}p with m ∈Z is
the least nonnegative residue of m modulo p.
2020 Mathematics Subject Classification. 11A15, 11A07, 11R11.
Funding. The first, second and third authors are supported by the National Natural Science Foundation of
China (grants 11771330, 12071208 and 11971222, respectively).
Manuscript received 8 March 2022, accepted 26 April 2022.
1. Introduction
The theory of quadratic residues modulo primes plays an important role in fundamental number
theory.
Let p be an odd prime and let a ∈Zwith p - a. By Gauss’ Lemma (cf. [4, p. 52]),(
a
p
)
= (−1)|{16k6 p−12 : {ka}p> p2 }|,
where
( ·
p
)
denotes the Legendre symbol, and we write {x}p for the least nonnegative residue of
an integer x modulo p.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1066 Qing-Hu Hou, Hao Pan and Zhi-Wei Sun
Let n be any positive odd integer, and let a ∈Z with gcd(a(1−a),n) = 1. In 2020, Z.-W. Sun [6]
proved the following new result:
(−1)|{16k6 n−12 : {ka}n>k}| =
(
2a(1−a)
n
)
,
where
( ·
n
)
is the Jacobi symbol.
Let p be an odd prime and let a,b ∈Zwith a(1−a) 6≡ 0 (mod p). By [5, Lemma 2.7], we have
|{x ∈ {0, . . . , p −1} : {ax +b}p > x}| = p −1
2
.
In 2019 Z.-W. Sun [5] employed Galois theory to prove that
(−1)|{16i< j6 p−12 : {i 2}p>{ j 2}p }| =
{
1 if p ≡ 3 (mod 8),
(−1)(h(−p)+1)/2 if p ≡ 7 (mod 8),
where h(−p) is the class number of the imaginary quadratic fieldQ(p−p).
Motivated by the above work, for an odd prime p and integers a and b, we introduce the
notation
Np (a,b) :=
∣∣∣∣{16 x 6 p −12 : {x2 +b}p > {ax2 +b}p
}∣∣∣∣ .
Example 1. We have N7(4,0) = 2 since
{12}7 < {4×12}7, {22}7 > {4×22}7 and {32}7 > {4×32}7.
Let p be a prime with p ≡ 1 (mod4). Then q2 ≡−1 (modp) for some integer q , hence for a, x ∈Z
we have {(qx)2}p > {a(qx)2}p if and only if {x2}p < {ax2}p . Thus, for each a = 2, . . . , p −1 there are
exactly (p −1)/4 positive integers x < p/2 such that {x2}p > {ax2}p . Therefore Np (a,0) = (p −1)/4
for all a = 2, . . . , p −1.
In this paper we establish the following novel theorem which was conjectured by the first and
third authors [3] in 2018.
Theorem 2. Let p > 3 be a prime, and let b be any integer. Set
S =
{
Np (a,b) : 1 < a < p and
(
a
p
)
= 1
}
and
T =
{
Np (a,b) : 1 < a < p and
(
a
p
)
=−1
}
.
Then |S| = |T | = 1 if p ≡ 1 (mod 4), and |S| = |T | = 2 if p ≡ 3 (mod 4). Moreover, the set S does not
depend on the value of b.
Example 3. Let’s adopt the notation in Theorem 2. For p = 5, we have S = {1} for any b ∈ Z, and
the set T depends on b as illustrated by the following table:
b 0 1 2 3 4
T {1} {0} {1} {2} {1}
.
For p = 7, we have S = {1,2} for any b ∈Z, and the set T depends on b as illustrated by the following
table:
b 0 1 2 3 4 5 6
T {0,1} {1,2} {2,3} {1,2} {2,3} {1,2} {0,1}
.
Qing-Hu Hou, Hao Pan and Zhi-Wei Sun 1067
2. Proof of Theorem 2
Lemma 4. For any prime p ≡ 3 (mod 4), we have
p−1∑
z=1
z
(
z
p
)
=−ph(−p),
where h(−p) is the class number of the imaginary quadratic fieldQ(p−p).
Remark 5. This is a known result of Dirichlet (cf. [1, Corollary 5.3.13]).
Lemma 6. For any prime p ≡ 3 (mod 4) with p > 3, there are x, y, z ∈ {1, . . . , p −1} such that(
x
p
)
=
(
x +1
p
)
= 1, −
(
y
p
)
=
(
y +1
p
)
= 1, and
(
z
p
)
=−
(
z +1
p
)
= 1.
Proof. By a known result (see, e.g., [2, pp. 64–65]), we have∣∣∣∣{x ∈ {1, . . . , p −2} : ( xp
)
=
(
x +1
p
)
= 1
}∣∣∣∣= p −34 > 0.
Hence∣∣∣∣{y ∈ {1, . . . , p −2} : −( yp
)
=
(
y +1
p
)
= 1
}∣∣∣∣= ∣∣∣∣{y ∈ {1, . . . , p −2} : ( y +1p
)
= 1
}∣∣∣∣− p −34
= p −1
2
−1− p −3
4
= p −3
4
> 0
and ∣∣∣∣{z ∈ {1, . . . , p −2} : ( zp
)
=−
(
z +1
p
)
= 1
}∣∣∣∣= ∣∣∣∣{z ∈ {1, . . . , p −2} : ( zp
)
= 1
}∣∣∣∣− p −34
= p −1
2
− p −3
4
= p +1
4
> 0.
Now the desired result immediately follows. 
Proof of Theorem 2. Let a ∈ {2, . . . , p −1}. For any x ∈Z, it is easy to see that{
ax2 +b
p
}
+
{
(1−a)x2
p
}
−
{
x2 +b
p
}
=
{
0 if {x2 +b}p > {ax2 +b}p ,
1 if {x2 +b}p < {ax2 +b}p ,
where {α} denotes the fractional part of a real number α. Thus
Np (a,b) =
(p−1)/2∑
x=1
(
1+
{
x2 +b
p
}
−
{
ax2 +b
p
}
−
{
(1−a)x2
p
})
= p −1
2
+
(p−1)/2∑
x=1
{
x2 +b
p
}
−
(p−1)/2∑
x=1
{
ax2 +b
p
}
−
(p−1)/2∑
x=1
{
(1−a)x2
p
}
= p −1
2
+
p−1∑
x=1
( xp )=1
{
x +b
p
}
−
p−1∑
y=1
( yp )=( ap )
{
y +b
p
}
−
p−1∑
z=1
( zp )=( 1−ap )
z
p
.
Suppose that ( ap ) = εwith ε ∈ {±1}. Then
Np (a,b) = p −1
2
+
p−1∑
x=1
( xp )=1
{
x +b
p
}
−
p−1∑
y=1
( yp )=ε
{
y +b
p
}
−
p−1∑
z=1
( zp )=δε
z
p
,
where δ= ( a(1−a)p ).
1068 Qing-Hu Hou, Hao Pan and Zhi-Wei Sun
If ε= 1, then
Np (a,b) = p −1
2
− 1
p
p−1∑
z=1
( zp )=δ
z
does not depend on b.
If p ≡ 1 (mod 4), then (−1p )= 1 and hence
p−1∑
z=1
( zp )=1
z =
p−1∑
z=1
( p−zp )=1
(p − z) = p p −1
2
−
p−1∑
z=1
( zp )=1
z,
thus
p−1∑
z=1
( zp )=1
z = p p −1
4
and
p−1∑
z=1
( zp )=−1
z =
p−1∑
z=1
z −p p −1
4
= p p −1
4
.
So, if p ≡ 1 (mod 4), then |S| = |T | = 1, and moreover
S =
{
p −1
2
− p −1
4
}
=
{
p −1
4
}
.
Now assume that p ≡ 3 (mod 4). We want to show that |S| = |T | = 2. By Lemma 4,
p−1∑
z=1
z
(
z
p
)
=−ph(−p) 6= 0.
Thus
p−1∑
z=1
( zp )=1
z =
p−1∑
z=1
z
1+ ( zp )
2
= p p −1
4
− p
2
h(−p)
and hence
p−1∑
z=1
( zp )=−1
z =
p−1∑
z=1
z −
p−1∑
z=1
( zp )=1
z = p p −1
4
+ p
2
h(−p).
By Lemma 6, for some a ∈ {2, . . . , p − 2} we have ( a−1p ) = ( ap ) = 1 and hence ( a(1−a)p ) = −1. For
a′ = p +1−a, we have (
a′
p
)
=−1 and
(
a′(1−a′)
p
)
=
(
(1−a)a
p
)
=−1.
By Lemma 6, for some a∗,b∗ ∈ {2, . . . , p −2} we have
−
(
a∗−1
p
)
=
(
a∗
p
)
= 1 and
(
b∗−1
p
)
=−
(
b∗
p
)
= 1.
Note that (
a∗(1−a∗)
p
)
= 1 =
(
b∗(1−b∗)
p
)
.
Now we clearly have |S| = |T | = 2. Moreover,
S =
{
p −1
2
−
(
p −1
4
± h(−p)
2
)}
=
{
p −1±2h(−p)
4
}
.
The proof of Theorem 2 is now complete. 
Qing-Hu Hou, Hao Pan and Zhi-Wei Sun 1069
References
[1] H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Springer,
1993.
[2] H. Davenport, The Higher Arithmetic. An Introduction to the Theory of Numbers, 8th ed., Cambridge University Press,
2008.
[3] Q.-H. Hou, Z.-W. Sun, “Sequence A320159 at OEIS (On-Line Encyclopedia of Integer Sequences)”, 2018, http://oeis.
org/A320159.
[4] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Graduate Texts in Mathematics,
vol. 84, Springer, 1990.
[5] Z.-W. Sun, “Quadratic residues and related permutations and identities”, Finite Fields Appl. 59 (2019), p. 246-283.
[6] ——— , “Quadratic residues and quartic residues modulo primes”, Int. J. Number Theory 16 (2020), no. 8, p. 1833-1858.

