The Rubin-Stark Conjecture For Imaginary Abelian Fields Of Odd Prime Power Conductor

Abstract

We build upon ideas developed in [9], as well as results of Greither on a strong form of Brumer's Conjecture ([2]--[4]), and prove Rubin's integral version of Stark's Conjecture for imaginary abelian extensions of Q of odd prime power conductor.

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Page 1

THE RUBIN–STARK CONJECTURE FOR IMAGINARY
ABELIAN FIELDS OF ODD PRIME POWER CONDUCTOR
Cristian D. Popescu1
Abstract.
a strong form of Brumer’s Conjecture ([2]–[4]), and prove Rubin’s integral version
of Stark’s Conjecture for imaginary abelian extensions of Q of odd prime power
conductor.
We build upon ideas developed in [9], as well as results of Greither on
INTRODUCTION
In the present paper, we prove Rubin’s integral version (Conjecture B, [10], §2.1)
of Stark’s general conjecture (see Conjecture 5.1 in [12], Chap. I), for abelian
extensions of type K/Q, where K is an abelian, imaginary number field of odd
prime power conductor ℓn. These extensions belong to the class of “nice” CM
extensions of totally real number fields, introduced by Greither in [4]. In [9], we used
results obtained by Greither in [4] on the “odd part” of a strong form of Brumer’s
Conjecture, and settled Rubin’s Conjecture up to a power of 2, for general “nice”
extensions. In this paper, we restrict ourselves to the subclass of “nice” extensions
K/Q described above and, by using ideas developed in [9], as well as results of
Greither ([2] and [3]) on the 2–part of the strong Brumer conjecture, we completely
settle Rubin’s Conjecture in this case.
Unlike in [9], where we attack the “odd” part of Rubin’s Conjecture “one prime
at a time”, all the arguments in this paper are global in nature. This is made
possible by the crucial observation that, for general abelian CM extensions of totally
real number fields K/k, one can state a conjecture equivalent to Rubin’s, over the
ring Z[Gal(K/k)]/(1 + j), rather than Z[Gal(K/k)] itself (see Conjecture B− in
§2 below), where j is the complex conjugation automorphism of K. As it turns
out, for K/Q as above, many arithmetically meaningful Z[Gal(K/Q)] –modules
which, in general, are not of finite projective dimension over Z[Gal(K/Q)], have
projective dimension at most 1 over Z[Gal(K/Q)]/(1 + j), after tensoring with
Z[Gal(K/Q)]/(1+j). We take full advantage of this fact and prove Conjecture B−
in the context described above, and conclude that the equivalent Conjecture B of
Rubin is also true.
The paper is organized as follows. In §1, we introduce the notations and main
definitions, state Rubin’s Conjecture B and list some of its functoriality properties.
In §2, we state the equivalent Conjecture B− for general abelian CM extensions
of totally real number fields. In §3, we describe results of Greither on the strong
Brumer Conjecture which are of relevance in this context, and draw several conclu-
sions which are used in the subsequent sections. In §4, we prove Conjecture B−for
1991 Mathematics Subject Classification. 11R42, 11R58, 11R27.
1Research on this project was partially supported by NSF grants DMS–9801267 and DMS–0196340
1

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2CRISTIAN D. POPESCU
extensions K/Q as described above, under certain additional hypotheses. In §5, we
use the results of §4 and some functoriality properties stated in §1 to prove Rubin’s
Conjecture B for the class of extensions K/Q under consideration.
Acknowledgements. We would like to thank Cornelius Greither for helpful
conversations and especially for sharing his results with us prior to publication.
1. PRELIMINARY CONSIDERATIONS
1.1 Notation. In what follows, Z(p)will denote the localization of Z at its prime
ideal pZ, for all prime numbers p.Let us assume that M is an arbitrary Z–
module. Then?
for any prime number p, M(p):= Z(p)M. If G is a finite, abelian group, then
?G denotes the set of its irreducible, complex–valued characters. For any χ ∈?G,
C[G]. Also, if M is a Z[G]–module, ∧r
M, will denote the quotient of M by its submodule of Z–torsion
points. If A is an arbitrary (commutative) ring, AM := A ⊗ZM. In particular,
eχ:= 1/|G|?
Let us assume that K/k is a finite, abelian extension of number fields, of Galois
group G := Gal(K/k). Let S be a finite set of primes in k, which contains the
infinite primes as well as the ones which ramify in K/k, and let SK be the set of
primes in K, sitting above primes in S. Then, USand ASwill denote respectively
the group of units and the ideal–class group of the ring of SK–integers OSof K. If
T is an additional finite, non–empty set of primes in k, disjoint from S, then US,T
denotes the subgroup of US consisting of S–units congruent to 1 modulo every
prime in TK, and AS,T denotes the (S,T)–ideal-class group of K, as defined in
[10],§1.1. For a finite prime w in K, we denote by K(w) its residue field. Then, as
pointed out in [10], §1.1, we have an exact–sequence of Z[G]–modules
σ∈Gχ(σ−1) · σ is the idempotent associated to χ in the group–ring
GM := ∧r
Z[G]M.
(0)0 − → US,T− → US
resT
−−→ ∆T− → AS,T− → AS− → 0,
where ∆T := ⊕w∈TKK(w)×, and resT(x) := (xmodw; w ∈ TK), for all x ∈ US. As
in [10], to any S, T and χ as above, one can associate a meromorphic L–function
LS(s,χ) of complex variable s, holomorphic away from s = 1, and an everywhere
holomorphic L–function LS,T(s,χ) :=?
v in T. As shown in [12], Chapitre I, §3, the orders of vanishing at s = 0 of these
L–functions, rχ,S:= ords=0LS(s,χ) = ords=0LS,T(s,χ), are given by
?card{v ∈ S |χ|Gv= 1Gv}, if χ ?= 1G
where Gv is the decomposition group of v relative to K/k. Now, we combine the
L–functions above, to obtain the associated Stickelberger functions
?
These are viewed as C[G]–valued, meromorphic functions, of complex variable s.
For any Z[G]–module M with trivial Z–torsion and any positive integer r, we let
v∈T(1 − χ(σ−1
v)Nv1−s) · LS(s,χ), where
Nv = |k(v)|, and σvis the Frobenius morphism associated to v in G, for all primes
(1)rχ,S:=
card(S) − 1, if χ = 1G
,
ΘS(s) :=
χ∈? G
LS(s,χ−1) · eχ,ΘS,T(s) :=
?
χ∈? G
LS,T(s,χ−1) · eχ.
Mr,S:= {x ∈ M |eχ· x = 0 in CM ,∀χ ∈?G, such that rχ,S?= r}.

