Comptes Rendus Mathématique

Nir Lazarovich

Arielle Leitner

Local Limits of Connected Subgroups of SL${}_3(\protect \mathbb{R})$

Mélodie Andrieu

A Rauzy fractal unbounded in all directions of the plane

K. Senthil Kumar

Comptes Rendus
Mathématique
K. Senthil Kumar
Linear dependence of quasi-periods over the rationals
Volume 359, issue 4 (2021), p. 409-414
<https://doi.org/10.5802/crmath.171>
© Académie des sciences, Paris and the authors, 2021.
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
2021, 359, n 4, p. 409-414
https://doi.org/10.5802/crmath.171
Linear dependence of quasi-periods over the
rationals
K. Senthil Kumara
a National Institute of Science Education and Research, HBNI, P.O.Jatni, Khurda,
Odisha-752 050, India
E-mail: senthil@niser.ac.in
Abstract. In this note we shall show that a lattice Zω1 +Zω2 in C has Q-linearly dependent quasi-periods if
and only if ω2/ω1 is equivalent to a zero of the Eisenstein series E2 under the action of SL2(Z) on the upper
half plane of C.
2020 Mathematics Subject Classification. 11J72, 11J89.
Manuscript received 23rd March 2020, revised 7th August 2020, accepted 17th December 2020.
1. Introduction
Let L = Zω1 +Zω2 be a lattice in C with ω2/ω1 ∈ H, the upper half plane of C. Let σ(z;ω1,ω2)
and ζ(z;ω1,ω2) respectively be the Weierstrass sigma and zeta functions associated to L. Let
g2 and g3 be the invariants of L. The numbers η1(L) = η(ω1) = 2ζ(ω1/2;ω1,ω2), η2(L) = η(ω2)
= 2ζ(ω2/2;ω1,ω2) are called the quasi-periods associated to L. When L is clear from the context,
we simply write η1,η2 instead of η1(L) and η2(L) respectively. One of the long standing open
problem in transcendental number theory is to find the dimension of the vector space VL
generated by
1,ω1,ω2, η1, η2, π (1)
over Q, the algebraic closure of Q. Starting from the work of Siegel [10], Schneider [9], Baker [1],
Coates [3,4] and finally by Masser [8], it is now known that for a lattice L with algebraic invariants
g2, g3, the vector space VL has dimension 4 in the CM case and 6 in the non-CM case. This is
because in the CM case, there are two linear relations among the numbers in (1). The first one is
τω1 −ω2 = 0
where τ=ω2/ω1 ∈Q and the other one is given by
Cη1 −τη2 −κω2 = 0, (2)
where C is the constant term of the minimal polynomial of τ over Q and κ ∈ Q(τ, g2, g3) (see [8,
Lemma 3.1] or [2, Theorem 8] for more details). Masser also proved that the number κ in (2)
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
410 K. Senthil Kumar
vanishes if and only if τ is congruent to i =p−1 or ρ = e2πi /3 under SL2(Z); and in that case, η1
and η2 are linearly dependent overQ(τ).
Apart from lattices with algebraic invariants, there are two more cases for which we know the
dimension of VL. For example, if ω1 = 1 and ω2 = i then by Siegel [10] at least one of the g2, g3 is
not algebraic. And by (2), the quotient η2/η1 = −i in this case. (Note that we used (2) to find the
ratio η2/η1; because, as we shall see later that, η2/η1 depends only on ω2/ω1 and not on g2, g3;
this ratio can also be obtained from (4) and (9) below by choosing an appropriate γ). Hence by
the Legendre’s relation [7, p. 241] the vector space VL has dimension two. Similarly, if ω1 = 1
and ω2 = ρ then in this case also at least one of the g2, g3 is not algebraic and by (2) we have
η2/η1 = ρ−1. Hence in this case also the vector space VL has dimension two. Except for these
cases the author is not aware of any other lattices L for which the dimension of VL is known.
In [6] Heins shown that a pair of complex numbers (z1, z2) occur as quasi-periods of some lattice
L if and only if |z1|+|z2| > 0. Thus there are lattices withQ-linearly dependent quasi-periods, and
therefore, for such latticesL the vector space VL has dimension at most five. Unfortunately, Heins
method does not allow us to determine the lattices withQ-linearly dependent quasi-periods. The
purpose of this note is to classify all such lattices. For τ ∈ H, the generalised Eisenstein series of
weight 2 is defined by
G2(τ) =
∑
c
∑
d
(cτ+d)−2 (3)
where the sum is over all integers c and d with |c|+ |d | > 0; while the normalised Eisenstein series
of weight 2 is defined by
E2(τ) = 3G2(τ)/π2 = 1−24
∞∑
n=1
σ1(n)q
n (4)
whereσ1(n) is the sum of all positive divisors of n, and q = e2πiτ. Our main result is the following.
Main Theorem. Let L=Zω1+Zω2 be a lattice inCwith τ= ω2ω1 ∈H. Then η1 and η2 areQ-linearly
dependent if and only if τ is congruent to a zero of E2(z) under SL2(Z).
The following corollary is immediate.
Corollary 1. Let L = Zω1 +Zω2 be a lattice in C with τ = ω2ω1 ∈ H is equivalent to a zero of E2(z)
under SL2(Z). Then VL has dimension at most 4 in the CM case and at most five in the non-CM
case.
We shall prove the Main Theorem in the next section. The proof relies on the formula express-
ing the quasi-periods in-terms of G2 (see Lemma 3) and the transformation formula of E2 given by
E2(γ.τ) = (cτ+d)2E2(τ)+ 6c
πi
(cτ+d) (5)
where τ ∈H and γ= (a bc d ) ∈ SL2(Z).
