OVERCONVERGENT GENERALISED EIGENFORMS OF WEIGHT ONE AND CLASS FIELDS OF REAL QUADRATIC FIELDS

HENRI DARMON, ALAN LAUDER AND VICTOR ROTGER

Abstract. This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigenforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark’s conjecture or its p-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli.

1. Introduction and statement of the main result

Fourier coefficients of modular forms often describe interesting arithmetic functions. Classical examples are the partition function, the divisor function, and representation numbers of quadratic forms, which are related to Fourier coefficients of the Dedekind eta function, Eisenstein series, and theta series, respectively. A more modern instance is the appearance of Frobenius traces of two-dimensional Galois representations as Fourier coefficients of normalised eigenforms. Just as germane to an understanding of the present work are the topological and arithmetic intersection numbers of special cycles arising in the formulae of Hirzebruch-Zagier, Gross-Kohnen-Zagier, and their vast generalisations growing out of the “Kudla program”.

The main theorem of this paper expresses the Fourier coefficients of certain p-adic modular forms of weight one as p-adic logarithms of algebraic numbers in ring class fields of real quadratic fields. It suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark’s (still unproved) conjecture [St] or Gross’s (more tractable) p-adic variant [Gr]. An analogy with the growing body of work on Fourier coefficients of incoherent Eisenstein series and weak harmonic Maass forms suggests that this approach is perhaps closer in spirit to the classical theory of singular moduli.

To set the stage for the main result, let K be a real quadratic field of discriminant D > 0 and let χ_{K} denote the even quadratic Dirichlet character associated to it. Let

ψ : G_{K} := Gal (K̄/K)⟶ℂ^{×}

be a ray class character (of order m, conductor 𝔣_{ψ} and central character χ_{ψ}) which is of mixed signature, i.e., which is even at precisely one of the infinite places of K and odd at the other. Hecke’s theta series g := θ_{ψ} attached to ψ is a holomorphic newform of weight one, level N and nebentype character χ with Fourier coefficients in L := ℚ(μ_{m}), where

N = D · Norm_{K/ℚ}𝔣_{ψ}, χ = χ_{K}χ_{ψ}.

Fix a prime p = 𝔭 𝔭′ which does not divide N and is split in K/ℚ. The eigenform g is said to be regular at p if the Hecke polynomial

x^{2} − a_{p}(g)x + χ(p) = (x − ψ(𝔭′))(x − ψ(𝔭)) =: (x − α)(x − β)

has distinct roots. Assume henceforth that this regularity hypothesis holds, and let

g_{α}(z) := g(z) − βg(pz), g_{β}(z) := g(z) − αg(pz)

be the two distinct p-stabilisations of g, which are eigenvectors for the U_{p} operator with eigenvalues α and β respectively. Note that these stabilisations are both ordinary, since α and β are roots of unity.

Let S_{k}(N, χ) (resp. S_{k}^{(p)}(N, χ)) denote the space of classical (resp. p-adic overconvergent) modular forms of weight k, level N and character χ, with coefficients in ℂ_{p}. The Hecke algebra 𝕋 of level Np generated over ℚ by the operators T_{ℓ} with ℓ ∤ Np and U_{q} with q|Np acts naturally on the spaces S_{k}(Np, χ) and S_{k}^{(p)}(N, χ). The normalised eigenform g_{α} ∈ S_{1}(Np, χ) gives rise to an algebra homomorphism φ_{g_{α}} : 𝕋⟶L satisfying

φ_{g_{α}}(T_{ℓ}) = a_{ℓ}(g) if ℓ ∤ Np, φ_{g_{α}}(U_{ℓ}) = \{ \begin{matrix}a_{ℓ}(g) & if ℓ|N; \\ α & if ℓ = p, \end{matrix}

and g_{α} generates the one-dimensional eigenspace S_{1}(Np, χ)[g_{α}] attached to this system of Hecke eigenvalues.

In [CV], Cho and Vatsal make the important observation that the Coleman-Mazur eigencurve is smooth but not étale over weight space at the points corresponding to g_{α} and g_{β}. (It is likely that the ramification degree of the weight map is always equal to two at these points under the regularity assumption: see [CV], [BD], and Adel Bettina’s forthcoming PhD thesis [Bet] for various results in this direction.) In particular, letting I_{g_{α}} be the kernel of φ_{g_{α}}, the subdmodule of S_{1}^{(p)}(N, χ) which is annihilated by I_{g_{α}}^{2}, denoted S_{1}^{(p)}(N, χ)[[g_{α}]], is two-dimensional and contains non-classical forms which do not lie in the image of the natural inclusion

S_{1}(Np, χ)[g_{α}] ↪ S_{1}^{(p)}(N, χ)[[g_{α}]].

