Some consequences of a formula of Mazur and Rubin for arithmetic local constants

Jan Nekovář

The main results of this article are the following two instances of the parity conjecture for Selmer groups (see [N2, §12.1] for a general discussion of this conjecture). Along the way we also prove slightly weaker results for Hilbert modular forms of parallel weight two with trivial character (Theorems 1.4 and 3.5) and for abelian varieties with real multiplication (Theorem 4.3).

Theorem A. Let E be an elliptic curve over a totally real number field F and let p be a prime number. The p-Selmer rank of E over F

s_{p}(E/F) := rk_{Z}E(F) + cork_{Z_{p}} � (E/F)[p^{∞}]

(which is also equal to the dimension dim_{Q_{p}} H_{f}^{1}(F, V_{p}(E)) of the Bloch-Kato Selmer group [BK, Def. 5.1] of the Galois representation V_{p}(E) = T_{p}(E) ⊗_{Z_{p}} Q_{p} over F) and the analytic rank of E over F

r_{an}(E/F) := ord_{s=1} L(E/F, s)

satisfy

s_{p}(E/F) ≡ r_{an}(E/F) (mod 2)

in each of the following cases:

(1) E does not have complex multiplication;

(2) E has complex multiplication and 2 ∤ [F : Q];

(3) E has complex multiplication by an imaginary quadratic field K′ and p splits in K′/Q.

Note that potential modularity of E [Wi, Thm. A.1] implies that the L-function L(E/F, s) has a meromorphic continuation to C and satisfies the expected functional equation ([T2, proof of Cor. 2.2]; [N2, 12.11.6]). As a result, the integer ord_{s=1} L(E/F, s) ∈ Z is well-defined.

Various special cases of Theorem A (for F ≠ Q) were proved in [N2], [Ki] and [N6].

If the p-primary part of � (E/F) is finite for some prime number p, then s_{p}(E/F) = rk_{Z}E(F) and the statement of Theorem A is the conjecture of Birch and Swinnerton-Dyer for E over F modulo 2.

Theorem B. Let g = ∑_{n=1}^{∞} a_{n}q^{n} ∈ S_{2r}(Γ_{0}(N)) (r ≥ 1) be a normalised (a_{1} = 1) newform, let L = Q(a_{1}, a_{2}, …) be the (totally real) number field generated by its coefficients. For any prime 𝔭 of L above a rational prime p ≠ 2, denote by V_{𝔭}(g) the two-dimensional representation of G_{Q} = Gal(\overline{Q}/Q) over L_{𝔭} attached to g:

det (1 − X Fr_{geom}(l) | V_{𝔭}(g)) = 1 − a_{l}X + l^{2r−1}X^{2} (l ∤ pN).

In the case when r > 1, assume that the residual representation of V_{𝔭}(g) is irreducible. Then:

dim_{L_{𝔭}} H_{f}^{1}(Q, V_{𝔭}(g)(r)) ≡ ord_{s=r} L(g, s) (mod 2).

If g is (the newform associated to) a twist of a p-ordinary eigenform, Theorem B was proved in [N2, Thm. 12.2.3], even for p = 2 and without the assumption on the residual representation.

The proofs of Theorem A and B combine the techniques developed in [N1-6] and [AN] (namely, a combination of suitable relative parity results involving two Selmer groups with an Euler system argument [N3] applied to a non-trivial Euler system [CV], [AN]) with a formula of Mazur and Rubin [MR2, Thm. 1.4]. This formula expresses the difference of the parities of ranks of Selmer groups corresponding to two self-dual Selmer structures on a given finite (self-dual) Galois module as a finite sum of terms depending on purely local data at a finite set of (finite) primes. In a motivic setting, when the two Selmer structures are obtained by propagation from the Bloch-Kato Selmer structures for two self-dual geometric Galois representations which are congruent modulo a prime ideal dividing p, these local terms are expected to mirror the local εfactors of the corresponding L-functions. Unfortunately, such a relation to ε-factors remains conjectural (in the required generality) even in the fairly simple situation relevant to us, when the two Galois representations come from two congruent Hilbert modular forms of parallel weight (as in Sect. 3). This means that we do not have at our disposal appropriate relative parity results in the generality we desire. To get around this problem we apply the formula of Mazur and Rubin in two different global situations for which the local data agree. We obtain a “bi-relative” global result (Theorem 2.2) for the parities of ranks of four different Selmer groups. If we are able to control three of them (in our case, Theorem 1.4 applies to two of them and the auxiliary global situation is chosen in such a way that the third Selmer group is trivial, by an application of another Euler system argument [Ka], [N7]), the sought for parity result for the remaining Selmer group follows. Note that the formula of Mazur and Rubin is used in the proofs of both Theorems 1.1 (on which Theorem 1.4 relies) and 2.2. This programme is carried out for Hilbert modular forms in Sect. 3; the results for abelian varieties with real multiplication are deduced in Sect. 4. The assumptions on E in Theorem A come from an application of [N7, Cor. Thm. B’].

This work owes its origin to the author’s stay at the Centre de Recerca Matemàtica at Universitat Autònoma de Barcelona in December 2009. He is grateful to the organisers of the programme “Arithmetic Geometry” for invitation and to the CRM for its hospitality. He would also like to thank the referee for helpful comments.

Notation and conventions

All representations (in particular, characters) of various Galois groups are assumed to be continuous. Given a number field F, a choice of an embedding \overline{F} ↪ \overline{F_{v}}, for each prime v of F, identifies G_{F_{v}} = Gal(\overline{F_{v}}/F_{v}) with a subgroup of G_{F} = Gal(\overline{F} /F). For each representation V of G_{F} we denote by V_{v} its restriction to G_{F_{v}}. Denote by S_{∞} (resp. by S_{p}) the set of all archimedean primes (resp. of all primes above a rational prime p) of F. For any R[G]-module M and a character χ : G ⟶ R^{×} we denote by M^{(χ)} = {m ∈ M | ∀g ∈ G g(m) = χ(g)m} the χ-eigenspace for the action of G on M.

1. A parity result for Hilbert modular forms of parallel weight two

1.1. Theorem (An abstract cohomological version of the case 𝔖 = ∅ of [MR2, Thm. 7.1]). Let F be a number field, let V be a geometric representation (in the sense of Fontaine and Mazur) of G_{F} with coefficients in a finite extension K of Q_{p}, where p ≠ 2. Assume that:

(1) There exists a non-degenerate skew-symmetric G_{F} -equivariant bilinear pairing ⟨ , ⟩ : V × V ⟶ K(1).

