p-ADIC RANKIN L-SERIES AND RATIONAL POINTS ON CM ELLIPTIC CURVES

MASSIMO BERTOLINI HENRI DARMON KARTIK PRASANNA

1. Introduction

The aim of this article is to present a new proof of a theorem of Karl Rubin (see [Ru] and Thm. 1 below) relating values of the Katz p-adic L-function of an imaginary quadratic field at certain points outside its range of classical interpolation to the formal group logarithms of rational points on CM elliptic curves. This theorem has been seminal in providing a motivation for Perrin-Riou’s formulation ([PR2], [PR3]) of the p-adic Beilinson conjectures. The new proof described in this work is based on the p-adic Gross-Zagier type formula of [BDP-gz], and only makes use of Heegner points (as opposed to the original proof which relied on on a comparison between Heegner points and elliptic units). Hence it should be adaptable to more general situations, for example to the setting of general CM fields.

Let A be an elliptic curve over ℚ with complex multiplication by the ring of integers of a quadratic imaginary field K. A classical result of Deuring identifies the Hasse-Weil L-series L(A, s) of A with the Lseries L(ν_{A}, s) attached to a Hecke character ν_{A} of K of infinity type (1, 0). When p is a prime which splits in K and does not divide the conductor of A, the Hecke L-function L(ν_{A}, s) has a p-adic analog, namely the Katz two-variable p-adic L-function attached to K. It is a p-adic analytic function, denoted ν ↦ ℒ_{p}(ν), on the space of Hecke characters equipped with its natural p-adic analytic structure. Section 3.1 recalls the definition of this L-function: the values ℒ_{p}(ν) at Hecke characters of infinity type (1 + j_{1}, −j_{2}) with j_{1}, j_{2} ≥ 0 are defined by interpolation of the classical L-values L(ν^{−1}, 0). Letting ν^{∗} := ν ∘ c, where c denotes complex conjugation on the ideals of K, it is readily seen by comparing Euler factors that L(ν, s) = L(ν^{∗}, s). A similar equality need not hold in the p-adic setting, because the involution ν ↦ ν^{∗} corresponds to the map (j_{1}, j_{2}) ↦ (j_{2}, j_{1}) on weight space and therefore does not preserve the lower right quadrant of weights of Hecke characters that lie in the range of classical interpolation. Since ν_{A} lies in the domain of classical interpolation, the p-adic L-value ℒ_{p}(ν_{A}) is a simple multiple of L(ν_{A}^{−1}, 0) = L(A, 1). Suppose that it vanishes. (This implies, by the Birch and Swinnerton-Dyer conjecture, that A(ℚ) is infinite.) The value ℒ_{p}(ν_{A}^{∗}) is a second, a priori more mysterious p-adic avatar of the leading term of L(A, s) at s = 1. Rubin’s theorem gives a formula for this quantity:

Theorem 1 (Rubin). Let ν_{A} be a Hecke character of type (1, 0) attached to an elliptic curve A/ℚ with complex multiplication. Then there exists a global point P ∈ A(ℚ) such that

(1.1)  ℒ_{p}(ν_{A}^{∗}) = Ω_{p}(A)^{−1} log_{ω_{A}}(P)^{2} (mod K^{×}),

where • Ω_{p}(A) is the p-adic period attached to A as in Section 2.3;

• ω_{A} ∈ Ω^{1}(A/ℚ) is a regular differential on A over ℚ, and log_{ω_{A}} : A(ℚ_{p})⟶ℚ_{p} denotes the p-adic

formal group logarithm with respect to ω_{A}. The point P is of infinite order if and only if L(A, s) has a simple zero at s = 1.

(For a more precise statement without the K^{×} ambiguity, see [Ru].) Formula (1.1) is peculiar to the padic world and suggests that p-adic L-functions encode arithmetic information that is not readily apparent in their complex counterparts.

The proof of Theorem 1 given in [Ru] breaks up naturally into two parts:

During the preparation of this article, KP was supported partially by NSF grants DMS-1015173 and DMS-0854900.

(1) Rubin exploits the Euler system of elliptic units to construct a global cohomology class κ_{A} belonging to a pro-p Selmer group Sel_{p}(A/ℚ) attached to A. The close connection between elliptic units and the Katz L-function is then parlayed into the explicit evaluation of two natural p-adic invariants attached to κ_{A}: the p-adic formal group logarithm log_{A,p}(κ_{A}) and the cyclotomic p-adic height ⟨κ_{A}, κ_{A}⟩:

(1.2)  log_{A,p}(κ_{A}) = (1 − β_{p}^{−1})^{−1}ℒ_{p}(ν_{A}^{∗})Ω_{p}(A),
(1.3)  ⟨κ_{A}, κ_{A}⟩ = (1 − α_{p}^{−1})^{−2}ℒ′_{p}(ν_{A})ℒ_{p}(ν_{A}^{∗}),

where

• α_{p} and β_{p} denote the roots of the Hasse polynomial x^{2} − a_{p}(A)x + p, ordered in such a way that ord_{p}(α_{p}) = 0 and ord_{p}(β_{p}) = 1;

• the quantity ℒ′_{p}(ν_{A}) denotes the derivative of ℒ_{p} at ν_{A} in the direction of the cyclotomic

character. If ℒ′_{p}(ν_{A}) is non-zero, then an argument based on Perrin-Riou’s p-adic analogue of the Gross-Zagier formula and the work of Kolyvagin implies that Sel_{p}(A/ℚ) ⊗ ℚ is a one-dimensional ℚ_{p}-vector space with κ_{A} as a generator. (Cf. Thm. 8.1 and Cor. 8.3 of [Ru].) Equations (1.2) and (1.3) then make it possible to evaluate the ratio

(1.4)  \frac{log_{A,p}^{2}(κ)}{⟨κ, κ⟩} = \frac{(1 − β_{p}^{−1})^{−2}ℒ_{p}(ν_{A}^{∗})Ω_{p}(A)^{2}}{(1 − α_{p}^{−1})^{−2}ℒ′_{p}(ν_{A})} ,

a quantity which does not depend on the choice of generator κ of the ℚ_{p}-vector space Sel_{p}(A/ℚ)⊗ℚ.

(2) Independently of the construction of κ_{A}, the theory of Heegner points can be used to construct a canonical point P ∈ A(ℚ), which is of infinite order when ℒ′_{p}(ν_{A}) ≠ 0. Its image κ_{P} ∈ Sel_{p}(A/ℚ) under the connecting homomorphism of Kummer theory supplies us with a second generator for Sel_{p}(A/ℚ) ⊗ ℚ. Furthermore, the p-adic analogue of the Gross-Zagier formula proved by Perrin-Riou in [PR1] shows that

(1.5)  ⟨κ_{P} , κ_{P} ⟩ = ℒ′_{p}(ν_{A})Ω_{p}(A)^{−1} (mod K^{×}).

Rubin obtains Theorem 1 by setting κ = κ_{P} in (1.4) and using (1.5) to eliminate the quantities involving ⟨κ_{P} , κ_{P} ⟩ and ℒ′_{p}(ν_{A}).

The reader will note the key role that is played in Rubin’s proof by both the Euler systems of elliptic units and of Heegner points. The new approach to Theorem 1 described in this paper relies solely on Heegner points, and requires neither elliptic units nor Perrin-Riou’s p-adic height calculations. Instead, the key ingredient in this approach is the p-adic variant of the Gross-Zagier formula arising from the results of [BDP-gz] which is stated in Theorem 3.12. This formula expresses p-adic logarithms of Heegner points in terms of the special values of a p-adic Rankin L-function attached to a cusp form f and an imaginary quadratic field K, and may be of some independent interest insofar as it exhibits a strong analogy with Rubin’s formula but applies to arbitrary—not necessarily CM—elliptic curves over ℚ. When f is the theta series attached to a Hecke character of K, Theorem 1 follows from the factorisation of the associated p-adic Rankin L-function into a product of two Katz L-functions, a factorisation which is a simple manifestation of the Artin formalism for these p-adic L-series.

One might expect that the statement of Theorem 1 should generalise to the setting where ν_{A} is replaced by an algebraic Hecke character ν of infinity type (1, 0) of a quadratic imaginary field K (of arbitrary class number) satisfying

(1.6)  ν|_{𝔸_{ℚ}} = ε_{K} · N,

where ε_{K} denotes the quadratic Dirichlet character associated to K/ℚ and N : 𝔸_{ℚ}^{×}⟶ℝ^{×} is the adèlic norm character. Chapter 3 treats this more general setting, which (although probably amenable as well to the original methods of [Ru]) is not yet covered in the literature. Assumption (1.6) implies that the classical functional equation relates L(ν, s) to L(ν, 2−s). Assume further that the sign w_{ν} in this functional equation satisfies

(1.7)  w_{ν} = −1,

so that L(ν, s) vanishes to odd order at s = 1. For technical reasons, it will also be convenient to make two further assumptions. Firstly, we assume that

(1.8)  The discriminant −D of K is odd.

Secondly, we note that assumption (1.6) implies that 𝔡_{K} := \sqrt{−D} necessarily divides the conductor of ν, and we further restrict the setting by imposing the assumption that

(1.9)  The conductor of ν is exactly divisible by 𝔡_{K}.

The statement of Theorem 2 below requires some further notions which we now introduce. Let E_{ν} be the subfield of ℂ generated by the values of the Hecke character ν, and let T_{ν} be its ring of integers. A general construction which is recalled in Sections 2.2 and 3.6 attaches to ν an abelian variety B_{ν} over K of dimension [E_{ν} : K], equipped with inclusions

T_{ν} ⊂ End_{K}(B_{ν}), E_{ν} ⊂ End_{K}(B_{ν}) ⊗ ℚ.

Given λ ∈ T_{ν}, denote by [λ] the corresponding endomorphism of B_{ν}, and set

(1.10)  Ω^{1}(B_{ν}/E_{ν})^{T_{ν}} := {ω ∈ Ω^{1}(B_{ν}/E_{ν}) such that [λ]^{∗}ω = λω, ∀λ ∈ T_{ν}} ,
(1.11)  (B_{ν}(K) ⊗ E_{ν})^{T_{ν}} := {P ∈ B_{ν}(K) ⊗_{ℤ} E_{ν} such that [λ]P = λP, ∀λ ∈ T_{ν}} .

The vector space Ω^{1}(B_{ν}/E_{ν})^{T_{ν}} is one-dimensional over E_{ν}. The results of Gross-Zagier and Kolyvagin, which continue to hold in the setting of abelian variety quotients of modular curves, also imply that (B_{ν}(K) ⊗ E_{ν})^{T_{ν}} is one-dimensional over E_{ν} when L(ν, s) has a simple zero at s = 1.

After fixing a p-adic embedding K ⊂ ℚ_{p}, the formal group logarithm on B_{ν} gives rise to a bilinear pairing

⟨ , ⟩ : Ω^{1}(B_{ν}/K) × B_{ν}(K) ⟶ ℚ_{p}
(ω, P) ↦ log_{ω} P,

satisfying ⟨[λ]^{∗}ω, P ⟩ = ⟨ω, [λ]P ⟩ for all λ ∈ T_{ν}. This pairing can be extended by E_{ν}-linearity to an E_{ν} ⊗ ℚ_{p}-valued pairing between Ω^{1}(B_{ν}/E_{ν}) and B_{ν}(K) ⊗ E_{ν}. When ω and P belong to these E_{ν}-vector spaces, we will continue to write log_{ω}(P) for ⟨ω, P ⟩.

Theorem 2. Let ν be an algebraic Hecke character of infinity type (1, 0) satisfying (1.6), (1.7), (1.8) and (1.9) above. Then there exists P_{ν} ∈ B_{ν}(K) such that

ℒ_{p}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}} (P_{ν})^{2} (mod E_{ν}^{×}),

where Ω_{p}(ν^{∗}) ∈ ℂ_{p} is the p-adic period attached to ν in Definition 2.13, and ω_{ν} is a non-zero element of Ω^{1}(B_{ν}/E_{ν})^{T_{ν}}. The point P_{ν} is non-zero if and only if L′(ν, 1) ≠ 0.

Remark 3. Assumptions (1.8) and (1.9) could certainly be relaxed with more work. For instance, (1.8) is needed since the main theorem of [BDP-gz] is only proved for imaginary quadratic fields of odd discriminant. Likewise, removing (1.9) would require generalizing the main result of loc. cit. to the case of Shimura curves over ℚ.

Remark 4. In [BDP-ch], we give a conjectural construction of rational points on CM elliptic curves (called Chow-Heegner points) using cycles on higher dimensional varieties. While this construction of points is contingent on a certain case of the Tate conjecture, the corresponding contruction at the level of cohomology classes can be made unconditionally. The results of this paper, combined with those of [BDP-gz], are used in [BDP-ch] to establish that these cohomology classes indeed correspond to global points via the Kummer map.

Remark 5. The methods used in the proof of Theorem 2 also give information about the special values ℒ_{p}(ν^{∗}) for Hecke characters ν of type (1 + j, −j) satisfying (1.6) with j ≥ 0. A discussion of this point will be taken up in future work. (See [BDP-co].)

2. Hecke characters and periods

Throughout this article, all number fields that arise are viewed as embedded in a fixed algebraic closure ℚ̄ of ℚ. A complex embedding ℚ̄⟶ℂ and p-adic embeddings ℚ̄⟶ℂ_{p} for each rational prime p are also fixed from the outset, so that any finite extension of ℚ is simultaneously realised as a subfield of ℂ and of ℂ_{p}.

2.1. Algebraic Hecke characters. We will recall briefly some key definitions regarding algebraic Hecke characters, mainly to fix notation. The reader is referred to [Scha] Ch. 0 for more details. Let K and E be number fields. Given a ℤ-linear combination

ϕ = ∑_{σ} n_{σ}σ ∈ ℤ[Hom(K, ℚ̄)]

of embeddings of K into ℚ̄, we define

α^{ϕ} := ∏_{σ}(σα)^{n_{σ}},

for all α ∈ K^{×}. Let I_{𝔣} denote the group of fractional ideals of K which are prime to a given integral ideal 𝔣 of K, and let

J_{𝔣} := {(α) such that α >> 0 and α − 1 ∈ 𝔣} ⊆ I_{𝔣}.

Definition 2.1. An E-valued algebraic Hecke character (or simply Hecke character) of K of infinity type ϕ and conductor dividing 𝔣 is a homomorphism

χ : I_{𝔣} → E^{×}

such that

(2.1)  χ((α)) = α^{ϕ}, for all (α) ∈ J_{𝔣}.

