Growth of Selmer groups of Hilbert modular forms over ring class fields

Jan Nekovář

0. Introduction

0.1. Fix an algebraic closure \overline{Q} of Q, a prime number p and embeddings i_{∞} : \overline{Q} ↪ C and i_{p} : \overline{Q} ↪ \overline{Q}_{p}.

Let F be a totally real number field and g ∈ S_{k}(𝔫, 1) a cuspidal Hilbert modular newform over F of parallel weight k, trivial character (which implies that k is even) and (exact) level 𝔫.

Let K be a totally imaginary quadratic extension of F and χ : A_{K}^{∗}/K^{∗}A_{F}^{∗} ⟶ C^{∗} a (continuous) character of finite order. Fix a number field L ⊂ \overline{Q} such that i_{∞}(L) contains all Hecke eigenvalues λ_{g}(v) of g and all values of χ; denote by 𝔭 the prime of L above p induced by i_{p}.

Let V (g) = V_{𝔭}(g) be the two-dimensional representation of G_{F} = Gal(\overline{Q}/F) with coefficients in L_{𝔭} attached to g: if v ∤ ∞p𝔫 is a prime of F, then V (g) is unramified at v and

det (1 − Fr(v)_{geom} X | V (g)) = 1 − i_{p}(λ_{g}(v))X + (Nv)^{k−1}X^{2}.

The Tate twist V = V (g)(k/2) is self-dual in the sense that there exists a skew-symmetric isomorphism

V ⟶^{∼} V^{∗}(1) = Hom_{L_{𝔭}}(V, L_{𝔭})(1).

Denote by η = η_{K/F} : A_{F}^{∗} /F^{∗}N_{K/F} A_{K}^{∗} ⟶^{∼} {±1} the quadratic character corresponding to the extension K/F. 0.2. Normalising the reciprocity map rec_{K} : A_{K}^{∗}/K^{∗} ⟶ Gal(K^{ab}/K) so that the local uniformisers correspond to geometric Frobenius elements, we identify χ with the coresponding Galois character χ : G_{K} = Gal(\overline{Q}/K) ⟶ O_{L}^{∗} ⊂ L_{𝔭}^{∗}; put K_{χ} = \overline{Q}^{Ker(χ)}. There are canonical isomorphisms

H_{f}^{1}(K, V ⊗ χ^{±1}) ⟶^{∼} (H_{f}^{1}(K_{χ}, V) ⊗ χ^{±1})^{Gal(K_{χ}/K)} = H_{f}^{1}(K_{χ}, V)^{(χ^{∓1})},

where H_{f}^{1}(−) are the Bloch-Kato Selmer groups and

M^{(χ)} = {m ∈ M | ∀σ ∈ Gal(K_{χ}/K) σ(m) = χ(σ)m},

for any O_{L}[Gal(K_{χ}/K)]-module M. The action of the complex conjugation ρ ∈ Gal(K_{χ}/F) on H_{f}^{1}(K_{χ}, V) interchanges the eigenspaces H_{f}^{1}(K_{χ}, V)^{(χ^{±1})}, which implies that their dimensions

h_{f}^{1}(K, V ⊗ χ^{±1}) := dim_{L_{𝔭}} H_{f}^{1}(K, V ⊗ χ^{±1})

are equal to each other: h_{f}^{1}(K, V ⊗ χ) = h_{f}^{1}(K, V ⊗ χ^{−1}). 0.3. Denote by π = π(g) the (irreducible, cuspidal) automorphic representation of GL_{2}(A_{F}) generated by g, and by θ_{χ} the automorphic representation of GL_{2}(A_{F}) generated by the theta series of χ. They are both self-dual, as the central character of π (resp., of θ_{χ}) is trivial (resp., is equal to η = η_{K/F}, and θ_{χ} ⊗ η = θ_{χ}).

The Rankin-Selberg L-function L(π × θ_{χ}, s) has Euler factors

L_{v}(π × θ_{χ}, s) = ∏_{w|v} det(1 − Fr(w)_{geom} (Nw)^{1/2−s} | (V ⊗ χ)^{I_{w}})^{−1},

where v ∤ ∞p is a prime of F and w a prime of K (cf. [N 1, 12.6.2.2]). We shall abuse the notation and write L_{v}(π × χ, s) instead of L_{v}(π × θ_{χ}, s). The complete L-function L(π × χ, s) = ∏_{v} L_{v}(π × χ, s) is equal to L(π × χ^{−1}, s), has holomorphic continuation to C and a functional equation of the form

L(π × χ, s) = ε(π × χ, s) L(π × χ, 1 − s),
ε(π × χ, s) = c(π × χ)^{1/2−s} ε(π × χ, \frac{1}{2}), ε(π × χ, \frac{1}{2}) ∈ \{±1\}.

Put

r_{an}(K, g, χ) := ord_{s=1/2} L(π × χ, s) = ord_{s=1/2} ∏_{v∤∞} L_{v}(π × χ, s)

(the Γ-factors L_{v}(π × χ, s) for v | ∞ take finite non-zero values at s = 1/2).

The conjectures of Bloch and Kato ([B-K], [F-PR]) predict that

(0.3.1)  r_{an}(K, g, χ) =^{?} h_{f}^{1}(K, V ⊗ χ).

The main result of the present article is the following theorem.

0.4. Theorem. Assume that g ∈ S_{k}(𝔫, 1) is potentially p-ordinary, i.e., that there exists a finite solvable extension of totally real number fields F′/F such that the base change BC_{F′/F} (π(g)) is equal to π(g′), where g′ is a p-ordinary (cuspidal) Hilbert eigenform over F′ (equivalently, that there exists a character of finite order φ : A_{F}^{∗} /F^{∗} ⟶ \overline{Q}^{∗} such that the newform associated to g ⊗ φ is p-ordinary; see [N 1, 12.5.10]). If g has complex multiplication by a totally imaginary quadratic extension K′ of F, assume, in addition, that p ≠ 2 and that K′ ⊄ K_{χ}. Then:

(1) If 2 ∤ r_{an}(K, g, χ), then 2 ∤ h_{f}^{1}(K, V ⊗ χ).

(2) If 2 | r_{an}(K, g, χ) and if there exists a prime v | p of F which does not split in K/F and for which π(g)_{v} = St ⊗ μ, μ : F_{v}^{∗} ⟶ {±1}, χ_{w} = μ ∘ N_{K_{w}/F_{v}}, where w is the unique prime of K above v (as g is potentially ordinary, this can occur only if k = 2; see [N 1, 12.5.4]), then 2 | h_{f}^{1}(K, V ⊗ χ). [The hypothesis in (2) can be interpreted as saying that the Euler factor at v of the p-adic counterpart of L(π × χ, s) has a trivial zero of odd order at the central point; cf. [N 1, 12.6.3.10] and [N 1, 12.6.4.3].]

0.5. Corollary. Let K[∞] ⊂ K^{ab} be the union of all ring class fields of K in the sense of [A-N, 1.1] (the Galois group Gal(K[∞]/K) is the quotient of Gal(K^{ab}/K) by rec_{K}(A_{F}^{∗})). Let K_{0}/K be a finite subextension of K[∞]/K. Assume that g ∈ S_{k}(𝔫, 1) is potentially p-ordinary; if g has complex multiplication by a totally imaginary quadratic extension K′ of F, assume, in addition, that p ≠ 2 and that K′ ⊄ K_{0}. Then

h_{f}^{1}(K_{0}, V) := dim_{L_{𝔭}} H_{f}^{1}(K_{0}, V) ≥ |X^{−}(g, K_{0})|,

where

X^{±}(g, K_{0}) = \{χ : Gal(K_{0}/K) ⟶ C^{∗} | ε(π(g) × χ, \frac{1}{2}) = ±1\}.

0.6. Example. Let E be an elliptic curve over F. It is expected that E is modular in the sense that there exists g ∈ S_{2}(𝔫, 1) such that L_{v}(E/F, s) = L_{v}(π(g), s + 1/2), for all primes v of F. If this is the case, assume that E has potentially ordinary reduction (= potentially good ordinary or potentially multiplicative reduction) at all primes of F above p. If E has complex multiplication by Q(\sqrt{−D}), assume, in addition, that p ≠ 2 and F (\sqrt{−D}) ⊄ K_{χ}. Theorem 0.4 then implies the following:

(1) If 2 ∤ ord_{s=1} L(E/K, χ, s), then 2 ∤ (dim_{L}(E(K_{χ}) ⊗ L)^{(χ^{−1})} + cork_{O_{L,𝔭}}(� (E/K_{χ}) ⊗ O_{L,𝔭})^{(χ^{−1})}).

(2) If 2 | ord_{s=1} L(E/K, χ, s) and if there exists a prime v | p of F which does not split in K/F and a character μ : F_{v}^{∗} ⟶ {±1} such that χ_{w} = μ ∘ N_{K_{w}/F_{v}} and the quadratic twist E ⊗ μ has split

multiplicative reduction at v, then 2 | (dim_{L}(E(K_{χ}) ⊗ L)^{(χ^{−1})} + cork_{O_{L,𝔭}}(� (E/K_{χ}) ⊗ O_{L,𝔭})^{(χ^{−1})}).