(2) After possibly multiplying ⟨ , ⟩ by an element of K^{×}, there exists a G_{F} -stable O_{K}-lattice T ⊂ V which is self-dual (i.e., for which the rescaled pairing defines an isomorphism T ⟶^{∼} T^{∗}(1)). [This is automatic if dim_{K}(V) = 2, for any T.] Let K/F be a quadratic extension, let K′ be a cyclic extension of K of p-power order, dihedral over F. Assume that no finite prime of K stable under Gal(K/F) ramifies in K′/K. Then, for each character χ : Gal(K′/K) ⟶ K^{×},

dim_{K} H_{f}^{1}(K′, V)^{(χ^{±1})} − dim_{K} H^{0}(K′, V)^{(χ^{±1})} ≡ dim_{K} H_{f}^{1}(K, V) − dim_{K} H^{0}(K, V) (mod 2).

Proof. Fix a finite set S of primes of F containing S_{∞} ∪ S_{p} such that V is unramified outside S. Fix a uniformiser t ∈ O = O_{K} and denote by k = O/tO the residue field of K. The K-subspaces H_{f}^{1}(F_{v}, V) ⊂ H^{1}(F_{v}, V) (v ∉ S_{∞}) define, by propagation [MR1, Ex. 1.1.2], a Selmer structure H_{f}^{1}(F_{v}, X) ⊂ H^{1}(F_{v}, X) on each X = T, V/T, T/t^{n}T, \overline{T} = T/tT, which is cartesian on \{T/t^{n}T \}_{n≤∞} [MR1, Lemma 3.7.1]. The exact sequences

0 ⟶ H^{0}(F, V/T) ⊗_{O} k ⟶ H_{f}^{1}(F, \overline{T}) ⟶ H_{f}^{1}(F, V/T)[t] ⟶ 0
0 ⟶ H^{0}(F_{v}, T) ⊗_{O} k ⟶ H^{0}(F_{v}, \overline{T}) ⟶ H_{f}^{1}(F_{v}, T)[t] ⟶ 0

imply that

dim_{k} H_{f}^{1}(F, V/T)[t] − dim_{K} H^{0}(F, V) = dim_{k} H_{f}^{1}(F, \overline{T}) − dim_{k} H^{0}(F, \overline{T}),  (1.1.1)
dim_{k} (H_{f}^{1}(F_{v}, \overline{T}) = H_{f}^{1}(F_{v}, T) ⊗_{O} k) = dim_{k} H^{0}(F_{v}, \overline{T}) + dim_{K} H_{f}^{1}(F_{v}, V) − dim_{K} H^{0}(F_{v}, V). (1.1.2)

So far we have not used the assumptions (1) and (2) of the Theorem, but we are going to do it now. The existence of a non-degenerate skew-symmetric bilinear pairing on H_{f}^{1}(F, V/T)/ (H_{f}^{1}(F, T) ⊗_{O} K/O) with values in K/O constructed in [Fl] (taking into account [BK, Prop. 3.8]) implies that

dim_{K} H_{f}^{1}(F, V) = cork_{O} (H_{f}^{1}(F, T) ⊗_{O} K/O) ≡ dim_{k} H_{f}^{1}(F, V/T)[t] (mod 2);

we deduce from (1.1.1) that

dim_{K} H_{f}^{1}(F, V) − dim_{K} H^{0}(F, V) ≡ dim_{k} H_{f}^{1}(F, \overline{T}) − dim_{k} H^{0}(F, \overline{T}) (mod 2).  (1.1.3)

The induced representation Ind_{Gal(K′/K)}^{Gal(K′/F)}(χ) has a natural model I[χ] (free of rank two) over O, which is equipped with a non-degenerate symmetric G_{F} -equivariant pairing I[χ]×I[χ] ⟶ O inducing an isomorphism I[χ] ⟶^{∼} I[χ]^{∗}. By Shapiro’s lemma,

H_{f}^{1}(F, V ⊗ I[χ]) = H_{f}^{1}(K, V ⊗ χ) = (H_{f}^{1}(K′, V) ⊗ χ)^{Gal(K′/K)} = H_{f}^{1}(K′, V)^{(χ^{−1})},
H^{j}(F, V ⊗ I[χ]) = H^{j}(K, V ⊗ χ) = H^{j}(K′, V)^{(χ^{−1})}.

As I[χ] ⟶^{∼} I[χ^{−1}], the above groups are isomorphic, respectively, to H_{f}^{1}(K′, V)^{(χ)} and H^{j}(K′, V)^{(χ)}.

The discussion leading to the formulas (1.1.1-3) applies to V ⊗ I[χ] and the self-dual lattice T ⊗_{O} I[χ]. Note that there is a canonical identification \overline{T} ⊗ I[χ] = \overline{T} ⊗ I[1], where we have denoted by “1” the trivial character of Gal(K′/K) (this notation, which occurs only in 1.1 and 1.2, should not to be confused with the Tate twist “(1)”). However, the Selmer structures H_{f,χ}^{1}(F_{v}, −) (resp. H_{f,1}^{1}(F_{v}, −)) on the G_{F} -module \overline{T} ⊗ I[χ] = \overline{T} ⊗ I[1] obtained by propagation of the subspaces H_{f}^{1}(F_{v}, V ⊗ I[χ]) ⊂ H^{1}(F_{v}, V ⊗ I[χ]) (resp. H_{f}^{1}(F_{v}, V ⊗ I[1]) ⊂ H^{1}(F_{v}, V ⊗ I[1])) are not necessarily the same. The formula [MR2, Thm. 1.4] applies in our case, since both Selmer structures H_{f,χ}^{1} and H_{f,1}^{1} are self-dual, thanks to [BK, Prop. 3.8]; it yields

dim_{k} H_{f,χ}^{1}(F, \overline{T} ⊗ I[χ]) − dim_{k} H_{f,1}^{1}(F, \overline{T} ⊗ I[1]) ≡ ∑_{v∈S−S_{∞}} δ_{v} (mod 2),  (1.1.4)

where

δ_{v} ≡ dim_{k} H_{f,1}^{1}(F, \overline{T} ⊗ I[1])/ (H_{f,1}^{1}(F, \overline{T} ⊗ I[1]) ∩ H_{f,χ}^{1}(F, \overline{T} ⊗ I[χ])) (mod 2).

Combining (1.1.4) with (1.1.3) for T ⊗_{O} I[χ] and T ⊗_{O} I[1], we obtain

χ_{f}(K, V ⊗ χ) − χ_{f}(K, V) ≡ ∑_{v∈S−S_{∞}} δ_{v} (mod 2),  (1.1.5)

where we have put

χ_{f}(K, W) := dim_{K} H_{f}^{1}(K, W) − dim_{K} H^{0}(K, W).  (1.1.6)

To conclude the proof, it remains to prove the following Lemma.