An overconvergent form in S_{1}^{(p)}(N, χ)[[g_{α}]] which is not a multiple of g_{α} is called a generalised eigenform attached to g_{α}, and is said to be normalised if its first Fourier coefficient is equal to zero. Such a normalised generalised eigenform, denoted

g_{α}^{♭} := ∑_{n=2}^{∞} a_{n}(g_{α}^{♭})q^{n},

is uniquely determined by g_{α} up to scaling, and the Hecke operators act on it by the rule

(1)  T_{ℓ}g_{α}^{♭} = a_{ℓ}(g_{α})g_{α}^{♭} + a_{ℓ}(g_{α}^{♭})g_{α},  U_{q}g_{α}^{♭} = a_{q}(g_{α})g_{α}^{♭} + a_{q}(g_{α}^{♭})g_{α},

for all primes ℓ ∤ Np and all q|Np. Theorem 1 below shows that the Fourier coefficients a_{n}(g_{α}^{♭}) are interesting arithmetic quantities with a bearing on explicit class field theory for K.

Let ψ′ denote the character deduced from ψ by composing it with the involution in Gal (K/ℚ). The ratio ψ_{♡} := ψ/ψ′ is a totally odd ring class character of K. Let H denote the ring class field of K which is fixed by the kernel of ψ_{♡}, and set G := Gal (H/K).

If ℓ ∤ N is any rational prime which is inert in K/ℚ, the corresponding prime ℓ of K splits completely in H/K, and the set Σ_{ℓ} of primes of H above ℓ is endowed with the structure of a principal G-set. Given λ ∈ Σ_{ℓ}, let u(λ) ∈ O_{H}[1/λ]^{×} ⊗ ℚ be any λ-unit of H satisfying ord_{λ}(u(λ)) = 1. While u(λ) is only defined up to units in O_{H}^{×}, the element

u(ψ_{♡}, λ) = ∑_{σ∈G} ψ_{♡}^{−1}(σ) ⊗ u(λ)^{σ} ∈ L ⊗ O_{H}[1/ℓ]^{×}

is independent of the choice of generator u(λ), since there are no genuine units in L ⊗ O_{H}^{×} in the eigencomponents for the totally odd character ψ_{♡}. The ℓ-unit u(ψ_{♡}, λ) does depend on the choice of λ ∈ Σ_{ℓ}. Section 2 below uses ψ to define a function η : Σ_{ℓ}⟶μ_{m} for which the element

(2)  u(ψ_{♡}, ℓ) := η(λ) ⊗ u(ψ_{♡}, λ) ∈ L ⊗ O_{H}[1/ℓ]^{×}

depends only on the inert prime ℓ and not on the choice of prime λ ∈ Σ_{ℓ} above it. Fix embeddings of L and of H into ℚ̄_{p}, and let

log_{p} : L ⊗ H^{×}⟶ℚ̄_{p}

be the resulting p-adic logarithm on H^{×}, extended to L ⊗ H^{×} by L-linearity. The main result of this paper is

Theorem 1. The normalised generalised eigenform g_{α}^{♭} attached to g_{α} can be scaled in such a way that, for all primes ℓ ∤ N,

a_{ℓ}(g_{α}^{♭}) = \{ \begin{matrix}0 & if χ_{K}(ℓ) = +1; \\ log_{p} u(ψ_{♡}, ℓ) & if χ_{K}(ℓ) = −1. \end{matrix}

More generally, for all n ≥ 2 with gcd(n, N) = 1,

(3)  a_{n}(g_{α}^{♭}) = \frac{1}{2} ∑_{ℓ|n} log_{p}u(ψ_{♡}, ℓ) · (ord_{ℓ}(n) + 1) · a_{n/ℓ}(g_{α}),

where the sum runs over primes ℓ that are inert in K.

The following two examples illustrate Theorem 1.