2. Quasi-periods and Laurent’s expansions
Let σ(z;τ) = σ(z;1,τ) and ζ(z;τ) = ζ(z;1,τ) respectively be the Weierstrass sigma and zeta
functions associated to the lattice Lτ = Z+Zτ with τ ∈ H. These two functions are connected
by the relation ζ(z;τ) = σ′(z;τ)σ(z;τ) .
For ω ∈Lτ \ {0}, we write
1
z −ω =−
1
ω
− z
ω2
− z
2
ω3
− z
3
ω4
− ·· ·
for z near the origin. Thus, we have
1
z −ω +
1
ω
+ z
ω2
=− z
2
ω3
− z
3
ω4
− ·· · .
C. R. Mathématique — 2021, 359, n 4, 409-414
K. Senthil Kumar 411
Now summing over all non-zero periods of Lτ and adding the term 1/z, we obtain
ζ(z;τ) = 1
z
−
∞∑
k=2
G2k z
2k−1 (6)
where G2k = G2k (τ) =
∑
ω∈Lτ \{0}ω
−2k for k ≥ 2 (the coefficients of even powers of z in (6) are zero,
since ζ(z;τ) is an odd function).
The next lemma gives a connection between quasi-periods and the values of generalized
Eisenstein series G2.
Lemma 2. Let η1 be the quasi-period associated to the period 1 of the lattice Lτ = Z+Zτ with
τ ∈H. Then η1 =G2(τ).
Proof. We follow the strategy as given in [7, Chapter 18]. Accordingly, we express the Laurent’s
expansion of ζ(z;τ) near the origin into two different ways and then comparing the correspond-
ing coefficients we obtain the required representation for η1. The first one is given by (6). For
obtaining the second representation, let qz = e2πi z . Consider the function
φ1(z) = (2πi )−1
(
qz −1
) ∞∏
n=1
(
1−qz+nτ
)(
1−qnτ−z
)(
1−qnτ
)2 . (7)
Since τ ∈H, we have |qnτ| < 1/2n for large values of n, and hence, for such values∣∣∣∣ qnτ(1−qnτ)2
∣∣∣∣< 1(2n −1)2 .
It follows that the series
∞∑
n=1
((
1−qz+nτ
)(
1−qnτ−z
)(
1−qnτ
)2 −1
)
(8)
converges absolutely and uniformly on compact subsets of C. Thus, the function φ1 is entire.
Moreover, it satisfying the following transformation formulas (see [7, p. 247] for more details):
φ1(z +1) =φ1(z) and φ1(z +τ) =− 1
qz
φ1(z).
On the other hand, the entire function
φ2(z) = e−
1
2 η1z
2
q1/2z σ(z;τ)
also satisfies
φ2(z +1) =φ2(z) and φ2(z +τ) =− 1
qz
φ2(z).
Therefore, the quotient φ1(z)/φ2(z) is elliptic. The product in (7) shows that both φ1 and
φ2 have a simple zero at each point of Z+Zτ and no other zeros. Hence φ1(z)/φ2(z) must be
constant. Taking limit z → 0 we see that the constant is 1, and therefore φ1(z) = φ2(z). We thus
have
σ(z;τ) = (2πi )−1e 12 η1z2 (q1/2z −q−1/2z ) ∞∏
n=1
(
1−qz+nτ
)(
1−qnτ−z
)(
1−qnτ
)2 .
Since the series in (8) converges absolutely and uniformly on compact subsets of C, taking
logarithmic derivative term by term on the right side of the above equation we obtain
ζ(z;τ) = η1z +πi
(
qz +1
qz −1
)
+2πi
∞∑
n=1
(
qnτ−z
1−qnτ−z
− qz+nτ
1−qz+nτ
)
.
C. R. Mathématique — 2021, 359, n 4, 409-414
412 K. Senthil Kumar
If we restrict the values of z such that |qτ| < |qz | < |q−1τ |, then we have
∞∑
n=1
(
qnτ−z
1−qnτ−z
− qnτ+z
1−qnτ+z
)
=
∞∑
n=1
∞∑
m=1
(
qmnτ−z −qmnτ+z
)
=
∞∑
m=1
(
q−mz −qmz
)( ∞∑
n=1
qmnτ
)
=
∞∑
m=1
(
qmτ
1−qmτ
)(
q−mz −qmz
)
.
Near the origin, we have
i
(
qz +1
qz −1
)
= cotπz =
∞∑
k=0
(−1)k 22k B2k (πz)2k−1
(2k)!
,
and
q−mz −qmz =−2i
∞∑
k=0
(−1)k (2πmz)
2k+1
(2k +1)!
where Br is the r th Bernoulli’s number. Thus we have,
ζ(z;τ) = η1z +π
∞∑
k=0
(−1)k 22k B2k (πz)2k−1
(2k)!
−4π
∞∑
m=1
∞∑
k=0
(−1)k
(
qmτ
1−qmτ
)
(2πmz)2k+1
(2k +1)! .
Now comparing the coefficients of z on the above equation with that of (6) we get
η1 = π
222B2
2
−8π2
∞∑
m=1
mqmτ
1−qmτ
= π
2
3
(
1−24
∞∑
m=1
mqmτ
1−qmτ
)
= π
2
3
(
1−24
∞∑
m=1
∞∑
`=1
mq`mτ
)
= π
2
3
(
1−24
∞∑
n=1
σ1(n)q
n
)
= G2(τ),
by (4). This completes the proof of the Lemma 2. 
There is a slight change in the notations used in the above lemma from that of [7, Chapter 18].