1.2. Lemma. Under the assumptions of Theorem 1.1 we have δ_{v} ≡ 0 (mod 2), for all v ∈ S − S_{∞}.

Proof. If there is a unique prime w | v in K, then χ_{w} (= the restriction of χ to G_{K_{w}}) is unramified by assumption, and therefore trivial [MR2, Lemma 6.5]. It follows that I[χ]_{v} = I[1]_{v}, hence H_{f,χ}^{1}(F_{v}, \overline{T} ⊗I[χ]) = H_{f,1}^{1}(F_{v}, \overline{T} ⊗ I[1]).

The case when v splits as vO_{K} = ww′ requires a more detailed argument. In this case K_{w} = F_{v} = K_{w′}, I[1]_{v} = 1 ⊕ 1 and I[χ]_{v} = χ_{w} ⊕ χ_{w}^{−1}. As

δ_{v} ≡ dim_{k} (\frac{Y ⊕ Y}{(Y ∩ Z_{+}) ⊕ (Y ∩ Z_{−})}) (mod 2),

where

Y = Im (H_{f}^{1}(F_{v}, T) ⊗_{O} k ↪ H^{1}(F_{v}, \overline{T})) ,
Z_{±} = Im (H_{f}^{1}(F_{v}, T ⊗ χ_{w}^{±1}) ⊗_{O} k ↪ H^{1}(F_{v}, \overline{T} ⊗ χ_{w}^{±1}) = H^{1}(F_{v}, \overline{T})) ,

we must show that

dim_{k}(Y ∩ Z_{+}) ≡^{?} dim_{k}(Y ∩ Z_{−}) (mod 2).

Firstly, the local duality

H^{1}(F_{v}, \overline{T}) × H^{1}(F_{v}, \overline{T}) ⟶ H^{2}(F_{v}, k(1)) ⟶^{∼} k

is a non-degenerate symmetric bilinear pairing under which Y^{⊥} = Y and Z_{±}^{⊥} = Z_{∓}, by [BK, Prop. 3.8]. Secondly, (1.1.2) applied to T ⊗ χ_{w}^{±1} yields (since \overline{T} ⊗ χ_{w}^{±1} = \overline{T})

dim_{k}(Z_{±}) − dim_{k} H^{0}(F_{v}, \overline{T}) = dim_{K} H_{f}^{1}(F_{v}, V ⊗ χ_{w}^{±1}) − dim_{K} H^{0}(F_{v}, V ⊗ χ_{w}^{±1}).

If v ∤ p, then the R.H.S. is equal to zero. If v | p, then the R.H.S. is equal, by [BK, Cor. 3.8.4], to

dim_{K} D_{dR}(V_{v} ⊗ χ_{w}^{±1})/F il^{0} = dim_{K} D_{dR}(V_{v})/F il^{0},

which does not depend on the sign ±. In either case,

dim_{k}(Z_{+}) = dim_{k}(Z_{−}) = \frac{1}{2} dim_{k} H^{1}(F_{v}, \overline{T}) = dim_{k}(Y)

and

dim_{k}(Y ∩ Z_{+}) = dim_{k}(Y) + dim_{k}(Z_{+}) − dim_{k}(Y + Z_{+}) = dim_{k} H^{1}(F_{v}, \overline{T}) − dim_{k}(Y + Z_{+})
= dim_{k}(Y + Z_{+})^{⊥} = dim_{k}(Y^{⊥} ∩ Z_{+}^{⊥}) = dim_{k}(Y ∩ Z_{−}),

as required. Lemma (and Theorem 1.1) is proved. 1.3. If V arises as a subquotient of H_{et}^{2r−1}(X ⊗_{F} \overline{F} , K)(r) for some proper and smooth scheme X over F, then H^{0}(L, V) = 0 for all finite extensions L/F, by Deligne’s proof of Weil’s conjectures. Theorem 1.1 in this case states that

dim_{K} H_{f}^{1}(K′, V)^{(χ^{±1})} ≡ dim_{K} H_{f}^{1}(K, V) (mod 2).  (1.3.1)

This remark applies, in particular, to V = V_{𝔭}(g)(r) as in Theorem B, and to any subrepresentation of V_{p}(A) ⊗_{Q_{p}} K, where A is an abelian variety over F.

1.4. Theorem (Generalisation of [N6, Thm. 1]). Let g ∈ S_{2}(𝔫, 1) be a cuspidal Hilbert modular newform of parallel weight two and trivial character over a totally real number field F. Let L be the (totally real) number field generated by its Hecke eigenvalues λ_{v}(g). For any prime 𝔭 of L above a rational prime p ≠ 2, denote by V_{𝔭}(g) the two-dimensional representation of G_{F} over L_{𝔭} attached to g:

det (1 − X Fr_{geom}(v) | V_{𝔭}(g)) = 1 − λ_{v}(g)X + N(v)X^{2} (v ∤ p𝔫).

Assume that at least one of the following three conditions is satisfied:

(a) 2 ∤ [F : Q];

(b) there exists a non-archimedean prime of F at which the local component of the automorphic representation π(g) of P GL_{2}(A_{F}) attached to g is a twist of the Steinberg representation;

(c) there exists a non-archimedean prime v_{0} of F at which the local component of π(g) is supercuspidal. Then:

dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(g)(1)) ≡ r_{an}(F, g) (mod 2),

where r_{an}(F, g) := ord_{s=1} L(g, s).

Proof. (a), (b) In the case when g corresponds to an elliptic curve defined over F this result was proved in [N6]. The argument of [loc. cit.] applies in general, with the following modifications: we replace the conductor of E by 𝔫 (the level of g) and use Theorem 1.1 instead of [MR2, Thm. 7.1]. As V_{𝔭}(g)(1) arises as a subrepresentation of V_{p}(A) ⊗_{Q_{p}} L_{𝔭}, where A is the Jacobian of a suitable Shimura curve, (1.3.1) applies in this case.