Example 1.1. Let ψ be the quadratic character of K = ℚ(\sqrt{21}) of conductor 𝔣_{ψ} := (3, \sqrt{21}) attached to the quadratic extension K(\sqrt{3 + \sqrt{21}}) of K, so that m = 2 and L = ℚ. The weight one modular form g attached to ψ is of level N = 63 and has for nebentype character the odd quadratic Dirichlet character χ_{7} of conductor 7. The ring class character ψ_{♡} is a genus character associated to K and the associated genus field is just the (narrow) Hilbert class field H = ℚ(\sqrt{−3}, \sqrt{−7}) of K.

The prime p = 5 is split in K, and the roots of the associated Hecke polynomial are α = 1 and β = −1. Hence the generalised eigenspace of g_{α} in S_{1}^{(5)}(63, χ_{7}) contains a normalised generalised eigenform g_{α}^{♭} which is unique up to scaling. The fast algorithms of [La] for calculating overconvergent modular forms were used to efficiently compute this generalised eigenform numerically with an accuracy of 50 significant 5-adic digits, producing a modular form whose first non-vanishing Fourier coefficient a_{2}(g_{α}^{♭}) is equal to 1.

For primes ℓ < 300 that split in K (including ℓ = p) it was observed that a_{ℓ}(g_{α}^{♭}) = 0. When χ_{K}(ℓ) = −1, it was observed that

(4)  a_{ℓ}(g_{α}^{♭}) = \frac{log_{p}(u(ψ_{♡}, ℓ))}{log_{p}(u(ψ_{♡}, 2))},

where u(ψ_{♡}, ℓ) denotes a suitable fundamental ℓ-unit of norm 1 in H/K, for all inert ℓ ≥ 2. (The logarithm of such a unit is unique up to sign.) At the ramified primes we observed

(5)  a_{3}(g_{α}^{♭}) = 0,  a_{7}(g_{α}^{♭}) = \frac{1}{2} · \frac{log_{p}(u(ψ_{♡}, 7))}{log_{p}(u(ψ_{♡}, 2))}.

Here the 2-unit is u(ψ_{♡}, 2) := (−3 + \sqrt{−7})/4. The numerical values of the first few non-zero coefficients a_{ℓ}(g_{α}^{♭}) for ℓ > 2 prime, and the values of u(ψ_{♡}, ℓ) verifying (4) and (5), are listed in the table below. The p-adic logarithms were calculated relative to the 5-adic embedding of H in which \sqrt{21} ≡ −1 mod 5.

\begin{matrix}ℓ & a_{ℓ}(g_{α}^{♭}) mod 5^{50} & u(ψ_{♡}, ℓ) \\ 7 & 20012844832326722144621655295530693 & (1 − 4\sqrt{−3})/7 \\ 11 & 9753260368539762436495550803834302 & (−3 − 4\sqrt{−7})/11 \\ 13 & 80089851328257507529397205421800237 & (−1 + 15\sqrt{−3})/26 \\ 19 & 5387308298676565974776485314728008 & (11 + 21\sqrt{−3})/38 \\ 23 & 7951947833969991753485495228957006 & (9 + 8\sqrt{−7})/23 \\ 29 & 59833461154145179184050173388767665 & (−27 − 4\sqrt{−7})/29 \\ 31 & 8202575240226165174943058721099781 & (−13 + 35\sqrt{−3})/62 \end{matrix}

Example 1.2. Let χ be a Dirichlet character of conductor 145 with order 4 at the prime 5 and order 2 at the prime 29. The space S_{1}(145, χ) is one-dimensional and spanned by the modular form

g = q + iq^{4} + iq^{5} + (−i − 1)q^{7} − iq^{9} + (−i + 1)q^{13} − q^{16} − q^{20} + · · · .

It is the theta series attached to a quartic character of K = ℚ(\sqrt{29}) ramified at one of the primes above (5). Level 145 is the smallest where one encounters weight one theta series attached to a character of a real quadratic field, but not to a character of any imaginary quadratic field. (There are two non-conjugate such forms, the other of which appears in [DLR, Example 4.1].)

The prime p = 13 is split in K and the roots of the Hecke polynomial for this prime are α = 1 and β = −i. We view g_{α} as a 13-adic modular form using the embedding of L = ℚ(i) into ℚ_{13} for which i ≡ 5 mod 13. The coefficients of the normalised eigenform g_{α}^{♭}, scaled so that a_{2}(g_{α})^{♭} = 1, are given in the second column of the table below for the inert primes ℓ = 3, 11, 17 and 19 of K.