In [7], lattices inC are written in the formZω1+Zω2 with the assumptionω1/ω2 ∈H. This implies
that the quasi-period associated to the period 1 of the lattice ω−12 (Zω1 +Zω2) is denoted by η2
in [7, Chapter 18]. Whereas, in our notation lattices in C are written in the form Zω1 +Zω2 with
the assumption ω2/ω1 ∈ H. This implies that the quasi-period associated to the period 1 of the
lattice ω−11 (Zω1 +Zω2) is denoted by η1.
The following lemma is the homogeneous version of Lemma 2.
Lemma 3. Let L=Zω1 +Zω2 be a lattice in Cwith τ= ω2ω1 ∈H. We have
η1 = G2(τ)
ω1
and η2 = τG2(τ)−2πi
ω1
. (9)
Proof. By the Legendre’s relation
ω2η1(L)−ω1η2(L) = 2πi ,
hence it is sufficient to show that η1(L) = G2(τ)ω1 . Since η1(L) is homogeneous of degree −1, it
is enough to prove this lemma when L = Z+Zτ with τ ∈ H. We are thus reduced to show that
for L = Z+Zτ with τ ∈ H, we have η1(L) = G2(τ); but, this is a consequence of Lemma 2. This
completes the proof. 
C. R. Mathématique — 2021, 359, n 4, 409-414
K. Senthil Kumar 413
3. Proof of the Main Theorem
Let L = Zω1 +Zω2 be a lattice in C with τ = ω2ω1 ∈ H. By Lemma 3, the quotient η2(L)/η1(L)
is a function of τ and we denote it by F (τ) (this function was first introduced and studied by
Heins [5]). Hence by (4) and (9) we have
F (τ) = τE2(τ)+6/πi
E2(τ)
. (10)
It follows from this identity that η1(L) and η2(L) areQ-linearly dependent if and only if F (τ) is
a rational number (it is convenient here to assume ∞ is a rational). Hence we are reduced to show
that F (τ) is a rational number if and only if there exists a zero τ′ of E2(z) and a matrix γ ∈ SL2(Z)
such that τ= γ.τ′.
If F (τ) = ∞, then we have E2(τ) = 0. If F (τ) = 0, then we have τE2(τ)+ 6/πi = 0; and hence
E2
(−1
τ
) = τ2E2(τ)+6τ/πi = 0. Suppose that F (τ) is a rational number which is neither 0 nor ∞,
say q/p, with (p, q) = 1. Then, by (10) we have(−pτ+q)E2(τ) = 6p
πi
. (11)
Choose r, s ∈Z such that pr −qs =−1. Then the matrix
γ=
(
s −r
−p q
)
∈ SL2(Z).
We set τ′ = γ.τ. Then by (5),
E2(τ
′) = (−pτ+q)
(
(−pτ+q)E2(τ)− 6p
πi
)
,
which is equal to zero by (11).
Conversely, let τ′ be a zero of E2(z), and let γ = ( a bc d ) be an element of SL2(Z). We shall show
that F (γ.τ′) is a rational number. If c = 0, thenγ= T b where T = ( 1 10 1 ). Thus E2(γ.τ′) = 0, and hence
F (γ.τ′) = ∞. If a = 0, then γ = ( 0 −11 d ), and hence γ.τ′ = −1τ′+d . It follows from (5) that F (γ.τ′) = 0.
Now let γ= ( a bc d ) be an element of SL2(Z) such that ac 6= 0. Then, by (5) we have
0 = E2(τ′) = E2
(
γ−1(γ.τ′)
)= (−c(γ.τ′)+a)2 E2(γ.τ′)− 6c
πi
(−c(γ.τ′)+a) .
Since τ′ is not a rational number we must have(
γ.τ′−a/c)E2(γ.τ′)+ 6
πi
= 0.
Again by (5), we have E2(γ.τ′) 6= 0, from this we conclude that F (γ.τ′) = a/c is a rational number,
and this completes the proof of the Main Theorem.
4. Concluding remarks
It is expected that the zeros of E2 are transcendental; but so far none of them is known to
be transcendental. One may ask whether transcendence of ω2/ω1 is a necessary condition for
Zω1+Zω2 to haveQ-linearly dependent quasi-periods? The answer is no. For example, the quasi-
periods associated to Z+Zi are Q-linearly dependent. It is interesting to classify all lattices with
Q-linearly dependent quasi-periods.
Acknowledgments
I would like to thank the referee for the careful reading and helpful suggestions on the earlier
version of the manuscript.
C. R. Mathématique — 2021, 359, n 4, 409-414
414 K. Senthil Kumar
References
[1] A. Baker, “On the quasi-periods of the Weierstrass ζ-function”, Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1969
(1970), p. 145-157.
[2] W. D. Brownawell, K. K. Kubota, “The algebraic independence of Weierstrass functions and some related numbers”,
Acta Arith. 33 (1977), p. 111-149.
[3] J. Coates, “Linear forms in the periods of the exponential and elliptic functions”, Invent. Math. 12 (1971), p. 290-299.
[4] ——— , “The transcendence of linear forms in ω1,ω2,η1,η2,2πi ,”, Am. J. Math. 93 (1971), p. 385-397.
[5] M. H. Heins, “On the pseudo-periods of the Weierstrass zeta-functions”, SIAM J. Numer. Anal. 3 (1966), p. 266-268.
[6] ——— , “On the pseudo-periods of the Weierstrass zeta functions. II.”, Nagoya Math. J. 30 (1967), p. 113-119.
[7] S. Lang, Elliptic functions, second ed., Graduate Texts in Mathematics, vol. 112, Springer, 1987.
[8] D. Masser, Elliptic functions and transcendence, Lecture Notes in Mathematics, vol. 437, Springer, 1975.
