Representing Ω(l∞) for real abelian fields
Jürgen Ritter and Alfred Weiss
∗
For real subfields K of a cyclotomic fields Q(ζ) we remove the tameness assumption
at a given odd prime number l, which was needed in [RW2] in order to establish
the equivalence of the Lifted Root Number Conjecture at l and an equivariant
main conjecture of Iwasawa theory for abelian extensions of totally real number
fields K/k a .
a
added in proof: the latter conjecture has meanwhile been confirmed for real K ⊂
Q(ζ), see our paper [manuscripta math. 109 (2002),131-146]
Let K/k be an extension of real number fields which is contained in a cyclotomic field Q(ζn ),
ζn a root of unity of order n, G the Galois group of K/k, and let l be an odd prime number.
When l is tamely ramified in K/k, Theorem F of [RW2] implies the equivalence of the Lifted
Root Number Conjecture at l (see [GRW]) and a “main conjecture” of equivariant Iwasawa
theory. A summary of the basic ideas is outlined in [RW2, §1] and will not be repeated here.
The object here is to remove the ramification assumption on l.
Both the Lifted Root Number Conjecture at l and the “main conjecture” of equivariant
Iwasawa theory are compatible with restriction and deflation. For the Lifted Root Number
Conjecture this is [GRW, p.70]; for the “main conjecture” it is proved in the appendix at the
end of this paper. As a consequence, we restrict ourselves to the situation
K = Q(ζn )+ is the maximal real subfield of Q(ζn ) with n = lm+1 n0 , m ≥ 0 , l - n0 ; k = Q .
The tameness assumption on l in the proof of [RW2, Theorem F] was used to get a representing homomorphism of a certain class Ω(l∞) in the Grothendieck group K0 T (Zl G) of the
finitely generated torsion Zl G-modules of finite projective dimension. This representing homomorphism is a preimage of Ω(l∞) under the canonical map HomGl (Rl (G), Fl× ) K0 T (Zl G)
(compare [GRW, Appendix A]), where, as usual, Fl is a Galois extension of Ql with Galois
group Gl , which houses the values of all characters of G, and where Rl (G) is the ring of
Fl -valued characters of G.
We first recall the construction of the class Ω(l∞) . Let S be a finite G-stable set of primes
of K, which contains all primes dividing l∞ and which is sufficiently large, and let G̃ denote
the Galois group of K/Q. Also, S∗ is a set of G-representatives of the elements in S. Define
(compare [RW2, §7])
∗
We acknowledge financial support provided by NSERC.
1
ξ
[K:Q]
ξ = (N K
K/Q ξK
)t/[K:Q] , where ξK is the Ramachandra number of K and where t is a
suitable multiple of [K : Q]
1
S-units a(p) ∈ K, for p ∈ S∗ , which are fixed by Gp , a(p) = pt/fp , where fp is the
residue degree of p with respect to K/Q, and a(p) ≡ 1 mod p0 for p0 6= p in S
a to be the Q-idèle having component 1 except at l where the component is u−1 with
u ∈ 1 + lZl defined by ζlu∞ = ζlγ∞ 2 for a lift γ ∈ G(Q(ζl∞ )/Q) of a fixed generator γQ of
the Galois group of the cyclotomic l-extension Q∞ ⊂ Q(ζl∞ ) of Q
L(Kp× ) to be the l-completion of the multiplicative group of the local field Kp .
From these data we obtain a unique ZG̃-homomorphism ZS →
(
a(p)a if p ∈ S∗ is finite
ξa
if p = p∞
Q
S
Kp×
which maps p to
.
Here, p∞ ∈ S∗ is the archimedean prime arising from ζn 7→ e2πi/n . The map induces a
Q
Zl G̃-injection ϕ(l∞) : Zl Sl∞ Sl∞ L(Kp× ) with Sl∞ = {p ∈ S : p|l or p is archimedean} 3 .
The local fundamental classes, for p ∈ Sl∞ , yield the exact sequence [RW2, (7.1)]
Y
L(Kp× ) Sl∞
Y
L(Vp ) →
Sl∞
Y
Zl Gp Zl Sl∞
Sl∞
with finitely generated Zl Gp -modules L(Vp ) having finite projective dimension 4 . By [RW2,
Lemma 2.5] the two ends of the sequence have the same Zl -rank. Therefore we can perform
a lifted-Ω construction (see [RW2, §1.4]) with the pair
Y
Sl∞
L(Kp× ) Y
L(Vp ) →
Y
Zl Gp Zl Sl∞
and ϕ(l∞) : Zl Sl∞ L(Kp× )
Sl∞
Sl∞
Sl∞
Y
and arrive at the element Ω(l∞) ∈ K0 T (Zl G).
If l is tamely ramified in K/k, then Ω(l∞) is represented by χ 7→
] logl al (χ,1G )
( [K:Q−2t
)
(−4t)[k:Q] dim Vχ L∗l (1, χ)
Q
G0
det(−Frp | Vχ p )
S∗
(>l )
×
Q
(t
G
dim Vχ p
G0
G
det(Frp − 1 | Vχ p /Vχ p )−1 )
Sl∗
Q
ρ(ψ)
16=ψ|χ
in HomGl (Rl (G), Fl× ). This is formula (8.5) in [RW2, §8] with al = u−1 . In it, Frp ∈ Gp /G0p
Q
denotes the Frobenius automorphism of p. Moreover, the factor 16=ψ|χ ρ(ψ), with the product
running through the non-trivial characters ψ of G̃ extending the given abelian character χ
of G, comes from a Ramachandra index formula and need not be explained here in detail
(but see e.g. [RW2, Lemma 8.1]). Finally, L∗l (1, χ) is the residue at 1 of the l-adic L-function
Ll (s, χ).
recall that ξK = NQ(ζQ
ξ and ξn = QI (1 − ζnnI )(1 − ζn−nI ) with I running through the proper subsets
n )/K n
s
of {1, . . . , s}, where n = i=1 pei i and nI = I pei i
2
meaning ζlun = ζlγn for all natural n ≥ 1
3
Lemma 7.1, its corollary, Lemma 7.2 and the first paragraph of §8 in [RW2] is the reference for this.
4
Gp is the decomposition subgroup of p in G. Its inertia subgroup is denoted by G0p .
1
Q
2
The purpose of this paper is to show, without assuming that l is tame in K = Q(ζn )+ , that
G0
the homomorphism χ 7→ (>l ) multiplied by the function χ 7→ det(−1 | Vχ l ) represents Ω(l∞) ;
here, l is the prime above l in S∗ (note that k = Q). In fact,
Theorem. Let K = Q(ζn )+ with ζn a primitive n th root of unity and n = lm+1 n0 , m ≥ 0 ,
l - n0 , and let k = Q. Assume that n0 is divisible by l − 1 when l 6= 3, and by 8 when
l = 3. Then Ω(l∞) is represented by χ 7→
] logl al (χ,1G )
( [K:Q−2t
)
(−4t)dim Vχ L∗l (1, χ)
G0
Q
S∗
Gl
G0
det(−Frp | Vχ p )
G0
× det(−1 | Vχ l ) · tdim Vχ det(Frl − 1 | Vχ l /VχGl )−1
5
Q
ρ(ψ) .
16=ψ|χ
From the theorem it readily follows that Theorem F in [RW2] remains valid without the
tameness assumption on l; we will explain this in §5.
For the proof of the theorem it will be convenient to modify the notation :
from now on, ζ is a primitive root of unity of order n0 and ζm one of order lm+1
l =ζ
satisfying ζm
m−1 .
Our hypothesis on n0 fixes the local picture (at l)
def
Lm = Ql (ζ, ζm ) is the completion of K at l ∈ S∗ above l
def
L = Ql (ζ), the completion of Q(ζ), is unramified over Ql and linearly disjoint from the
def
totally ramified extension km = Ql (ζm ) ; Lm = Lkm .
Only the first statement requires an argument. Denote the complex conjugation by c. Let
L | l be primes of Q(ζ, ζm ) respectively K above l, and let G̃L , Gl be the corresponding
decomposition groups. Then the natural surjection G̃L Gl is an isomorphism if, and only if,
c 6∈ G̃L . Because of the canonical identification of decomposition groups and the Galois groups
of the respective completions, the condition c 6∈ G̃L is equivalent to Lm = Ql (ζ, ζm ) = Kl .
Since n0 is divisible by l − 1, each primitive root ζ 0 of unity of order l − 1 lies in Q(ζ); also
ζ 0 ∈ Ql . Thus, c ∈ G̃L would imply that complex conjugation on Q(ζ 0 ) would belong to the
trivial decomposition group of the restriction of L to Q(ζ 0 ), which is a contradiction except
when l = 3. However, in this case 8 divides n0 , hence a primitive 8 th root of unity ζ 00 is then
contained in Q(ζ), and c ∈ G̃L restricted to Q(ζ 00 ) would lie in the decomposition group of
the prime of Q(ζ 00 ) below L, which is not so.
We finally outline the basic steps in the proof of the theorem. Since Ω(l∞) is essentially a
local object, we first concentrate on the contribution that comes from the local situation at
l. To that end we exploit, in section 1, Coleman’s power series for getting special elements
×
of L×
m which, in section 2, are used to define a map fm : Rm Lm such that the push-out
def
along fm of the natural relation module sequence for the Galois group Gl = G(Lm /Ql ) 6 ,
Rm ZGl ⊕ ZGl → ZGl Z ,
5
6
Q
Since G̃ = G, 16=ψ|χ ρ(ψ) equals ρ(χ) or 1 according as χ 6= 1 or χ = 1.
so Gl = Gl = G̃L
3
tensored over Z with Zl , yields a sequence L(L×
m ) L(Vm ) → Zl Gl Zl representing the
local fundamental class. These special elements are constructed by imitating generators of
Rm . The computation of coker fm closes this section. It is perhaps worth mentioning that
this is done simultaneously for all m. The next section, §3, concerns the transition to the
map ϕ(l∞) and §4 then the actual proof of the theorem.
At some stages of our proof, particularly in §2, similarities to work of Greither [Gr] can
be observed. Greither proves Chinburg’s second conjecture whereas we are concerned with
a lifted version of Chinburg’s third conjecture and need to verify the specific formulas of
[RW2, §8] without the tameness assumption made there. For this, it would neither shorten
nor ease our approach if we combined it with [Gr]. Also, the paper [BB] by Bley and Burns
overlaps to some extent with ours. However, our results have been achieved independently
from that paper and concentrate on the special situation introduced in [RW2], which is
different from the setup in [BB].
1 . Coleman power series
This section is based on [Co] with the simplification that the Lubin-Tate formal group F is
the multiplicative group. We recall the definitions of the trace operator S, the norm operator
N, and the operator Ξ in this case 7 . Set o = the ring of integers in L and σ = the Frobenius
generator of G(L/Ql ), and let η run through the l th roots of unity.
S : o[[T ]] → o[[T ]] , S(g)( (1 + T )l − 1 ) =
P
η
g( η(1 + T ) − 1 )
(compare [Co, Theorem 11])
N : 1 + T 2 o[[T ]] → 1 + T 2 o[[T ]] , N(g)( (1 + T )l − 1 ) =
Q
η
g( η(1 + T ) − 1 )
(compare [Co, Theorem 4])
Ξ : T 2 o[[T ]] → T 2 o[[T ]] , Ξ(g)(T ) =
∞
P
i=0
1
(σ i g)( (1
li
i
+ T )l − 1 )
(compare [Co, p.110]) .
The two operators S and Ξ are related by formula
(1.1a)
on T 2 o[[T ]] .
SΞ = S + σΞ
To see this, let g ∈ T 2 o[[T ]] :
(SΞg)( (1 + T )l − 1 ) =
(Ξg)( η(1 + T ) − 1 )
Pη P∞ 1 i
i
=
(σ g)( (1 + η(1 + T ) − 1)l − 1 )
li
Pη Pi=0
i
i
∞ 1
i
l
l
=
i (σ g)( η (1 + T ) − 1 )
Pη i=0 l
P∞ 1
i+1 g)((1 + T )li+1 − 1)
=
η g(η(1 + T ) − 1) + l i=0 li+1 (σ
P
= (Sg)((1 + T )l − 1) + σ
P∞
i=0
(σ i g)
((1
li
+ T )l
i+1
− 1) .
Here the second term is (Ξg)((1 + T )l − 1), which establishes (1.1a).
We need some more notation. Let
β ∈ o generate an integral normal basis of L/Ql and have trace 1 in Ql
ULs = 1 + psL be the group of units of level s of a field L ⊃ Ql with prime ideal pL
7
note that the fields K, H there are Ql , Ql (ζl ) here
4
G(Ql (ζl∞ )/Ql ) = hτ, γi with τ, γ generating the subgroup of order l − 1 respectively the
Sylow-l subgroup of G(Ql (ζl∞ )/Ql )
κ be the character G(Ql (ζl∞ )/Ql ) → Zl × defined by κ(τ ) = t ∈ Zl × and κ(γ) = u ∈ 1 + lZl ,
where ζlτ∞ = ζlt∞ and ζlγ∞ = ζlu∞ .
