Documenta Math. 191
Erratum for
“On the Parity of Ranks of Selmer Groups III”
cf. Documenta Math. 12 (2007), 243–274
Jan Nekov´aˇr
Received: March 31, 2008
Communicated by Ulf Rehmann
Abstract. In the manuscript “On the Parity of Ranks of Selmer Groups III” Documenta Math. 12 (2007), 243–274, [1], Remark 4.1.2(4) and the treatment of archimedeanε-factors in 4.1.3 are incor-rect. Contrary to what is stated in 0.3, the individual archimedean
ε-factorsεu(M) (u| ∞) cannot be expressed, in general, in terms of
Mp, but their product can.
2000 Mathematics Subject Classification: 11G40 11R23
To motivate the corrections below, consider a motive M (pure of weight w) over F with coefficients in L. Set Se∞ ={τ : F ֒→ C}, Sep ={σ: F ֒→Qp} and denote byr∞:Se∞−→S∞,rp:Sep −→Sp the canonical surjections. Fix an embedding ι : L ֒→ C and an isomorphism λ : Qp −→∼ C such that p is induced by ιp=λ−1◦ι:L ֒→Qp. To eachv∈Sp then corresponds a subset
S∞(v) ={r∞(λ◦σ)|rp(σ) =v} ⊂S∞
such that
X
w∈S∞(v)
[Lw:R] = [Fv :Qp].
For eachτ ∈Se∞, the Betti realizationMB,τ is an L-vector space and there is a Hodge decomposition
MB,τ⊗L,ιC= M
i∈Z
(ιMτ)i,w−i.
192 Jan Nekov´aˇr
The corresponding Hodge numbers
hi,w−i(ιMu) :=hi,w−i(ιMτ) = dimC(ιMτ)i,w−i
depend only onu=r∞(τ)∈S∞. The de Rham realizationMdRis a freeL⊗Q
F-module; its Hodge filtration is given by submodulesFrM
dR(not necessarily free) which correspond, under the de Rham comparison isomorphism
MdR⊗L⊗QF,ι⊗τC−→∼ MB,τ⊗L,ιC,
to
(FrMdR)⊗L
⊗QF,ι⊗τC−→∼ M
i≥r
(ιMτ)i,w−i,
hence
dimC (grFiMdR)⊗L⊗QF,ι⊗τC=hi,w−i(ιMτ).
The p-adic realization Mp of M is isomorphic, as an Lp-vector space, to
MB,τ⊗LLp (for anyτ∈Se∞). For eachv∈Sp,DdR(Mp,v) is a freeLp⊗QpF
v-module equipped with a filtration satisfying
DrdR(Mp,v)
∼
−→FrMdR⊗L⊗QF (Lp⊗QpFv). This implies that, for eachi∈Z, the dimension
div(Mp) := dimLp D
i
dR(Mp,v)/DdRi+1(Mp,v)
is equal to
dimLp gr
i
FMdR⊗L⊗QF(Lp⊗QpFv)
= dimQ
p gr
i
FMdR⊗L⊗QF,ιp⊗incl(Qp⊗QpFv)
= X
σ:Fv֒→Qp
dimQ
p gr
i FMdR
⊗L⊗QF,ιp⊗σQp
= X
u∈S∞(v)
[Fu:R]hi,w−i(ιMu),
hence
d−
v(Mp) :=
X
i<0 i di
v(Mp) =
X
u∈S∞(v)
[Fu:R]d−(ιMu),
d−(ιMu) :=X i<0
i hi,w−i(ιMu). (⋆)
Erratum for “On the Parity of Ranks . . .” 193
Corrections to §4.1 and §5.1: firstly, 4.1.2(4) and 5.1.2(9) should be deleted. Secondly,§4.1.3 should be reformulated as follows: we assume thatV
satisfies 4.1.2(1)-(3). For eachv∈Spwe define
d−
definition gives the correct product of archimedeanε-factors. The formula (4.1.3.6) should be replaced by
∀v∈Sp εe(Vv) = (−1)d
wherer2(F) denotes the number of complex places ofF.
Corrections to Theorem 5.3.1 and its proof: the statement should say that, under the assumptions 5.1.2(1)-(8), the quantity
(−1)h1f(F,V)/ε(V) = (−1)eh
In the proof, a reference to (4.1.3.7) should be replaced by that to (4.1.3.7’), which yields
194 Jan Nekov´aˇr
Corrections to §5.3.3: the first question should ask whether (−1)d−v(V)ε(W D(Vv)N−ss) (v∈Sp)
depends only on Vv?
Corrections to §5.3.4-5: it is often useful to use a slightly more general version of Example 5.3.4 with Γ = Γ0×∆, where Γ0 is isomorphic toZp and ∆ is finite (abelian). Given a characterα: ∆−→ O∗
p, set
R=Op[[Γ0]], T = T⊗OpOp[[Γ
+]]⊗
Op[∆],αOp,
T+
v = Tv+⊗OpOp[[Γ+]]
⊗Op[∆],αOp (v∈Sp).
As in 5.3.4(2)-(3), T is an R[GF,S]-module equipped with a skew-symmetric
R-bilinear pairing (, ) :T × T −→R(1) inducing an isomorphism
T ⊗Q−→∼ HomR(T, R(1))⊗Q.
In 5.3.4(5) we have to replaceβ: Γ−→Lp(β) byβ : Γ0−→Lp(β); then
TP/̟PTP = IndGGF,SF
0,S(V ⊗(β×α)).
In 5.3.5, we set, for anyLp[Γ]-moduleM,
M(β×α)={x∈M⊗L
pLp(β)| ∀σ∈Γ σ(x) = (β×α)(x)};
then
Hf1(F,TP/̟PTP) =Hf1(F0, V ⊗(β×α))
= Hf1(Fβ, V)⊗(β×α)
Gal(Fβ/F0)
=Hf1(Fβ, V)(β
−1
×α−1 )
and
τ:Hf1(Fβ, V)(β
−1
×α−1 ) ∼
−→Hf1(Fβ, V)(β×α).
Applying Corollary 5.3.2, we obtain, for any pair of characters of finite order
β, β′ : Γ
0−→L
∗
p, that
(−1)h1f(F0,V⊗(β×α))/ ε(F0, V ⊗(β×α))
= (−1)h1
f(F0,V⊗(β
′
×α))/ ε(F0, V ⊗(β′×α)). (5.3.5.1′)
References
[1] Jan Nekov´aˇr, On the Parity of Ranks of Selmer Groups III, Documenta Math. 12 (2007) 243–274