[9] T. Schneider, Einführung in die transzendenten Zahlen, Grundlehren der Mathematischen Wissenschaften, vol. 81,
Springer, 1957.
[10] C. L. Siegel, “Über die Perioden elliptischer Funktionen”, J. Reine Angew. Math. 167 (1932), p. 62-69.
C. R. Mathématique — 2021, 359, n 4, 409-414


Linear dependence of quasi-periods over the rationals

Hai-Liang Wu

Comptes Rendus
Mathématique
Hai-Liang Wu
Determinants concerning Legendre symbols
Volume 359, issue 6 (2021), p. 651-655
<https://doi.org/10.5802/crmath.205>
© Académie des sciences, Paris and the authors, 2021.
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
2021, 359, n 6, p. 651-655
https://doi.org/10.5802/crmath.205
Combinatorics, Number theory / Combinatoire, Théorie des nombres
Determinants concerning Legendre symbols
Hai-LiangWua
a School of Science, Nanjing University of Posts and Telecommunications, Nanjing
210023, People’s Republic of China
E-mail: whl.math@smail.nju.edu.cn
Abstract. The evaluations of determinants with Legendre symbol entries have close relation with combina-
torics and character sums over finite fields. Recently, Sun [9] posed some conjectures on this topic. In this
paper, we prove some conjectures of Sun and also study some variants. For example, we show the following
result:
Let p = a2 +4b2 be a prime with a,b integers and a ≡ 1 (mod 4). Then for the determinant
S(1, p) := det
[(
i 2 + j 2
p
)]
1≤i , j≤ p−12
,
the number S(1, p)/a is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
2020 Mathematics Subject Classification. 11C20, 11L10, 11R18.
Funding. This research is supported by the National Natural Science Foundation of China (Grant No.
11971222) and by NUPTSF (Grant No. NY220159).
Manuscript received 8th January 2021, revised 27th March 2021, accepted 2nd April 2021.
1. Introduction
Given an n × n complex matrix M = [ai j ]1≤i , j≤n , we often use det M or |M | to denote the
determinant of M . The evaluation of determinants with Legendre symbol entries is a classical
topic in number theory, combinatorics and finite fields. Krattenthaler’s survey papers [7, 8]
introduce many concrete examples and advanced techniques on determinant calculation.
Let p be an odd prime and let
( ·
p
)
denote the Legendre symbol. Carlitz [2] studied the following
(p −1)× (p −1) matrix
Dp :=
[(
i − j
p
)]
1≤i , j≤p−1
.
He obtained that the characteristic polynomial of Dp is precisely∣∣xIp−1 −Dp ∣∣= (x2 − (−1) p−12 p) p−32 (x2 − (−1) p−12 ),
where Ip−1 is the (p −1)× (p −1) identity matrix.
Along this line, Chapman [3] further investigated the following matrices:
Cp (x) :=
[
x +
(
i + j −1
p
)]
1≤i , j≤ p−12
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
652 Hai-Liang Wu
and
C∗p (x) :=
[
x +
(
i + j −1
p
)]
1≤i , j≤ p+12
,
where x is a variable. In the case p ≡ 1 (mod 4), let εp > 1 and h(p) be the fundamental unit
and class number of the real quadratic field Q(
p
p) respectively and let εh(p)p = ap +bppp with
2ap ,2bp ∈Z. Chapman proved that
detCp (x) =
{
(−1)(p−1)/42(p−1)/2(bp −ap x) if p ≡ 1 (mod 4),
−2(p−1)/2x if p ≡ 3 (mod 4),
and that
detC∗p (x) =
{
(−1)(p−1)/42(p−1)/2(pbp x −ap ) if p ≡ 1 (mod 4),
−2(p−1)/2 if p ≡ 3 (mod 4).
Moreover, Chapman [4] posed a conjecture concerning the determinant of the p+12 ×
p+1
2 matrix
C =
[(
j − i
p
)]
1≤i , j≤ p+12
.
Due to the difficulty of the evaluation on this determinant, he called it “evil” determinant. Finally
this conjecture was confirmed completely by Vsemirnov [11, 12].
Recently Sun [9] studied various determinants of matrices involving Legendre symbol entries.
Let p be a prime and d be an integer with p - d . Sun defined
S(d , p) := det
[(
i 2 +d j 2
p
)]
1≤i , j≤ p−12
.
In the same paper, Sun also studied some properties of the above determinant. For example, he
showed that −S(d , p) is always a quadratic residue modulo p if ( dp ) = 1 and that S(d , p) = 0 if( d
p
)=−1. Moreover, Sun posed the following conjecture:
Conjecture 1 (Sun). Let p ≡ 3 (mod 4) be a prime. Then −S(1, p) is an integral square.
This conjecture was later confirmed by Alekseyev and Krachun by using some algebraic
number theory. In the case p ≡ 1 (mod 4), Cohen, Sun and Vsemirnov also posed the following
conjecture.
Conjecture 2 (Cohen, Sun and Vsemirnov). Let p = a2 +4b2 be a prime with a,b integers and
a ≡ 1 (mod 4). Then S(1, p)/a is an integral square.
For example, if p = 5 = 12 + 4× 12, then S(1,5) = 1 = 1× 12. If p = 13 = (−3)2 + 4× 12, then
S(1,13) =−27 =−3×32.
As the first result of this paper, by considering some character sums over finite fields, we
confirm this conjecture and obtain the following result. For convenience, for each d ∈Zwe set
ε(d) =
{
−1 if ( dp )= 1 and d is not a biquadratic residue modulo p,
1 otherwise .
