Counting ideals in polynomial rings

0.17 MB·

by  Lenny Fukshansky

COUNTING IDEALS IN POLYNOMIAL RINGS
7 LENNYFUKSHANSKY,STEFANKU¨HNLEIN,ANDREBECCASCHWERDT
1
0
2 Abstract. We investigate properties of zeta functions of polynomial rings
and their quotients, generalizing and extending some classical results about
n Dedekind zeta functions of number fields. By an application of Delange’s
a versionoftheIkeharaTauberianTheorem,wearethenabletodeterminethe
J
asymptotic orderoftheidealcountingfunctioninsuchrings. Asaresult,we

7 produce counting estimates on ideal lattices of bounded determinant coming
1 from fixed number fields, as well as density estimates for any ideal lattices
amongallsublatticesofZd. Weconcludewithsomemoregeneralspeculations

] andopenquestions.
T
N
.
h
1. Introduction
t
a
m A classical arithmetic problem in the theory of finitely generated groups and
rings is the study of the asymptotic order of growth of the number of subgroups
[
of bounded index. A common approach to this problem involves studying the

1
analytic properties of a corresponding zeta function (a Dirichlet series generating
v
function) and then applying a Tauberian theorem to deduce information about
3
3 the number in question, represented by the coefficients of this zeta function. This
6 research direction received a great deal of attention over the years as can be seen
4 from [11], [8], [6], [7], [3], [17] and the references within. In the recent years, a
0
similar approach has also been applied to the more geometric setting of counting
.
1 sublattices in lattices, e.g. [15], [10], [9], [1], [14].
0 In this note, we consider some special cases of the following generalsetting. Let
7
R be a commutative ring with identity such that for every natural number n the
1
setofidealsinRofindexnisafinite number,callthisnumbera (R).Oneisoften
: n
v interested in the asymptotic behaviour of the sequence a (R) or the summatory
n
i
X sequence AN(R):= n≤Nan(R). To that end one canintroduce the zeta function
ar P ζ(R,s):= an(R)
ns
n
X
and calculate its abscissa of convergence, so that one might apply a Tauberian
theorem.
We will make use of the following well-known result, which is a consequence of
Delange’s extended version of Ikehara’s Tauberian Theorem ([5]):
Theorem 1.1. Let (a ) be any sequence of non-negative real numbers and
n n≥1
∞
a
n
Z(s):= .
ns
n=1
X
2010 Mathematics Subject Classification. 11R42,11M41,11M45,20E07,11H06.
Key words and phrases. Dedekindzeta-function, ideallattices.
ThefirstauthorwaspartiallysupportedbytheNSAgrantH98230-1510051.
1
2 LENNYFUKSHANSKY,STEFANKU¨HNLEIN,ANDREBECCASCHWERDT
Assume that Z(s) has abscissa of convergence σ > 0 and admits a meromorphic
extension tosome neighborhood of the line ℜ(s)=σ with a pole of order w at s=σ
and no other singularity. Then
N
a ∼c·Nσ·(logN)w−1
n
n=1
X
holds for c= Res(Z,σ), where Res(Z,σ) stands for the residue of Z(s) at s=σ and
σ·Γ(w)
Γ(w) is the value of gamma function at w.
The classical situation is that of rings of integers in number fields, where the
zeta function ζ(R,s) is Dedekind’s zeta function which convergesfor ℜ(s)>1 and
is meromorphic on C with a simple pole at s=1. In this situation A (R)∼c·N,
N
the constant being the residue of ζ(R,s) at s=1.
One of our general results is the calculation of the zeta-function for polynomial
rings:
Theorem 1.2. Let K be a number field with ring of integers O . Then we have
K
∞
ζ(O [X],s)= ζ (d(s−1)),
K K
d=1
Y
where ζ :=ζ(O ,·) is the Dedekind zeta-function of the number field K.
K K
In particular, this function has abscissa of convergence σ = 2, meromorphic
extensiontoℜ(s)>1andsimplepolesats= d+1, d∈N.Asthepolesaccumulate
d
towards1,ζ(O [X],s)cannotbeextendedmeromorphicallybeyondℜ(s)>1.The
K
largest pole is at s = 2 and therefore the number of ideals in O [X] of index less
K
than N grows like a multiple of N2.
