J. Ramanujan Math. Soc. 20, No.2 (2005) 1–10
A variation on the solvable case of the Dedekind
conjecture
Joshua M. Lansky1 and Kevin M. Wilson2
1
American University, Washington, DC 20016 USA
2
University of Maryland, College Park, MD 20742 USA
Communicated by: V. Kumar Murty
Received:
Abstract. Let G be the Galois group of a solvable Galois extension K/F
of number fields. In this note, we demonstrate the holomorphy of certain
Artin L-functions attached to K/F , generalizing results of M. R. Murty
and A. Raghuram [1]. We also give a bound (generalizing one in [1]) on
the orders of certain Artin L-functions at an arbitrary point in the complex
plane by the order of a corresponding quotient of Dedekind zeta functions.
We deduce some corollaries on the possible orders of zeros of such quotients.
1991 Mathematics Subject Classification: Primary 11M06; Secondary
11M20
1. Introduction
Let E/F be a finite extension of number fields. A well-known conjecture of
Dedekind states that the quotient of Dedekind zeta functions ζE (s)/ζF (s) is
entire. The conjecture is known in the case where E/F is Galois due to the
work of Aramata and Brauer (see [3],[2]). In addition, Uchida [5] and van der
Waall [6] have given proofs of it if the Galois closure of E/F is solvable.
Moreover, the Dedekind conjecture is known to follow easily from Artin’s
conjecture on the holomorphy of Artin L-functions as ζE (s)/ζF (s) can be
factored into a product of such L-functions.
In [1], M. R. Murty and A. Raghuram prove several results that are closely
related to the Dedekind conjecture in the solvable closure case. For instance,
if K/F is a solvable extension and ψ is a one-dimensional character of a
subgroup H ⊂ Gal(K/F ), they establish the holomorphy of the quotient of
m(χ)
the Artin L-function L(s, IndG
, where χ is
H (ψ), K/F ) by L(s, χ, K/F )
any one-dimensional character of G, and m(χ) is a constant depending on χ
1
2
Joshua M. Lansky and Kevin M. Wilson
that is equal to 1 or 0. As discussed in Section 3, this generalizes the above
result of Uchida and van der Waall. They also prove
of the
Q the holomorphy
m(χ)
quotient of L(s, IndG
(ψ),
K/F
)
by
the
product
L(s,
χ,
K/F
)
as χ
χ
H
varies over all one-dimensional characters of G.
Let Gi denote the ith term in the derived series of G. Given an irreducible
character χ of G, Murty and Raghuram define the level of χ to be
l(χ) = min {i : χ|Gi = χ (1) · 1Gi } ,
where 1Gi denotes the trivial character of Gi . They conjecture that the
preceding holomorphy result generalizes to the quotient of the function
L(s, IndG
H (ψ), K/F ) by the analogous product over all irreducible characters
of G of level ≤ i. (Note that the preceding result constitutes the special case
of this conjecture in which i = 1.) They prove the conjecture for certain characters ψ. In Section 3, we give a fully general proof. We also present a similar
result on the product over characters of level equal to i, as well as a corollary
on the possible order of ζK (s)/ζK i (s) at s = s0 ∈ C, where K i is the fixed
field of Gi .
Murty and Raghuram use the i = 1 case of the above result to prove a bound
on the orders of non-abelian L-functions at a fixed point in C by the order of a
certain quotient of zeta functions. Let s0 ∈ C. Continuing our assumption that
K/F is solvable, let n(G, χ ) denote the order of L(s, χ, K/F ) at s0 . Then
Theorem 5.3 in [1] states that
2
X
ζK (s)
2
n(G, χ ) ≤ ords0
,
(1)
ζK ab (s)
b
χ ∈G
l(χ)>1
b is the set of irreducible characters of G and K ab is the maximal abelian
where G
subextension of K/F . This improves on a similar bound due to Foote and
Murty [7, §3], which involves a sum over all irreducible characters, but which
is valid more generally for K/F Galois (see Section 4). Murty and Raghuram
conjecture an inequality analogous to (1) with a sum over irreducible characters
of level greater than i. In Section 4, we prove a refinement of this conjecture
using the results of Section 3. We also state some corollaries generalizing those
in [1].