The smallest integral ideal 𝔤 such that χ can be extended to a Hecke character of conductor dividing 𝔤 is called the conductor of χ, and is denoted 𝔣_{χ}.

The most basic examples of algebraic Hecke characters are the norm characters on ℚ and on K respectively, which are given by

N((a)) = |a|, N_{K} := N ∘ N_{ℚ}^{K}.

Note that the infinity type ϕ of a Hecke character χ must be trivial on all totally positive units congruent to 1 mod 𝔣. Hence the existence of such a χ implies there is an integer w(χ) (called the weight of χ or of ϕ) such that for any choice of embedding of \overline{ℚ} into ℂ,

n_{σ} + nσ̄ = w(χ), for all σ ∈ Hom(K, ℚ̄).

Let U_{𝔣} ⊂ U′_{𝔣} ⊂ 𝔸_{K}^{×} be the subgroups defined by

U′_{𝔣} := \{(x_{v}) ∈ 𝔸_{K}^{×} such that \begin{matrix}x_{v} ≡ 1 (mod 𝔣), for all v|𝔣, \\ x_{v} > 0, for all real v \end{matrix} \} ,

and

U_{𝔣} := {(x_{v}) ∈ U′_{𝔣} such that x_{v} ∈ O_{K_{v}}^{×}, for all non-archimedean v}.

A Hecke character χ of conductor dividing 𝔣 may also be viewed as a character on 𝔸_{K}^{×}/U_{𝔣} (denoted by the same symbol by a common abuse of notation),

(2.2)  χ : 𝔸_{K}^{×}/U_{𝔣} → E^{×},  satisfying  χ|_{K^{×}} = ϕ.

To wit, given x ∈ 𝔸_{K}^{×}, we define χ(x) by choosing α ∈ K^{×} such that αx belongs to U′_{𝔣}, and setting

(2.3)  χ(x) = χ(i(αx))ϕ(α)^{−1},

where the symbol i(x) denotes the fractional ideal of K associated to x. This definition is independent of the choice of α by (2.1). In the opposite direction, given a character χ as in (2.2), we can set

χ(𝔞) = χ(x), for any x ∈ U′_{𝔣} such that i(x) = 𝔞.

The subfield of E generated by the values of χ on I_{𝔣} is easily seen to be independent of the choice of 𝔣 and will be denoted E_{χ}.

Definition 2.2. The central character ε_{η} of a Hecke character η of K is the finite order character of ℚ given by

η|_{𝔸_{ℚ}^{×}} = ε_{η} · N^{w(η)}.

The infinity type ϕ defines a homomorphism Res_{K/ℚ}(G_{m}) → Res_{E/ℚ}(G_{m}) of algebraic groups and therefore induces a homomorphism

ϕ_{𝔸} : 𝔸_{K}^{×} → 𝔸_{E}^{×}

on adelic points. Given a Hecke character χ with values in E and a place λ of E (either finite or infinite), we may use ϕ_{𝔸} to define an idèle class character

χ_{λ} : 𝔸_{K}^{×}/K^{×} → E_{λ}^{×},

by setting

χ_{λ}(x) = χ(x)/ϕ_{𝔸}(x)_{λ}.

If λ is an infinite place, the character χ_{λ} is a Grossencharacter of K of type A_{0}. If λ is a finite place, then χ_{λ} factors through G_{K}^{ab} and gives a Galois character (denoted ρ_{χ,λ}) valued in E_{λ}^{×}, satisfying

ρ_{χ,λ}(Frob_{𝔭}) = χ(𝔭)

for any prime ideal 𝔭 of K not dividing 𝔣λ.

Let 𝔤 be any integral ideal of K. The L-function (and L-function with modulus 𝔤) attached to χ are defined by

L(χ, s) = ∏_{𝔭} (1 − \frac{χ(𝔭)}{N𝔭^{s}})^{−1} , L_{𝔤}(χ, s) = ∏_{𝔭∤𝔤} (1 − \frac{χ(𝔭)}{N𝔭^{s}})^{−1} .

Note that L(χ, s) = L_{𝔣_{χ}}(χ, s).

The following definition will only be used in Sec. 3.6.

Definition 2.3. Let E = ∏_{i} E_{i} be a product of number fields. An E-valued algebraic Hecke character of conductor dividing 𝔣 is a character

χ : I_{𝔣} → E^{×}

whose projection to each component E_{i} is an algebraic Hecke character in the sense defined above.

2.2. Abelian varieties associated to characters of type (1, 0). In this section, we limit the discussion to the case where K is an imaginary quadratic field. Let τ : K ↦ ℂ be the given complex embedding of K. A Hecke character of infinity type ϕ = n_{τ} τ + nτ̄τ̄ will also be said to be of infinity type (n_{τ} , nτ̄).

Let ν be a Hecke character of K of infinity type (1, 0) and conductor 𝔣_{ν}, let E_{ν} ⊃ K denote the subfield of ℚ̄ generated by its values, and let T_{ν} be the ring of integers of E_{ν}. The Hecke character ν gives rise to a compatible system of one-dimensional ℓ-adic representations of G_{K} with values in (E_{ν} ⊗ ℚ_{ℓ})^{×}, denoted ρ_{ν,ℓ}, satisfying

ρ_{ν,ℓ}(σ_{𝔞}) = ν(𝔞), for all 𝔞 ∈ I_{𝔣_{ν}ℓ},

where σ_{𝔞} ∈ Gal(K̄/K) denotes Frobenius conjugacy class attached to 𝔞. The theory of complex multiplication realises the representations ρ_{ν,ℓ} on the division points of CM abelian varieties:

Definition 2.4. A CM abelian variety over K is a pair (B, E) where (1) B is an abelian variety over K;

(2) E is a product of CM fields equipped the structure of a K-algebra and an inclusion

i : E⟶ End_{K}(B) ⊗ ℚ,

satisfying dim_{K}(E) = dim B;

(3) for all λ ∈ K ⊂ E, the endomorphism i(λ) acts on the cotangent space Ω^{1}(B/K) as multiplication by λ.

The abelian varieties (B, E) over K with complex multiplication by a fixed E form a category denoted CM_{K,E} in which a morphism from B_{1} to B_{2} is a morphism j : B_{1}⟶B_{2} of abelian varieties over K for which the diagrams

\xymatrix{B_{1} \ar[r]^{j} \ar[d]^{e} & B_{2} \ar[d]^{e} \\ B_{1} \ar[r]^{j} & B_{2}}

commute, for all e ∈ E which belong to both End_{K}(B_{1}) and End_{K}(B_{2}). An isogeny in CM_{K,E} is simply a morphism in this category arising from an isogeny on the underlying abelian varieties.

If (B, E) is a CM abelian variety, its endomorphism ring over K contains a finite index subring T^{0} of the integral closure T of ℤ in E. After replacing B by the K-isogenous abelian variety Hom_{T_{0}}(T, B), we can assume that End_{K}(B) contains T. This assumption, which is occasionally convenient, will consistently be made from now on.

Let (B, E) be a CM-abelian variety with E a field, and let E′ ⊃ E be a finite extension of E with ring of integers T′. The abelian variety B ⊗_{T} T′ is defined to be the variety whose L-rational points, for any L ⊃ K, are given by

(B ⊗_{T} T′)(L) = (B(ℚ̄) ⊗_{T} T′)^{Gal(ℚ̄/L)}.

This abelian variety is equipped with an action of T′ by K-rational endomorphisms, described by multiplication on the right, and therefore (B ⊗_{T} T′, E′) is an object of CM_{K,E′}. Note that B ⊗_{T} T′ is isogenous to t := dim_{E}(E′) copies of B, and that the action of T on B ⊗_{T} T′ agrees with the “diagonal” action of T on B^{t}.

Let ℓ be a rational prime. For each CM abelian variety (B, E), let

T_{ℓ}(B) := lim_{←,n} B[ℓ^{n}](K̄), V_{ℓ}(B) := T_{ℓ}(B) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}

be the ℓ-adic Tate module and ℓ-adic representation of G_{K} attached to B. The ℚ_{ℓ}-vector space V_{ℓ}(B) is a free E ⊗ ℚ_{ℓ}-module of rank one via the action of E by endomorphisms. The natural action of G_{K} := Gal(K̄/K) on V_{ℓ}(B) commutes with this E ⊗ ℚ_{ℓ}-action, and the collection {V_{ℓ}(B)} thus gives rise to a compatible system of one-dimensional ℓ-adic representations of G_{K} with values in (E ⊗ℚ_{ℓ})^{×}, denoted ρ_{B,ℓ}. We note in passing that for any extension E′ ⊃ E where T′ is the integral closure of T in E′, we have

T_{ℓ}(B ⊗_{T} T′) = T_{ℓ}(B) ⊗_{T} T′, V_{ℓ}(B ⊗_{T} T′) = V_{ℓ}(B) ⊗_{E} E′.

The following result is due to Casselman (cf. Theorem 6 of [Shi]).

Theorem 2.5. Let ν be a Hecke character of K of type (1, 0) as above, and let ρ_{ν,ℓ} be the associated one-dimensional ℓ-adic representation with values in (E_{ν} ⊗ ℚ_{ℓ})^{×}. Then

(1) There exists a CM abelian variety (B_{ν}, E_{ν}) satisfying

ρ_{B_{ν},ℓ} ≃ ρ_{ν,ℓ}.

(2) The CM abelian variety B_{ν} is unique up to isogeny over K. More generally, if (B, E) is any CM abelian variety with E ⊃ E_{ν} satisfying ρ_{B,ℓ} ≃ ρ_{ν,ℓ} ⊗_{E_{ν}} E as (E ⊗ ℚ_{ℓ})[G_{K}]-modules, then there is an isogeny in CM_{K,E} from B to B_{ν} ⊗_{T_{ν}} T.

Let ψ be a Hecke character of infinity type (1, 0), and let χ be a finite order Hecke character of K, so that ψχ^{−1} also has infinity type (1, 0). In comparing the abelian varieties B_{ψ} and B_{ψχ^{−1}}, it is useful to introduce a CM abelian variety B_{ψ,χ} over K, which we now describe.

Let E_{χ} denote the field generated by K and the values of χ. We denote by E_{ψ,χ} the compositum of E_{ψ} and E_{χ}, and by T_{ψ,χ} ⊂ E_{ψ,χ} its ring of integers. We also write H_{χ} for the abelian extension of K which is cut out by χ viewed as a Galois character of G_{K}. Consider first the abelian variety over K with endomorphisms by T_{ψ,χ}:

B_{ψ,χ}^{0} := B_{ψ} ⊗_{T_{ψ}} T_{ψ,χ}.

The natural inclusion i_{ψ} : T_{ψ}⟶T_{ψ,χ} induces a morphism

(2.4)  i : B_{ψ}⟶B_{ψ,χ}^{0}

with finite kernel, which is compatible with the T_{ψ}-actions on both sides and is given by

i(P) = P ⊗ 1.

Lemma 2.6. Let F be any number field containing E_{ψ,χ}. With notations as in equation (1.10) of the Introduction, the restriction map i^{∗} induces an isomorphism

(2.5)  i^{∗} : Ω^{1}(B_{ψ,χ}^{0}/F)^{T_{ψ,χ}}⟶Ω^{1}(B_{ψ}/F)^{T_{ψ}}

of one-dimensional F -vector spaces.

Proof. The fact that B_{ψ} and B_{ψ,χ}^{0} are CM abelian varieties over F implies that the spaces Ω^{1}(B_{ψ}/F) and Ω^{1}(B_{ψ,χ}^{0}/F) of regular differentials over F are free of rank one over T_{ψ} ⊗_{O_{K}} F and T_{ψ,χ} ⊗_{O_{K}} F respectively. In particular, the source and target in (2.5) are both one-dimensional over F. The space Ω^{1}(B_{ψ,χ}^{0}/F) is canonically identified with Hom_{T_{ψ}}(T_{ψ,χ}, Ω^{1}(B_{ψ}/F)), and under this identification, the pullback

i^{∗} : Ω^{1}(B_{ψ,χ}^{0}/F)⟶Ω^{1}(B_{ψ}/F)

corresponds to the natural restriction

Hom_{T_{ψ}}(T_{ψ,χ}, Ω^{1}(B_{ψ}/F))⟶Ω^{1}(B_{ψ}/F)

sending the function φ to φ(1). (To see this, consider the map i_{∗} on tangent spaces and dualize.) It follows directly from this description that ker(i^{∗}) ∩ Ω^{1}(B_{ψ,χ}^{0}/F)^{T_{ψ,χ}} = 0, and hence that the restriction of i^{∗} to the one-dimensional F -vector space Ω^{1}(B_{ψ,χ}^{0}/F)^{T_{ψ,χ}} is injective. □

We define ω_{ψ,χ}^{0} ∈ Ω^{1}(B_{ψ,χ}^{0}/ℚ̄)^{T_{ψ,χ}} by

(2.6)  i^{∗}(ω_{ψ,χ}^{0}) = ω_{ψ}, where ω_{ψ} ∈ Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}}.

It follows from Lemma 2.6 that ω_{ψ,χ}^{0} exists and is unique (once ω_{ψ} has been chosen), and that ω_{ψ,χ}^{0} belongs to Ω^{1}(B_{ψ,χ}^{0}/E_{ψ,χ}).

The character χ^{−1} : Gal(H_{χ}/K)⟶T_{χ}^{×} can be viewed as a one-cocycle in

H^{1}(Gal(H_{χ}/K), T_{ψ,χ}^{×}) ⊂ H^{1}(Gal(H_{χ}/K), Aut_{K}(B_{ψ,χ}^{0})).

Let

(2.7)  B_{ψ,χ} := (B_{ψ,χ}^{0})^{χ^{−1}}

denote the twist of B_{ψ,χ}^{0} by this cocycle. There is a natural identification B_{ψ,χ}^{0}(K̄) = B_{ψ,χ}(K̄) of sets, arising from an isomorphism of varieties over H_{χ}, where H_{χ} is the extension of K cut out by χ. The actions of G_{K} on B_{ψ,χ}^{0}(K̄) and B_{ψ,χ}(K̄), denoted ∗_{0} and ∗ respectively, are related by

(2.8)  σ ∗ P = (σ ∗_{0} P) ⊗ χ^{−1}(σ),  for all σ ∈ G_{K}.