The ring class field H of conductor 5 is a cyclic quartic extension of K given by

H = K(\sqrt{5}, δ) where δ^{2} = \frac{\sqrt{145} − 15}{32}.

Let σ be the generator of Gal (H/K) defined by

σ(\sqrt{5}) = −\sqrt{5}, σ(δ) = −\frac{1}{4}(3\sqrt{5} + \sqrt{29})δ.

We embed H in the quartic unramified extension

ℚ_{13^{4}} = ℚ(α) where α^{4} + 3α^{2} + 12α + 2 = 0

of ℚ_{13}, in such a way that

\sqrt{29} ≡ 9, \sqrt{5} ≡ 8α^{3} + 2α^{2} + 7α + 10, δ ≡ α^{3} + 5α^{2} + 6α + 10 (mod 13).

For ℓ = 3, 11, 17 and 19, it was verified that

a_{ℓ}(g_{α}^{♭}) = \frac{log_{13}(u(ψ_{♡}, ℓ))}{log_{13}(u(ψ_{♡}, 2))}

to 20-digits of 13-adic precision, where (denoting the group operation in L ⊗ H^{×} additively)

u(ψ_{♡}, ℓ) := u_{ℓ} + i ⊗ σ(u_{ℓ}) − σ^{2}(u_{ℓ}) − i ⊗ σ^{3}(u_{ℓ}),

for a suitable ℓ-unit u_{ℓ} of H. The 2-unit u_{2} is given by

u_{2} := \frac{1}{2}(−\sqrt{5} − \sqrt{29} + 6)δ + \frac{1}{8}(\sqrt{29} − 7)\sqrt{5} + \frac{1}{8}(\sqrt{29} + 1),

and the others are listed in the last column of the table below.

ℓ  a_{ℓ}(g_{α}^{♭}) mod 13^{20}  u_{ℓ}
3 12915196799386050150007 (\sqrt{5} + \sqrt{29} − 4)δ + \frac{1}{4}(\sqrt{29} − 4)\sqrt{5} + \frac{1}{4}(2\sqrt{29} − 13)
11 3524143318627577732842 (\frac{1}{4} ((\sqrt{29} + 1)\sqrt{5} + (−\sqrt{29} + 11)) δ + \frac{1}{4} (\sqrt{5} − 1))^{4}
17 229407992393437964510 ((16\sqrt{29} + 84)\sqrt{5} + (36\sqrt{29} + 200)) δ
+\frac{1}{4}(11\sqrt{29} + 63)\sqrt{5} + \frac{1}{4}(15\sqrt{29} + 83)
19 15142834827825079965585 (\frac{1}{4} ((3\sqrt{29} − 13)\sqrt{5} + (−15\sqrt{29} + 85)) δ
+\frac{1}{8}(3\sqrt{29} − 15)\sqrt{5} + \frac{1}{8}(7\sqrt{29} − 35))^{2}

Remark 1.3. Theorem 1 was inspired by the work of Bellaiche and Dimitrov [BD] on the geometry of the Coleman-Mazur eigencurve at classical weight one points. Theorem 1.1. of loc.cit. asserts that this eigencurve is étale over weight space at any classical weight one point for which p is regular, unless g is a theta series of a real quadratic field in which p splits. This explains why Theorem 1 above focusses on this setting, the only “regular at p” scenario where a non-trivial generalised eigenform co-exists with its classical weight one counterpart.

Remark 1.4. Every (totally odd) ring class character of K can be written as ψ′/ψ for a suitable ray class character ψ (of mixed signature). (Cf. for example Lemma 6.7 of [DR].) Hence all ring class fields of K can be generated by exponentials of the Fourier coefficients of g_{α}^{♭} for suitable real dihedral newforms g of weight one attached to K, in much the same way that the ring class fields of a quadratic imaginary K can be generated by values of the modular function j(z) at arguments z ∈ K. The existence of the generalised eigenforms g_{α}^{♭} of Theorem 1 can therefore be envisioned as an eventual pathway to “explicit class field theory” for real quadratic fields.