The group Gl has three generators σm (= σ), τm (= τ ), γm which are chosen so that σm
generates the Galois group of the unramified extension Lm /km , and τm and γm generate
the subgroups of order l − 1 respectively lm of G(Lm /L). Also, the natural identification of
G(Lm /L) and G(km /Ql ) sends τm , γm to the respective restrictions of τ, γ on km . We take
the liberty of just writing σ, τ rather than σm , τm , and of viewing σ also as an element in
G(L/Ql ) and τ, γm also as elements in G(km /Ql ). We finally define τ̂ =
l−2
P
τ j and similarly
j=0
σ̂ , γ̂m .
Moreover, if Σ is a subgroup of hσi, then Lm and L denote the fixed fields of Σ in Lm
respectively L, σ is the Frobenius automorphism here, and β is the trace of β in L. Whenever
we construct a particular element x in L, or a power series in L[[T ]], then x is meant to be
the correspondingly defined element in L, respectively power series in L[[T ]].
In this section, by means of power series, we construct elements of L×
m which allow us to
define an injective map Rm → L×
with
a
computable
cokernel.
In
the
course of doing so it
m
×
will be advantageous to split Lm into its nontrivial and trivial eigenspaces with respect to the
τ -action. We indicate this by adding the indices 0 and 1, respectively, at appropriate stages.
The existence of the following power series is taken from [Co]. We recall that the Galois
action of G(L(ζ∞ )/L) ' G(Ql (ζ∞ )/Ql ) on a power series f (T ) ∈ L[[T ]] is given by (f (T ))α =
f ( (1 + T )κ(α) − 1 ). Note that the evaluation at T = ζm − 1 of power series f (T ) ∈ 1 + T o[[T ]]
yields 1-units in L.
1. ∃ W∞,0 (T ) ∈ 1 + T 2 Zl [[T ]] satisfying
log W∞,0 =
∞
X
1
i=0
li
i
i
(1 + T )l κ(τ γ) − 1 − κ(τ γ)( (1 + T )l − 1 )
and wm,0 = W∞,0 (T )l−1−τ̂ |T =ζm −1 ∈ Uk2m has
log wm,0 = (l − 1 − τ̂ )( τ γm − κ(τ γ) )
m
li − 1
X
ζm
i=0
li
.
2. ∃ V∞,0 (T ) ∈ 1 + T 2 o[[T ]] satisfying
log V∞,0 =
∞
X
1
i=0
l
i
i
σ i (β) (1 + T )l κ(τ γ) − 1 − κ(τ γ)( (1 + T )l − 1 )
i
m
σ
and vm,0
= V∞,0 (T )(l−1−τ̂ )(τ γ−1) |T =ζm −1 ∈ UL2 m has
log vm,0 = (l − 1 − τ̂ )(τ γm − 1)( τ γm − κ(τ γ) )σ −m
m
X
σ i (β)
i=0
5
li
i
l
(ζm
− 1) .
In fact, the existence of W∞,0 with log W∞,0 = Ξ((τ γ − κ(τ γ))T ) follows from [Co, Theorem
24] because ( τ γ − κ(τ γ) )T = (1 + T )κ(τ γ) − 1 − κ(τ γ)T ∈ T 2 Zl [[T ]] .
For V∞,0 start with β( τ γ − κ(τ γ) )T .
Lemma 1.1. The elements wm,0 , vm,0 are compatible with the norms in the towers km , Lm ,
τ γm −1
σ̂
Σ̂ = v
m ≥ 0, respectively. Moreover, vm,0
= wm,0
and vm,0
m,0 .
l−1−τ̂
For the first assertion we use [Co, Theorems 4, 11 and Corollary 12] and arrive at N(W∞,0
)=
l−1−τ̂
N(W∞,0 )
and log(NW∞,0 ) = S(log W∞,0 ). Therefore, in order to prove that
!
l−1−τ̂
l−1−τ̂
N(W∞,0
) = W∞,0
(which implies the claim for wm,0 by [Co, Theorem 16]), we need to check whether
1. (l − 1 − τ̂ ) S(log W∞,0 ) − log W∞,0 =0
2. N(W∞,0 ) and W∞,0 both have constant term 1 .
l−1−τ̂
l−1−τ̂
Indeed, by 1., N(W∞,0
) and W∞,0
only differ by a root of unity in L, since the logarithm
l−1−τ̂
of their quotient (NW∞,0 /W∞,0 )
∈ 1+T o[[T ]] vanishes. Hence, 2. guarantees the desired
!
equality =.
For 1., we employ formula (1.1a) to compute S log W∞,0 = SΞ( (τ γ − κ(τ γ))T ) =
S( (τ γ − κ(τ γ))T ) + σΞ( (τ γ − κ(τ γ))T ) = S( (τ γ − κ(τ γ))T ) + log W∞,0 since σ acts trivially. Hence it suffices to show that l − 1 − τ̂ annihilates S( (τ γ − κ(τ γ)T ). This follows from
S( (τ γ − κ(τ γ))T ) = (τ γ − κ(τ γ))S(T ) (by [Co, Corollary 5]) and S(T )((1 + T )l − 1) =
P
η (η(1 + T ) − 1) = −l .
Regarding 2., we conclude from the above and from W∞,0 (0) = 1 that (NW∞,0 )l−1−τ̂ (0) is a
root of unity. Now, with η as before,
l−1−τ̂
(NW∞,0 )l−1−τ̂ ( (1 + T )l − 1 ) =
η W∞,0 ( η(1 + T ) − 1 )
Q
l−1−τ̂
2
(NW∞,0 )l−1−τ̂ (0) =
η W∞,0 (η − 1) ≡ 1 mod (ζ0 − 1)
Q
in Ql (ζ0 ) .
The congruence is due to W∞,0 (T ) ∈ 1 + T 2 Zl [[T ]]. Thus the root of unity in question is =1.
We next turn to vm,0 and follow the same reasoning as with wm,0 . However, here we need to
(l−1−τ̂ )(τ γ−1)
(l−1−τ̂ )(τ γ−1)σ
take in the automorphism σ, since we have to show that NV∞,0
= V∞,0
to apply [Co, Theorem 16]. As before
S(log V∞,0 ) − σ log V∞,0 = SΞ( β(τ γ − κ(τ γ))T ) − σΞ( β(τ γ − κ(τ γ))T ) =
S( β(τ γ − κ(τ γ))T ) = β(τ γ − κ(τ γ))S(T ) = β(κ(τ γ) − 1)l
is annihilated by (l − 1 − τ̂ )(τ γ − 1). This implies that our power series differ by a root of
unity, which is 1 because (NV∞,0 )(0) ≡ 1 mod (ζ0 − 1)2 in L(ζ0 ), again as before.
The second assertion is a consequence of V∞,0 (T )σ̂ = W∞,0 (T ), which holds because logarithms and leading terms agree (as σ̂(β) = 1). Applying (l − 1 − τ̂ )(τ γ − 1) and evaluating
τ γm −1
σ̂
Σ̂
at ζm − 1 yields vm,0
= wm,0
. Similarly Σ̂(β) = β implies vm,0
= v m,0 and the proof of
Lemma 1.1 is complete.
In the next lemma Coleman’s map Θ occurs (see [Co, p.108]). Recall also the definition of
l−1
u ∈ 1 + lZl , and set y = exp( − l−σ
(β)τ̂ (log u) ).
6
Lemma 1.2. 1. l − σ is a unit in Zl Gl , and y ∈ UL1 has y σ̂ = u1−l , y Σ̂ = y.
2. T τ̂ (γ−1) = ul−1 g(T ) with g(T ) ∈ 1 + T 2 Zl [[T ]]
3. ΞβΘ log g = log h with h(T ) ∈ 1 + T 2 o[[T ]]
def
m
σ
4. V∞,1 (T ) = y −1 h(T ) evaluated at T = ζm − 1 produces the 2-unit vm,1
∈ Lm
satisfying
σ̂
Σ̂ = v
a) vm,1
= (ζm − 1)τ̂ (γm −1) , vm,1
m,1
τd
γm
b) vm,1
= y 1−σ
5. The elements vm,1 are compatible with the norms in the tower Lm , m ≥ 0 .
Proof. For 1., l − σ = −(1 − lσ −1 )σ has inverse −σ −1
∞
P
(lσ −1 )i . The rest is obvious.
i=0
2. follows from
T
τ̂ (γ−1)
=
(1 + T )u − 1 τ̂
T
=
∞
X
u
!
T i−1
i
i=1
τ̂
= (u(1 + g̃(T )))τ̂ = ul−1 g(T )
with g(T ) ∈ 1 + T 2 Zl [[T ]] resulting from the modulo degree 2 computation
(1 + g̃(T ))τ̂ ≡ (1 + a1 T )τ̂ =
l−2
Y
h
1 + a1 (1 + T )κ(τ
j)
i
−1
≡ 1 + a1
l−2
X
j=0
κ(τ j ) T = 1 ,
j=0
since κ 6= 1 on hτ i.
By Lemma 21 (ii) of [Co] (with h(T ) = (1 + T )l − 1), Θ log g ∈ Zl [[T ]]. Moreover, Θ log g is
divisible by T 2 , since g − 1 is so. Hence we can apply Ξ to βΘ log g and obtain log h with
some h(T ) ∈ 1 + T 2 o[[T ]], by [Co, Theorem 24]. This is 3.
Claim : hσ̂ = g .
The claim together with 2. implies the first assertion of 4a), because y σ̂ = u1−l by 1.
The claim itself is once again a consequence of Coleman’s results. Indeed, σ̂ log h = σ̂ΞβΘ log g
= ΞΘ log g and so Θσ̂ log h = ΘΞΘ log g = Θ log g (see [Co, p.110]). Now, in the notation of
[Co, p.108] this becomes, firstly, ΘF (hσ̂ − 1) = ΘF (g − 1) and, secondly, by [Co, p.109/110],
σ̂
log hg = a log(1 + T ) for some a ∈ Ql . But the left hand side is ≡ 0 mod degree 2 and the
right hand side is ≡ aT , so a = 0 and
hσ̂
g
is a root of unity, whence =1, as h, g ∈ 1 + T 2 o[[T ]].
The second assertion of 4a) follows analogously from hΣ̂ = h.
For 4b), observe that there is no nontrivial root of unity in UL1 , so it suffices to verify
that τd
γm log vm,1 = (1 − σ)y. For this we start from σ m log vm,1 = log V∞,1 (ζm − 1) =
l−1
l−σ (β)τ̂ (log u) + (log h)(ζm − 1). Since
∞
P
log h(T ) = ΞβΘ log(g) =
(1.2a)
i=0
∞
P
i=0
σ i (β)
li
σ i (β)
(Θ log g)( (1
li
li
i
+ T )l − 1 ) =
(log g)( (1 + T ) − 1 ) − 1l (log g)( (1 + T )l
i+1
− 1)
has most terms vanishing at T = ζm − 1 (as g(0) = 1), we have
m−1 i+1
(β)
σ i (β)
li − 1) − σ −1 P σ
li+1
(log g)(ζm
(log g)(ζm
li
li+1
i=0
i=0
m
P
(1−σ −1 )σ i (β)
li − 1) .
1) +
(log
g)(ζ
m
li
i=1
(log h)(ζm − 1) =
β(log g)(ζm −
m
P
7
− 1) =
But from g(T ) = (u−1 T γ−1 )τ̂ this becomes
(log h)(ζm −1)=β τ̂ (log u−1 +(γm −1) log(ζm −1))+τ̂
m
P
i=1
(σ−1)σ i−1 (β)
( log u−1 +(γm −1) log(ζm−i −1) ) .
li
Combining with the first equation gives
σ m log vm,1 =τ̂ (log u)
σ−1
(β)−
l−σ
m
P
i=1
because
l−1
l−σ
−1=
σ−1
l−σ .
(σ−1)σ i−1 (β)
li
+τ̂ (γm −1) β log(ζm −1)+
m
P
i=1
(σ−1)σ i−1 (β)
li
log(ζm−i −1)
Acting by τd
γm this is
σ τd
γm (log vm,1 ) = lm (l − 1)τ̂ (log u)(σ − 1)
m
m 1
1X
σ i−1 (β)
−
l−σ
l i=1 l
since the first term is fixed by τ and γm while τd
γm (γm −1) = 0. Summing the geometric series
σm
σ−1
−m
m
we then get τd
γm (log vm,1 ) = σ l (l − 1)τ̂ (log u) σ−1
l−σ lm (β) = (l − 1)τ̂ (log u)( l−σ )(β) =
l−1
(σ − 1)τ̂ (log u) l−σ
(β) = −(σ − 1) log y, as required.
We are left with verifying 5. In analogy with the corresponding part in the proof of Lemma
!
σ . We first need
1.1, it suffices to show NV∞,1 = V∞,1
!
S log V∞,1 = σ log V∞,1 = σ
l − 1
l−σ
(β)τ̂ log u + log h .
l−1
l−1
The left hand side equals S( l−σ
(β)τ̂ log u + log h) = l l−σ
(β)τ̂ log u + S log h, since S transforms a constant function c into lc. Therefore, we need to show
(50 )
! (l−1)(σ−l)
(β)τ̂
l−σ
S log h − σ log h =
log u = (l − 1)β τ̂ log u−1 .
