Compatibility of arithmetic and algebraic local constants (the case ℓ ≠ p)

Jan Nekovář

0. Introduction

Let M be a motive (in an old-fashioned sense) over a number field F with coefficients in another number field L. For every finite prime 𝔭 | p of L the 𝔭-adic realisation V = M_{𝔭} of M is a representation of the Galois group G_{F} = Gal(\overline{F} /F), which is finite-dimensional over K = L_{𝔭} and geometric in the sense of Fontaine and Mazur [FoM].

The L-function L(M, s) = ∑_{n≥1} a_{n}n^{−s} (if well defined) is a Dirichlet series with coefficients in L. It gives rise to complex-valued L-functions L(ιM, s) = ∑_{n≥1} ι(a_{n})n^{−s} for various embeddings ι : L ↪ C.

The conjectures of Bloch and Kato [BK] (generalised by Fontaine and Perrin-Riou [FoPR]) predict that the order of vanishing

r_{an}(ιM) := ord_{s=0}L(ιM, s)

should be equal to

χ_{f}(F, V^{∗}(1)) := h_{f}^{1}(F, V^{∗}(1)) − h^{0}(F, V^{∗}(1)) = dim_{K} H_{f}^{1}(F, V^{∗}(1)) − dim_{K} H^{0}(F, V^{∗}(1)),

where H_{f}^{1} is the Bloch-Kato Selmer group [BK, Def. 5.1] and V^{∗}(1) = Hom_{K}(V, Z_{p}(1) ⊗_{Z_{p}} K).

If the motive M is self-dual in the sense that L is totally real and there exists a skew-symmetric L-linear isomorphism M ≃ M^{∨}(1) (inducing a skew-symmetric isomorphism of K[G_{F}]-modules V ≃ V^{∗}(1)), then the completed L-function of ιM is expected to satisfy a symmetric functional equation

(L_{∞} · L)(ιM, s) = a^{s} ε(M)(L_{∞} · L)(ιM, −s),  (0.0)

where ε(M) = ∏_{v} ε_{v}(M) is a product of local constants ε_{v}(M) = ±1 (ε_{v}(M) does not depend on ι if v is a finite prime of F; the product ε_{∞}(M) = ∏_{v|∞} ε_{v}(M) does not depend on ι either). The functional equation (0.0) implies that

ε(M) = (−1)^{r_{an}(ιM)+r_{∞}(ιM)}, r_{∞}(ιM) := ord_{s=0}L_{∞}(ιM, s).

In particular, in the presence of (0.0) the (mod 2)-version of the Bloch-Kato conjecture

r_{an}(ιM) ≡^{?} χ_{f}(F, V) (mod 2)  (0.1)

is equivalent to

ε(M)(−1)^{r_{∞}(ιM)} =^{?} (−1)^{χ_{f} (F,V)}.  (0.2)

In the special case when the motive M is pure (necessarily of weight −1), then

r_{∞}(ιM) = 0, χ_{f}(F, V) = h_{f}^{1}(F, V),

the local constant ε_{v}(M) = ε_{v}(V) for v ∤ ∞ depends only on the restriction V |_{G_{v}} of V to the local Galois group G_{v} = Gal(\overline{F}_{v}/F_{v}) and the archimedean term ε_{∞}(M) depends only on V |_{G_{w}} for all w | p (see [N2, Erratum]). Consequently, in the pure case both sides of the conjectural equality (0.2), which boils down to

ε(V) =^{?} (−1)^{h_{f}^{1} (F,V)},  (0.3)

depend only on V.

The author’s research was supported in part by the grant ANR-BLAN-0114

Several authors ([DD1-2], [MR2-3], [N2]) have tried to establish a relative version of (0.2), namely, an equality

(−1)^{χ_{f} (F,V)}/(−1)^{χ_{f} (F,V′)} =^{?} (−1)^{r_{∞}(ιM)−r_{∞}(ιM′)} ε(M)/ε(M′)  (0.4)

or its special case when M and M′ are pure

(−1)^{h_{f}^{1} (F,V)}/(−1)^{h_{f}^{1} (F,V′)} =^{?} ε(V)/ε(V′),  (0.5)

for suitably related pairs of self-dual motives M and M′.

If p ≠ 2 and if there are G_{F} -stable self-dual lattices T ⊂ V, T′ ⊂ V′ whose reductions \overline{T} = T/𝔭T and \overline{T}′ = T′/𝔭T′ are (symplectically) isomorphic G_{F} -modules, a formula of Mazur and Rubin [MR2, Thm. 1.4] (combined with Flach’s generalisation of the Cassels–Tate pairing [Fl]) expresses

χ_{f}(F, V) − χ_{f}(F, V′) ≡ ∑_{v} d(ℱ_{v}, ℱ′_{v}) (mod 2) ∈ Z/2Z

as a sum of local terms, each of which depends only on the restrictions T |_{G_{v}} and T′|_{G_{v}} (only finite primes v of F dividing p or those at which V or V′ are ramified contribute to the formula). Mazur and Rubin expect these local terms (“arithmetic local constants”) to reflect the local decomposition of

ε(M)/ε(M′) = ∏_{v} ε_{v}(M)/ε_{v}(M′).

Our first main result (Theorem 2.17) confirms this intuition. We show that ε_{v}(V)/ε_{v}(V′) = (−1)^{d(ℱ_{v},ℱ′_{v})} for all v ∤ p. In fact, we prove a purely local version of this statement for pairs of self-dual representations of the local Galois group G_{v}. A special case of this result was proved by Chetty [Ch], who also treated some cases when v | p.

The case of v | p is much more complicated, in general. In Section 3 we give an example of two situations in which arithmetic local constants at v | p can be shown to match ε_{v}(V)/ε_{v}(V′).

In Section 4 we make explicit the relations (0.4), (0.5) in two global cases whose local behaviour at primes above p is covered by results of Section 3.

In Section 5 we combine one of the main results of Section 4 (Theorem 4.12) with techniques of [N5] in order to relax the assumptions of [N5, Theorem A] as follows.

Theorem. If E is an elliptic curve defined over a totally real number field F and if End\overline{Q}(E) ⊗ Q ≠ Q(i), Q(\sqrt{−3}), then the p-parity conjecture

ord_{s=1}L(E/F, s) ≡ rk_{Z}E(F) + cork_{Z_{p}} � (E/F)[p^{∞}] (mod 2)

holds for all primes p ≠ 2.

Corollary. Under the same assumptions the p-parity conjecture

ord_{s=1}L(E/F′, s) ≡ rk_{Z}E(F′) + cork_{Z_{p}} � (E/F′)[p^{∞}] (mod 2)

holds (if p ≠ 2) for any tower of finite extensions F ⊂ F_{1} ⊂ F′, where F_{1}/F is abelian and F′/F_{1} is a Galois extension of odd degree.

See 5.12-13 for a more precise version of these statements (which also covers certain cases when End\overline{Q}(E) ⊗ Q = Q(i), Q(\sqrt{−3}) or p = 2).

Notation and conventions. All representations (in particular, characters) are continuous. For a number field F we denote by h_{F} (resp. Cl_{F}) the class number (resp. the ideal class group) of the ring of integers O_{F} of F. We abbreviate ⊗_{Z} as ⊗. For an abelian group A we let Â = A ⊗ Ẑ, where Ẑ is the pro-finite completion of Z.

1. Bilinear algebra

(1.1) Let O be a discrete valuation ring with fraction field K, uniformiser π and residue field k = O/πO of characteristic char(k) ≠ 2. Let X be an O-module of finite length equipped with a non-degenerate symmetric bilinear pairing

( , ) : X × X ⟶ K/O.

(1.2) The k-vector space X[π] has a canonical decreasing exhaustive filtration

F^{0} = X[π] ⊃ F^{1} = πX[π^{2}] ⊃ F^{2} = π^{2}X[π^{3}] ⊃ ⋯

whose graded quotients gr_{F}^{i} = F^{i}/F^{i+1} (i ≥ 0) are equipped with non-degenerate symmetric pairings

( , )_{i} : gr_{F}^{i} × gr_{F}^{i} ⟶ k, (π^{i}x, π^{i}y)_{i} = π^{i}(x, y) (mod πO)

(which depend not only on ( , ) but also on π). (1.3) The filtration F^{i} is stable by any O-linear automorphism f : X ⟶ X. If f is an isometry (i.e., if (f(x), f(y)) = (x, y) for all x, y ∈ X), then the induced automorphisms gr^{i}(f) : gr_{F}^{i} ⟶ gr_{F}^{i} are also isometries.

(1.4) Proposition. If f : X ⟶ X is an isometry, then:

(i) dim_{k} Im (X^{f=1} ⟶ X/πX) ≡ ∑_{i≥0} dim_{k} Ker(gr^{i}(f) − 1) (mod 2). (ii) ([N2, Lemma 2.2.2]) If πX = 0 (in other words, if (X, ( , )) is a quadratic space over k), then

(−1)^{dim_{k}(X^{f=1})} = det(−f).