There is an obvious variant of this statement when E is replaced by an abelian variety A_{0} with O_{L_{0}} ↪ End_{F} (A_{0}), where L_{0} is a totally real number field of degree [L_{0} : Q] = dim(A_{0}). 0.7. Example [D]. Let q ≠ 2 be a prime number and F_{0} a totally real number field such that F_{0}∩Q(μ_{q^{∞}}) = Q. Let a ∈ O_{F_{0}} − {0} be an element satisfying a ∉ F_{0}^{∗q} and, for each finite prime v_{0} of F_{0} not dividing q, ord_{v_{0}}(a) < q. For each r ≥ 1, put F_{r} = F_{0}Q(μ_{q^{r}})^{+}, K_{r} = F_{0}(μ_{q^{r}}); then Gal(K_{r}/F_{0}) ⟶^{∼} (Z/q^{r}Z)^{∗} and Gal(K_{r}( ^{q^{r}} \sqrt{a})/K_{r}) ⟶^{∼} Z/q^{r}Z. Fix an injective character χ_{r} : Gal(K_{r}( ^{q^{r}} \sqrt{a})/K_{r}) ⟶ \overline{Q}^{∗}; then χ_{r} (more precisely, χ_{r} ∘ rec_{K_{r}}) factors through A_{K_{r}}^{∗}/K_{r}^{∗}A_{F_{r}}^{∗}.

Let g_{0} ∈ S_{k}(𝔫_{0}, 1) be a cuspidal Hilbert newform over F_{0} of parallel weight k, trivial character and level 𝔫_{0}; put V = V_{𝔭}(g_{0})(k/2). For r ≥ 1, denote by g_{r} the corresponding base change newform over F_{r} (i.e., BC_{F_{r}/F} (π(g_{0})) = π(g_{r})). An easy exercise in group theory (see §3.1 below) shows that, for each r ≥ 1,

H_{f}^{1}(F_{0}( ^{q^{r}} \sqrt{a}), V) = H_{f}^{1}(F_{0}, V) ⊕ ⨁_{s=1}^{r} H_{f}^{1}(K_{s}, V ⊗ χ_{s}).

0.8. Theorem. In the notation of 0.7, assume that (∀v_{0}|q) π(g_{0})_{v_{0}} is a principal series representation and that (∀v_{0}|a) π(g_{0})_{v_{0}} is not supercuspidal. Put 𝔫_{0}^{(aq)} = 𝔫_{0}/(𝔫_{0}, (aq)^{∞}) and d = (−1)^{[F_{0}:Q]}N(𝔫_{0}^{(aq)}) ∈ Z − {0}. Then:

(1) For each r ≥ 1, ε(π(g_{r}) × χ_{r}, \frac{1}{2}) = (\frac{d}{q}). [See [D, Thm. 6] for a special case.]

(2) Assume that g_{0} is potentially p-ordinary. If g_{0} has complex multiplication by a totally imaginary quadratic extension K′_{0} of F_{0}, assume that p ≠ 2 and that (q ≡ 1 (mod 4) or K′_{0} ≠ F_{0}(\sqrt{−q})) [the latter condition is automatically satisfied if 2 ∤ [F_{0} : Q]]. If (\frac{d}{q}) = −1, then

∀r ≥ 0  h_{f}^{1}(F_{0}( ^{q^{r}} \sqrt{a}), V) − h_{f}^{1}(F_{0}, V) ≥ r,  h_{f}^{1}(F_{0}( ^{q^{r}} \sqrt{a}), V) − h_{f}^{1}(F_{0}, V) ≡ r (mod 2),
h_{f}^{1}(F_{0}(μ_{q^{r}}, ^{q^{r}} \sqrt{a}), V) − h_{f}^{1}(F_{0}, V) ≥ q^{r} − 1.

0.9. Corollary. In the notation of 0.7, assume that E is a modular elliptic curve over F_{0} such that for each prime v_{0} of F_{0} above q (resp., dividing a) there exists a quadratic twist of E with good reduction (resp., semistable reduction) at v_{0}. If E has complex multiplication, assume that p ≠ 2 and that (q ≡ 1 (mod 4) or F_{0} ⊗ End\overline{Q}(E) ≠ F_{0}(\sqrt{−q})) [the latter condition is automatically satisfied if 2 ∤ [F_{0} : Q]]. Assume, finally,

that (\frac{d}{q}) = −1, where d = (−1)^{[F_{0}:Q]}N(cond(E)^{(aq)}). Then, for each prime number p such that E has potentially ordinary reduction at all primes of F_{0} above p, the ranks

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

satisfy

∀r ≥ 0  s_{p}(E/F_{0}( ^{q^{r}} \sqrt{a})) − s_{p}(E/F_{0}) ≥ r,  s_{p}(E/F_{0}( ^{q^{r}} \sqrt{a})) − s_{p}(E/F_{0}) ≡ r (mod 2),
s_{p}(E/F_{0}(μ_{q^{r}}, ^{q^{r}} \sqrt{a})) − s_{p}(E/F_{0}) ≥ q^{r} − 1.

0.10. Remarks. (1) The proof of Theorem 0.4 does not use any explicit formula for the global root number ε(π(g) × χ, \frac{1}{2}) (in fact, such a formula is not available in general). Instead, the local root numbers enter the picture indirectly, through their representation-theoretical characterisation ([T], [W, Thm. 2], [S]) that appears in a refinement of the method of Cornut and Vatsal ([A-N]).

(2) A suitable strenghtening of the methods of [M-R] or [D-D2] would allow us to drop the assumption of potentially ordinary reduction at p in Example 0.6 (if the order of χ is a power of p, this can be done, in many cases, by combining [M-R, Thm. 6.4], [D-D2, Thm. 4.3] and [N 4, Thm. 1]).

(3) In the special case p = q there are more elementary approaches to Corollary 0.9 ([M-R, Thm. 7.1], [D-D1, Prop. 4.13], [C-F-K-S, Thm. 4.2, 4.6]), which do not require E to be modular but instead assume that s_{p}(E/F_{0}(μ_{p})) ≡ ord_{s=1}L(E/F_{0}(μ_{p}), s) (mod 2), which under fairly general circumstances follows from [N 4, Thm. 1].

(4) If χ^{2} = 1, then χ = φ ∘ N_{K/F}, where φ : A_{F}^{∗} /F^{∗} ⟶ {±1}, and

H_{f}^{1}(K, V ⊗ χ) = H_{f}^{1}(F, V ⊗ φ) ⊕ H_{f}^{1}(F, V ⊗ φη_{K/F}).

In many cases [N 1, Thm. 12.2.3] applies to g ⊗ φ and g ⊗ φη_{K/F}, yielding

r_{an}(K, g, χ) ≡ h_{f}^{1}(K, V ⊗ χ) (mod 2).

1. Proof of Theorem 0.4

In this section we prove Theorem 0.4., following closely the arguments in [N 1, §12.9]. The main difference is that the non-vanishing results of [C-V, §4] are replaced by their strenghtening proved in [A-N]. We are going to use the notation from [A-N] and [N 1, ch. 12].

Assume that g ∈ S_{k}(𝔫, 1) is potentially p-ordinary and, if g has complex multiplication by a totally imaginary quadratic extension K′ of F, that p ≠ 2 and K′ ⊄ K_{χ}. Put π = π(g), η = η_{K/F}, ε_{v} = ε(π_{v} ×χ_{v}, \frac{1}{2}) ∈ \{±1\} and ε = ε(π ×χ, \frac{1}{2}) = ∏_{v} ε_{v}. Note that ε_{v} does not depend on the choice of an additive character, by the self-duality of V ⊗ Ind_{K/F} (χ) ([N 3, 2.2.1(1)]).

1.1. Reduction steps

1.1.1. Let Σ_{p} (resp., Σ_{∞} = {τ_{1}, …, τ_{d}}, d = [F : Q]) be the set of all primes of F above p (resp., above ∞). Put

Σ_{χ} = Σ_{χ}(g) = {v ∈ Σ_{p} | v does not split in K/F, π_{v} = St ⊗ μ, μ : F_{v}^{∗} ⟶ {±1}, χ_{w} = μ ∘ N_{K_{w}/F_{v}}}

(where w is the unique prime of K above v). As g is potentially p-ordinary, π_{v} is not supercuspidal for any v ∈ Σ_{p}, which implies that

(1.1.1.1)  ∀v ∈ Σ_{p}  η_{v}(−1) ε_{v} = \{ \begin{matrix}−1, & v ∈ Σ_{χ} \\ 1, & v ∉ Σ_{χ} \end{matrix}

(see [N 1, 12.6.2.4]). At the archimedean primes,

(1.1.1.2)  ∀v ∈ Σ_{∞}  η_{v}(−1) = −1,  ε_{v} = 1.

1.1.2. Reduction to the case k = 2. Assume that k > 2 and that Theorem 0.4 has been proved for k = 2. As in [N 1, 12.9.5.1], let g′ ∈ S_{2}(𝔫′, 1) be a weight 2 specialisation of the twisted Hida family containing a twist of the p-stabilisation of a p-ordinary twist of g; put π′ = π(g′), V′ = V_{𝔭}(g′)(1). As k > 2, we have Σ_{χ}(g) = ∅ ([N 1, Lemma 12.5.4]). According to [N 1, 12.7.14.5(v),(ii),(iii)] (combined with (2.2.3.1) below),

1.1.2.1. g has complex multiplication by K′ ⟺ g′ has complex multiplication by K′; 1.1.2.2. r_{an}(K, g, χ) = r_{an}(K, g, χ) + |Σ_{χ}(g)| ≡ r_{an}(K, g′, χ) + |Σ_{χ}(g′)| (mod 2). 1.1.2.3. h_{f}^{1}(K, V ⊗ χ) = h_{f}^{1}(K, V ⊗ χ) + |Σ_{χ}(g)| ≡ h_{f}^{1}(K, V′ ⊗ χ) + |Σ_{χ}(g′)| (mod 2).