(c) Thanks to (a) we can assume that 2 | [F : Q]. In addition, we can assume, as in [N6, Step 3] (after replacing F by a suitable cyclic extension of odd degree), that there exists a prime P | p in F, P ≠ v_{0}. Let K be any totally imaginary quadratic extension of F in which P splits and which satisfies the properties of Lemma 1.5 below (and such that g does not have CM by K). As in [N5, 1.2-1.5] (for χ = 1, Σ = {P }, c = 1), the generalisation of [CV, Thm. 4.1] proved in [AN, Thm. 4.3.1] combined with [N3, Thm. 3.2] implies that there is a finite cyclic subextension K′/K of the ring class field extension K[P^{∞}]/K and a character χ of Gal(K′/K) for which 2 ∤ dim_{K} H_{f}^{1}(K′, V_{𝔭}(g)(1))^{(χ)}, where K = L_{𝔭}(χ). Theorem 1.1 then yields

2 ∤ dim_{L_{𝔭}} H_{f}^{1}(K, V_{𝔭}(g)(1)) = dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(g)(1)) + dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(g ⊗ α)(1)),  (⋆)

where α is the quadratic character associated to K/F. We can now vary K as in the endgame of [N1]:

If 2 ∤ r_{an}(F, g), then 2 | r_{an}(F, g ⊗ α) for any α as in Lemma 1.5 below. According to [Wa, Thm. 4] and [FH, Thm. B.1] there exists such an α satisfying r_{an}(F, g⊗α) = 0, which implies that H_{f}^{1}(F, V_{𝔭}(g⊗α)(1)) = 0, by [N7, Thm. B(b)]; thus 2 ∤ dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(g)(1)), by (⋆).

If 2 | r_{an}(F, g), then 2 ∤ r_{an}(F, g ⊗ α) for any α as in Lemma 1.5. The previous argument applies to g ⊗ α, yielding 2 ∤ dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(g ⊗ α)(1)). Applying (⋆) again, we obtain 2 | dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(g)(1)).

1.5. Lemma. Let g be as in Theorem 1.4(c). If 2 | [F : Q], then there exists a character μ : G_{F_{v_{0}}} ⟶ {±1} such that, for any character α : G_{F} ⟶ {±1} satisfying

α_{v_{0}} = μ,  ∀v | 𝔫, v ≠ v_{0} α_{v} = 1,  ∀v ∈ S_{∞} α_{v}(−1) = −1,

the corresponding quadratic extension K = \overline{F}^{Ker(α)} of F is totally imaginary and 2 ∤ r_{an}(F, g)+r_{an}(F, g ⊗α).

Proof. See [N7, Prop. 2.10.2].

2. A relative parity result with a twist

2.1. Assume that V satisfies the assumption (1) of Theorem 1.1. For each non-archimedean prime v of F we write, as in [N4, Prop. 2.2.1(1)],

ε_{v}(V) = ε_{v}(V_{v}) = ε(W D(V_{v}), ψ, dx_{ψ}) ∈ {±1},

where ψ is any non-trivial additive character of F_{v} and dx_{ψ} the corresponding self-dual Haar measure on F_{v}, and W D(V_{v}) is the representation of the Weil-Deligne group of F_{v} attached to V_{v} if v ∤ p, resp. to D_{pst}(V_{v}) if v | p (see [D, 8.4], [Fo], [FoPR, I.1.3.2]).

2.2. Theorem. Let F and K be as in Theorem 1.1 (in particular, p ≠ 2). Let V and V′ be geometric representations of G_{F} with coefficients in K which satisfy the assumptions (1) and (2) of Theorem 1.1. Let T ⊂ V and T′ ⊂ V′ be G_{F} -stable O-lattices, self-dual with respect to the corresponding pairings ⟨ , ⟩ : T × T ⟶ O(1) and ⟨ , ⟩′ : T′ × T′ ⟶ O(1). Assume that there exists an isomorphism of k[G_{F}]- modules u : \overline{T}′ = T′ ⊗_{O} k ⟶^{∼} \overline{T} = T ⊗_{O} k compatible with the pairings induced by ⟨ , ⟩ (resp. ⟨ , ⟩′) on \overline{T} (resp. on \overline{T}′). Let S be a finite set of primes of F containing S_{∞} ∪ S_{p} and all primes at which V or V′ is ramified. If α : G_{F} ⟶ {±1} is a character such that α_{v} = 1 for all v ∈ S − S_{∞}, then (using the notation from (1.1.6)):

χ_{f}(F, V) − χ_{f}(F, V′) ≡ χ_{f}(F, V ⊗ α) − χ_{f}(F, V′ ⊗ α) (mod 2),
∀v ∉ S_{∞} ε_{v}(V)/ε_{v}(V′) = ε_{v}(V ⊗ α)/ε_{v}(V′ ⊗ α).

Proof. As remarked in the course of the proof of Theorem 1.1, the Selmer structure H_{f}^{1}(F_{v}, \overline{T}) (resp. H_{f′}^{1}(F_{v}, \overline{T})) obtained by propagation of H_{f}^{1}(F_{v}, V) ⊂ H^{1}(F_{v}, V) (resp. by propagation of H_{f}^{1}(F_{v}, V′) ⊂ H^{1}(F_{v}, V′) composed with the isomorphism H^{1}(F_{v}, \overline{T}′) ⟶^{∼} H^{1}(F_{v}, \overline{T}) induced by u) is self-dual. Combining [MR2, Thm. 1.4] with (1.1.3) we obtain

χ_{f}(F, V) − χ_{f}(F, V′) ≡ dim_{k} H_{f}^{1}(F, \overline{T}) − dim_{k} H_{f′}^{1}(F, \overline{T}) ≡ ∑_{v∈S−S_{∞}} δ_{v}(T_{v}, T′_{v}) (mod 2),  (2.2.1)

where

δ_{v}(T_{v}, T′_{v}) ≡ dim_{k} H_{f}^{1}(F_{v}, \overline{T})/ (H_{f}^{1}(F_{v}, \overline{T}) ∩ H_{f′}^{1}(F, \overline{T})) (mod 2).

Set S(α) = S ∪ {v | α_{v} is ramified}. We claim that

∀v ∈ S(α) − S ∀j = 0, 1, 2 H^{j}(F_{v}, \overline{T} ⊗ α) = 0.  (2.2.2)

Indeed, H^{0}(F_{v}, \overline{T} ⊗ α) ⊂ (\overline{T} ⊗ α)^{I_{v}} = 0 (since p ≠ 2) and H^{2}(F_{v}, \overline{T} ⊗ α) = H^{0}(F_{v}, (\overline{T} ⊗ α)^{∗}(1))^{∗} = H^{0}(F_{v}, \overline{T} ⊗ α)^{∗} = 0, by local duality. Finally, H^{1}(F_{v}, \overline{T} ⊗ α) = 0 by the local Euler characteristic formula.

