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Michael Rosen

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Juliusz Brzeziński

Juliusz BRZEZIŃSKI
Corrigendum to “On traces of the Brandt-Eichler matrices”
Tome 29, no 1 (2017), p. 321-325.
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Journal de Théorie des Nombres
de Bordeaux 29 (2017), 321–325
Corrigendum to “On traces of the Brandt-Eichler
matrices”
par Juliusz BRZEZIŃSKI
Résumé. Cette note est une correction de mon article [1]. Elle
corrige une formule dans la proposition 2.2. D’après le résultat
corrigé, le nombre ι(n,m) d’idéaux principaux à gauche de norme
qm dans l’ordre de Eichler de niveau n sur un anneau de valuation
discrète R dont le corps résiduel est de cardinalité q est ι(n,m) =
(m+ 1)qm si m < n et
ι(n,m) = (n+ 1)qm + 2qm−1 + · · ·+ 2qn + qn−1
lorsque m ≥ n. La démonstration de la proposition n’était pas
donnée dans mon article (étant “pénible mais sans obstacle”).
Malheureusement, certains coefficients dans le second cas étaient
erronés. Une démonstration complète suit ci-dessous.
Abstract. This is a correction to my paper [1]. It corrects
a formula in Proposition 2.2. The corrected result says that the
number ι(n,m) of principal left ideals with norm qm in the Eichler
order of level n over a discrete valuation ring R with residue field
of cardinality q is ι(n,m) = (m+ 1)qm if m < n and
ι(n,m) = (n+ 1)qm + 2qm−1 + · · ·+ 2qn + qn−1
when m ≥ n. The proof of the Proposition was not given in my
paper (as “tedious but straightforward”). Unfortunately, some co-
efficients in the second case were erroneous. A complete proof
follows below.
Let R be a discrete valuation ring with maximal ideal (π). Assume that
the residue ring R/(π) is finite and let q = |R/(π)| be its cardinality. The
discrete valuation defined by R will be denoted by v. Thus v(πm) = m. Let
Λn =
(
R R
πnR R
)
be an Eichler order in the matrix algebra M2(K) over the quotient field K
of R.
Mathematics Subject Classification. 11S45, 11R52.
Mots-clefs. Eichler order, ideal norm.
I am grateful to John Voight for informing me that his numerical computations of the number
of ideals deviated from the formula given in Proposition 2.2 of [1]. I am also grateful to the Referee
for several valuable comments.
322 Juliusz Brzeziński
Our purpose is to find the number of principal left ideals in Λn with
norm qm (m > 0), which will be denoted by ι(n,m). Since each such ideal
has form ΛnM , where M ∈ Λn and the determinant of M is a generator
of (πm), we want to find a set of representatives for the orbits Λ∗nM of the
unit group Λ∗n of Λn acting on the set of matrices in Λn having norm qm
(recall that the norm of ΛnM is the cardinality of R/(det(M))).
We denote by
ε =
(
e11 e12
πne21 e22
)
the elements of Λ∗n, and by
M =
(
m11 m12
πnm21 m22
)
the matrices in Λn.
Our purpose is to choose a “canonical” set of representatives of the orbits
Λ∗nM .
We split the orbits into 3 types: Type 1 are those having a representant
withm11 = 0, Type 2 those having a representant withm21 = 0 and Type 3
those which can not be represented by a matrix with m11m21 = 0.
First we give a canonical choice of matrices representing each type and
when this is done, we count the number of orbits by counting the number
of representatives of each kind.
Type 1. If a matrix M has m11 = 0, then detM = πnm21m12 6= 0, so
multiplying M from the left by a suitable diagonal unit matrix, we may
assume that