We will, however, be more generally interested in ideals in quotient rings of
O [X], which are related to lattices in Od for some d. To that end, we will need
K K
some more machinery. Let R again be any commutative ring. If I ≤R is an ideal
of index mn with coprime m and n, then by the Chinese Remainder TheoremR/I
is a direct product of two rings of orders m and n, respectively, and therefore I is
the intersection of two ideals of indices m and n, respectively. These are uniquely
determined by I, and therefore
a (R)=a (R)·a (R).
mn m n
This means that the sequence (a (R)) is multiplicative, and so ζ(R,s) formally
n n
has a decomposition as an Euler product:
∞
a (R)
pk
ζ(R,s)= E (R,s), where E (R,s):= .
p p pks
p∈P k=0
Y X
If R˜ ⊆R is a subring of finite index, then for almost all prime numbers the Euler-
factorsE (R,s)andE (R˜,s)coincide,andthereforethezetafunctionsζ(R,s)and
p p
ζ(R˜,s) have the same convergence behaviour, if these finitely many Euler factors
do not behave very abnormally. We will have to control this behaviour, when it
comes to applications of this observation.
This paper is organized as follows. In Section 2, we describe the construction
oflattices from ideals in rings of integers of number fields andquotient polynomial
rings via the coefficient embedding. We then use the Dedekind zeta function to
COUNTING IDEALS IN POLYNOMIAL RINGS 3
obtaincountinganddensityestimatesonthenumbersofideallatticescomingfrom
a fixed ring of algebraic integers, improving on previous results [2] in some cases.
Our main result in this section (Theorem 2.3) is an asymptotic estimate on the
number of ideals in the quotient polynomial ring Z[X]/(f), where f(X) ∈ Z[X]
is a monic separable polynomial. We also briefly comment on the situation when
f(X)is not separable. In Section3, we discuss the questionof whichsublattices of
Zd arise as ideal lattices from quotient polynomial rings Z[X]/(f) for some monic
polynomial f of degreed. Specifically, we investigate the analytic propertiesof the
corresponding zeta function (Theorem 3.2) and, as a consequence of this theorem,
show that the number of such ideal sublattices of index ≤N grows asymptotically
like O(N2) (Corollary 3.3). Finally in Section 4, we prove Theorem 1.2 and use it
to deduce that the proportion of ideal lattices of index ≤N among all sublattices
of Zd for d≥3 tends to 0 as N →∞ (Corollary 4.2).
Remark 1.1. After finishing our calculations we became aware of the so-called
K¨ahler zeta-functions, in particular the results of Lustig in [12]. With some ad-
ditionalwork,Lustig’sresultscanbe usedtoproveourTheorem1.2. Hisapproach
however is purely local, while we employ more global methods. Furthermore, our
result is more specific and neither our Theorem 3.2 nor Remark 4.1 (b) seem to
have a parallel there.
2. Ideal lattices from fixed rings
In this section, we motivate our subsequent questions by looking at rings of
integers in number fields.
Let K be a number field of degree d > 1 with r real and r pairs of complex
1 2
conjugateembeddings(sod=r +2r ),andwriteO foritsringofintegers. There
1 2 K
exists θ ∈ O such that K = Q(θ). Fix this θ, then for each α ∈ K there exist
K
a ,...,a ∈Q such that
0 d−1
d−1
α= a θn.
n
n=0
X
NoticethatZ[θ] is afinite-index subringofO ,possibly proper. Define anembed-
K
ding ρ :K →Qd by
K
ρ (α)=(a ,...,a ),
K 0 d−1
then for every nonzero ideal I ⊆ O the image ρ (I) is a full-rank lattice in Rd,
K K
and for every ideal I ⊆ Z[θ] the image ρ (I) is a finite index sublattice of Zd.
K
Furthermore,ifsomefiniteindexsublatticeΛ⊆Zd isequaltoρ (I)forsomeideal
K
I ⊆O , we will say that Λ is an O -ideal lattice, or just an ideal lattice when the
K K
choiceof K and θ is fixed. The firstobservation(see equation(1) of [2]) is that for
each ideal I ⊆O , its norm is given by
K
ν(I)=D−1det(ρ (I)),
K K
where D :=det(ρ (O )).