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THE RUBIN-STARK CONJECTURE3
1.2 Rubin’s Conjecture. Let K/k, S and T be as in §1.1 and let r be a positive
integer. Assume that the set of data (K/k, S, T, r) satisfies the following group of
extended hypotheses.


Under these extra–hypohteses, equalities (1) above imply right away that rχ,S≥ r,
for all χ ∈?G. Let
viewed as an element in C[G]. In this context, the main purpose of Stark’s Conjec-
ture and its integral version due to Rubin, is an interpretation of the (analytically
defined) Θ(r)
S,T(0) in terms of the arithmetic of K/k.
For every r–tuple (φ1,...,φr) ∈ HomZ[G](US,T, Z[G])r, one can view each φi
as an element of HomC[G](CUS,T,C[G]), and define a C[G]–morphism
(H)







S contains all the infinite primes of k.
S contains all the primes which ramify in K/k.
S contains at least r primes which split completely in K/k.
|S| ≥ r + 1.
T ?= ∅,S ∩ T = ∅,US,T∩ µK= {1}.
Θ(r)
S,T(0) := lim
s→0s−rΘS,T(s),
C ∧r
GUS,T
φ1∧···∧φr
− − − − − − → C[G] ,
such that, for every u1,...,ur∈ US,T, one has
φ1∧ ··· ∧ φr(u1∧ ··· ∧ ur) :=det
1≤i,j≤r(φi(uj)) .
Definition 1.2.1. Assuming that (K/k, S, T, r) satisfies hypotheses (H), let ΛS,T
be the Z[G]–submodule of Q ∧r
GUS,T, defined by


If r ≥ 1, let us fix an r–tuple V := (v1,v2,...,vr) of r–distinct primes in S,
which split completely in K/k, and let W := (w1,w2,...,wr), with wi prime in
K, sitting above vi, for all i = 1,...,r. We associate to every wi an element
λwi∈ HomC[G](US,T, C[G]), defined as
?
for all u ∈ US,T. Here, for a prime w in K, | · |w denotes the absolute value
associated to w, normalized as in [10], §1.1. Rubin’s C[G]–linear regulator
ΛS,T=



{ε ∈ (Q ∧r
GUS,T)r,S|(φ1∧ ··· ∧ φr)(ε) ∈ Z[G] ,
∀φ1,...,φr∈ HomZ[G](US,T, Z[G])}, if r ≥ 1,
Z[G]0,S, if r = 0.
λwi(u) := −
σ∈G
log|uσ−1|wi· σ ,
RW : C ∧r
GUS,T
RW
−→ C[G] ,
is defined by
RW :=
?λw1∧ ··· ∧ λwr, if r ≥ 1
1C∧0
GUS,T= 1C[G], if r = 0
Remark 1. As pointed out in [7, §1.6, Remark 2], RW induces an isomorphism
RW|(C∧r
GUS,T)r,S: (C ∧r
GUS,T)r,S
∼
−→ C[G]r,S.