As Theorem 0.4 holds for g′ by assumption, the statements 1.1.2.1-3 imply that it also holds for g (in fact, one can always choose g′ satisfying Σ_{χ}(g′) = ∅, by [N 1, 12.7.10]). 1.1.3. Further reduction in the case k = 2 (cf. [N 1, 12.10.1]). From now on, until the end of §1, k = 2. Assume that Theorem 0.4 has been proved for k = 2 under the following additional assumption:

(1.1.3.1)  ∃ Σ ⊆ Σ_{p}, Σ ≠ ∅  (−1)^{|Σ∩Σ_{χ}|−1} = ε(π × χ, \frac{1}{2}) = ε.

There is precisely one case in which the assumptions of Theorem 0.4 (ε = −1 or |Σ_{χ}| ≠ 0) are satisfied, but (1.1.3.1) is not:

|Σ_{χ}| = |Σ_{p}| = 1, ε = −1.

In this case fix a totally real cyclic extension F_{1}/F of odd order n > 1 in which the unique prime v ∈ Σ_{p} splits completely and put K_{1} = F_{1}K, χ_{1} = χ ∘ N_{K_{1}/K} : A_{K_{1}}^{∗}/K_{1}^{∗}A_{F_{1}}^{∗} ⟶ C^{∗} (thus (K_{1})_{χ_{1}} = F_{1}K_{χ}), π_{1} = BC_{F_{1}/F} (π) = π(g_{1}), Δ = Gal(F_{1}/F) = Gal(K_{1}/K), Δ̂ = Hom(Δ, C^{∗}). As g is potentially p-ordinary, so is the base change form g_{1} over F_{1}. Furthermore, if g has complex multiplication by K′ ([K′ : F] = 2), then g_{1} has complex multiplication by F_{1}K′, and

(1.1.3.2)  K′ ⊄ K_{χ} ⟺ F_{1}K′ ⊄ F_{1}K_{χ} = (K_{1})_{χ_{1}}.

We have

ε(π_{1} × χ_{1}, \frac{1}{2}) = ∏_{α∈_{Δ}} ε(π × (θ_{χ} ⊗ α), \frac{1}{2}), h_{f}^{1}(K_{1}, V ⊗ χ_{1}) = ∑_{α∈_{Δ}} h_{f}^{1}(K, V ⊗ χα).

As

ε(π × (θ_{χ} ⊗ α^{−1}), \frac{1}{2}) = ε(π × (θ_{χ} ⊗ α), \frac{1}{2})^{−1}

by the functional equation, it follows that

(1.1.3.3)  r_{an}(K, g_{1}, χ_{1}) ≡ r_{an}(K, g, χ) (mod 2).

Similarly,

\begin{aligned}h_{f}^{1}(K, V ⊗ χα) & = h_{f}^{1}(K, V^{∗}(1) ⊗ χ^{−1}α^{−1}) = h_{f}^{1}(K, V ⊗ χ^{−1}α^{−1}) ([N1, 12.5.9.5(iv)]) \\ & = h_{f}^{1}(K, V ⊗ χα^{−1}) \end{aligned}

(the last equality follows from the fact that the action of the complex conjugation ρ ∈ Gal(F_{1}K_{χ}/F) interchanges the χα- and χ^{−1}α-eigenspaces for the action of Gal(F_{1}K_{χ}/K) on H_{f}^{1}(F_{1}K_{χ}, V)), which implies that

(1.1.3.4)  h_{f}^{1}(K_{1}, V ⊗ χ_{1}) ≡ h_{f}^{1}(K, V ⊗ χ) (mod 2).

Finally,

(1.1.3.5)  Σ_{χ_{1}}(g_{1}) = {v_{1} | v, v ∈ Σ_{χ}(g)}.

Putting (1.1.3.2-5) together, we obtain that Theorem 0.4 holds for g iff it holds for g_{1}. However, g_{1} satisfies (1.1.3.1), as there are n > 1 primes above p in F_{1}. This concludes the proof of the reduction step to the case when (1.1.3.1) holds.

1.2. Passage to a Shimura curve

1.2.1. Thanks to 1.1, we can (and will) assume that k = 2 and that there exists a non-empty subset Σ = {P_{1}, …, P_{s}} ⊆ Σ_{p} for which (1.1.3.1) holds; fix such Σ (in fact, we can always take s = 1 or s = 2). Our goal is to show that r_{an}(K, g, χ) ≡ h_{f}^{1}(K, V ⊗ χ) (mod 2). 1.2.2. Choice of a quaternion algebra. As

ε = ∏_{v} ε_{v} = ∏_{v} η_{v}(−1) ε_{v} = ∏_{v∈Σ_{∞}} ∏_{v∈Σ} ∏_{v∉Σ∪Σ_{∞}} η_{v}(−1) ε_{v} = (−1)^{[F:Q]}(−1)^{|Σ∩Σ_{χ}|} ∏_{v∉Σ∪Σ_{∞}} η_{v}(−1) ε_{v}

(by (1.1.1.1-2)), the condition (1.1.3.1) implies that

∏_{v∉Σ∪Σ_{∞}} η_{v}(−1) ε_{v} = (−1)^{[F:Q]−1}.

This means that there exists a (unique) quaternion algebra B over F with local invariants

inv_{v}(B) = \{ \begin{matrix}1, & v ∈ Σ ∪ \{τ_{1}\} \\ −1, & v ∈ Σ_{∞} − \{τ_{1}\} \\ η_{v}(−1) ε_{v}, & v ∉ Σ ∪ Σ_{∞}. \end{matrix}

1.2.3. Proposition. (1) There exists a (unique) irreducible (cuspidal) automorphic representation π′ of B_{A}^{∗} such that π = JL(π′) is associated to π′ by the Jacquet-Langlands correspondence. The representation π′ has trivial central character, π′_{v} = 1 if v ∈ Σ_{∞} − {τ_{1}}, and π′_{τ_{1}} is the weight 2 holomorphic discrete series.

(2) There exists an F -embedding t : K ↪ B; fix such t.

(3) ∀v ∉ Σ ∪ Σ_{∞} there exists a non-zero t_{v}(K_{v}^{∗})-invariant linear form ℓ_{v} : π′_{v} ⟶ C(χ_{v}^{−1}).

Proof. (1) At each finite prime v of F at which π_{v} is a principal series representation we have η_{v}(−1) ε_{v} = 1 ([N 1, 12.6.2.4(i)]), hence B is unramified at v, by construction. This implies that π lies in the image of the Jacquet-Langlands correspondence. The central characters of π and π′ coincide and the archimedean behaviour of π′ follows from the coresponding local correspondence.

(2) At each finite prime v of F which splits in K/F we have η_{v}(−1) ε_{v} = 1 ([N 1, 12.6.2.4]); thus B is ramified only at primes of F that do not split in K/F, which implies the existence of t.

(3) By construction, inv_{v}(B) = η_{v}(−1) ε_{v} for each v ∉ Σ ∪ Σ_{∞}; we apply the fundamental results of [T] and [W, Thm. 2] (see also [S]) (note that, if inv_{v}(B) = 1, then dim(π′_{v}) = dim(π_{v}) = ∞).

1.2.4. Proposition (choice of an order). There exists an O_{F} -order R ⊂ B such that ∀v ∉ Σ∪Σ_{∞} ℓ_{v}((π′_{v})^{R_{v}^{∗}}) ≠ 0; ∀v ∈ Σ R_{v} is an Eichler order of level o(π′_{v}) = o(π_{v}) (hence dim(π′_{v})^{R_{v}^{∗}} = 1).

Proof. Fix any O_{F} -order R_{0} ⊂ B. There exists a finite set S_{0} ⊃ Σ_{∞} of primes of F such that, for each v ∉ S_{0}, the following conditions hold: R_{0,v} is a maximal O_{F,v}-order in B_{v} ⟶^{∼} M_{2}(F_{v}), t_{v}^{−1}(R_{0,v}) = O_{K,v}, χ_{v} is unramified and o(π′_{v}) = o(π_{v}) = 0 ( ⟺ π_{v} is an unramified principal series representation). This implies, by [G-P, 2.3], that ℓ_{v}((π′_{v})^{R_{v}^{∗}}) ≠ 0 for all v ∉ S_{0}. On the other hand, for each v ∈ S_{0} − (Σ ∪ Σ_{∞}), there exists a vector x_{v} ∈ π′_{v} such that ℓ_{v}(x_{v}) ≠ 0. As the central character of π′_{v} is trivial, there exists an O_{F,v}-order R(v) ⊂ B_{v} such that x_{v} ∈ (π′_{v})^{R(v)^{∗}}. For v ∈ Σ, let R(v) ⊂ B_{v} be any Eichler order of level o(π_{v}). The O_{F} -order R ⊂ B given by its localisations

R_{v} = \{ \begin{matrix}R_{0,v}, & v ∉ S_{0} \\ R(v), & v ∈ S_{0} − Σ_{∞} \end{matrix}

has the required properties. 1.2.5. Shimura curve. Fix an O_{F} -order R ⊂ B satisfying Proposition 1.2.4 and put H = R̂^{∗}. From now on, we shall work with the Shimura curve N_{H}^{∗} (in the notation of [N 2, 1.4] and [A-N, 2.1.2]). By construction,

π^{′H} = π′_{∞} ⊗ ⨂′_{v∤∞}(π′_{v})^{R_{v}^{∗}} ≠ 0.

1.3. CM points

1.3.1. As in [A-N, 2.2], fix an isomorphism B_{τ_{1}} ⟶^{∼} M_{2}(R) and denote by z ∈ C, Im(z) > 0 the unique point fixed by t(K^{∗}) ⊂ B^{∗} ⊂ B_{τ_{1}}^{∗} ⟶^{∼} GL_{2}(R). Let

C = {[z, b_{1} ⋯ b_{s}]_{H} | b_{i} ∈ B_{P_{i}}^{∗}} ⊂ CM(N_{H}, K)

be the (P_{1} ⋯ P_{s})-isogeny class of the CM point [z, 1]_{H} on the curve N_{H}^{∗}.