K K K
For T ∈R , define
>0
N (T) = Λ⊆Zd :Λ=ρ (I) for some ideal I ⊆O , detΛ≤T
K K K
(1) ≤ (cid:12)(cid:8)I ⊆OK :ν(I)≤TDK−1 . (cid:9)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:8) (cid:9)(cid:12)
(cid:12) (cid:12)
4 LENNYFUKSHANSKY,STEFANKU¨HNLEIN,ANDREBECCASCHWERDT
Anupper boundonN (T)hasbeen obtainedinTheorem2 of[2],which wasthen
K
used in Corollary 1 of [2] to establish a density estimate on ideal lattices from O
K
among all sublattices of Zd. In the special case when O = Z[θ], one can easily
K
use the standard analytic method to prove an asymptotic formula for N (T) as
K
T →∞ and also deduce a more precise density estimate.
Lemma 2.1. Suppose that O =Z[θ]. As T →∞,
K
2r1+r2πr2h R
K K
(2) N (T)∼ T,
K
ω |∆ |
K K
where h is the class number, R the regulator, ∆ the discriminant, and ω the
K K p K K
number of roots of unity of the number field K.
Proof. Note that D = since O = Z[θ]. Furthermore, every ideal lattice from
K K
O isasublatticeofZd,andsothereisequalityinthe inequalityof (1). Lets∈C
K
and define the Dedekind zeta-function of K by
∞
ζ (s)= ν(I)−s = a n−s,
K n
I⊆XOK nX=1
where a is the number of ideals of norm n in O , and so, by (1),
n K
[T]
N (T)= a .
K n
n=1
X
It is well-known that ζ (s) is analytic in the half-plane ℜ(s) > 1 and has only a
K
simple pole at s=1 with residue given by the analytic class number formula:
2r1+r2πr2h R
K K
lim(s−1)ζ (s)= .
K
s→1 ω |∆ |
K K
Combining this formula with the Tauberian Theorem 1.1 yields the result. (cid:3)
p
Corollary 2.2. Define
M(T)= Λ⊆Zd :detΛ≤T .
Suppose that OK =Z[θ]. As T →(cid:12)∞(cid:8) , (cid:9)(cid:12)
(cid:12) (cid:12)
N (T) 2r1+r2πr2h R d
K ∼ K K T1−d.
M(T) ω |∆ | d ζ(n)
K K n=2
Proof. The zeta-function of all finitepindex sQublattices of Zd is
ζZd(s)= (detΛ)−s.
ΛX⊆Zd
It is a well-known fact (see, for instance p. 793 of [6]) that
ζZd(s)=ζ(s)ζ(s−1)···ζ(s−d+1),
where ζ(s) is the Riemann zeta-function. Hence ζZd(s) is analyticin the half-plane
ℜ(s)>d and has a simple pole at s=d with the residue
ζ(d)ζ(d−1)···ζ(2).
Combining this formula with Theorem 1.1 implies that
d
ζ(n)
M(T)∼ n=2 Td.
d
Q
COUNTING IDEALS IN POLYNOMIAL RINGS 5
This estimate together with Lemma 2.1 yields the result. (cid:3)
In the special case when O = Z[θ] there is also another way to look at ideal
K
lattices from O . Let f(X) be the minimal polynomial of θ, which is a monic
K
irreducible polynomial in Z[X] of degree d. Then OK ∼= Zf := Z[X]/(f(X)) and
ideal lattices from O correspondto ideal lattices in Z , which is a special case of
K f
theideallatticeconstructionof[13]. DefineaZ-moduleisomorphismρ :Z →Zd,
f f
given by
d−1
ρ a xn =(a ,...,a ).
f n 0 d−1
!
n=0
X
Let I ⊆ Z be an ideal, then it is known that ρ(I) is a finite index sublattice of
f
ρ(Z ) = Zd (Lemma 3 of [13]). A counting estimate on such ideal sublattices is
f
then given by our Lemma 2.1 above.