A new theorem on quadratic residues modulo primes

Laurent Clozel

Comptes Rendus
Mathématique
Laurent Clozel
Minoration universelle de certaines intégrales quadratiques des
fonctions L automorphes
Volume 360 (2022), p. 1087-1092
https://doi.org/10.5802/crmath.327
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1087-1092
https://doi.org/10.5802/crmath.327
Algèbre / Algebra
Minoration universelle de certaines intégrales
quadratiques des fonctions L automorphes
Laurent Clozel∗, a
a Mathématiques, Université Paris-Sud, Bâtiment 307, rue Michel Magat, 91405 Orsay
Cedex, France
Courriel : clozel@math.u-psud.fr
Résumé. Soit π une représentation cuspidale unitaire de GL(m,A), où A désigne l’anneau des adèles du
corps Q des rationnels. Soit L(s,π) sa fonction L. On introduit une minoration universelle des intégrales∫ +∞
−∞ | L(1/2+i t ,π)1/2+i t−s |2d t , où s est égal à 0 ou à un zéro de L(s,π) dans la bande critique. On esquisse la
démonstration pour m ≤ 2 et sous quelques hypothèses sur π. Celle–ci repose sur la transformation de
Mellin. On donne une application à l’abscisse de convergence de la série de Dirichlet L(s,π). Des résultats
plus complets seront donnés dans un article à paraître ainsi que son Appendice.
Manuscrit reçu le 8 décembre 2021, révisé le 11 janvier 2022, accepté le 12 janvier 2022.
Abridged version
Let π be a cuspidal, unitary representation of GL(m,A), whereA denotes the ring of adèles of the
rational fieldQ. Let L(s,π) be its L–function. In this section, we exclude the case where L(s,π) has
a pole, i.e., where m=1 and π is a character of the idèle group factoring through the idèle norm.
The paper is concerned with the following universal inequalities:∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t − s
∣∣∣∣2 d t > 2π log2 (A)
where s is a zero of L(s,π) in the critical strip.∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t
∣∣∣∣2 d t > π2 . (C)
The Note studies the case where these integrals are known to be finite, i.e., m ≤ 2.
The proof relies on a relation between these quotients of L(s,π) and a certain function Hs (x).
Let L(s,π) =∑∞1 ann−s ; we define, for x ≥ 0 and π satisfying our assumptions:
As (x) =
∑
n≤x
ann
−s , (1)
Hs (x) = xs−1
∑
n≤x
ann
−s . (2)
∗Auteur correspondant.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1088 Laurent Clozel
The basic relation is, M f (w) denoting the Mellin transform of a function f on R+:
M Hs (w) = L(1−w,π)
1−w − s (3)
where s = 0 (identity (C )) or s is a zero of L(s,π) in the critical strip (identity (A)).
In order to make sense of this identity for Re(w) = 12 , we need a growth estimate for As when
x → ∞. If s = 0, such estimates are due to Landau (1915) and to Friedlander and Iwaniec [2].
In the paper [1] announced in this Note, we extend the Friedlander–Iwaniec method to the case
where s is in the critical strip. If furthermore L(s,π) = 0, we are then able for m ≤ 2 to obtain the
identity (3) in a vertical strip containing the critical line. In both cases, the inequalities (A) and (C)
then follow from the Parseval equality for the Mellin transform.
The estimate on (A) can also be used to show that the abscissa of convergence σc of L(s,π)
verifies σc ≤ 1− 2m+1 (for any m).
1. Introduction
1.1.
Soit π une représentation unitaire cuspidale de GL(m,A) où A est l’anneau des adèles de Q. Soit
L(s,π) sa fonction L. Plus généralement, on peutconsidérer une représentation π=π1 ×π2 ×·· ·×
πr , oùm = m1 +m2 + ·· · +mr , πi est une représentation cuspidale unitaire de GL(mi ,A), et ×
désigne l’induction à partir du parabolique de blocs (m1, . . . , mr ). Ainsi
L(s,π) = L(s,π1) · · ·L(s,πr ).
Nous supposons que L(s,π) a au plus un pôle simple, en s = 1. Ainsi il y a plus une occurence
de mi = 1, πi (x) = [x|c (x ∈A×) ; et nous supposons alors que c = 0, de sorte que L(s,πi ) = ζ(s). On
dira alors que L(s,π) contient un facteur ζ.
Rappelons que L(s,π) peut être complétée par un facteur archimédien L(s,π∞), qui peut être
écrit
L(s,π∞) = c(π∞)π
ms
2
m∏
i=1
Γ
( s + ci
2
)
où c(π∞) est une constante. On associe aussi à π un conducteur D , entier positif. Voir Jacquet [4]
ainsi que [5]. On pose
Λ(s,π) = D s/2L(s,π∞)L(s,π) ;
alorsΛ(s,π) vérifie l’équation fonctionnelle
Λ(s,π) = ε(π)Λ (1− s, π̃)
où |ε(π)| = 1 et π̃ est la représentation duale.
Nous énonçons les théorèmes annoncés dans cette Introduction sous l’hypothèse suivante.
Hypothèse H.
(i) m ≤ 2
(ii) Le facteur archimédien π∞ de π est autodual
(iii) π vérifie la conjecture de Ramanujan : les facteurs πv de π sont tempérés (en toutes les
places v de Q).
Si m = 1, π est donc un caractère de Dirichlet. Si m = 2, L(s,π) est le produit de deux fonctions
L de Dirichlet (dont peut–être la fonction ζ) ou π est cuspidale tempérée, π∞ étant autoduale.
Soit s = 12 + iτ un zéro de L(s,π) sur la droite critique.
Laurent Clozel 1089
Theorem A. (L(s,π) sans facteur ζ)∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t − s
∣∣∣∣2 d t > 2π log2.
(on ne confondra pas la représentation π et le nombre π= 3,141592. . .)
Si L(s,π) contient un facteur ζ, le résultat est différent :
Theorem B. ∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t − s
∣∣∣∣2 d t > 2π(log2− 2|κ|1− s
)
(i)
où κ est le résidu en s = 1 de L(s,π).∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t − s
∣∣∣∣2 d t >π log2. (ii)
Les deux résultats qui suivent ont été suggérés par Peter Sarnak. On remarquera que s = 0 n’est
pas en général un zéro de L(s,π).
Theorem C. (L(s,π) sans facteur ζ).∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t
∣∣∣∣2 d t >π
Theorem D. Dans le cas général ∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t
∣∣∣∣2 d t > π2 .
1.2.
Sous l’hypothèse que m ≤ 2, les intégrales figurant dans les théorèmes A–D sont finies. Ceci
résulte de théorèmes classiques quand les fonctions L sont des fonctions L de Dirichlet, et
d’un théorème de Michel et Venkatesh [6] pour L(s,π), π étant une repréentation cuspidale de
GL(2). En général, la finitude résulterait évidemment d’une approximation de l’hypothèse de
Lindelöf généralisée, cf. [3, Corollary 5.20]. On peut cependant considérer que les théorèmes A–D
sont vrais si les intégrales sont infinies. Avec cette interprétation, Peter Sarnak a remarqué qu’ils
pouvaient être démontrés, sans hypothèse sur π autre que d’être induite de cuspidales unitaires.
La démonstration figurera dans l’Appendice de [1].
2. Esquisse de démonstrations (m = 2)
2.1.
Nous nous limitons dans cette Note au cas où π est une représentation de GL(2,A) vérifiant
l’hypothèse H 1. Pour s = 0, ou Re(s) ∈]0,1[, posons pour x ≥ 0
As (x) =
∑
n≤x
ann
−s (4)
Hs (x) = xs−1
∑
n≤x
ann
−s − κ
1− s . (5)
1Le cas où m = 1 est évidemment plus facile. Voir [1, § 1.4, § 3.2].
1090 Laurent Clozel
On considère la transformée de Mellin de Hs :
M Hs (w) =
∫ ∞
0
Hs (x)x
w−1 d x. (6)
Pour définir M Hs , nous avons besoin d’estimer la croissance de Hs (x).
Considérons tout d’abord H0. Dans ce cas (pour m arbitraire, m > 1, et π vérifiant nos
conditions), Friedlander et Iwaniec ont démontré le théorème suivant [2].
Théorème 1 (Friedlander–Iwaniec). On suppose m > 1.
A0(x)−κx =Oε
(
x1−
2
m+1 +ε
)
(i)
H0(x) =Oε
(
x−
2
m+1 +ε
)
. (ii)
(Les constantes implicites ne dépendent que de ε, du conducteur arithmétique D de π et du
paramètre de Langlands de π∞). Un résultat analogue, sous des hypothèses différentes sur π∞,
est dû à Landau (1915) ; voir [1, Theorem 2.1].
Pour m = 2, on voit donc que H0 ∈ L2(R+), et que M H0(w) est défini et holomorphe pour 0 <
Re w < 23 . Il résulte alors de la formule de Parseval pour la transformée de Mellin [7, Theorem 71]
que M H0(
1
2 + i t ) est une fonction L2 de t et que
1
2π
∫ +∞
−∞
∣∣∣∣M H0 (12 + i t
)∣∣∣∣2 d t = ∫ ∞
0
|H0(x)|2 d x. (7)
Il reste à calculer
M H0(w) =
∫ ∞
0
(
x−1
∑
n≤x
an −κ
)
xw−1d x.
Le calcul est élémentaire, en utilisant l’estimée du théorème 1 On obtient
M H0(w) = L(1−w,π)
1−w .
Il résulte alors de (7) que
1
2π
∫ +∞
−∞
∣∣∣∣L(1/2+ i t ,π)1/2+ i t
∣∣∣∣2 d t = ∫ ∞
0
|H0(x)|2 d x.
Si κ= 0, l’intégrale de droite est minorée par ∫ 21 x−2d x = 12 ; en général, un calcul simple montre
que
∫ 1
0 |κ|2d x +
∫ 2
1 |x−1 −κ|2d x ≥ 14 . D’où les théorèmes C et D.
Remarque. Il résulte en particulier de ce calcul que L(1/2+i t ,π)1/2+i t est une fonction L
2 de t . On a en
fait, d’après un résultat de sous–convexité de Michel et Venkatesh [6], |L(1/2+ i t )| = O(|t |1/2−δ)
pour δ> 0.
Remarque. En fait, on l’a vu, M H0 est définie et holomorphe pour τ = Re(w) < 23 , de plus
H0(x)xτ+i t−1 ∈ L2(R). Il résulte de nouveau des propriétés de la transformée de Mellin [7] que∫ +∞
−∞ | L(τ+i t ,π)τ+i t |2d t <∞ si τ > 13 (par le changement de variable w 7→ 1−w). On peut vérifier que
ceci n’est pas impliqué par le résultat de sous–convexité.
2.2.
Pour démontrer, de façon analogue, les théorèmes A et B, il nous faut obtenir une estimée pour
As (x) et Hs (x). Nous sommes donc amené à reprendre la démonstration de Friedlander–Iwaniec.
Le résultat est le suivant. Nous revenons au cas de m arbitraire (> 1) et nous supposons que π
vérifie l’hypothèse H (hormis m ≤ 2.)
Théorème 2. Soit s =σ+ iτ, 0 <σ< 1. On suppose m > 1. Alors, pour x assez grand,
As (x) = κ
1− s x
1−s +L(s,π)+O
(
x1−
2
m+1 −σ+ε
)
.
Laurent Clozel 1091
La démonstration est donnée dans [1, Theorem 2.2]. (Pour x supérieur à une constante X de
l’ordre de D1/2, la constante implicite ne dépend que de ε, D , τ et des paramètres de π∞.)
Corollaire. Sous les mêmes hypothèses, si L(s,π) = 0,
Hs (x) =O
(
x−
2
m+1 +ε
)
.
On a donc la même majoration que pour H0. Revenons au cas m = 2. Alors M Hs (w) est, de
nouveau, défini et holomorphe pour 0 < Re(w) < 23 . La fonction Hs (x) appartient à L2(R+) ; on
calcule explicitement sa transformée de Mellin ; si L(s,π) = 0,
M Hs (w) = L(1−w,π)
1− s −w .
En particulier on a alors
1
2π
∫ +∞
−∞
∣∣∣∣∣L(
1
2 + i t ,π)
1
2 + i t − s
∣∣∣∣∣
2
d t =
∫ ∞
0
|Hs (x)|2 d x. (8)
Si L(s,π) n’a pas de pôle, le membre de droite est minoré par
∫ 2
1 x
2σ−2d x = log2 si σ = 12 ; d’où
le théorème A (noter qu’on obtient une minoration explicite même si s n’est pas sur la droite
critique). Si κ 6= 0, un calcul simple montre que ∫ 20 |Hs (x)|2d x est minoré, si σ= 12 , par 12 log2. On
a aussi
∫ 2
1 |xs−1 − κ1−s |2 > log2− 2|κ||1−s| , estimée meilleure pour |s| grand.
Comme dans le cas où s = 0, on peut étendre l’identité (8), en remplaçant l’abscisse τ= 12 par
τ> 13 (cf. les remarques suivant le théorème 1).
3. Applications et extrapolations du théorème 2
3.1.
Pour m arbitraire, supposons π cuspidale (et non triviale si m = 1), ainsi que l’hypothèse H,
hormis m ≤ 2.
D’après le théorème 2, on a alors∑
n≤x
ann
−s = L(s,π)+O
(
x1−
2
m+1 −σ+ε
)
.
On en déduit :
Théorème 3. Pour π cuspidale (vérifiant l’hypothèse H) et non triviale, la série de Dirichlet L(s,π)
est convergente pour σ> 1− 2m+1 .
On peut affirmer davantage sous une conjecture de Friedlander–Iwaniec [2, Conjecture 2] :
Conjecture (Friedlander–Iwaniec). Pour x assez grand, A0(x) ¿ x m−12m +ε.
En admettant que ceci donne, comme dans le théorème 2, la majoration As (x) ¿ x m−12m −σ+ε,
on peut alors conjecturer, pour l’abscisse de convergence σc de L(s,π),
σc ≤ 1
2
− 1
2m
. (9)
Un argument simple de convexité montre d’ailleurs (inconditionnellement) que
σc ≥ 1
2
− 1
m
. (10)
(en particulierσc ≥ 0 si m ≥ 2, et aussi bien sûr si m = 1). Je ne sais pas quelle devrait être la valeur
correcte de σc .
En supposant la conjecture précédente, et l’amélioration correspondante du reste dans le
théorème 2, on peut alors reprendre l’argument de la remarque 2.1. On trouve (grâce à la
transformation de Mellin) que L(τ+i t )τ+i t est une fonction L
2 de t si 12 − 12m < τ ≤ 12 . Il est facile de
voir que ceci est impliqué par l’hypothèse de Lindelöf généralisée.
1092 Laurent Clozel
4. Questions
On peut évidemment se demander si, par exemple, les minorations du théorème A et du théo-
rème C sont optimales. Dans le cas du théorème A ou B, on peut aussi demander si l’intégrale
quadratique peut rester bornée quand le zéro s tend vers l’infini. Ceci semble peu probable. Pour
la fonction ζ de Riemann, on a par exemple le résultat suivant.
Soit µ un ordre de ζ(s) sur la droite σ= 1/2 : |ζ(1/2+ i t |¿ |t |µ+ε (|t |→∞) ; par exemple µ= 16 .
Théorème 4. Supposons que, pour une suite (s) de zéros de ζ sur la droite critique,
I (s) =
∫ +∞
−∞
∣∣∣∣ ζ(1/2+ i t )1/2+ i t − s
∣∣∣∣2
reste borné. Soit µ un ordre de ζ sur la droite critique.
Alors, pour de tels s = 1/2+ iτ,
ζ′(s) ¿ τµ/3+ε.
Bien que la conclusion soit probablement vraie, il semble peu plausible qu’elle puisse être
ainsi démontrée.
Terminons en remarquant que pour un corps global F de fonctions, on a un analogue évident
du théorème C; dans ce cas on trouve que la borne inférieure peut être approchée arbitrairement
si m, F et π sont convenablement choisis.
Références
[1] L. Clozel, P. Sarnak, « A universal lower bound for certain quadratic integrals of automorphic L-functions », https:
//arxiv.org/abs/2203.12475, 2022.
[2] J. B. Friedlander, H. Iwaniec, « Summation formulae for coefficients of L-functions », Can. J. Math. 57 (2005), n 3,
p. 494-505.
[3] H. Iwaniec, E. Kowalski, Analytic number theory, Colloquium Publications, vol. 53, American Mathematical Society,
2004.
[4] H. Jacquet, « Principal L-functions of the linear group », in Automorphic forms, representations and L-functions,
Proceedings of Symposia in Pure Mathematics, vol. 33, American Mathematical Society, 1979, p. 63-86.
[5] H. Jacquet, I. I. Piatetski-Shapiro, J. A. Shalika, « Conducteur des représentations du groupe linéaire », Math. Ann. 256
(1981), n 2, p. 199-214.
[6] P. Michel, A. Venkatesh, « The subconvexity problem for GL2 », Publ. Math., Inst. Hautes Étud. Sci. 111 (2010), p. 171-
271.
[7] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, third éd., Chelsea Publishing, 1986.