Remark 1.5. Comparing (3) with the formula for the coefficient denoted a_{n}(ϕ) in Thm. 1 of [KRY] reveals a strong analogy between Theorem 1 and a fundamental result of Kudla, Rapoport, and Yang on the Fourier coefficients of central derivatives of incoherent Eisenstein series of weight one. When Eisenstein series are replaced by a weight one cuspidal newform g, the notion of mock modular forms provides a fruitful substitute for Kudla’s incoherent Eisenstein series. The mock modular form attached to g, denoted g^{♯}, is the holomorphic part of a Harmonic weak Maass form mapping to g under an appropriate differential operator. Recent work of Bill Duke, Stephan Ehlen, Yingkun Li, and Maryna Viazovska relates the Fourier coefficients of g^{♯} to the complex logarithms of algebraic numbers belonging to the field cut out by the adjoint Ad_{g} of the two-dimensional Artin representation attached to g. The articles [DuLi], [E1], [E2], and [Vi] focus largely on the case where g is a theta series of an imaginary quadratic field. The algebraic numbers whose logarithms arise in the Fourier expansion of g^{♯} then belong to abelian extensions of imaginary rather than real quadratic fields, and the proofs in loc.cit. rely crucially on the theory of complex multiplication. Such a theory is unavailable for real quadratic fields, and the techniques exploited in the present work, based on deformations of Galois representations, are thus quite different, substantially simpler, and fundamentally p-adic in nature.

Remark 1.6. The concluding section of [DuLi] makes an experimental study of g^{♯} when g is an octahedral newform of level 283, in which the Stark conjecture for the Artin L-function attached to Ad_{g} (as described, for instance, in §6 of [St]) plays an essential role. The case where g is the theta series of a character of a real quadratic field is treated extensively in [Li], where it is explained that the Fourier coefficients of g^{♯} are expected to be logarithms of algebraic numbers belonging to K rather that to some non-trivial ring class field of K. This (disappointing, at least for explicit class field theory) feature of the archimedean context can be traced to the fact that the L-function of the representation induced from ψ_{♡} does not vanish at s = 0, and that there are no non-trivial Stark units in the ψ_{♡}-isotypic part of the unit group of H. As pointed out in [CV] and [BD], it is precisely this phenomenon which leads to the existence of the overconvergent generalised eigenform g_{α}^{♭} of Theorem 1.

Remark 1.7. There are other instances where the properties of weak harmonic Maass forms resonate with those of overconvergent modular forms like g_{α}^{♭}. For example, Bruinier and Ono study the Fourier coefficients of the holomorphic part of the weak harmonic Maass form of weight 1/2 attached to a classical modular form g of weight 3/2 whose Shimura lift has rational coefficients and hence corresponds to an elliptic curve over ℚ. The main result of [BO] relates these Fourier coefficients to Heegner points on the elliptic curve defined over a varying collection of quadratic fields. The p-adic logarithms of the same Heegner points are realised in [DT] as the Fourier coefficients of a “modular form of weight 3/2+ε” arising as an infinitesimal p-adic deformation of g over weight space. It would be interesting to flesh out the rather tantalising analogy between weak harmonic Maass forms and p-adic deformations of classical eigenforms. To what extent can the latter be envisaged as non-archimedean counterparts of the former?

Remark 1.8. The ℓ-units u(ψ_{♡}, ℓ) are precisely the Gross-Stark units studied in [DD], which proposes a conjectural analytic formula for their ℓ-adic logarithms refining Gross’s ℓ-adic analogue of the Stark conjectures. Theorem 1 above holds unconditionally and concerns the p-adic logarithms of the same ℓ-units, for primes p ≠ ℓ. It therefore bears no direct connection with the Gross-Stark conjecture, even though its proof, like that of the Gross-Stark conjecture given in [DDP] and [Ve], relies crucially on the deformation theory of p-adic Galois representations.

We close the introduction with the following corollary of Theorem 1, which shows that a naive version of the q-expansion principle fails for the generalised eigenspace S_{1}^{(p)}(N, χ)[[g_{α}]].

Corollary 2. Let S ⊂ S_{1}^{(p)}(N, χ)[[g_{α}]] be a ℚ̄-vector space which is stable under all the Hecke operators. Then either S is contained in S_{1}(Np, χ)[g_{α}], or it is infinite-dimensional over ℚ̄.