In particular, for any L ⊃ K, we have:

(2.9)  B_{ψ,χ}(L) = {P ∈ B_{ψ}(ℚ̄) ⊗_{T_{ψ}} T_{ψ,χ} such that σP = P ⊗ χ(σ), ∀σ ∈ Gal(ℚ̄/L)} .

Likewise, the natural actions of G_{K} on Ω^{1}(B_{ψ,χ}^{0}/K̄) and on Ω^{1}(B_{ψ,χ}/K̄) are related by

(2.10)  σ ∗ ω = [χ^{−1}(σ)]^{∗}(σ ∗_{0} ω)  for all σ ∈ G_{K}.

The isomorphism of B_{ψ,χ}^{0} and B_{ψ,χ} as CM abelian varieties over H_{χ} gives natural identifications

Ω^{1}(B_{ψ,χ}^{0}/H_{χ}) = Ω^{1}(B_{ψ,χ}/H_{χ}), Ω^{1}(B_{ψ,χ}^{0}/E′_{ψ,χ})^{T_{ψ,χ}} = Ω^{1}(B_{ψ,χ}/E′_{ψ,χ})^{T_{ψ,χ}},

where E′_{ψ,χ} denotes the subfield of ℚ̄ generated by H_{χ} and E_{ψ,χ}. Let ω_{ψ,χ}^{0} and ω_{ψ,χ} be E_{ψ,χ} vector space generators of Ω^{1}(B_{ψ,χ}^{0}/E_{ψ,χ})^{T_{ψ,χ}} and Ω^{1}(B_{ψ,χ}/E_{ψ,χ})^{T_{ψ,χ}} respectively, the former being chosen to satisfy (2.6) above. Since they both generate Ω^{1}(B_{ψ,χ}/E′_{ψ,χ})^{T_{ψ,χ}} as an E′_{ψ,χ}-vector space, they necessarily differ by a non-zero scalar in E′_{ψ,χ}.

To spell out the relation between ω_{ψ,χ}^{0} and ω_{ψ,χ} more precisely, it will be useful to introduce the notion of a generalised Gauss sum attached to any finite order character χ of G_{K}. Given such a character, let

E{χ} := {λ ∈ E_{χ}H_{χ} such that λ^{σ} = χ(σ)λ, ∀σ ∈ Gal(E_{χ}H_{χ}/E_{χ})}.

This set is a one-dimensional E_{χ}-vector space in a natural way. It is not closed under multiplication, but

(2.11)  E{χ_{1}} · E{χ_{2}} = E{χ_{1}χ_{2}} (mod (E_{χ_{1}}E_{χ_{2}})^{×}).

Definition 2.7. An E_{χ}-vector space generator of E{χ} is called a Gauss sum attached to the character χ, and is denoted 𝔤(χ).

By definition, the Gauss sum 𝔤(χ) belongs to E{χ} ∩ (E_{χ}H_{χ})^{×}, but is only well-defined up to multiplication by E_{χ}^{×}. It follows from (2.11) that

(2.12)  𝔤(χ_{1}χ_{2}) = 𝔤(χ_{1})𝔤(χ_{2}) (mod (E_{χ_{1}}E_{χ_{2}})^{×}),  𝔤(χ^{−1}) = 𝔤(χ)^{−1} (mod E_{χ}^{×}).

The following lemma pins down the relationship between the differentials ω_{ψ,χ}^{0} and ω_{ψ,χ}.

Lemma 2.8. For all Hecke characters ψ and χ as above,

ω_{ψ,χ} = 𝔤(χ)ω_{ψ,χ}^{0} (mod E_{ψ,χ}^{×}).

Proof. Let λ ∈ (H_{χ}E_{ψ,χ})^{×} be the scalar satisfying

(2.13)  ω_{ψ,χ} = λω_{ψ,χ}^{0}.

Since ω_{ψ,χ} is an E_{ψ,χ}-rational differential on B_{ψ,χ}, for all σ ∈ Gal(K̄/E_{ψ,χ}) we have

(2.14)  ω_{ψ,χ} = σ ∗ ω_{ψ,χ} = [χ^{−1}(σ)]^{∗}σ ∗_{0} ω_{ψ,χ} = χ^{−1}(σ)λ^{σ}ω_{ψ,χ}^{0},

where the second equality follows from equation (2.10) and the last from the fact that the differential ω_{ψ,χ}^{0} belongs to Ω^{1}(B_{ψ,χ}^{0}/E_{ψ,χ})^{T_{ψ,χ}}. Comparing (2.13) and (2.14) gives λ^{σ} = χ(σ)λ, and hence λ = 𝔤(χ) (mod E_{ψ,χ}^{×}). □

The following lemma relates the abelian varieties B_{ψ,χ} and B_{ν}, where ν = ψχ^{−1}.

Lemma 2.9. There is an isogeny defined over K:

i_{ν} : B_{ψ,χ}⟶B_{ν} ⊗_{T_{ν}} T_{ψ,χ}

which is compatible with the action of T_{ψ,χ} by endomorphisms on both sides.

Proof. The pair (B_{ψ,χ}^{0}, E_{ψ,χ}) is a CM abelian variety having ψ (viewed as taking values in E_{ψ,χ}) as its associated Hecke character. The Hecke character attached to the Galois twist B_{ψ,χ} is therefore ψχ^{−1} = ν. The second part of Theorem 2.5 implies that B_{ψ,χ} and B_{ν} ⊗_{T_{ν}} T_{ψ,χ} are isogenous over K as CM abelian varieties. □

2.3. Complex periods and special values of L-functions. This section recalls certain periods attached to the quadratic imaginary field K and to Hecke characters of this field. We begin by fixing:

(1) An elliptic curve A with complex multiplication by O_{K}, defined over a finite extension F of K. (Note that F necessarily contains the Hilbert class field of K.)

(2) A regular differential ω_{A} ∈ Ω^{1}(A/F).

(3) A non-zero element γ of H_{1}(A(ℂ), ℚ).

The complex period attached to this data is defined by

(2.15)  Ω(A) := \frac{1}{2πi} ∫_{γ} ω_{A}  (mod F^{×}).

Note that Ω(A) depends on the pair (ω, γ). A different choice of ω or γ has the effect of multiplying Ω(A) by a scalar in F^{×}, and therefore Ω(A) can be viewed as a well-defined element of ℂ^{×}/F^{×}.

For any Hecke character ψ of K, recall that ψ^{∗} is the Hecke character defined as in the Introduction by ψ^{∗}(x) = ψ(x̄). Suppose that ψ is of infinity type (1, 0), and as before let E_{ψ} ⊂ ℚ̄ ⊂ ℂ denote the field generated by the values of ψ (or, equivalently, ψ^{∗}). Choose (arbitrary) non-zero elements

ω_{ψ} ∈ Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}}, γ ∈ H_{1}(B_{ψ}(ℂ), ℚ),

where B_{ψ} is the CM abelian variety attached to ψ by Theorem 2.5, and Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}} is defined in equation (1.10) of the Introduction. The period Ω(ψ^{∗}) attached to ψ^{∗} is defined by setting

Ω(ψ^{∗}) = \frac{1}{2πi} ∫_{γ} ω_{ψ} (mod E_{ψ}^{×}).

Note that the complex number Ω(ψ^{∗}) does not depend, up to multiplication by E_{ψ}^{×}, on the choices of ω_{ψ} and γ that were made to define it.

Lemma 2.10. If ψ is a Hecke character of infinity type (1, 0), and χ is a finite order character, then

(2.16)  Ω(ψ^{∗}χ) = Ω(ψ^{∗})𝔤(χ^{∗})^{−1} (mod E_{ψ,χ}^{×}).

Proof. Choose a non-zero generator γ of H_{1}(B_{ψ,χ}^{0}(ℂ), ℚ) = H_{1}(B_{ψ,χ}(ℂ), ℚ) (viewed as a one-dimensional E_{ψ,χ} vector space via the endomorphism action). By definition,

Ω((ψχ^{−1})^{∗}) = ∫_{γ} ω_{ψ,χ} = 𝔤(χ) ∫_{γ} ω_{ψ,χ}^{0} = 𝔤(χ)Ω(ψ^{∗}) (mod E_{ψ,χ}^{×}),

where the second equality follows from Lemma 2.8. The result now follows after substituting χ^{∗−1} for χ. □

As in [Scha] §1.8, one can also attach a period Ω(ψ) to an arbitrary Hecke character ψ of K; these satisfy the following relations:

Proposition 2.11. Let ψ be a Hecke character of infinity type (k, j). Then (1) The ratio

\frac{Ω(ψ^{∗})}{(2πi)^{j}Ω(A)^{k−j}}

is algebraic.

(2) For all ψ and ψ′,

Ω(ψψ′) = Ω(ψ)Ω(ψ′) (mod E_{ψ,ψ′}^{×}),

where E_{ψ,ψ′} is the subfield of ℚ̄ generated by E_{ψ} and E_{ψ′}.

The following theorem is due to Goldstein and Schappacher [GS] in certain cases and Blasius [Bl] in the general case (even CM fields).

Theorem 2.12. Suppose that ψ has infinity type (k, j) with k > j, and that m is a critical integer for L(ψ^{−1}, s). Then

\frac{L(ψ^{−1}, m)}{(2πi)^{m}Ω(ψ^{∗})} belongs to E_{ψ},

and for all τ ∈ Gal(E_{ψ}/K),

(\frac{L(ψ^{−1}, m)}{(2πi)^{m}Ω(ψ^{∗})})^{τ} = \frac{L((ψ^{−1})^{τ} , m)}{(2πi)^{m}Ω((ψ^{∗})^{τ})}.

2.4. p-adic periods. Fix a prime p that splits in K. We will need p-adic analogs of the periods Ω(A) and Ω(ν^{∗}). The p-adic analogue Ω_{p}(A) of Ω(A) is obtained by considering the base change A_{ℂ_{p}} of A to ℂ_{p} (via our fixed embedding of F into ℂ_{p}). Assume that A has good reduction at the maximal ideal of O_{ℂ_{p}}, i.e., that A_{ℂ_{p}} extends to a smooth proper model A_{O_{ℂp}} over O_{ℂ_{p}}. The p-adic completion Â_{O_{ℂp}} of A along its special fiber is isomorphic to Ĝ_{m}. Following [deS] II, §4.4, choose an isomorphism ι_{p} : Â⟶Ĝ_{m} over O_{ℂ_{p}}, and define Ω_{p}(A) ∈ ℂ_{p}^{×} by the rule

(2.17)  ω_{A} = Ω_{p}(A) · ι_{p}^{∗}(du/u),

where u is the standard coordinate on Ĝ_{m}. The invariant Ω_{p}(A) ∈ ℂ_{p}^{×} thus defined depends on the choices of ω_{A} and ι_{p}, but only up to multiplication by a scalar in F^{×}. Observe also that Ω(A) and Ω_{p}(A) each depend linearly in the same way on the choice of the global differential ω_{A}.

The p-adic period Ω_{p}(A) can be used to define p-adic analogs of the complex periods that appear in the statement of Theorem 2.12.

Definition 2.13. Let ν be a Hecke character of K of type (1, 0). The p-adic period Ω_{p}(ν^{∗}) is defined by

Ω_{p}(ν^{∗}) := Ω_{p}(A) · \frac{Ω(ν^{∗})}{Ω(A)}.

More generally, for any character ν of infinity type (k, j), we define

Ω_{p}(ν^{∗}) := Ω_{p}(A)^{k−j} · \frac{Ω(ν^{∗})}{(2πi)^{j}Ω(A)^{k−j}} .

It can be seen from this definition that the period Ω_{p}(ν^{∗}), like its complex counterpart Ω(ν^{∗}), is welldefined up to multiplication by a scalar in E_{ν}^{×}. The following p-adic analog of Lemma 2.10 is a direct consequence of this lemma combined with the definition of Ω_{p}(ψ):

Lemma 2.14. If ψ is a Hecke character of infinity type (1, 0), and χ is a finite order character, then

(2.18)  Ω_{p}(ψ^{∗}χ) = Ω_{p}(ψ^{∗})𝔤(χ^{∗})^{−1} (mod E_{ψ,χ}^{×}).

Likewise, Proposition 2.11 implies:

Proposition 2.15. Let ψ be a Hecke character of infinity type (k, j). Then (1) The ratio

\frac{Ω_{p}(ψ^{∗})}{(2πi)^{j}Ω_{p}(A)^{k−j}}

is algebraic.

(2) For all ψ and ψ′,

(2.19)  Ω_{p}(ψψ′) = Ω_{p}(ψ)Ω_{p}(ψ′) (mod E_{ψ,ψ′}^{×}).

3. p-adic L-functions and Rubin’s formula

3.1. The Katz p-adic L-function. Throughout this chapter, we will fix a prime p that is split in K. Let 𝔠 be an integral ideal of K which is prime to p, and let Σ(𝔠) denote the set of all Hecke characters of K of conductor dividing 𝔠. Denote by 𝔭 the prime above p corresponding to the chosen embedding K ↪ ℚ̄_{p}.

A character ν ∈ Σ(𝔠) is called a critical character if L(ν^{−1}, 0) is a critical value in the sense of Deligne, i.e., if the Γ-factors that arise in the functional equation for L(ν^{−1}, s) are non-vanishing and have no poles at s = 0. The set Σ_{crit}(𝔠) of critical characters can be expressed as the disjoint union

Σ_{crit}(𝔠) = Σ_{crit}^{(1)}(𝔠) ∪ Σ_{crit}^{(2)}(𝔠),

where

Σ_{crit}^{(1)}(𝔠) = {ν ∈ Σ(𝔠) of type (ℓ_{1}, ℓ_{2}) with ℓ_{1} ≤ 0, ℓ_{2} ≥ 1} ,
Σ_{crit}^{(2)}(𝔠) = {ν ∈ Σ(𝔠) of type (ℓ_{1}, ℓ_{2}) with ℓ_{1} ≥ 1, ℓ_{2} ≤ 0} .

The possible infinity types of Hecke characters in these two critical regions are sketched in Figure 1. Note in particular that when 𝔠 = 𝔠̄, the regions Σ_{crit}^{(1)}(𝔠) and Σ_{crit}^{(2)}(𝔠) are interchanged by the involution ν ↦ ν^{∗}. The set Σ_{crit}(𝔠) is endowed with a natural p-adic topology as described in Section 5.2 of [BDP-gz]. The subsets Σ_{crit}^{(1)}(𝔠) and Σ_{crit}^{(2)}(𝔠) are dense in the completion Σ̂_{crit}(𝔠) relative to this topology.