Proof. (i) There is an orthogonal decomposition X = X_{1} ⊕ X_{1}^{⊥}, where f − 1 is nilpotent (resp. invertible) on X_{1} (resp. on X_{1}^{⊥}):

X_{1} = ⋃_{n≥1} Ker (f − 1)^{n}, X_{1}^{⊥} = ⋂_{n≥1} Im (f − 1)^{n}.

Statement (i) is trivial for X_{1}^{⊥}; we can replace X by X_{1} and assume that f = 1 + N, where N : X ⟶ X is nilpotent. The formula

{x, y} = (Nx, y) + (Nx, Ny)/2 = (Nx, (1 + N/2)y)  (x, y ∈ X)  (1.4.1)

defines a skew-symmetric (hence alternating, since 2 ∈ O^{×} by assumption) bilinear form X × X ⟶ K/O with kernel equal to Ker(N) = X^{f=1}. The structure theory of symplectic O-modules of finite length implies that there is an exact sequence

0 ⟶ X^{f=1} ⟶ X ⟶ Z ⊕ Z ⟶ 0,

for a suitable O-module Z. As a result,

dim_{k} Im (X^{f=1} ⟶ X/πX) = dim_{k}(X/πX) − 2 dim_{k}(Z/πZ) ≡ dim_{k}(X/πX) =
= dim_{k} X[π] = ∑_{i≥0} dim_{k} gr_{F}^{i} (mod 2).

Finally, applying statement (ii) to the unipotent isometry gr^{i}(f) of gr_{F}^{i} we obtain

dim_{k} gr_{F}^{i} ≡ dim_{k} Ker(gr^{i}(f) − 1) (mod 2).

(ii) In [N2, Lemma 2.2.2] we reproduced a proof of this statement due to J. Oesterlé (it relied on a decomposition of f into a product of reflections). Alternatively, one can use the orthogonal decomposition into generalised eigenspaces of f (after replacing k by its algebraic closure \overline{k} and X by X ⊗_{k} \overline{k})

X = ⨁_{λ} X_{λ}, X_{λ} = ⋃_{n≥1} Ker (f − λ)^{n}.

The orthogonality relation X_{λ} ⊥ X_{μ} for λμ ≠ 1 implies that dim_{k} X_{λ} = dim_{k} X_{1/λ}, hence

det(−f) = det(−f | X_{1}) ∏_{λ≠1}(−λ)^{dim_{k} X_{λ}} = det(−f | X_{1}) = (−1)^{dim_{k} X_{1}}.

As f − 1 is nilpotent on X_{1}, formula (1.4.1) defines a skew-symmetric bilinear form X_{1} × X_{1} ⟶ k with kernel X_{1}^{f=1}. It follows that

dim_{k} X_{1} ≡ dim_{k} X_{1}^{f=1} = dim_{k} X^{f=1} (mod 2).

(1.5) Corollary. If f : X ⟶ X is an isometry, then

(−1)^{dim_{k} Im(X^{f=1}⟶X/πX)} = det(−f | X[π]).

Proof. The R.H.S. is equal to ∏_{i≥0} det(−gr^{i}(f)); the result follows from combining (i) with (ii) applied to each gr_{F}^{i}.

2. Arithmetic local constants (the case ℓ ≠ p)

(2.1) Let O be the ring of integers in a finite extension K of Q_{p}, where p ≠ 2. Fix a uniformiser π ∈ O and denote by k = O/πO the residue field of K. (2.2) Let K be a finite extension of Q_{ℓ}. In 2.6–2.17 we assume that ℓ ≠ p. Denote by G = Gal(\overline{K}/K) ⊃ I ⊃ I_{w} the absolute Galois group, the inertia group and the wild inertia group of K, respectively. (2.3) Let V be a finite-dimensional K-vector space equipped with a continuous linear action of G and a non-degenerate G-equivariant skew-symmetric pairing

⟨ , ⟩ : V × V ⟶ K(1) = K ⊗_{Z_{p}} Z_{p}(1).

(2.4) For any G-stable O-lattice T ⊂ V denote by \overline{T} = T/πT its reduction modulo π. We say that T is selfdual with respect to ⟨ , ⟩ if ⟨ , ⟩ induces a pairing T × T ⟶ O(1) whose reduction \overline{T} × \overline{T} ⟶ k(1) = μ_{p} ⊗ k is non-degenerate. (2.5) Note that T is self-dual with respect to a multiple of ⟨ , ⟩ by a suitable scalar a ∈ K^{×} if dim_{K}(V) = 2 (or if \overline{T} is an irreducible k[G]-module). (2.6) From now on until the end of §2 we assume that T is self-dual with respect to ⟨ , ⟩ and that ℓ ≠ p.

According to Tate’s local duality the pairings

H^{i}(\overline{T}) × H^{2−i}(\overline{T}) ⟶^{∪} H^{2}(k(1)) ≃ k (H^{i}(−) = H^{i}(G, −))

are non-degenerate (and symmetric if i = 1). The local Euler characteristic formula implies that the dimensions h^{i}(\overline{T}) = dim_{k} H^{i}(\overline{T}) satisfy

h^{1}(\overline{T}) = h^{0}(\overline{T}) + h^{2}(\overline{T}) = 2 h^{0}(\overline{T}).

The unramified cohomology H_{ur}^{1}(\overline{T}) = H^{1}(G/I, \overline{T}^{I}) ⊂ H^{1}(\overline{T}) is an isotropic subspace of the quadratic space (H^{1}(\overline{T}), ∪) (since cd_{p}(G/I) = 1) of dimension h_{ur}^{1}(\overline{T}) = h^{0}(\overline{T}) = h^{1}(\overline{T})/2. It follows that H_{ur}^{1}(\overline{T}) is a Lagrangian subspace of H^{1}(\overline{T}), i.e., a subspace equal to its own orthogonal complement (which is equivalent to being an isotropic subspace of dimension h^{1}(\overline{T})/2). (2.7) Similarly, the unramified cohomology H_{ur}^{1}(V) = H^{1}(G/I, V^{I}) ⊂ H^{1}(V) is a Lagrangian subspace of H^{1}(V). The space H^{1}(V) is equipped with a non-degenerate symmetric pairing ∪ : H^{1}(V) × H^{1}(V) ⟶ H^{2}(K(1)) ≃ K and its dimension is equal to dim_{K} H^{1}(V) = 2 dim_{K} H^{0}(V).

By propagation ([MR1, Example 1.1.2]), H_{ur}^{1}(V) defines subspaces

H_{f}^{1}(T) = Ker(H^{1}(T) ⟶ H^{1}(V)/H_{ur}^{1}(V)) = H_{ur}^{1}(T) + H^{1}(T)_{tors} ⊂ H^{1}(T), H_{ur}^{1}(T) = H^{1}(G/I, T^{I}), ℱ = Im(H_{f}^{1}(T) ⟶ H^{1}(\overline{T})) ⊂ H^{1}(\overline{T}).

The subspace ℱ is again Lagrangian in H^{1}(\overline{T}). If V = V^{I} is unramified, then ℱ = H_{ur}^{1}(\overline{T}). (2.8) Definition. For Lagrangian subspaces L, L′ ⊂ H^{1}(\overline{T}), define

d(L, L′) = dim_{k}(L/L ∩ L′) (mod 2) ∈ Z/2Z

(of course, d(L, L′) = d(L′, L), since dim_{k}(L) = dim_{k}(L′) = h^{0}(\overline{T})). (2.9) Proposition [KMR, Cor. 2.5]. If L_{0}, L_{1}, L_{2} ⊂ H^{1}(\overline{T}) are Lagrangian subspaces, then

d(L_{0}, L_{1}) − d(L_{0}, L_{2}) + d(L_{1}, L_{2}) = 0 ∈ Z/2Z.

(2.10) Denote by

H_{/ur}^{1}(−) = H^{1}(−)/H_{ur}^{1}(−) (− = V, T, \overline{T})

the “ramified part” of H^{1}(−). The restriction map induces an isomorphism

H_{/ur}^{1}(−) ≃ H^{1}(I, −)^{f=1} (− = V, T, \overline{T}),

where f ∈ G is any lift of the geometric Frobenius element generating G/I topologically. (2.11) Proposition. There is a canonical isomorphism

ℱ / (ℱ ∩ H_{ur}^{1}(\overline{T})) ≃ Im (H_{/ur}^{1}(T)_{tors} ⟶ H_{/ur}^{1}(\overline{T})) .

Proof. The L.H.S. is isomorphic to

Im (ℱ ⟶ H_{/ur}^{1}(\overline{T})) = Im (H_{f}^{1}(T) ⟶ H_{/ur}^{1}(T) ⟶ H_{/ur}^{1}(\overline{T})) .

On the other hand,

Im (H_{f}^{1}(T) ⟶ H_{/ur}^{1}(T)) = H_{f}^{1}(T)/H_{ur}^{1}(T) = (H_{ur}^{1}(T) + H^{1}(T)_{tors})/H_{ur}^{1}(T)
= Im (H^{1}(T)_{tors} ⟶ H_{/ur}^{1}(T)) .