Recall that the conductor of a CM point x = [z, b]_{H} ∈ CM(N_{H}, K) is the non-zero ideal c(x) ⊂ O_{F} satisfying

∀v ∉ Σ_{∞} t_{v}^{−1}(b_{v}R_{v}b_{v}^{−1}) = O_{F,v} + c(x)O_{K,v}.

Similarly, the conductor of χ is the smallest non-zero ideal c(χ) ⊂ O_{F} (w.r.t. divisibility) such that

∀v ∉ Σ_{∞} χ_{v}((O_{F,v} + c(χ)O_{K,v})^{∗}) = 1.

1.3.2. Lemma. ∀v ∉ Σ ∪ Σ_{∞} ∀x ∈ C ord_{v}(c(χ)) ≤ ord_{v}(c(x)) (the R.H.S. depends only on C, by [A-N, (2.2.4.1)]).

Proof. There exists x_{v} ∈ (π′_{v})^{R_{v}^{∗}} such that ℓ_{v}(x_{v}) ≠ 0. As

∀a ∈ K_{v}^{∗} ℓ_{v}(π′_{v}(t_{v}(a)) x_{v}) = χ_{v}^{−1}(a) ℓ_{v}(x_{v}),

it follows that χ_{v}(t_{v}^{−1}(R_{v}^{∗})) = 1; but t_{v}^{−1}(R_{v}) = O_{F,v} + c(x)O_{K,v}, by definition.

1.3.3. Corollary-Definition. Put c_{0} = c(χ)/(c(χ), (P_{1} ⋯ P_{s})^{∞}) and c = c(x)/(c(x), (P_{1} ⋯ P_{s})^{∞}) (which is independent of the choice of x ∈ C). Then c_{0} | c and c(χ) = c_{0}P_{1}^{n_{1}} ⋯ P_{s}^{n_{s}} (n_{i} ∈ N). As c_{0} | c, χ is a character of G^{(c)} = Gal(K[cP^{∞}]/K), in the notation of [A-N, 1.2.2]; let χ_{0} : G_{0}^{(c)} ⟶ C^{∗} be the restriction of χ to the torsion subgroup G_{0}^{(c)} = (G^{(c)})_{tors} of G^{(c)}.

1.3.4. Lemma. For each m ∈ N there exists a CM point x_{m} ∈ C which is good in the sense of [A-N, 4.2.3] and satisfies (P_{1} ⋯ P_{s})^{m} | c(x_{m}).

Proof. Let

X : ∏_{i=1}^{s} P_{a_{i}}(T_{i}) ↠ C

be the ∏_{i=1}^{s} K_{P_{i}}^{∗}-equivariant surjection from [A-N, (4.2.2.3)]. For each i = 1, …, s choose a path γ_{i} = (y_{0} → ⋯ → y_{a_{i}}) ∈ P_{a_{i}}(T_{i}) such that ℒ_{i}(y_{0}) = m and ℒ_{i}(y_{a_{i}}) = m + a_{i}; the CM point x_{m} = X(γ_{1}, …, γ_{s}) has the required properties.

1.4. Quotient of the Jacobian of the Shimura curve

1.4.1. As in [N 2, 1.16], denote by J(N_{H}^{∗}) the Jacobian of the curve N_{H}^{∗}. As recalled in [A-N, 2.1.7] and [N 2, 1.18], there exists an isogeny J(N_{H}^{∗}) ⟶ A_{0}^{a} × A_{1} (defined over F), where: A_{0} is an F -simple abelian variety with End_{F} (A_{0}) = O_{L_{0}}, L_{0} is a totally real number field of degree [L_{0} : Q] = dim(A_{0}), there is a unique embedding σ_{0} : L_{0} ↪ R such that the L-function L(σ_{0} A_{0}/F, s) = L(σ_{0} h^{1}(A_{0}), s) (in the notation of [N 3, 0.0]) is equal (Euler factor by Euler factor) to L(π′, s − 1/2) = L(π, s − 1/2), a = dim ⨂′_{v∤∞}(π′_{v})^{R_{v}^{∗}} ≥ 1 and Hom_{F} (A_{0}, A_{1}) = 0 (in the notation of [A-N, 2.1.7], π′ = π(σ_{0})). As σ_{0}(L_{0}) is the field generated over Q by the Hecke eigenvalues of g, there is a unique embedding i : L_{0} ↪ L satisfying i_{∞}|_{L} ∘ i = σ_{0}.

For j = 1, …, a, denote by pr_{j} : A_{0}^{a} ⟶ A_{0} the projection on the j-th factor. 1.4.2. Put S = {v | cd_{K/F} , v ≠ P_{1}, …, P_{s}}. According to Proposition 1.2.4,

(1.4.2.1)  ∀v ∈ S  ℓ_{v}((π′_{v})^{R_{v}^{∗}}) ≠ 0.

Denote by

ℓ_{S} = ⨂_{v∈S} ℓ_{v} : ⨂_{v∈S} π′_{v} ⟶ C(χ_{S}^{−1})

the tensor product of the linear forms ℓ_{v} (v ∈ S); above, χ_{S} = ∏_{v∈S} χ_{v} : ∏_{v∈S} K_{v}^{∗} ⟶ C^{∗}. 1.4.3. As in [A-N, 2.4.3], put A = O_{L} ⊗_{O_{L_{0}}} A_{0}. As explained in [A-N, 2.1.7], Γ(A, Ω_{A/F}^{1} ) is a free L ⊗_{Q} F -module of rank one and the projection J(N_{H}^{∗}) ⟶ A_{0}^{a} × A_{1} ⟶ A_{0}^{a} induces an isomorphism

Γ(A, Ω_{A/F}^{1} )^{a} ⊗_{F,τ_{1}} C ⟶^{∼} ⨁_{σ:L↪C} ⨂′_{v∤∞}(^{σ}π′_{v})^{R_{v}^{∗}}.

In particular, there is a natural B̂^{∗}-equivariant projection on the term corresponding to σ = i_{∞}|_{L}:

r : Γ(A, Ω_{A/F}^{1} )^{a} ⊗_{F,τ_{1}} C ↠ ⨂′_{v∤∞}(π′_{v})^{R_{v}^{∗}}.

The linear form ℓ_{S} induces a linear map

ℓ_{S} ⊗ id : ⨂′_{v∤∞} π′_{v} ⟶ (⨂′_{v∉Σ_{∞}∪S} π′_{v}) ⊗ χ_{S}^{−1}

and it follows from (1.4.2.1) that there exists j ∈ {1, …, a} such that

(1.4.3.1)  (ℓ_{S} ⊗ id) ∘ r ∘ pr_{j}^{∗} (Γ(A, Ω_{A/F}^{1} ) ⊗_{F,τ_{1}} C) ≠ 0.

In the notation of [A-N, 2.1.3], let α : N_{H}^{∗} ⟶ A_{0} be the morphism (defined over F) given by

α : N_{H}^{∗} ⟶^{ι_{H}} J(N_{H}^{∗}) ⟶ A_{0}^{a} × A_{1} ⟶ A_{0}^{a} ⟶^{pr_{j}} A_{0}.

1.5. End of the proof

1.5.1. Using Lemma 1.3.4, choose a sequence of good CM points {x_{m}} ⊆ C satisfying (P_{1} ⋯ P_{s})^{m} | c(x_{m}) for all m ∈ N. By (1.4.3.1), the main non-vanishing result from [A-N] (Theorem 4.3.1) applies to the sequence of points {α(x_{m})} ⊂ A_{0}(K[cP^{∞}]) and the character χ_{0}: if m ∈ N is large enough, then there exists a character χ_{m} : G^{(c)} ⟶ \overline{Q}^{∗} of finite order with the following properties:

1.5.1.1. the restriction of χ_{m} to G_{0}^{(c)} is equal to χ_{0}; 1.5.1.2. c(χ_{m})c_{0}^{−1} = c(x_{m})c^{−1} (hence (P_{1} ⋯ P_{s})^{m} | c(χ_{m})); 1.5.1.3. e_{\overline{χ}_{m}}(1 ⊗ α(x_{m})) ≠ 0 ∈ \overline{Q} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}]).

The assumptions in Theorem 0.4 imply that g does not have complex multiplication by any totally imaginary quadratic extension of F contained in K_{χ_{m}}, which means that the main result of [N 2] (Thm 3.2) applies to the non-torsion point e_{\overline{χ}_{m}}(1 ⊗ α(x_{m})), yielding

(1.5.1.4)  h_{f}^{1}(K, V ⊗ χ_{m}^{±1}) = 1.

As m is large enough, we also have

(1.5.1.5)  ε(π × χ_{m}, \frac{1}{2}) = −1,

by [A-N, 2.4.12(3)], hence

r_{an}(K, g, χ_{m}) ≡ 1 ≡ h_{f}^{1}(K, V ⊗ χ_{m}) (mod 2).

Finally, as the restrictions of χ and χ_{m} to the torsion subgroup of G^{(c)} coincide, we have

r_{an}(K, g, χ) − h_{f}^{1}(K, V ⊗ χ) ≡ r_{an}(K, g, χ_{m}) − h_{f}^{1}(K, V ⊗ χ_{m}) ≡ 0 (mod 2),

by Proposition 2.1.2(5) below (which was earlier proved in [N 1, 12.6.4.7(v)] in the special case when (c, (p)) = (1)). This finishes the proof of Theorem 0.4.