The pairings ⟨ , ⟩, ⟨ , ⟩′ and the isomorphism u induce the same data for T ⊗ α and T′ ⊗ α. Applying (2.2.1) to these twisted modules, we obtain

\begin{aligned}χ_{f}(F, V ⊗ α) − χ_{f}(F, V′ ⊗ α) & ≡ ∑_{v∈S(α)−S_{∞}} δ_{v}((T ⊗ α)_{v}, (T′ ⊗ α)_{v}) ≡ ∑_{v∈S−S_{∞}} δ_{v}((T ⊗ α)_{v}, (T′ ⊗ α)_{v}) \\ & ≡ ∑_{v∈S−S_{∞}} δ_{v}(T_{v}, T′_{v}) ≡ χ_{f}(F, V) − χ_{f}(F, V′) (mod 2), \end{aligned}

where the second (resp. the third) congruence follows from (2.2.2) (resp. from the fact that α_{v} = 1 for all v ∈ S − S_{∞}).

Let us now prove the statement about local ε-constants. For v ∈ S − S_{∞} there is nothing to prove, as (W ⊗ α)_{v} = W_{v} (W = V, V′), hence ε_{v}(W ⊗ α) = ε_{v}(W). For v ∉ S(α) all four ε-constants are equal to 1. Finally, for v ∈ S(α) − S, ε_{v}(W) = 1 (W = V, V′). It follows from (2.2.2) that (W ⊗ α)^{I_{v}} = 0, which implies that ε_{v}(W ⊗ α) = ε_{0,v}(W ⊗ α). As the local ε_{0}-constants at primes not dividing p are compatible with congruences modulo p [D, Thm. 6.5], the isomorphism \overline{T}′ ⊗ α ⟶^{∼} \overline{T} ⊗ α implies that ε_{v}(V ⊗ α), ε_{v}(V′ ⊗ α) ∈ {±1} are congruent modulo p, therefore they are equal to each other. 2.3. In practice, we are often given a slightly different set of data: 2.3.1. representations V and V′ which satisfy the assumption (1) of Theorem 1.1; 2.3.2. a G_{F} -stable O-lattice T ⊂ V which is self-dual with respect to ⟨ , ⟩ : T × T ⟶ O(1), 2.3.3. and for which \overline{T} = T ⊗_{O} k is an absolutely irreducible representation of G_{F}; 2.3.4. a dense set of elements g ∈ G_{F} for which Tr(g | V) ≡ Tr(g | V′) (mod tO).

The condition 2.3.4 implies that, for any G_{F} -stable O-lattice T′ ⊂ V′, the semi-simplification \overline{T}^{′ss} of \overline{T}′ is isomorphic to \overline{T}^{ss}, which is in turn equal to \overline{T}, by 2.3.3. It follows that there is an isomorphism u : \overline{T}′ ⟶^{∼} \overline{T} of k[G_{F}]-modules, which is unique up to a scalar in k^{×} (again by 2.3.3). Irreducibility of \overline{T}′ implies that any G_{F} -stable O-lattice in V′ is of the form aT′ for some a ∈ K^{×}; as a result, T′ satisfies the assumption (2) of Theorem 1.1. Finally, the pairings induced on \overline{T} by ⟨ , ⟩ (resp. by ⟨ , ⟩′ and u) coincide up to a multiplicative factor b ∈ k^{×} (by 2.3.3). After multiplying ⟨ , ⟩′ by a suitable element of O^{×}, we obtain b = 1. In other words, the conditions 2.3.1–2.3.4 give rise to the data required in Theorem 2.2.

3. Two applications of Theorem 2.2 to modular forms

3.1. Let F be a totally real number field. If g ∈ S_{k}(𝔫, 1) is a cuspidal Hilbert newform over F of level 𝔫, trivial character and parallel weight k (necessarily even), then its completed L-function coincides, up to a shift, with the L-function of the automorphic representation π(g) of P GL_{2}(A_{F}) associated to g:

(L_{∞} · L) (g, s) = L(π(g), s − (k − 1)/2), L_{∞}(g, s) = Γ_{C}(s)^{[F:Q]}.

As the Γ-factor L_{∞}(g, s) has no zero nor pole at the central point s = k/2 of the functional equation, the parity of the analytic rank of g over F

r_{an}(F, g) := ord_{s=k/2} L(g, s)

can be read off from the corresponding ε-constant in the functional equation:

L(π(g), s) = ε(π(g), s) L(π(g), 1 − s), (−1)^{r_{an}(F,g)} = ε(π(g), \frac{1}{2}) = ∏_{v} ε_{v}(π(g)_{v}, \frac{1}{2}).

If L, L_{𝔭} and V_{𝔭}(g) are as in Theorem B (with an appropriate modification if F ≠ Q; see Theorem 1.4 in the case k = 2), then the Galois representation V = V_{𝔭}(g)(k/2) satisfies the assumption (1) of Theorem 1.1. The conjectures of Bloch and Kato ([BK], [FoPR]) predict that

dim_{L_{𝔭}} H_{f}^{1}(F, V) =^{?} r_{an}(F, g).

We are interested in this conjecture modulo 2:

dim_{L_{𝔭}} H_{f}^{1}(F, V) ≡^{?} r_{an}(F, g) (mod 2).

3.2. Let g ∈ S_{k}(𝔫, 1) be as in 3.1. If F′/F is a quadratic extension and α : Gal(F′/F) ⟶^{∼} {±1} the corresponding quadratic character, then we have

H_{f}^{1}(F′, V) = H_{f}^{1}(F, V) ⊕ H_{f}^{1}(F, V ⊗ α)  (3.2.1)

and

L(g ⊗ F′, s) = L(g, s)L(g ⊗ α, s),  r_{an}(F′, g) = r_{an}(F, g) + r_{an}(F, g ⊗ α),  (3.2.2)

where we have denoted, somewhat abusively, by g′ = g ⊗ F′ the base change of g to an automorphic form on P GL_{2}(A_{F′}) and by r_{an}(F′, g) the analytic rank r_{an}(F′, g ⊗ F′) (strictly speaking, it is the automorphic representation of P GL_{2}(A_{F′}) attached to g′ which is the base change of π(g)). 3.3. Proof of Theorem B. The statement for r = 1 is a special case of Theorem 1.4(a). If r > 1, then it follows from [R, Thm. 2.1, Thm. 2.2, Cor. 3.2] (the author would like to thank F. Diamond for pointing out this reference) and our assumption about the residual representation of V_{𝔭}(g) that there exists a normalised newform g_{1} ∈ S_{2}(N_{1}, ω^{2−2r}) of level N_{1} dividing pN whose coefficients lie in a number field L′ ⊃ L and which satisfies, for a suitable prime 𝔭′ | 𝔭 of L′,

∀g ∈ G_{Q} Tr(g | V_{𝔭′}(g_{1})) ≡ Tr(g | V_{𝔭}(g) ⊗_{L_{𝔭}} L′_{𝔭′}) (mod 𝔭′).