The statement of the proposition follows from the fact that Im(H^{1}(T)_{tors} ⟶ H_{/ur}^{1}(T)) = H_{/ur}^{1}(T)_{tors} (the inclusion “⊂” is automatic; the opposite inclusion “⊃” is a consequence of the fact that H^{1}(T)/H_{f}^{1}(T) is torsion-free). (2.12) Proposition. There exists a non-degenerate symmetric bilinear pairing

( , ) : H^{1}(I, T)_{tors} × H^{1}(I, T)_{tors} ⟶ K/O

for which f is an isometry. Proof. This is a local version of the Cassels–Tate pairing, attached to the duality isomorphism RΓ(I, T) ≃ RHom_{O}(RΓ(I, T), O[−1]) (cf. the discussion in [N1, 10.1]). It can be described explicitly as follows. There is an orthogonal decomposition (with respect to ⟨ , ⟩) T = T_{tame} ⊕ T_{wild}, in which T_{tame} = T^{I_{w}}, T_{wild}^{I_{w}} = 0 and H^{i}(I, T) = H^{i}(I/I_{w}, T_{tame}). We can replace T by T_{tame} and assume that T is a representation of the group G/I_{w}, which is topologically generated by f and any fixed topological generator t of the tame inertia group I/I_{w}. If q denotes the cardinality of the residue field of K (which is prime to p, by assumption), then tf = ft^{q}.

Consider ⟨ , ⟩ as a pairing ⟨ , ⟩ : T × T ⟶ O satisfying

∀x, y ∈ T ⟨tx, ty⟩ = ⟨x, y⟩, ⟨fx, fy⟩ = q^{−1}⟨x, y⟩.

If c, c′ ∈ Z^{1}(I/I_{w}, T) are 1-cocycles whose cohomology classes [c], [c′] lie in H^{1}(I/I_{w}, T)[π^{m}] (m ≥ 0), then π^{m}c(t) = (t − 1)a for some a ∈ T. We define

([c], [c′]) = (π^{−m}⟨a, c′(t)⟩) (mod O) ∈ π^{−m}O/O ⊂ K/O.

It is an elementary exercise to check that this formula defines a pairing with the required properties.

(2.13) Proposition-Definition. The invariant

e(T) := d (ℱ , H_{ur}^{1}(\overline{T})) = dim_{k} ℱ / (ℱ ∩ H_{ur}^{1}(\overline{T})) (mod 2) ∈ Z/2Z

satisfies

(−1)^{e(T)} det(−f | V^{I}) ≡ det(−f | \overline{T}^{I}) (mod πO).

Proof. According to Proposition 2.11 we have

ℱ / (ℱ ∩ H_{ur}^{1}(\overline{T})) ≃ Im (X^{f=1} ⟶ H^{1}(I, \overline{T})) = Im (X^{f=1} ⟶ X/πX) , X = H^{1}(I, T)_{tors}

(since both maps X/πX ⟶ H^{1}(I, T)/πH^{1}(I, T) ⟶ H^{1}(I, \overline{T}) are injective). Combining Proposition 2.12 with Corollary 1.5 we obtain

(−1)^{e(T)} = det(−f | X[π]) ∈ {±1} ⊂ k^{×},

hence

(−1)^{e(T)} det(−f | V^{I}) ≡ (−1)^{e(T)} det(−f | T^{I}/πT^{I}) = det(−f | T^{I}/πT^{I}) det(−f | X[π])
= det(−f | \overline{T}^{I}) (mod πO),

where the last equality follows from the exact sequence

0 ⟶ T^{I}/πT^{I} ⟶ \overline{T}^{I} ⟶ X[π] ⟶ 0.

(2.14) Deligne [D] defined local constants ε(V, ψ, dx) depending on V (more precisely, on the corresponding representation of the Weil–Deligne group of K), a non-trivial additive character ψ of K and a Haar measure dx on K. If dx_{ψ} is the Haar measure which is self-dual with respect to ψ, then the value of ε(V) = ε(V, ψ, dx_{ψ}) does not depend on ψ and is equal to ±1 ([N2, Prop. 2.2.1]).

Deligne ([D, (5.1)]) also introduced modified local constants

ε_{0}(V) = ε(V) det(−f | V^{I}) ∈ O^{×}

and showed ([D, Thm. 6.5]) that the value of ε_{0}(V) (mod πO) ∈ k^{×} depends only on \overline{T}; we denote it by ε_{0}(\overline{T}). The congruence in Proposition 2.13 can be restated as follows.

(2.15) Corollary. (−1)^{e(T)}ε(V) ≡ det(−f | \overline{T}^{I}) ε_{0}(\overline{T})^{−1} (mod πO).

(2.16) Let V′ be another representation of G with coefficients in K, equipped with a non-degenerate Gequivariant skew-symmetric pairing ⟨ , ⟩′ : V′ × V′ ⟶ K(1). Assume that T′ ⊂ V′ is a self-dual lattice with respect to ⟨ , ⟩′ and that there exists an isomorphism of k[G]-modules \overline{T} ≃ \overline{T}′ = T′/πT′ compatible with the pairings \overline{T} × \overline{T} ⟶ k(1) and \overline{T}′ × \overline{T}′ ⟶ k(1) induced by ⟨ , ⟩ and ⟨ , ⟩′, respectively.

Fix such an isomorphism \overline{T} ≃ \overline{T}′; it induces an isomorphism H^{1}(\overline{T}) ≃ H^{1}(\overline{T}′) under which

ℱ′ = Im (H_{f}^{1}(T′) ⟶ H^{1}(\overline{T}′)) ⊂ H^{1}(\overline{T}′)

becomes a Lagrangian subspace of H^{1}(\overline{T}). The invariant d(ℱ , ℱ′) ∈ Z/2Z is the arithmetic local constant of Mazur and Rubin [MR2].

(2.17) Theorem (Compatibility of arithmetic and algebraic local constants in the case ℓ ≠ p). In the situation of 2.16 we have (−1)^{d(ℱ,ℱ′)} = ε(V)/ε(V′).

Proof. Applying Corollary 2.15 to T and T′ we obtain

(−1)^{e(T)}ε(V) ≡ (−1)^{e(T′)}ε(V′) (mod πO),

hence (−1)^{e(T)}ε(V) = (−1)^{e(T′)}ε(V′), since 1 ≢ −1 (mod πO). We conclude by noting that e(T) − e(T′) + d(ℱ , ℱ′) = 0 ∈ Z/2Z, thanks to Proposition 2.9.

3. Arithmetic local constants (the case ℓ = p)

(3.1) Assume now that ℓ = p (≠ 2) in the situation of 2.2 and that we are given T ⊂ V and T′ ⊂ V′ as in 2.16 (with T self-dual, too). (3.2) Tate’s local duality still holds; the corresponding quadratic space (H^{1}(\overline{T}), ∪) has (even) dimension h^{1}(\overline{T}) = 2 h^{0}(\overline{T}) + [K : Q_{p}] dim_{k} \overline{T}. In order to produce suitable Lagrangian subspaces of H^{1}(\overline{T}) we need to assume that both V and V′ are potentially semistable representations of G.

The Bloch-Kato subspaces

H_{f}^{1}(−) = Ker (H^{1}(−) ⟶ H^{1}(− ⊗_{Q_{p}} B_{cris})) (− = V, V′)

are then Lagrangian subspaces of H^{1}(V) and H^{1}(V′), respectively ([BK, Prop. 3.8]). They define, by propagation, subspaces

H_{f}^{1}(T) = Ker (H^{1}(T) ⟶ H^{1}(V)/H_{f}^{1}(V)) ⊂ H^{1}(T)

and H_{f}^{1}(T′) ⊂ H^{1}(T′). Their images

ℱ = Im (H_{f}^{1}(T) ⟶ H^{1}(\overline{T})) ⊂ H^{1}(\overline{T})
ℱ′ = Im (H_{f}^{1}(T′) ⟶ H^{1}(\overline{T}′)) ⊂ H^{1}(\overline{T}′) ≃ H^{1}(\overline{T})

are Lagrangian subspaces of H^{1}(\overline{T}). (3.3) One can attach to Fontaine’s module D_{pst}(V) a representation W D(V) of the Weil–Deligne group of K ([Fo], [FoPR, I.1.3.2]), which is self-dual in the same way as V is. The local constant ε(V) := ε(W D(V), ψ, dx_{ψ}) ∈ {±1} is again independent of ψ (and similarly for ε(V′)).

It would be highly desirable to relate ε(V)/ε(V′) to the arithmetic local constant d(ℱ , ℱ′). We are going to discuss two special cases in which this can be done, but it is not clear in what generality one can expect such a relation to hold. (3.4) In the first case the local conditions are given by flat cohomology, as in the pioneering work of Mazur [M]. Assume that Q_{p} ⊂ K_{1} ⊂ K is a subfield and that T and T′ have the following properties.