We thank M. Ram Murty and A. Raghuram for helpful communications.
We also thank the referee for corrections and useful suggestions.
2. Preliminaries and notation
Let G be a finite group. We will denote the set of irreducible characters of G by
b If H is any subgroup of G and ψ a character of H , then IndG ψ will denote
G.
H
The Dedekind conjecture
3
the representation of G induced from ψ. Let 1G denote the trivial character of
G. We recall the elementary fact that
X
IndG
1
=
χ(1)χ.
(2)
{1}
{1}
b
χ ∈G
More generally, a character χ of a group G is called monomial if χ = IndG
H (ψ)
for some one-dimensional character ψ of a subgroup H of G.
If f, f 0 : G → C, we let
hf, f 0 i =
1 X
f (g)f 0 (g)
|G| g∈G
1/2
be the usual inner product. Let |f | = hf, f iG .
Denote by Gi the ith term in the derived series of G. That is, G0 = G, and
for all i ≥ 1, Gi = [Gi−1 , Gi−1 ]. Recall that G is solvable if and only if this
series terminates with the trivial subgroup. In this case, as in [1], we define the
level of an irreducible character χ of G to be
l(χ) = min {i : χ |Gi = χ (1) · 1Gi } .
We denote by S i the set of irreducible characters of G of level less than or
equal to i.
We record here a useful result from [1] on characters of solvable groups.
Lemma 2.1. Let G be a solvable group, and let H be a subgroup of G. Then
for all i ≥ 1,
X
G
IndG
IndG
H 1H = IndH Gi 1H Gi +
Hj ψj ,
j
where j runs over some possibly empty indexing set, the Hj are subgroups of
G, and ψj is a one-dimensional character of Hj .
Let K/F be a Galois extension of number fields with Galois group G. For
any subgroup H of G, we will denote the fixed field of H by K H . If K/F is
i
solvable, we will write K i for K G .
Let χ be a character of G. We denote by L(s, χ, K/F ) the Artin L-function
associated to χ . We now give a brief review of the basic properties of Artin
L-functions (see [3]).
Proposition 2.2. (E. Artin). Let K/F be a Galois extension of number fields
with Galois group G. Let H be a subgroup of G. Let χ , χ1 , and χ2 be characters
of G. Let ψ be a character of H .
4
Joshua M. Lansky and Kevin M. Wilson
(i) L(s, χ1 ⊕ χ2 , K/F ) = L(s, χ1 , K/F )L(s, χ2 , K/F ) for characters χ1
and χ2 of G.
(ii) L(s, ψ, K/K H ) = L(s, IndG
H (ψ), K/F ) for any character ψ of H .
(iii) If χ is a non-trivial one-dimensional character of G, then L(s, χ, K/F )
extends to an entire function.
(iv) If χ is a monomial character not containing the trivial character then
L(s, χ, K/F ) extends to an entire function of s.
Note that (ii) and (iii) (which follows from Artin reciprocity) immediately
imply (iv). Also, observe that the Dedekind zeta function
ζF (s) = L(s, 1G , K/F ).
(3)
ζK (s) = L(s, IndG
{1} 1{1} , K/F ),
(4)
and so by (2) and Proposition 2.2 (i), we have
Y
ζK (s) =
L(s, χ , K/F )χ(1) ,
(5)
By (ii),
b
χ ∈G
the Artin-Takagi factorization.
3. Generalization of a theorem of Uchida and van der Waall
Uchida [5] and van der Waall [6] independently proved the Dedekind conjecture in the case where E/F has solvable Galois closure. In [1], Murty and
Raghuram prove the following generalization of the theorem of Uchida and
van der Waall.
Theorem 3.1. (M. R. Murty–A. Raghuram). Let G be the Galois group of a
solvable Galois extension K/F of number fields. Let χ be a one-dimensional
character of G. Then for every subgroup H of G and for every one-dimensional
character ψ of H , the quotient
L(s, IndG
H (ψ), K/F )
G
L(s, χ , K/F )hχ ,IndH ψiG
(6)
has no poles at s = s0 6= 1.