Recall that 𝔭 is the prime above p induced by our chosen embedding of K into ℂ_{p}. The following theorem on the existence of the p-adic L-function is due to Katz. The statement below is a restatement of [deS] (II, Thm. 4.14) with a minor correction, and restricted to characters unramified at p. We remark that since our characters are unramified at p, the Gauss sum in the interpolation formula in loc. cit. is equal to 1.

Theorem 3.1. There exists a p-adic analytic function ν ↦ ℒ_{p,𝔠}(ν) (valued in ℂ_{p}) on Σ̂_{crit}(𝔠) which is determined by the interpolation property:

(3.1)  \frac{ℒ_{p,𝔠}(ν)}{Ω_{p}(A)^{ℓ_{1}−ℓ_{2}}} = (\frac{\sqrt{D}}{2π} )^{ℓ_{2}} (ℓ_{1} − 1)!(1 − ν(𝔭)/p)(1 − ν^{−1}(𝔭̄))\frac{L_{𝔠}(ν^{−1}, 0)}{Ω(A)^{ℓ_{1}−ℓ_{2}}},

for all critical characters ν ∈ Σ_{crit}^{(2)}(𝔠) of infinity type (ℓ_{1}, ℓ_{2}).

The right hand side of (3.1) belongs to ℚ̄, by Part 1 of Proposition 2.11 and Theorem 2.12 with m = 0. Equation (3.1) should be interpreted to mean that the left hand side also belongs to ℚ̄, viewed as a subfield of ℂ_{p} under the chosen embeddings, and agrees with the right hand side. Note that although both sides of (3.1) depend on the choice of the differential ω_{A} that was made in the definition of the periods Ω(A) and Ω_{p}(A), the quantity ℒ_{p,𝔠}(ν), just like its complex counterpart L_{𝔠}(ν^{−1}, 0), does not depend on this choice.

Figure 1. Critical infinity types for the Katz p-adic L-function

Remark 3.2. Once a choice of 𝔠 is fixed, we shall often drop the subscript 𝔠 and simply write ℒ_{p} for the p-adic L-function.

The following corollary is the p-adic counterpart of Thm. 2.12.

Corollary 3.3. Suppose that ν ∈ Σ_{crit}^{(2)}(𝔠). Then

\frac{ℒ_{p,𝔠}(ν)}{Ω_{p}(ν^{∗})} belongs to E_{ν}.

Proof. Suppose that ν has infinity type (ℓ_{1}, ℓ_{2}). By the definition of Ω_{p}(ν^{∗}) and the the interpolation property of the Katz p-adic L-function in Thm. 3.1, we have

\frac{ℒ_{p,𝔠}(ν)}{Ω_{p}(ν^{∗})} = \frac{ℒ_{p,𝔠}(ν)}{Ω_{p}(A)^{ℓ_{1}−ℓ_{2}}} × \frac{(2πi)^{ℓ_{2}}Ω(A)^{ℓ_{1}−ℓ_{2}}}{Ω(ν^{∗})}
= \sqrt{−D}^{ℓ_{2}} · (ℓ_{1} − 1)!(1 − ν(𝔭))(1 − ν^{−1}(𝔭̄))\frac{L_{𝔠}(ν^{−1}, 0)}{Ω(ν^{∗})}.

The result is now a direct consequence of Theorem 2.12 with m = 0. □

Cor. 3.3 expresses ℒ_{p,𝔠}(ν) as an E_{ν}-multiple of a p-adic period Ω_{p}(ν^{∗}), when ν lies in the range Σ_{crit}^{(2)}(𝔠) of classical interpolation for the Katz p-adic L-function. On the other hand, the characters in Σ_{crit}^{(1)}(𝔠) are outside the range of interpolation, and so Cor. 3.3 does not directly say anything about these values, and indeed the main goal of this paper is to obtain analogous results for certain characters in Σ_{crit}^{(1)}(𝔠). It turns out that the methods of this paper only allow us to study ℒ_{p,𝔠}(ν) for characters ν in Σ_{crit}^{(2)}(𝔠) satisfying the following auxiliary (but not unnatural) condition:

(3.2)  ν is a self-dual Hecke character with ε_{ν} = ε_{K}.

For the benefit of the reader, we now recall this key definition.

Definition 3.4. A Hecke character ν ∈ Σ_{crit}(𝔠) is said to be self-dual or anticyclotomic if

νν^{∗} = N_{K}.

The reason for the terminology in Definition 3.4 is that the functional equation for the L-series L(ν^{−1}, s) relates L(ν^{−1}, s) to L(ν^{−1}, −s), and therefore s = 0 is the central critical point for this complex L-series. Note that a self-dual character is necessarily of infinity type (1+j, −j) for some j ∈ ℤ. Also the conductor of a self-dual character is clearly invariant under complex conjugation. If 𝔠 is an integral ideal such that 𝔠 = 𝔠̄, we denote by Σ_{sd}(𝔠) the set of self-dual Hecke characters of conductor exactly 𝔠, and write

Σ_{sd}^{(1)}(𝔠) = Σ_{crit}^{(1)}(𝔠) ∩ Σ_{sd}(𝔠), Σ_{sd}^{(2)}(𝔠) = Σ_{crit}^{(2)}(𝔠) ∩ Σ_{sd}(𝔠).

In particular, the possible infinity types of characters in Σ_{sd}^{(2)}(𝔠) correspond to the black dots in Figure 1.

For convenience, we restate Thm. 3.1 for self-dual characters.

Proposition 3.5. For all characters ν ∈ Σ_{sd}^{(2)}(𝔠) of infinity type (1 + j, −j) with j ≥ 0,

(3.3)  \frac{ℒ_{p,𝔠}(ν)}{Ω_{p}(A)^{1+2j}} = (1 − ν^{−1}(𝔭̄))^{2} × \frac{j!(2π)^{j}L_{𝔠}(ν^{−1}, 0)}{\sqrt{D}^{j}Ω(A)^{1+2j}}.

Remark 3.6. In the proposition above, we could equally write L(ν^{−1}, 0) instead of L_{𝔠}(ν^{−1}, 0) since ν has conductor exactly equal to c.

Remark 3.7. The central character of such a ν is very restricted. Indeed, for any Hecke character ν it is clear that εν̄ = \overline{ε_{ν}}, while ε_{ν^{∗}} = ε_{ν}. If further ν is a self-dual character, it follows that for any x ∈ 𝔸_{K}^{×},

ν(N_{ℚ}^{K}(x)) = ν(xx̄) = (νν^{∗})(x) = N_{K}(x) = N(N_{ℚ}^{K}(x)).

Hence

ν|_{N_{ℚ}^{K}𝔸_{K}^{×}} = N  and  ε_{ν}|_{N_{ℚ}^{K}𝔸_{K}^{×}} = 1.

This implies that the central character ε_{ν} of a self-dual character ν is either 1 or ε_{K}, where ε_{K} denotes the quadratic Dirichlet character corresponding to the extension K/ℚ. Conversely, it is easy to see that if ν is a Hecke character with w(ν) = 1 and ε_{ν} = 1 or ε_{K}, then ν is a self-dual character.

We define:

(3.4)  Σ_{sd}(𝔠)^{+} := {ν ∈ Σ_{sd}(𝔠); ε_{ν} = 1}, Σ_{sd}(𝔠)^{−} := {ν ∈ Σ_{sd}(𝔠); ε_{ν} = ε_{K}}.

The sets Σ_{sd}^{(1)}(𝔠)^{±} and Σ_{sd}^{(2)}(𝔠)^{±} are defined similarly.

Our approach to studying ℒ_{p,𝔠}(ν) for characters ν satisfying (3.2), i.e., those ν lying in Σ_{sd}^{(1)}(𝔠)^{−} for some 𝔠, relies on a different kind of p-adic L-function. This latter p-adic L-function is attached to Rankin-Selberg L-series and is recalled in the following section.

3.2. p-adic Rankin L-series. In this section, we consider p-adic L-functions obtained by interpolating special values of Rankin-Selberg L-series associated to modular forms and Hecke characters of a quadratic imaginary field K of odd discriminant. We briefly recall the definition of this p-adic L-function that is given in Sec. 5 of [BDP-gz], referring the reader to loc.cit. for a more detailed description.

Let S_{k}(Γ_{0}(N), ε) denote the space of cusp forms of weight k ≥ 2 and character ε on Γ_{0}(N). Let f ∈ S_{k}(Γ_{0}(N), ε) be a normalized newform and let E_{f} denote the subfield of ℂ generated by its Fourier coefficients.

Definition 3.8. The pair (f, K) is said to satisfy the Heegner hypothesis if O_{K} contains a cyclic ideal of norm N, i.e., an integral ideal 𝔑 of O_{K} with O_{K}/𝔑 = ℤ/Nℤ.

Assume from now on that (f, K) satisfies the Heegner hypothesis, and let 𝔑 be a cyclic O_{K}-ideal of norm N. We write 𝔑_{ε} for the unique ideal dividing 𝔑 of norm N_{ε}.

Definition 3.9. A Hecke character χ of K of infinity type (ℓ_{1}, ℓ_{2}) is said to be central critical for f if

ℓ_{1} + ℓ_{2} = k  and  ε_{χ} = ε.

The reason for the terminology of Definition 3.9 is that when χ satisfies these hypotheses, the complex Rankin L-series L(f, χ^{−1}, s) is self-dual and s = 0 is its central (critical) point.

Definition 3.10. Let c be a rational integer prime to pN. Then Σ_{cc}(c, 𝔑, ε) is defined to be the set of Hecke characters χ of K such that

(1) χ is central critical for f.

(2) 𝔣_{χ} = c · 𝔑_{ε}.

(3) The local sign ε_{q}(f, χ^{−1}) = +1 for all finite primes q.

It is easily checked that this agrees with the definition of Σ_{cc}(c, 𝔑, ε) given in [BDP-gz] §4.1 where this is just denoted Σ_{cc}(𝔑). Further, as in loc. cit., given conditions (1) and (2) above, condition (3) is automatic except possibly for primes q lying in the set S_{f} defined by:

S_{f} := {q : q | (N, D), q ∤ N_{ε}}.

The set Σ_{cc}(c, 𝔑, ε) can be expressed as a disjoint union

Σ_{cc}(c, 𝔑, ε) = Σ_{cc}^{(1)}(c, 𝔑, ε) ∪ Σ_{cc}^{(2)}(c, 𝔑, ε),

where Σ_{cc}^{(1)}(c, 𝔑, ε) and Σ_{cc}^{(2)}(c, 𝔑, ε) denote the subsets consisting of characters of infinity type (k + j, −j) with 1 − k ≤ j ≤ −1 and j ≥ 0 respectively. We shall also denote by Σ̂_{cc}(c, 𝔑, ε) the completion of Σ_{cc}(c, 𝔑, ε) relative to the p-adic compact open topology on Σ_{cc}(c, 𝔑, ε) which is defined in Section 5.2 of [BDP-gz]. The infinity types of Hecke characters in Σ_{cc}^{(1)}(c, 𝔑, ε) and Σ_{cc}^{(2)}(c, 𝔑, ε) correspond respectively to the white and black dots in the shaded regions in Figure 2. We note that the set Σ_{cc}^{(2)}(c, 𝔑, ε) of classical central critical characters “of type 2” is dense in Σ̂_{cc}(c, 𝔑, ε).

For all χ ∈ Σ_{cc}^{(2)}(c, 𝔑, ε) of infinity type (k+j, −j) with j ≥ 0, let E_{f,χ} denote the subfield of ℂ generated by E_{f} and the values of χ, and let E_{f,χ,ε} be the field generated by E_{f,χ} and by the abelian extension of ℚ cut out by ε. The algebraic part of L(f, χ^{−1}, 0) is defined by the rule

(3.5)  L_{alg}(f, χ^{−1}, 0) := w(f, χ)^{−1}C(f, χ, c) · \frac{L(f, χ^{−1}, 0)}{Ω(A)^{2(k+2j)}} ,

where w(f, χ)^{−1} ∈ E_{f,χ,ε} and C(f, χ, c) are respectively the scalar (of complex norm 1) and the explicit real constant defined in equation (5.1.11) and Theorem 4.6 of [BDP-gz]; we have

(3.6)  C(f, χ, c) = \frac{2^{k+2j−2}π^{k+2j−1}j!(k + j − 1)!w_{K}}{\sqrt{D}^{k+2j−1}c^{k+2j−1}} ∏_{q|c} \frac{q − ε_{K}(q)}{q − 1},

where w_{K} = #O_{K}^{×} is the number of roots of unity in K. Thm. 5.5 and Thm. 5.10 of loc. cit. show respectively that the values L_{alg}(f, χ^{−1}, 0) belongs to \overline{ℚ}, and that they interpolate p-adically:

Proposition 3.11. Let χ ↦ L_{p}(f, χ) be the function on Σ_{cc}^{(2)}(c, 𝔑, ε) defined by

(3.7)  L_{p}(f, χ) := Ω_{p}(A)^{2(k+2j)}(1 − χ^{−1}(𝔭̄)a_{p}(f) + χ^{−2}(𝔭̄)ε(p)p^{k−1})^{2}L_{alg}(f, χ^{−1}, 0),

for χ of infinity type (k + j, −j) with j ≥ 0. This function extends (uniquely) to a p-adically continuous function on Σ̂_{cc}(c, 𝔑, ε).

The function χ ↦ L_{p}(f, χ) on Σ̂_{cc}(c, 𝔑, ε) will be referred to as the p-adic Rankin L-function attached to the cusp form f.

3.3. A p-adic Gross-Zagier formula. In this section, we specialise to the case where the newform f is of weight k = 2, and assume that χ is a finite order Hecke character of K satisfying

χN_{K} belongs to Σ_{cc}^{(1)}(c, 𝔑, ε).

In particular, the character χN_{K} lies outside the domain Σ_{cc}^{(2)}(c, 𝔑, ε) of classical interpolation defining L_{p}(f, −). The p-adic Gross-Zagier formula alluded to in the title of this section relates the special value L_{p}(f, χN_{K}) to the formal group logarithm of a Heegner point on the modular abelian variety attached to f.