Thisleadsustothemoregeneralquestion: iff(X)∈Z[X]isamonicpolynomial
of degree d, what is the asymptotic behaviour of the number of ideals of index at
most N in Z :=Z[X]/(f) as N goes to infinity? We again define
f
ζ(Z ,s)= ν(I)−s,
f
IX⊆Zf
where the sum goes over finite index ideals and ν(I):=|Z /I|.
f
Theorem2.3. Letf ∈Z[X]bemonicandassumethatf =g ···g withg ,...,g
1 k 1 k
in Z[X] irreducible, monic and pairwise distinct. Then the zeta-function ζ (s)
f
converges for ℜ(s)>1, has a meromorphic extension to the halfplane
{s∈C|ℜ(s)>0},
and has apole of order k at s=1. In particular, A (Z )∼cN(logN)k−1 for some
N f
constant c.
Proof. For 1≤i≤k, let θ ∈C be a root of g and O ⊆C be the integral closure
i i i
ofZ[θ ]. We firstobservethat bya variantofthe Chinese RemainderTheorem,the
i
ring Z is isomorphic to a subring of finite index in S :=O ×···×O . If d is the
f 1 k
degree of f, then both rings have an additive group isomorphic to Zd. Let M be
the index of Z in S. We denote by a (Z ) the number of ideals of index n in Z
f n f f
and by a (S) the number of ideals of index n in S respectively.
n
Whenever I ⊆S is an ideal of index n in S, then MI ⊆Z is an ideal of index
f
Md−1·n in Z . As multiplication by M is injective on S, this multiplication map
f
from ideals in S to ideals in Z is injective, and for every natural number N
f
NMd−1 N
a (Z )≥ a (S).
n f n
n=1 n=1
X X
Thereforetheabscissaofconvergenceofζ(Z ,s)isatleastaslargeasthatofζ(S,s).
f
This only depends on the fact, that the rings have an additive group isomorphic
to Zd and therefore holds for all subrings of finite index in S. Furthermore, as
remarked in Section 1 the Euler-factors of both zeta-functions do coincide for all
prime numbers not dividing the index.
We nowconsiderEuler-factorsofthe ring Z+MS,andshowthatthey havethe
same convergence behaviour as Euler-factors for S. Since
Z+MS ⊆Z ⊆S,
f
6 LENNYFUKSHANSKY,STEFANKU¨HNLEIN,ANDREBECCASCHWERDT
it would follow that Euler-factors of Z must also have the same convergence be-
f
haviour. Writing M =p ·p ···p as a product of prime numbers, we see that
1 2 l
S ⊇Z+p S ⊇Z+p (Z+p S)=Z+p p S ⊇··· ,
1 2 1 1 2
and this shows by the same argument as above that it suffices to treat the case
M =p and the Euler-factor at the prime p.
Therefore let S be a ring additively isomorphic to Zd, p a prime number and
R=Z+pS.ForeveryidealI ofindexpk inR,SI isanidealofp-powerindexinS.
AspS ⊆R,weseethatpSI ⊆I andthattheindexofSI inS isatleastp−d·pk.For
two ideals I ,I in R the equality SI = SI implies pI ⊆ pSI = pSI ⊆ I and
1 2 1 2 2 2 1 1
by symmetry pI ⊆ I . Hence p2I ⊆ pI ⊆ I . The cardinality of the fibre of the
1 2 1 2 1
map I 7→ SI therefore is at most the number of subgroups in I /p2I ∼= Zd/p2Zd,
1 1
which is finite and independent of I . Call this number γ.
1
Then
K K+d
1
a (S)≥ a (Z+pS),
pk γ pk
k=0 k=0
X X
thereby showing that the Euler-factor in ζ(Z+pS,s) converges whenever that of
ζ(S,s) does. Coming back to our statement, the Euler-factors in the zeta-function
forO ×···×O arejusttheproductsoftheEuler-factorsofζ(O ,s),...,ζ(O ,s),
1 k 1 k
which all converge for ℜ(s)>0. This completes the proof. (cid:3)
Remark 2.1. What if f(X) is not separable? Here are some speculations. Going
through the proof of Theorem 2.3, it is sufficient to understand the zeta-functions
for rings of the type Z[X]/(fe), where f is monic and irreducible. We calculated
this for f(X)=X,e=2 and obtained
ζ(Z[X]/(X2),s)=ζ(s)·ζ(2s−1),
which again has a double pole at s=1. We expect
ζ(Z[X]/(Xe),s)=ζ(s)·ζ(2s−1)·ζ(3s−2)···ζ(es−(e−1)),
which has a pole of order e at s = 1. More generally, we would expect that the
order of the pole of ζ(Z[X]/(f),s) at s = 1 is the sum of the multiplicities of the
irreducible monic factors of f.