The left hand side in (50 ) is, by (1.1a),
SΞ log g − σΞβΘ log g = SβΘ log g = β SΘ log g .
h
However, Θ log g = Θτ̂ log(u−1 T γ−1 ) = τ̂ Θ(log u−1 + log T γ−1 ) = τ̂ log u−1 −
1
l
log u−1
i
+ log T γ−1 − 1l log( (1 + T )l − 1 )γ−1 , and applying S gives, with η running through the lth
roots of unity,
τ̂ (l − 1) log u−1 + τ̂
P log(η(1 + T ) − 1)γ−1 − 1l log( (1 + η(1 + T ) − 1)l − 1 )γ−1
η
h
Q
Q
i
= τ̂ (l − 1) log u−1 + τ̂ log
h
= τ̂ (l − 1) log u−1 + τ̂ log((1
γ−1 − 1 log
η (η(1 + T ) − 1)
l
l
γ−1
+ T ) − 1)
− log((1 + T )l
−
l
γ−1
η ((1 + T ) − 1)
i
1)γ−1 = τ̂ (l − 1) log u−1 .
This establishes (50 ).
It remains to check whether
y l−σ
1)2 ,
NV∞,1
σV∞,1
(0) = 1. But NV∞,1 (0) =
y −1 h(T )
Q
η
V∞,1 (η −1), hence
mod (ζ0 −
because V∞,1 (T ) =
with h(T ) ∈ 1 +
1-unit in L, hence ≡ 1 mod (ζ0 − 1)2 . Lemma 1.2 is proved.
T 2 o[[T ]].
NV∞,1
σV∞,1 (0) ≡
But y l−σ is a
τ = v . Since v
−1
Remark. v0,1
0,1
m,1 is the value of V∞,0 (T ) = y h(T ) at T = ζm − 1 we need
τ
τ
to show h(T ) = h(T ). Clearly g(T ) = g(T ), so τ fixes the right hand side of the equality
(1.2a) above, hence it fixes log h(T ). Now h(T ) ∈ 1 + T 2 o[[T ]] implies the claim.
8
× with w
τ̂
wm = wm,0 wm,1 ∈ km
m,1 = (ζm − 1) .
vm = vm,0 vm,1 ∈ UL2 m ,
Definition.
Combining our relations we now get
σ−1 ,
y τ γm −1 = 1 = wm
(1.2b)
σ̂ = w τ γm −1 ,
vm
m
τd
γm = y 1−σ ,
vm
τd
γm
the last one because τd
γm (τ γm − 1) = 0 implies vm,0
= 1.
The elements wm , vm , y are precisely the elements which we need for a map fm : Rm → L×
m.
However, in order to calculate its cokernel we still need one more element, rm ∈ Lm , which
we now define.
Lemma 1.3. If R∞ (T ) ∈ 1 + T o[[T ]] is such that
log R∞ (T ) = β log(1 + T ) + (1 − σ)
∞
X
(1 + T )li − 1
i
σ (β)
i=0
then
σm = R (T )l−1−τ̂ |
1
rm
∞
T =ζm −1 ∈ ULm
1. log rm = (l − 1 − τ̂ )(1 − σ)
m
P
i=0
li
− log(1 + T ) ,
satisfies
−1
σ i−m (β) ζm−i
li
2. The elements rm are compatible with the norms in the tower Lm , m ≥ 0 .
σ̂ is a root of unity of order lm+1 , r Σ̂ = r
3. rm
m
m
(τ γm −1)(τ γm −κ(τ γ))
4. rm
1−σ
= vm,0
Proof. The power series R∞ (T ) is taken from [Co, p.110]. Since R∞ (T )−1 is in the maximal
ideal of o[[T ]] we have R∞ (0) ≡ 1 mod l. Putting T = 0 in the equation for log R∞ (T ) we
get log R∞ (0) = 0 hence R∞ (0) = 1. Differentiating the equation for log R∞ (T ) and putting
0 (0)
R∞
0 (0) = β. Thus R (T ) = 1 + βT + . . . .
T = 0 then gives R
= β, hence R∞
∞
∞ (0)
l−1−τ̂ =
Claim 1. of Lemma 1.3 is due to log ζm = 0. For 2. we need to check whether NR∞
(l−1−τ̂ )σ
R∞
, which is equivalent to
and NR∞ l−1−τ̂ (0) = R∞ (l−1−τ̂ )σ (0) .
S(l − 1 − τ̂ ) log R∞ = (l − 1 − τ̂ )σ log R∞
This will be confirmed now. As before, η runs through the lth roots of unity. We use log η = 0
P
and η η = 0.
(S log R∞ )( (1 + T )l − 1 ) =
=
P h
η
log R∞ (η(1 + T ) − 1)
P
η
β log(η(1 + T )) + (1 − σ)
= lβ log(1 + T ) +
P
η (1
P∞
i
i=0 σ (β)
i
− log(η(1 + T ))
i
(1+T )l −1
∞
i
i=1 σ (β) i
li
hP
− σ)
i
(η(1+T ))l −1
li
− log(1 + T )
+ β( η(1 + T ) − 1 − log(1 + T ) )
hP
li
i
(1+T ) −1
∞
i
= lβ log(1 + T ) + l(1 − σ)
− log(1 + T )
i=1 σ (β)
li
− l(1 − σ)(β) − l(1 − σ)(β)
log(1 + T )
= lσ(β)( 1 + log(1 + T ) ) − lβ + l log R∞ − (1 − σ)(β)(T − log(1 + T )) − β log(1 + T )
= lσ(β)( 1 + log(1 + T ) ) − lβ + l log R∞ − l(1 − σ)(β)T
+ l(1 − σ)(β) log(1 + T ) − lβ log(1 + T )
= l(σ − 1)(β)(1 + T ) + l log R∞ .
9
On the other hand,
σ log R∞ ( (1 + T )l − 1 ) = σ(β) log(1 + T )l + σ(1 − σ)
= lσ(β) log(1 + T ) + l(1 − σ)
P∞
i+1 (β)
i=0 σ
P∞
i
i=0 σ (β)
i+1
(1+T )l
−1
li+1
i+1
(1+T )l
li
−1
− log(1 + T )l
− log(1 + T )
= lσ(β) log(1 + T ) + l log R∞ − (1 − σ)(β)(T − log(1 + T )) − β log(1 + T )
= lσ(β) log(1 + T ) + l log R∞ − l(1 − σ)(β)T + l(1 − σ)(β) log(1 + T ) − lβ log(1 + T )
= l log R∞ + l(σ − 1)(β)T .
Putting the two results together shows that S log R∞ − σ log R∞ = l(σ − 1)(β), which is
annihilated by l − 1 − τ̂ . This gives the first relation by [Co, Corollary 5].
(l−1−τ̂ )σ
Regarding the values at zero, we know now that (NR∞ )l−1−τ̂ /R∞
(0) is a root of unity
Q
l−1−τ̂
l−1−τ̂
l(l−1−τ̂
)
in L. By (NR∞ )
(0) = η R∞ (η−1)
≡ R∞ (0)
mod (ζ0 −1) and R∞ (0) = 1 ,
this root of unity is ≡ 1 mod (ζ0 − 1), hence 1 because in L.
σ̂ is a root of unity in k .
For 3. we use that σ̂ log rm = 0 by 1., since σ̂(1 − σ) = 0. Thus rm
m
Now
−m
(1.3a)
rm ≡ (1 + β(ζm − 1))σ (l−1−τ̂ ) mod (ζm − 1)2
σ̂
l−1 /ζ τ̂ = ζ l−1 ,
rm
≡ ( 1 + (σ̂β)(ζm − 1) )l−1−τ̂ = (1 + ζm − 1)l−1−τ̂ = ζm
m
m
because the fixed field of τ in km does not contain l-power roots of unity 6= 1. Reading the
σ̂ is a power of ζ ; then reading it mod (ζ − 1)2 gives
congruence mod (ζm − 1) implies rm
m
m
a
the first part of 3. since ζm ≡ 1 mod (ζm − 1)2 implies l divides a. The second part is done
analogously.
For 4. we use the identity R∞ (T )τ γ−κ(τ γ) = V∞,0 (T )1−σ : both sides are in 1 + T o[[T ]] and
they have the same logarithm directly from the definition. Raising this identity to the power
σ −m (τ γ − 1)(l − 1 − τ̂ ) and substituting T = ζm − 1 gives the result.
2 . The basic sequence L(L×
m ) L(Vm ) ∆Gl
This section starts from a relation module Rm for Gl , given in terms of generators and
2
relations. We define a map fm : Rm → L×
m so that the push-out of Rm (ZGl ) → ZGl Z
×
0
along fm gives a sequence Lm Vm → ZGl Z with extension class l − 1 times the
2
×
fundamental class uLm /Ql of Lm /Ql in H 2 (Gl , L×
m ) = ExtZGl (Z, Lm ). We then l-complete,
0
getting L(L×
m ) L(Vm ) → Zl Gl Zl , and pushout along multiplication by 1/l − 1 on
×
L(Lm ). This gives the l-completion of the fundamental class sequence. Finally, we determine
the cokernel of the map Zl ⊗ Rm → L(L×
m ) induced by fm .
Lemma 2.1. Let Rm be the ZGl -module with generators am , bm , cm and relations
(τ γm − 1)bm = 0 = (σ − 1)cm , σ̂am = (τ γm − 1)cm , τd
γm am = (1 − σ)bm .
Then the sequence
Rm ZGl ⊕ ZGl → ZGl Z ,
in which the first map is defined by
am 7→ (1 − σ, τ γm − 1) , bm 7→ (τd
γm , 0) , cm 7→ (0, σ̂) ,
the second map takes (x, y) ∈ ZGl ⊕ ZGl to x(τ γm − 1) + y(σ − 1) ∈ ZGl and the last
map is the augmentation map, is exact.
10
The lemma can be checked directly or looked up in [HS, p.168]. Rm is called a relation
module for Gl .
Lemma 2.2. Let U be a subgroup of Gl and set G = Gl /U . Then taking U -invariants respectively U -coinvariants transforms Rm ZGl ⊕ ZGl → ZGl Z into
Rm U
(ZGl ⊕ ZGl )U
'↑
ZG ⊕ ZG
→ (ZGl )U
'↑
→
ZG
Z
with the vertical arrows induced by the norm map Û . In particular, the resulting sequence
Rm U ZG ⊕ ZG → ZG Z
shows that Rm U is a relation module for G.
For the proof see [RW2, Lemma 3.3]. We will use the special cases :
1. U = Σ ≤ hσi. We identify the corresponding Rm of Lemma 2.1 with Rm U by means of
am = Û am , bm = Û bm , cm = cm
[Gl :U ]−1
= Û (0,
X
σi) .
i=0
i
l i with i ≤ m. Then G = hσ, τ γ
2. U = hγm
m−i i is the Galois group of Lm−i /Ql , and
Rm = Rm−i is identified with Rm U by means of
am = Û am , bm = bm
[Gl :U ]−1
= Û (
X
(τ γm )i , 0) , cm = Û cm .
i=0
Definition.
fm : Rm → L×
m is the ZGl -map defined by
am 7→ vm , bm 7→ y , cm 7→ wm
where vm , y, wm are the elements introduced in the previous section.
Observe that, by (1.2b), fm indeed respects the relations in Rm and is compatible with
passing to quotients G of Gl as in 1. or 2. above.
The diagram below defines the push-out of Rm ZGl ⊕ ZGl → ZGl Z along fm :
Rm
(2.2a)
fm ↓
L×
m
ZGl ⊕ ZGl
↓
Vm0
→ ZGl Z
k
k
→ ZGl Z .
Its bottom sequence has extension class a multiple, zuLm /Ql , of the fundamental class uLm /Ql
2
×
in huLm /Ql i = H 2 (Gl , L×
m ) = ExtZGl (Z, Lm ). Since the fundamental class respects deflation,
the analogous diagram for the level G = Gl /hσi yields
Rm
fm ↓
×
km
ZG ⊕ ZG
↓
0
Vm
11
→ ZG Z
k
k
→ ZG Z
in which the bottom sequence has extension class zukm /Ql . However, G is cyclic and Rm =
× V 0 → ZG Z as the push-out along
Zbm ⊕ ZGcm ' Z ⊕ ZG. This allows us to view km
m
1 7→ bm 7→ f m (bm ) = y σ̂ of the top sequence in
→ ZG Z
↓
k
k
ZG ⊕ ZG → ZG Z
↓
k
k
0
Vm
→ ZG Z .
ZG
Z
↓
Rm
fm ↓
×
km
Of course, in the top sequence 1Z is sent to τd
γm and 1ZG to τ γm − 1.
By Lemma 1.3, y σ̂ = u1−l . This together with [RW1, Lemma 3.1] and [Se, p.227], see also
[Sn, pp.52-53], gives zukm /Ql = (l − 1)ukm /Ql . Our discussion does not depend on a special
choice of the level m. Hence, starting at level m0 ≥ m produces a z 0 and the congruence
0
0
z0
z 0 ≡ l − 1 mod [km0 : Ql ] = (l − 1)lm . As a first consequence, l − 1 | z 0 and l−1
≡ 1 mod lm .