Minoration universelle de certaines intégrales quadratiques des fonctions $L$ automorphes

Anis  M. Haddouche

Wei Lu

Comptes Rendus
Mathématique
Anis M. Haddouche and Wei Lu
A unified approach for covariance matrix estimation under Stein loss
Volume 360 (2022), p. 1093-1098
https://doi.org/10.5802/crmath.356
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1093-1098
https://doi.org/10.5802/crmath.356
Statistics / Statistiques
A unified approach for covariance matrix
estimation under Stein loss
Anis M. Haddouche ∗, a andWei Lua
a INSA Rouen, Normandie Univ, UNIROUEN, UNIHAVRE, LITIS and LMI, avenue de
l’Université, BP 8, Saint-Étienne-du-Rouvray, 76801, France
E-mails: mohamed.haddouche@insa-rouen.fr (A. M. Haddouche),
wei.lu@insa-rouen.fr (W. Lu)
Abstract. In this paper, we address the problem of estimating a covariance matrix of a multivariate Gaussian
distribution, from a decision theoretic point of view, relative to a Stein type loss function. We investigate the
case where the covariance matrix is invertible and the case when it is non–invertible in a unified approach.
Résumé. Dans cet article, nous abordons le problème de l’estimation d’une matrice de covariance d’une
distribution gaussienne multivariée, du point de vue de la théorie de la décision, par rapport à une fonction de
coût de type Stein. Nous étudions dans une approche unifiée le cas où la matrice de covariance est inversible
et le cas où elle n’est pas inversible.
Mathematical subject classification (2010). 62H12, 62F10, 62C99.
Manuscript received 13 July 2021, revised 4 January 2022, accepted 13 March 2022.
1. Introduction
Let X be an observed p ×n matrix of the form
X = B Z , (1)
where B is a p × r matrix of unknown parameters, with r ≤ p, and Z is a r × n random
matrix. Assume that r is known and that the columns of Z are identically and independently
distributed as the r -dimensional multivariate normal distribution Nr (0r , I r ). Then the columns
of X are identically and independently distributed from the p-dimensional multivariate normal
Np (0p ,Σ), where Σ= B B T is the unknown p ×p covariance matrix with
rank(Σ) = r ≤ p .
It follows that the p × p sample covariance matrix S = X X T has a singular Wishart distribution
(see Srivastava [9]) such that
rank(S) = min(n,r ) = q ≤ p ,
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1094 Anis M. Haddouche and Wei Lu
with probability one. Note that, as the matrices S and Σ are non–invertible, we denote in the
following by S+ and Σ+ the Moore-Penrose inverses of S and Σ respectively.
We consider the problem of estimating the covariance matrix Σ under the Stein type loss
function
L
(
Σ̂,Σ
)= tr(Σ+Σ̂)− ln∣∣Λ(Σ+Σ̂)∣∣−q , (2)
where Σ̂ estimates Σ and Λ(Σ+Σ̂) is the diagonal matrix of the q positives eigenvalues of Σ+Σ̂.
Note that this loss function is based on the Kullback–Leibler divergence between two multivariate
normal distributions with the same means and with covariances Σ̂ and Σ. It is an extension of the
usual Stein loss function (see Stein [10]). Thus, the corresponding risk function is given by
R
(
Σ̂,Σ
)= E [L (Σ̂,Σ)] ,
where E(·) denotes the expectation with respect to the model (1). Note that the loss function (2)
is an adaptation of the original Stein loss function (see Stein [11]) to the context of the model (1)
(see Tsukuma [13] for more details).
As related by Ledoit and Wolf [8], the difficulty of covariance estimation is commonly charac-
terized by the ratio p/n. In fact, the usual estimators of the form
Σ̂a = a S with a > 0, (3)
perform poorly when n, p → ∞ with p/n → c > 0. Indeed, the larger (smaller) eigenvalues of
Σ are overestimated (underestimated) by those estimators. Hence, in this asymptotic context,
alternative estimators dominating (with lower risk functions) these usual estimators are needed.
A possible approach is to regularize the eigenvalues of Σ̂a , which gives rise to the class of
orthogonally invariant estimators (see Takemura [12]) as those in (5) below.
Note that, as we consider the model (1), we deal in a unified approach, with the following
cases.
(i) n < r = p: Σ is invertible of rank p and S is non–invertible of rank n;
(ii) r = p ≤ n: Σ and S are invertible;
(iii) r < p ≤ n: Σ and S are non–invertible of rank r ;
(iv) r ≤ n < p: Σ and S are non–invertible of rank r ;
(v) n < r < p: Σ and S are non–invertible of ranks r and n respectively.
The class of orthogonally invariant estimators was considered by various authors. For instance,
see Stein [11], Dey and Srivastava [2] and Haff [5] for the case (i), Konno [6] and Haddouche et
al. [4] for the cases (i) and (ii). See also Chételat and Wells [1] for the cases (iii) and (iv). Recently,
Tsukuma and Kubokawa [13] extended the Stein [11] estimator to the five possible cases, cited
above, in a unified approach. Similarly, we extend here the class of Haff [5] estimators to the
context of the model (1).
The rest of this paper is organized as follows. In Section 2, we derive the domination result of
the Haff estimators over the usual estimators. We study numerically the behavior of the proposed
estimators in Section 3.
2. Main result
Improving the class of the natural estimators in (3) relies on improving the optimal estimator
among this class, that is, the one which minimizes the loss function (2).
Proposition 1 (Tsukuma [13]). Under the Stein type loss function (2), the optimal estimator
among the class (3) is given by
Σ̂ao = ao S, where ao =
1
m
and m = max(n,r ). (4)
Anis M. Haddouche and Wei Lu 1095
As mentioned in Section 1, we consider the class of orthogonally invariant estimators as
alternative estimators. Let S = H L H> be the eigenvalue decomposition of S where H is a p × q
semi–orthogonal matrix of eigenvectors and L = diag(l1, . . . , lq ), with l1 >, . . . ,> lq , is the diagonal
matrix of the q positive corresponding eigenvalues (see Kubokawa and Srivastava [7] for more
details). The class of orthogonally invariant estimators is of the form
Σ̂Ψ = ao
(
S +H LΨ(L) H>) (5)
with Ψ(L) = diag(ψ1(L), . . . ,ψq (L)), where ψi (L) (i = 1, . . . , q) is a differentiable function of L.
More precisely, we consider an extension of the empirical Bayes Haff type estimators in [5], to the
context of the model (1), defined as
Σ̂α = ao
(
S +HLΨ(L)H>) where Ψ(L) = b L−α
tr(L−α)
, with α≥ 1 and b > 0, (6)
where ao is given in (4). Note that, for α = 1, this is the estimator considered by Konno [6],
who deals with the cases (i) and (ii) under a quadratic loss. An extension to elliptical setting
was considered recently by Haddouche et al. [4]. Note also that Tsukuma [13] considered also
empirical Bayes estimators but not the one in (6). We give in the following theorem our main
result.
Theorem 2. Let Ψ(L) = diag(ψ1(L), . . . ,ψq (L)) where, for any i = 1, . . . , q, ψi (L) = blαi /tr(L−α)
with α ≥ 1 and b > 0, be a differentiable function of L. The Haff type estimators in (6) dominates
the optimal estimator in (4), under the loss function (2), as soon as
0 < b ≤ bo =
2
(
q −1)
m −q +1 .
Proof. The proof consists to showing that the risk difference between the Haff type estimators
in (6) and the optimal estimator in (4), namely,
∆(α,ao ) = R
(
Σ̂α,Σ
)−R (Σ̂ao ,Σ) , (7)
is non–positive. Note that Σ̂α can be written as
Σ̂α = ao HLΦ(L)H> with Φ(L) = I q +Ψ(L).
The risk of these estimators under the Stein type loss function (2) is given by
R
(
Σ̂α,Σ
)= E (tr(Σ+Σ̂α))−E (ln∣∣Λ(Σ+Σ̂α)∣∣)−q . (8)
First, dealing with E(tr(Σ̂αΣ+)), we apply in Tsukuma [13, Lemma A.2] in order to get rid of the
unknown parameter Σ+. It follows that,
E
(
tr
(
Σ+Σ̂α
))= aoE
(
q∑
i=1
{(
m −q +1)ϕi +2li ∂ϕi
∂li
+2
q∑
j > i
liϕi − l jϕ j
li − l j
})
(9)
where, for any i = 1, . . . , q ,
φi = 1+b
l−αi
tr(L−α)
,
∂ϕi
∂li
= bα 1− tr(L
−α) lαi
tr2 (L−α) l 1+2αi
and
q∑
j > i
liϕi − l jϕ j
li − l j
=
q∑
j > i
(
1+ b
tr(L−α)
{
l 1−αi − l 1−αj
li − l j
})
.
Using the fact, for any j > i , l j > li , it can be shown that
q∑
j > i
liϕi − l jϕ j
li − li
≤ (q − i ) . (10)
1096 Anis M. Haddouche and Wei Lu
Therefore, using (10), we obtain
E
(
tr
(
Σ+Σ̂α
))≤ ao E
(
q∑
i=1
{(
m −q +1)(1+b li −α
tr(L−α)
)
+2bα 1− tr(L
−α) lαi
tr2 (L−α) l 2αi
+2(q − i )})
= ao m q +ao b E
(
q∑
i=1
{(
m −q +1) li −α
tr(L−α)
+2α1− tr(L
−α) lαi
tr2 (L−α) l 2αi
})
= ao
(
m q +b (m −q +1))+2αE ( tr(L−2α)
tr2 (L−α)
−1
)
.
Now, from the submultiplicativity of the trace for semi–definite positive matrices, we have
tr(L−2α) ≤ tr2(L−α). Then, an upper bound for (9) is given by
E
(
tr
(
Σ+Σ̂α
))≤ ao (m q +b (m −q +1)) . (11)
Secondly, dealing with E(ln |Λ(Σ̂αΣ+)|) in (8), it can be shown that
Λ
(
Σ+Σ̂α
)= aoΛ(Σ+HLΦ(L)H>)
= aoΛ
(
L1/2H>Σ+HL1/2Φ(L)
)
.
Note that L1/2H>Σ+HL1/2 andΦ(L) are full rank q ×q matrices. It follows that∣∣Λ(Σ̂αΣ+)∣∣= aqo ∣∣L1/2H>Σ+HL1/2Φ(L)∣∣ .
Therefore
E
(
ln
∣∣Λ(Σ̂αΣ+)∣∣)= q ln(ao)+E (ln ∣∣L1/2H TΣ+HL1/2∣∣)+E (ln |Φ(L)|) . (12)
Using the fact that ln(1+x) ≥ 2 x/(2+x), for any positive constant x, we have
ln |Φ(L)| = ln
∣∣∣∣I q + b L−αtr(L−α)
∣∣∣∣
=
q∑
i=1
ln
(
1+ b l
−α
i
tr(L−α)
)
≥
q∑
i=1
2b l−αi /tr(L
−α)
2+b l−αi /tr(L−α)
.
Then
ln |Φ(L)| ≥ 2b
2+b , (13)
since, for any i = 1, . . . , q , l−αi ≤ tr(L−α). Consequently, thanks to (13), a lower bound for (12) is
given by
E
(
ln
∣∣Λ(Σ̂αΣ+)∣∣)≥ q ln(ao)+E (ln ∣∣L1/2H TΣ+HL1/2∣∣)+ 2b
2+b . (14)
Now, relying on the proof of [13, Proposition 2.1] in Tsukuma, it can be shown that
R
(
Σ̂ao ,Σ
)=−q ln(ao)−E (ln∣∣L1/2H TΣ+HL1/2∣∣) . (15)
Finally, combining (11), (14) and (15), an upper bound for the risk difference (7) is given by
∆(α,ao ) ≤ ao
(
m −q +1) b − 2b
2+b
= b
b +2
( (
m −q −1) (b +2)
m
−2
)
,
since ao = 1/m, which is non-positive as soon as
0 < b ≤ bo =
2
(
q −1)
m −q +1 . 
Anis M. Haddouche and Wei Lu 1097
3. Numerical study
We study here numerically the performance of the Haff type estimators in (6). As shown numeri-
cally by Fourdrinier et al. [3] that the best constant b is bo . So that, we consider in this section the
following Haff type estimators
Σ̂α = ao
(
S +HLΨ(L)H>) where Ψ(L) = bo L−α
tr(L−α)
,
with α≥ 1 and bo =
2
(
q −1)
m −q +1 . (16)
As for the population covariance matrixΣ, we consider the following structures: (i) the identity
matrix I p and (ii) an autoregressive structure with coefficient 0.9. We set their p − r smallest
eigenvalues to zero in order to construct matrices of rank r ≤ p.
To assess the performance of the proposed estimators, we compute the Percentage Reduction
In Average Loss (PRIAL), for some values of p, n, r and α, defined as
PRIAL
(
Σ̂α
)= R (Σ̂ao ,Σ)−R (Σ̂α,Σ)
R
(
Σ̂ao ,Σ
) ×100,
where Σ̂ao and Σ̂α are respectively defined in (4) and (3).
Table 1. Effect of α on PRIAL’s for the structures(i) and (ii) of Σ.
Σ (p,n) r Σ̂1 Σ̂2 Σ̂3 Σ̂4 Σ̂5
(i)
(30,50)
10 6.85 12.45 15.53 16.95 17.53
20 9.20 13.91 14.88 14.68 12.47
30 11.81 14.33 13.41 12.43 11.71
(50,30)
20 18.31 19.65 17.75 16.44 15.63
40 17.12 16.33 14.07 12.78 12.02
50 11.80 14.23 13.29 12.30 11.59
(150,30)
20 18.17 19.69 17.87 16.56 15.71
40 17.08 16.31 14.07 12.76 11.99
60 8.88 12.27 12.33 11.75 11.19
150 2.83 5.09 6.50 7.25 7.61
(ii)
(30,50)
10 6.06 8.70 9.62 9.89 9.92
20 8.81 11.66 12.12 11.93 11.61
30 11.46 13.15 12.35 11.48 10.82
(50,30)
20 17.18 17.45 15.85 14.76 14.07
40 16.34 15.18 13.19 12.00 11.28
50 11.33 12.82 12.02 11.19 10.57
(150,30)
20 17.30 18.00 16.38 15.19 14.42
40 16.27 15.04 13.04 11.84 11.13
60 8.48 10.43 10.29 9.81 9.35
150 2.73 4.03 4.61 4.86 4.95
Table 1 shows that the proposed estimators improve over Σ̂ao for any possible ordering of
p,n and r . Compared to other cases, the Haff type estimators Σ̂α (for α = 1, . . . ,5) have better
performances in the case where p > n > r , with PRIAL’s higher than 14.07% for both structures (i)
and (ii) ofΣ. We report that the optimal value ofα, which maximizes the PRIAL’s, depends on p,n
and r .
1098 Anis M. Haddouche and Wei Lu
We then compare the Prial’s of the Haff type estimators in and the well known James–Stein
type estimator Σ̂JS considered by Tsukuma [13], given by
Σ̂JS = HLD JSq H>,
where D JSq = diag(d
JS
1 , . . . , d
JS
q ) with, for i = 1, . . . , q , d JSi = (n + r −2i +1)−1.
Table 2. PRIAL’s of the Haff type estimators and the Stein type estimator for the struc-
tures (ii) of Σ.
(p,n) r Σ̂1 Σ̂2 Σ̂3 Σ̂4 Σ̂JS
(30,40)
7 9.06 12.34 13.36 13.68 9.87
14 9.60 13.09 13.78 13.74 20.45
(40,30)
7 11.87 15.57 16.41 16.55 16.04
14 13.91 16.41 16.13 15.55 26.35
(20,100)
5 4.04 5.61 6.14 6.29 4.68
15 3.67 5.45 6.38 7.12 11.80
Table 2 shows that in some cases the PRIALs of the Haff-type estimators are higher than
those of the Stein-type estimator, while in other cases the Stein-type estimator has a better
performance. Consequently, their risk functions intersect and thus, in our opinion, a theoretical
dominance results between these two estimators cannot be derived.
References
[1] D. Chételat, M. T. Wells, “Improved second order estimation in the singular multivariate normal model”, J. Multivari-
ate Anal. 147 (2016), p. 1-19.
[2] D. K. Dey, C. Srinivasan, “Estimation of a covariance matrix under Stein’s loss”, Ann. Stat. 13 (1985), p. 1581-1591.
[3] D. Fourdrinier, A. M. Haddouche, F. Mezoued, “Covariance matrix estimation under data-based loss”, Stat. Probab.
Lett. 177 (2021), article no. 109160.
[4] A. M. Haddouche, D. Fourdrinier, F. Mezoued, “Scale matrix estimation of an elliptically symmetric distribution in
high and low dimensions”, J. Multivariate Anal. 181 (2021), article no. 104680.
[5] L. R. Haff, “Empirical Bayes estimation of the multivariate normal covariance matrix”, Ann. Stat. 8 (1980), no. 3,
p. 586-597.
[6] Y. Konno, “Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions
under an invariant quadratic loss”, J. Multivariate Anal. 100 (2009), no. 10, p. 2237-2253.
[7] T. Kubokawa, M. S. Srivastava, “Estimation of the precision matrix of a singular Wishart distribution and its applica-
tion in high-dimensional data”, J. Multivariate Anal. 99 (2008), no. 9, p. 1906-1928.
[8] O. Ledoit, M. Wolf, “A well-conditioned estimator for large-dimensional covariance matrices”, J. Multivariate Anal.
88 (2004), no. 2, p. 365-411.
[9] M. S. Srivastava, “Singular Wishart and multivariate Beta distributions”, Ann. Stat. 31 (2003), no. 5, p. 1537-1560.
[10] C. M. Stein, “Estimation of a covariance matrix”, in Rietz Lecture, 39th Annual Meeting of the IMS, Atlanta, GA, 1975,
1975.
[11] ——— , “Lectures on the theory of estimation of many parameters”, J. Sov. Math. 34 (1986), no. 1, p. 1373-1403.
[12] A. Takemura, “An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal
population”, Tsukuba J. Math. 8 (1984), p. 367-376.
[13] H. Tsukuma, “Estimation of a high-dimensional covariance matrix with the Stein Loss”, J. Multivariate Anal. 148
(2016), p. 1-17.