M =
(
0 πs
πn+r m22
)
.
Now, we can take a product
εM =
(
1 0
πnq 1
)
M =
(
0 πs
πn+r m22 + qπn+s
)
,
so we can assume that m22 is reduced modulo πn+s. Thus we arrive to the
description of the matrices of Type 1:
M (1)r,s,c =
(
0 πs
πn+r c
)
,
where c is reduced modulo πn+s. We check easily that two matrices M (1)r,s,c
and M (1)r′,s′,c′ define the same orbit if and only if r = r′, s = s′ and c = c′
(mod πn+s).
The number ι1(n,m) of matrices of Type 1 with norm qm is equal to
all possible choices of r, s, c such that r + s + n = m and c is reduced
On traces of the Brandt-Eichler matrices 323
modulo πn+s. Thus, we have such matrices only if m ≥ n and for each
s = 0, . . . ,m− n we have qs+n such matrices, that is,
(0.1) ι1(n,m) = qn + · · ·+ qm.
Type 2. Since this time, we have detM = m11m22 6= 0, we can multiply
M by a diagonal unit matrix, so that
M =
(
πr m12
0 πs
)
.
Now, we can take a product
εM =
(
1 q
0 1
)
M =
(
πr m12 + qπs
0 πs
)
,
so we can assume that m12 is reduced modulo πs. Thus, the matrices of
Type 2 are
M (2)r,s,c =
(
πr c
0 πs
)
,
where c is reduced modulo πs. We check easily that two matrices M (2)r,s,c
and M (2)r′,s′,c′ define the same orbit if and only if r = r′, s = s′ and c = c′
(mod πs).
The number ι2(n,m) of matrices of Type 2 with norm qm is equal to all
possible choices of r, s, c such that r + s = m and c is reduced modulo πs.
Thus, for each s = 0, . . . ,m we have qs such matrices, that is,
(0.2) ι2(n,m) = 1 + q + · · ·+ qm.
Type 3. This time, we assume that there is no representant of the orbit
ΛnM with m11m21 = 0, so we can start with a representant
M =
(
πk m12
πN m22
)
,
where N ≥ n, which we obtain multiplying M by a suitable diagonal unit
matrix. First of all, we will show that it is possible to choose a representant
of the orbit Λ∗nM such that N − k ∈ {1, . . . , n− 1} (that is, n+ k > N). In
fact, if N − k ≥ n, then we can multiply M by the matrix( 1 0
−πN−k 1
)
,
which gives a representant with 0 in the left lower position.
Let now r = v(m22 − πN−km12), so detM = πk(m22 − πN−km12) and
v(detM) = k + r. It is easy to check that two matrices
M =
(
πk m12
πN m22
)
and M ′ =
(
πk
′
m′12
πN
′
m′22
)
,
324 Juliusz Brzeziński
with r = v(m22 − πN−km12), r′ = v(m′22 − πN
′−k′m′12) represent the same
orbit of Λ∗n if and only if (N, k, r) = (N ′, k′, r′) and m12 = m′12 (mod πr)
and m22 = m′22 (mod π(n+k)−N+r). Moreover, a product εMN,k,r, where
ε ∈ Λ∗n never has 0 in the first column, that is, these matrices can not
represent an orbit of Type 1 or 2. Thus the matrices of Type 3 are
M
(3)
N,k,r,a,b =
(
πk b
πN a
)
,
where N ≥ n, N − n < k < N , r = v(a − πN−kb), a is reduced modulo
π(n+k)−N+r and b is reduced modulo πr. Notice that v(detM (3)N,k,r,a,b) =
k + r.
In the sequel, we will use the following observations: If r < N − k, then
a = πra0, π - a0. If r ≥ N − k, then a = πN−ka0, πr−(N−k) | a0 − b and
πr−(N−k)+1 - a0 − b.
Now we want to compute the number ι3(n,m) of matrices of Type 3 with
norm qm. This is a little more complicated than the corresponding task in
cases 1 and 2. First of all, notice that fixing n,m, we have n ≤ N < n+m
(that is, N assumes m values). In fact, if N ≥ m+n, then k > N −n ≥ m,
which is impossible, since k + r = m. If we fix N , then the number of
possible pairs (k, r) such that k + r = m and N − n < k < N will be
denoted by cN . It is easy to check that
cN = n−max (1, N −m).
In order to compute ι3(n,m), we start counting the contribution coming