Let g′ ∈ S_{2}(N′, 1) be the newform associated to g_{1} ⊗ω^{r−1} (of level dividing N multiplied by a suitable power of p); set K = L′_{𝔭′}, O = O_{K}, V = V_{𝔭}(g)(r) ⊗_{L_{𝔭}} K and V′ = V_{𝔭′}(g′)(1) = V_{𝔭′}(g_{1})(1) ⊗ ω^{r−1}.

The representations V, V′ satisfy 2.3.1 and 2.3.4 (note that Z_{p}(r) and Z_{p}(1)⊗ω^{r−1} have the same residual representation F_{p}(r)). Fix any G_{Q}-stable O-lattice T ⊂ V. It satisfies 2.3.3 (irreducibility implies absolute irreducibility, as the action of the complex conjugation on \overline{T} has two distinct eigenvalues ±1 contained in k = O/tO) and, after rescaling the symplectic form ⟨ , ⟩ : V × V ⟶ K(1), also 2.3.2. The discussion in 2.3 implies that the assumptions of Theorem 2.2 are satisfied. Using, in addition, 1.3, we deduce that

dim_{K} H_{f}^{1}(Q, V) − dim_{K} H_{f}^{1}(Q, V′) ≡ dim_{K} H_{f}^{1}(Q, V ⊗ α) − dim_{K} H_{f}^{1}(Q, V′ ⊗ α) (mod 2),  (3.3.1)

whenever α : G_{Q} ⟶ {±1} is a character satisfying

∀l | pN  α_{l} = 1.  (3.3.2)

According to Theorem 1.4(a),

dim_{K} H_{f}^{1}(Q, V′) ≡ r_{an}(Q, g′) (mod 2),  dim_{K} H_{f}^{1}(Q, V′ ⊗ α) ≡ r_{an}(Q, g′ ⊗ α) (mod 2).  (3.3.3)

Combining (3.3.1) and (3.3.3) with Lemma 3.4 below, we obtain

dim_{K} H_{f}^{1}(Q, V) − r_{an}(Q, g) ≡ dim_{K} H_{f}^{1}(Q, V ⊗ α) − r_{an}(Q, g ⊗ α) (mod 2).  (3.3.4)

It follows from the non-vanishing results of [Wa, Thm. 4] and [FH, Thm. B.1] that there exists a character α satisfying (3.3.2) for which r_{an}(Q, g ⊗ α) = 0. A fundamental result of Kato [Ka, Thm. 14.2(2)] then implies that H_{f}^{1}(Q, V ⊗ α) = 0. The congruence (3.3.4) for this particular α becomes

dim_{K} H_{f}^{1}(Q, V) ≡ r_{an}(Q, g) (mod 2),

which proves Theorem B.

3.4. Lemma. For any character α satisfying (3.3.2) we have

r_{an}(Q, g) − r_{an}(Q, g′) ≡ r_{an}(Q, g ⊗ α) − r_{an}(Q, g′ ⊗ α) (mod 2).

Proof. To simplify the notation we write ε_{v}(h) = ε_{v}(π(h)_{v}, \frac{1}{2}) for the corresponding local ε-constants. It is enough to show that, for any prime v of Q,

ε_{v}(g)/ε_{v}(g′) =^{?} ε_{v}(g ⊗ α)/ε_{v}(g′ ⊗ α).  (3.4.1)

Firstly, ε_{∞}(h) = ε_{∞}(h ⊗ α) (h = g, g′), since the twist by α does not change the weight. Secondly, if l is a prime number dividing pN, then (3.3.2) implies that π(h ⊗ α)_{l} = π(h)_{l} (h = g, g′), hence ε_{l}(h ⊗ α) = ε_{l}(h). Finally, if l does not divide pN, then π(g)_{l} = π(μ, μ^{−1}) and π(g′)_{l} = π(μ′, μ^{′−1}) are unramified principal series representations with trivial central characters; it follows that π(g ⊗ α) = π(μα_{l}, μ^{−1}α_{l}), π(g′ ⊗ α) = π(μ′α_{l}, μ^{′−1}α_{l}) and

ε_{l}(g) = μ(−1) = 1 = μ′(−1) = ε_{l}(g′), ε_{l}(g ⊗ α) = (μα_{l})(−1) = α_{l}(−1) = (μ′α_{l})(−1) = ε_{l}(g′ ⊗ α),

which completes the proof of (3.4.1).

3.5. Theorem. Let g ∈ S_{2}(𝔫, 1), L and 𝔭 | p (p ≠ 2) be as in Theorem 1.4. Assume that 2 | [F : Q], that the residual representation T_{𝔭}(g)/𝔭T_{𝔭}(g) (where T_{𝔭}(g) ⊂ V_{𝔭}(g) is a G_{F} -stable O_{L,𝔭}-lattice) is an irreducible G_{F} -module and that one of the following two conditions holds:

(1) g has no complex multiplication and V_{𝔭}(g) is not quaternionic (in the sense of 3.6 below);

(2) g has complex multiplication: g is the theta series attached to an algebraic Hecke character A_{K(g)}^{×} ⟶ L^{′×}, where K(g) (resp. L′) is a totally imaginary quadratic extension of F (resp. of L), 𝔭 splits in L′/L and V_{𝔭}(g)|_{G_{K(g)}} = ψ_{1} ⊕ ψ_{2}, where ψ_{i} : G_{K(g)} ⟶ L_{𝔭}^{×} are characters for which ψ_{2}(Ker(ψ_{1})) is infinite. Then:

dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(g)(1)) ≡ r_{an}(F, g) (mod 2).