Theorem 3. Let p = a2 + 4b2 be a prime with a,b integers and a ≡ 1 (mod 4) and let d be an
integer. Then ε(d)S(d , p)/a is an integral square. In particular, when d = 1 the number S(1, p)/a is
an integral square.
Sun [9] also made the following conjecture.
Conjecture 4 (Sun). Let S∗(1, p) denote the determinant obtained from S(1, p) via replacing the
entries
(
12+ j 2
p
) (
j = 1, . . . , p−12
)
in the first row by
( j
p
) (
j = 1, . . . , p−12
)
respectively. Then −S∗(1, p) is
an integral square if p ≡ 1 (mod 4).
C. R. Mathématique — 2021, 359, n 6, 651-655
Hai-Liang Wu 653
As an application of Theorem 3, we confirm this conjecture.
Corollary 5. Let p ≡ 1 (mod 4) be a prime. Then −S∗(1, p) is an integral square.
For example, S∗(1,5) =−12, S∗(1,13) =−32 and S∗(1,17) =−212.
The proofs of our main results will be given in Section 2.
2. Proofs of the main results
We begin with the following permutation involving quadratic residues (readers may refer to [5,10]
for details on the recent progress on permutations over finite fields). Let p ≡ 1 (mod 4) be a prime
and let d ∈Zwith ( dp )= 1. If we write p = 2n +1, then clearly the sequence
d ·12 mod p, . . . ,d ·n2 mod p
is a permutation πp (d) of the sequence
12 mod p, . . . ,n2 mod p.
Let sgn(πp (d)) be the sign of πp (d). We first have the following result:
Lemma 6. Let p ≡ 1 (mod 4) be a prime, and let d ∈Z be a quadratic residue modulo p. Then
sgn(πp (d)) =
{
1 if d is a biquadratic residue modulo p,
−1 otherwise .
Proof. It is clear that
sgn(πp (d)) ≡
∏
1≤i< j≤n
d j 2 −di 2
j 2 − i 2 (mod p).
By this we obtain
sgn(πp (d)) ≡
(
d
p−1
4
)n−1 ≡ d p−14 (mod p).
This implies the desired result. 
We also need the following known result concerning eigenvalues of a matrix.
Lemma 7. Let M be an m × m complex matrix. Let µ1, . . . ,µm be complex numbers, and let
u1, . . . ,um be m-dimensional column vectors. Suppose that Muk = µk uk for each 1 ≤ k ≤ m
and that u1, . . . ,um are linear independent. Then µ1, . . . ,µm are exactly all the eigenvalues of M
(counting multiplicities).
Before the proof of Theorem 3, we first introduce some notation. In the remaining part of this
section, we let p = a2+4b2 be a prime with a,b ∈Z and a ≡ 1 (mod 4), and let n = p−12 . In addition,
we let χ(Z/pZ) denote the group of all multiplicative characters on the finite fieldZ/pZ= Fp , and
let χp be a generator of χ(Z/pZ), i.e.,
χ(Z/pZ) = {χkp : k = 1,2, . . . , p −1}.
Readers may refer to [6, Chapter 8] for a detailed introduction to characters on finite fields. Also,
given any matrix M , the symbol M T denotes the transpose of M .
Now we are in a position to prove our first theorem.
Proof of Theorem 3. Throughout this proof, we define
Mp :=
[(
i 2 + j 2
p
)]
1≤i , j≤n
.
C. R. Mathématique — 2021, 359, n 6, 651-655
654 Hai-Liang Wu
We first determine all the eigenvalues of Mp . For k = 1,2, . . . ,n, we let
λk :=
∑
1≤ j≤n
(
1+ j 2
p
)
χkp ( j
2). (1)
We claim that λ1, . . . ,λn are exactly all the eigenvalues of Mp (counting multiplicities). In fact, for
any 1 ≤ i ,k ≤ n we have∑
1≤ j≤n
(
i 2 + j 2
p
)
χkp ( j
2) = ∑
1≤ j≤n
(
1+ j 2/i 2
p
)
χkp ( j
2/i 2)χkp (i
2)
= ∑
1≤ j≤n
(
1+ j 2
p
)
χkp ( j
2)χkp (i
2) =λkχkp (i 2).
This implies that for each k = 1, . . . ,n, we have
Mp vk =λk vk ,
where
vk :=
(
χkp (1
2),χkp (2
2), . . . ,χkp (n
2)
)T .
Since ∣∣∣∣∣∣∣∣∣∣
χ1p (1
2) χ2p (1
2) . . . χnp (1
2)
χ1p (2
2) χ2p (2
2) . . . χnp (2
2)
...
...
. . .
...
χ1p (n
2) χ2p (n
2) . . . χnp (n
2)
∣∣∣∣∣∣∣∣∣∣
=± ∏
1≤i< j≤n
(
χp ( j
2)−χp (i 2)
) 6= 0,
the vectors v1, . . . ,vn are linear independent. By Lemma 7 our claim holds. Hence we have
S(1, p) = det Mp =
∏
1≤k≤n
λk =
∏
1≤k≤n
( ∑
1≤ j≤n
(
1+ j 2
p
)
χkp ( j
2)
)
. (2)
Now we turn to the last product. When k = n, by [6, Chapter 5, Exercise 8] we have
λn =
∑
1≤ j≤n
(
1+ j 2
p
)
χnp ( j
2) = ∑
1≤ j≤n
(
1+ j 2
p
)
=−1. (3)
When k = n/2, by [1, Theorem 6.2.9] we have
λn/2 =
∑
1≤ j≤n
(
1+ j 2
p
)
χn/2p ( j
2) = ∑
1≤ j≤n
(
1+ j 2
p
)(
j
p
)
=−a. (4)
As Mp is a real symmetric matrix, every eigenvalue λk of Mp is real. Hence for any 1 ≤ k ≤ p−54 we
have λk =λn−k . Let
f (x) := det(xIn −Mp )
be the characteristic polynomial of Mp . By the above we observe that all roots of f (x) apart from
λn =−1 andλn/2 =−a are of even multiplicity. Using unique factorisation inZ[x], one can obtain
that
f (x) = (x +1)(x +a)g (x)2,
where g (x) is a polynomial with integer coefficients. Therefore we obtain that S(1, p)/a = g (0)2 is
an integral square.