1.5.2. In fact, it is enough to appeal to a weaker non-vanishing result [A-N, Thm. 2.5.1], which does not require the CM points x_{m} to be good and which yields, for m large enough, a character χ_{m} : G^{(c)} ⟶ \overline{Q}^{∗} satisfying 1.5.1.1 and 1.5.1.3, but not necessarily 1.5.1.2. The equality (1.5.1.5) in this case follows from [A-N, Prop. 2.6.2(2)].

2. Iwasawa theory

Throughout this section we assume that g ∈ S_{k}(𝔫(g), 1) is a potentially p-ordinary Hilbert modular newform of level 𝔫(g); let π = π(g). We also fix K as in 0.1 and put η = η_{K/F}. Fix a non-empty subset Σ = {P_{1}, …, P_{s}} ⊆ Σ_{p} and an ideal c ⊂ O_{F} relatively prime to P_{1} ⋯ P_{s}. As in [A-N, 1.2.2], set K[cP^{∞}] = ⋃_{m≥1} K[cP_{1}^{m} ⋯ P_{s}^{m}], G^{(c)} = Gal(K[cP^{∞}]/K) and G_{0}^{(c)} = (G^{(c)})_{tors}.

2.1. ε-factors

2.1.1. For a character χ : G^{(c)} ⟶ C^{∗} of finite order, let

ε_{v} = ε(π_{v} × χ_{v}, \frac{1}{2}) ∈ \{±1\},  ε̃_{v} = ε̃(π_{v} × χ_{v}, \frac{1}{2}) ∈ \{±1\},

where

ε̃_{v} = ε_{v} × \{ \begin{matrix}−1, & v ∈ Σ_{χ}(g) \\ 1, & v ∉ Σ_{χ}(g) \end{matrix}

is the modified local ε-factor defined in [N 1, 12.6.3.10] (if k > 2, then Σ_{χ}(g) = ∅ and ε̃_{v} = ε_{v} for all v). If χ′ : G^{(c)} ⟶ C^{∗} is another character of finite order, denote the corresponding local ε-factors by ε′_{v} and ε′_{v}.

2.1.2. Proposition. Let χ, χ′ : G^{(c)} ⟶ C^{∗} be characters of finite order such that χ|_{G_{0}^{(c)}} = χ′|_{G_{0}^{(c)}}. If v is a prime of F, then:

(1) If v | ∞ or if v ∤ ∞ splits in K/F, then ε_{v} = ε′_{v} = ε̃_{v} = ε′_{v} = 1.

(2) If v = P_{i} (i = 1, …, s), then ε̃_{v} = ε′_{v} = η_{v}(−1).

(3) If v | cd_{K/F}, v ≠ P_{1}, …, P_{s}, then ε_{v} = ε′_{v} and ε̃_{v} = ε′_{v}.

(4) If v ∤ ∞cd_{K/F}, then ε_{v} = ε′_{v} (more precisely, χ_{v} = χ′_{v} = 1 if v does not split in K/F) and

ε̃_{v} = ε′_{v} = \{ \begin{matrix}1, & if π_{v} is not supercuspidal \\ η_{v}(v)^{o(π_{v})}, & if π_{v} is supercuspidal. \end{matrix}

(5) r_{an}(K, g, χ) − h_{f}^{1}(K, V ⊗ χ) ≡ r_{an}(K, g, χ′) − h_{f}^{1}(K, V ⊗ χ′) (mod 2).

Proof. (1) [N 1, 12.6.2.2.1, 12.6.2.4]. (2) As v | p and g is potentially p-ordinary, π_{v} is not supercuspidal; apply [N 1, 12.6.2.4(i),(ii)]. (3) Let S = {v | cd_{K/F} , v ≠ P_{1}, …, P_{s}}. By [A-N, 1.2.7], G_{0}^{(c)} has a subgroup G_{1}^{(c)} ⟶^{∼} U′(c, P_{1} ⋯ P_{s}) = ∏_{v∈S} U′_{v}, where each U′_{v} is isomorphic to a quotient of (K_{v}^{∗})^{∘}/F_{v}^{∗}, via the reciprocity map rec_{K}. If v ∈ S does not split in K/F, then (K_{v}^{∗})^{∘} = K_{v}^{∗}, which means that χ_{v} depends only on χ|_{G_{1}^{(c)}}; thus χ_{v} = χ′_{v}, which implies that ε_{v} = ε′_{v} and ε̃_{v} = ε′_{v}. If v ∈ S splits in K/F, we conclude by (1).

(4) If π_{v} is not supercuspidal, then ε̃_{v} = ε′_{v} = η_{v}(−1) = 1 (recall that v is unramified in K/F, by assumption), by [N 1, 12.6.2.4(i),(ii)]. If π_{v} is supercuspidal, then ε_{v} = ε̃_{v} = ε′_{v} = ε′_{v} = η_{v}(v)^{o(π_{v})}, by [N 1, 12.6.2.4(iii)]. If v is inert in K/F, then χ_{v} = χ′_{v} = 1, as both χ_{v} and χ′_{v} are unramified and K_{v}^{∗} = O_{K,v}^{∗}F_{v}^{∗}. If v splits in K/F, then ε_{v} = ε′_{v}, by (1).

(5) As ε̃:= ∏_{v} ε̃_{v} is equal to ε′ := ∏_{v} ε′_{v}, by (1)–(4), we deduce from [N 1, 12.6.4.3] that

h̃_{f}^{1}(K, V ⊗ χ) − h_{f}^{1}(K, V ⊗ χ) + r_{an}(K, g, χ) ≡ h̃_{f}^{1}(K, V ⊗ χ′) − h_{f}^{1}(K, V ⊗ χ′) + r_{an}(K, g, χ′) (mod 2),

where h̃_{f}^{1}(K, V ⊗ χ) denotes the dimension of the “extended Selmer group” H̃_{f}^{1}(K, V ⊗ χ) defined in [N 1, 12.5.9.2]. On the other hand, [N 1, Thm. 10.7.17] implies that

h̃_{f}^{1}(K, V ⊗ χ) ≡ h̃_{f}^{1}(K, V ⊗ χ′) (mod 2),

as explained in the proof of [N 1, 12.6.4.7(v)]. Combining the two congruences, we obtain the desired result.

2.1.3. Proposition-Definition. (1) ε_{lim}(π × χ_{0}, \frac{1}{2}) := ε(π × χ, \frac{1}{2}) ∏_{i=1}^{s} η_{P_{i}}(−1) ε_{P_{i}} ∈ \{±1\} depends only on π and χ_{0} = χ|_{G_{0}^{(c)}}.

(2) If k > 2, then ε_{lim}(π × χ_{0}, \frac{1}{2}) = ε(π × χ, \frac{1}{2}) for any χ satisfying χ|_{G_{0}^{(c)}} = χ_{0}.

(3) If n ∈ N is large enough and (P_{1} ⋯ P_{s})^{n} | c(χ), then ε(π × χ, \frac{1}{2}) = ε_{lim}(π × χ_{0}, \frac{1}{2}).

Proof. (1) By Proposition 2.1.2, the individual terms in

ε(π × χ, \frac{1}{2}) ∏_{i=1}^{s} η_{P_{i}}(−1) ε_{P_{i}} = ∏_{v∤P_{1}···P_{s}} η_{v}(−1) ε_{v}

are the same for χ and χ′.

(2) If k > 2, then each π_{P_{i}} is a principal series representation [N 2, 12.5.4], hence η_{P_{i}}(−1) ε_{P_{i}} = 1 ([N 2, 12.6.2.4(i)]).

(3) If n is large enough, then each term η_{P_{i}}(−1) ε_{P_{i}} is equal to 1, by [J-L, Prop. 3.8] and [J, Thm. 20.6].

2.2. Ranks

2.2.1. Fix a character χ_{0} : G_{0}^{(c)} ⟶ O_{L}^{∗}, where L is a number field such that i_{∞}(L) contains all Hecke eigenvalues of g. Let 𝔭 | p be as in 0.1; put O = O_{L,𝔭}, and Λ = O[[G^{(c)}]]. For any O[G_{0}^{(c)}]-module M, define

M_{(χ_{0})} := M ⊗_{O[G_{0}^{(c)}],χ_{0}} O;

then

Λ_{(χ_{0})} ⟶^{∼} O[[G^{(c)}/G_{0}^{(c)}]] ⟶^{∼} O[[X_{1}, …, X_{r}]], r = ∑_{i=1}^{s}[F_{P_{i}} : Q_{p}].

2.2.2. We say that a prime v of F is exceptional if v | p, v ∤ P_{1} ⋯ P_{s} and π_{v} = St ⊗ μ (μ^{2} = 1). Denote by M(g, K; c, P_{1}, …, P_{s}; χ_{0}) the set of all exceptional primes of F that do not split in K/F and satisfy

\{ \begin{matrix}(χ_{0})_{w} = μ ∘ N_{K_{w}/F_{v}}, & if v | cd_{K/F} \\ ord_{v}(𝔫(g)) = 1, & if v ∤ cd_{K/F} \end{matrix}

(where w is the unique prime of K above v). If k > 2, then there are no exceptional primes.