A unified approach for covariance matrix estimation under Stein loss

Zhuo Liu

Hui Yang

Xiangyu Zhou

New Properties of Multiplier Submodule Sheaves

Henry Fallet

Comptes Rendus
Mathématique
Henry Fallet
Cherednik algebra for the normalizer
Volume 360 (2022), p. 47-52
<https://doi.org/10.5802/crmath.281>
© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2022, 360, p. 47-52
https://doi.org/10.5802/crmath.281
Representation theory / Théorie des représentations
Cherednik algebra for the normalizer
Henry Falleta
a 33 Rue St Leu, 80000 Amiens, LAMFA, UMR 7352 CNRS-UPJV, France
E-mail: henry.fallet@u-picardie.fr
Abstract. Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient
category of the category O for the rational Cherednik algebra and the category of finite dimension modules
of the Hecke algebra of a complex reflection group W . We announce a generalization of this result to the
extension of the Hecke algebra associated to the normalizer of a reflection subgroup.
2020 Mathematics Subject Classification. 20C08.
Manuscript received 29th April 2021, revised 6th July 2021 and 11th October 2021, accepted 9th October 2021.
Version française abrégée
Dans [14] est définie une extension de l’algèbre de Hecke d’un groupe fini de reflexions com-
plexes W . On la note H(W,W0), elle est réalisée à partir du normalisateur d’un sous groupe de
réflexions W0 de W . Le but de cet article est d’établir un résultat équivalent à celui de [9, théo-
rème 5.12] pour la catégorie des H(W0,W )-modules de dimension fini. Pour cela nous allons dé-
finir différents objets : l’algèbre de Cherednik associée à H(W,W0), éléments d’Euler, Opérateurs
de Dunkl–Opdam, catégorie O . Puis nous construisons explicitement le « foncteur K Z » qui va
réaliser l’équivalence de catégorie. Nous donnons ensuite une construction similaire permettant
d’omettre le groupe de réflexions W .
1. Introduction
Let V be a C-vector space of finite dimension n. Let W < GL(V ) be a finite complex reflection
group. Let W0 <W be a reflection subgroup of W . According to [3], we can associate to W a braid
group B(W ) and a Hecke algebra H(W ). In [14] is introduced an extension of H(W ) as an algebra
associated to the normalizer NW (W0), called its Hecke algebra and denoted H(W,W0), for more
results see [10, 11].
We refer to [3] for the general definitions used below. We have a surjection π : B(W ) → W
sending a braided reflection of hyperplane H to the distinguished reflection sH of hyperplane H .
We denote A the hyperplane arrangement of W . Let B̂0 := π−1(NW (W0)) and J be the two-sided
ideal of CB̂0 generated by 〈σmHH = 1, H ∈ A \ A0 and σmHH =
∑mH−1
k=0 aH ,kσ
k
H , H ∈ A0〉 where mH
is the order of the pointwise stabilizer of H in W , denoted WH and the scalars (aH ,k )k∈{0,...mH−1}
are complex numbers invariant under the action of NW (W0), ∀ w ∈ NW (W0), aw(H),k = aH ,k for
all k ∈ {1, . . . ,mH − 1}. As in [14] we define the Hecke algebra of the normalizer as the quotient
algebra of CB̂0 by the ideal J .
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
48 Henry Fallet
There is a second equivalent definition. Let K := Ker(π1(X /W ) → π1(X0/W0)) and B̃0 := B̂0K
where X :=V \⋃H∈A H and X0 =V \⋃H∈A0 H . Then
H(W,W0) ' CB̃0〈σmHH =
∑mH−1
k=0 aH ,kσ
k
H , H ∈A0〉
We introduce Cherednik algebras in this new context, and we prove
Theorem 1. There exists an equivalence of categories between the quotient category O/Otor
and the category of H(W,W0)-modules of finite dimension, where O is a highest weight category
associated to the Cherednik algebra of the pair (W0,W ).
2. Construction of the K Z0 -functor
2.1. The Cherednik algebra of the pair (W0,W )
We denote A(W0,W ) this algebra, and we define it as an algebra admitting a triangular decom-
position in the sense of [13]. As a vector space, A(W0,W ) is C[V ]⊗CNW (W0)⊗C[V ∗] and we add
the following relations on the generators of C[V ], C[V ∗] and CNW (W0),
[x ′, x] = 0 for all (x, x ′) ∈V ∗×V ∗
[y, y ′] = 0 for all (y, y ′) ∈V ×V
[y, x] = t x(y)+ ∑
H∈A0
αH (y)x(vH )
αH (vH )
mH−1∑
j=0
mH (kH , j+1 −kH , j )εH , j
where εH , j = 1mH
∑
w∈WH \{id} det(w)
j w is a primitive orthogonal idempotent of CWH , αH ∈ V ∗
such that Ker(αH ) = H . The vector vH ∈ V is such that C.vH is a WH -stable complement of H .
The set (kH , j ) j∈{0,...,mH−1} is a set of complex number such that kw(H), j = kH , j and t ∈ C. In order
to define a K Z functor, we need to assume t 6= 0. Therefore, up to renormalization we can assume
t = 1 which we do from now on.
As noticed by the referee, this algebra is a special case of a symplectic reflection algebra as
in [6], for NW (W0) acting on V ⊕V ∗ in natural way.
2.2. Dunkl–Opdam operators
We denote by D(X ) the algebra of differential operators over X . In [11] is introduced a differential
1-form, NW (W0)-equivariant and integrable,
ω0 =
∑
H∈A0
aH
dαH
αH
∈Ω1(X )⊗CW0
where aH = ∑mH−1j=0 mH kH , j εH , j . We build a connection on a trivial vector bundle over X , by
∇ := d +ω0. This connection is flat and NW (W0)-equivariant. The covariant derivative of this
connection in the direction of y ∈ V is a differential operator called Dunkl–Opdam operator,
noted Ty .
Proposition 2. For all y ∈ V , Ty := ∂y + ∑H∈A0 αH (y)αH aH ∈ D(X ) o NW (W0). This family of
differential operators satisfies two properties ∀ (y, y ′) ∈V ×V ,
[Ty ,Ty ′ ] = 0
and ∀ y ∈V , ∀ w ∈ NW (W0), w.Ty .w−1 = Tw(y).
We introduce the algebra A(W,W0)r eg = C[X ]⊗C[V ] A(W,W0). We can define a faithful repre-
sentation of A(W0,W ).
C. R. Mathématique — 2022, 360, 47-52
Henry Fallet 49
Theorem 3 (Dunkl embedding).
(1) Φ : A(W0,W ) −→ D(X )oNW (W0)
x ∈V ∗ 7−→ x
w ∈ NW (W0) 7−→ w
y ∈V 7−→ Ty
is an injective morphism of algebras.
(2) By localization, the morphism Φ becomes an isomorphism of algebra. We note Φr eg the
isomorphism between A(W0,W )r eg and D(X )oNW (W0).
2.3. The category O-
Let eu0 = ∑y∈B y∗.y −∑H∈A0 aH , where B is a basis of V . This operator is called the Euler
element. It induces an inner graduation on A(W0,W ), A(W0,W )i := {a ∈ A(W0,W ) | [eu0, a] = i a}
for all i ∈Z, because [eu0, x] = x, [eu0, y] =−y , [eu0, w] = 0.
For every simple CNW (W0) module E ,
∑
H∈A0 aH ∈ Z (CNW (W0)) acts on E by multiplication
by a scalar cE . We define a partial ordering on Irr(NW (W0)): E < E ′ if cE − cE ′ ∈Z>0.
For each E ∈ Irr(NW (W0)) we define a A(W0,W ) module called standard object or Verma
module,
∆(E) = IndA(W0,W )
C[V ∗]⊗CNW (W0) E
The category O is a full sub category of the category of A(W0,W ) modules, where the modules
are finitely generated, locally nilpotent for the action of C[V ∗] and isomorphic to the direct sum
of the generalized eu0-eigenspaces. According to [1,2,9], the category O is Abelian, Artinian. The
object ∆(E) is indecomposable. The category O is highest weight with {∆(E)}E∈Irr(NW (W0)) as the
set of standard object. Every standard object∆(E) admits a simple head L(E). Every simple object
in O is isomorphic to some L(E) and L(E) admits a projective cover. Every object M of O admits
a finite composition series. The B.G.G reciprocity law is satisfied inside O .
2.4. Functor K Z0
Let δ := ∏H∈A αH ∈ C[V ]. Let (A(W0,W )-mod)tor be the subcategory of A(W0,W )-mod with δ-
torsion, i.e. M ∈ A(W0,W )-mod, Mtor := {m ∈ M | ∃ n ≥ 0 δn .m = 0}, then M ∈ (A(W0,W )-mod)tor
if Mtor = M . Let Otor :=O ∩ (A(W0,W )-mod)tor
We have a localization functor,
Loc : A(W0,W )-mod −→ A(W0,W )r eg -mod
M 7−→ A(W0,W )r eg ⊗A(W0,W ) M
This functor induces a fully faithful functor OOtor → A(W0,W )r eg -mod.
The Dunkl embedding gives an equivalence of categories between A(W0,W )r eg -modules
and D(X )o NW (W0)-modules. We also have the following equivalence of categories between
D(X ) o NW (W0)-modules and e.(D(X ) o NW (W0)).e-modules and with D(X )-modules where
e = 1|NW (W0)|
∑