0
z0
− 1)uLm /Ql has
If now m0 is such that lm is the precise l-power dividing [Lm : Ql ], then ( l−1
2 (G , L(L× )). Consequently, the
order prime to l and thus vanishes under H 2 (Gl , L×
)
→
H
l
m
m
bottom sequence in the diagram below
Rm
fm ↓
L×
m
↓
L(L×
m)
1
↓
l−1
L(L×
m)
ZGl ⊕ ZGl
ZGl
ˇ↓
↓
→
L(Vm0 )
Z
has the image of uLm /Ql
in H 2 (Gl , L(L×
m )) as its
extension class.
ˇ↓
→ Zl Gl Zl
Z ⊗fm
1/(l−1)
l
×
Notation. The composite map Zl ⊗Z Rm −→
L(L×
m ) −→ L(Lm ) will be denoted by fm,l .
0
Lemma 2.3. The short exact sequence L(L×
m ) L(Vm ) ∆Gl , induced by the bottom sequence in the above diagram, coincides with the l-completion of the short exact sequence
0
L×
m Vm ∆Z Gl having extension class uLm /Ql . In particular, L(Vm ) and L(Vm ) can
be identified and are cohomologically trivial.
This follows directly from [RW2, 3. and 4. of Lemma 2.1], since L(L×
m ) L(Vm ) ∆Gl and
1
×
0
L(Lm ) L(Vm ) ∆Gl have the same extension class in ExtZl Gl (∆Gl , L(L×
m )).
Lemma 2.4.
fm,l is injective and coker fm,l ' Zl Gl /( τ γm − κ(τ γ) )
l
Remark. Zl Gl /(τ γm − κ(τ γ)) ' ind G
G(Lm /L) hζm i.
τ γm −1 induces a homomorphism
Proof of Lemma : With rm as in §1, sending 1 to rm
αm : Zl Gl /(τ γm − κ(τ γ)) → coker fm,l
(τ γm −1)(τ γm −κ(τ γ))
because rm
1−σ
= vm,0
∈ im fm,l by 4. of Lemma 1.3.
We observe that once we know αm is surjective then the finiteness of coker fm,l and the
injectivity of fm,l follow immediately. The first assertion holds because τ γm having order
m
m
m
lm (l − 1) implies that κ(γ)l (l−1) − 1 = κ(τ γ)l (l−1) − (τ γm )l (l−1) ∈ Zl Gl (τ γm − κ(τ γ)),
12
m
hence the nonzero element κ(γ)l (l−1) − 1 of Zl annihilates Zl Gl /(τ γm − κ(τ γ)). This implies
the second assertion because Zl ⊗ Rm and L(L×
m ) both have Zl -rank |Gl | + 1.
The remainder of the proof splits into two cases. In both we first establish that αm is
surjective.
Step 1 . m = 0
Now Gl is the group G(L0 /Ql ) and γ0 = 1. The properties of elements r0 , v0,0 , y, w0,1 used
below are taken from §1.
Consider the Zl Gl -module p20 , where p0 is the prime ideal in L0 . Write
εi =
1
l−1j
X
κi (τ −j )τ j
mod (l−1)
for the idempotent of Zl hτ i associated to the character κi , i mod (l − 1), and put
a = (τ − κ(τ γ))(ζ0 − 1) ∈ k0 .
Since (τ − κ(τ γ))εi = (ti − tu)εi , we find that εi (a) = t(ti−1 − u)εi (ζ0 − 1) has k0 -valuation
i for 2 ≤ i ≤ l : for the k0 -valuation of εi (ζ0 − 1) can be determined from that of the Gauß
×
sum for κi ([La, p.7]) and u−1
l ∈ Zl . As εi , 2 ≤ i ≤ l, is an o-basis of ohτ i it follows that
p20 = ohτ i · a. But o = Zl hσiβ, as in §1, hence aβ is a Zl Gl -basis of p20 (because σ fixes a and
τ fixes β).
1
1
1−l log u
2
From l−1
log v0,0 = (τ − 1)(1 − ε0 )(aβ) , l−1
log y = l−σ
l lβ it follows that (1 − ε0 )p0 =
1−l
are units in Zl hτ i/(τ̂ ),
hlog v0,0 iZl Gl , ε0 p20 = lo = Zl hσi · lβ = hlog yiZl hσi because τ − 1, l−σ
2
2
2
Zl hσi respectively (note that ε0 = τ̂ /|τ |). Therefore p0 = (1−ε0 )p0 +ε0 p0 = hlog v0,0 , log yiZl Gl .
By the isomorphism log : UL2 0 → p20 in the tame extension L0 /Ql we thus have UL2 0 =
hv0,0 , yiZl Gl .
The map x 7→ ζx−1
induces an isomorphism UL1 0 /UL2 0 → L0 of hσi-modules, where L0 is the
0 −1
residue field of L0 . This respects the Gl -structure on making τ act on L0 by multiplication
l−1
by t, since κ(τ ) = t; then τ̂ acts by multiplication by t t−1−1 = 0, so the isomorphism takes
r0 UL2 0 to −β, by (1.3a), and thus r0τ −1 UL2 0 to (1 − t)β. But L0 = Fl hσiβ and (1 − t) ∈ Fl hσi× ,
so we conclude that UL1 0 = hr0τ −1 , UL2 0 iZl Gl = hr0τ −1 , w0,1 , v0,0 iZl Gl .
Finally, w0,1 = (ζ0 − 1)τ̂ has L0 -valuation l − 1, hence
τ −1
1
L(L×
, w0,1 , v0,0 , yiZl Gl .
0 ) = hw0,1 , UL0 iZl Gl = hr0
On the other hand, im (f0,l ) = hw0 , v0 , yiZl Gl by definition, hence im (f0,l ) = hw0 , v0,0 , yiZl Gl
l−1
by v0 = v0,0 v0,1 with v0,1 ∈ UL2 0 = hv0,0 , yi and v0l−1−τ̂ = v0,0
, which follows from the Remark
2
following Lemma 1.2. Similarly w0,0 ∈ UL0 then implies that im (f0,l ) = hw0,1 , v0,0 , yi.
Combining the above we see that r0τ −1 generates coker f0,l ' L0 : for UL2 0 ⊂ im (f0,l ) and
Zl
2
1
2
L(L×
0 )/UL0 = w0,1 × UL0 /UL0 . It follows that f0,l is injective, by the earlier argument, which
also shows that Zl Gl /(τ − κ(τ γ)) is annihilated by (κ(γ)l−1 − 1)Zl = lZl , hence is isomorphic
to Fl hσi. But then α0 is an isomorphism by L0 ' Fl hσi.
Step 2 . m ≥ 1
Set U = G(Lm /L0 ) = hγm i. Deflating the l-completion of diagram (2.2a) from Gm to
Gm /U = G0 , as in the discussion there, we observe that (fm,l )U = f0,l up to the identification
13
×
× U
of Zl ⊗ RU
m with Zl ⊗ R0 (and L(Lm ) with L(L0 )) in the second special case of Lemma 2.2.
By naturality of the Tate cohomology groups H −1 , H 0 we get the commutative diagram
Zl ⊗ (Rm )U
Û
H 0 (U, Zl ⊗ Rm )
(fm,l )U ↓
(fm,l )U ↓
L(L×
m )U
Zl ⊗ RU
m
Û
'↓
L(L×
0)
H 0 (U, L(L×
m ))
because H −1 (U, Zl ⊗ Rm ) = 0 = H −1 (U, L(L×
m )) as U is cyclic. Moreover, the right vertical
arrow is an isomorphism because it dimension shifts in the l-completed diagram (2.2a) to the
identity map on H −2 (U, Zl ) 8 . Since (fm,l )U = f0,l , Step 1 implies that (fm,l )U is injective,
so the snake lemma gives an isomorphism
(coker fm,l )U ' coker f0,l
τ γm −1 to r τ −1 by 2. of Lemma 1.3.
taking (the class of) rm
0
Taking U -coinvariants of
αm
Zl Gl /(τ γm − κ(τ γ)) −→ coker fm,l coker αm
therefore shows that (αm )U = α0 , up to the identifications above, hence (coker αm )U '
coker α0 = 0 by Step 1. Now Nakayama’s lemma gives coker αm = 0.
By our observations before Step 1, fm,l is thus injective. Diagram (2.2a) gives the exact
sequence
0
Zl Gl ⊕ Zl Gl L(Vm
) coker fm,l ,
so Lemma 2.3 implies that coker fm,l is cohomologically trivial. Taking U -coinvariants of
αm
ker αm Zl Gm /(τ γm − κ(τ γ)) coker fm,l
then gives (ker αm )U = 0, because (αm )U = α0 is an isomorphism by Step 1 and
H −2 (U, coker fm,l ) = 0. Finally, Nakayama’s lemma implies ker αm = 0. Lemma 2.4 is
established.
3 . Transition to Ramachandra’s ϕ
Recall our task, namely to find a representing homomorphism of
Ω(l∞) ∈ K0 T (Zl G)
in HomGl (Rl (G), Fl× ) .
Recall also from the introduction that Ω(l∞) arises from the lifted-Ω construction with respect
to the data
Y
Sl∞
L(Kp× ) Y
Sl∞
L(Vp ) →
Y
Zl Gp Zl Sl∞
and ϕ(l∞) : Zl Sl∞ Sl∞
Y
L(Kp× ) .
Sl∞
If l | l, then, in our particular situation Kl = Lm and the l-part of the above 4-term sequence
is the sequence
L(L×
m ) L(Vm ) → Zl Gl ⊕ Zl Gl Zl ⊕ Zl Gl
8
Similarly H −1 (U, Zl ⊗ Rm ) ' H −1 (U, L(L×
m )) when U is not cyclic ... .
14
which naturally originates in the basic sequence studied in §2. In order to get the map ϕ(l∞)
involved we first define a map ιm : Z ⊕ ZGl Rm which, tensored with Zl over Z and
composed with fm,l , induces
×
×
ind (fm,l ιm,l ) : ind G
Gl (Zl ⊕ Zl Gl ) ⊕x L(Lm ) = ⊕x L(Kxl ) ,
with x ∈ G running through a system of coset representatives of the decomposition group Gl
of l in G. Of course, we identify Gl with Gl . Observe that, since k = Q and K is real,
ind G
Gl (Zl ⊕ Zl Gl ) = Zl G/Gl ⊕ Zl G ' Zl Sl∞ .
Now the two maps, ind (fm,l ιm,l ) and ϕ(l∞) are related by the triangle
g
Ql G/Gl ⊕ Ql G
ind (fm,l ιm,l ) &
←−
Ql G/Gl ⊕ Ql G
Ql ⊗Zl ⊕x L(Kx×l )
. ϕ(l∞)
which also defines the map g.
By [GRW, Proposition 1] we have Ω(l∞) = Ωind (fm,l ιm,l ) + ∂[Ql G/Gl ⊕ Ql Gl , g] with ∂ the
∂
connecting homomorphism K1 (Ql G) → K0 T (Zl G). In Lemma 3.1 we turn to a representing
homomorphism of Ωind (fm,l ιm,l ) and in Lemma 3.2 to one of ∂[Ql G/Gl ⊕ Ql G, g]. This provides
first information on Ω(l∞) which then will be exploited in the following section.
Definition.
ιm : Z ⊕ ZGl → Rm is the ZGl -map given by
ιm (1, 0) = σ̂bm , ιm (0, 1) = am + τd
γm cm .
On composing ιm with the embedding Rm ZGl ⊕ ZGl of Lemma 2.1 it follows readily that
ιm is injective. There is a lifted-Ω construction (see [RW2, §1.4]) associated to ιm , namely
from the data
Zl ⊗ Rm Zl Gl ⊕ Zl Gl → Zl Gl ⊕ Zl Gl Zl ⊕ Zl Gl and ιm,l = Zl ⊗ ιm : Zl ⊕ Zl Gl Zl ⊗Z Rm ,
where the first sequence is obtained from Rm ZGl ⊕ ZGl → ZGl Z (see Lemma 2.1) by
=
adding ZGl → ZGl to its right end and then tensoring with Zl over Z.
(
Lemma 3.1.
1. Ωιm,l is represented
9
by χ 7→
|Gl |2 if χ = 1
1
if χ 6= 1
in HomGl (Rl (Gl ), Fl× ).
^
2. [coker fm,l
ιm,l ] = [coker ιg
m,l ] + [coker fm,l ] in K0 T (Zl Gl ).
3. χ 7→ χ(τ γm ) − κ(τ γ) represents [coker fm,l ].
The notation in 2. is defined in [RW2, §1.4]; it will also be recalled in the proof. Throughout,
we drop all indices (except for the l in Zl and the m in Lm , Vm ) 10 and just write R rather
than Zl ⊗Z Rm . Moreover, elements in R, ∆G, Zl , Zl G are denoted by r, d, z, a, respectively,
and a(1) is the augmentation of a.