Proof. As in the proof of Theorem B, the G_{F} -modules V_{𝔭}(g)(1) ⊃ T_{𝔭}(g)(1) satisfy 2.3.1–2.3.3. The level raising machinery [T1] together with [DS, Lemme 6.11] imply that there exists a newform g′ ∈ S_{2}(𝔫′, 1) of level 𝔫′ satisfying 𝔮 | 𝔫′ | 𝔫𝔮 (for a suitable prime 𝔮 ∤ 𝔫) whose Hecke eigenvalues lie in a number field L′ ⊃ L and satisfy

∀v ∤ p𝔫𝔮 λ_{v}(g′) ≡ λ_{v}(g) (mod 𝔭′),

for a suitable prime 𝔭′ | 𝔭 of L′. It follows from the Čebotarev density theorem that the representations V = V_{𝔭}(g)(1) ⊗_{L_{𝔭}} K, T = T_{𝔭}(g)(1) ⊗_{O_{L,𝔭}} O_{K} (K = L′_{𝔭′}) and V′ = V_{𝔭′}(g′)(1) satisfy 2.3.1–2.3.4. Applying Theorem 2.2 and taking into account 1.3, we obtain, for any character α : G_{F} ⟶ {±1} satisfying

∀v | p𝔫𝔮  α_{v} = 1,  (3.5.1)

that

dim_{K} H_{f}^{1}(F, V) − dim_{K} H_{f}^{1}(F, V′) ≡ dim_{K} H_{f}^{1}(F, V ⊗ α) − dim_{K} H_{f}^{1}(F, V′ ⊗ α) (mod 2).  (3.5.2)

As ord_{𝔮}(𝔫′) = 1, the local representation π(g′)_{𝔮} is the twist of the Steinberg representation by an unramified character of order one or two. As a result, Theorem 1.4(b) applies to g′ and its quadratic twists:

dim_{K} H_{f}^{1}(F, V′) ≡ r_{an}(F, g′) (mod 2),  dim_{K} H_{f}^{1}(F, V′ ⊗ α) ≡ r_{an}(F, g′ ⊗ α) (mod 2).  (3.5.3)

The argument used in the proof of Lemma 3.4 applies, yielding

r_{an}(F, g) − r_{an}(F, g′) ≡ r_{an}(F, g ⊗ α) − r_{an}(F, g′ ⊗ α) (mod 2).  (3.5.4)

Combining (3.5.2)–(3.5.4), we obtain

dim_{K} H_{f}^{1}(F, V) − r_{an}(F, g) ≡ dim_{K} H_{f}^{1}(F, V ⊗ α) − r_{an}(F, g ⊗ α) (mod 2),  (3.5.5)

for any quadratic character α satisfying (3.5.1). As in 3.3, it follows from [Wa, Thm. 4] and [FH, Thm. B.1] that there exists α satisfying (3.5.1) such that r_{an}(F, g ⊗ α) = 0. A generalisation of [L, Thm. C] proved in [N7, Thm. B] implies that H_{f}^{1}(F, V ⊗ α) = 0 (this is where the assumptions (1) and (2) come in, by [N7, B.5.5(2)] and [N7, B.6.5(2)], respectively). The congruence (3.5.5) for this α yields the desired result. 3.6. (Non)-quaternionic representations. If g from Theorem 3.5 does not have complex multiplication, recall from [N7, App. B.3] that there exists a finite abelian group Γ ⊂ Aut(L/Q) of exponent at most two and a quaternion algebra D over L^{Γ} such that, for each finite prime 𝔭 of L, the Lie algebra of the Galois image

Im (G_{F} ⟶ Aut_{L_{𝔭}}(V_{𝔭}(g)) ⟶^{∼} GL_{2}(L_{𝔭}))

is equal to

{x ∈ D_{𝔭_{Γ}} ⊂ M_{2}(L_{𝔭}) | Trd(x) ∈ Q_{p}},

where 𝔭_{Γ} is the prime of L^{Γ} ⊂ L below 𝔭 and D_{𝔭_{Γ}} = D ⊗_{L^{Γ}} (L^{Γ})_{𝔭_{Γ}}.

As in [N7, B.4.7] we say that V_{𝔭}(g) is quaternionic if D_{𝔭_{Γ}} is a division algebra (which can happen only for finitely many 𝔭).

According to [N7, B.4.8(1)], if the extension L_{𝔭}/(L^{Γ})_{𝔭_{Γ}} is unramified and the residual representation T_{𝔭}(g)/𝔭T_{𝔭}(g) is an irreducible G_{F} -module, then V_{𝔭}(g) is not quaternionic. In particular, the condition “V_{𝔭}(g) is not quaternionic” can be omitted in Theorem 3.5(1) if L_{𝔭}/(L^{Γ})_{𝔭_{Γ}} is unramified.

4. Parity results for abelian varieties with real multiplication

4.1. Let F and L be totally real number fields, let A be an abelian variety over F satisfying

dim(A) = [L : Q],  O_{L} = End_{F} (A).  (4.1.1)

For each finite prime 𝔭 of L the two-dimensional L_{𝔭}-representation V_{𝔭}(A) := T_{p}(A)⊗_{O_{L}⊗Z_{p}} L_{𝔭} of G_{F} satisfies the assumptions of Theorem 1.1 (with K = L_{𝔭}).

Recall that A is modular (over F) if there exists a cuspidal Hilbert modular newform g ∈ S_{2}(𝔫, 1) whose field of Hecke eigenvalues is equal to ι(L) ⊂ C (for some embedding ι : L ↪ C) and which satisfies

V_{𝔭}(A) ⟶^{∼} V_{𝔭}(g)(1)

for one ( ⟺ for each) finite prime 𝔭 of L. This is, in turn, equivalent to an equality of L-functions

L(ιA/F, s) = L(g, s)

(Euler factor by Euler factor), which implies that

∀σ ∈ Aut(C) L(σιA/F, s) = L(^{σ}g, s).

4.2. The potential automorphy results of [BLGGT] (Theorems 4.5.1 and 5.3.1) imply that every abelian variety A satisfying (4.1.1) is potentially modular in the following sense: for each finite extension M/F there exists a totally real finite extension F′/F which is linearly disjoint from M/F such that A ⊗_{F} F′ is modular over F′.

As in [N2, 12.11.6] and [N6, Step 4], a minor improvement (use of Solomon’s induction theorem [CR, Thm. 15.10] instead of the usual Brauer theorem) of an argument of Taylor [T2, proof of Cor. 2.2] implies that there exist intermediate fields F ⊂ F_{i} ⊂ F′ and integers n_{i} with the following properties:

(4.2.1) A is modular over each F_{i}: there exists a Hilbert modular newform g_{i} of parallel weight 2 over F_{i} such

that L(ιA/F_{i}, s) = L(g_{i}, s) and V_{𝔭}(A)|_{G_{F_{i}}} ⟶^{∼} V_{𝔭}(g_{i})(1) for each finite prime 𝔭 of L. (4.2.2) L(ιA/F, s) = ∏_{i} L(ιA/F_{i}, s)^{n_{i}} = ∏_{i} L(g_{i}, s)^{n_{i}}. (4.2.3) V_{𝔭}(A) = ⨁_{i} n_{i} Ind_{G_{F_{i}}}^{G_{F}} (V_{𝔭}(A)|_{G_{F_{i}}} ) = ⨁_{i} n_{i} Ind_{G_{F_{i}}}^{G_{F}} (V_{𝔭}(g_{i})(1)) in the Grothendieck ring of L_{𝔭}[G_{F}]-

modules.