from matrices of Type 3 with fixed N . For each r (and k = m − r), we
count the number of corresponding matrices of Type 3.
We have two cases. First we consider the case r < N − k. As we know,
we have qr possibilities for b. As regards a, we have a = πra0, π - a0, so
the number of possibilities for a is given by the number of possible residues
a0 modulo π(n+k)−N , which are invertible in R/(π(n+k)−N ), that is, we get
q(n+k)−N−q(n+k)−N−1 possibilities for a0. The number of possible matrices
is
qr(q(n+k)−N − q(n+k)−N−1) = qr+(n+k)−N − qr+(n+k)−N−1
= q(n+m)−N − q(n+m)−N−1.
(0.3)
The second case is r ≥ N − k. This time, we have as before that b
is reduced modulo πr, but a = πN−ka0, where πr−(N−k) | a0 − b and
πr−(N−k)+1 - a0−b. Since a is reduced modulo πn+k−N+r, we have to count
the number of pairs (a0, b) such that a0 is reduced modulo πn+2k−2N+r, b
is reduced modulo πr and πr−(N−k) | a0 − b, πr−(N−k)+1 - a0 − b. These
configuration seems to be rather messy, but the situation is essentially very
simple: we have residues modulo some πx and πy and we have to count
On traces of the Brandt-Eichler matrices 325
the number of pairs of residues whose difference is divisible by πz and not
divisible by πz+1 for some z ≤ min(x, y). If we use q as before, then the
answer is
qxqy−z − qxqy−z−1 = qx+y−z − qx+y−z−1,
which can be easily checked (take qx residues a0 = 0, 1, . . . , qx − 1 and for
each of them, those residues among b = 0, 1, . . . , qy− 1 for which qz divides
a0 − b; then repeat the counting looking at those for which qz+1 divides
a0 − b and subtract their number).
In our case, we have x = n + 2k − 2N + r, y = r, z = r + k −N . Thus
x+ y − z = n+ k −N + r = n+m−N , which means that the number of
matrices is exactly the same as in the case r < N − k and is given by (0.3).
Now, it remains to compute the number ι(n,m) of all matrices of Types 1,
2, 3. We have
ι(n,m) = ι1(n,m) + ι2(n,m) + ι3(n,m)
=
m∑
i=n
qi +
m∑
i=0
qi +
n+m−1∑
N=n
cN (qn+m−N − qn+m−N−1),
where the first sum is 0 when m < n.
If m < n, then it is easy to check that cN = m + n − N for N =
n, . . . , n+m− 1. Thus, we have the sum:
ι(n,m) =
m∑
i=0
qi +
n+m−1∑
N=n
(n+m−N)(qn+m−N − qn+m−N−1) = (m+ 1)qm.
If m ≥ n, then cN = n − 1 for N = n, . . . ,m and cN = n + m − N for
N = m+ 1, . . . , n+m− 1. Hence, we have
ι(n,m) =
m∑
i=n
qi +
m∑
i=0
qi +
m∑
N=n
(n− 1)(qn+m−N − qn+m−N−1)
+
n+m−1∑
N=m+1
(n+m−N)(qn+m−N − qn+m−N−1)
= (n+ 1)qm + 2qm−1 + · · ·+ 2qn + qn−1.
References
[1] J. Brzeziński, “On traces of the Brandt-Eichler matrices”, Journal de Théorie des Nombres
de Bordeaux 10 (1998), no. 2, p. 273-285.
Juliusz Brzeziński
Mathematical Sciences
University of Gothenburg and Chalmers
S-412 96 Göteborg, Sweden
E-mail: jub@chalmers.se
URL: http://www.chalmers.se/sv/personal/Sidor/jub.aspx


Corrigendum to “On traces of the Brandt-Eichler matrices”

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Realizable Classes and Embedding Problems

A Geometric Proof of Hermite’s Theorem in Function Fields

Special functions and twisted $L$ -series

Rigid $\tau$ -crystals

On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$ -extensions of algebraic number fields

The size function for quadratic extensions of complex quadratic fields

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Binary quadratic forms as dessins

Corrigendum to “On traces of the Brandt-Eichler matrices”

A characterization of class groups via sets of lengths II

1,235

1,351