(3.4.1) The ramification index of K_{1}/Q_{p} satisfies e(K_{1}/Q_{p}) < p − 1. (3.4.2) The action of G on T and T′ extends to an action of G_{1} = Gal(\overline{K}/K_{1}). (3.4.3) All the data ⟨ , ⟩, ⟨ , ⟩′ and \overline{T} ≃ \overline{T}′ are G_{1}-equivariant. (3.4.4) There exist π-divisible O-modules H and H′ over the ring of integers O_{K_{1}} ⊂ K_{1} and O[G_{1}]-equivariant

isomorphisms T ≃ T_{π}H(\overline{K}), T′ ≃ T_{π}H′(\overline{K}). (3.5) Assumptions (3.4.2-4) imply that V and V′ are crystalline representations of G_{1}, hence W D(V), W D(V′) are unramified representations of the Weil–Deligne group of K_{1} (which are self-dual, up to the Tate twist). As a result, the local constants of both V and V′ over any extension of K_{1} are equal to 1.

(3.6) Proposition. Under assumptions (3.4.1-4) we have ℱ = ℱ′ and d(ℱ , ℱ′) = 0.

Proof. In this situation \overline{T} = H[π](\overline{K}), \overline{T}′ = H′[π](\overline{K}) and

ℱ = Im (H_{fppf}^{1}(O_{K}, H[π]) ⟶ H^{1}(\overline{T})) ,  ℱ′ = Im (H_{fppf}^{1}(O_{K}, H′[π]) ⟶ H^{1}(\overline{T}′))  (3.6.1)

(see [N4, Prop. A.2.6]). Assumption (3.4.1) implies, by Raynaud’s theorem [Ra, Cor. 3.3.6], that the fixed isomorphism \overline{T} ≃ \overline{T}′ in (3.4.3) comes from a unique isomorphism H[π] ≃ H′[π] of finite flat k-vector space schemes over O_{K_{1}}. The statement of the theorem then follows from the functoriality of the morphisms in (3.6.1). (3.7) In the second case we fix two representations ρ, ρ′ : G ⟶ GL_{n}(O) with open kernel whose reductions \overline{ρ} = \overline{ρ}′ : G ⟶ GL_{n}(O/πO) = GL_{n}(k) modulo π coincide. In order to simplify the notation, write ρ_{+} = ρ, ρ′_{+} = ρ′ and denote by ρ_{−} = ρ^{∗}, ρ′_{−} = (ρ′)^{∗} : G ⟶ GL_{n}(O) the dual representations. Instead of the pair T, T′ as in (3.1) we consider the pair T ⊗ (ρ_{+} ⊕ ρ_{−}) and T ⊗ (ρ′_{+} ⊕ ρ′_{−}). (3.8) The subspaces

ℱ ⊂ H^{1}(\overline{T} ⊗ (\overline{ρ}_{+} ⊕ \overline{ρ}_{−})) = H^{1}(\overline{T} ⊗ (\overline{ρ}′_{+} ⊕ \overline{ρ}′_{−})) ⊃ ℱ′

are respectively equal to ℱ = Y_{+} ⊕ Y_{−} and ℱ′ = Y′_{+} ⊕ Y′_{−}, where

\begin{aligned}Y_{±} & = Im (H_{f}^{1}(T ⊗ ρ_{±}) ⊗_{O} k ↪ H^{1}(\overline{T} ⊗ \overline{ρ}_{±})) , \\ Y′_{±} & = Im (H_{f}^{1}(T ⊗ ρ′_{±}) ⊗_{O} k ↪ H^{1}(\overline{T} ⊗ \overline{ρ}′_{±}) = H^{1}(\overline{T} ⊗ \overline{ρ}_{±})) , \end{aligned}

hence

d(ℱ , ℱ′) ≡ dim_{k} (Y_{+}/(Y_{+} ∩ Y′_{+})) + dim_{k} (Y_{−}/(Y_{−} ∩ Y′_{−})) (mod 2).

The following proposition is a generalisation of [N5, Lemma 1.2].

(3.9) Proposition. In the situation of (3.7) we have ε(V ⊗ (ρ ⊕ ρ^{∗})) = ε(V ⊗ (ρ′ ⊕ ρ^{′∗})) and d(ℱ , ℱ′) = 0.

Proof. Tate’s local duality gives rise to a perfect pairing

X_{+} × X_{−} = H^{1}(\overline{T} ⊗ \overline{ρ}_{+}) × H^{1}(\overline{T} ⊗ \overline{ρ}_{−}) ⟶ H^{2}(k(1)) ≃ k

(thus dim_{k}(X_{+}) = dim_{k}(X_{−})) under which Y_{+}^{⊥} = Y_{−} and (Y′_{+})^{⊥} = Y′_{−} ([BK, Prop. 3.8]). As in ([N5], proof of Lemma 1.2) we have

dim_{k}(Y_{±}) − dim_{k} H^{0}(\overline{T} ⊗ \overline{ρ}_{±}) = dim_{K} H_{f}^{1}(V ⊗ ρ_{±}) − dim_{K} H^{0}(V ⊗ ρ_{±}) =
= dim_{K} D_{dR}(V ⊗ ρ_{±})/F il^{0},

which does not depend on the sign ±. Together with an analogous formula for ρ′_{±} this implies that dim_{k}(Y_{±}) = dim_{k}(X_{±})/2 = dim_{k}(Y′_{±}). As a result,

dim_{k}(Y_{+} ∩ Y′_{+}) = dim_{k}(Y_{+}) + dim_{k}(Y′_{+}) − dim_{k}(Y_{+} + Y′_{+}) = dim_{k}(X_{+}) − dim_{k}(Y_{+} + Y′_{+})
= dim_{k}(Y_{+} + Y′_{+})^{⊥} = dim_{k}(Y_{+}^{⊥} ∩ (Y′_{+})^{⊥}) = dim_{k}(Y_{−} ∩ Y′_{−}),

hence d(ℱ , ℱ′) = 0. The local constants satisfy

ε = ε(V ⊗ (ρ ⊕ ρ^{∗})) = det(V ⊗ ρ)(−1) ≡ det(V ⊗ ρ′)(−1) = ε(V ⊗ (ρ′ ⊕ ρ^{′∗})) = ε′ (mod πO),

which implies that ε = ε′ (since p ≠ 2).

4. The global case

(4.1) Let Q_{p} ⊂ K ⊃ O ⟶ O/πO = k be as in (2.1) (in particular, p ≠ 2). (4.2) Let F be a number field and S a finite set of primes of F containing all infinite primes and all primes above p. Denote by G_{F,S} = Gal(F_{S}/F) the Galois group of the maximal extension of F unramified outside S. (4.3) Consider a global version of 2.16: let T (resp. T′) be a free O-module of finite rank equipped with a continuous O-linear action of G_{F,S} and a skew-symmetric bilinear G_{F,S}-equivariant pairing ⟨ , ⟩ : T × T ⟶ O(1) (resp. ⟨ , ⟩′ : T′ × T′ ⟶ O(1)) whose reduction \overline{T} × \overline{T} ⟶ k(1) (resp. \overline{T}′ × \overline{T}′ ⟶ k(1)) modulo π is non-degenerate. Assume that there exists an isomorphism of k[G_{F,S}]-modules \overline{T} ≃ \overline{T}′ compatible with the above pairings; we fix it and use it to identify various cohomology groups of \overline{T}′ with those of \overline{T}. Set V = T ⊗_{O} K, V′ = T′ ⊗_{O} K and assume that V and V′ are potentially semistable representations of G_{v} = Gal(\overline{F}_{v}/F_{v}), for all v | p. (4.4) For finite primes v ∤ p (resp. v | p) of F we have the submodules H_{f}^{1}(G_{v}, T) ⊂ H^{1}(G_{v}, T) and H_{f}^{1}(G_{v}, T′) ⊂ H^{1}(G_{v}, T′) defined in 2.7 (resp. in 3.2) and their images ℱ_{v}, ℱ′_{v} ⊂ H^{1}(G_{v}, \overline{T}), which are Lagrangian subspaces of H^{1}(G_{v}, \overline{T}). For v ∉ S the representations V and V′ are unramified, hence ℱ_{v} = ℱ′_{v} = H_{ur}^{1}(G_{v}, \overline{T}).

In other words, these subspaces define two self-dual Selmer structures ([MR2, 1.2]) ℱ = {ℱ_{v}} and ℱ′ = \{ℱ′_{v}\} for the self-dual k[G_{F,S}]-module \overline{T}. (4.5) The corresponding Selmer groups are

\begin{aligned}H_{ℱ}^{1} (F, \overline{T}) & = Ker(H^{1}(G_{F,S}, \overline{T}) ⟶ ⨁_{v∈S_{f}} H^{1}(G_{v}, \overline{T})/ℱ_{v}) \\ H_{ℱ′}^{1} (F, \overline{T}) & = Ker(H^{1}(G_{F,S}, \overline{T}) ⟶ ⨁_{v∈S_{f}} H^{1}(G_{v}, \overline{T})/ℱ′_{v}), \end{aligned}

where S_{f} = S \ \{v | ∞\}. Denote by h_{ℱ}^{1} (F, \overline{T}) (resp. h_{ℱ′}^{1} (F, \overline{T})) their respective dimensions over k and let h^{0}(F, \overline{T}) = dim_{k} H^{0}(G_{F,S}, \overline{T}).