9
on irreducible characters
Hence Gl will be denoted by G until the end of proof. This will not cause confusion because the actual G
will not appear here.
10
15
In order to get a representing homomorphism of Ωι we first build the pull-back diagram
R ⊕ ∆G
∆G
α̃ ↓
|G| ↓
e
R Zl G ⊕ Zl G ∆G
R where e : R → Zl G ⊕ Zl G is the map
occurring in Lemma 2.1,
k
then build the push-out diagram
∆G
|G| ↓
∆G
Zl G ⊕ Zl G
d 7→ (d, 0) , (a1 , a2 ) 7→ (a1 (1), a2 )
β̃ ↓
k
(Zl ⊕ Zl G) ⊕ ∆G Zl ⊕ Zl G
and define
Zl ⊕ Zl G
with
d 7→ (0, d) , (z, a, d) 7→ (z, a)
ι̃ : Zl G ⊕ Zl G → Zl G ⊕ Zl G by ι̃ = α̃(ι ⊕ 1)β̃ .
It is easily checked that above we can take
for α̃ the map α̃(r, d) = (Ĝs)(d) + e(r) with s : ∆G → Zl G ⊕ Zl G a Zl -splitting of the
bottom sequence in the pull-back diagram
and for β̃ the map β̃(a1 , a2 ) = (a1 (1), a2 , (|G| − Ĝ)a1 ) .
Thus ι̃ takes
(1, 0)
to
(Ĝs)(|G| − Ĝ) + σ̂b
and
(0, 1)
to a + τcγc .
We use a calculation of [RW2, §6.3] for getting the 2 × 2 matrix in (Zl G)2×2 of ι̃ :
Because G = hσ, τ γi, we can choose s : ∆G → Zl G ⊕ Zl G to be the map
s (τ γ)i σ j − 1 =
(τ γ)i − 1
τγ − 1
, (τ γ)i
σj − 1 .
σ−1
Now, (Ĝs)(|G| − Ĝ) = g∈G g −1 s(g|G| − Ĝ)
P
P
= |G| g g −1 s(g − 1) + Ĝs(|G| − Ĝ) = (|G| − Ĝ) g g −1 s(g − 1)
P
(τ γ)i −1
i σ j −1
τ γ−1 , (τ γ) σ−1
i −1
P −j σ j −1
Ĝ) i,j σ −j (τ γ)−i (ττγ)
,
σ
γ−1
σ−1
P
P
−1
Ĝ) σ̂ i≥1 ( (τ γ) + · · · + (τ γ)−i , |τ γ| j≥1 (σ −1
= (|G| − Ĝ)
= (|G| −
= (|G| −
−j
−i
i,j σ (τ γ)
P
+ · · · + σ −j ) ) .
Consequently, by Lemma 2.1 ι̃ has the matrix
−1 + · · · + (τ γ)−i )
i≥1 ( (τ γ)
P
Ĝ)|τ γ| j≥1 (σ −1 + · · · + σ −j )
Ĝ + (|G| − Ĝ)σ̂
(|G| −
P
1−σ
!
τ γ − 1 + Ĝ
Its determinant equals
|G|Ĝ + (τ γ − 1)(|G| − Ĝ)σ̂
P|τ γ|−1
i=1
( (τ γ)−1 + · · · + (τ γ)−i )−
|σ|−1
−(|G| − Ĝ)|τ γ| (1 − σ)
X
(σ −1 + · · · + σ −j )
j=1
|
{z
=σ̂−|σ|
}
= |G|Ĝ + (|G| − Ĝ)σ̂(|τ γ| − τcγ) − (|G| − Ĝ)|τ γ|(σ̂ − |σ|)
= |G|Ĝ + (|G| − Ĝ)(σ̂|τ γ| − |τ γ|σ̂ + |G|) = |G|2 .
16
Therefore, χ 7→ |G|2 is a representing homomorphism
( of [coker ι̃] and Ωι is represented as
1
if χ = 1
claimed, since 2∂[∆G, |G|)] is represented by χ 7→
|G|2 if χ 6= 1 .
The assertion 2. compares two lifted-Ω constructions in which the 2-extension for Ωf ι is
obtained from that for Ωι by pushout along f as in (the l-completion of) diagram (2.2a)
=
(with Zl G → Zl G added to its right end). This implies the commutative diagram
→ (Zl ⊕ Zl G) ⊕ ∆G →
R ⊕ ∆G
→ Zl G ⊕ Zl G
k
k
(f, 1) ↓
↓
×
f
f ι : Zl G ⊕ Zl G → (Zl ⊕ Zl G) ⊕ ∆G → L(Lm ) ⊕ ∆G →
L(Vm )
,
ι̃ :
Zl G ⊕ Zl G
in short
(Zl G)2 (Zl G)2
ˇ↓
k
(Zl G)2 L(Vm )
↓
ˇ
coker f
coker ι̃
ˇ↓
fι
coker f
which gives 2. by the snake lemma.
Finally, Lemma 2.4 implies 3. The lemma is proved.
Corollary. Ω(l∞) − ∂[Ql G/Gl ⊕ Ql G, g] is represented by
(
χ 7→
|Gl |2 (χ(τ γm ) − κ(τ γ)) if res G
Gl (χ) = 1
χ(τ γm ) − κ(τ γ)
if res G
Gl (χ) 6= 1 .
This follows from the equality Ω(l∞) = Ωind (fm,l ιm,l ) + ∂[Ql G/Gl ⊕ Ql G, g] mentioned earlier,
from
^
Ωfm,l ιm,l = [coker fm,l
ιm,l ] − 2∂[∆Gl , |Gl |] = Ωιm,l + [coker fm,l ]
and from the commutative diagram (see [Fr, p.63])
HomGl (Rl (G), Fl× )
ind G
Gl ↑
∂
−→
∂
HomGl (Rl (Gl ), Fl× ) −→
K0 T (Zl G)
ind G
Gl ↑
K0 T (Zl Gl )
.
For the next lemma recall vm ∈ UL1 m from §1 and the elements ξ, a(l) ∈ K , t ∈ N as well as
the Q-idèle a from the introduction. We repeat that
a(l) ∈ L(Kl× ) is fixed by Gl and, when viewed in L(Ql × ) ⊂ L(Kl× ), is = lt/|Gl | uz for
some z ∈ Zl , as u generates 1 + lZl
a has component 1 except at l, where the component is u−1 .
In order to ease our notation we regard K as a subfield of Lm = Kl ; in particular, log ξ is an
unambiguous element of Lm 11 . Finally, the set {x} is always a set of coset representatives
of Gl in G, which contains 1 ∈ G. Each x defines an isomorphism Kl → Kxl by l - lim κi 7→
i→∞
xl - lim κxi for κi ∈ K.
i→∞
11
×
log : L×
m → Lm , with log(l) = 0, induces log : L(Lm ) → Lm .
17
Lemma 3.2. For an irreducible character χ ∈ Rl (G) set χl = res G
Gl χ, eχ =
eχl =
1
|Gl |
1
|G|
P
χ(h−1 )h,
h∈G
χl (h−1 )h. Let g be the map defined by the commutative triangle
P
h∈Gl
g
Ql G/Gl ⊕ Ql G
←−
ind (fm,l ιm,l ) &
Ql G/Gl ⊕ Ql G
Ql ⊗Zl ⊕x L(Kx×l ) .
. ϕ(l∞)
Then ∂[Ql G/Gl ⊕ Ql G, g] is represented by
χ 7→
−t|G|
+ |Gt[ξ|χ)
2
|Gl |2
l | log u
t[ξ|χ)
|Gl |2 log u
(l−1)[ξ|χ)
|Gl |(eχl log vm )









in HomGl (Rl (G), Fl× ) , where [ξ | χ) =
P
if χ = 1
if χ 6= 1 , χl = 1
if χl 6= 1
g∈G χ(g
−1 ) log(ξ g ) .
Proof. We begin by evaluating g on the two generators (1, 0) and (0, 1) of Ql G/Gl ⊕ Ql G.
The respective images are denoted by (ν, c) and (µ, d) where
ν = x νx x , µ = x µx x ∈ Ql G/Gl (and x 7→ x ∈ G/Gl )
P
P
c = x γx x , d = x δx x ∈ Ql G , so γx , δx ∈ Ql Gl .
P
P
Now, the south-east arrow in the triangle takes
1
(1, 0) to (u−1 , 1, . . . , 1), since fm,l (σ̂bm ) = (y σ̂ ) l−1 = u−1 (compare §2 and Lemma
1.2)
1
1
l−1
τd
γm ) l−1 with w = w
(0, 1) to (vm
l, 1, . . . , 1) , since fm,l (am +τd
γm cm ) = (vm wm
m
m,0 wm,1 ,
τ̂
τ
d
γ
m
wm,0 = 1 and (ζm − 1)
= l.
¿From the definition of ϕ(l∞) , given in the introduction, we see that the south-west arrow
takes
(1, 0) to a(l)a and (0, 1) to ξa .
Putting things together we obtain, in L(L×
m ), the equations
−1
γx /l−1 γx
l = a(l)x u−1 ,
u−νx vm
−1
δx /l−1 δx
u−µx vm
l = ξ x u−1 .
Since (ν, c), like (1, 0), is fixed by Gl , all γx are multiples of Ĝl . However, the left equation
above implies that γx has augmentation t/|Gl | or 0 according as x = 1 or x =
6 1. Therefore,
γ1 = Ĝl
t
|Gl |2
and γx = 0 for x 6= 1 .
Next we work with the right equation. In it every term except for possibly lδx is a unit, so
−1
δ /l−1
lδx = 1, i.e., δx has augmentation 0, and vmx
= ξ x uµx −1 remains. We apply Ĝl and
−1
get 1 = ξ Ĝl x u|Gl |(µx −1) , whence
µx = 1 −
Ĝl
−1
log(ξ x )/ log u ;
|Gl |
18
we also apply log directly and get
δx
−1
log vm − log(ξ x ) = (µx − 1) log u .
l−1
A representing homomorphism of ∂[Ql G/Gl ⊕ Ql G, g], by [GRW, p.75], is obtained from
evaluating det g | HomFl G (Vχ , Fl [G/Gl ] ⊕ Fl G) , where Vχ = (Fl G)eχ is an Fl G-module
affording χ. To do so we use the basis
1. eχ 7→ (eχ , 0) , eχ 7→ (0, eχ )
if χl = 1
of
2. eχ 7→ (0, eχ )
if χl 6= 1 .
HomFl G (Vχ , Fl [G/Gl ] ⊕ Fl G).
1. Case χl = 1, so χ(x) = χ(x) :
From g(eχ , 0) = (eχ ν, eχ c) , g(0, eχ ) = (eχ µ, eχ d) and from δx having augmentation 0 we
derive the matrix
!
P
P
x νx χ(x)
x µx χ(x)
|Gl | |Gtl |2
0
the determinant of which equals
− |Gtl |
P
P µx χ(x) = − |Gtl | x 1 −
x
P
P
= − |Gtl |
− |Gtl |
=
x χ(x) −
P
x χ(x)
−
1
|Gl | log u
Ĝl
|Gl |
χ(x) χ(h) log(ξ x
x; h∈Gl
1
|Gl | log u [ξ
−1
log(ξ x )/ log u χ(x)
| χ) =
−1 h−1
)
| {z }
 =1
 − t|G|2 +
|G |
if χ = 1

if χ 6= 1 .
l
t[ξ|χ)
|Gl |2 log u
t[ξ|χ)
|Gl |2 log u
2. Case χl 6= 1 :
Now g(0, eχ ) = (eχ µ, eχ d) = (0, eχ d) as µ is fixed by Gl , and
eχ d = χ(d)eχ =
X
χ(δx )χ(x)eχ =
x
X
χl (δx )χ(x)eχ .
x
Because χl 6= 1, the formula relating δx and µx yields
−1
χl (δx /l − 1)(eχl log vm ) − eχl log(ξ x ) = eχl (µx − 1) log u = 0 ,
−1
so χl (δx ) =
(l−1)eχl log(ξ x
eχl log vm
)
and
−1
P (l−1)eχl log(ξ x )
χ(x)
x χl (δx )χ(x) =
x
eχl log vm
P
P
−1
−1
= eχ l−1
χ(h−1 )h log(ξ x )
x χ(x)|Gl |
log
v
m
l
h∈Gl
P
−1
[ξ|χ)
l−1
−1
= |Gl |(eχ log vm ) x,h χ(xh ) log(ξ x h ) = eχ l−1
log vm |Gl |
l
l
P
.
The proof of Lemma 3.2 is finished. Combining the lemma with the previous corollary gives
Corollary.
Ω(l∞) is represented by



 (−t|G| +



χ 7→
t[ξ|χ)
log u )(1
− κ(τ γ))
t
log u (1 − κ(τ γ))[ξ | χ)
(l−1)[ξ|χ)
|Gl |(eχl log vm ) (χ(τ γm ) −
19
if χ = 1
if χ =
6 1 , χl = 1
κ(τ γ))) if χl 6= 1 .