3. Ideal lattices in dimension d
In this section, we aim to understand which subgroups of Zd arise as images of
finite index ideals in Z :=Z[X]/(f) under ρ for some monic polynomial f(X) of
f f
degree d. Again, we fix the group isomorphism ρ :Z →Zd, given by
f f
d−1
ρ a Xn =(a ,...,a ).
f n 0 d−1
!
n=0
X
LikeeveryfiniteindexsubgroupofZd,suchasubgroupisgeneratedbythecolumns
of an integral full-rank upper triangular matrix
a a ... a
0,0 0,1 0,d−1
0 a ... a
1,1 1,d−1
A:= ... ... ... ... ,
 
 0 ... 0 ad−1,d−1
 
COUNTING IDEALS IN POLYNOMIAL RINGS 7
the j-th column corresponding to a generator of degree j in the ideal.
Multiplication by X is an endomorphism of the ideal. This shows that for each
0 ≤ j ≤ d−2 the j-th column of A shifted down by one belongs to the subgroup
generated by the 0-th to (j +1)-th column. For j = d−1 it shows, that there
exists some monic poynomial f of degree d such that the (d−1)-th column shifted
by one (in Zd+1) belongs to the subgroup generated by the columns of A and the
column containing the coefficients of f. This last demand is always satisfied, and
we therefore only have to deal with the first set of conditions!
Wecallsuchamatrixanidealizing matrix,andtwoidealizingmatricesarecalled
indifferent,whenevertheycorrespondtothesameideal. Onecancheckbyinduction
on d that an idealizing upper triangular matrix satisfies:
(3) ∀1≤j ≤d−1:a divides a whenever 0≤i,k≤j.
j,j i,k
Details of this calculation can be found in [16]. The idealizing matrices for which
in everyi-throw the entries are between0 anda −1 give a set of representatives
i,i
for the indifferency-classes. We callthese matrices reduced idealizing matrices, and
now count them by induction in the following way.
Lemma 3.1. Let A ∈ Zd×d be a reduced idealizing matrix. Then the number of
β ∈Zd such that
A β
A˜ := ∈Zd+1×d+1
0 1
(cid:18) (cid:19)
is a reduced idealizing matrix is ad .
d−1,d−1
Proof. As the first d−1 columns of A˜ already satisfy the conditions imposed on
idealizing matrices, we only have to care about the condition that the shifted d-
th column is contained in the Z-span of the columns of (δ) with δ ∈ Zd−1,a =
a
a ∈Z.
d−1,d−1
Then our condition is
0
−aβ ∈AZd.
δ
(cid:18) (cid:19)
Note that by (3) a divides every entry of A and that therefore we get
1 0
β ∈ +AZd .
a δ
(cid:18)(cid:18) (cid:19) (cid:19)
Two different values of β define the same indifferency class if and only if their
difference is in AZd. As the index of AZd in 1AZd is ad, we getexactly ad possible
a
indifferency classes of matrices A˜. (cid:3)
Theorem3.2. Letd∈Nbesomenaturalnumber,anddenotebyc(d) thenumberof
n
subgroups in Zd of index n which are in the image of ρ for some monic polynomial
f
f ∈Z[X] of degree d. Then
∞ c(d) d−1
ζ(d)(s):= n = ζ(i(s−1)) ·ζ(ds),
ns
" #
n=0 i=1
X Y
where ζ =ζ(1) is the Riemann zeta-function.
Proof. Theproofisbyinductionond. We havetocountindifferency classesofma-
triceswithgivendeterminantn. Thediagonalentriesofsuchamatrixarenumbers
8 LENNYFUKSHANSKY,STEFANKU¨HNLEIN,ANDREBECCASCHWERDT
a ,...,a ,whereeacha divides thea withj <i.Itismoreconvenienttowrite
0 d−1 i j
these numbers as products
b b ···b , b ···b , ..., b b , b ,
1 2 d 2 d d−1 d d
where b ,...,b are any natural numbers.