The existence of Flach’s generalisation of the Cassels–Tate pairing [Fl] implies [N5, (1.1.3)] that

\begin{aligned}χ_{f}(F, V) & ≡ h_{ℱ}^{1} (F, \overline{T}) − h^{0}(F, \overline{T}) (mod 2) \\ χ_{f}(F, V′) & ≡ h_{ℱ′}^{1} (F, \overline{T}) − h^{0}(F, \overline{T}) (mod 2), \end{aligned}

hence

χ_{f}(F, V) − χ_{f}(F, V′) ≡ h_{ℱ}^{1} (F, \overline{T}) − h_{ℱ′}^{1} (F, \overline{T}) (mod 2).  (4.5.1)

(4.6) The R.H.S. of (4.5.1) is given by a formula of Mazur and Rubin [MR2, Thm. 1.4]:

h_{ℱ}^{1} (F, \overline{T}) − h_{ℱ′}^{1} (F, \overline{T}) ≡ ∑_{v∈S_{f}} d(ℱ_{v}, ℱ′_{v}) (mod 2),  (4.6.1)

where d(ℱ_{v}, ℱ′_{v}) ≡ dim_{k} ℱ_{v}/(ℱ_{v} ∩ ℱ′_{v}) (mod 2), as in 2.8. Combining (4.6.1) with (4.5.1), we obtain

χ_{f}(F, V) − χ_{f}(F, V′) ≡ ∑_{v∈S_{f}} d(ℱ_{v}, ℱ′_{v}) (mod 2).  (4.6.2)

(4.7) For each finite prime v of F, there are local constants ε_{v}(V) = ε(V |_{G_{v}}) = ±1 and ε_{v}(V′) = ε(V′|_{G_{v}}) = ±1, equal to 1 for v ∉ S. If V and V′ are 𝔭-adic realisations of self-dual motives M and M′ over F with isomorphic real Hodge realisations, then the archimedean L- and ε-factors of M and M′ coincide and the conjectural equality (0.4) becomes

∏_{v∈S_{f}} ε_{v}(V)/ε_{v}(V′) =^{?} (−1)^{χ_{f} (F,V)−χ_{f} (F,V′)} = ∏_{v∈S_{f}}(−1)^{d(ℱ_{v},ℱ′_{v})} ,  (4.7.1)

which is equivalent, thanks to Theorem 2.17, to

∏_{v|p} ε_{v}(V)/ε_{v}(V′) =^{?} ∏_{v|p}(−1)^{d(ℱ_{v},ℱ′_{v})} .  (4.7.2)

If the discussion in Section 3 applies at each v | p, then (4.7.2) holds term by term and we can go backwards from (4.7.2) to (4.7.1) and (0.4). Let us make this explicit in two cases corresponding to Propositions 3.6 and 3.9, respectively. The first case is covered in paragraphs 4.8 to 4.14 and the second case in paragraphs 4.15 and 4.16. (4.8) The first case involves abelian varieties with real multiplication. Let L be a totally real number field and A an abelian variety over F equipped with a ring morphism O_{L} ↪ End_{F} (A). For each rational prime p the decomposition O_{L} ⊗ Z_{p} = ∏_{𝔭|p} O_{L,𝔭} induces decompositions

T_{p}(A) = ∏_{𝔭|p} T_{𝔭}(A), V_{p}(A) = ∏_{𝔭|p} V_{𝔭}(A)

(V_{p}(A) is a free L ⊗ Q_{p}-module). An O_{L}-linear polarisation λ : A ⟶ A^{t} defines skew-symmetric Weil pairings

T_{𝔭}(A) × T_{𝔭}(A) ⟶ T_{𝔭}(A) ⊗_{O_{L,𝔭}} T_{𝔭}(A) ⟶ Z_{p}(1),

hence O_{L,𝔭}-bilinear skew-symmetric pairings

⟨ , ⟩ : T_{𝔭}(A) × T_{𝔭}(A) ⟶ Hom_{Z_{p}}(O_{L,𝔭}, Z_{p})(1) ≃ O_{L,𝔭}(1).

If Ker(λ)[𝔭] = 0, then T_{𝔭}(A) is self-dual with respect to ⟨ , ⟩ in the sense of 2.4. (4.9) We consider M = h^{1}(A)(1) as a (self-dual) motive with coefficients in L. Its 𝔭-adic realisations are M_{𝔭} = V_{𝔭}(A)^{∗}(1) ≃ V_{𝔭}(A) and the Euler factors of its L-function

L_{v}(M, s) = det(1 − Fr_{geom}(v)(Nv)^{−s} | M_{𝔭}^{I_{v}})^{−1} (v ∤ p)

have coefficients in L and do not depend on 𝔭. The L-functions L(ιM, s) = L(ιA/F, s + 1) for various embeddings ι : L ↪ R are defined for Re(s) > 1/2 and are related to the usual L-function of A (when we consider h^{1}(A) as a motive with coefficients in Q) by

L(A/F, s) = ∏_{ι:L↪R} L(ιA/F, s).  (4.9.1)

The archimedean L- and ε-factors of L(ιM, s) are equal to

L_{∞}(ιM, s) = ((2π)^{−s}Γ(s))^{[F:Q] dim(A)/[L:Q]} , ε_{∞}(ιM) = (−1)^{(r_{1}(F)+r_{2}(F)) dim(A)/[L:Q]}.

The global ε-factor ε(ιA/F) := ε(M) ∈ {±1} does not depend on ι, but we keep ι in the notation in order not to confuse ε(ιA/F) with the ε-factor of the product L-function (4.9.1).

(4.10) The representation V_{𝔭}(A) of G_{F} is pure of weight -1 at each finite prime of F, which implies that h^{0}(F′, V_{𝔭}(A)) = 0, for every finite extension F′/F. The rank of the Bloch-Kato Selmer group of V_{𝔭}(A) is equal to

h_{f}^{1}(F′, V_{𝔭}(A)) = rk_{O_{L}}A(F′) + cork_{O_{L,𝔭}} � (A/F′)[𝔭^{∞}].

(4.11) Fix a prime 𝔭 of L dividing a rational prime p ≠ 2 and let K = L_{𝔭}, O = O_{L,𝔭} and k = O/𝔭. Assume that

(4.11.1) A and A′ are abelian varieties over F of the same dimension equipped with embeddings O_{L} ↪

End_{F} (A), End_{F} (A′); (4.11.2) there are O_{L}-linear polarisations λ : A ⟶ A^{t} and λ′ : A′ ⟶ A^{′t} such that Ker(λ)[𝔭] = Ker(λ′)[𝔭] = 0.

As in 4.8, they give rise to self-dual O[G_{F,S}]-modules (for suitable S) T ×T ⟶ O(1), T′ ×T′ ⟶ O(1), where T = T_{𝔭}(A) ⊂ V = V_{𝔭}(A) and T′ = T_{𝔭}(A′) ⊂ V′ = V_{𝔭}(A′);

(4.11.3) there exists an isomorphism of k[G_{F,S}]-modules \overline{T} = T/𝔭T ≃ \overline{T}′ = T′/𝔭T′ compatible with the

reductions modulo 𝔭 of the pairings from (4.11.2); (4.11.4) the absolute ramification index of each prime v | p of F satisfies e(v | p) < p − 1; (4.11.5) A and A′ have good reduction at all primes v | p of F.

These assumptions imply that, for each v | p, the restrictions of T and T′ to G_{v} satisfy (3.4.1-4) with K_{1} = K = F_{v}.

(4.12) Theorem. In the situation of (4.11), let F′ be a finite extension of F and α : G_{F′} = Gal(\overline{F} /F′) ⟶ {±1} a quadratic (or trivial) character. We have, for each ι : L ↪ R,

(−1)^{h_{f}^{1} (F′,V_{𝔭}(A)⊗α)}/ε(ιA ⊗ α/F′) = (−1)^{h_{f}^{1} (F′,V_{𝔭}(A′)⊗α)}/ε(ιA′ ⊗ α/F′),

where we have denoted by A ⊗ α the quadratic twist of A ⊗_{F} F′ by α (and similarly for A′).

Proof. Let c(F′, α) be the quotient of the L.H.S. by the R.H.S. If α ≠ 1, then c(F′, α) = c(F^{′(α)}, 1)/c(F′, 1), where F^{′(α)} = \overline{F}^{Ker(α)}. It follows that it is sufficient to prove the claim for α = 1 (and varying F′). As explained in 4.7, this follows from Theorem 2.17 and Proposition 3.6 applied to T and T′ over various completions of F′.

(4.13) If, in addition, dim(A) = dim(A′) = [L : Q] and the field F′ is totally real, then the abelian varieties A and A′ (as well as their quadratic twists) are potentially modular over F′ [BLGGT, Cor. 5.4.2]. In particular, their L-functions over F′ have meromorphic continuation to C and satisfy the expected functional equations [BLGGT, Cor. 5.4.3]

(L_{∞} · L)(ιA ⊗ α/F′, s) = a^{s−1} ε(ιA ⊗ α/F′)(L_{∞} · L)(ιA ⊗ α/F′, 2 − s),
(L_{∞} · L)(ιA′ ⊗ α/F′, s) = b^{s−1} ε(ιA′ ⊗ α/F′)(L_{∞} · L)(ιA′ ⊗ α/F′, 2 − s).