Observe that the theorem of Uchida and van der Waall follows from the special case where K/F is the (solvable) Galois closure of E/F , H = Gal(K/E),
and both ψ and χ are trivial. For then, (6) is equal to ζE (s)/ζF (s) by Equations
(3) and (4). (Of course, the simple poles of ζE (s) and ζF (s) at s = 1 cancel.)
The Dedekind conjecture
5
Murty and Raghuram show further that holomorphy at s0 6= 1 still holds
for the quotient of L(s, IndG
H (ψ), K/F ) by the product of the functions
G
L(s, χ, K/F )hχ ,IndH ψiG as χ ranges over the set of all one-dimensional characters of G, i.e., characters of level not greater than 1. This result is used to
prove bounds on orders of certain Artin L-functions (see Section 4). Moreover,
they conjecture that holomorphy also holds for the quotient by the analogous
product over all characters of level not greater than i. They prove this conjecture when ψ extends to a character of G. We deal with the general case in
Theorem 3.4.
Lemma 3.2. Let G be a solvable group. Let ψ be a one-dimensional character of a subgroup H of G such that ψ|H ∩Gi is trivial. Let ψ 0 be the unique
extension of ψ to a character of H Gi that is trivial on Gi . Then, for any irreducible character χ of G,
(
hχ, IndG
H (ψ)iG , if l(χ ) ≤ i,
G
0
hχ, IndH Gi (ψ )iG =
0,
if l(χ ) > i.
Proof. A straightforward Mackey theory argument shows that the restriction
of IndG
(ψ 0 ) to Gi is a multiple of the trivial character 1Gi . This proves the
H Gi
lemma in the case l(χ) > i.
Now suppose l(χ) ≤ i. By Frobenius reciprocity, it suffices to show
hχ |H Gi , ψ 0 iH Gi = hχ |H , ψiH .
(7)
Let (σ, V ) be a representation of G with character χ . Since both σ and ψ 0 are
trivial on Gi , it follows that
HomH Gi (σ |H Gi , ψ 0 ) = HomH (σ |H , ψ).
Equation (7) now follows by taking the dimensions of these two spaces over
C.
2
For the following corollary, we recall the set S i of irreducible characters of
G of level less than or equal to i.
Corollary 3.3. Let d be the greatest common divisor of the degrees of the
b r S i . Then ords0 (ζK (s)/ζK i (s)) = kd for some nonnegative
characters in G
integer k.
Proof. By Proposition 2.2 (i) and (ii), and Lemma 3.2 (with H = {1}),
Q
Q
χ (1)
χ(1)
b L(s, χ , K/F )
b L(s, χ, K/F )
ζK (s)
χ ∈G
χ∈G
Q
.
=
=
χ(1)
ζK i (s)
L(s, IndG
(1 i ), K/F )
χ∈S i L(s, χ, K/F )
Gi G
6
Joshua M. Lansky and Kevin M. Wilson
Thus
ords0
ζK (s)
ζK i (s)
X
=
χ(1) ords0 (L(s, χ, K/F ))
b i
χ ∈GrS
is a multiple of the greatest common divisor of the degrees χ (1) (l(χ ) > 1). It
must be a nonnegative multiple by the theorem of Aramata and Brauer.
2
Theorem 3.4. Let K/F be a solvable extension of number fields, and let
G = Gal(K/F ). Let ψ be a one-dimensional character of a subgroup H of
G. Then
Y
G
G
L(s, IndH (ψ), K/F )
L(s, χ, K/F )hχ,IndH (ψ)iG
(8)
b
χ ∈G
l(χ)≤i
is an entire function of s.
Proof. First assume that ψ is trivial on H ∩ Gi . Then ψ extends uniquely to
a character ψ 0 of H Gi that is trivial on Gi . Lemma 3.2 then implies that the
denominator in (8) is equal to
0
L(s, IndG
H Gi (ψ ), K/F ).
Hence, by Proposition 2.2 (iv), it suffices to show that
G
0
IndG
H (ψ) − IndH Gi (ψ )
is a sum of monomial characters not containing the trivial representation.