1 d
For d = 1, we just have one indifferency class of matrices of determinant n for
every n, hence
ζ(1)(s)=ζ(s).
For d = 2 and every value of b , by Lemma 3.1, we obtain exactly b classes of
1 1
matrices
b β
1 ,
0 1
(cid:18) (cid:19)
andeverymatrix witharbitraryb ,b is b times oneofthese, thereforewe haveb
1 2 2 1
matrices with fixed b . This gives
1
b 1 1
ζ(2)(s)= 1 = · =ζ(s−1)×ζ(2s).
bs·b2s bs−1 b2s
bX1,b2 1 2 Xb1 1 Xb2 2
Going onlike this, using Lemma 3.1repeatedly, we find that for fixed b ,...,b we
1 d
have b ·b2·b3·bd−1 indifferency classes of idealizing matrices with diagonal
1 2 3 d−1
(b ···b , b ···b , ..., b b , b ),
1 d 2 d d−1 d d
and this leads to
∞ b ·b2·b3···bd−1 d−1
ζ(d)(s)= 1 2 3 d−1 = ζ(i(s−1))×ζ(ds).
(b ·b2·b3···bd−1·bd)s
b1,.X..,bd=1 1 2 3 d−1 d iY=1
(cid:3)
Corollary 3.3. For d ≥ 2, the number A of subgroups in Zd of index less than
N
or equal to N, which are in the image of ρ for some monic polynomial f ∈ Z[X]
f
of degree d, grows asymptotically as cN2, where
Res(ζ(d),2) d−1
c= = ζ(i) ·ζ(2d).
2
" #
i=2
Y
Proof. Apply the Tauberian Theorem 1.1 to ζ(d). The abscissa of convergence is 2
and Γ(2) = 1; use the fact that the Riemann zeta-function ζ(s) has a simple pole
at s=1 and no other poles in C. (cid:3)
4. Zeta-functions of Polynomial Rings
In this section we prove Theorem 1.2. We start with its versionfor K =Q, and
then explain how to generalize it to any number field.
Proposition 4.1. We have
∞
ζ(Z[X],s)= ζ(i(s−1)).
i=1
Y
COUNTING IDEALS IN POLYNOMIAL RINGS 9
Proof. Every ideal of finite index in Z[X] contains some monic polynomial, and
therefore a (R) = lim ad. The discussion in the previous section shows that
n d→∞ n
for a fixedn the sequence (ad) becomes constantfor d>n (very roughestimate).
n d
This leads to
ζ(Z[X],s)= lim ζ(d)(s),
d→∞
which gives the desired result. The infinite product on the right hand side in fact
converges absolutely for ℜ(s)>2. (cid:3)
Corollary 4.2. For d≥3, the proportion of the ideal lattices of index ≤N in Zd
among all subgroups of index ≤N tends to zero as N goes to infinity.
Proof. The zeta-function ζ(Z[X],s) converges for ℜ(s) > 2 and has a simple pole
at s=2. Theorem1.1 says that the number of ideal lattices has quadratic growth,
while that for all subgroups has growth of order Nd. (cid:3)
Proof of Theorem 1.2. Wenowexplainhowthecalculationsintheproofofthelast
Propositioncan be extended to rings of integers in arbitrary number fields instead
of Z.
(a) Let R be an infinite principal ideal domain such that every nonzero ideal
in R has finite index. Here, the method using idealizing matrices can be
extendeddirectly. Repeatingmutatismutandistheargumentsfromthelast
section shows the formal identity
∞
ζ(R[X],s)= ζ(R,i(s−1)).
i=1
Y
(b) If O is the ring of integers in a fixed number field K of finite degree over
K
Q, we use the Euler-product
ζ(OK[X],s)= ζ(Z(p)⊗ZOK[X],s),
p∈P
Y
as the ideals of p-power index in O [X] correspond bijectively and index
K
preservingly to ideals of finite index in Z(p)⊗ZOK. But this last ring is a
Dedekind domain with finitely many prime ideals only (i.e. semilocal) and
thereforeisaprincipalidealdomainwhichsatisfiesthe conditionfrompart
(a) of this proof. Hence
∞
ζ(OK[X],s)= ζ(Z(p)⊗ZOK,i(s−1)).
p∈Pi=1
YY
For every prime number p, the factor ζ(Z(p) ⊗Z OK,i(s−1)) is the pro-
duct of the “true” Euler factors of ζ(O ,i(s−1)) for prime ideals in O
K K
containingp,andtherefore,usingtheEuler-decompositionoftheDedekind
zeta-function ζ(O ,·), we arrive at the assertion of Theorem 1.2.