In particular, the analytic rank

r_{an}(ιA ⊗ α/F′) := ord_{s=1}L(ιA ⊗ α/F′, s) ∈ Z

is defined and satisfies

(−1)^{r_{an}(ιA⊗α/F′)} = ε(ιA ⊗ α/F′)

(and similarly for A′). The statement of Theorem 4.12 can be rewritten, therefore, as follows:

r_{an}(ιA ⊗ α/F′) − h_{f}^{1}(F′, V_{𝔭}(A) ⊗ α) ≡ r_{an}(ιA′ ⊗ α/F′) − h_{f}^{1}(F′, V_{𝔭}(A′) ⊗ α) (mod 2).  (4.13.1)

(4.14) Theorem. If, in the situation of Theorem 4.12, dim(A) = dim(A′) = [L : Q], the field F′ is totally real and A′ does not have potentially good reduction everywhere, then we have, for each ι : L ↪ R,

r_{an}(ιA ⊗ α/F′) ≡ h_{f}^{1}(F′, V_{𝔭}(A) ⊗ α) (mod 2).

Proof. The assumption on A′ implies, by [N5, Thm. 4.3(b)], that the R.H.S. of (4.13.1) is equal to 0 ∈ Z/2Z. (4.15) We now consider a global situation in which Proposition 3.9 applies. Let T be as in 4.3 and let F ⊂ F′ ⊂ F_{1} be Galois extensions with Galois groups Gal(F_{1}/F) = D_{2n} ⊃ Gal(F_{1}/F′) = C_{n} (dihedral and cyclic group, respectively).

Assume that χ, χ′ : Gal(F_{1}/F′) ⟶ O^{×} are characters such that (χ′/χ)^{p^{m}} = 1 for some m ≥ 0; then \overline{χ}′ = \overline{χ}. Consider the induced representations I(χ) = Ind_{C_{n}}^{D_{2n}}(χ) and I(χ′) as representations G_{F,S} ⟶ GL_{2}(O), for suitable S. They are self-dual in the sense that there exist symmetric isomorphisms I(χ) ≃ I(χ)^{∗} and I(χ′) ≃ I(χ′)^{∗}. Again, \overline{I(χ)} = \overline{I(χ′)}.

The following theorem is a generalisation of [N5, Thm. 1.1].

(4.16) Theorem. In the situation of 4.15, assume that for each prime w | p of F′ which is stable by Gal(F′/F), the character χ′/χ is unramified at w. Then we have

(−1)^{h_{f}^{1} (F,V ⊗I(χ))}/ ∏_{v∤∞} ε_{v}(V ⊗ I(χ)) = (−1)^{h_{f}^{1} (F,V ⊗I(χ′))}/ ∏_{v∤∞} ε_{v}(V ⊗ I(χ′)).

Proof. We apply the discussion in 4.7 to the pair of representations T ⊗ I(χ) and T ⊗ I(χ′) of G_{F,S}. As before, we need to analyse the individual terms for v | p in (4.7.2). If there is only one prime w | v in F′, our assumption implies that χ′/χ is trivial on the inertia group of w in F_{1}/F′, hence on the full decomposition group of w in F_{1}/F′, by [MR2, Lemma 6.5(i)]. Thus I(χ)|_{G_{v}} = I(χ′)|_{G_{v}}, which implies that ℱ_{v} = ℱ′_{v} and ε_{v}(V ⊗ I(χ)) = ε_{v}(V ⊗ I(χ′)). If vO_{F′} = ww′, then F′_{w} = F_{v} = F′_{w′} and I(χ)|_{G_{v}} = χ_{w} ⊕ χ_{w}^{−1} (where χ_{w} = χ|_{G_{w}}), I(χ′)|_{G_{v}} = χ′_{w} ⊕ χ_{w}^{′−1}. Proposition 3.9 then yields d(ℱ_{v}, ℱ′_{v}) = 0 and ε_{v}(V ⊗ I(χ)) = ε_{v}(V ⊗ I(χ′)). As a result, equality (4.7.2) holds in our situation, hence so does (4.7.1), as claimed.

5. Elliptic curves with complex multiplication

(5.1) Let \overline{Q} be the algebraic closure of Q in C, let c ∈ G_{Q} = Gal(\overline{Q}/Q) be the complex conjugation. Fix an imaginary quadratic field K ⊂ \overline{Q}. (5.2) Denote by H (resp. by H_{g} = H ∩ Q^{ab}) the Hilbert class field (resp. the genus field) of K. If we write the discriminant of K in the form D_{K} = −2^{a}p_{1} ⋯ p_{r}q_{1} ⋯ q_{s}, where p_{j} ≡ 1 (mod 4) and q_{k} ≡ 3 (mod 4) are distinct primes, then

H_{g} = \{ \begin{matrix}H_{0}, & if a = 0, \\ H_{0}(i), & if a = 2, \\ H_{0}(\sqrt{(−1)^{s−1}2}), & if a = 3, \end{matrix} H_{0} = Q(\sqrt{p_{1}}, …, \sqrt{p_{r}}, i\sqrt{q_{1}}, …, i\sqrt{q_{s}}).

The set of finite primes of the maximal real subfield H_{g}^{+} of H_{g} which are ramified in H_{g}/H_{g}^{+} is equal to

Ram_{f}(H_{g}/H_{g}^{+}) = \{ \begin{matrix}\{v | 2\}, & if s = 0 (⟹ a ≠ 0) \\ \{v | q_{1}\}, & if s = 1 (⟹ a ≠ 2) \\ ∅, & if s ≥ 2. \end{matrix}

(5.3) The Galois group Gal(H/Q) is a semi-direct product Gal(H/K) ⋊ {1, c} ≃ Cl_{K} ⋊ {1, c}, where c^{2} = 1 and cgc^{−1} = g^{−1} for each g ∈ Gal(H/K). This implies that the following properties are equivalent:

H is a CM field ⟺ c ∈ Z(Gal(H/Q)) ⟺ Cl_{K}^{2} = {1} ⟺
⟺ Gal(H/Q) is abelian ⟺ H = H_{g}.

There are 65 known imaginary quadratic fields K for which H = H_{g}, with the largest value of |D_{K}| being equal to 5460 = 4 · 3 · 5 · 7 · 13. There is at most one additional K with H = H_{g}, but its existence would contradict the generalised Riemann hypothesis [We].

(5.4) Proposition. If E_{1} is a elliptic curve defined over a subfield F of \overline{Q} for which O = End_{Q}(E_{1}) is an order in K, then there exists an isogeny E_{1} ⟶ E defined over F such that End_{Q}(E) = O_{K}.

Proof. If O = Z + fO_{K} is the order of conductor f ≥ 1, then E = E_{1}/\{P ∈ E_{1}(\overline{Q}) | (fO_{K}) · P = 0\} has the required property. (5.5) If E is an elliptic curve over F ⊂ \overline{Q} such that End\overline{Q}(E) = O_{K} and j(E) ∈ R, then F K ⊃ K(j(E)) = H, F ⊃ H^{+} := Q(j(E)) and there exists an elliptic curve E_{0} over H^{+} such that E_{0} ⊗_{H^{+}} \overline{Q} ≃ E ⊗_{F} \overline{Q}.

The set of H^{+}-isomorphism classes of such curves E_{0} (modulo quadratic twists with respect to the quadratic extension H/H^{+}) is in a bijection with the set of algebraic Hecke characters

ψ : A_{H}^{×} ↠ A_{H}^{×}/(H ⊗ R)^{×} = Ĥ^{×} ⟶ K^{×},  ψ|_{H^{×}} = N_{H/K}  (5.5.1)

which are c-equivariant (ψ ∘ c = c ∘ ψ). See [S, Cor. Thm. 10] and [G, Thm. 10.1.3, Thm. 10.2.1].

Each of the two curves corresponding to ψ has bad reduction at a prime v of H^{+} iff v is ramified in H/H^{+} or there is w | v in H at which ψ is ramified (i.e. ψ_{w}(O_{H,w}^{×}) ≠ {1}).

Note that a prime v ramified in H/H^{+} always lies above a prime in Ram_{f}(H_{g}/H_{g}^{+}), since H^{+}∩H_{g} = H_{g}^{+} and H/H_{g} is unramified.

(5.6) Proposition. Let E be as in 5.5. If K ≠ Q(i), Q(\sqrt{−3}) and if p is a rational prime not dividing 2D_{K}, then there exists E_{0} as in 5.5 (which is equivalent to E being a quadratic twist of E_{0} ⊗_{H^{+}} F, since Aut\overline{Q}(E) = \{±1\}) with good reduction at all primes above p.

