Erratum for

“On the Parity of Ranks of Selmer Groups III”

cf. Documenta Math. 12 (2007), 243–274

Jan Nekovář

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 archimedean ε-factors in 4.1.3 are incorrect. Contrary to what is stated in 0.3, the individual archimedean ε-factors ε_{u}(M) (u | ∞) cannot be expressed, in general, in terms of M_{𝔭}, 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 S̃_{∞} = \{τ : F ↪ C\}, S̃_{p} = \{σ : F ↪ \overline{Q}_{p}\} and denote by r_{∞} : S̃_{∞} ⟶ S_{∞}, r_{p} : S̃_{p} ⟶ S_{p} the canonical surjections. Fix an embedding ι : L ↪ C and an isomorphism λ : \overline{Q}_{p} ⟶^{∼} C such that 𝔭 is induced by ι_{p} = λ^{−1} ∘ ι : L ↪ \overline{Q}_{p}. To each v ∈ S_{p} then corresponds a subset

S_{∞}(v) = {r_{∞}(λ ∘ σ) | r_{p}(σ) = v} ⊂ S_{∞}

such that

∑_{w∈S_{∞}(v)}[L_{w} : R] = [F_{v} : Q_{p}].

For each τ ∈ S̃_{∞}, the Betti realization M_{B,τ} is an L-vector space and there is a Hodge decomposition

M_{B,τ} ⊗_{L,ι} C = ⨁_{i∈Z}(ιM_{τ})^{i,w−i}.

The corresponding Hodge numbers

h^{i,w−i}(ιM_{u}) := h^{i,w−i}(ιM_{τ}) = dim_{C} (ιM_{τ})^{i,w−i}

depend only on u = r_{∞}(τ) ∈ S_{∞}. The de Rham realization M_{dR} is a free L⊗_{Q} F -module; its Hodge filtration is given by submodules F^{r}M_{dR} (not necessarily free) which correspond, under the de Rham comparison isomorphism

M_{dR} ⊗_{L⊗_{Q}F,ι⊗τ} C ⟶^{∼} M_{B,τ} ⊗_{L,ι} C,

to

(F^{r}M_{dR}) ⊗_{L⊗_{Q}F,ι⊗τ} C ⟶^{∼} ⨁_{i≥r}(ιM_{τ})^{i,w−i},

hence

dim_{C} ((gr_{F}^{i} M_{dR}) ⊗_{L⊗_{Q}F,ι⊗τ} C) = h^{i,w−i}(ιM_{τ}).

The 𝔭-adic realization M_{𝔭} of M is isomorphic, as an L_{𝔭}-vector space, to M_{B,τ} ⊗_{L} L_{𝔭} (for any τ ∈ S̃_{∞}). For each v ∈ S_{p}, D_{dR}(M_{𝔭,v}) is a free L_{𝔭} ⊗_{Q_{p}} F_{v}- module equipped with a filtration satisfying

D_{dR}^{r}(M_{𝔭,v}) ⟶^{∼} F^{r}M_{dR} ⊗_{L⊗_{Q}F} (L_{𝔭} ⊗_{Q_{p}} F_{v}).

This implies that, for each i ∈ Z, the dimension

d_{v}^{i}(M_{𝔭}) := dim_{L_{𝔭}} (D_{dR}^{i}(M_{𝔭,v})/D_{dR}^{i+1}(M_{𝔭,v}))

is equal to

dim_{L_{𝔭}} (gr_{F}^{i} M_{dR}) ⊗_{L⊗_{Q}F} (L_{𝔭} ⊗_{Q_{p}} F_{v})

= dim_{\overline{Q}_{p}} (gr_{F}^{i} M_{dR}) ⊗_{L⊗_{Q}F,ι_{p}⊗incl} (\overline{Q}_{p} ⊗_{Q_{p}} F_{v})

= ∑_{σ:F_{v}↪\overline{Q}_{p}} dim_{\overline{Q}_{p}} (gr_{F}^{i} M_{dR}) ⊗_{L⊗_{Q}F,ι_{p}⊗σ} \overline{Q}_{p}

= ∑_{u∈S_{∞}(v)}[F_{u} : R] h^{i,w−i}(ιM_{u}),

hence

d_{v}^{−}(M_{𝔭}) := ∑_{i<0} i d_{v}^{i}(M_{𝔭}) = ∑_{u∈S_{∞}(v)}[F_{u} : R] d^{−}(ιM_{u}),

d^{−}(ιM_{u}) := ∑_{i<0} i h^{i,w−i}(ιM_{u}). (⋆)

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 that V satisfies 4.1.2(1)-(3). For each v ∈ S_{p} we define

d_{v}^{−}(V) := ∑_{i<0} i d_{v}^{i}(V),  d_{v}^{i}(V) = dim_{L_{𝔭}} (D_{dR}^{i}(V_{v})/D_{dR}^{i+1}(V_{v})) (4.1.3.1′)

and

∏_{u∈S_{∞}(v)} ε(V_{u}) := (−1)^{d_{v}^{−}(V)} ∏_{u∈S_{∞}(v),F_{u}=C}(−1)^{dim_{L𝔭} (V)/2}  (4.1.3.2′)