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4CRISTIAN D. POPESCU
In [10], §2.1, Rubin states the following.
Conjecture B(K/k, S, T, r) (Rubin). Let us assume that the data (K/k,S,T,r)
satisfies hypotheses (H). Then, for any choice of V and W, there exists a unique
εS,T,W∈ ΛS,T, such that RW(εS,T,W) = Θ(r)
K/k,S,T(0).
Remark 2. As pointed out in Remark 2, §5.1 of [7], if (K/k, S, T, r) is given,
then the truth of Rubin’s Conjecture does not depend on the particular choice of V
and W. This is why we will suppress W from the notation εS,T,W. Also, Remark
1 above shows that the uniqueness in the conjecture above is automatic.
In [9], we prove the following functoriality properties for Conjecture B.
Lemma 1.2.2. Let us assume that the set of data (K/k, S, T, r) satisfies hypothe-
ses (H). Then, the following hold true.
(i) If S ⊆ S′and (K/k,S′,T,r) satisfies hypotheses (H), then
B(K/k, S, T, r) =⇒ B(K/k,S′,T,r).
(ii) Let S′:= S ∪ {vr+1,...,vr′}, with vr+1,...,vr′ distinct primes in k, which
do not belong to S and split completely in K/k. If T ∩ S′= ∅, then
B(K/k,S′,T,r′) =⇒ B(K/k, S, T, r) .
(iii) Let T′be a finite set of primes in k, such that T ⊆ T′and S ∩T′= ∅. Then
B(K/k, S, T, r) =⇒ B(K/k,S,T′,r).
Proof. See Proposition 2.3(i), (ii), (v), in §3 of [9].
?
2. AN EQUIVALENT CONJECTURE
Throughout this section, K/k will denote an abelian extension of Galois group G,
such that k is a totally real and K is a CM number field. We will denote by j the
(unique) automorphism of K induced by complex conjugation. Obviously, since k
is totally real, j is an element (of order 2) in G. The main goal of this section is to
formulate a conjecture B−(K/k,S,T,r), equivalent to B(K/k,S,T,r), but stated
in terms of the ring Z[G]−:= Z[G]/(1 + j) and modules over it, rather than the
ring Z[G] itself.
2.1 The ring R−. As in [9], §4.1, let R := Z[G] and R−:= Z[G]/(1 + j). For
every R–module M, let
M−:= M/(1 + j)M (
∼
−→ M ⊗RR−),M−:= {x ∈ M |(1 + j) · x = 0},
and let πM : M ։ M− be the canonical projection. In particular, let π := πR.
Obviously, both M−and M−come endowed with natural R−–module structures.

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THE RUBIN-STARK CONJECTURE5
Lemma 2.1.1. Let H be the subgroup of G generated by j, and let M be an R–
module. Let us assume that M is H–cohomologically trivial. Then,
(1) One has an equality, respectively isomorphism of R−–modules
M−= (1 − j)M ,M−
∼
− − − − − →
(1−j)M
M−,
where (1 − j)M takes (x mod (1 + j)M) into (1 − j) · x.
(2) The inverse of the isomorphism (1 − j)M in (1) is given by
1
2πM: M−− → M−,
defined as1
2πM((1 − j)x) := πM(x), for all x in M.
Proof (sketch). The equality in (1) is a direct consequence of?H
?H
trivial modules: R, R ⊗ZA = A[G], where A is an arbitrary subring of C, and
N ⊗ZQ, N ⊗ZC, N(p), where N is an arbitrary R–module and p is an odd prime.
−1(H,M) = 0.
The isomorphism in (1) is a consequence of the equalities M−= (1 − j)M and
0(H,M) = 0. (2) can be proved by direct calculation.
?
In particular, Lemma 2.1.1 can be applied to the following H–cohomologically
Corollary 2.1.2. Let N be an R module and A a subring of C. Then, one has an
isomorphism of abelian groups
?
φ −→ φ−,
HomA[G]
(N ⊗ A)−, A[G]
?
∼
− → HomA[G]−
?
(N ⊗ A)−, A[G]−
?
where φ−:=1
2πA[G]◦ φ, and φ = (1 − j)A[G]◦ φ−, for all A[G]–linear maps φ in
(M ⊗ A)−,A[G].
HomA[G]
??
Proof. Let us first remark that A[G]–linearity implies that any φ as above satisfies
Im(φ) ⊆ A[G]−. Now, the isomorphism in Corollary 2.1.2 is a consequence of
Lemma 2.1.1, applied to M := A[G], and the observation that one always has an
equality
?
HomA[G]
(N ⊗ A)−, A[G]−
?
= HomA[G]−
?
(N ⊗ A)−, A[G]−
?
.
?
Remark 1. In general, the quotient of a group ring by one of its ideals, and in
particular R−, is not a group ring any longer. However, let us assume that
the 2–Sylow subgroup G2 of G is cyclic. (This will be the case throughout
sections §§3–5 of this paper). Then, R−happens to be a group ring still. Indeed,
let |G2| = 2t, for some t ≥ 1. Let G = G2× G′, where G′has odd order. Then, as
Greither remarks in [3], since G is cyclic and consequently G2is cyclic of order 2t,
we have isomorphisms
(2)
R−= Z[G2]/(1 + j)[G′]
∼
−→ Z[X]/(X2t− 1,X2t−1+ 1)[G′]
∼
−→ Z[ζ2t][G′].