K
(cid:3)
Remark 4.1. We conclude this paper with some remarks.
(a) The first factor in ζ(O [X],s) is ζ(O ,s− 1), which is the Hasse-Weil
K K
zeta-function of the affine line A1 .
OK
10 LENNYFUKSHANSKY,STEFANKU¨HNLEIN,ANDREBECCASCHWERDT
This corresponds to those ideals in O [X] where the quotient ring has
K
squarefree characteristic. It would be interesting to understand the zeta-
function we have calculated from this point of view and perhaps describe
it as that of some deformation of the affine line.
Thereisanotherwaytodiscoverζ(O ,s−1)asafactorinζ(O [X],s),
K K
namely by counting only ideals which contain a linear monic polynomial.
(b) If a (G) denotes the number of isomorphism classes of abelian groups of
n
order n, then an argumentbasedon the structure theorem for finitely gen-
erated abelian groups shows, that
∞ a (G) ∞
n = ζ(ds)=ζ(Z[X],s+1).
ns
n=1 d=1
X Y
Namely, for every choice b ,...,b ∈ N there is exactly one abelian group
1 d
with elementary divisors b ,b b ,b b b ,...,b ·...·b . This group
d d d−1 d d−1 d−2 d 1
has order b ·b2·...·bd, and therefore we get
1 2 d
∞ a (G) 1
n
= lim
nX=1 ns d→∞b1X,...,bd (b1·b22···bdd)s
= lim ζ(s)·ζ(2s)···ζ(ds).
d→∞
This is closely related to the Cohen-Lenstra heuristics, cf. [4], although
there the product ζ(s+i) plays a more important role. However,the
i≥1
residue of our ζ(O [X],s) at s = 2 is the number C on page 35 of loc.
K ∞
Q
cit. We thank Ernst-Ulrich Gekeler for kindly guiding us towards [4].
Maybe it is possible by such means to obtain a nice interpretation of
ζ(Z[X],s)asthezeta-functioncountingfiniteabeliangroupsendowedwith
some endomorphism satisfying certain properties. The number a (G) is
n
the number of orbits of subgroups of index n in Z∞ = Z under the
i∈N
action of the automorphism group of Z∞. It would also be interesting to
L
count the number of orbits of Aut(Z[X]) acting on ideals of index n. We
did not yet carry out the corresponding calculations.
(c) We should mention the short note [18] of Witt. His results imply that the
sum formally defining the ζ-function for the polynomial ring Z[X,Y] does
not converge anywhere in the complex plane.
References
[1] M.Baake,R.Scharlau,andP.Zeiner.Well-roundedsublatticesofplanarlattices.ActaArith.,
166(4):301–334, 2014.
[2] J.A.Buchmann andR.Lindner.Densityof ideallattices. InJ. A.Buchmann, J.Cremona,
andM.E.Pohst,editors,AlgorithmsandNumberTheory,number09221inDagstuhlSeminar
Proceedings,Dagstuhl,Germany,2009.SchlossDagstuhl-Leibniz-ZentrumfuerInformatik,
Germany.
[3] C. Bushnell and I. Reiner. Zeta functions of arithmetic orders and Solomon’s conjectures.
Math. Z.,173(2):135–161, 1980.
[4] H.CohenandH.W.Lenstra.Heuristicsonclassgroupsofnumberfields.InNumberTheory,
Proceedings Journees Arithmetiques. Noodwijkerhout/Neth. 1983, Lect. Notes Math. 1068,
p.33-62.,1984.
[5] H. Delange. G´en´eralisation du th´eor`eme de ikehara. Annales scientifiques de l E´.N.S. 3e
s´erie, tome 71, no 3,pages 213–242, 1954.