Proof. If S is not contained in S_{1}(Np, χ)[g_{α}], it contains a non-zero (not necessarily normalised) generalised eigenform h of the form g_{α}^{♭} + λg_{α}, where g_{α}^{♭} is a normalised generalised eigenform and λ ∈ ℂ_{p}. By (1) combined with the stability of S under the Hecke operators,

(T_{ℓ} − a_{ℓ}(g_{α}))h = a_{ℓ}(g_{α}^{♭})g_{α}

also belongs to S, for all primes ℓ ∤ Np. Theorem 1 implies that the forms Ω log_{p}(u(ψ_{♡}, ℓ))g_{α} (for a suitable Ω ≠ 0) belong to S, for all ℓ ∤ Np which are inert in K. The corollary follows from the linear independence over ℚ̄ of the p-adic logarithms of algebraic numbers.

Acknowledgements. The third author was supported by Grant MTM2012-34611.

2. Proof of Theorem 1

The theta series g = θ_{ψ} corresponds to an odd, irreducible, two-dimensional Artin representation

ϱ : G_{ℚ}⟶GL_{2}(L),

obtained by inducing ψ from G_{K} to G_{ℚ}. The two-dimensional L-vector space underlying ϱ decomposes as a direct sum of one-dimensional representations ψ and ψ′ when restricted to G_{K}. Fix an element τ_{0} in the complement G_{ℚ} \ G_{K} of G_{K} in G_{ℚ}, and let e_{1} and e_{2} be eigenvectors for the G_{K}-action attached to ψ and ψ′ respectively, chosen so that e_{1} = ϱ(τ_{0})e_{2}. Relative to this basis,

(6)  ϱ_{|G_{K}} = ( \begin{matrix}ψ & 0 \\ 0 & ψ′ \end{matrix} ) ,  ϱ_{|G_{ℚ}\G_{K}} = ( \begin{matrix}0 & η′ \\ η & 0 \end{matrix} )

where η and η′ are L-valued functions on G_{ℚ} \ G_{K} given by the rule η(τ) := ψ(τ_{0}τ) and η′(τ) := ψ(ττ_{0}^{−1}).

Let L_{p} denote a p-adic completion of L and let L_{p}[ε] denote the ring of dual numbers, for which ε^{2} = 0. The theorems of Cho-Vatsal [CV] and Bellaiche-Dimitrov [BD] show that the tangent space H_{ord}^{1}(ℚ, Ad^{0}(ϱ)) of the universal ordinary deformation space attached to ϱ with constant determinant, denoted as t_{D′} in [BD, Definition 2.1], is one-dimensional over L_{p}. This means that there is a unique (up to conjugation, and replacing ε by a non-zero multiple) ordinary lift

ϱ̃: G_{ℚ}⟶GL_{2}(L_{p}[ε])

of ϱ satisfying

(7)  det(ϱ̃) = det(ϱ).

We begin by observing that ϱ̃ can be written in the form

(8)  ϱ̃_{|G_{K}} = ( \begin{matrix}ψ & ψ′κ′ · ε \\ ψκ · ε & ψ′ \end{matrix} ) ,  ϱ̃_{|G_{ℚ}\G_{K}} = ( \begin{matrix}d_{1} · ε & η′ \\ η & d_{2} · ε \end{matrix} ) .

The fact that the diagonal entries of ϱ̃_{|G_{K}} remain “constant”, i.e., belong to L and are equal to those of ϱ, follows from (7) and the fact that this is true of the lower right-hand matrix entry, which is unramified at p by the ordinarity of ϱ̃, since there are no non-trivial homomorphisms from G_{K} to L_{p} that are unramified at a prime above p. It likewise ensures that the antidiagonal entries of ϱ̃_{|G_{ℚ}\G_{K}} are equal to those of ϱ_{|G_{ℚ}\G_{K}}, i.e., are described by the functions η and η′.

Lemma 2.1. The functions κ and κ′ belong to H^{1}(K, L_{p}(ψ_{♡}^{−1})) and to H^{1}(K, L_{p}(ψ_{♡})) respectively. Their restrictions to G_{H} are related by the rule

(9)  κ′(σ) = \frac{η′(τ)}{η(τ)}κ(τστ^{−1}),

for any τ ∈ G_{ℚ} \ G_{K}. The class κ is unramified at 𝔭, and the class κ′ is unramified at 𝔭′.