2.2.3. Let χ : G^{(c)} ⟶ O^{′∗} be a character of finite order, where O′ is the ring of integers in a finite extension of L_{𝔭}. The extended Selmer group of V ⊗ χ ([N 1, 12.5.9.2]) sits in an exact sequence

0 ⟶ ⨁_{w|v|p} H^{0}(K_{w}, V_{v}^{−} ⊗ χ_{w}) ⟶ H̃_{f}^{1}(K, V ⊗ χ) ⟶ H_{f}^{1}(K, V ⊗ χ) ⟶ 0,

where v (resp., w) is a prime of F (resp., of K) and

0 ⟶ V_{v}^{+} ⟶ V ⟶ V_{v}^{−} ⟶ 0

is an exact sequence of L_{𝔭}[Gal(\overline{F}_{v}/F_{v})]-modules arising from the potential p-ordinarity of g. Denote by h^{0}, h̃_{f}^{1} and h_{f}^{1} the dimensions of the respective cohomology groups over the fraction field of O′; then

(2.2.3.1)  h̃_{f}^{1}(K, V ⊗ χ) − h_{f}^{1}(K, V ⊗ χ) = ∑_{w|v|p} h^{0}(K_{w}, V_{v}^{−} ⊗ χ_{w}),

where

h^{0}(K_{w}, V_{v}^{−} ⊗ χ_{w}) = \{ \begin{matrix}1, & π_{v} = St ⊗ μ (μ^{2} = 1), χ_{w} = μ ∘ N_{K_{w}/F_{v}} \\ 0, & otherwise \end{matrix}

([N 1, 12.5.8]). In particular, h^{0}(K_{w}, V_{v}^{−} ⊗ χ_{w}) = 0 and h̃_{f}^{1}(K, V ⊗ χ) = h_{f}^{1}(K, V ⊗ χ) if k > 2.

2.2.4. Proposition. Asume that k = 2. Let χ : G^{(c)} ⟶ O^{′∗} be a character of finite order satisfying χ|_{G_{0}^{(c)}} = χ_{0} and such that (P_{1} ⋯ P_{s})^{n} | c(χ), for large enough n ∈ N. Then:

(1) h̃_{f}^{1}(K, V ⊗ χ) − h_{f}^{1}(K, V ⊗ χ) ≡ |M(g, K; c, P_{1}, …, P_{s}; χ_{0})| (mod 2).

(2) If there exists an exceptional prime of F that splits in K/F, assume, in addition, that there is i ∈ {1, …, s} such that F_{P_{i}} = Q_{p}. Then

h̃_{f}^{1}(K, V ⊗ χ) − h_{f}^{1}(K, V ⊗ χ) = |M(g, K; c, P_{1}, …, P_{s}; χ_{0})|.

Proof. If w is a prime of K dividing P_{1} ⋯ P_{s}, then χ_{w}^{2} ≠ 1 if n is large enough, which means that w does not contribute to the R.H.S. of (2.2.3.1).

Let v be an exceptional prime of F. If v does not split in K/F and v | cd_{K/F}, then χ_{w} = (χ_{0})_{w} (note that the decomposition group of v in G^{(c)} is finite, as v does not split in K/F) and

h^{0}(K_{w}, V_{v}^{−} ⊗ χ_{w}) = h^{0}(K_{w}, V_{v}^{−} ⊗ (χ_{0})_{w}) = \{ \begin{matrix}1, & v ∈ M(g, K; c, P_{1}, …, P_{s}; χ_{0}) \\ 0, & v ∉ M(g, K; c, P_{1}, …, P_{s}; χ_{0}). \end{matrix}

If v does not split in K/F and v ∤ cd_{K/F}, then v is inert in K/F (vO_{K} = w) and χ_{w} is unramified; thus χ_{w} = 1 and (using [N 1, (12.3.9.1)])

h^{0}(K_{w}, V_{v}^{−} ⊗χ_{w}) = \{ \begin{matrix}1, & μ is unramified ⟺ ord_{v}(𝔫(g)) = 1 \\ 0, & otherwise \end{matrix} = \{ \begin{matrix}1, & v ∈ M(g, K; c, P_{1}, …, P_{s}; χ_{0}) \\ 0, & v ∉ M(g, K; c, P_{1}, …, P_{s}; χ_{0}). \end{matrix}

If v splits in K/F (vO_{K} = ww′), then the h^{0}-terms in (2.2.3.1) for w and w′ are the same; this proves (1). Under the assumptions of (2), there is i such that Gal(K[P_{i}^{∞}]/K) is isomorphic to a product of Z_{p} by a finite abelian group. As the decomposition groups of w and w′ in Gal(K[P_{i}^{∞}]/K) are infinite, it follows that χ_{w}^{2}, χ_{w′}^{2} ≠ 1 if n is large enough; thus w and w′ do not contribute to the R.H.S. of (2.2.3.1).

2.2.5. Proposition. Fix a G_{F} -stable O-lattice T ⊂ V. Let w|v|p be as in 2.2.3. Define

Z_{w} = \underrightarrow{lim}_{α} ⨁_{w_{α}|w} H^{0}((K_{α})_{w_{α}}, (V/T)_{v}^{−}),

where and K_{α}/K runs through all finite subextensions of K[cP_{1}^{∞} ⋯ P_{s}^{∞}]/K and (V/T)_{v}^{−} is the image of V_{v}^{−} in V/T. Then

cork_{Λ_{(χ_{0})}} ⨁_{w|v} (Z_{w})_{(χ_{0})} = \{ \begin{matrix}1, & v ∈ M(g, K; c, P_{1}, …, P_{s}; χ_{0}) \\ 0, & v ∉ M(g, K; c, P_{1}, …, P_{s}; χ_{0}). \end{matrix}

Proof. If the decomposition group of w in K[cP_{1}^{∞} ⋯ P_{s}^{∞}]/K is infinite, then the L.H.S. is equal to zero ([N 1, 12.6.4.10]). We can assume, therefore, that v is exceptional and does not split in K/F. Choose an isomorphism G^{(c)} ⟶^{∼} G_{0}^{(c)} × (G^{(c)}/G_{0}^{(c)}); then χ_{0} becomes a character of G^{(c)} and D ((Z_{w})_{(χ_{0})}) =

D (H^{0}(K_{w}, (V/T)_{v}^{−} ⊗ (χ_{0}^{−1})_{w})) ⊗_{O} Λ_{(χ_{0})}, where D(−) denotes the Pontrjagin dual (cf. [N 1, 9.6.5]); thus

cork_{Λ_{(χ_{0})}} (Z_{w})_{(χ_{0})} = h^{0}(K_{w}, V_{v}^{−} ⊗ (χ_{0}^{−1})_{w}) = \{ \begin{matrix}1, & v ∈ M(g, K; c, P_{1}, …, P_{s}; χ_{0}) \\ 0, & v ∉ M(g, K; c, P_{1}, …, P_{s}; χ_{0}), \end{matrix}

by the proof of Proposition 2.2.4. 2.2.6. Fix a finite set S of primes of F containing Σ_{p} ∪ Σ_{∞} and all primes dividing 𝔫(g), and a subset Σ′ ⊆ S − (Σ_{p} ∪ Σ_{∞}). As in [N 1, 12.5.9.1], consider the corresponding Greenberg’s local conditions for T and V/T. In order to simplify the notation, put K_{∞} = K[cP_{1}^{∞} ⋯ P_{s}^{∞}].

2.2.7. Proposition. (1) The (co)-ranks

∀j = 1, 2 rk_{Λ_{(χ_{0})}} H̃_{f,Iw}^{j}(K_{∞}/K, T)_{(χ_{0}^{±1})} = cork_{Λ_{(χ_{0})}} H̃_{f}^{j}(K_{S}/K_{∞}, V/T)_{(χ_{0}^{±1})} = r(g)

(where H̃_{f,Iw}^{j}(K_{∞}/K, T) is the Λ-module defined in [N 1, 8.8.5]) are equal to the same integer r(g) and do not depend on S and Σ′.

(2) Assume that ε_{lim}(π × χ_{0}, \frac{1}{2}) = −1. If g has CM by a totally imaginary quadratic extension K′ of F, assume that p ≠ 2 and K′ ⊄ K_{∞}^{Ker(χ_{0})}. Then r(g) ≡ 1 + |M(g, K; c, P_{1}, …, P_{s}; χ_{0})| (mod 2).

Proof. (1) See the proof of [N 1, 12.6.4.12].

(2) Let χ : G^{(c)} ⟶ O^{′∗} be a character of finite order such that χ|_{G_{0}^{(c)}} = χ_{0} and (P_{1} ⋯ P_{s})^{n} | c(χ), for large enough n ∈ N. We have

r(g) ≡ h̃_{f}^{1}(K, V ⊗ χ) (mod 2)  [N1, 10.7.17]
≡ h_{f}^{1}(K, V ⊗ χ) + |M(g, K; c, P_{1}, …, P_{s}; χ_{0})| (mod 2) Proposition 2.2.4(1)
≡ 1 + |M(g, K; c, P_{1}, …, P_{s}; χ_{0})| (mod 2) Theorem 0.4(1)

(Theorem 0.4(1) applies, as ε(π × χ, \frac{1}{2}) = ε_{lim}(π × χ_{0}, \frac{1}{2}) = −1, by Proposition-Definition 2.1.3(3)).

2.2.8. Proposition. Assume that k = 2 and ε_{lim}(π × χ_{0}, \frac{1}{2}) = −1. If g has CM by a totally imaginary quadratic extension K′ of F, assume that p ≠ 2 and K′ ⊄ K_{∞}^{Ker(χ_{0})}. If there exists an exceptional prime of F that splits in K/F, assume, in addition, that there is i ∈ {P_{1}, …, P_{s}} such that F_{P_{i}} = Q_{p}. Then:

(1) The (co)-ranks in Proposition 2.2.7(1) are equal to r(g) = 1 + |M(g, K; c, P_{1}, …, P_{s}; χ_{0})|.

(2) cork_{Λ_{(χ_{0})}} S_{V/T}^{str} (K_{∞})_{(χ_{0}^{±1})} = 1, where S_{V/T}^{str} (K_{∞}) ⊂ H^{1}(K_{∞}, V/T) is Greenberg’s strict Selmer group [N 1, 9.6.5].