Since H Gi is solvable, Proposition 2.1 implies that
X
Gi
Gi
IndH
IndH
H (1H ) = 1H Gi +
Hj (ψj )
j
where the Hj are subgroups of H Gi and ψj is a nontrivial one-dimensional
character of Hj . Tensoring both sides of this equation by ψ 0 and using the fact
that the tensor product “commutes” with induction, we obtain
X
Gi
0
Gi
0
IndH
(ψ)
=
ψ
+
IndH
H
Hi (ψi · ψ |Hj ).
j
Inducing from H Gi to G yields
G
0
IndG
H (ψ) = IndH Gi (ψ ) +
X
j
0
IndG
Hj (ψj · ψ |Hj ).
(9)
It remains to show that none of the terms in the above summation contain the
trivial character.
The Dedekind conjecture
7
If ψ 6= 1H , then
h1G , IndG
H (ψ)iG = h1H , ψiH = 0,
and thus 1G does not occur in (9). If ψ = 1H , then Lemma 3.2 implies that
1G occurs in IndG
(ψ 0 ) with the same multiplicity with which it occurs in
H Gi
G
IndH (ψ). Hence, it follows again that the summation in (9) does not contain
1G . This proves the theorem in the case where ψ|H ∩Gi is trivial.
Assume now that ψ is nontrivial on H ∩ Gi . For x ∈ G, denote by ψ x
the character of xH x −1 ∩ Gi given by g 7→ ψ(x −1 gx). Then, by Mackey’s
theorem and Frobenius reciprocity,
X
i
x
i
i
hIndG
hIndG
H (ψ)|Gi , 1Gi iGi =
xH x −1 ∩Gi (ψ ), 1G iG
x∈Gi /G\H
X
=
hψ x , 1xH x −1 ∩Gi ixH x −1 ∩Gi
x∈Gi /G\H
X
=
hψ, 1H ∩Gi iH ∩Gi
x∈Gi /G\H
= 0.
Hence IndG
H (ψ) contains no characters of level less than or equal to i. It follows that the quotient (8) is equal to L(s, IndG
H (ψ), K/F ), which is entire by
Proposition 2.2 (iv).
2
Corollary 3.5. Let ψ0 be a one-dimensional character of a subgroup H of
G, and let S be the set of irreducible characters of level i occuring in IndG
H (ψ0 ).
Then, the product of L-functions
Y
G
L(s, χ , K/F )hχ,IndH (ψ0 )iG
χ ∈S
is entire.
Proof. If ψ0 is nontrivial on H ∩ Gi , then, by the last paragraph of the proof of
Theorem 3.4, we have hχ, IndG
H (ψ0 )iG = 0 for any χ ∈ S, and the corollary
follows. Otherwise, ψ0 extends uniquely to a character ψ of H Gi trivial on
Gi . Then Theorem 3.4 implies that
Y
G
G
L(s, IndH Gi (ψ), K/F )
L(s, χ, K/F )hχ,IndH Gi (ψ)iG
χ ∈S i−1
is entire. By Lemma 3.2 and Proposition 2.2 (i), this is equal to
Y
Y
G
hχ ,IndG
(ψ0 )iG
H
L(s, χ , K/F )
L(s, χ, K/F )hχ,IndH (ψ0 )iG
χ∈S i
χ ∈S i−1
=
Y
χ ∈S
G
L(s, χ, K/F )hχ,IndH (ψ0 )iG .
2
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Joshua M. Lansky and Kevin M. Wilson
4. A bound on the orders of L-functions
Let K/F be a Galois extension of number fields with Galois group G. Fix
b let n(G, χ ) be the order of L(s, χ, K/F ) at s0 . In [7],
s0 ∈ C. For χ ∈ G,
R. Foote and V. K. Murty prove the following bound on the orders of Artin
L-functions at a point in the complex plane.
Theorem 4.1. (R. Foote–V. K. Murty). If K/F a Galois extension of number
fields with Galois group G, and s0 ∈ C, then
X
2
n(G, χ )2 ≤ ords0 ζK (s) .
b
χ ∈G
The proof of this theorem makes use of the Heilbronn character 2G , a virtual
character of G defined by
X
2G (g) =
n(χ , G)χ(g) (g ∈ G).
b
χ ∈G
In [1], M. R. Murty and A. Raghuram use a truncated version of 2G (whose
b r S 1 ) to obtain the inequality
index of summation runs over G
2
X
ζK (s)
2
n(G, χ ) ≤ ords0
.