g∈NW (W0) g it is an idempotent of CNW (W0). From the results of [4] we get an
isomorphism of algebras D(X )NW (W0) ' D(X /NW (W0)), thanks to the fact that NW (W0) acts
without fixed points on X .
Let us examine the structure of D(X )oNW (W0)-modules for the case of a localized standard
object. The localized Verma module ∆(E)r eg is a free C[X ]-module of dimension dim(E), so
it corresponds to an algebraic vector bundle over X . We endow this vector bundle with a
connection by considering the action of Ty on an element of ∆(E)r eg . This leads to the formula
∇y (P ⊗ v) := ∂y (P ⊗ v) = ∂y (P )⊗ v +
∑
H∈A0
αH (y)
αH
.(P ⊗aH v)
C. R. Mathématique — 2022, 360, 47-52
50 Henry Fallet
Proposition 4. ∇y is a flat, NW (W0)-equivariant connection with regular singularities over V .
Since this property is true for every standard object, it is also true for every object in O . Apply-
ing the Riemann–Hilbert–Deligne correspondance, we get a horizontal sections functor OOtor →
Cπ1(X /NW (W0))-mod, M → ((M NW (W0)r eg )an)∇. According to [11, Proposition 2.6], this action by
monodromy factorizes through H(W,W0). So we get a functor K Z0 :
O
Otor
→ H(W,W0)-mod which
is exact and fully-faithful. From classical results (see [15]), we get that K Z0 is representable by a
projective object noted PK Z0 . We prove the following
Theorem 5. K Z0 is fully faithful and essentially surjective from the category
O
Otor
to the category
of H(W,W0)-modules of finite dimension.
3. Forgetting W
In this section we provide a related result involving only W0, and not the ambient group W . The
general setting is as follows. Let G be a finite subgroup of GL(V ). Let G0 be a normal subgroup
of G generated by reflexions. Let R0 be the set of reflexions of G0 and A0 the arrangement of
reflecting hyperplanes of G0. The first goal is to build up a Hecke algebra for G from the Hecke
algebra of G0 generalizing H(W0,W ) for G = NW (W0).
Let X + be the subspace of V on which G acts freely and let X0 be the subspace of V on which G0
acts freely. The manifold X0 \ X + is of codimension > 2 then π1(X +) ' π1(X0) [12, Theorem 2.3].
Since G0 acts freely on X0, it also acts freely on X + therefore the projection maps X0 → X0/G0 and
X + → X +/G0 are covering maps and we get two short exact sequences.
1 // π1(X +) // π1(X +/G0) // G0 // 1
1 //
=
OO
π1(X0) //
'
OO
π1(X0/G0) //
OO
G0 //
=
OO
1
=
OO
The exactness and the commutativity of the diagram together imply π1(X +/G0) 'π1(X0/G0).
The braid group B0 of G0 is a normal subgroup of B :=π1(X +/G), we get a short exact sequence
1 // B0 :=π1(X0/G0) // π1(X +/G) // G/G0 // 1
Let I be the ideal of CB0 generated by the relations σ
mH
H =
∑mH−1
k=0 aH ,kσ
k
H for σH a braided
reflection associated to H ∈ A0. Then the Hecke algebra of G0 is the quotient H0 := CB0I .
According to the now proven BMR freeness conjecture (see the references of [11] or its weaker
version in Characteristic 0 [5]) it is an algebra finitely generated of dimension |G0|. Let I+ =
CB ⊗CB0 I be the ideal which define the Hecke algebra of G , H(G) := CBI+ ' CB ⊗CB0 H0 is of
dimension |G|.
Let us make a link between this new algebra and the algebra H(W0,W ). We defined H(W0,W )
as a quotient of the algebra CB̃ 0. We defined B̃ 0 as the quotient of π1(X /NW (W0)) by K :=
Ker(π1(X ) →π1(X0)). Since X0 \ X + has codimension > 2
K = Ker(π1(X ) −→π1(X0)) ' Ker(π1(X /NW (W0)) −→π1(X +/NW (W0)))
And B̃ 0 'π1(X +/NW (W0)) is our group π1(X +/G) =: B . As a result, the algebra H(W0,W ) is the
same as H(G).
Let us consider the category O0tor the full subcategory of O of module annihilated by a power
of δ0 :=∏H∈A0 αH . We have
Theorem 6. K Z0 is fully faithful and essentially surjective from the category
O
O0tor
to the category
of finite dimension H(G)-modules.
C. R. Mathématique — 2022, 360, 47-52
Henry Fallet 51
A priori Otor and O0tor are differents. Actually, we can prove that these two categories are the
same. Let M ∈O0tor then Loc(M) =C[X ]⊗C[X0] (C[X0]⊗C[V ] M)︸ ︷︷ ︸
=0
, so M ∈Otor .
Conversely, let M be a module inside Otor , we would like to prove Mr eg 0 :=C[X0]⊗C[V ] M = 0.
Let i : X + ,→ X0 be a continuous injection of the open set X + inside X0. We denote by OX +
the structural sheaf of X + and OX0 the structural sheaf of OX0 . We denote by DX0 the sheaf of
algebraic differential operators over X0 and DX + the sheaf of algebraic differential operators over
X +, [8, definitions 2.1.5 and 2.1.12].
We have a morphism of ringed space (i , i ]) : (X +,OX + ) → (X0,OX0 ) where i ] : i−1OX0 → OX + is
the identity map, then i ]x : OX0,x →OX +,x is the identity too. The pull back functor is
i∗ : DX0 -mod −→ DX +-mod
(M ,∇0) 7−→ (i∗M , i∗∇0)
We have two functors A(NW (W0))0r eg -mod →DX0 -mod, M → (M ,∇0) and A(NW (W0))r eg -mod →
DX +-mod, M → (M ,∇0).
We need to prove i∗Mr eg 0 = 0. We have for all x ∈ X ⊂ X0, Mr eg ,x = 0 it is due to M ∈ Otor .
Since Mr eg 0 and Mr eg are locally free OX0 -module, respectively OX0 -module, (i
∗Mr eg 0 )x ' Mr eg ,x .
Therefore, (i∗Mr eg 0 )x ' Mr eg 0,x 'OnX0,x ' 0 then n = 0.
Since i∗Mr eg 0 is a locally free OX module, there exists an open affine covering (Ui )i∈I of X such
that (i∗Mr eg 0 )|Ui ' (OX|Ui )
n = 0, thus i∗Mr eg 0 = 0 so Mr eg 0 = 0 then M ∈O0tor . The categories Otor
and O0tor are equals. The proof of the equivalence of categories induced by K Z0 uses the same
arguments as for 5.
Acknowledgements
These results are part of my PhD-Thesis [7] at University Picardie Jules Verne under the supervi-
sion of Prof. Ivan Marin. I would like to thank the referee, which suggested forgetting the ambient
group W .
References
[1] G. Bellamy, U. Thiel, “Highest weight theory for finite-dimensional graded algebras with triangular decomposition”,
Adv. Math. 330 (2018), p. 361-419.
[2] C. Bonnafé, R. Rouquier, “Cherednik algebras and Calogero-Moser cells”, https://arxiv.org/abs/1708.09764, 2017.
[3] M. Broué, G. Malle, R. Rouquier, “Complex reflection groups, braid groups, Hecke algebras”, J. Reine Angew. Math.
1998 (1998), no. 500, p. 127-190.
[4] R. C. Cannings, M. P. Holland, “Differential operators on varieties with a quotient subvariety”, J. Algebra 170 (1994),
no. 3, p. 735-753.
[5] P. Etingof, “Proof of the Broué–Malle–Rouquier conjecture in characteristic zero (after I. Losev and I. Marin-G.
Pfeiffer)”, Arnold Math. J. 3 (2017), no. 3, p. 445-449.
[6] P. Etingof, V. Ginzburg, “Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra
homomorphism”, Invent. Math. 147 (2002), no. 2, p. 243-348.
[7] H. Fallet, “Opérateurs de Dunkl–Opdam, Catégorie O , Algèbres de Cherednik”, PhD thesis in preparation at Univer-
sité Picardie Jule-Verne.
[8] V. Ginzburg, “Lectures on D-modules”, Online lecture notes, available at Sabin Cautis’ webpage http://www.math.
columbia.edu/~scautis/dmodules/dmodules/ginzburg.pdf, with collaboration of Baranovsky, V. and Evens S, 1998.
[9] V. Ginzburg, N. Guay, E. Opdam, R. Rouquier, “On the category O for rational Cherednik algebras”, Invent. Math. 154
(2003), no. 3, p. 617-651.
[10] T. Gobet, A. Henderson, I. Marin, “Braid groups of normalizers of reflection subgroups”, https://arxiv.org/abs/2002.
05468, to appear in Ann. Inst. Fourier, 2020.
[11] T. Gobet, I. Marin, “Hecke algebras of normalizers of parabolic subgroups”, https://arxiv.org/abs/2006.09028, 2020.
[12] C. Godbillon, Eléments de topologie algébrique, Editions Hermann, 1971.
C. R. Mathématique — 2022, 360, 47-52
52 Henry Fallet
[13] R. R. Holmes, D. K. Nakano, “Brauer-type reciprocity for a class of graded associative algebras”, J. Algebra 144 (1991),
no. 1, p. 117-126.
[14] I. Marin, “Artin groups and Yokonuma–Hecke algebras”, Int. Math. Res. Not. 2018 (2018), no. 13, p. 4022-4062.
[15] J. J. Rotman, An introduction to homological algebra, Springer, 2008.
C. R. Mathématique — 2022, 360, 47-52