(3) If K ⊂ K′_{∞} ⊂ K_{∞} is an intermediate field such that χ_{0} factors through χ′_{0} : Gal(K′_{∞}/K)_{tors} ⟶ O_{L}^{∗}, then the (co)-ranks

∀j = 1, 2 rk_{Λ′_{(χ′_{0})}} H̃_{f,Iw}^{j}(K′_{∞}/K, T)_{(χ_{0}^{′±1})} = cork_{Λ′_{(χ′_{0})}} H̃_{f}^{j}(K_{S}/K′_{∞}, V/T)_{(χ_{0}^{′±1})} = r′(g)

(where Λ′ = O[[Gal(K′_{∞}/K)]]) are all equal, do not depend on S and Σ′, and satisfy

r′(g) ≥ r(g), r′(g) ≡ r(g) (mod 2).

Proof (cf. [N 1, 12.6.4.12, 12.9.8]). (1),(2) Denote the coranks in (2) by s_{±}(g). The exact sequence

H^{0}(K_{∞}, V/T)_{(χ_{0}^{±1})} ⟶ ⨁_{w|v|p}(Z_{w})_{(χ_{0}^{±1})} ⟶ H̃_{f}^{1}(K_{S}/K_{∞}, V/T)_{(χ_{0}^{±1})} ⟶ S_{V/T}^{str} (K_{∞})_{(χ_{0}^{±1})} ⟶ 0

([N 1, (9.6.5.1)]) combined with Proposition 2.2.5 and 2.2.7 implies that

r(g) = s_{±}(g) + |M(g, K; c, P_{1}, …, P_{s}; χ_{0})|.

For m ∈ N large enough, let χ_{m} be as in 1.5.1; then

h^{1}(K, V ⊗ χ_{m}) = 1,  h̃^{1}(K, V ⊗ χ_{m}) = 1 + |M(g, K; c, P_{1}, …, P_{s}; χ_{0})|,

by (1.5.1.4) and Proposition 2.2.4(2). Finally,

h̃^{1}(K, V ⊗ χ_{m}) ≥ r(g),  h̃^{1}(K, V ⊗ χ_{m}) ≡ r(g) (mod 2)

by [N 1, 10.7.17], which implies that

s_{±}(g) = 1, r(g) = 1 + |M(g, K; c, P_{1}, …, P_{s}; χ_{0})|.

(3) This follows from [N 1, 8.10.11(ii)] and [N 1, 10.7.17(iii)].

2.2.9. Proposition. Let R be the quotient of the ordinary Hecke algebra defined in [N 1, 12.7.7] (Spec(R) is the branch of the cyclotomic Hida family of parallel weight passing through the p-stabilisation f^{0} of a p-ordinary twist of g). Let \overline{R} = R[[G^{(c)}]] and \overline{𝔮} = \overline{R} · Ker(χ_{0} : O[G_{0}^{(c)}] ⟶ O) (\overline{𝔮} is a minimal prime of \overline{R}). Let T be the big Galois representation defined in [N 1, 12.7.15.3] (then T_{P}/PT_{P} ⟶^{∼} V, where P is the arithmetic point of R corresponding to f^{0}). Assume that ε_{lim}(π × χ_{0}, \frac{1}{2}) = −1. If g has CM by a totally imaginary quadratic extension K′ of F, assume that p ≠ 2 and K′ ⊄ K_{∞}^{Ker(χ_{0})}. Then:

(1) The ranks

∀j = 1, 2 rk_{\overline{R}\overline{𝔮}} H̃_{f,Iw}^{j}(K_{∞}/K, T )\overline{𝔮} = rk_{\overline{R}\overline{𝔮}} (H̃_{f,Iw}^{j}(K_{∞}/K, T )^{ι})\overline{𝔮} = 1.

(2) For all but finitely many arithmetic points P′ of R, r(g_{P′}) = 1, where g_{P′} ∈ S_{k′}(𝔫′, 1) satisfies

V (g_{P′})(k′/2) ⟶^{∼} T_{P′}/P′T_{P′}.

Proof (cf. [N 1, 12.9.8, 12.9.11]). The four ranks in (1) are equal, by [N 1, 12.7.15.7(i), 12.7.16.8(i)]; denote their common value by h. Let g′ = g_{P′} be as in (2). According to [N 1, 12.7.15.8(iii)],

h ≤ r(g′), h ≡ r(g′) (mod 2).

Choose P′ such that k′ = 2 and π(g′)_{v} is a principal series representation for all v | p (cf. [N 1, 12.7.10]); then M(g, K; c, P_{1}, …, P_{s}; χ_{0}) = ∅. Applying (a slight variant of) Proposition 2.2.8(1) to g′, we obtain r(g′) = 1, hence h = 1. The statement (2) follows from the fact that h = r(g′) for almost all P′.

3. Proof of Theorem 0.8

3.1. Kummer theory

3.1.1. As in 0.7, let q ≠ 2 be a prime number and F_{0} a totally real number field such that F_{0} ∩Q(μ_{q^{∞}}) = Q. Let a ∈ O_{F_{0}} − {0} be an element satisfying a ∉ F_{0}^{∗q} and, for each finite prime v_{0} of F_{0} not dividing q, ord_{v_{0}}(a) < q. For r ≥ 1, put F_{r} = F_{0}Q(μ_{q^{r}})^{+}, K_{r} = F_{0}(μ_{q^{r}}).

3.1.2. Let r ≥ 1. As F_{0} ∩ Q(μ_{q^{∞}}) = Q,

Gal(K_{r}/F_{0}) ⟶^{∼} Gal(Q(μ_{q^{r}})/Q) ⟶^{∼} (Z/q^{r}Z)^{∗}.

The map

F_{0}^{∗} ⊗ Z/q^{r}Z = H^{1}(F_{0}, μ_{q^{r}}) ⟶^{res} H^{1}(K_{r}, μ_{q^{r}}) = K_{r}^{∗} ⊗ Z/q^{r}Z

is injective, as

|Ker(res)| = |H^{1}(K_{r}/F_{0}, μ_{q^{r}})| = |Ĥ^{0}(K_{r}/F_{0}, μ_{q^{r}})| ≤ |H^{0}(F_{0}, μ_{q^{r}})| = 1;

thus a ∉ K_{r}^{∗q}, which yields, by Kummer theory, an isomorphism

Gal (K_{r}( ^{q^{r}} \sqrt{a})/K_{r}) ⟶^{∼} μ_{q^{r}}, g ↦ g( ^{q^{r}} \sqrt{a})/ ^{q^{r}} \sqrt{a}.

A choice of a compatible system of roots ^{q^{r}} \sqrt{a} ∈ \overline{Q} such that ( ^{q^{r}} \sqrt{a})^{q} = ^{q^{r−1}} \sqrt{a} determines a compatible system of isomorphisms

Gal (K_{r}( ^{q^{r}} \sqrt{a})/F_{0}) ⟶^{∼} μ_{q^{r}} ⋊ (Z/q^{r}Z)^{∗},

under which Gal (K_{r}( ^{q^{r}} \sqrt{a})/F_{0}( ^{q^{r}} \sqrt{a})) corresponds to (Z/q^{r}Z)^{∗}.

3.1.3. Proposition. Let H be a finite group acting on a finite abelian group A; let G = A ⋊ H. Let L be a field of characteristic zero containing all roots of unity of order equal to the exponent of A. Then the L[G]-module L[G/H] decomposes into simple L[G]-modules as follows:

L[G/H] = ⨁_{[χ:A⟶L^{∗}]} Ind_{A⋊H_{χ}}^{A⋊H}(χ ⊗ 1),

where χ runs through a set of representatives of H\Hom(A, L^{∗}) and H_{χ} ⊂ H is the stabiliser of χ.

Proof. Easy exercise. 3.1.4. For a subfield k ⊂ \overline{Q}, put G_{k} = Gal(\overline{Q}/k). If k′/k is a finite subextension of \overline{Q}/k, denote by

Res_{k′/k} : (G_{k} − Mod) ⟶ (G_{k′} − Mod) , Ind_{k′/k} : (G_{k′} − Mod) ⟶ (G_{k} − Mod)

the corresponding restriction and induction functors, respectively.

In the situation of 3.1.2, apply Proposition 3.1.3 to

A = Gal (K_{r}( ^{q^{r}} \sqrt{a})/K_{r}) ⟶^{∼} μ_{q^{r}}, H = Gal (K_{r}( ^{q^{r}} \sqrt{a})/F_{0}( ^{q^{r}} \sqrt{a})) ⟶^{∼} (Z/q^{r}Z)^{∗};

we obtain, for any field L ⊃ Q(μ_{q^{r}}),

(3.1.4.1)  Ind_{F_{0}( ^{q^{r}} \sqrt{a})/F_{0}}(Z) ⊗ L = L[A ⋊ H/H] = ⨁_{s=0}^{r} ρ_{s},  ρ_{s} = Ind_{K_{s}/F_{0}}(χ_{s}),

where χ_{s} : Gal (K_{s}( ^{q^{s}} \sqrt{a})/K_{s}) ↪ L^{∗} is an injective character (of order q^{s}) and ρ_{s} is an absolutely irreducible representation of G of dimension φ(q^{s}). If s ≥ 1, denote by ρ the non-trivial element of Gal(K_{s}/F_{s}) ⟶^{∼} Gal (Q(μ_{q^{s}})/Q(μ_{q^{s}})^{+}); then ^{ρ}χ_{s} = χ_{s}^{−1}. As the order of χ_{s} is odd, it follows that χ_{s} (more precisely, χ_{s} ∘ rec_{K_{s}}) factors through A_{K_{s}}^{∗}/K_{s}^{∗}A_{F_{s}}^{∗}.