Proof. The first assertion is standard, and follows directly from the fact that ϱ is a homomorphism on G_{K}. The second follows by a similar argument, from a direct calculation of the anti-diagonal entries of ϱ̃(τστ^{−1}) = ϱ̃(τ)ϱ̃(σ)ϱ̃(τ)^{−1}, using the fact that ψ(σ) = ψ′(σ) when σ ∈ G_{H} to simplify the calculation. Finally, the fact that ϱ is ordinary at the rational prime p (relative to a p-adic embedding of H which sends K to its completion at 𝔭) implies that κ is unramified at 𝔭. The relation (9) between κ and κ′ implies that κ′ is unramified at 𝔭′, since any τ ∈ G_{ℚ} \ G_{K} interchanges the primes 𝔭 and 𝔭′ of K above p. □

The space H_{𝔭′}^{1}(K, L_{p}(ψ_{♡})) of global classes that are unramified at 𝔭′ is one-dimensional over L_{p}, and restriction to G_{H} gives an isomorphism

(10)  H_{𝔭′}^{1}(K, L_{p}(ψ_{♡}))⟶ hom_{𝔭}′(G_{H}, L_{p}(ψ_{♡}))^{Gal (H/K)}.

The target of this restriction map can be described explicitly in terms of global class field theory, which identifies hom_{𝔭}′(G_{H}, L_{p}) with the continuous homomorphisms from the group 𝔸_{H}^{×} of idèles of H which are trivial on principal elements and on O_{H_{v}}^{×} for all primes v ∤ 𝔭 of H. The space hom_{𝔭}′(G_{H}, L_{p}) is of dimension t := \frac{[H:K]}{2}. To describe it more concretely, let σ_{∞} denote complex conjugation in G = Gal (H/K) (which is well-defined, independently of the choice of a complex embedding of H). Given a choice σ_{1}, …, σ_{t} of coset representatives for ⟨σ_{∞}⟩ in G, let κ_{j} denote the idèle class character whose restriction to O_{H_{v}}^{×} is trivial for all v ∤ 𝔭 and whose restriction to H_{𝔭}^{×} := (H ⊗_{K} K_{𝔭})^{×} is equal to

κ_{j}(a) = log_{p}(a^{σ_{j}}/a^{σ_{j}σ_{∞}}).

The elements κ_{1}, …, κ_{t} form an L_{p} basis for hom_{𝔭}′(G_{H}, L_{p}), and the one-dimensional target of (10) is spanned by the function

κ_{ψ_{♡}} := ∑_{j=1}^{t} ψ_{♡}^{−1}(σ_{j})κ_{j},

whose restriction to H_{𝔭}^{×} is given by

(11)  κ_{ψ_{♡}} (a) = ∑_{σ∈G} ψ_{♡}^{−1}(σ) · log_{p}(a^{σ}).

Lemma 2.2. For all primes ℓ of ℚ that are inert in K, and all λ ∈ Σ_{ℓ},

κ_{ψ_{♡}} (Frob_{λ}) = log_{p}(u(ψ_{♡}, λ)).

Proof. By global class field theory, the value of κ_{ψ_{♡}} (Frob_{λ}) is equal to the image of κ_{ψ_{♡}} on the idèle which is equal to the inverse of a local uniformiser of H_{λ} at λ, and to 1 everywhere else. This idèle class agrees, modulo the kernel of κ_{ψ_{♡}}, with the idèle which is trivial at all places of H except 𝔭 and equal to u(λ) in H_{𝔭}^{×}. The result now follows from (11). □

Assume from now on that κ′ has been scaled so that it is equal to the class κ_{ψ_{♡}} of (11). The Galois representation ϱ̃ comes from an overconvergent weight one eigenform

g̃_{α} = g_{α} + g_{α}^{♭} · ε

with coefficients in the ring L_{p}[ε] of dual numbers. The modular form g_{α}^{♭} is a generalised eigenform in the sense of the introduction, and its Fourier coefficients acquire a Galois-theoretic interpretation via the identity

(12)  trace(ϱ̃(σ_{ℓ})) = a_{ℓ}(g_{α}) + a_{ℓ}(g_{α}^{♭}) · ε,

where ℓ ∤ N is any rational prime and σ_{ℓ} ∈ G_{ℚ} is a Frobenius element at ℓ (attached to an arbitrary embedding of ℚ̄ into ℚ̄_{ℓ}). Note that although ϱ̃ is always ramified at p, (8) ensures that the trace of ϱ̃(σ_{p}) does not depend on the choice of Frobenius element at p.