Cherednik algebra for the normalizer

Feng Qi

Decreasing properties of two ratios defined by three and four polygamma functions

Jiu Liu

Jia-Feng Liao

Hui-Lan Pan

Chun-Lei Tang

Comptes Rendus
Mathématique
Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang
A bifurcation-type result for Kirchhoff equations
Volume 360 (2022), p. 247-254
<https://doi.org/10.5802/crmath.294>
© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2022, 360, p. 247-254
https://doi.org/10.5802/crmath.294
Partial differential equations / Équations aux dérivées partielles
A bifurcation-type result for Kirchhoff
equations
Jiu Liua, Jia-Feng Liaob, Hui-Lan Panc and Chun-Lei Tang∗, d
a School of Mathematics and Statistics, Qiannan Normal University for Nationalities,
Duyun, Guizhou 558000, People’s Republic of China
b School of Mathematics and Information, China West Normal University, Nanchong,
Sichuan 637002, People’s Republic of China
c School of Science, Chongqing University of Posts and Telecommunications,
Chongqing 400065, People’s Republic of China
d School of Mathematics and Statistics, Southwest University, Chongqing 400715,
People’s Republic of China
E-mail: tangcl@swu.edu.cn
Abstract. This paper deals with a class of Kirchhoff type elliptic Dirichlet boundary value problems where the
combined effects of Kirchhoff term and nonlinear term allow us to establish a bifurcation-type result as the
positive parameter varies.
2020 Mathematics Subject Classification. 35J60, 35J20.
Funding. Supported by National Natural Science Foundation of China (11861052,11971393), Science and
Technology Foundation of Guizhou Province ([2019]1300, [2019]5653), Innovation group project of Education
Department of Guizhou ([2019]067) and Science and technology research project of Chongqing Education
Commission (KJQN201800636).
Manuscript received 31st July 2021, revised 18th September 2021, accepted 27th October 2021.
1. Introduction and main result
Let Ω be a smooth bounded domain in RN , N ≥ 1, and 2∗ =
{
2N
N−2 , N ≥ 3,
+∞, N = 1,2. We consider the
following Kirchhoff type boundary value problems−
(
a +b
∫
Ω
|∇u|2dx
)
∆u = g (x,u) inΩ,
u = 0 on ∂Ω,
(1)
where a > 0,b > 0 and g :Ω×R→R is a suitable continuous function.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
248 Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang
When N = 1,Ω= (0,L), a = 1 and g (x,u) = g (x), solutions of Eq. (1) are related to the stationary
states of the hyperbolic equation
∂2u
∂t 2
−
(
1+b
∫ L
0
∣∣∣∣∂u∂x
∣∣∣∣2 dx) ∂2u∂x2 = g , u(0, t ) = u(L, t ) = 0,
which was proposed by Kirchhoff in 1883 to describe the small transversal oscillations of an
elastic clamped string. For this reason, Eq. (1) is often called Kirchhoff equation.
Over the past 20 years Eq. (1) has been extensively studied by using variational methods, see
for example, [1, 2, 6–10, 12] and the references therein. See also [11] and references therein for a
broad survey.
In particular, Ambrosetti and Arcoya investigated the following special case of Eq. (1)−
(
a +b
∫
Ω
|∇u|2dx
)
∆u = |u|p−2u in Ω,
u = 0 on ∂Ω,
(2)
where 2 0 such that for any b ∈ (0,γ0), Ib has a mountain pass critical point u1
with Ib(u1) > 0, as well as a global minimum u2 with Ib(u2) < 0, see [2, Theorem 4.4],
where Ib is the Euler functional associated with Eq. (2);
(II) if b is large, then Eq. (2) has no nontrivial solution, see [2, Remark 4.5(ii)].
Conclusions (I) and (II) motivate us to study bifurcation-type results on Eq. (2) as the param-
eter b varies. This is the object of this paper to which we give a positive answer.
Define
Sp = inf
u∈H\{0}
‖u‖2
|u|2p
.
where ‖u‖2 = ∫Ω |∇u|2dx and |u|pp = ∫Ω |u|p dx denote the usual norms of u in H := H 10 (Ω) and
Lp (Ω) respectively. Then ∫
Ω
|u|p dx ≤ Ap‖u‖p for all u ∈ H , (3)
where Ap = S−
p
2
p . Define
Λ= p −2
2
A
2
p−2
p
(
4−p
2a
) 4−p
p−2
.
By using the variational methods, we obtain the following bifurcation-type theorem.
Theorem 1. Suppose that a > 0,b > 0,2 Λ;
(ii) Eq. (2) has at least two positive solutions for any b <Λ;
(iii) Eq. (2) has at least a positive solution for b =Λ.
According to Remark 7 below, we can give the rough graphs of the mountain pass value cb and
the local minimum value mb .
From [2] we know that for any b ∈ (0,γ0), Ib possesses a global minimum. But if b ∈ (0,Λ) is
close toΛ, then 0 is a global minimum. So we need to select an appropriate constraint in order to
obtain a local minimum.
The Euler functional associated with Eq. (2) is
Ib(u) =
a
2
‖u‖2 + b
4
‖u‖4 − 1
p
∫
Ω
|u|p dx.
Obviously, Ib is of class C
1 and
〈I ′b(u), v〉 =
(
a +b‖u‖2)∫
Ω
∇u ·∇vdx −
∫
Ω
|u|p−2uvdx
C. R. Mathématique — 2022, 360, 247-254
Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang 249
Figure 1.
for all u, v ∈ H .
2. Proof of Theorem 1
To prove Theorem 1, five lemmas are in order.
Lemma 2. Suppose that b > 0 and 2 < p < min{4,2∗}. Then Ib is coercive on H.
Proof. From (3) we have
Ib(u) =
a
2
‖u‖2 + b
4
‖u‖4 − 1
p
∫
Ω
|u|p dx
≥ a
2
‖u‖2 + b
4
‖u‖4 − Ap
p
‖u‖p .
Since 2 0,0 < b <Λ,2 < p < min{4,2∗} and
Tb =
[
(p −2)Ap
2b
] 1
4−p
.
Then there exist tb ∈ (0,Tb) and t ′b ∈ (Tb ,+∞) such that all nontrivial solution of Eq. (2) belongs to
the set {u ∈ H : tb ≤ ‖u‖ ≤ t ′b}. Moreover, if 0 < b1 < b2 < Λ, then tb1 < tb2 and Mb2 ⊂ Mb1 , where
Mb := {u ∈ H : ‖u‖ ≥ tb}.
Proof. Suppose that u ∈ H is a nontrivial solution of Eq. (2). Then combining with (3) we have
a‖u‖2 +b‖u‖4 =
∫
Ω
|u|p dx ≤ Ap‖u‖p .
Thus
b‖u‖2 − Ap‖u‖p−2 +a ≤ 0.
For t > 0, define hb(t ) = bt 2 − Ap t p−2 +a. Then h′b(Tb) = 2bTb − (p −2)Ap T
p−3
b = 0, h′b(t ) < 0 for
all t ∈ (0,Tb) and h′b(t ) > 0 for all t ∈ (Tb ,+∞).
When b < Λ, hb(Tb) = T p−2b (bT
4−p
b − Ap ) + a < 0. Combining with hb(0) = a > 0 and
limt→+∞ hb(t ) = +∞ implies that there exist tb ∈ (0,Tb) and t ′b ∈ (Tb ,+∞) such that hb(tb) =
hb(t
′
b) = 0, hb(t ) > 0 for all t ∈ (0, tb)∪ (t ′b ,+∞) and hb(t ) < 0 for all t ∈ (tb , t ′b). So tb ≤ ‖u‖ ≤ t ′b .
If 0 < b1 < b2 <Λ, then hb1 (tb2 ) < hb2 (tb2 ) = 0. So there exists tb1 ∈ (0, tb2 ) such that hb1 (tb1 ) = 0
and hb1 (t ) > 0 for all t ∈ (0, tb1 ). Hence Mb2 ⊂ Mb1 . 
From Lemma 2, mb := infu∈Mb Ib(u) is well defined. We consider the constraint problem
mb = infu∈Mb Ib(u) in order to obtain a local minimum solution of Eq. (2).
C. R. Mathématique — 2022, 360, 247-254
250 Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang
Lemma 4. Suppose that a > 0,0 0 and vb ∈ H such that ‖vb‖ > tb and inf‖u‖=tb Ib(u) ≥ αb > Ib(vb).
Moreover, Ib(vb) ≥ mb .
Proof. From (3) we have
inf
‖u‖=tb
Ib(u) = inf‖u‖=tb
(
a
2
‖u‖2 + b
4
‖u‖4 − 1
p
∫
Ω
|u|p dx
)
≥ inf
‖u‖=tb
(
a
2
‖u‖2 + b
4
‖u‖4 − Ap
p
‖u‖p
)
= at
2
b
2
+ bt
4
b
4
−
Ap t
p
b
p
.
For t > 0, define fb(t ) = at
2
2 + bt
4
4 −
Ap t p
p . Then f
′
b(t ) = at +bt 3 − Ap t p−1 = thb(t ) > 0 for all t ∈
(0, tb)∪ (t ′b ,+∞) and f ′b(t ) < 0 for all t ∈ (tb , t ′b). Thus fb(tb) > fb(0) = 0 and fb(tb) > fb(Tb) > fb(t )
for all t ∈ (Tb , t ′b].
The infimum Sp can be achieved by a positive function e ∈ H and
∫
Ω |e|p dx = Ap‖e‖p . Let
vb = Tb‖e‖e. Then ‖vb‖ = Tb > tb and
Ib(vb) =
aT 2b
2
+ bT
4
b
4
−
Ap T
p
b
p
= fb(Tb).
Hence inf‖u‖=tb Ib(u) ≥αb := fb(tb) > fb(Tb) = Ib(vb) ≥ mb . 
Lemma 5. Suppose that a > 0,0 < b <Λ and 2 < p < min{4,2∗}. Then there exists {un} ⊂ Mb such
that Ib(un) → mb and I ′b(un) → 0.
Proof. By Ekeland’s variational principle [5], there exists un ∈ Mb such that
mb ≤ Ib(un) ≤ mb +
1
n
and for any v ∈ Mb ,
Ib(v)− Ib(un) ≥−
1
n
‖un − v‖.
Thus Ib(un) → mb . From Lemma 4 we know that for n large enough, ‖un‖ > tb . For any h ∈ H and
‖h‖ = 1, let v = un + th for t small enough, we have
〈I ′b(un),h〉 = limt→0+
Ib(un + th)− Ib(un)
t
≥− 1
n
.
Similarly,
〈I ′b(un),−h〉 = limt→0+
Ib(un − th)− Ib(un)
t
≥− 1
n
.
Then
|〈I ′b(un),h〉| ≤
1
n
= o(1).
According to the arbitrariness of h we have I ′b(un) → 0. 
Lemma 6. Suppose that a > 0,b > 0 and 2 < p < min{4,2∗}. If {un} satisfies supn Ib(un) <+∞ and
I ′b(un) → 0, then {un} contains a convergent subsequence.
C. R. Mathématique — 2022, 360, 247-254
Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang 251
Proof. Since {un} satisfies supn Ib(un) <+∞, from Lemma 2 we have {un} is bounded in H . Then
there exists u ∈ H such that up to a subsequence, un * u in H , un → u in Lp (Ω) and un(x) → u(x)
a.e. in Ω. Combining with the Hölder inequality implies∣∣∣∣∫
Ω
|un |p−2un(un −u)dx
∣∣∣∣≤ ∫
Ω
|un |p−1|un −u|dx
≤ |un |p−1p |un −u|p
= o(1).
Note that ∫
Ω
∇un ·∇udx =
∫
Ω
|∇u|2dx +o(1)
and
o(1) = 〈I ′b(un),un −u〉 = (a +b‖un‖2)
∫
Ω
∇un ·∇(un −u)dx −
∫
Ω
|un |p−2un(un −u)dx.
Thus ‖un‖→‖u‖. Therefore, un → u in H . 
Proof of Theorem 1.
Conclusion (i). Eq. (2) has no nontrivial solutions for any b >Λ.
Suppose that u ∈ H is a nontrivial solution of Eq. (2). Then using (3) and the Young inequality
we have
a‖u‖2 +b‖u‖4 =
∫
Ω
|u|p dx
≤ Ap‖u‖p
=
(
2a
4−p
) 4−p
2 ‖u‖4−p Ap
(
4−p
2a
) 4−p
2 ‖u‖2p−4
≤ 4−p
2
2a
4−p ‖u‖
2 + p −2
2
A
2
p−2
p
(
4−p
2a
) 4−p
p−2 ‖u‖4
= a‖u‖2 + p −2
2
A
2
p−2
p
(
4−p
2a
) 4−p
p−2 ‖u‖4.
(4)
So b ≤Λ. Conclusion (i) holds.
Conclusion (ii). Eq. (2) has at least two positive solutions for any b <Λ.
From Lemma 5 and Lemma 6 we have that there exists u ∈ Mb such that Ib(u) = mb and
I ′b(u) = 0. Note that Ib(|u|) = Ib(u) = mb and ‖|u|‖ = ‖u‖ > tb . For any h ∈ H ,
〈I ′b(|u|),h〉 = limt→0+
Ib(|u|+ th)− Ib(|u|)
t
≥ 0
and
〈I ′b(|u|),−h〉 = limt→0+
Ib(|u|− th)− Ib(|u|)
t
≥ 0.
Then I ′b(|u|) = 0. That |u| > 0 follows from the strong maximum principle. In fact, |u| is a ground
state solution. Indeed, from Lemma 3 one has v ∈ Mb for any v ∈ {v ∈ H\{0} : I ′b(v) = 0} and then
Ib(v) ≥ mb = Ib(|u|).
According to Lemma 4, we define
cb = inf
γ∈Γb
max
t∈[0,1]
Ib(γ(t )),
where
Γb = {γ ∈C ([0,1], H) : γ(0) = 0,γ(1) = vb}.
Then cb ≥ αb > mb and combining with Lemma 6, we know that cb is a critical value by the
mountain pass lemma [3]. From Theorem 10 in [4] we want to state positivity property of the
C. R. Mathématique — 2022, 360, 247-254
252 Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang
mountain pass solution. We take p(u) = |u| in [4, Theorem 10] and obtain a critical point u ≥ 0.
The strong maximum principle implies u > 0. Conclusion (ii) holds.
Conclusion (iii). Eq. (2) has at least a positive solution for b =Λ.
Because Sp can be achieved by some positive normalized function v ∈ H , i.e., Sp = ‖v‖2 and
|v |p = 1, S
1
p−2
p v is a positive solution of the following semi-linear elliptic equation{
−∆u = |u|p−2u in Ω,
u = 0 on ∂Ω. (5)
By scaling the function w := ( 4−p2a ) 12−p S 1p−2p v solves the following equation−
2a
4−p∆u = |u|
p−2u in Ω,
u = 0 on ∂Ω.
Moreover, by calculating one has a+b‖w‖2 = 2a4−p . It shows that w is a positive solution of Eq. (2).
Conversely, if w is a positive solution of Eq. (2), then (a +b‖w‖2) 12−p w is the one of Eq. (5). Then
when the positive solution of Eq. (5) is unique, the one of Eq. (2) is also unique, for example, when
Ω is ball or p is close to 2. Conclusion (iii) holds. 
Remark 7. We prove some properties of critical values mb and cb .
(i) mb and cb are monotone increasing functions on (0,Λ);
(ii) mΛ := limb→Λ− mb and cΛ := limb→Λ− cb are critical values of Eq. (2) with b =Λ;
(iii) mΛ = cΛ = p−24p ( 24−p )
2
p−2 (aSp )
p
p−2 = IΛ(u) for all u ∈ {u ∈ H\{0} : I ′Λ(u) = 0};
(iv) limb→0+ mb =−∞ and c0 := limb→0+ cb > 0.
Proof of (i). For any b1,b2 ∈ (0,Λ) and b1 < b2, from Lemma 3 we have Mb2 ⊂ Mb1 . Then
mb1 = infu∈Mb1
Ib1 (u) ≤ infu∈Mb2
Ib1 (u) ≤ infu∈Mb2
Ib2 (u) = mb2 .
Fix b2 ∈ (0,Λ). For any γ ∈ Γb2 , max0≤t≤1 Ib2 (γ(t )) ≥ fb2 (tb2 ) > fb2 (Tb2 ) > fb2 (t ) for all t ∈
(Tb2 , t
′
b2
], where the symbols are from Lemma 3 and 4. According to the definition of Tb , there
exists b1 ∈ (0,b2) such that Tb2 < Tb1 < t ′b2 . For any b ∈ (b1,b2), we define
γ̃(t ) =