Now we consider S(d , p). If p | d , then clearly S(d , p) = 0. If ( dp )=−1, then by [9, Theorem 1.2]
we know that S(d , p) = 0. Suppose now that d is a quadratic residue modulo p. Then clearly we
have
S(d , p) = sgn(πp (d))S(1, p).
Now our desired result follows from Lemma 6. 
We now prove our next result.
C. R. Mathématique — 2021, 359, n 6, 651-655
Hai-Liang Wu 655
Proof of Corollary 5. By [1, Theorem 6.2.9] for any 1 ≤ i , j ≤ n we have∑
1≤i≤n
(
i 2 + j 2
p
)(
i
p
)
=−a
(
j
p
)
and hence
− ∑
2≤i≤n
(
i 2 + j 2
p
)(
i
p
)
−a
(
j
p
)
=
(
1+ j 2
p
)
. (5)
By this we have
S∗(1, p) = −1
a
∣∣∣∣∣∣∣∣∣∣∣
−a( 1p ) −a( 2p ) . . . −a( np )( 22+12
p
) ( 22+22
p
)
. . .
( 22+n2
p
)
...
...
. . .
...( n2+12
p
) ( n2+22
p
)
. . .
( n2+n2
p
)
∣∣∣∣∣∣∣∣∣∣∣
=−S(1, p)/a.
The last equality follows from (5). Now our desired result follows from Theorem 3. 
Acknowledgments
We thank the referee for helpful comments.
References
[1] B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs
and Advanced Texts, John Wiley & Sons, 1998.
[2] L. Carlitz, “Some cyclotomic matrices”, Acta Arith. 5 (1959), p. 293-308.
[3] R. Chapman, “Determinants of Legendre symbol matrices”, Acta Arith. 115 (2004), no. 3, p. 231-244.
[4] ——— , “My evil determinant problem”, preprint, available at http://empslocal.ex.ac.uk/people/staff/rjchapma/
etc/evildet.pdf, 2012.
[5] X.-d. Hou, “Permutation polynomials over finite fields – a survey of recent advances”, Finite Fields Appl. 32 (2015),
p. 82-119.
[6] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, vol. 84,
Springer, 1990.
[7] C. Krattenthaler, “Advanced determinant calculus”, Sémin. Lothar. Comb. (1999), article no. B42q (67 pages).
[8] ——— , “Advanced determinant calculus: a complement”, Linear Algebra Appl. 411 (2005), p. 68-166.
[9] Z.-W. Sun, “On some determinants with Legendre symbol entries”, Finite Fields Appl. 56 (2019), p. 285-307.
[10] ——— , “Quadratic residues and related permutations and identities”, Finite Fields Appl. 59 (2019), p. 246-283.
[11] M. Vsemirnov, “On the evaluation of R. Chapman’s “evil determinant””, Linear Algebra Appl. 436 (2012), no. 11,
p. 4101-4106.
[12] ——— , “On R. Chapman’s “evil determinant”: case p ≡ 1 (mod 4)”, Acta Arith. 159 (2013), no. 4, p. 331-344.
C. R. Mathématique — 2021, 359, n 6, 651-655


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Comptes Rendus
Mathématique
Akaki Tikaradze
Generic simplicity of quantum Hamiltonian reductions
Volume 359, issue 6 (2021), p. 739-742
<https://doi.org/10.5802/crmath.214>
© Académie des sciences, Paris and the authors, 2021.
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
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Comptes Rendus
Mathématique
2021, 359, n 6, p. 739-742
https://doi.org/10.5802/crmath.214
Algebra / Algèbre
Generic simplicity of quantum Hamiltonian
reductions
Akaki Tikaradzea
a University of Toledo, Department of Mathematics & Statistics, Toledo, OH 43606,
USA.
E-mail: tikar06@gmail.com
Abstract. Let a reductive group G act on a smooth affine complex algebraic variety X . Let g be the Lie algebra
of G and µ : T∗(X ) → g∗ be the moment map. If the moment map is flat, and for a generic character χ : g→C,
the action of G on µ−1(χ) is free, then we show that for very generic characters χ the corresponding quantum
Hamiltonian reduction of the ring of differential operators D(X ) is simple.
Manuscript received 16th September 2020, revised 1st April 2021 and 12th April 2021, accepted 13th April 2021.
Let a reductive algebraic group G act on a smooth affine algebraic variety X over C. Let g be
the Lie algebra of G . Let µ : T ∗(X ) → g∗ be the corresponding moment map. We will assume that
this map is flat, and for generic G-invariant character χ ∈ g∗ the action of G on µ−1(χ) is free.
Given a G-invariant character χ ∈ g∗, denote by Uχ(G , X ) the quantum Hamiltonian reduction
of D(X ) with respect to χ. So,
Uχ(G , X ) =
(
D(X )/D(X )gχ
)G ,
where gχ = {g − χ(g ) ∈ D(X ), g ∈ g}. The usual filtration on D(X ) by the order of differential
operators induces the corresponding filtration on Uχ(G , X ). Then it follows from the flatness of
the moment map that
grUχ(G , X ) =O
(
µ−1(0)//G
)
.