4 . Ω(l∞)
Recall that K is the fixed field of c = complex conjugation in Q(ζ, ζm ). If K (l) is the fixed
field of Gl in K, then Q(ζ, ζm ) = K (l) (ζ) · K (l) (ζm ) and
G(Q(ζ, ζm )/K (l) ) = G(Q(ζ, ζm )/K (l) (ζ)) × G(Q(ζ, ζm )/K (l) (ζm )) .
−
Gl
−
Q(ζ, ζm )
@
@
K @
K (l) (ζ)
K (l) (ζm )
@
@
@
@
@
@ Q(ζm )
K (l) @
@
@
Q
Ql (ζ, ζm ) = Lm
@
@
@ Ql (ζm )
Ql (ζ)
@
@
@
Ql
−
Gl
−
The characters of G, or of Gl , will always be viewed as characters of G(Q(ζ, ζm )/Q), respectively of G(Q(ζ, ζm )/K (l) ), by means of inflation. Thus, if χl ∈ Rl (Gl ) is irreducible, then χl
can naturally be decomposed into the (tensor) product χl = χζ · χm with χζ inflated from
G(K (l) (ζ)/K (l) ) and χm inflated from G(K (l) (ζm )/K (l) ).
In order to comply with the notation of [RW2, §7], it would perhaps be appropriate here to
choose an embedding s : Qc → Ql c of the field Qc of all algebraic complex numbers into an
algebraic closure Ql c of Ql containing Fl and Lm such that s singles out the component Lm in
0
m+1
Ql ⊗Q K ⊂ Ql ⊗Q Ql c . Here, K ⊂ Qc by means of ζ = e2πi/n , ζm = e2πi/l
. Applying s−1
to characters χ of G with values in Fl provides characters with values in C to which a Galois
Gauß sum τ (s−1 χ) can be associated. However, there does not seem to be a real danger of
misunderstanding the notation τ (χ) which is just short for sτ (s−1 χ).
Observe that χm can viewed as a character of G(Q(ζm )/Q) = G(Ql (ζm )/Ql ) as well.
Definition. The conductor of χm is denoted by ljm +1 with jm ≥ −1.
In this connection we state the first lemma, the proof of which is clear.
Lemma 4.1. Let χl be an irreducible Fl -character of Gl and write, as before, χl = χζ · χm .
Let eχl , eχζ , eχm be the corresponding primitive idempotents. Furthermore, let dζ , dm be
elements in Ql (ζ), Ql (ζm ), respectively, and set d = dζ dm ∈ Lm . Then
eχl (d) = eχζ (dζ ) · eχm (dm )
in
Ql c .
The representing homomorphism of Ω(l∞) , which is displayed in the corollary at the end of
the previous section, includes the terms eχl log vm and [ξ | χ). The next lemma elaborates
on eχl log vm . Recall that Gl = hσ, τ, γm i; recall also the element β ∈ L. The scalar product
between two characters χ1 , χ2 of a group is denoted by (χ1 , χ2 ).
Lemma 4.2. Let χ ∈ Rl (G) be irreducible and set res G
Gl χ = χl = χζ · χm .
1. If χl = 1, then eχl log vm = 0.
20
2. If χl (τ ) 6= 1, then eχl log vm,1 = 0 and
eχl log vm,0 = l−m (χ(τ γm ) − 1)( χ(τ γm ) − κ(τ γ) )χ−jm (σ)(eχζ β)τ (χm )
12 .
3. If χl 6= 1 but χl (τ ) = 1, then eχl log vm,0 = 0 and
(
eχl log vm,1 =
1−χ(σ) G1 (0)
eχζ (β)( 1−χ(σ)/l
) lm ,
χm = 1
m −1))
τ (χm ) ,
−χ(σ)−jm eχζ (β) G1 (χ(γ
lm
χm 6= 1
where G1 (T ) ∈ Zl [[T ]] is the Deligne-Ribet power series defined by Ll (1 − s, 1) =
G1 (us − 1) / us − 1 , with 1 here the trivial character of G(Q(ζm )/Q).
We start the proof of the lemma from log vm = log vm,0 + log vm,1 and the formulas
log vm,0 = (l − 1 − τ̂ )(τ γm − 1)( τ γm − κ(τ γ) )σ −m m
i=0
σ−1
−m
log vm,1 = σ τ̂ l−σ (β) log u + β(γm − 1) log(ζm − 1)+
P
m
P
+
i=1
(σ−1)σ i−1 (β)
( log u−1
li
σ i (β) li
(ζm
li
− 1)
+ (γm − 1) log(ζm−i − 1) )
from §1.
1. follows from e1 log vm =
1
|Gl |
Ĝl ) =
log(vm
1
|Gl |
log 1 = 0 .
2. eχl τ̂ = 0 yields eχl log vm,1 = 0. On the other hand, eχm (ζm−i − 1) = eχm ζm−i , as χm 6= 1.
Thus, on account of Lemma 4.1,
eχl log vm,0 = (l − 1)(χ(τ γm ) − 1)(χ(τ γm ) − κ(τ γ))
m
X
χ(σ)−m+i
i=0
li
(eχζ β)(eχm ζm−i ) .
Reading χm as a character of (Z/lm+1 )× ' G(Q(ζm )/Q), we see that
eχm ζm−i =
1
(l − 1)lm
vanishes if m − i > jm , since the factor
a
χm (a−1 )ζm−i
X
a∈(Z/lm+1 )×
P
a∈G(Q(ζm )/Q(ζjm ))
a
ζm−i
= tr Q(ζm )/Q(ζjm ) ζm−i = 0 splits
off. However, if m − i < jm , then eχm ζm−i = 0 as well, by [La, Theorem 1.1, p.71]. So we
only need to consider
eχm ζjm =
1
(l − 1)lm
X
a∈(Z/lm+1 )×
χm (a−1 )ζjam =
lm−jm
(l − 1)lm
X
χm (a−1 )ζjam =
a∈(Z/ljm +1 )×
1
τc (χ̌m ) ,
(l − 1)ljm
√
def
where τc (χ̌m ) = Wχ̌m ljm +1 iδ is the classical Gauß sum of the contragredient character χ̌m :
see [Wa, p.29]. Here, Wχ̌m is the Artin root number of χ̌√
m , and δ is 0 or 1 according as
jm +1 iδ in the notation of [Ma,
χm (−1) = 1 or = −1. This classical Gauß sum is W (χ̌
)
√m l
p.14]. Passing to the Galois Gauß sum τ (χm ) = W (χ̌m ) ljm +1 W∞ (χm )−1 , as defined in [Ma,
1
p.48], with W∞ (χm ) = i−n(χm ,∞) = i−δ [Ma, p.24], we arrive at eχm ζjm = (l−1)l
jm τ (χm ) ,
which is 2.
For later reference we add that, by τc (χm )τc (χ̌m ) = ljm +1 χm (−1),
12
Here τ (χm ) is the Galois Gauß sum of the character χm of G(Q(ζm )/Q) as in the Definition. Admittedly,
the two occurrences of τ (as element in Gl and as Galois Gauß sum) is also a bit confusing.
21
τ (χm ) =
(4.2a)
ljm +1
τc (χm ) χm (−1) .
Finally, with respect to 3., we observe that χm (τ ) = 1 implies χm (−1) = 1, so χm is an even
character. Moreover, the factor l − 1 − τ̂ yields eχl log vm,0 = 0. On the other hand, and
again by Lemma 4.1,
h
eχl log vm,1 = (l−1)χ(σ)−m (eχζ β) χ(σ−1)
l−χ(σ) (χm , 1) log u+χ(γm −1)(eχm log(ζm −1))
i
m
(4.2b)
i−1
P χ(σ−1)χ(σ)
−1 + χ(γ − 1)(e
log(ζ
−
1))
.
(χ
,
1)
log
u
+
m−i
m
m
χ
i
m
l
i=1
When χm = 1 the claimed formula for eχ log vm,1 follows easily from
log u =
(4.2c)
which results from res s=1 Ll (s, 1) = 1 −
1
l
l
1−l G1 (0) ,
and G1 (us − 1) =
u1−s −1
s−1
· (s − 1)Ll (s, 1) .
We may thus assume χm 6= 1, hence jm ≥ 0, and compute
eχm log(ζm−i − 1) =
=
1
(l−1)lm

 0

li
(l−1)lm
P
a∈(Z/lm−i+1 )×
P
a∈(Z/lm+1 )×
a
χm (a−1 ) log(ζm−i
− 1)
a
χm (a−1 ) log(ζm−i
if m − i < jm
− 1) if m − i ≥ jm .
Above, the first equality is due to [Wa, Lemma 8.4, p.147], and the second follows from Galois
theory. By [Wa, Lemma 8.6, p.148], with fχ = F, g = l, t = lm−i−jm , and [Wa, 5.18, p.63]
the last expression equals
li
(l − 1)lm
χm (a−1 ) log(ζjam − 1) =
X
a∈(Z/ljm +1 )×
ljm +1 1
−
L
(1,
χ
)
.
m
l
(l − 1)lm−i
τc (χm )
Note that χm (l) = 0 by jm ≥ 0. We continue and use formulas (4.2a) and 2. prior to [RW2,
Proposition 5.4]. If 0 ≤ i ≤ m − jm , then
eχm log(ζm−i − 1) = −
1
Gχm (0)
1
G1 (χ(γm ) − 1)
τ (χm ) = −
τ (χm ) .
(l − 1)lm−i χ(γm − 1)
(l − 1)lm−i χ(γm − 1)
Substituting this into (4.2b) we get eχl log vm,1 =
(χ(γm )−1) τ (χm )
(l − 1)χ(σ)−m eχζ (β)χ(γm − 1) G1χ(γ
1−l
m −1)
=
−χ(σ)−m eχζ (β)G1 (χ(γm
−
m)
1)) τ (χ
lm
h
h
1
lm
+ χ(σ − 1)
1 + (χ(σ) − 1)
m−j
Pm
i=1
m−j
Pm
χ(σ)i−1 1
li
lm−i
i
i
χ(σ)i−1 ,
i=1
completing the proof of Lemma 4.2.
On putting the information from Lemma 4.2 together with that from the corollary at the end
of the previous section we obtain that Ω(l∞) is represented by χ 7→















−t(l−1)
G1 (0)l (1
−lm (l−1)
|Gl |
− κ(τ γ))[ξu−1 | χ)
if χl = 1
G0
(χ(γm )−κ(τ γ))[ξu−1 |χ) det(σ jm |Vχ /Vχ l )
G0
σ−1
eχζ (β)G1 (χ(γm −1))τ (χm ) det( 1−σ/l
|Vχ l )
χ(σ jm )[ξu−1 |χ)
lm (l−1)
|Gl | eχζ (β)τ (χm )(χ(τ γm )−1)
22
if χl 6= 1 , χ(τ ) = 1
if χ(τ ) 6= 1 ,
with G0l the inertia subgroup of Gl . Here the first two cases of the Corollary ending §3 are
unified by using [ξu−1 , χ), which equals [ξ, χ) unless χ = 1, and log u is eliminated by (4.2c).
The last case splits into two cases in Lemma 4.2, with 3. there unified by the relation
σ−1
G0
G0
det(
| Vχ l ) det(σ −jm | Vχ /Vχ l ) =
1 − σ/l
(
σ−1
χ( 1−σ/l
) , χm = 1
−j
m
χ(σ
) , χm 6= 1 .
Next we change this representing homomorphism by a factor in im (K1 (Zl G) → K1 (Ql G)) 13 ,
namely by
χ 7→ χ − G1 (γm − 1)(γm − κ(τ γ))−1 eτ + (τ γm − 1)(1 − eτ )
where eτ = τ̂ /l − 1. Observe that −G1 (γm − 1)(γm − κ(τ γ))−1 eτ + (τ γm − 1)(1 − eτ ) ∈
Zl G = Zl Geτ ⊕ Zl G(1 − eτ ) is indeed a unit. This follows readily by applying the irreducible
characters χ of G to it : if χ(τ ) 6= 1, then χ(τ γm ) − 1 ≡ χ(τ ) − 1 mod (ζm − 1); if χ(τ ) = 1,
×
then χ(γm −κ(τ γ)) ≡ 1−κ(τ ) mod (ζm −1). Finally, G1 (T ) satisfies G1 (0) = 1−l
l log u ∈ Zl ,
hence G1 (T ) ∈ Zl [[T ]]× by the Weierstraß preparation theorem.
We deduce that Ω(l∞) is represented by













t(l−1)
−1
l [ξu
χ 7→
| χ)
if χl = 1
G0
[ξu−1 |χ) det(σ jm |Vχ /Vχ l )
G0
σ−1
|σ|eχζ (β)τ (χm ) det( 1−σ/l
|Vχ l )
χ(σ jm )[ξu−1 |χ)
|σ|eχζ (β)τ (χm )
if χl 6= 1 , χ(τ ) = 1
if χ(τ ) 6= 1
because |Gl | = |σ|lm (l − 1). These cases are combined to give
Proposition 4.3. Ω(l∞) is represented by χ 7→
G0
Gl
G0
tdim Vχ [ξu−1 | χ) det(σ jm | Vχ /Vχ l ) det(1 − σ/l | Vχ l )
G0
|σ|eχζ (β)τ (χm ) det(σ − 1 | Vχ l /VχGl )
.