If M is an L[G_{F_{0}}]-module, the projection formula

Ind_{k′/k} (X ⊗_{L} Res_{k′/k}(Y)) = Ind_{k′/k}(X) ⊗_{L} Y

together with (3.1.4.1) imply that

(3.1.4.2)  Ind_{F_{0}( ^{q^{r}} \sqrt{a})/F_{0}} ∘ Res_{F_{0}( ^{q^{r}} \sqrt{a})/F_{0}}(M) = ⨁_{s=0}^{r} Ind_{K_{s}/F_{0}} (Res_{K_{s}/F_{0}}(M) ⊗ χ_{s}) .

3.2. ε-factors

3.2.1. In the situation of 3.1.1, let g_{0} ∈ S_{k}(𝔫_{0}, 1) be a cuspidal Hilbert modular newform over F_{0} of parallel weight k, trivial central character and level 𝔫_{0}; put V = V_{𝔭}(g_{0})(k/2). For r ≥ 1, denote by g_{r} the base change of g_{0} over F_{r} (i.e., π(g_{r}) = BC_{F_{r}/F} (π(g_{0}))). 3.2.2. It follows from (3.1.4.2) and Shapiro’s Lemma that

(3.2.2.1)  ∀r ≥ 1  H_{f}^{1}(F_{0}( ^{q^{r}} \sqrt{a}), V) = H_{f}^{1}(F_{0}, V) ⊕ ⨁_{s=1}^{r} H_{f}^{1}(K_{s}, V ⊗ χ_{s})

(taking L ⊃ Q(μ_{q^{r}}) in 0.1), hence

(3.2.2.2)  ∀r ≥ 1  h_{f}^{1}(F_{0}( ^{q^{r}} \sqrt{a}), V) − h_{f}^{1}(F_{0}( ^{q^{r−1}} \sqrt{a}), V) = h_{f}^{1}(K_{r}, V ⊗ χ_{r}).

3.2.3. Proposition. Assume that (∀v_{0}|a) π(g_{0})_{v_{0}} is not supercuspidal and that (∀v_{0}|q) π(g_{0})_{v_{0}} is a principal series representation. Then:

∀r ≥ 1  ε(π(g_{r}) × χ_{r}, \frac{1}{2}) = (\frac{d}{q}) ,  d = (−1)^{[F_{0}:Q]}N(𝔫_{0}^{(aq)}),

where 𝔞^{(𝔟)} = 𝔞/(𝔞, 𝔟^{∞}) is the prime-to-𝔟 part of 𝔞, for any non-zero ideals 𝔞, 𝔟 in the ring of integers of a given number field.

Proof. Fix r ≥ 1. Let v be a finite prime of F_{r}. Our assumptions imply that

(3.2.3.1) v ramifies in K_{r}/F_{r} ⟺ v | q v | a, v ∤ q ⟹ v | c(χ_{r}) ⟹ v | aq.

It follows that the condition H(K_{r}, χ_{r}) from [N 1, 12.6.3.5] is satisfied. Applying [N 1, 12.6.3.9], we obtain

ε(π(g_{r}) × χ_{r}, \frac{1}{2}) = (−1)^{[F_{r}:Q]} η_{K_{r}/F_{r}}(𝔫(g_{r})^{(q)}) (−1)^{|R(1)^{0}∩\{v∤c(χ_{r})\}|} (−1)^{|R(1)^{−}∩\{v|c(χ_{r})\}|},

where 𝔫(g_{r}) ⊂ O_{F_{r}} is the level of g_{r} and R(1)^{0} (resp., R(1)^{−}) contains those elements of Σ_{1}(g_{r}) which are ramified (resp., inert) in K_{r}/F_{r} (see [N 1, 12.6.3.2]). By (3.2.3.1), we have R(1)^{0} = ∅ and

R(1)^{−} ∩ {v | c(χ_{r})} = {v | a; v is inert in K_{r}/F_{r}, π(g_{r}) = St ⊗ μ, μ : F_{r,v}^{∗} ⟶ {±1} is unramified}
= {v | a; v is inert in K_{r}/F_{r}, π(g_{r}) = St ⊗ μ, μ : F_{r,v}^{∗} ⟶ {±1}, o(π(g_{r})_{v}) = 1}
= {v | a; v is inert in K_{r}/F_{r}, π(g_{r}) = St ⊗ μ, μ : F_{r,v}^{∗} ⟶ {±1}, 2 ∤ o(π(g_{r})_{v})}

(as π(g_{r})_{v} is not supercuspidal), hence

ε(π(g_{r}) × χ_{r}, \frac{1}{2}) = (−1)^{[F_{r}:Q]} η_{K_{r}/F_{r}}(𝔫(g_{r})^{(aq)}).

As F_{r}/F is unramified outside q, it follows that

𝔫(g_{r})^{(q)} = 𝔫_{0}^{(q)}O_{F_{r}}, 𝔫(g_{r})^{(aq)} = 𝔫_{0}^{(aq)}O_{F_{r}}.

Let F′ = Q(μ_{q^{r}})^{+}, K′ = Q(μ_{q^{r}}). If I_{0} ⊂ O_{F_{0}} is an ideal prime to q, let I_{r} = I_{0}O_{F_{r}}; then

η_{K_{r}/F_{r}}(I_{r}) = η_{K′/F′}(N_{F_{r}/F′}(I_{r})) ≡ N(I_{r}) (mod q^{r}) ≡ N(I_{0})^{φ(q^{r})/2} ≡ N(I_{0})^{(q−1)/2} (mod q),

hence

η_{K_{r}/F_{r}}(I_{0}O_{F_{r}}) = (\frac{N(I_{0})}{q}) , (−1)^{[F_{r}:Q]} η_{K_{r}/F_{r}}(I_{0}O_{F_{r}}) = (\frac{(−1)^{[F_{0}:Q]}N(I_{0})}{q}) .

Taking I_{0} = 𝔫_{0}^{(aq)}, we obtain the desired result. 3.2.4. The proof of Proposition 3.2.3 shows that the formula for ε(π(g_{r})×χ_{r}, \frac{1}{2}) still holds if we replace the assumption “(∀v_{0}|q) π(g_{0})_{v_{0}} is a principal series representation” by “(∀v_{0}|q) π(g_{0})_{v_{0}} is not supercuspidal, and the extension K_{r}( ^{q^{r}} \sqrt{a})/K_{r} is ramified at all primes above q”, as R(1)^{0} ∩ \{v ∤ c(χ_{r})\} = ∅ in this case.

3.3. End of the proof

3.3.1. The statement of Theorem 0.8(1) follows from Proposition 3.2.3. 3.3.2. By (3.2.2.2), the first line in the statement of Theorem 0.8(2) will follow from Theorem 0.4(1) applied to each g_{s} and χ_{s} (1 ≤ s ≤ r). As g_{0} is potentially p-ordinary, so is g_{s}. By 3.3.1, 2 ∤ r_{an}(K_{s}, g_{s}, χ_{s}), so it remains to check that if g_{s} has CM by a totally imaginary quadratic extension K′ of F_{s}, then K′ is not contained in K_{s}( ^{q^{s}} \sqrt{a}). If K′ ⊂ K_{s}( ^{q^{s}} \sqrt{a}), then K′ = K_{s} and g_{0} has CM by a totally imaginary quadratic extension K′_{0} of F_{0} such that F_{r}K′_{0} = K_{s}. However, the only quadratic extension of F_{0} contained in K_{s} is F_{0}(\sqrt{q^{∗}}), where q^{∗} = (−1)^{(q−1)/2}q; thus q ≡ 3 (mod 4) and K′_{0} = F_{0}(\sqrt{−q}), which contradicts the assumptions.

It remains to show is that, in the case when 2 ∤ [F_{0} : Q], g_{0} cannot have CM by K′_{0} = F_{0}(\sqrt{−q}), q ≡ 3 (mod 4). If that were the case, then each finite prime v_{0} of F_{0} ramified in K′_{0}/F_{0} would divide q, hence π(g_{0})_{v_{0}} would not be supercuspidal; thus π(g_{0})_{v_{0}} = π(μ, μη′_{v_{0}}), where η′ = η_{K′_{0}/F_{0}} ([N 1, 12.6.1.2.3]). As the central character of g_{0} is trivial, η′_{v_{0}} = μ^{−2}, hence η′_{v_{0}}(−1) = 1. This would imply that 1 = ∏_{w_{0}} η′_{w_{0}}(−1) = (−1)^{[F_{0}:Q]}, which contradicts our assumption. 3.3.3. Let us now prove the second line in the statement of Theorem 0.8(2). For each s = 0, …, r, ρ_{s} = Ind_{K_{s}/F_{0}}(χ_{s}) is an absolutely irreducible L[G_{F_{0}}]-module of dimension φ(q^{s}). If 1 ≤ s ≤ r, then ρ_{s} ⊗_{L} L_{𝔭} occurs in H_{f}^{1}(K_{r}( ^{q^{r}} \sqrt{a}), V) with odd multiplicity, by Shapiro’s Lemma combined with Theorem 0.4(1) applied to each g_{s} and χ_{s}. As the trivial representation ρ_{0} occurs with multiplicity h_{f}^{1}(F_{0}, V), it follows that

h_{f}^{1}(K_{r}( ^{q^{r}} \sqrt{a}), V) − h_{f}^{1}(F_{0}, V) ≥ ∑_{s=1}^{r} dim(ρ_{s}) = ∑_{s=1}^{r} φ(q^{s}) = q^{r} − 1.

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