In what follows by a very generic subset we mean a complement of a union of countably many
proper closed Zariski subsets. Under these assumptions we have the following result.
Theorem 1. For very generic values of a G-invariant character χ ∈ g∗, the corresponding quantum
Hamiltonian reduction Uχ(G , X ) is simple. Moreover, if f ∈ gZ/[gZ,gZ] is so that G acts freely on
µ−1(χ) whenever χ( f ) 6= 0, then Uχ(G , X ) is simple for all χ such that χ( f ) ∉Q.
The proof is be based on the reduction modulo pn technique for a large prime p.
At first, we recall that given a ring R such that p is not a zero divisor, then the center of
its reduction modulo p,Rp = R/pR acquires a natural Poisson bracket, to be referred to as the
reduction modulo p Poisson bracket, defined as follows. Given central elements x, y ∈ Z (Rp ), let
x ′, y ′ ∈ R be their lifts. Then
{x, y} =
(
1
p
[
x ′, y ′
])
mod p ∈ Z (Rp) .
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
740 Akaki Tikaradze
We use the following result [5, Corollary 8].
Lemma 2. Let k be a perfect field of characteristic p. Let A be a p-adically complete topologically
free W (k)-algebra, such that A1 = A/p A is an Azumaya algebra over its center Z1. Assume that
Spec(Z1) is a smooth symplectic k-variety under the reduction modulo p Poisson bracket. Then
A[p−1] is topologically simple.
Next we need to recall some results and notations associated with quantum Hamiltonian
reduction of the ring of crystalline differential operators in characteristic p from [1].
Let X be a smooth affine variety over an algebraically closed field k of characteristic p, and
G be a reductive algebraic group over k with the Lie algebra g. Denote by D(X ) the ring of
crystalline differential operators on X . As before, we have the moment map µ : T ∗(X ) → g∗ and
the algebra homomorphism U (g) → D(X ). Now recall that the p-center of U (g), denoted by Zp (g),
is generated by g p − g [p], g ∈ g. We get an isomorphism
i : Sym(g)(1) → Zp (g).
On the other hand, the center of D(X ) is generated by O(X )p and ξp −ξ[p],ξ ∈ TX and this leads
to an isomorphism
O
(
T ∗(X )
)(1) → Z (D(X )).
We have η′ : Zp (g) → Z (D(X )) and the corresponding homomorphism
η : Sym(g)(1) →O (T ∗(X ))(1) .
Given χ ∈ g∗, then χ[1] ∈ g∗ is defined as follows:
χ[1](g ) =χ(g )p −χ(g [p]) , g ∈ g.
Using the above homomorphisms it follows that the center of Uχ(G , X ) contains O(µ−1(χ[1])//G).
In this setting the following holds.
Lemma 3 ([1]). Let χ ∈ (g∗)G be a character. Then Uχ(G , X ) is a finite algebra over µ−1(χ[1])//G. If
G acts freely of µ−1(χ[1]), then Uχ(G , X ) is an Azumaya algebra over µ−1(χ[1])//G.
We need the following criterion of simplicity of certain filtered quantizations.
Lemma 4. Let S ⊂C be a finitely generated ring, and let R be a filtered S-algebra, such that gr(R) is
a finitely generated commutative ring over S. Assume that for all large enough primes p the algebra
Rp = R/pR is an Azumaya algebra over its center Zp , moreover Spec(Zp ) is a smooth symplectic
variety over Sp under the reduction modulo p Poisson bracket. Let F be the field of fractions of S.
Then RF = R ⊗S F is a simple ring.
Proof. Let I be a nonzero two sided ideal of R such that (R/I )F 6= 0. After localizing S further,
we may assume using the generic flatness theorem that gr(R/I ) and R/I ,R are free S-modules.
Hence for p À 0, Īp (the p-adic completion of I ) is a topologically free nontrivial two-sided ideal
of R̄p (the p-adic completion of R). Now Lemma 2 yields a contradiction. 
Next we state a result implying that taking quantum Hamiltonian reduction and reducing
modulo a large prime commute. The statement and its proof were kindly provided by W. van der
Kallen (via mathoverflow.org.) Possible mistakes in the proof below are solely due to the author.
Theorem 5 (van der Kallen). Let S be a commutative Noetherian ring of finite homological
dimension, let R be a commutative S-algebra flat over S. Let G be a split reductive group over
S acting on R. Then for all p À 0 and a base change to a characteristic p field S → k, the map
RG ⊗S k → RGkk is surjective.
C. R. Mathématique — 2021, 359, n 6, 739-742
Akaki Tikaradze 741
Proof. At first, recall that there exists an integer n ≥ 1 so that H i (G ,S[ 1n ]) = 0 for all i [3,
Theorem 33]. This implies H i (G ,S[ 1n ] ⊗S N ) = 0 for any S-module N with the trivial G-action
(since S has a finite global dimension). Let D, N be respectively the image and kernel of the map
S[ 1n ] → k. As H 1(G ,S[ 1n ]⊗S N ) = 0, we get that (R⊗S S[ 1n ])G → (R⊗S D)G is surjective. Now flatness
of k over D yields that
(R ⊗S D)G ⊗D k = RGkk .
Therefore, we obtain the desired surjectivity RG ⊗S k → RGkk 
Proof of Theorem 1. Recall that some 0 6= f ∈ O((g∗)G ) has the property that for any χ ∈
(g/[g,g])∗ such that f (χ) 6= 0, the action of G on µ−1(χ) is free. Let S ⊂C be a large enough finitely
generated subring over which X , f and the action of G on X are defined. Let U ⊂ (g/[g,g])∗S denote
the complement of the zero locus of f . Thus, GS acts freely onµ−1(U ). Localizing S further and us-
ing the generic flatness theorem, we may assume thatO(µ−1(0)//G) andO(µ−1(0))/O(µ−1(0)//G)
is a flat S-module.