If ℓ ∤ N is a prime of K for which χ_{K}(ℓ) = 1, and hence factors as a product of two primes λ and λ′ of K, then the Frobenius element Frob_{ℓ} attached to ℓ belongs to G_{K}, and hence by (8),

trace(ϱ̃(Frob_{ℓ})) = ψ(λ) + ψ′(λ) = a_{ℓ}(g_{α}).

The first assertion in Theorem 1 when ℓ is a split prime follows in light of (12).

Let now ℓ ∤ N be a prime of K for which χ_{K}(ℓ) = −1. Let H_{ϱ} be the abelian extension of K which is fixed by the kernel of ϱ. The function η gives rise to a function on the primes of H_{ϱ} above ℓ by setting η(λ) := η(σ_{λ}), where σ_{λ} ∈ Gal (H_{ϱ}/ℚ) is the ℓ-power Frobenius automorphism attached to the prime λ. Observe that, if λ_{1} = σλ_{2} for some σ ∈ Gal (H_{ϱ}/K), then

(13)  η(λ_{1}) = η(σ_{λ_{1}}) = η(σσ_{λ_{2}}σ^{−1}) = ψ_{♡}^{−1}(σ)η(λ_{2}).

In particular, the value η(λ) depends only on the restriction of the ideal λ to the ring class field H ⊂ H_{ϱ}, and therefore η can also be viewed as a function on the sets Σ_{ℓ} of primes of H that were described in the introduction.

Fix now an embedding ι_{ℓ} : ℚ̄ ↪ ℚ̄_{ℓ}. It determines a Frobenius element σ_{ℓ} ∈ G_{ℚ}, whose square is the Frobenius element in G_{K} attached to the prime ℓ of K. Let λ ∈ Σ_{ℓ} denote the prime of H above ℓ determined by ι_{ℓ}. By (12) and (8),

(14)  trace(ϱ̃(σ_{ℓ})) = (d_{1}(σ_{ℓ}) + d_{2}(σ_{ℓ})) · ε = a_{ℓ}(g_{α}^{♭}) · ε,

while a direct calculation of ϱ̃(σ_{ℓ}^{2}) using (8) shows that

( \begin{matrix}ψ(σ_{ℓ}^{2}) & ψ′(σ_{ℓ}^{2})κ′(σ_{ℓ}^{2}) · ε \\ ψ(σ_{ℓ}^{2})κ(σ_{ℓ}^{2}) · ε & ψ′(σ_{ℓ}^{2}) \end{matrix} ) = ( \begin{matrix}η(σ_{ℓ})η′(σ_{ℓ}) & η′(σ_{ℓ})(d_{1}(σ_{ℓ}) + d_{2}(σ_{ℓ})) · ε \\ η′(σ_{ℓ})(d_{1}(σ_{ℓ}) + d_{2}(σ_{ℓ})) · ε & η(σ_{ℓ})η′(σ_{ℓ}) \end{matrix} ) .

Comparing the upper right hand corners in this equality of matrices yields

ψ′(σ_{ℓ}^{2})κ′(σ_{ℓ}^{2}) = η′(σ_{ℓ})(d_{1}(σ_{ℓ}) + d_{2}(σ_{ℓ})),

and hence by (14),

a_{ℓ}(g_{α}^{♭}) = η(σ_{ℓ})κ′(σ_{ℓ}^{2}) = η(λ)κ_{ψ_{♡}} (Frob_{λ}) = η(λ) log_{p} u(ψ_{♡}, λ) = log_{p} u(ψ_{♡}, ℓ),

where the penultimate equality follows from Lemma 2.2, and the last from the definition of u(ψ_{♡}, ℓ) given in (2). The first assertion in Theorem 1 follows. The second assertion is a direct consequence of the multiplicativity properties of the Fourier coefficients of g̃_{α}, now that the values of a_{ℓ}(g̃_{α}) for ℓ prime are known.

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H. D.: McGill University, Montreal, Canada

E-mail address: darmon@math.mcgill.ca

A. L.: University of Oxford, U. K.

E-mail address: lauder@maths.ox.ac.uk

V. R.: Universitat Politècnica de Catalunya, Barcelona, Spain

E-mail address: victor.rotger@upc.edu