ζK 1 (s)
b 1
χ ∈GrS
This is an improvement upon Theorem 4.1 in the solvable case if one is interested in the holomorphy of Artin L-functions since the L-functions omitted
in the above sum are precisely the abelian ones, which are known to be entire
(Proposition 2.2 (iii)). Murty and Raghuram conjecture an analogous bound
involving the quotient ζK (s)/ζK i (s). Below, we state and prove a refinement
of this conjecture.
Theorem 4.2. Let K/F be a solvable extension of number fields, and let
G = Gal(K/F ). Then for i ≥ 1,
2
X
ζK i (s)
2
n(G, χ ) ≤ ords0
.
(10)
ζK i−1 (s)
b
χ ∈G
l(χ)=i
Proof. The proof is analogous that of Theorem 5.3 in [1]. Let
X
2iG =
n(G, χ)χ.
b
χ ∈G
l(χ)=i
Then the left-hand side of (10) is equal to
1 X i
|2iG |2 =
|2 (g)|2 .
|G| g∈G G
(11)
The Dedekind conjecture
9
For g ∈ G,
|2iG (g)| ≤
X
n(G, χ )|χ (g)|
b
χ ∈G
l(χ)=i
≤
X
χ(1)n(G, χ)
b
χ ∈G
l(χ)=i


Y

L(s, χ, K/F )χ(1)  .
= ord 
s=s0
(12)
b
χ ∈G
l(χ)=i
Now by Lemma 3.2, Proposition 2.2, and Equation (5), we have
Y
L(s, χ , K/F )χ (1) = ζK (s)/ζK i (s).
χ ∈S
/ i
An analogous equality holds for S i−1 and K i−1 . Since S i r S i−1 is the set of
b of level i, it follows that (12) is equal to
characters in G
 Y

L(s, χ , K/F )χ (1)
 χ ∈S

ζK (s)/ζK i−1 (s)
/ i−1


Y
ord
= ords0
s=s0 
ζK (s)/ζK i (s)
L(s, χ , K/F )χ (1) 
χ ∈S
/ i
= ords0
ζK i (s)
ζK i−1 (s)
The theorem now follows from (11).
2
We conclude this section with the following corollaries.
Corollary 4.3.
Let i ≥ 1. With notation as above, if
ords0 (ζK i (s)) = ords0 (ζK i−1 (s)),
then n(G, χ) = 0 for every irreducible character χ of G of level i.
Corollary 4.4. Let i ≥ 2. With notation as above, ζK i (s)/ζK i−1 (s) cannot
have any poles or simple zeros.
Proof. The proof of this corollary is analogous to that of [1, Cor. 5.1], which
concerns the quotient ζK (s)/ζK 1 (s). The corollary also follows directly from
the latter by applying it to the extension K i /K i−2 (for i ≥ 2).
2
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Joshua M. Lansky and Kevin M. Wilson
References
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J. Ramanujan Math. Soc., 15 (2000) no. 4, 225–245.
[2] E. de Shalit, Artin L-functions, An Introduction to the Langlands Program, 73–87,
Birkhäuser, 2003, ed. by J. Bernstein and S. Gelbart, ch. 4, Boston.
[3] J. Martinet, Character theory and Artin L-functions, Algebraic number fields: Lfunctions and Galois properties, 1–87, (1977) Proc. Symp., Univ. Durham, Durham,
1975, Academic Press, London.
[4] H. Heilbronn, On real zeros of Dedekind ζ -functions, Canadian J. Math., 25 (1973)
870–873.
[5] K. Uchida, On Artin L-functions, Tohuku Math. J., 27 (1975) no. 2, 75–81.
[6] R. van der Waall, On a conjecture of Dedekind on zeta functions, Indag. Math., 37
(1975) 83–86.
[7] R. Foote and V. K. Murty, Zeros and poles of Artin L-functions, Math. Proc. Camb.
Phil. Soc., 105 (1989) 5–11.