This is easily checked case by case, observing that |σ|eχζ (β) = σ̂(β) = 1 and τ (χm ) = 1 when
χl = 1.
We next modify the formula in Proposition 4.3 to facilitate comparison with (>l ). For this
purpose, we write our χ, inflated to G(Q(ζ, ζm )/Q) = G(Q(ζ, ζm )/Q(ζm ))×G(Q(ζ, ζm )/Q(ζ)),
as χ = χ̃ζ · χ̃m , with χ̃ζ inflated from G(Q(ζ)/Q) ' G(Q(ζ, ζm )/Q(ζm )) and χ̃m inflated from
G(Q(ζm )/Q) ' G(Q(ζ, ζm )/Q(ζ)). We let Sζ be the set of finite primes of Q(ζ) obtained by
restricting those in Q(ζ, ζm ) above S to Q(ζ). The symbol ≈ indicates an equality up to a
factor det(u | Vχ ) with a unit u of Zl G which is independent of χ (as at the end of §7.2 in
[RW2]).
Lemma 4.4.
log u (χ,1)
)
(−2t)τ (χ)L∗l (1, χ) det(1 −
1. [ξu−1 | χ) ≈ ( |G|2t
2. |σ|eχζ (β) ≈ τ (χ̃ζ )
Q
p∈Sζ,∗
D0
det(−Frp | V(χ̃ζp)p )−1
σ
l
note that the elements in this image represent the trivial element in K0 T (Zl G)
23
Q
ρ(ψ)
16=ψ|χ
with (χ̃ζ )p denoting the restriction
of χ̃ζ to the decomposition group Dp of p in G(Q(ζ)/Q).
13
G0
| Vχ l )−1
3. τ (χm ) = τ (χ̃m ) and τ (χ) = τ (χ̃ζ )τ (χ̃m )χ(σ −jm −1 )χm (fχ˜ζ )−1 .
The assertion 1. is [RW2, Corollary in §8
det(
X
14 ],
with al = u−1 , because G̃ = G and
sx log(ξu−1 )x−1 | Vχ ) = det(
x∈G
X
logl (s(ξ x )u−1 )x | Vχ )
x∈G
is [ξu−1 | χ) in our present notation in which s is omitted and logl is written log (compare
with the notation before Lemmas 7.8 and 8.1 of [RW2]).
For 2., we apply [RW2, Theorem E 15 ] to the extension Q(ζ)/Q (in place of K/k there). Then
χ̃ζ restricts to χζ ,
an admissible S 0 .
Q
p∈Sl∗
Nsp kp /Ql (sp (bp )|χ0p ) becomes
|σ|−1
P
i=0
χζ (σ −i )σ i (β) = |σ|eχζ (β) and Sζ is
For 3. we first note that χ̃m is the inflation of χm 16 under G(Q(ζ, ζm )/Q) → G(Q(ζm )/Q). So
the first equality is due to the invariance of our Galois Gauß sums with respect to inflation,
as follows from [Ma, pp.18,22,48], and from the invariance of W∞ (ω) = 1 or i−1 according
as ω(−1) = 1 or = −1 (see [Ma, p.24]), which is preserved for ω = χm since the complex
conjugation on Q(ζ, ζm ) has nontrivial restriction to Q(ζm ).
Actually we also need to observe that the above, with ω = χ, χ̃ζ , also applies to the invariance
of the Galois Gauß sums in 1. and 2.
Since the conductors fχ̃ζ and ljm +1 of χ̃ζ , χ̃m , respectively, are coprime, we have
τ (χ) = τ (χ̃ζ )τ (χ̃m )χ̃ζ (ljm +1 )−1 χ̃m (fχ̃ζ )−1 .
Observing that σ ∈ Gl induces the Frobenius automorphism ζ 7→ ζ l on Q(ζ)/Q, hence
χ̃ζ (l) = χ(σ), and that χ̃m inflates χm , the lemma is proved.
Substituting 1.,2. of the lemma into Proposition 4.3 and taking account of the cancellations
G0
provided by 3. and by det(σ | Vχ /Vχ l ) = χ(σ)1−(χm ,1) , jm (χm , 1) = −(χm , 1), we obtain the
χ 7→
Corollary. Ω(l∞) is represented by
Gl
log u (χ,1)
tdim Vχ ( |G|2t
)
(−2t)L∗l (1, χ)χ(σ)(χm ,1)−1 χm (fχ̃ζ )−1
G0
× det(σ − 1 | Vχ l /VχGl )−1
Q
p∈Sζ,∗
D0
det(−Frp | V(χ̃ζp)p )
Q
ρ(ψ) .
16=ψ|χ
This is to be compared with the χ-value in formula (>l ) in the introduction, which in our
special case k = Q , K = Q(ζn )+ reads
G0
Gl
(>0l )
log u (χ,1)
tdim Vχ ( |G|2t
)
(−4t)L∗l (1, χ) det(σ − 1 | Vχ l /VχGl )−1
×
Q
G0
det(−Frp | Vχ p )
p∈S∗
Q
ρ(ψ)
16=ψ|χ
G0
because al = u−1 , Sl∗ = {l} and σ = Frl on Vχ p . We analyze the difference in these formulas.
14
in which the first logl should read log
whose proof neither requires K to be real nor χ to be even
16
in the notation of 2. of Lemma 4.2
15
24
Lemma 4.5. For prime numbers p 6= l, let Fp denote the automorphism ζ 7→ ζ , ζm 7→ ζm p
of Q(ζ, ζm ). Then
Q
1.
D0
p∈Sζ,∗
Q
p∈S∗
det(−Frp |V(χ̃ p) )
ζ p
G0
det(−Frp |Vχ p )
2. χm (fχ̃ζ ) ≈
= χ(−σ)1−(χm ,1)
Q
G0
det(Fp | Vχ p )−1
l6=p∈S∗
G0
det(Fp | Vχ p )−1
Q
l6=p∈S∗
Choose a prime P of Q(ζ, ζm ) above each
p ∈ S∗ and set Pζ = P ∩ Q(ζ). Putting
G0 = G(Q(ζ, ζm )/Q) we have
hci 0
G
res G
G0p χ = 1 ⇐⇒ res (G0 )0 χ = 1
G0
G
∪
∪
(G0 )0P G0p
P
because χ is real (inflated to G0 ).
For 1., note that p ↔ Pζ is a bijection S∗ ↔ Sζ,∗ , so we may study the contribution of each
Q
Q
p to our ratio Sζ,∗ / S∗ .
0
G
If p 6= l, then (G0 )0P ≤ G((Q(ζ, ζm )/Q(ζm )) hence [ res G
G0p χ = 1 ⇐⇒ res (G0 )0P χ̃ζ = 1 ], and for
such p we have
0
DP
G0
det(−Frp | Vχ p ) = −χ(Frp ) = −χ̃ζ (FrP )χ̃m (FrP ) = det(−FrPζ | V(χ̃ζ )ζP )χ̃(Fp )
ζ
because FrP and Fp agree on Q(ζm ); thus the p-contribution to the ratio is χ̃m (Fp )−1 =
G0
p −1
χ(Fp )−1 in this case. In the other case, res G
G0p χ 6= 1, it is 1, which is what det(Fp | Vχ )
gives in both cases.
When p = l, then (G0 )0P = G(Q(ζ, ζm )/Q(ζ)) so [ res G
G0p χ = 1 ⇐⇒ χ̃m = 1 ]. If χ̃m = 1
Q
Q
then χ(−Frl ) = χ̃ζ (−FrPζ ) hence l contributes 1 to the ratio Sζ,∗ / S∗ . If χ̃m 6= 1 then l
contributes 1 to
Q
S∗ , but the contribution of Pζ to
Q
0
DP
Sζ,∗
is still det(−FrPζ | V(χ̃)Pζ ) = χ(−σ),
ζ
because σ is FrPζ on Q(ζ) and the identity on Q(ζm ).
Q
For assertion 2., factor n0 = p pnp , noting that each p 6= l and that only such p|n0 can
contribute to fχ̃ζ . Correspondingly, each p ∈ S∗ not dividing ln0 has G0p = 1, hence det(Fp |
G0
Vχ p ) = χ(Fp ) ≈ 1 because the image of Fp in Zl G is a unit. So we may again focus on the
G0
(p)
p-parts of our formula for p|n0 , and must show χm (fχ̃ζ ) ≈ det(Fp | Vχ p )−1 .
np
The inertia subgroup (G0 )0P of P|p is G(Q(ζ, ζm )/Q(ζ p , ζm )) ' (Z/pnp Z)× and the ν th
ramification group (G0 )νP corresponds to {a ∈ (Z/pnp Z)× : a ≡ 1 mod pν } for 1 ≤ ν ≤ np
([Se, p.86]). Define
\
0 )ν ∈ Z G0 , 1 ≤ ν ≤ n
e0p,ν = pν−np (G
p
l
P
n
ε0p = Fp p +
Pnp −1
ν=1
e0p,ν (Fpν − Fpν+1 ) .
These ε0p are units in Zp G0 : applying an irreducible character χ0 of G0 gives
χ0 (ε0p ) = χ0 (Fptp )
0
with tp = tp (χ0 ) ≥ 1 minimal with res G
t
(G0 )Pp
χ0 = 1.
25
G0
G0
Letting εp be the image of ε0p in (Zp G)× , it follows that det(εp | Vχ p ) = det(Fp | Vχ p ) and
that


1
, res G
G0p
G0p χ = 1
det(εp | Vχ /Vχ ) =
tp
 χ(Fp ) , res G
G0 χ 6= 1 .
p
G0
(p)
0
G
This implies χm (fχ̃ζ ) = det(εp | Vχ /Vχ p ). For [ res G
G0 χ 6= 1 ⇐⇒ res (G0 )0 χ 6= 1 ], and in
p
this case
(p)
fχ̃ζ
=
ptp
([Se, pp.109,81/82]), hence
G0
G0
P
(p)
χm (fχ̃ζ )
t
tp
= χm (Frp ) = χ(Fpp ) = det(εp |
(p)
Vχ /Vχ p ). In the other case, Vχ p = Vχ and fχ̃ζ = 1 since χ trivial on (G0 )0P implies χ̃ζ trivial
on the inertia group of Pζ in Q(ζ)/Q.
G0
(p)
But χ 7→ det(εp | Vχ ) is ≈ 1, by εp ∈ (Zl G)× , hence χ(fχ̃ζ ) ≈ det(εp | Vχ p )−1 = det(Fp |
G0
Vχ p )−1 and the lemma is proved.
We finally prove the Theorem. Dividing the formula of the last corollary by that of (>0l ) and
applying the lemma gives
D0
Q
1
(χm ,1)−1 χ (f )−1
m χ̃ζ
2 χ(σ)
Sζ,∗
det(−Frp |V(χ̃ p) )
ζ p
G0
det(−Frp |Vχ p )
S∗
(−1)(χm ,1)
1−(χm ,1)
Q
≈ 21 χ(σ)(χm ,1)−1 χ(−σ)
=
−2
≈ (−1)(χm ,1) .
G0
Since det(−1 | Vχ l ) = (−1)(χm ,1) we are done.
5 . Theorem F without tameness at l
In our special case Q(ζn )+ /Q, our Theorem gives the formula (8.5) of [RW2] with an extra
G0
factor det(−1 | Vχ l ), but without assuming tameness of l 6= 2 in Q(ζn )+ /Q. Based on (8.5),
the remaining calculation proving Theorem F in [RW2] does not again invoke the tameness
hypothesis, so we may just repeat the calculation including the extra factor. This yields the
desired result in our special case.
It may now seem that the new and old (8.5) contradict each other: but in the tame case the
G0
factor χ 7→ det(−1 | Vχ l ) is ≈ 1 by Lemma 8.4. Indeed in [RW2], this was used at the end
of the proof to eliminate the extra factor.
For a general extension K/k of totally real cyclotomic fields we choose an n so that K ⊂ Q(ζn )+
and reduce to the special case Q(ζn )+ /Q in the following way. The assertion of Theorem F is
that the class ω (l) (K/k) in K0 T (Zl G(K/k)) is represented by χ 7→ χ(Υ)
Now (by [GRW, Theorem 30 ])
Q
G0
p
S∗−
((Npk )− dim Vχ .
defl : K0 T (Zl G(Q(ζn )+ /Q)) → K0 T (Zl G(K/Q)) takes ω (l) (Q(ζn )+ /Q) to ω (l) (K/Q)
and res : K0 T (Zl G(K/Q)) → K0 T (Zl G(K/k)) takes ω (l) (K/Q) to ω (l) (K/k) .
G0
p
The class c(K/k) ∈ K0 T (Zl G(K/k)) represented by χ 7→ S∗− (Npk )− dim Vχ behaves in the
same way : in view of [Fr, p.63] the proof of this is contained in the inflation/induction
properties of Euler factors of Artin L-functions. So it remains to show that b(K/k) ∈
K0 T (Zl G(K/k)) represented by χ 7→ χ(Υ) has the deflation/restriction properties : for these
Q
26
can then be applied to the relation ω (l) (Q(ζn )+ /Q) = b(Q(ζn )+ /Q) + c(Q(ζn )+ /Q) of the
special case to deduce the general one.