It follows that, for each σ ∈ Aut(C), the L-function

L(σιA/F, s) = ∏_{i} L(^{σ}g_{i}, s)^{n_{i}}

has a meromorphic continuation to C and satisfies the expected functional equation. In particular, the analytic rank

r_{an}(σιA/F) := ord_{s=1} L(σιA/F, s) ∈ Z

is defined. As the ε-constant in the functional equation of L(^{σ}g_{i}, s) does not depend on σ, the parity

r_{an}(τA/F) (mod 2) ∈ Z/2Z

of the analytic rank r_{an}(τA/F) does not depend on the embedding τ : L ↪ C.

4.3. Theorem. Let A, F and L be as in (4.1.1). Let 𝔭 be a prime of L above a rational prime p ≠ 2. Assume that at least one of the following conditions holds:

(a) A is modular over F and 2 ∤ [F : Q].

(b) A does not have potentially good reduction everywhere.

(c) A does not have complex multiplication, A[𝔭] is an irreducible G_{F} -module and the simple algebra C := End\overline{F} (A) ⊗ Q satisfies C ⊗_{Z(C)} Z(C)_{𝔭_{C}} ⟶^{∼} M_{n}(Z(C)_{𝔭_{C}}), where 𝔭_{C} is the prime of Z(C) ⊂ L below 𝔭 [the latter condition follows from the irreducibility of A[𝔭] if L_{𝔭}/Z(C)_{𝔭_{C}} is unramified].

(d) A has complex multiplication by a totally imaginary quadratic extension L′ of L (defined over a totally imaginary quadratic extension K(A) of F), A[𝔭] is an irreducible G_{F} -module, 𝔭 splits in L′/L and the image of G_{K(A)} in Aut_{L′⊗_{L}L_{𝔭}}(V_{𝔭}(A)) = L_{𝔭}^{×} × L_{𝔭}^{×} contains an open subgroup of Z_{p}^{×} × Z_{p}^{×}.

(e) A[𝔭] is a reducible G_{F} -module, L_{𝔭}/Q_{p} is unramified and p > 2[L_{𝔭} : Q_{p}] + 1. Then the Selmer rank

dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(A)) = rk_{O_{L}}A(F) + cork_{O_{L,𝔭}} � (A/F)[𝔭^{∞}]

satisfies

dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(A)) ≡ r_{an}(τA/F) (mod 2),

for each embedding τ : L ↪ C.

Proof. The case (a) follows from Theorem 1.4(a). In the cases (b)-(e) we have, thanks to 4.2,

dim_{L_{𝔭}} H_{f}^{1}(F, V_{𝔭}(A)) − r_{an}(τA/F) ≡ ∑_{i} n_{i} (dim_{L_{𝔭}} H_{f}^{1}(F_{i}, V_{𝔭}(g_{i})(1)) − r_{an}(F_{i}, g_{i})) (mod 2),

which means that we can replace F by F_{i} and assume that A is modular over F (taking M = F (A[𝔭]) in 4.2 we ensure that A[𝔭] is irreducible as a G_{F_{i}}-module in the case (c) or (d)). The case (b) then follows from Theorem 1.4(b) and the cases (c) and (d) from Theorem 3.5 (using [N7, B.6.5(2)]). In the case (e) we can assume, thanks to Theorem 1.4(c), that π(g) is a principal series representation at each finite prime of F, which implies that A acquires locally at each completion of F (hence also globally, by [AT, ch. 10, Thm. 5]) good reduction over a suitable cyclic extension. The result then follows from an O_{L,𝔭}-equivariant version of the proof of [CFKS, Thm. 2.1]. 4.4. Proof of Theorem A. As in the proof of Theorem 4.3, potential modularity of E [Wi, Thm. A.1] together with (4.2.2-3) imply that we can write s_{p}(E/F) − r_{an}(E/F) as an integral linear combination of s_{p}(E/F_{i}) − r_{an}(E/F_{i}), for suitable totally real extensions F_{i}/F over which E is modular. It is enough, therefore, to replace F by F_{i} and consider only the case when E is modular over F (which is automatic if E has complex multiplication).

Assume first that p = 2. It follows from [Wa, Thm. 4] and [FH, Thm. B.1] that there exists a non-trivial quadratic character α : G_{F} ⟶ {±1} such that r_{an}(E ⊗ α/F) = 0. This implies, by [N7, Cor. of Thm. B’], that s_{2}(E ⊗ α/F) = 0. Let F′/F be the quadratic extension corresponding to α. As

s_{2}(E/F′) ≡ r_{an}(E/F′) (mod 2)

by [DD2, Cor. 4.8], we conclude by the following analogue of (3.2.1-2):

s_{p}(E/F′) = s_{p}(E/F) + s_{p}(E ⊗ α/F), r_{an}(E/F′) = r_{an}(E/F) + r_{an}(E ⊗ α/F).

If p ≠ 2, we can assume that 2 | [F : Q], in view of [N6, Thm. 1(a)]. Theorem 4.3(c),(d) (resp. (e)) then implies the desired result if E[p] is an irreducible G_{F} -module (resp. when E[p] is reducible and p > 3). The remaining case when p = 3 and E[3] is a reducible G_{F} -module is treated in [DD2, Cor. 5.8]. 4.5. Further absolute parity results (it would be too cumbersome to list them all here) follow from a combination of Theorem A with the relative parity results proved in [MR2, Thm. 6.4, 7.1], [MR3, Thm. 1.1], [DD1, Thm. 4.3, 4.5], [DD2, Prop. 6.12], [G, §11.8] and [Ro, Thm. 2.1]. 4.6. Note that our proof of Theorem A in the case when E[p] is a reducible G_{F} -module uses Theorem 1.4(c), which relies on several very recent technical advances: [AN], [N7] and [YZZ] (used in the proof of [N7, Thm. B(b)]). It would be desirable to have a more direct proof in the reducible case. 4.7. The conclusion of Theorem A also holds in the case when E has complex multiplication (hence is modular over F), p ≠ 2 and the conductor of E is not a square, by Theorem 1.4(c) (conductors are preserved under the local Langlands correspondence and the conductor of any principal series representation of P GL_{2}(F_{v}) is a square).

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