Figure 2. Critical infinity types for the p-adic Rankin L-function

The Eichler-Shimura construction associates to f an abelian variety B_{f} with endomorphism by an order in the ring of integers T_{f} ⊂ E_{f}, and a surjective morphism

Φ_{f} : J_{1}(N)⟶B_{f}

of abelian varieties over ℚ, called the modular parametrisation, which is well-defined up to a rational isogeny. Let

ω_{f} = 2πif(τ)dτ ∈ Ω^{1}(X_{1}(N)/E_{f})

be the differential form on X_{1}(N) attached to f; we use the same symbol ω_{f} to denote the associated one-form on J_{1}(N). Let ω_{B_{f}} ∈ Ω^{1}(B_{f}/E_{f})^{T_{f}} be the unique one-form satisfying

(3.8)  Φ_{f}^{∗}(ω_{B_{f}}) = ω_{f}.

Let A′ be an elliptic curve with endomorphism ring isomorphic to the order O_{c} = ℤ+cO_{K} of conductor c, defined over the ring class field H_{c} of conductor c. The pair (A′, A′[𝔑]) corresponds to a point on X_{0}(N)(H_{c}). Let t be any generator of A′[𝔑]. Then the triple (A′, A′[𝔑], t) corresponds to a point in X_{1}(N), whose field of definition H_{c,𝔑} is an abelian extension of K, independent of the choice of t, and the finite order Hecke character χ can be viewed as a character

χ : Gal(H_{c,𝔑}/K)⟶E_{χ}.

Fix a cusp ∞ of X_{1}(N) which is defined over ℚ, and let

(3.9)  Δ = [A′, A′[𝔑], t] − (∞) ∈ J_{1}(N)(H_{c,𝔑}).

To the pair (f, χ) we associate a Heegner point by letting G = Gal(H_{c,𝔑}/K) and setting

(3.10)  P_{f}(χ) := ∑_{σ∈G} χ^{−1}(σ)Φ_{f}(Δ^{σ}) ∈ B_{f}(H_{c,𝔑}) ⊗_{T_{f}} E_{f,χ}.

Note that, since P_{f}(χ)^{σ} = P_{f}(χ) for any σ ∈ Gal(H_{c,𝔑}/H_{χ}), the point P_{f}(χ) lies in the subspace B_{f}(H_{χ}) ⊗_{T_{f}} E_{f,χ}. The embedding of ℚ̄ into ℂ_{p} that was fixed from the outset allows us to consider the formal group logarithm

log_{ω_{B_{f}}} : B_{f}(H_{c,𝔑})⟶ℂ_{p}.

We extend this function to B_{f}(H_{c,𝔑}) ⊗_{T_{f}} E_{f,χ} by E_{f,χ}-linearity.

Theorem 3.12. With notations and assumptions as above,

L_{p}(f, χN_{K}) = (1 − χ^{−1}(𝔭̄)p^{−1}a_{p}(f) + χ^{−2}(𝔭̄)ε(p)p^{−1})^{2} log_{ω_{B_{f}}}^{2} (P_{f}(χ)).

Proof. Let

ℰ (f, χ) := (1 − χ^{−1}(𝔭̄)p^{−1}a_{p}(f) + χ^{−2}(𝔭̄)ε(p)p^{−1})^{2} ∈ E_{f,χ}^{×}

be the Euler factor appearing in the statement of Theorem 3.12. Let F′ denote the p-adic completion of H_{c,𝔑}. Theorem 5.13 of [BDP-gz] in the case k = 2 and r = j = 0, with χ replaced by χN_{K}, gives

(3.11)  L_{p}(f, χN_{K}) = ℰ (f, χ) × (∑_{σ∈G} χ^{−1}(σ) · AJ_{F′}(Δ^{σ})(ω_{f}))^{2} .

Note that in this context, the p-adic Abel-Jacobi map AJ_{F′} that appears in (3.11) is related to the formal group logarithm by

AJ_{F′}(Δ)(ω_{f}) = log_{ω_{f}} (Δ).

Therefore,

(3.12)  L_{p}(f, χN_{K}) = ℰ (f, χ) (∑_{σ∈G} χ^{−1}(σ) log_{ω_{f}} (Δ^{σ}))^{2} .

Theorem 3.12 follows from this formula and the fact that, by (3.8),

log_{ω_{f}} (Δ) = log_{Φ_{f}^{∗}(ω_{B_{f}})}(Δ) = log_{ω_{B_{f}}} (Φ_{f}(Δ)).
□

In the special case where f has rational Fourier coefficients and χ = 1 is the trivial character, the abelian variety B_{f} is an elliptic curve quotient of J_{0}(N) and the Heegner point P_{f} := P_{f}(1) belongs to B_{f}(K). Theorem 3.12 implies in this case that

(3.13)  L_{p}(f, N_{K}) = (\frac{p + 1 − a_{p}(f)}{p})^{2} log^{2}(P_{f}),

where log : B_{f}(K_{𝔭})⟶K_{𝔭} is the formal group logarithm attached to a rational differential on B_{f}/ℚ. Equation (3.13) exhibits a strong analogy with Theorem 1 of the Introduction, although it applies to arbitrary (modular) elliptic curves and not just elliptic curves with complex multiplication.

The remainder of Chapter 3 explains how Theorem 3.12 can in fact be used to prove Theorems 1 and 2 of the Introduction. The key to this proof is a relation between the Katz p-adic L-function of Section 3.1 and the p-adic Rankin L-function L_{p}(f, χ) of Section 3.2 in the special case where f is a theta series attached to a Hecke character of the imaginary quadratic field K. This explicit relation is described in the following section.

3.4. A factorisation of the p-adic Rankin L-series. This section focuses on the Rankin L-function L_{p}(f, χ) of f and K in the special case where f is a theta series associated to a Hecke character of the same imaginary quadratic field K.

More precisely, let ψ be a fixed Hecke character of K of infinity type (k − 1, 0) with k = r + 2 ≥ 2. Consider the associated theta series:

θ_{ψ} := ∑_{𝔞} ψ(𝔞)q^{N𝔞} = ∑_{n=1}^{∞} a_{n}(θ_{ψ})q^{n},

where the first sum is taken over integral ideals of K. The Fourier coefficients of θ_{ψ} generate a number field E_{θ_{ψ}} which is clearly contained in E_{ψ}.

The following classical proposition is due to Hecke and Schoenberg. (Cf. [Ogg] or Sec. 3.2 of [Za]).

Proposition 3.13. The theta series θ_{ψ} belongs to S_{k}(Γ_{0}(N), ε), where

(1) The level N is equal to DM, with M = N_{ℚ}^{K}𝔣_{ψ},

(2) The Nebentypus character ε is equal to ε_{K}ε_{ψ}.

Lemma 3.14. If the conductor 𝔣_{ψ} of ψ is a cyclic ideal 𝔪 of norm M prime to D, then f := θ_{ψ} satisfies the Heegner hypothesis relative to K.

Proof. In this case, the modular form θ_{ψ} is of level N = DM, by Proposition 3.13. But then the ideal

(3.14)  𝔑 := 𝔡_{K}𝔪,

with 𝔡_{K} := (\sqrt{−D_{K}}) is a cyclic ideal of K of norm N. □

We will assume from now on that the condition in Lemma 3.14 is satisfied. Furthermore, we will always take 𝔑 to be the ideal in (3.14).

The goal of this section is to factor the p-adic Rankin L-function L_{p}(θ_{ψ}, ·) as a product of two Katz p-adic L-functions. As a preparation to stating the main result we record the following lemma:

Lemma 3.15. Let c be an integer prime to pN and let χ be any character in Σ_{cc}(c, 𝔑, ε).

(1) If χ belongs to Σ_{cc}^{(2)}(c, 𝔑, ε), then ψ^{−1}χ belongs to Σ_{sd}^{(2)}(c𝔡_{K})^{−} and ψ^{∗−1}χ belongs to Σ_{sd}^{(2)}(c𝔡_{K}M)^{−}.

(2) If χ belongs to Σ_{cc}^{(1)}(c, 𝔑, ε), then ψ^{−1}χ belongs to Σ_{sd}^{(1)}(c𝔡_{K})^{−} and ψ^{∗−1}χ belongs to Σ_{sd}^{(2)}(c𝔡_{K}M)^{−}.

Proof. We first note that when χ is of type (k + j, −j) then ψ^{−1}χ is of infinity type (1 + j, −j) and ψ^{∗−1}χ is of infinity type (k + j, 1 − (k + j)). Since χ ∈ Σ_{cc}(c, 𝔑, ε), we have

ε_{χ} = ε = ε_{ψ} · ε_{K}.

Thus ε_{ψ^{−1}χ} equals ε_{K} and the same holds for ε_{ψ^{∗}^{−1}χ} since ε_{ψ^{∗}} = ε_{ψ}. It follows then from Remark 3.7 that ψ^{−1}χ and ψ^{∗}^{−1}χ are self-dual characters.

Let q be a rational prime dividing M. Since 𝔪 is a cyclic O_{K}-ideal, it follows that q = 𝔮𝔮̄ must be split in K, and exactly one of 𝔮,𝔮̄ divides 𝔪. From this it is easy to see that ε_{ψ} has conductor exactly M, hence ε has conductor exactly N and 𝔑_{ε} = 𝔑. Thus 𝔣_{χ} = c𝔑 = c𝔡_{K}𝔪 and 𝔣_{ψ^{∗}^{−1}χ} = c𝔡_{K}𝔪𝔪̄ = c𝔡_{K}M. On the other hand, since ε_{χ} = ε_{ψ}ε_{K}, it follows that 𝔣_{ψ^{−1}χ} = c𝔡_{K}.

The preceding remarks imply that if χ is in Σ_{cc}(c, 𝔑, ε), then ψ^{−1}χ lies in Σ_{sd}(c𝔡_{K})^{−} and ψ^{∗}^{−1}χ lies in Σ_{sd}(c𝔡_{K}M)^{−}. To finish we note that if j ≥ 0, then both ψ^{−1}χ and ψ^{∗}^{−1}χ lie in Σ_{sd}^{(2)}, while if −(k − 1) ≤ j ≤ −1, then ψ^{∗}^{−1}χ is in Σ_{sd}^{(2)} while ψ^{−1}χ lies in Σ_{sd}^{(1)}. □

Theorem 3.16. For all χ ∈ Σ_{cc}(c, 𝔑, ε),

(3.15)  L_{p}(θ_{ψ}, χ) = \frac{w(θ_{ψ}, χ)^{−1}w_{K}}{2c^{k+2j−1}} ∏_{q|c} \frac{q − ε_{K}(q)}{q − 1} × ℒ_{p,c𝔡_{K}} (ψ^{−1}χ) × ℒ_{p,c𝔡_{K}M}(ψ^{∗−1}χ).

Proof. Since Σ_{cc}^{(2)}(c, 𝔑, ε) is dense in Σ̂_{cc}^{(2)}(c, 𝔑, ε), it suffices to prove the formula for the characters χ in this range, where it follows directly from the interpolation properties defining the respective p-adic L-functions. More precisely, by (3.7),

(3.16)  \frac{L_{p}(θ_{ψ}, χ)}{Ω_{p}(A)^{2(k+2j)}} = ((1 − ψχ^{−1}(𝔭̄))(1 − ψ^{∗}χ^{−1}(𝔭̄))^{2} L_{alg}(θ_{ψ}, χ^{−1}, 0).

Let δ_{c} := ∏_{q|c} \frac{q−ε_{K}(q)}{q−1}. By the definition of L_{alg}(θ_{ψ}, χ^{−1}, 0) given in (3.5) and (3.6),

L_{alg}(θ_{ψ}, χ^{−1}, 0) = w(θ_{ψ}, χ)^{−1}C(θ_{ψ}, χ, c)\frac{L(θ_{ψ}, χ^{−1}, 0)}{Ω(A)^{2(k+2j)}}
(3.17)  = w(θ_{ψ}, χ)^{−1}w_{K}δ_{c} \frac{2^{r+2j}π^{k+2j−1}j!(k + j − 1)!}{\sqrt{D}^{k+2j−1}c^{k+2j−1}} × \frac{L(ψχ^{−1}, 0)L(ψ^{∗}χ^{−1}, 0)}{Ω(A)^{2(k+2j)}}

= \frac{w(θ_{ψ}, χ)^{−1}w_{K}δ_{c}}{2c^{k+2j−1}} (\frac{j!(2π)^{j}L(ψχ^{−1}, 0)}{\sqrt{D}^{j}Ω(A)^{1+2j}}) × (\frac{(k + j − 1)!(2π)^{k+j−1}L(ψ^{∗}χ^{−1}, 0)}{\sqrt{D}^{k+j−1}Ω(A)^{1+2(k+j−1)}}) .

Combining (3.16) and (3.17) with the interpolation property of the Katz p-adic L-function given in Proposition 3.5, we obtain

(3.18)  \frac{L_{p}(θ_{ψ}, χ)}{Ω_{p}(A)^{2(k+2j)}} = \frac{w(θ_{ψ}, χ)^{−1}w_{K}δ_{c}}{2c^{k+2j−1}} × \frac{ℒ_{p,c𝔡_{K}} (ψ^{−1}χ)}{Ω_{p}(A)^{1+2j}} × \frac{ℒ_{p,c𝔡_{K}M}(ψ^{∗−1}χ)}{Ω_{p}(A)^{1+2(k+j−1)}} .

Clearing the powers of Ω_{p}(A) on both sides gives the desired result. □

The Nebentypus character ε can be viewed as a finite order Galois character of G_{ℚ}. Recall that E_{ψ,χ,ε} denotes the smallest extension of E_{ψ,χ} containing the field through which this character factors.

Corollary 3.17. For all χ ∈ Σ_{cc}(c, 𝔑, ε),

L_{p}(θ_{ψ}, χ) = ℒ_{p,c𝔡_{K}} (ψ^{−1}χ) × ℒ_{p,c𝔡_{K}M}(ψ^{∗−1}χ) (mod E_{ψ,χ,ε}^{×}).

Proof. This follows from Theorem 3.16 in light of the fact that the constant that appears on the right hand side of (3.15) belongs to E_{ψ,χ,ε}^{×}. □

3.5. Proof of Rubin’s Theorem. The goal of this section is to prove Theorem 2 of the Introduction. Let 𝔠 = 𝔠̄ be an integral ideal in O_{K} invariant under complex conjugation and let ν ∈ Σ_{sd}(𝔠)^{−} be a Hecke character of K of infinity type (1, 0). Since ε_{ν} = ε_{K}, it follows that 𝔡_{K}|𝔠. We will also assume that ν satisfies the following additional conditions:

(i) The sign w_{ν} of the functional equation of the L-function L(ν, s) is −1.