Proof. Let E_{1} be any elliptic curve over H^{+} such that E_{1}⊗_{H^{+}} \overline{Q} ≃ E ⊗_{F} \overline{Q}. Denote by ψ the Hecke character (5.5.1) corresponding to E_{1}. Its p-component ψ_{p} : (H ⊗Q_{p})^{×} ⟶ K^{×} maps (O_{H} ⊗Z_{p})^{×} to (K^{×})_{tors} = {±1}, hence factors through (O_{H} ⊗ F_{p})^{×} ⟶ {±1} (since p ≠ 2). It is also trivial on ^{1−c}(O_{H} ⊗ F_{p})^{×}, which coincides, by Hilbert’s Theorem 90, with the kernel of the surjective norm map N_{H/H^{+}} : (O_{H} ⊗ F_{p})^{×} ↠ (O_{H^{+}} ⊗ F_{p})^{×} (the assumption p ∤ D_{K} implies that p is unramified in H/Q). It follows that the restriction of ψ_{p} to (O_{H} ⊗ Z_{p})^{×} factors as η_{p} ∘ N_{H/H^{+}} for some η_{p} : (O_{H^{+}} ⊗ F_{p})^{×} ⟶ {±1}. There exists η : A_{H^{+}}^{×}/(H^{+})^{×} ⟶ {±1} whose p-component is equal to η_{p}. The character ψ · (η ∘ N_{H/H^{+}}) then corresponds to an elliptic curve E_{0} with the desired properties. (5.7) It may be worthwhile to recall that there are much sharper results than Proposition 5.6 (even though they will not be needed in the proof of Theorem 5.10 below). Let E and K be as in Proposition 5.6. (A) ([S, Example 3], [G, Thm. 11.2.4]). If there is a prime q | D_{K} such that q ≡ 3 (mod 4), then qO_{K} = Q^{2} and the character

φ : (Ô_{K})^{×} ↠ O_{K,Q}^{×} ↠ (O/Q)^{×} = (Z/qZ)^{×} ⟶^{χ_{q}} {±1}

(where χ_{q}(a) = (\frac{a}{q}) is the Legendre symbol) satisfies

φ ∘ c = φ, φ|_{(_{O_{K}})^{×}∩K^{×}} = φ|_{O_{K}^{×}} = φ|_{{±1}} = id.

It extends, therefore (in a unique way) to a morphism φ : K^{×}(Ô_{K})^{×} ⟶ K^{×} such that φ|_{K^{×}} = id (and φ ∘ c = φ, by uniqueness). The Hecke character

ψ = φ ∘ N_{H/K} : A_{H}^{×} ↠ Ĥ^{×} ⟶^{N_{H/K}} K^{×}(Ô_{K})^{×} ⟶^{φ} K^{×}  (5.7.1)

then corresponds to an elliptic curve E_{0} over H^{+} (together with its quadratic twist with respect to H/H^{+}) as in 5.5, which has good reduction at all primes v ∤ q of H^{+}.

A more refined argument [Ro] shows that if D_{K} is divisible by two distinct primes q_{1}, q_{2} ≡ 3 (mod 4), then there exists E_{0} as in 5.5 with good reduction at all primes of H^{+}. (B) If 8 | D_{K}, then 2O_{K} = P^{2} and K_{P} = Q_{2}(x) with x^{2} = −2 or −10. We claim that there exists a character

φ : (Ô_{K})^{×} ↠ O_{K,P}^{×} ↠^{α} O_{K,P}^{×} /^{1−c}(O_{K,P}^{×})(O_{K,P}^{×})^{2} ⟶ {±1}

such that φ(−1) = −1. It is enough to check that α(−1) ≠ 1, which follows from the fact that

1−c(O_{K,P}^{×}) (mod 8) = {1, 1 + 4x, −3 ± 2x (mod 8)}
(O_{K,P}^{×})^{2} (mod 8) = {1, 1 + 4x, −1 ± 2x (mod 8)}.

As in (A), it follows that φ extends in a unique way to φ : K^{×}(Ô_{K})^{×} ⟶ K^{×} such that φ|_{K^{×}} = id. Formula (5.7.1) then defines a Hecke character corresponding to a pair of elliptic curves E_{0} as in 5.5 with good reduction at all primes v ∤ 2 of H^{+}. (C) Both constructions (A) and (B) yield Q-curves, but no such curves exist if D_{K} = −4p_{1} ⋯ p_{r}, where p_{j} ≡ 1 (mod 4) are distinct primes ([G, §11], [Na, Prop. 5]). For reasons explained in 5.8 below we are particularly interested in imaginary quadratic fields K for which H = H_{g}. Among the 65 known fields of this type there are four whose discriminant is of the form D_{K} = −4p_{1} ⋯ p_{r}, namely,

\{ \begin{matrix}D_{K} = −4p,  p ∈ \{5, 13, 37\}; \\ D_{K} = −4 · 5 · 17. \end{matrix}

In each of the first three cases D_{K} = −4p we have H^{+} = Q(\sqrt{p}), H = H^{+}(i), h_{K} = 2 and h_{H^{+}} = 1, which implies that h_{H} = 1 ([FT, Thm. 74, Cor.1]). The group of units is equal to O_{H}^{×} = μ_{4} · ε^{Z}, where ε = (1 + \sqrt{5})/2, (3 + \sqrt{13})/2 and 6 + \sqrt{37}, respectively. The prime 2 remains inert in H^{+}/Q and Ram_{f}(H/H^{+}) = {(2)}: 2O_{H} = P^{2}.

We claim that there is a c-equivariant Hecke character (5.5.1) of the form

ψ : A_{H}^{×} ↠ Ĥ^{×} = H^{×}(Ô_{H})^{×} ⟶^{φ} K^{×},  (5.7.2)

where φ|_{H^{×}} = N_{H/K} and

φ|_{(_{O_{H}})^{×}} : (Ô_{H})^{×} ↠ O_{H,P}^{×} ↠^{α} O_{H,P}^{×} /^{1−c}O_{H,P}^{×} (O_{H,P}^{×})^{2}μ_{4} ⟶^{N_{H/H^{+}}} N_{H/H^{+}}(O_{H,P}^{×})/N_{H/H^{+}}(O_{H,P}^{×})^{2} ⟶^{\overline{φ}} \{±1\}.

It is enough to check that the composite map β = \overline{φ} ∘ N_{H/H^{+}} ∘ α : O_{H,P}^{×} ⟶ \{±1\} satisfies, for a suitable choice of \overline{φ},

β|_{(_{O_{H}})^{×}∩H^{×}} = β|_{O_{H}^{×}} = N_{H/K}|_{O_{H}^{×}}.

This is equivalent to N_{H/H^{+}}(α(ε)) ≠ 1, since N_{H/K}(ε) = −1 and N_{H/K}(μ_{4}) = {1}.

If N_{H/H^{+}}(α(ε)) = 1, then ε^{2} ∈ N_{H/H^{+}}(O_{H,P}^{×})^{2} and there exists u = ±1 such that uε ∈ N_{H/H^{+}}(O_{H,P}^{×}). However, the quadratic Hilbert symbols (uε, −1)_{v} over various completions of H^{+} are given by

(±ε, −1)_{v} = \{ \begin{matrix}1, & if v ∤ 2∞ \\ ±1, & if v = ∞_{1} \\ ∓1, & if v = ∞_{2}. \end{matrix}

The quadratic reciprocity law over H^{+} then implies that (±ε, −1)_{(2)} = −1, hence ±ε ∉ N_{H/H^{+}}(O_{H,P}^{×}). It follows that there exists \overline{φ} such that β(ε) = −1. This yields a Hecke character ψ as in (5.7.2), which corresponds to a pair of elliptic curves E_{0} as in 5.5 with good reduction at all primes v ∤ 2 of H^{+}. (5.8) Assume that E is an elliptic curve defined over a totally real field F ⊂ \overline{Q} and End\overline{Q}(E) = O_{K}. The field H^{+} = Q(j(E)) ⊂ F is then totally real and H = K(j(E)) is a CM field, hence H = H_{g} and H^{+} = H_{g}^{+} (in particular, if 2 ∤ [F : Q], then H^{+} = Q and h_{K} = 1).

The discussion in 5.7 implies that if D_{K} ≠ −3, −4 is not equal to −4 · 5 · 17 (nor to the mythical 66^{th} discriminant) and if there exists a prime q ≡ 3 (mod 4) dividing D_{K} (resp. if no such prime q exists), then there exists a quadratic twist of E of the form E_{0} ⊗_{H_{g}^{+}} F, where E_{0} is an elliptic curve over H_{g}^{+} with good reduction outside primes above q (resp. with good reduction outside primes above 2). However, this statement will not be used in the proof of Theorem 5.10 below (Proposition 5.6 will suffice).