γ(2t ), 0 ≤ t ≤ 1
2
,
[Tb2 + (2t −1)(Tb −Tb2 )]e
‖e‖ ,
1
2
≤ t ≤ 1.
Then γ̃ ∈ Γb , Tb2 ≤ Tb2 + (2t −1)(Tb −Tb2 ) ≤ Tb < Tb1 < t ′b2 for all t ∈
[ 1
2 ,1
]
. Thereby, one has
max
1
2 ≤t≤1
Ib2 (γ̃(t )) = max1
2 ≤t≤1
fb2 (Tb2 + (2t −1)(Tb −Tb2 )) ≤ fb2 (Tb2 ) < max0≤t≤1 Ib2 (γ(t ))
and then
max
0≤t≤1 Ib2 (γ(t )) = max0≤t≤ 12
Ib2 (γ̃(t )) = max0≤t≤1 Ib2 (γ̃(t )) > max0≤t≤1 Ib(γ̃(t )) ≥ cb .
In view of the arbitrariness of γ, we have cb2 ≥ cb .
Proof of (ii). For any b ∈ (Λ2 ,Λ), there exists a positive function ub ∈ H such that Ib(ub) = mb and
I ′b(ub) = 0. Combining with (3) implies
Λ
2
‖ub‖4 ≤ a‖ub‖2 +b‖ub‖4 =
∫
Ω
|ub |p dx ≤ Ap‖ub‖p .
C. R. Mathématique — 2022, 360, 247-254
Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang 253
Thus ‖ub‖ ≤
( 2Ap
Λ
) 1
4−p . From
mb = Ib(ub) ≤
a
2
‖ub‖2 +
Λ
4
‖ub‖4
it follows that mb has a upper bound on
(
Λ
2 ,Λ
)
. We set mΛ = supΛ
2 <b<Λmb . For any ε > 0, there
exists bε ∈
(
Λ
2 ,Λ
)
such that mΛ− ε < mbε ≤ mΛ. Take δ = Λ−bε. When Λ−δ < b < Λ, by using
the monotonicity we have mΛ − ε < mbε ≤ mb ≤ mΛ < mΛ + ε, i.e., mb is left continuous at Λ
(mΛ = limb→Λ− mb). Let {bn} ⊂
(
Λ
2 ,Λ
)
be an increasing sequence and bn → Λ. Then mbn → mΛ
and there exists a positive sequence {un} ⊂ H such that Ibn (un) = mbn and I ′bn (un) = 0. So
‖un‖ ≤
( 2Ap
Λ
) 1
4−p and then
IΛ(un) = Ibn (un)+
Λ−bn
4
‖un‖4 = mbn +
Λ−bn
4
‖un‖4 → mΛ,
‖I ′Λ(un)‖∗ = sup‖v‖=1
〈I ′Λ(un), v〉 = sup‖v‖=1
(Λ−bn)‖un‖2
∫
Ω
∇un ·∇vdx → 0.
From Lemma 6, we know that there exists a nonnegative u ∈ H such that up to a subsequence,
un → u in H . Hence IΛ(u) = mΛ and I ′Λ(u) = 0. The strong maximum principle implies that u is
positive.
The case of cb is proved to be completely similar.
Proof of (iii). Let I ′Λ(u) = 0, then from (4) we have
a‖u‖2 +Λ‖u‖4 =
∫
Ω
|u|p dx = Ap‖u‖p .
Let h(t ) =Λt 2 − Ap t p−2 +a for t > 0. Then h′(T ) = 0, h′(t ) < 0 for all t ∈ (0,T ) and h′(t ) > 0 for all
t ∈ (T,+∞), where
T =
[
(p −2)Ap
2Λ
] 1
4−p
.
Since h(T ) = T p−2(ΛT 4−p −Ap )+a = 0, T is a unique positive solution of equationΛt 2−Ap t p−2+
a = 0. Thereby, for any u ∈ {u ∈ H\{0} : I ′Λ(u) = 0} we have ‖u‖ = T and
IΛ(u) = IΛ(u)− 1
p
〈I ′Λ(u),u〉
= (p −2)a
2p
T 2 − (4−p)Λ
4p
T 4
= p −2
4p
(
2
4−p
) 2
p−2
(aSp )
p
p−2 .
Then
mΛ = cΛ = p −2
4p
(
2
4−p
) 2
p−2
(aSp )
p
p−2 .
Proof of (iv). From Lemma 3 we know that tb is a monotone increasing function on (0,Λ). Fix
w ∈ H , ‖w‖ = 1. For any L > 0 there exists B > t Λ
2
such that
I0(B w) = aB
2
2
‖w‖2 − B
p
p
∫
Ω
|w |p dx <−L−1.
Then there exists b0 ∈
(
0, Λ2
)
such that
Ib0 (B w) = I0(B w)+
b0B 4
4
‖w‖4 <−L.
C. R. Mathématique — 2022, 360, 247-254
254 Jiu Liu, Jia-Feng Liao, Hui-Lan Pan and Chun-Lei Tang
So for any b ∈ (0,b0), we have ‖B w‖ = B > t Λ
2
> tb0 > tb and then
mb = inf
u∈Mb
Ib(u) ≤ Ib(B w) ≤ Ib0 (B w) <−L,
i.e., limb→0+ mb =−∞.
Since cb is a monotone increasing function on (0,Λ) and cb > 0 for all b ∈ (0,Λ), we obtain
that limb→0+ cb exists. In the following, we prove c0 := limb→0+ cb > 0. We adopt the symbols of
Lemma 3 and Lemma 4. Define t0 =
( a
Ap
) 1
p−2 . Then t0 < Tb and hb(t0) = bt 20 > 0. Since tb < Tb ,
hb(tb) = 0 and h′b(t ) < 0 for all t ∈ (0,Tb), we have t0 < tb . Combining with f ′b(t ) > 0 for all t ∈ (0, tb)
implies fb(tb) > fb(t0). Then we get
inf
‖u‖=tb
Ib(u) ≥αb = fb(tb) > fb(t0) ≥ t 20
(
a
2
− Ap
p
t p−20
)
= p −2
2p
a
p
p−2
A
2
p−2
p
> 0.
Thereby cb ≥αb ≥ p−22p a
p
p−2
A
2
p−2
p
. Hence c0 ≥ p−22p a
p
p−2
A
2
p−2
p
> 0.
Acknowledgments
The authors express their gratitude to the reviewer who made many helpful suggestions and
pointed out kindly to us a simple proof of Theorem 1(iii) which led to an improvement of the
original manuscript.
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C. R. Mathématique — 2022, 360, 247-254