(ii) 𝔡_{K}||𝔠. Thus 𝔠 = (c)𝔡_{K} for a unique positive rational integer c that is prime to D. Let p be a rational prime split in K that is prime to c.

Definition 3.18. A pair (ψ, χ) of Hecke characters is said to be good for ν if it satisfies the following conditions.

(1) The character ψ is of type (1, 0) and has conductor 𝔪, where 𝔪 is a cyclic O_{K}-ideal prime to pD. Thus θ_{ψ} is a newform in S_{2}(Γ_{0}(N), ε) where N = MD and ε = ε_{ψ}ε_{K} is a Dirichlet character of conductor exactly N. Let 𝔑 := 𝔪𝔡_{K}.

(2) The character χ is of finite order, and χN_{K} belongs to Σ_{cc}^{(1)}(c, 𝔑, ε). This implies (on account of Lemma 3.15 applied to χN_{K}) that ψ^{−1}χN_{K} lies in Σ_{sd}^{(1)}(𝔠) and ψ^{∗}^{−1}χN_{K} lies in Σ_{sd}^{(2)}(𝔠M).

(3) The character ψχ^{−1} is equal to ν, i.e., ψ^{−1}χN_{K} = ν^{∗}.

(4) The classical L-value L(ψ^{∗}χ^{−1}N_{K}^{−1}, 0) is non-zero, i.e., ℒ_{p,𝔠M}(ψ^{∗−1}χN_{K}) ≠ 0.

Remark 3.19. Suppose that a pair (ψ, χ) satisfies (1) and (3) above with 𝔪 prime to 𝔠. Then such a pair automatically satisfies (2) also. Indeed, the character χN_{K} = ψν^{∗} is of type (1, 1) and its central character is equal to

ε_{χ} = ε_{ψ}ε_{ν^{∗}} = ε_{ψ}ε_{K} = ε,

where ε is the nebentype character attached to θ_{ψ}. Further, 𝔣_{χ} = 𝔣_{ψ}𝔣_{ν^{∗}} = 𝔪 · c𝔡_{K}. It follows that the character χN_{K} belongs to Σ_{cc}(c, 𝔑, ε) with 𝔑 = 𝔡_{K}𝔪. (The set S_{θ_{ψ}} in the discussion below Defn. 3.10 is empty since D | N_{ε}.)

Remark 3.20. Suppose that a pair (ψ, χ) satisfies conditions (1), (2) and (3) above. Since χN_{K} lies in Σ_{cc}^{(1)}(c, 𝔑, ε), the sign in the functional equation of L(θ_{ψ}, χ^{−1}, s) is −1. As seen previously, this L-function factors as

L(θ_{ψ}, (χN_{K})^{−1}, s) = L(ψχ^{−1}N_{K}^{−1}, s)L(ψ^{∗}χ^{−1}N_{K}^{−1}, s) = L(νN_{K}^{−1}, s)L(ψ^{∗}χ^{−1}N_{K}^{−1}, s).

The normalization here is such that the central point is s = 0. Since the sign of L(ν, s) is −1, it follows that the sign of L(ψ^{∗}χ^{−1}N_{K}^{−1}, s) is +1. Hence condition (4) would be expected to hold generically.

The modular abelian variety B_{θ_{ψ}} attached to ψ is a CM abelian variety in the sense of Definition 2.4. Hence it is K-isogenous to the CM abelian variety B_{ψ} constructed in Section 2.2. In particular, the modular parametrisation Φ_{ψ} := Φ_{θ_{ψ}} can be viewed as a surjective morphism of abelian varieties over K

(3.19)  Φ_{ψ} : J_{1}(N)⟶B_{ψ}.

Given a good pair (ψ, χ), recall the Heegner divisor Δ ∈ J_{1}(N)(H_{c,𝔑}) that was constructed in Section 3.3, and the Heegner point

(3.20)  P_{ψ}(χ) := P_{θ_{ψ}}(χ) = ∑_{σ∈G} χ^{−1}(σ)Φ_{ψ}(Δ^{σ}) ∈ B_{ψ}(H_{χ}) ⊗_{T_{ψ}} E_{ψ,χ}

that was defined in equation (3.10) of that section with f = θ_{ψ}. Recall also that ω_{ψ} is an E_{ψ}-vector space generator of Ω^{1}(B_{ψ}/E_{ψ})^{T_{ψ}}. Viewing the point P_{ψ}(χ) as a formal linear combination of elements of B_{ψ}(H_{χ}) with coefficients in E_{ψ,χ}, we define the expression log_{ω_{ψ}}(P_{ψ}(χ)) by E_{ψ,χ}-linearity.

In the rest of this section, we will denote by E′_{ψ,χ} the subfield of ℚ̄ generated by E_{ψ}, E_{χ}, and the abelian extension H′_{χ} of K cut out by the finite order characters χ and χ^{∗}. The motivation for singling out good pairs for a special definition lies in the following proposition.

Proposition 3.21. For any pair (ψ, χ) which is good for ν,

(3.21)  ℒ_{p,𝔠}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ψ}}^{2}(P_{ψ}(χ)) (mod (E′_{ψ,χ})^{×}),

where Ω_{p}(ν^{∗}) is the p-adic period from Definition 2.13.

Proof. By Theorem 3.12 applied to f = θ_{ψ},

(3.22)  L_{p}(θ_{ψ}, χN_{K}) = log_{ω_{ψ}}^{2}(P_{ψ}(χ)) (mod E_{ψ,χ}^{×}).

On the other hand, since E′_{ψ,χ} contains E_{ψ,χ,ε}, Corollary 3.17 implies that

L_{p}(θ_{ψ}, χN_{K}) = ℒ_{p,𝔠}(ψ^{−1}χN_{K})ℒ_{p,𝔠M}(ψ^{∗−1}χN_{K}) (mod (E′_{ψ,χ})^{×})
(3.23)  = ℒ_{p,𝔠}(ν^{∗})ℒ_{p,𝔠M}(ψ^{∗−1}χN_{K}) (mod (E′_{ψ,χ})^{×}),

where the second equality follows from condition 3 in the definition of a good pair. The value ℒ_{p,𝔠M}(ψ^{∗−1}χN_{K}) is non-zero by condition 4 in the definition of a good pair. Therefore, by Cor. 3.3,

(3.24)  ℒ_{p,𝔠M}(ψ^{∗−1}χN_{K}) = Ω_{p}(ψ^{−1}χ^{∗}N_{K}) (mod E_{ψ,χ}^{×}).

Since ψχ^{−1} = ν, we have

(3.25)  Ω_{p}(ψ^{−1}χ^{∗}N_{K}) = Ω_{p}(ν^{−1}χ^{−1}χ^{∗}N_{K}) = Ω_{p}(ν^{∗} · χ^{∗}/χ) = Ω_{p}(ν^{∗}) (mod (E′_{ψ,χ})^{×}),

where the last equality follows from from Lemma 2.14. The proposition now follows by combining the equations (3.22) through (3.25). □

To go further, we will analyse the expression log_{ω_{ψ}}(P_{ψ}(χ)) and relate it to quantities depending solely on ν and not on the good pair (ψ, χ). It will be useful to view the point P_{ψ}(χ) appearing in (3.21) as an element of B_{ψ,χ}^{0}(H_{c,𝔑}) or as a K-rational point on the abelian variety B_{ψ,χ} that was introduced in Section 2.2. More precisely, after setting

(3.26)  P_{ψ}(χ) := ∑_{σ∈G} Φ_{ψ}(Δ^{σ}) ⊗ χ^{−1}(σ) ∈ B_{ψ}(K̄) ⊗_{T_{ψ}} T_{ψ,χ} = B_{ψ,χ}^{0}(K̄),

we observe that, for all τ ∈ Gal(K̄/K),

τ ∗_{0} P_{ψ}(χ) = ∑_{σ∈G} Φ_{ψ}(Δ^{τσ}) ⊗ χ^{−1}(σ)
= ∑_{σ∈G} Φ_{ψ}(Δ^{σ}) ⊗ χ^{−1}(στ^{−1}) = P_{ψ}(χ)χ(τ).

The point P_{ψ}(χ) therefore belongs to B_{ψ,χ}(K) by (2.9). For the following lemmas, recall the differentials ω_{ψ,χ}^{0} ∈ Ω^{1}(B_{ψ,χ}^{0}/E_{ψ,χ})^{T_{ψ,χ}} and ω_{ψ,χ} ∈ Ω^{1}(B_{ψ,χ}/E_{ψ,χ})^{T_{ψ,χ}}.

Lemma 3.22. For all good pairs (ψ, χ) attached to ν = ψχ^{−1},

log_{ω_{ψ}}(P_{ψ}(χ)) = log_{ω_{ψ,χ}^{0}}(P_{ψ}(χ)).

Proof. Let G = Gal(H_{c,𝔑}/K) and let P = Φ_{ψ}(Δ). Also, let i be the map defined in (2.4). Then

log_{ω_{ψ}}(P_{ψ}(χ)) = ∑_{σ∈G} χ(σ)^{−1} log_{ω_{ψ}}(P^{σ}) = ∑_{σ∈G} χ(σ)^{−1} log_{i^{∗}(ω_{ψ,χ}^{0})}(P^{σ})
= ∑_{σ∈G} χ(σ)^{−1} log_{ω_{ψ,χ}^{0}}(P^{σ} ⊗ 1) = ∑_{σ∈G} log_{χ(σ)^{−1}ω_{ψ,χ}^{0}}(P^{σ} ⊗ 1)

= ∑_{σ∈G} log_{ω_{ψ,χ}^{0}}(P^{σ} ⊗ χ(σ)^{−1}) = log_{ω_{ψ,χ}^{0}} (∑_{σ∈G} P^{σ} ⊗ χ(σ)^{−1}) = log_{ω_{ψ,χ}^{0}}(P_{ψ,χ}).
□

Lemma 3.23.

log_{ω_{ψ,χ}^{0}}(P_{ψ}(χ)) = log_{ω_{ψ,χ}}(P_{ψ}(χ)) (mod (E′_{ψ,χ})^{×}),

Proof. This follows from Lemma 2.8 since the Gauss sum 𝔤(χ) lies in (E′_{ψ,χ})^{×}. □

Lemma 3.24. There exists P_{ν} ∈ B_{ν}(K) and ω_{ν} ∈ Ω^{1}(B_{ν}/E_{ν})^{T_{ν}} such that

log_{ω_{ψ,χ}}(P_{ψ}(χ)) = log_{ω_{ν}} (P_{ν}) (mod (E′_{ψ,χ})^{×}).

Proof. Recall from Lemma 2.9 that there is a K-rational isogeny

B_{ν} ⊗_{T_{ν}} T_{ψ,χ}⟶B_{ψ,χ}.

Composing it with the natural morphism B_{ν}⟶B_{ν} ⊗_{T_{ν}} T_{ψ,χ}, we obtain a T_{ν}-equivariant morphism j : B_{ν}⟶B_{ψ,χ} defined over K with finite kernel. The fact that L(ν, s) has a simple zero at s = 1 implies that B_{ν}(K) ⊗ ℚ is one-dimensional over E_{ν}, and therefore that B_{ψ,χ}(K) ⊗ ℚ is one-dimensional over E_{ψ,χ}. In particular, if P_{ν} is any generator of B_{ν}(K) ⊗ ℚ, we may write

P_{ψ}(χ) = λj(P_{ν})

for some non-zero scalar λ ∈ E_{ψ,χ}^{×}. But letting

ω_{ν} = j^{∗}(ω_{ψ,χ}) ∈ Ω^{1}(B_{ν}/E′_{ψ,χ})^{T_{ν}} ,

we have

log_{ω_{ψ,χ}}(P_{ψ}(χ)) = log_{ω_{ψ,χ}}(λj(P_{ν})) = log_{λ^{∗}ω_{ψ,χ}}(j(P_{ν})) = λ log_{ω_{ψ,χ}}(j(P_{ν}))
= λ log_{j^{∗}ω_{ψ,χ}}(P_{ν}) = λ log_{ω_{ν}} (P_{ν}).

The lemma now follows after multiplying ω_{ν} by an appropriate scalar in (E′_{ψ,χ})^{×} so that it belongs to Ω^{1}(B_{ν}/E_{ν})^{T_{ν}}. □

Proposition 3.25. There exists ω_{ν} ∈ Ω^{1}(B_{ν}/E_{ν})^{T_{ν}} and P_{ν} ∈ B_{ν}(K) such that

(3.27)  ℒ_{p,𝔠}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}}^{2} (P_{ν}) (mod (E′_{ψ,χ})^{×}),

for all good pairs (ψ, χ) attached to ν.

Proof. This follows immediately from Proposition 3.21 and Lemmas 3.22 through 3.24. □

While Proposition 3.25 brings us close to Theorem 2 of the Introduction, it is somewhat more vague in that both sides of the purported equality may differ a priori by a non-zero element of the typically larger field E′_{ψ,χ}. The alert reader will also notice that this proposition is potentially vacuous for now, because the existence of a good pair for ν has not yet been established! The next proposition repairs this omission, and directly implies Theorem 2 of the Introduction.

Proposition 3.26. The set S_{ν} of pairs (ψ, χ) that are good for ν is non-empty. Furthermore,

(3.28)  ⋂_{(ψ,χ)∈S_{ν}} E′_{ψ,χ} = E_{ν}.

The proof of Proposition 3.26 rests crucially on a non-vanishing result of Rohrlich and Greenberg ([Ro], [Gre]) for the central critical values of Hecke L-series. In order to state it, we fix a rational prime ℓ which is split in K and let

K_{∞}^{−} = ∪_{n≥0}K_{n}^{−}

be the so-called anti-cyclotomic ℤ_{ℓ} extension of K; it is the unique ℤ_{ℓ}-extension of K which is Galois over ℚ and for which Gal(K_{∞}^{−}/ℚ) = ℤ_{ℓ} ⋊ (ℤ/2ℤ) is a generalised dihedral group.

Lemma 3.27 (Greenberg, Rohrlich). Let ψ_{0} be a self-dual Hecke character of K of infinity type (1, 0). Assume that the sign w_{ψ_{0}} in the functional equation of L(ψ_{0}, s) is equal to 1. Then there are infinitely many finite-order characters χ of Gal(K_{∞}^{−}/K) for which L(ψ_{0}χ, 1) ≠ 0.