(5.9) If E is an elliptic curve defined over a totally real field F ⊂ \overline{Q}, then E is potentially modular [Wi, Thm. A.1] and the analytic rank

r_{an}(E/F) = ord_{s=1}L(E/F, s) ∈ Z

is well defined (cf. the discussion in [N5, 4.2]). For each rational prime p the rank of the Bloch-Kato Selmer group h_{f}^{1}(F′, V_{p}(E)) over any finite extension F′ of F coincides with

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

The (mod 2)-version of the Bloch-Kato conjecture (0.1) boils down to

r_{an}(E/F) ≡ s_{p}(E/F) (mod 2)  (⋆)

(the “p-parity conjecture” for E over F). As shown in [N5, Thm. A], the congruence (⋆) holds in the following cases. (5.9.1) E does not have complex multiplication. (5.9.2) E has complex multiplication and 2 ∤ [F : Q] (this can happen only if the class number of K = End\overline{Q}(E) ⊗ Q is equal to 1).

(5.9.3) p ≠ 2 and E does not acquire good reduction everywhere over any cyclic extension of F ([N5, Thm. 1.4(c)]). (5.9.4) E has complex multiplication by K = End\overline{Q}(E) ⊗ Q and p splits in K/Q.

(5.9.5) E has complex multiplication by K = End\overline{Q}(E) ⊗ Q and p is ramified in K/Q. As (5.9.5) was not stated explicitly in [N5], here is the argument. The conjecture (⋆) is invariant under isogeny, so we can assume that End\overline{Q}(E) = O_{K}, by Proposition 5.4. The p-torsion E[p] is a reducible F_{p}[G_{F}]-module, since the cyclic subgroup E[P] (where pO_{K} = P^{2}) is stable under G_{F}. However, in the reducible case the p-parity conjecture is known: [DD2, Cor. 5.8] if p = 2, 3; [CFKS, Thm. 2.1] if p > 3 and the reduction type of E at primes above p is not too bad; the remaining cases follow from (5.9.3) (or from [Č]).

The case p = 2 is somewhat special. The argument in [N5] used in a crucial way the fact proved in [DD2, Cor. 4.8] that (⋆) holds for p = 2 over any quadratic extension of F.

We are now going to prove (⋆) in most of the remaining cases (for p ≠ 2).

(5.10) Theorem. Let F be a totally real number field and E an elliptic curve defined over F with complex multiplication by K = End\overline{Q}(E) ⊗ Q ≠ Q(i), Q(\sqrt{−3}). Then the p-parity conjecture

r_{an}(E/F) ≡ s_{p}(E/F) (mod 2)

holds for all primes p ∤ 2D_{K} (note that the case p | D_{K} is covered by (5.9.5)).

Proof. Thanks to Proposition 5.4 we can assume that End\overline{Q}(E) = O_{K}. Recall from 5.8 that H = H_{g}. According to Proposition 5.6 there exists an elliptic curve E_{0} over the real abelian field F_{0} = H^{+} = H_{g}^{+} with good reduction at all primes above p such that E is the twist of E_{0} ⊗_{F_{0}} F by a suitable character α : G_{F} ⟶ {±1}. The p-torsion E_{0}[p] is an absolutely irreducible F_{p}[G_{F_{0}}]-module, since Im(G_{F_{0}} ⟶ Aut(E_{0}[p]) ≃ GL_{2}(F_{p})) is the normaliser of a Cartan subgroup of GL_{2}(F_{p}).

We now apply the level-raising machinery to the cuspidal Hilbert modular eigenform with complex multiplication g_{0} over F_{0} attached to E_{0}. It has parallel weight 2, trivial character and level 𝔫 prime to p. If v is a finite prime of F_{0} not dividing p𝔫, the Hecke operator T (v) acts on g_{0} with an eigenvalue λ_{g_{0}}(v) ∈ Z satisfying

det(1 − X Fr_{geom}(v) | V_{p}(E_{0})(−1)) = 1 − λ_{g_{0}}(v)X + (Nv)X^{2}.  (5.10.1)

According to [Ri, Thm. 1] if F_{0} = Q (resp. [T] combined with [DS, Lemme 6.11] and the Jacquet–Langlands correspondence if 2 | [F_{0} : Q]), there exist a prime v_{0} of F_{0} not dividing p𝔫 and a cuspidal eigenform g of parallel weight 2, trivial character and level dividing v_{0}𝔫, which is new at v_{0} and whose Hecke eigenvalues satisfy

∀v ∤ pv_{0}𝔫  λ_{g}(v) ≡ λ_{g_{0}}(v) (mod 𝔭),  (5.10.2)

for some prime 𝔭 | p in the (totally real) number field L generated by the Hecke eigenvalues of g. Moreover, the 𝔭-adic Galois representation attached to g is of the form V_{𝔭}(A′)(−1), where A′ is an abelian variety over F_{0} with O_{L} ⊂ End_{F_{0}}(A′) and dim(A′) = [L : Q] (A′ arises as a simple quotient of the Jacobian of a suitable modular resp. Shimura curve).

Combining (5.10.2) with (5.10.1) and its analogue for g, we deduce from the Čebotarev density theorem that

∀g ∈ G_{F_{0}}  Tr(g | V_{p}(E_{0})) ≡ Tr(g | V_{𝔭}(A′)) (mod 𝔭).  (5.10.3)

The representations V = V_{p}(E_{0}) ⊗_{Q_{p}} L_{𝔭} and V′ = V_{𝔭}(A′) of G_{F_{0}} are two-dimensional over K = L_{𝔭} and self-dual in the sense of 2.3. After rescaling the corresponding pairings ⟨ , ⟩ : V × V ⟶ K(1) and ⟨ , ⟩′ : V′ × V′ ⟶ K(1) we can assume that there exist G_{F_{0}}-stable self-dual lattices T ⊂ V and T′ ⊂ V′ (cf. 2.5).

The congruence (5.10.3) implies that the k[G_{F_{0}}]-modules (where k = O_{L}/𝔭) \overline{T} = T/𝔭T and \overline{T}′ = T′/𝔭T′ have isomorphic semi-simplifications. However, \overline{T}^{ss} ≃ E_{0}[p]^{ss} ⊗ k = E_{0}[p] ⊗ k is an absolutely irreducible k[G_{F_{0}}]-module. It follows that \overline{T} ≃ \overline{T}′ ≃ E_{0}[p] ⊗ k and that T (resp. T′) is unique up to a scalar multiple, hence is homothetic to T_{p}(E_{0}) ⊗_{Z_{p}} O_{L,𝔭} (resp. to T_{𝔭}(A′)).

Consequently, the abelian varieties A = E_{0} ⊗ O_{L} and A′ defined over the field F_{0} satisfy (4.11.1-2). Moreover, the isomorphism \overline{T} ≃ \overline{T}′ is compatible with the reductions modulo 𝔭 of the pairings ⟨ , ⟩ and ⟨ , ⟩′, possibly after multiplying one of them by a suitable element of O^{×} = O_{L,𝔭}^{×}, since the space of k[G_{F_{0}}]-equivariant skew-symmetric isomorphisms \overline{T} ≃ \overline{T}^{∗}(1) is one-dimensional over k. Thus (4.11.3) is also satisfied. Finally, p ∤ D_{K} is unramified in F_{0}/Q and both A and A′ have good reduction at all primes of F_{0} above p, since the levels of both g_{0} and g are prime to p. As a result, (4.11.4-5) also hold.

The form g is new at v_{0}, which implies that the abelian variety A′ has totally toric reduction at v_{0}. It follows that the assumptions of Theorem 4.14 are satisfied for A and A′ (with F/F_{0} replacing the extension F′/F from 4.13). The statement of Theorem 4.14 then becomes (⋆), since

L(ιA ⊗ α/F, s) = L(E/F, s), h_{f}^{1}(F, V_{𝔭}(A) ⊗ α) = s_{p}(E/F).

(5.11) The argument in the proof of Theorem 5.10 also works for K = Q(i) (resp. K = Q(\sqrt{−3})), provided that p ≠ 2 (resp. p > 3) and that there exists an elliptic curve E_{0} over Q with good reduction at p such that E is a quadratic twist of E_{0} ⊗_{Q} F.

(5.12) Putting together 5.9–5.11, we see that the conjecture (⋆) for totally real number fields F has been established in all non-CM cases and in at least (1 − 2/65 × 1/4) > 99% of CM cases, namely, in all cases except the following two:

(5.12.1) p ≠ 2, E has complex multiplication by K = Q(i) or Q(\sqrt{−3}), p is inert in K/Q, 2 | [F : Q], E acquires good reduction everywhere over a cyclic extension of F, and no quadratic twist of E descends to an elliptic curve defined over Q which has good reduction at p.

(5.12.2) p = 2, E has complex multiplication by K, 2 is inert in K/Q, 2 | [F : Q], and E does not descend to any subfield F_{1} ⊂ F such that [F : F_{1}] = 2.

(5.13) Note that if (⋆) holds for E and all its quadratic twists over F, then

r_{an}(E/F′) ≡ s_{p}(E/F′) (mod 2)

holds for any tower of finite extensions F ⊂ F_{1} ⊂ F′, where F_{1}/F is abelian and F′/F_{1} is a Galois extension of odd degree ([N1, 12.11.7-8]).

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Université Pierre et Marie Curie (Paris 6) Institut de Mathématiques de Jussieu Théorie des Nombres, Case 247 4, place Jussieu F-75252 Paris cedex 05 FRANCE