A bifurcation-type result for Kirchhoff equations

Theo McKenzie

The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph

Fabian Reede

Comptes Rendus
Mathématique
Fabian Reede
Smooth components on special iterated Hilbert schemes
Volume 360 (2022), p. 425-429
<https://doi.org/10.5802/crmath.307>
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2022, Vol. 360, p. 425-429
https://doi.org/10.5802/crmath.307
Algebraic geometry / Géométrie algébrique
Smooth components on special iterated
Hilbert schemes
Fabian Reede∗, a
a Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1,
30167 Hannover, Germany
E-mail: reede@math.uni-hannover.de
Abstract. Let S be a smooth projective surface with pg = q = 0. We show how to use derived categorical
methods to study the geometry of certain special iterated Hilbert schemes associated to S by showing that
they contain a smooth connected component isomorphic to S.
Mathematical subject classification (2010). 14F08, 14J28, 14J29.
Manuscript received 10 November 2021, revised and accepted 29 November 2021.
1. Introduction
Hilbert schemes are ubiquitous in modern algebraic geometry. But even in good situations these
schemes can behave badly. This became clear with Mumford’s famous example, which shows
that there is an irreducible component of the Hilbert scheme of smooth irreducible curves in
P3 of degree 14 and genus 24 that is generically non-reduced, see [17]. More exactly Mumford
constructs a 56-dimensional irreducible family of such curves, such that the tangent space at
each point of this component has dimension 57 and proves that this family is not contained in
any other irreducible family of dimension > 56.
In this note we prove that there are certain iterated Hilbert schemes which contain at least one
smooth connected component. More exactly the main result of this note is:
Theorem 1. Assume S is a smooth projective surface with pg = q = 0 and let S[n] be the Hilbert
scheme of length n subschemes of S. Then the universal family Z in S ×S[n] can be understood as
a family of codimension two subschemes in S[n] with common Hilbert polynomial p(t ) classified
by S such that the classifying morphism identifies S with a smooth connected component of the
Hilbert scheme Hilbp(t )(S[n]).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
426 Fabian Reede
This theorem has its origin in a result by Lange and Newstead, who proved a similar result
for curves and moduli spaces of stable vector bundles on these curves in [15]. More exactly they
show that if M is a fine moduli space of stable vector bundles on a smooth projective curve C of
genus g Ê 2 with universal family U on C ×M , then for any c ∈ C the vector bundle Uc on M is
stable. Furthermore they show that for c 6= c ′ we have Uc 6∼=Uc ′ . Together with a previous result by
Narasimhan and Ramanan these results imply that C embeds as a smooth connected component
in a moduli spaces of stable vector bundles o M .
The main input into the proof of the main result of this note is a result by Krug and Sosna
which states that the integral functor Φ : Db(S) → Db(S[n]) with kernel the universal ideal sheaf
IZ is fully faithful. This result allows to reduce the computation of certain Ext-groups on S[n] to
the computation of easier Ext-groups on S.
All objects in this note are defined over the field of complex numbers C.
Acknowledgement
I thank Pieter Belmans for informing me about the fully faithfulness results in [3, 12] as well as
Ziyu Zhang for many useful conversations.
2. Proof of the Main Theorem
Let S be a smooth projective surface with pg = q = 0, that is we have
H1 (S,OS ) = H2 (S,OS ) = 0.
Remark 2. Around the year 1870 Max Noether posed the question if surfaces with pg = q = 0
are necessarily rational, see [2, § 3, Question 1]. By now it is known that the answer to this
question is negative. In fact besides rational surfaces there is a huge class of surfaces satisfying
these conditions which are not rational, most classically Enriques surfaces which have Kodaira
dimension zero. But there are also surfaces of general type satisfying these conditions for example
Godeaux surfaces, Campedelli surfaces or Beauville surfaces, see [1, 7] for more information and
examples.
In the following we denote the Hilbert scheme of length n subschemes of S by S[n], that is we
have as sets:
S[n] = {[Z ] ∣∣ Z ⊂ S is a zero-dimensional subscheme with h0 (Z ,OZ ) = n} .
It is well known that S[n] is smooth and that dim(S[n]) = 2n. Using this notation we have the
universal subscheme
Z = {(s, [Z ]) ∈ S ×S[n] ∣∣ s ∈ supp(Z )}⊂ S ×S[n] (1)
coming with the corresponding universal ideal sheaf IZ ,→OS×S[n] .
Remark 3. Recall that the universal family Z is flat over S[n]. Indeed, using definition (1) one can
see that the restriction of p : S ×S[n] → S[n] to Z is finite and flat of degree n. But as a matter of
fact Z is also flat over S due to [13, Theorem 2.1].
As we use integral functors in the following, we quickly recall their definition: let X and Y be
smooth projective varieties and denote their bounded derived categories of coherent sheaves by
Db(X ) and Db(Y ) respectively, then the integral functor with kernel K ∈ Db(X ×Y ) is defined by
ΦK : D
b(X ) → Db(Y ), E 7→ Rp∗
(
q∗E ⊗L K )
where p and q are the projections X ×Y → Y resp. X ×Y → X , see [10, § 5].
The description of the image of an integral transform is rather easy for a skyscraper sheaf Ox
of a closed point x ∈ X , which we will collect as
Fabian Reede 427
Example 4 ([10, Examples 5.4 (vi)]). Assume the kernel of the integral functor ΦK is in fact a
coherent sheaf on X ×Y flat over X , then we have for every closed point x ∈ X
ΦK (Ox ) ∼=Kx ,
where the fiber Kx :=K|{x}×Y is considered as a sheaf on Y via the second projection {x}×Y → Y .
Interpreting the universal ideal sheaf IZ as an element in Db(S ×S[n]) (as a complex concen-
trated in degree zero), we can look at the integral functor with kernel given by IZ :
Φ : Db(S) → Db (S[n]) , E 7→ Rp∗ (q∗E ⊗IZ ) .
In our case this integral functor has some very good properties. The main input into this note is
the following very useful fact discovered by Krug and Sosna:
Theorem 5 ([12, Theorem 1.2]). Let S be a smooth projective surface which satisfies pg = q = 0,
then the integral functor Φ is fully faithful.
As an application of Theorem 5 we have the following
Corollary 6. Assume S is a smooth projective surface with pg = q = 0 then for all E ,F ∈ Db(S) and
i Ê 0 there is an isomorphism
Exti
S[n]
(
Φ(E),Φ(F )
)∼= ExtiS (E ,F ).
Remark 7. There are also fully faithfulness results for universal families of moduli spaces
of stable bundles of rank two and degree one on smooth projective curves of genus g Ê 2
by Narasimhan as well as Fonarev and Kuznetsov, see [9, 18, 19]. Recently these results were
generalized to higher rank and degree by Belmans and Mukhopadhyay as well as Lee and Moon,
see [5, 16]. These results can be used to give a proof of the result of Lange and Newstead in the
spirit of this note.
Proof of Theorem 1. Remark 3 shows that the universal family Z ⊂ S ×S[n] is flat over S. Denote
the fiber over a closed point s ∈ S by Zs and its image in S[n] (via the second projection
{s}×S[n] ∼= S[n], which is an isomorphism) by Fs . This identification together with Example 4 gives
isomorphisms
IFs
∼= (IZ )s ∼=Φ (Os ) . (2)
Remark 8. By the definition of Z given in (1) we have
Fs =
{
[Z ] ∈ S[n] ∣∣ s ∈ supp(Z )}⊂ S[n].
That is, Fs is the subscheme of S[n] classifying all length n subschemes containing the closed
point s ∈ S in its support. Note that dim(Fs ) = 2n −2, that is Fs is a subscheme of codimension
two in S[n].
Since S is integral the Hilbert polynomial of Fs does not depend on s ∈ S by [11, Proposi-
tion 2.1.2], call it p(t ). We thus have a well defined classifying morphism
ϕ : S → Hilbp(t ) (S[n]) , s 7→ [Fs ] .
Remark 9. Here we can choose any ample line bundle L ∈ Pic(S[n]) to define the Hilbert
polynomial p(t ) of Fs , as there is no distinguished ample line bundle on S[n]. The choice of a
different ample line bundle L would give rise to a different Hilbert polynomial p(t ), but it would
not change the proof of the main theorem.
A more conceptual way would be to choose a n-very ample line bundle M on S, then by [6]
there is a closed embedding
S[n] → Gr(n,H0(S, M)∗) , [Z ] 7→ H0(S, M ⊗OZ )∗.
428 Fabian Reede
Composing this morphism with the Plücker embedding of the Grassmannian, we get a closed
embedding S[n] → PN for some N and we can pullback OPN (1) to get an ample line bundle L on
S[n].
We claim that the morphism ϕ identifies S with a smooth connected component of
Hilbp(t )(S[n]). To see this we have to show that the morphism is injective on closed points and
that for every closed point s ∈ S we have
dim
(
T[Fs ] Hilb
p(t ) (S[n]))= 2.
We start by picking two closed points s1 6= s2 ∈ S and note that using equation (2) as well as
Corollary 6, we get:
HomS[n]
(
IFs1 , IFs2
)∼= HomS[n] (Φ(Os1) ,Φ(Os2))∼= HomS (Os1 ,Os2)= 0. (3)
If [Fs1 ] = [Fs2 ] ∈ Hilbp(t )(S[n]), then we would have an isomorphism OFs1 ∼= OFs2 and the exact
sequences (for i = 1,2)
0 IFsi OS[n] OFsi 0
would give rise to a commutative diagram of short exact sequences (with the identity between
OS[n] ) which shows that there is a nontrivial morphism between IFs1 and IFs2 . But this is impossi-
ble by 3. So the classifying morphism ϕ is indeed injective on closed points.
To find dim(T[Fs ] Hilb
p(t )(S[n])) we remark that
T[Fs ] Hilb
p(t ) (S[n])∼= HomS[n] (IFs ,OFs ) ,
see for example [11, Proposition 2.2.7].
As q = 0 we have Pic0(S) = 0, but by [8, Theorem 5.4.] we also have an isomorphism
Pic0(S)
∼=−→ Pic0 (S[n])
and thus Pic0(S[n]) = 0. So we can use [14, Lemma B.5.6.] which gives an isomorphism
HomS[n]
(
IFs ,OFs
)∼= Ext1S[n] (IFs , IFs ) .
We find, using again equation (2) and Corollary 6:
Ext1
S[n]
(
IFs , IFs
)∼= Ext1S[n] (Φ (Os ) ,Φ (Os ))∼= Ext1S (Os ,Os)∼= Ts S.
Putting all results together shows dim(T[Fs ] Hilb
p(t )(S[n])) = 2 as desired. 
Remark 10. Theorem 5 can be generalized in the case n = 2 to all smooth projective varieties
X having the property Hi (X ,OX ) = 0 for i Ê 1 (that is OX is exceptional). By [3, Theorem A] the
integral functorΦ : Db(X ) → Db(X [2]) with kernel the universal ideal sheaf IZ is also fully faithful
in these cases. Thus our proof of the main result is also valid in these cases.
Remark 11. In the case of surfaces, the proof of the main result only works for those surfaces
with pg = q = 0, since for n Ê 2 the integral functor Φ : Db(S) → Db(S[n]) is fully faithful if and
only if pg = q = 0 by [4, Theorem A]. But there are similar results for K3 surfaces as well as abelian
surfaces, see [20]. In these cases the integral functor Φ is a so-called Pn-functor, which again
allows to reduce cohomological computations on S[n] to computations on S.
Fabian Reede 429
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