Proof. Let 𝔠′ be the conductor of ψ_{0}. In light of the hypothesis that w_{ψ_{0}} = 1, Theorem 1 of [Gre] implies that the Katz p-adic L-function (with p = ℓ) does not vanish identically on any open ℓ-adic neighbourhood of ψ_{0} in Σ_{sd}(𝔠′). (Cf. the discussion in the first paragraph of the proof of Proposition 1 on p. 93 of [Gre].) If U is any sufficiently small such neighbourhood, then

(1) The restriction to U of the Katz p-adic L-function is described by a power series with p-adically bounded coefficients, and therefore admits only finitely many zeros by the Weierstrass preparation theorem.

(2) The region U contains a dense subset of points of the form ψ_{0}χ, where χ is a finite order character

of Gal(K_{∞}^{−}/K). Lemma 3.27 follows directly from these two facts. □

Proof of Proposition 3.26. Let S̄_{ν} ⊃ S_{ν} be the set of pairs satisfying conditions 1-3 in the definition of a good pair, but without necessarily requiring the more subtle fourth condition. The proof of Proposition 3.26 will be broken down into four steps.

Step 1. The set S̄_{ν} is non-empty.

To see this, let ψ be any Hecke character of K of infinity type (1, 0) and conductor 𝔪, where 𝔪 is a cyclic O_{K}-ideal prime to 𝔠. Setting χ = ψν^{−1}, the pair (ψ, χ) satisfies conditions 1 and 3 by construction, and 2 as well on account of Remark 3.19. Therefore, the pair (ψ, χ) belongs to S̄_{ν}.

Step 2. Given (ψ, χ) ∈ S̄_{ν}, there exist (ψ_{1}, χ_{1}) and (ψ_{2}, χ_{2}) ∈ S_{ν} with E′_{ψ_{1},χ_{1}} ∩ E′_{ψ_{2},χ_{2}} ⊂ E′_{ψ,χ}.

To see this, let ℓ = λλ̄ be a rational prime which splits in K and is relatively prime to the class number of K and the conductors of ψ and χ, and which is unramified in E′_{ψ,χ}/ℚ. For such a prime, let

K_{∞} = ∪_{n≥0}K_{n}, K′_{∞} = ∪_{n≥0}K′_{n}

be the unique ℤ_{ℓ}-extensions of K which are unramified outside of λ and λ̄ respectively, with [K_{n} : K] = ℓ^{n} and likewise for K′_{n}. The condition that ℓ does not divide the class number of K implies that the fields K_{n} and K′_{n} are totally ramified at λ and λ̄ respectively. If α is any character of Gal(K_{∞}/K), the pair (ψ_{1}, χ_{1}) := (ψα, χα) still belongs to S̄_{ν}, with 𝔪 in condition 1 replaced by 𝔪λ^{n} for a suitable n ≥ 0. Furthermore,

(3.29)  L(ψ_{1}^{∗}χ_{1}^{−1}N_{K}^{−1}, 0) = L(ψ^{∗}χ^{−1}N_{K}^{−1} · (α^{∗}/α), 0).

The character α^{∗}/α is an anticyclotomic character of K of ℓ-power order and conductor, and all such characters can be obtained by choosing α appropriately. The fact that (ψ, χ) satisfies conditions (1),(2) and (3) of a good pair implies (see Remark 3.20) that the sign w_{ψ^{∗}χ^{−1}} is equal to +1. Hence, by Lemma 3.27, there exists a choice of α for which the L-value appearing on the right of (3.29) is non-vanishing. The corresponding pair (ψ_{1}, χ_{1}) belongs to S_{ν} and satisfies

E′_{ψ_{1},χ_{1}} ⊂ E′_{ψ,χ,ℓ,n} := E′_{ψ,χ}ℚ(ζ_{ℓ^{n}})K_{n}K′_{n}

for some n. Note that the extension E′_{ψ,χ,ℓ,n}/E′_{ψ,χ} has degree dividing ℓ^{∞}(ℓ − 1). Repeating the same construction with a different rational prime ℓ′ in place of ℓ such that ℓ′ − 1 is prime to ℓ yields a second pair (ψ_{2}, χ_{2}) ∈ S_{ν} and a corresponding extension E′_{ψ,χ,ℓ′,n′}, whose degree over E′_{ψ,χ} divides ℓ′^{∞}(ℓ′ − 1), and such that

E′_{ψ_{2},χ_{2}} ⊂ E′_{ψ,χ,ℓ′,n′}.

Let

E′′ := E′_{ψ,χ,ℓ,n} ∩ E′_{ψ,χ,ℓ′,n′}.

We see then using degrees that E′′/E′_{ψ,χ} has degree dividing (ℓ − 1), hence E′′ ⊆ E′_{ψ,χ}ℚ(ζ_{ℓ}). Since ℓ is unramified in E′_{ψ,χ}/ℚ, the extension E′′/E′_{ψ,χ} must be totally ramified at the primes above ℓ. On the other hand being a subextension of E′_{ψ,χ,ℓ′,n′}/E′_{ψ,χ}, it is also unramified at the primes above ℓ, hence E′′ = E′_{ψ,χ}. It follows that E′_{ψ_{1},χ_{1}} ∩ E′_{ψ_{2},χ_{2}} ⊂ E′_{ψ,χ}.

Thanks to Step 2, we are reduced to showing that

(3.30)  ⋂_{(ψ,χ)∈S̄_{ν}} E′_{ψ,χ} = E_{ν}.

The next step shows that the fields E′_{ψ,χ} can be replaced by E_{ψ,χ} in this equality.

Step 3. For all (ψ, χ) ∈ S̄_{ν}, there exists a finite order character α of G_{K} such that the pair (ψα, χα) belongs to S̄_{ν} and

(3.31)  E′_{ψ,χ} ∩ E′_{ψα,χα} ⊆ E_{ψ,χ}.

To see this, note that the finite order character χ has cyclic image, isomorphic to ℤ/nℤ say. Pick α such that conditions (i)-(iii) below are satisfied:

(i) α has order n and is ramified at a single prime λ of K which lies over a rational prime ℓ that is split in K.

(ii) λ is prime to the conductors of χ and χ^{∗}.

(iii) ℓ is unramified in E′_{ψ,χ}/ℚ. Conditions (i) and (ii) imply:

(iv) The field H_{χα}/K is totally ramified at λ and unramified at λ^{∗} while H_{χ^{∗}α^{∗}} is unramified at λ and totally ramified at λ^{∗}.

Taking

L = E_{ψ,χ}, M_{1} = H_{χ}H_{χ^{∗}}, M_{2} = H_{χα}H_{χ^{∗}α^{∗}},

we see from (iii) and (iv) that

(v) LM_{1}/L is unramified at all primes above ℓ, and

(vi) Any subextension of LM_{2}/L is ramified at some prime above λ or λ^{∗}.

Thus, LM_{1} ∩LM_{2} = L. But LM_{1} = E_{ψ,χ}H_{χ}H_{χ^{∗}} = E′_{ψ,χ}. Also since α has order n, we have E_{ψ′,χ′} = E_{ψ,χ} and

LM_{2} = E_{ψ,χ}H_{χα}H_{χ^{∗}α^{∗}} = E_{ψ′,χ′}H_{χα}H_{χ^{∗}α^{∗}} = E′_{ψ′,χ′},

so (3.31) follows.

Step 4. We are now reduced to showing

(3.32)  ⋂_{(ψ,χ)∈S̄_{ν}} E_{ψ,χ} = E_{ν}.

We will do this by showing

(3.33)  There exists a pair (ψ, χ) ∈ S̄_{ν} such that E_{ψ,χ} = E_{ν}.

We begin by choosing an ideal 𝔪_{0} of O_{K} with the property that O_{K}/𝔪_{0} = ℤ/Mℤ is cyclic, and an odd quadratic Dirichlet character ε_{M} of conductor dividing M. Let ψ_{0} be any Hecke character satisfying

ψ_{0}((a)) = ε_{M}(a mod 𝔪_{0})a

on principal ideals (a) of K. Such a ψ_{0} satisfies condition 1 in Definition 3.18, and therefore, after letting χ_{0} be the finite order character satisfying

ν^{∗} = ψ_{0}^{−1}χ_{0}N_{K},

it follows that (ψ_{0}, χ_{0}) belongs to S̄_{ν}. Furthermore, the restriction of ψ_{0} to the group of principal ideals of K takes values in K, and therefore

(3.34)  χ_{0}(σ) ∈ E_{ν},  for all σ ∈ G_{H} := Gal(K̄/H).

The character ψ_{0} itself takes values in a CM field of degree [H : K], denoted E_{0}, which need not be contained in E_{ν} in general. To remedy this problem, let H_{0} be the abelian extension of the Hilbert class field H cut out by the character χ_{0}. Next, let H′_{0} be any abelian extension of K containing H such that

(1) There is an isomorphism u : Gal(H′_{0}/K)⟶Gal(H_{0}/K) of abstract groups such that the diagram

(3.35)  [figure: 0; Gal(H′_{0}/H); Gal(H′_{0}/K); Gal(H/K); 0; 0; Gal(H_{0}/H); Gal(H_{0}/K); Gal(H/K); 0],

commutes, where the dotted arrows indicate the isomorphisms induced by u and the other arrows are the canonical maps of Galois theory.

(2) The relative discriminant of H′_{0} over K is relatively prime to its conjugate (and therefore to the

discriminant of K, in particular). If the bottom exact sequence of groups in (3.35) is split, then the extension H′_{0} is readily produced, using class field theory. To handle the general case, we follow an approach that is suggested by the proof of Prop. 2.1.7 in [Se]. Let Φ̃ := Gal(H_{0}/K) and let Ψ : G_{K}⟶Φ̃ be the homomorphism attached to the extension H_{0}. Since H is everywhere unramified over K, the restriction Ψ_{v} of Ψ to a decomposition group at any prime v of K maps the inertia subgroup I_{v} to ℭ := Gal(H_{0}/H). After viewing ℭ as a module of finite cardinality endowed with the trivial action of G_{K}, let H_{S}^{1}(K, ℭ) := Hom(G_{K,S}, ℭ) denote the group of homomorphisms from G_{K} to ℭ which are unramified outside a given finite set S of primes of K, and let H_{[S]}^{1}(K, ℭ^{∗}) denote the dual Selmer group attached to H_{S}^{1}(K, ℭ) in the sense of Theorem 2.18 of [DDT] for example. Here ℭ^{∗} := Hom(ℭ, G_{m}) is the Kummer dual of ℭ, which is isomorphic to μ_{n} when ℭ = ℤ/nℤ is cyclic of order n. Kummer theory (along with the non-degeneracy of the local Tate pairing) identifies H_{[S]}^{1}(K, μ_{n}) with the subgroup of K^{×}/(K^{×})^{n} consisting of elements α for which

ord_{v}(α) = 0 (mod n) for all v, res_{v}(α) ∈ (K_{v}^{×})^{n} for all v ∈ S.

Let S be any finite set of primes of K at which Ψ is unramified, satisfying the further conditions

(3.36)  v ∈ S ⇒ v̄ ∉ S,  and H_{[S]}^{1}(K, ℭ^{∗}) = 0.

The existence of such a set S follows from the statement that for any α ∈ K^{×} − (K^{×})^{n}, there is a set of primes v of K of positive Dirichlet density for which the image of α in K_{v}^{×} is not an n-th power. (This statement follows in turn from the Chebotarev density theorem applied to the extension K(μ_{n}, α^{1/n}).) Now let T be any finite set of places which is disjoint from S. Comparing the statement of Theorem 2.18 of [DDT] in the case M = ℭ and ℒ = S and ℒ = S ∪ T respectively, and noting that both H_{[S]}^{1}(K, ℭ^{∗}) and (a fortiori) H_{[S∪T]}^{1}(K, ℭ^{∗}) are trivial, gives

\frac{#H_{S∪T}^{1} (K, ℭ)}{#H_{S}^{1}(K, ℭ)} = ∏_{v∈T} \frac{#H^{1}(K_{v}, ℭ)}{#ℭ} = ∏_{v} # Hom(I_{v}, ℭ).

It follows that the rightmost arrow in the tautological exact sequence

0⟶H_{S}^{1}(K, ℭ)⟶H_{S∪T}^{1} (K, ℭ)⟶ ∏_{c∈T} Hom(I_{v}, ℭ)

is surjective. Letting T be the set of places at which Ψ is ramified, it follows that there is a homomorphism ϵ : G_{K}⟶ℭ satisfying

ϵ_{v} = Ψ_{v} on I_{v}, for all v ∉ S.

After possibly enlarging the set S satisfying (3.36) and translating ϵ by a suitable homomorphism unramified outside S, we may further assume that the homomorphism Ψϵ^{−1} maps G_{K} surjectively onto Φ̃; the field H′_{0} can then be obtained as the fixed field of the kernel of the homomorphism Ψϵ^{−1}.

With the extension H′_{0} in hand, let α : Gal(H′_{0}/K)⟶E_{χ}^{×} be the finite order Hecke character given by

α(σ) = χ_{0}(u(σ))^{−1},

and set (ψ, χ) = (ψ_{0}α, χ_{0}α). By construction, (ψ, χ) belongs to S̄_{ν}. We claim that χ and ψ take values in E_{ν}. Since ν^{∗} = ψ^{−1}χN_{K}, it is enough to prove this statement for χ. Observe that, for all integral ideals 𝔞 prime to the conductors of χ_{0}, χ, and ψ, we have

χ(𝔞) = χ_{0}(σ_{𝔞})/χ_{0}(u(σ_{𝔞})) = χ_{0}(σ_{𝔞}u(σ_{𝔞})^{−1}).

But the element σ_{𝔞}u(σ_{𝔞})^{−1} belongs to Gal(H_{0}/H) by construction, and hence χ_{0}(σ_{𝔞}^{−1}u(σ_{𝔞})) belongs to E_{ν} by (3.34). It follows that ψ and χ are E_{ν}–valued, and therefore E_{ψ,χ} = E_{ν}, as claimed in (3.33). □

3.6. Elliptic curves with complex multiplication. Theorem 2 of the Introduction admits an alternate formulation involving algebraic points on elliptic curves with complex multiplication rather than K-rational points on the CM abelian varieties B_{ν} of Theorem 2.5. The goal of this section is to describe this variant. As in the introduction, we just write ℒ_{p} for the p-adic L-function ℒ_{p,𝔠}, where 𝔠 is the conductor of ν.