(even though we are unable to define the individual ε(V_{u})). If V = M_{𝔭}, where M ⟶^{∼} M^{∗}(1) is pure (of weight −1), it follows from (⋆) and (2.3.1) that this definition gives the correct product of archimedean ε-factors. The formula (4.1.3.6) should be replaced by

∀v ∈ S_{p}  ε̃(V_{v}) = (−1)^{d_{v}^{−}(V)}(det V_{v}^{+})(−1) = ε(W D(V_{v})^{N−ss}), (4.1.3.6′)

which implies that

ε̃(V_{v}) ∏_{u∈S_{∞}(v)} ε(V_{u}) = (det V_{v}^{+})(−1) ∏_{u∈S_{∞}(v),F_{u}=C}(−1)^{dim_{L𝔭} (V)/2},

hence

∏_{v∈S_{p}∪S_{∞}} ε̃(V_{v}) = (−1)^{r_{2}(F) dim_{L𝔭} (V)/2} ∏_{v∈S_{p}}(det V_{v}^{+})(−1),  (4.1.3.7′)

where r_{2}(F) denotes the number of complex places of F. 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)^{h_{f}^{1} (F,V)}/ε(V) = (−1)^{_{h_{f}^{1}} (F,V)}/ε̃(V) =

= (−1)^{_{h_{f}^{1}} (F,V)}(−1)^{r_{2}(F) dim_{ℒ} (V)/2} ∏_{v∈S_{p}}(det V_{v}^{+})(−1) ∏_{v∉S_{p}∪S_{∞}} ε(V_{v})

depends only on V and V_{v}^{+} (v ∈ S_{p}). In the proof, a reference to (4.1.3.7) should be replaced by that to (4.1.3.7’), which yields

ε̃(V) = ∏_{v∈S_{p}∪S_{∞}} ε̃(V_{v}) ∏_{v∉S_{p}∪S_{∞}} ε(V_{v}) = (−1)^{r_{2}(F) dim_{L𝔭} (V)/2} ∏_{v∈S_{p}}(det V_{v}^{+})(−1) ∏_{v∉S_{p}∪S_{∞}} ε(V_{v}).

Corrections to §5.3.3: the first question should ask whether

(−1)^{d_{v}^{−}(V)}ε(W D(V_{v})^{N−ss}) (v ∈ S_{p})

depends only on V_{v}? Corrections to §5.3.4-5: it is often useful to use a slightly more general version of Example 5.3.4 with Γ = Γ_{0} × Δ, where Γ_{0} is isomorphic to Z_{p} and Δ is finite (abelian). Given a character α : Δ ⟶ O_{𝔭}^{∗}, set

R = O_{𝔭}[[Γ_{0}]], T = (T ⊗_{O_{𝔭}} O_{𝔭}[[Γ^{+}]]) ⊗_{O_{𝔭}[Δ],α} O_{𝔭},
T_{v}^{+} = (T_{v}^{+} ⊗_{O_{𝔭}} O_{𝔭}[[Γ^{+}]]) ⊗_{O_{𝔭}[Δ],α} O_{𝔭} (v ∈ S_{p}).

As in 5.3.4(2)-(3), T is an R[G_{F,S}]-module equipped with a skew-symmetric R-bilinear pairing ( , ) : T × T ⟶ R(1) inducing an isomorphism

T ⊗ Q ⟶^{∼} Hom_{R}(T , R(1)) ⊗ Q.

In 5.3.4(5) we have to replace β : Γ ⟶ L_{𝔭}(β) by β : Γ_{0} ⟶ L_{𝔭}(β); then

T_{P} /ϖ_{P} T_{P} = Ind_{G_{F_{0},S}}^{G_{F,S}}(V ⊗ (β × α)).

In 5.3.5, we set, for any L_{𝔭}[Γ]-module M,

M^{(β×α)} = {x ∈ M ⊗_{L_{𝔭}} L_{𝔭}(β) | ∀σ ∈ Γ σ(x) = (β × α)(x)};

then

H_{f}^{1}(F, T_{P} /ϖ_{P} T_{P}) = H_{f}^{1}(F_{0}, V ⊗ (β × α))
= (H_{f}^{1}(F_{β}, V) ⊗ (β × α))^{Gal(F_{β}/F_{0})} = H_{f}^{1}(F_{β}, V)^{(β^{−1}×α^{−1})}

and

τ : H_{f}^{1}(F_{β}, V)^{(β^{−1}×α^{−1})} ⟶^{∼} H_{f}^{1}(F_{β}, V)^{(β×α)}.

Applying Corollary 5.3.2, we obtain, for any pair of characters of finite order β, β′ : Γ_{0} ⟶ \overline{L}_{𝔭}^{∗}, that

(−1)^{h_{f}^{1} (F_{0},V ⊗(β×α))}/ ε(F_{0}, V ⊗ (β × α))

= (−1)^{h_{f}^{1} (F_{0},V ⊗(β′×α))}/ ε(F_{0}, V ⊗ (β′ × α)). (5.3.5.1′)

References

[1] Jan Nekovář, On the Parity of Ranks of Selmer Groups III, Documenta Math. 12 (2007) 243–274