Let e1, · · · , el be a basis of gZ/[gZ,gZ] over Z. Let S → k be a base change to a characteristic p
field k, let χ̄ denote the image of χ in g∗k . Then
χ̄[1](ēi ) =
(
χ̄(ēi )
)p − χ̄(ēi ).
Let W ⊂ (g/[g,g])∗ be the set of all χ so that χ(ei ) are algebraically independent over S. Clearly
W is a very generic subset. We will show that for any χ ∈W algebra Uχ(G , X ) is simple.
Put R = Uχ(G , X ). We verify that R satisfies assumptions in Lemma 4. Indeed, let S → k be
a base change to an algebraically closed field k of characteristic À 0, let χ̄ denote the base
change of χ. Recall that R is equipped with the filtration so that gr(R) = O(µ−1(0)//G). In
particular, R is a free S-module. Similarly, Uχ̄(Gk, Xk) is equipped with the filtration such that
gr(Uχ̄(Gk, Xk)) is a subring of O(µ−1k (0)//Gk). Now applying Theorem 5 to the action of G on
O(µ−1(0)), we conclude that O(µ−1(0)//G)⊗S k surjects onto O(µ−1k (0)//Gk). On the other hand,
since O(µ−1(0))/O(µ−1(0)//G) is flat over S, we get that
O
(
µ−1(0)//G
)⊗S k →O (µ−1(0))⊗S k
is injective. So, the restriction map
O
(
µ−1(0)//G
)⊗S k →O (µ−1k (0)//Gk)
is an isomorphism. Therefore, gr(R)⊗S k ∼= gr(Uχ̄(Gk, Xk)). Now flatnest of gr(R) over S implies
that gr(Rk) = gr(R)⊗S k. Hence we conclude that Rk ∼=Uχ̄(Gk, Xk).
Since χ(e1), · · · , χ(en) are algebraically independent over S, we get that f̄ (χ[1]) 6= 0 for all p À 0
and an appropriate base change S → k. Hence χ̄ ∈Uk. As GS acts freely on µ−1(U ), we conclude
that Gk acts freely on µ
−1(χ̄). So Lemma 3 implies that Uχ̄(Gk, Xk) is an Azumaya algebra over a
symplectic variety under the reduction modulo p Poisson bracket. So, conditions of Lemma 4 are
met. Hence we have shown that algebra Uχ(G , X ) is simple for very generic values of χ.
Now suppose there exists a nonzero f ∈ gZ/[gZ,gZ] such that G acts freely on µ−1(χ) when
χ( f ) 6= 0. Let S be a finitely generated subring containing χ(ei ) satisfying conditions as above.
Write f =∑i fi ei , fi ∈Z. Then given a base change S → k, we have
χ̄[1]( f̄ ) =∑
i
f̄i
((
χ̄(ēi )
)p − χ̄(ēi ))= χ̄( f̄ )p − χ̄( f̄ ).
Let χ be so that χ( f ) is irrational. Then it follows from the Chebotarev density theorem that
there are arbitrarily large primes p and a base change S → k to an algebraically closed field k
of characteristic p, such that χ̄( f̄ ) ∉ Fp . Hence χ[1]( f̄ ) is nonzero in k. So, Gk acts freely on µ−1(χ̄),
and arguing just as above we may conclude that the algebra Uχ(G , X ) is simple. 
C. R. Mathématique — 2021, 359, n 6, 739-742
742 Akaki Tikaradze
We may apply the above result to certain filtered quantizations of quiver varieties as follows.
Let Q be a quiver with n vertices, let α be a its positive root. Then G = ∏GLαi /C∗ acts on
the space of α-dimensional representations Rep(Q,α) giving rise to the moment map mα :
T ∗(Rep(Q,α)) → g∗. We will identify (g∗)G withλ ∈Cn such thatλ·α= 0. From now on we assume
that the moment map mα is flat. The set of such dimension vectors α was fully described by
Crawly-Boevey in [2, Theorem 1.1]. Denote by Aλ(Q,α) the corresponding quantum Hamiltonian
reduction of the ring of differential operators D(Rep(Q,α)) with respect to the character λ.
We have the following direct corollary of Theorem 1. Remark that stronger results on generic
simplicity follows from the works of Losev on quantizations of quiver varieties (see for example [4,
Theorem 1.4.2].)
Theorem 6. Let α be a positive root as above. Let λ ·α = 0 be such that λ ·β ∉ Q for any positive
root β<α. Then Aλ(Q,α) is simple.
References
[1] R. Bezrukavnikov, M. Finkelberg, V. Ginzburg, “Cherednik algebras and Hilbert schemes in characteristic p”, Repre-
sent. Theory 10 (2006), p. 254-298.
[2] W. Crawley-Boevey, “Geometry of the moment map for representations of quivers”, Compos. Math. 126 (2001), no. 3,
p. 257-293.
[3] V. Franjou, W. van der Kallen, “Power reductivity over an arbitrary base of change”, Doc. Math. Extra Vol., Andrei A.
Suslin’s Sixtieth Birthday (2010), p. 171-195.
[4] I. Losev, “Completions of symplectic reflection algebras”, Sel. Math., New Ser. 18 (2012), no. 1, p. 179-251.
[5] A. Tikaradze, “Ideals in deformation quantizations over Z/pnZ”, J. Pure Appl. Algebra 221 (2017), no. 1, p. 229-236.
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