We begin by reviewing the explicit construction of B_{ν} in terms of CM elliptic curves. The reader is referred to §4 of [GS], whose treatment we largely follow, for a more detailed exposition. Let F be any abelian extension of K for which

(3.37)  ν_{F} := ν ∘ N_{F/K}

becomes K-valued. There exists an elliptic curve A/F with complex multiplication by O_{K} whose associated Grossencharacter is ν_{F}. (Cf. Thm. 6 of [Shi] and its corollary on p. 512.) Let

B := Res_{F/K}(A).

It is an abelian variety over K of dimension d := [F : K]. Let G := Gal(F/K) = Hom_{K}(F, ℚ̄), where the natural identification between these two sets arises from the distinguished embedding of F into ℚ̄ that was fixed from the outset. By definition of the restriction of scalars functor, there are natural isomorphisms

B_{/F} = ∏_{σ∈G} A^{σ}, B(K̄) = A(K̄ ⊗_{K} F) = ∏_{σ∈G} A^{σ}(K̄)

of algebraic groups over F and abelian groups respectively. In particular, a point of B(K̄) is described by a d-tuple (P_{τ})_{τ∈G}, with P_{τ} ∈ A^{τ} (K̄). Relative to this identification, the Galois group G_{K} acts on B(K̄) on the left by the rule

ξ(P_{τ})_{τ} = (ξP_{τ})_{ξτ} , for all ξ ∈ G_{K}.

Consider the “twisted group ring”

(3.38)  T := ⊕_{σ∈G} Hom_{F} (A, A^{σ}) = {∑_{σ∈G} a_{σ}σ, with a_{σ} ∈ Hom_{F} (A, A^{σ})} ,

with multiplication given by

(3.39)  (a_{σ}σ)(a_{τ} τ) = a_{σ}a_{τ}^{σ}στ,

where the isogeny a_{τ}^{σ} belongs to Hom_{F} (A^{σ}, A^{στ}) and the composition of isogenies in (3.39) is to be taken from left to right. The right action of T on B(K̄) defined by

(3.40)  (P_{τ})_{τ} ∗ (a_{σ}σ) := (a_{σ}^{τ}(P_{τ}))_{τσ}

commutes with the Galois action described in (3.6), and corresponds to a natural inclusion T ↪ End_{K}(B). The K-algebra E := T ⊗_{ℤ} ℚ is isomorphic to a finite product

E = ∏_{i} E_{i}

of CM fields, and dim_{K}(E) = dim(B). Therefore, the pair (B, E) is a CM abelian variety in the sense of Definition 2.4. The compatible system of ℓ-adic Galois representations attached to (B, E) corresponds to an E-valued algebraic Hecke character ν̃ in the sense of Definition 2.3, satisfying the relation

(3.41)  σ_{𝔞}(P) = P ∗ ν̃(𝔞), for all 𝔞 ∈ I_{𝔣ℓ} and P ∈ B(K̄)_{ℓ^{∞}},

where σ_{𝔞} ∈ G_{K}^{ab} denotes as before the Artin symbol attached to 𝔞 ∈ I_{𝔣ℓ}.

The element ν̃(𝔞) ∈ T is of the form φ_{𝔞}σ_{𝔞}, where

(3.42)  φ_{𝔞} : A → A^{σ_{𝔞}},

is an isogeny of degree N𝔞 satisfying

(3.43)  φ_{𝔞}(P) = P^{σ_{𝔞}},

for any P ∈ A[𝔤] with (𝔤, 𝔞) = 1. Note that the isogenies φ_{𝔞} satisfy the following cocycle condition:

(3.44)  φ_{𝔞𝔟} = φ_{𝔟}^{σ_{𝔞}} ∘ φ_{𝔞}.

The following proposition relates the Hecke characters ν̃ and ν.

Proposition 3.28. Given any homomorphism j ∈ Hom_{K}(E, ℂ), let ν_{j} := j ∘ ν̃ be the corresponding ℂ-valued Hecke character of K of infinity type (1, 0). The assignment j ↦ ν_{j} gives a bijection from Hom_{K}(E, ℂ) to the set Σ_{ν,F} of Hecke characters ν′ of K (of infinity type (1, 0)) satisfying

ν′ ∘ N_{F/K} = ν ∘ N_{F/K}.

Proposition 3.28 implies that there is a unique homomorphism j_{ν} ∈ Hom_{K}(E, ℂ) satisfying j_{ν} ∘ ν̃ = ν. In particular, j_{ν} maps E to E_{ν} and T to a finite index subring of T_{ν}. The abelian variety B_{ν} attached to ν in Theorem 2.5 can now be defined as the quotient B ⊗_{T,j_{ν}} T_{ν}. In subsequent constructions, it turns out to be more useful to realise B_{ν} as a subvariety of B, which can be done by setting

(3.45)  B_{ν} := B[ker j_{ν}].

The natural action of T on B_{ν} factors through the quotient T/ ker(j_{ν}), an integral domain having E_{ν} as field of fractions.

Consider the inclusion

(3.46)  i_{ν} : B_{ν}(K) ↪ B(K) = A(F),

where the last identification arises from the functorial property of the restriction of scalars. The following Proposition gives an explicit description of the image of (B_{ν}(K) ⊗ E_{ν})^{T_{ν}} in A(F) ⊗_{O_{K}} E_{ν} under the inclusion i_{ν} obtained from (3.46).

Proposition 3.29. Let Ẽ be any field containing E_{ν}. The inclusion i_{ν} of (3.46) identifies (B_{ν}(K)⊗Ẽ)^{T_{ν}} with

(A(F) ⊗_{O_{K}} Ẽ)^{ν} := {P ∈ A(F) ⊗_{O_{K}} Ẽ such that φ_{𝔞}(P) = ν(𝔞)P^{σ_{𝔞}}, for all 𝔞 ∈ I_{𝔣}} .

Proof. It follows from the definitions that B(K) is identified with the set of (P_{τ}) with P_{τ} ∈ A^{τ} (K̄) satisfying

(3.47)  ξP_{τ} = P_{ξτ} , for all ξ ∈ G_{K}.

Furthermore, if such a (P_{τ}) belongs to (B_{ν}(K) ⊗ E_{ν})^{T_{ν}}, then after setting ν̃(𝔞) = φ_{𝔞}σ_{𝔞} as in (3.42), we also have

(3.48)  (φ_{𝔞}^{τ}(P_{τ}))_{τσ_{𝔞}} = (P_{τ})_{τ} ∗ ν̃(𝔞) = (ν(𝔞)P_{τ})_{τ} .

Equating the σ_{𝔞}-components of these two vectors gives

φ_{𝔞}(P_{1}) = ν(𝔞)P_{σ_{𝔞}} = ν(𝔞)σ_{𝔞}P_{1},

where 1 is the identity embedding of F and the last equality follows from (3.47). The Proposition follows directly from this, after noting that the identification of B(K) with A(F) is simply the one sending (P_{τ})_{τ} to P_{1}. □

Given a global field F as in (3.37), let F_{ν} denote the subfield of ℚ̄ generated by F and E_{ν}. Recall that ω_{A} ∈ Ω^{1}(A/F) is a non-zero differential and that Ω_{p}(A) is the associated p-adic period.

Theorem 3.30. There exists a point P_{A,ν} ∈ (A(F) ⊗_{O_{K}} E_{ν})^{ν} such that

ℒ_{p}(ν^{∗}) = Ω_{p}(A)^{−1} log_{ω_{A}}^{2}(P_{A,ν}) (mod F_{ν}^{×}).

The point P_{A,ν} is non-zero if and only if L′(ν, 1) ≠ 0.

Proof. Theorem 2 of the Introduction asserts that

(3.49)  ℒ_{p}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}}^{2} (P_{ν}),

for some point P_{ν} ∈ B_{ν}(K) ⊗ ℚ which is non-trivial if and only if L′(ν, 1) ≠ 0. By Lemma 2.14, we find

(3.50)  Ω_{p}(ν^{∗})^{−1} = Ω_{p}(A)^{−1} (mod F_{ν}^{×}).

Furthermore, by Prop. 3.29, we can view P_{ν} as a point P_{A,ν} ∈ (A(F) ⊗_{O_{K}} E_{ν})^{ν}, and we have

(3.51)  log_{ω_{ν}} (P_{ν}) = log_{ω_{A}}(P_{A,ν}) (mod F_{ν}^{×}).

Theorem 3.33 now follows by rewriting (3.49) using (3.50) and (3.51). □

3.7. A special case. This section is devoted to a more detailed and precise treatment of Theorem 3.30 under the following special assumptions:

(1) The quadratic imaginary field K has class number one, odd discriminant, and unit group of order two. This implies that K = ℚ(\sqrt{−D}) where D := − Disc(K) belongs to the finite set

S := {7, 11, 19, 43, 67, 163}.

(2) ψ_{0} is the Hecke character of K of infinity type (1, 0) given by the formula

(3.52)  ψ_{0}((a)) = ε_{K}(a mod 𝔡_{K})a.

The character ψ_{0} determines (uniquely, up to an isogeny) an elliptic curve A/ℚ satisfying

End_{K}(A) = O_{K}, L(A/ℚ, s) = L(ψ_{0}, s).

After fixing A, we will also write ψ_{A} instead of ψ_{0}. It can be checked that the conductor of ψ_{A} is equal to 𝔡_{K}, and that

ψ_{A}^{∗} = ψ̄_{A},  ψ_{A}ψ_{A}^{∗} = N_{K},  ε_{ψ_{A}} = ε_{K}.

Remark 3.31. The rather stringent assumptions on K that we have imposed exclude the arithmetically interesting, but somewhat idiosyncratic, cases where K = ℚ(\sqrt{−3}), ℚ(i), and ℚ(\sqrt{−2}).

With the above assumptions, the character ψ_{A} can be used to give an explicit description of the set Σ_{sd}(c𝔡_{K}):

Lemma 3.32. Let c be an integer prime to D, and let ν be a Hecke character in Σ_{sd}(c𝔡_{K}). Then ν is of the form

ν = ψ_{A}χ^{−1},

where χ is a finite order ring class character of K of conductor c.

Proof. The fact that ν and ψ_{A} both have central character ε_{K} implies that χ is a ring class character that is unramified at 𝔡_{K}, hence has conductor exactly c. □

Given a ring class character χ of conductor c as above with values in a field E_{χ}, let

(3.53)  (A(H_{c}) ⊗_{O_{K}} E_{χ})^{χ} := {P ∈ A(H_{c}) ⊗_{O_{K}} E_{χ} such that σP = χ(σ)P, ∀σ ∈ Gal(H_{c}/K)}.

Finally, choose a nonzero differential ω_{A} ∈ Ω^{1}(A/K), and write Ω_{p}(A) for the p-adic period attached to this choice as in Section 3.1. Since A = B_{ψ_{0}} is the abelian variety attached to ψ_{0}, it follows that Ω_{p}(ψ_{A}^{∗}) = Ω_{p}(A).

The following theorem is a more precise variant of Theorem 3.30.

Theorem 3.33. Let χ be a ring class character of K of conductor prime to 𝔡_{K}. Then there exists a point P_{A}(χ) ∈ (A(H_{χ}) ⊗_{O_{K}} E_{χ})^{χ} such that

ℒ_{p}(ψ_{A}^{∗}χ) = Ω_{p}(A)^{−1}𝔤(χ) log_{ω_{A}}^{2}(P_{A}(χ)) (mod E_{χ}^{×}).

The point P_{A}(χ) is non-zero if and only if L′(ψ_{A}χ^{−1}, 1) ≠ 0.

Proof. By Theorem 2 of the Introduction,

(3.54)  ℒ_{p}(ψ_{A}^{∗}χ) = ℒ_{p}(ν^{∗}) = Ω_{p}(ν^{∗})^{−1} log_{ω_{ν}}^{2} (P_{ν}) (mod E_{ν}^{×}),

for some point P_{ν} ∈ B_{ν}(K) ⊗ ℚ which is non-trivial if and only if L′(ψ_{A}χ^{−1}, 1) ≠ 0. Since χ^{∗−1} = χ and E_{ν} = E_{χ}, we find from Lemma 2.14 that

(3.55)  Ω_{p}(ν^{∗})^{−1} = Ω_{p}(ψ_{A}^{∗}χ^{∗−1})^{−1} = Ω_{p}(A)^{−1}𝔤(χ)^{−1} (mod E_{χ}^{×}).

After noting that (as in equation (2.7)) B_{ν} = B_{ψ,χ} = (A ⊗_{O_{K}} T_{χ})^{χ^{−1}} as abelian varieties over K, we observe that ω_{ν} = ω_{ψ,χ} and that the point P_{ν} ∈ B_{ν}(K) can be written as

P_{ν} = ∑_{σ∈G} P^{σ} ⊗ χ^{−1}(σ),

for some P ∈ A(H_{c}) ⊗ ℚ. Letting P_{A,χ} be the corresponding element in A(H_{c}) ⊗_{O_{K}} E_{χ} given by

P_{A,χ} = ∑_{σ∈G} χ^{−1}(σ)P^{σ},

we have

(3.56)  log_{ω_{ν}}^{2} (P_{ν}) = log_{ω_{ψ,χ}}^{2}(P_{ν}) = 𝔤(χ)^{2} log_{ω_{ψ,χ}^{0}}^{2}(P_{ν}) = 𝔤(χ)^{2} log_{ω_{A}}^{2}(P_{A,χ}) (mod E_{χ}^{×}),

where the second equality follows from Lemma 2.8 and the last from Lemma 3.22. Theorem 3.33 now follows by rewriting (3.54) using (3.55) and (3.56). □

In the special case where χ is a quadratic ring class character of K, cutting out an extension L = K(\sqrt{a}) of K, we obtain

(3.57)  ℒ_{p}(ψ_{A}^{∗}χ) = Ω_{p}(A)^{−1}\sqrt{a} log_{ω_{A}}^{2}(P_{A,L}^{−}) (mod K^{×}),

where P_{A,L}^{−} is a K-vector space generator of the trace 0 elements in A(L) ⊗ ℚ. Since in this case ψ_{A}χ is the Hecke character attached to a CM elliptic curve over ℚ, from (3.57) one recovers Rubin’s Theorem 1 of the Introduction.

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