To discuss b(K/k), let K∞ be the cyclotomic l-extension of K and write G = G(K/k) , G∞ =
G(K∞ /k). There are “deflation” maps ρ (from [RW2, Appendix 4B])
∂
K1 (R−1 Zl [[G∞ ]]) −→ K0 R(Zl [[G∞ ]])
ρ↓
ρ↓
K1 (Ql G)
∂
−→
K0 T (Zl G)
making the square above commute and, by definition of Υ ([RW2, Proposition 5.10]), it
follows that b(K/k) is the image of fS − ∂(ΘS ) under ρ : K0 R(Zl [[G∞ ]]) → K0 T (Zl G). We
will show, in §6, that the classes fS − ∂(ΘS ), at the top of the Iwasawa tower, do have
deflation and restriction. The desired properties of b(K/k) then follow from the commutative
squares below
defl
res
K0 R(Zl [[G∞ ]]) −→ K0 R(Zl [[G0∞ ]])
ρ↓
ρ0 ↓
K0 R(Zl [[G∞ ]]) −→ K0 R(Zl [[G∞ ]])
ρ↓
ρ↓
K0 T (Zl G)
defl
−→
K0 T (Zl G)
compatible with the notation
17
K0 T (Zl G)
res
−→
K0 T (Zl G0 )
of §6 (and verified via [RW2, Lemma 4B.1])
6 . Appendix
This section is actually an addendum to [RW2, §4, §5.1] and the notation is taken from
there 18 . We discuss how f ∈ K0 R(Zl G∞ ) and Θ ∈ K1 (R−1 Zl G∞ ) behave with respect to
deflation and restriction.
Recall the basic situation : l is a fixed odd prime number, k a totally real finite extension of
Q, k∞ the cyclotomic l-extension of k and K∞ a finite extension of k∞ which is Galois, with
group G∞ , over k.
Recall also that the elements f and Θ depend only on a fixed sufficiently large finite set of
primes of k, which contains all divisors of l∞, and on a fixed generator γk of Γk = G(k∞ /k)
([RW2, pp.8,9]). Moreover, the definitions of f and Θ in [RW2] require Leopoldt’s conjecture
and G∞ abelian, respectively, which will be tacitly assumed at the appropriate places below.
Lemma A1. Let G∞ = G∞ /N with N a finite normal subgroup of G∞ , and put K ∞ = K∞ N .
Then
1. defl: K0 R(Zl G∞ ) → K0 R(Zl G∞ ) takes f(K∞ /k) to f(K ∞ /k) ,
2. defl: K1 (R−1 Zl G∞ ) → K1 (R−1 Zl G∞ ) takes Θ(K∞ /k) to Θ(K ∞ /k) .
The first assertion is [RW2, Proposition 4.8] and the second follows, via the identification
K1 (R−1 Zl G∞ ) = (R−1 Zl G∞ )× ([RW2, Lemma 5.5]), from [RW2, Proposition 5.4] which
characterizes Θ already in M ∩ (R−1 Zl G∞ )× by χ(Θ) = Gχ,S (0) for all abelian l-adic characters χ of G∞ with open kernel. Indeed, given such a character χ of G∞ /N , then
χ(Θ) = (infl χ)(Θ) = Ginfl χ (0) = Gχ (0)
with the last equality due to Ll (s, infl χ) = Ll (s, χ) . Lemma A1 is proved.
17
18
here G is intended to be any common factor group of G and G∞ for which ρ is defined
except for often omitting an index S
27
Lemma A2. Let G0∞ be an open subgroup of G∞ and k 0 its fixed field. Let γk0 be the unique
0 /k 0 ) which restricts to γ m in Γ , where m = [k 0 ∩ k
generator of Γk0 = G(k∞
∞ : k].
k
k
Then
1. res : K0 R(Zl G∞ ) → K0 R(Zl G0∞ ) takes f(K∞ /k) to f(K∞ /k 0 ) ,
2. res : K1 (R−1 Zl G∞ ) → K1 (R−1 Zl G0∞ ) takes Θ(K∞ /k) to Θ(K∞ /k 0 ) ,
with the set S 0 for K∞ /k 0 consisting of the primes of k 0 above those in the set S for
K∞ /k.
For the proof we let H be the kernel of G∞ → Γk , observing that H 0 = H ∩ G0∞ is then the
kernel of G0∞ → Γk0 . We should also point out that the multiplicative set R ([RW2, p.32]) is
not as specific as it may seem: for if, in the notation there, K/k ⊂ K̃/k are finite Galois inside
K∞ /k, then taking the determinant of multiplication by x ∈ Zl ΓK on the free Zl ΓK̃ -module
Zl ΓK gives norm(x) ∈ Zl ΓK̃ which has x as a factor in Zl ΓK ; inverting norm(x) thus inverts
x. This allows us to take R inside Zl G0∞ below.
Three commutative diagrams are basic for the proof of 1.
∆G0∞
ˇ↓
∆G∞
↓
ˇ
F
=
Zl G0∞
ˇ↓
Zl G∞
↓
ˇ
F
Zl
k
Zl
X∞
k
X∞
0
Y∞
ˇ↓
Y∞
↓
ˇ
F
∆G0∞
ˇ↓
∆G∞
↓
ˇ
=
F
Zl G0∞
ˇ↓
Zl G∞
↓
ˇ
F
(2)
0
Y∞
ˇ↓
Y∞
↓
ˇ
F
coker Ψ0
(1) ˇ
↓
coker Ψ
↓
ˇ
C
All occurring modules are viewed as Zl G0∞ -modules and all non-marked maps are the natural
ones. The module F is, of course, free. The middle diagram arises from the translation
functor in analogy to [RW2, (4.∞)]. The right diagram depends on a compatible choice
of maps Ψ0 and Ψ, as defined prior to [RW2, Proposition 4.5], to make the top left square
0 to form a
commute. To construct these we choose a lift γ 0 ∈ G0∞ of γk0 and any c0∞ , d0∞ , y∞
0
0
m
Ψ . We then choose a lift γ ∈ G∞ of γk , observing that then γ = γ h0 with h0 ∈ H, and
define
c∞ = c0∞ [(1 + γ + · · · + γ m−1 )e + (γ 0 − 1)(e0 − e) + (1 − e0 )]
in the natural notation. From e0 e = e , h0 e = e it follows that d∞ = c∞ ( (γ − 1)e + (1 − e) )
0 under Y 0 → Y
is d0∞ , hence the image y∞ of y∞
∞ in the middle diagram permits us to form
∞
a compatible Ψ.
This gives the top half of the right diagram, and then the full diagram by applying the snake
lemma. In it, the arrow (1) is injective since its kernel, which is R-torsion ([RW2, Proposition
4.5]), embeds in the free Zl G0∞ -module F . This also implies injectivity of arrow (2), which is
easily checked, from the middle diagram, to be right multiplication by d∞ = d0∞ .
Now the bottom row of the right diagram shows C ' F/F d0∞ , hence
[C] = ∂ 0 (F, d0∞ ) = ∂ 0 (Zl G∞ , d0∞ ) − ∂ 0 (Zl G0∞ , d0∞ )
by the left column. And the right column together with the definition of f = f(K∞ /k) , f0 =
f(K∞ /k 0 ) gives
res f − f0 = ∂(Zl G0∞ , c0∞ ) + [C] − ∂ 0 (Zl G∞ , c∞ ) .
19
19
where the R−1 after
∂0(
is suppressed
28
Substituting [C], with d0∞ = d∞ on Zl G∞ , we get
res f − f0 = ∂ 0 (Zl G0∞ , c0∞ ) − ∂ 0 (Zl G0∞ , d0∞ ) + ∂ 0 (Zl G∞ , d∞ ) − ∂ 0 (Zl G∞ , c∞ )
= ∂ 0 (Zl G∞ , (γ − 1)e + (1 − e)) − ∂ 0 (Zl G0∞ , (γ 0 − 1)e + (1 − e0 )) = ∂ 0 res [Zl G∞ , (γ − 1)e + (1 − e)] − [Zl G0∞ , (γ 0 − 1)e + (1 − e0 )]
= ∂ 0 infl res [Zl Γk , γk − 1] − [Zl Γk0 , γk0 − 1] ,
because of the commutative square
in which the lower restriction in the diagram comes from the inclusion Γk0 → Γk
which is compatible with G0∞ ,→ G∞ , and
therefore takes γk0 to γkm .
res
K1 (R−1 Zl G∞ ) −→ K1 (R−1 Zl G0∞ )
infl ↑
infl ↑
res
−1
K1 (R Zl Γk ) −→ K1 (R−1 Zl Γk0 )
To compare [Zl Γk0 , γk0 −1] and res [Zl Γk , γk −1] we use the isomorphism det : K1 (R−1 Zl Γk0 ) →
(R−1 Zl Γk0 )× ([RW2, Lemma 5.5]) which sends the first to γk0 − 1 and the second to the
determinant of the R−1 Zl Γk0 -endomorphism γk −1 of R−1 Zl Γk : using the
basis 1, (γk −1), · · · ,
P
m
i
(γk − 1)m−1 the relation (γk − 1)m−1 · (γk − 1) = (γkm − 1) · 1 − m−1
(γ
k − 1) implies that
i=1
i
m−1
this determinant is (−1)
(γk0 − 1). These agree since m, which is a power of l, is odd.
−1
−1
×
∞
We now turn to 2. of Lemma A2. The map res G
G0∞ takes Θ ∈ K1 (R Zl G∞ ) = (R Zl G∞ )
to the determinant of the R−1 Zl G0∞ -endomorphism x 7→ xΘ of R−1 Zl G∞ . Now lr Θ ∈ Zl G∞
for some r (by Θ ∈ M as in the proof of Lemma A1) and Zl G∞ free over Zl G0∞ implies
m
that the determinant of lr Θ is in Zl G0∞ , hence lr res Θ ∈ Zl G0∞ . It follows from the second
paragraph of the proof of [RW2, Proposition 5.4] that res Θ is characterized by the values
χ0 (res Θ) with χ0 varying over l-adic characters of G0∞ with open kernel.
Every such χ0 factors through some finite quotient G0 of G0∞ and ρ0 res Θ has representing homomorphism
0
χ0 7→ χ0 (res Θ) = (ind G
G0 χ )(ρΘ) =
Y
res
K1 (R−1 Zl G∞ ) −→ K1 (R−1 Zl G0∞ )
ρ↓
ρ0 ↓
res
K1 (Ql G)
−→
K1 (Ql G0 )
χ0 (Θ)
χ|χ0
([Fr, p.63]), where χ runs through all abelian characters of G∞ extending χ0 .
Thus χ0 (res Θ) = χ|χ0 χ(Θ) for all χ0 , and res Θ must agree with Θ0 = Θ(K∞ /k 0 ) if we
Q
can show that χ0 (Θ0 ) = χ|χ0 χ(Θ) for all χ0 . This follows from the induction property
Q
0
χ|χ0 Ll,S (1 − s, χ) = Ll,S 0 (1 − s, χ ) of (truncated) l-adic L-functions. Namely, substituting
the Deligne-Ribet power series into this (compare [RW2] after the proof of Proposition 5.3)
and noting that γk0 acts by um if γk acts by u on ζl∞ we get
Q
Q
(∗)
χ|χ0
Gχ,S (us −1)
Gχ0 ,S 0 (ums −1)
Q
=
χ|χ0
Hχ (us −1)
Hχ0 (ums −1)
(
=
(−1)m−1 if χ0|H 0 = 1
1
if χ0|H 0 6= 1 .
For χ0|H 0 6= 1 implies χ|H 6= 1 for all χ|χ0 ; and if χ0|H 0 = 1 then there are m characters χ|χ0 with
χ|H = 1, which can be numbered
χ0 , · · · , χm−1 with χi (γk ) = χ1 (γk )ζ i , with ζ a primitive
Q
m-th root of unity. Then
χ|χ0
Hχ
Hχ (us −1)
0 (ums −1)
Qm
=
( χ0 (γk )ζ i us −1 )
i=1
0
χ (γk0 )ums −1
(−1)m−1 . Since m is odd, substituting s = 0 gives
Q
χ(Θ)
χ|χ0
χ0 (Θ0 )
χ (γ m )ums −1
= (−1)m−1 χ00 (γ k0 )ums −1 =
k
= 1 and Lemma A2 is proved.
Remark. When l = 2 , f − ∂Θ still has deflation and restriction and an easy modification
of f, Θ would also recover Lemma A1 and A2.
29
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[BB]
[Co]
[Fr]
[Gr]
[GRW]
[GW]
[HS]
[La]
[Ma]
[RW1]
[RW2]
[Se]
[Sn]
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Institut für Mathematik · Universität Augsburg · 86135 Augsburg · Germany
Department of Mathematics · University of Alberta · Edmonton T6G 2G1 · Canada
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