FIRST ORDER p-ADIC DEFORMATIONS OF WEIGHT ONE NEWFORMS

HENRI DARMON, ALAN LAUDER AND VICTOR ROTGER

Abstract. This article studies the first-order p-adic deformations of classical weight one newforms, relating their fourier coefficients to the p-adic logarithms of algebraic numbers in the field cut out by the associated projective Galois representation.

Contents

Introduction 1

Part A. The regular setting 2 1. The general case 3 2. CM forms 10 3. RM forms 17

Part B. The irregular setting 19 4. Generalised eigenspaces 20 5. The general case 22 6. CM forms 25 7. RM forms 28 References 30

Introduction

Let g be a classical cuspidal newform of weight one, level N and nebentypus character χ : (ℤ/Nℤ)^{×} → ℚ_{p}^{×}, with fourier expansion

g(q) = ∑_{n=1}^{∞} a_{n}q^{n}.

The p-stabilisations of g attached to a rational prime p ∤ N are the eigenforms of level Np defined by

(1)  g_{α}(q) := g(q) − β · g(q^{p}),  g_{β}(q) := g(q) − α · g(q^{p}),

where α and β are the (not necessarily distinct) roots of the Hecke polynomial

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

The forms g_{α} and g_{β} are eigenvectors for the Atkin U_{p} operator, with eigenvalues α and β respectively. Since α and β are roots of unity, these eigenforms are both ordinary at p.

An important feature of classical weight one forms is that they are associated to odd, irreducible, two-dimensional Artin representations, via a construction of Deligne-Serre. Let ϱ_{g} : G_{ℚ}⟶GL_{2}(ℂ) denote this Galois representation, and write V_{g} for the underlying representation space.

A fundamental result of Hida asserts the existence of a p-adic family of ordinary eigenforms specialising to g_{α} (or to g_{β}) in weight one. Bellaiche and Dimitrov [BD] later established the uniqueness of this Hida family, under the hypothesis that g is regular at p , i.e., that α ≠ β, or equivalently, that the frobenius element at p acts on V_{g} with distinct eigenvalues. In the intriguing special case where g is the theta series of a character of a real quadratic field F in which the prime p is split, the result of Bellaiche-Dimitrov further asserts that the unique ordinary first-order infinitesimal p-adic deformation of g is an overconvergent (but not classical) modular form of weight one. In [DLR2], the Fourier coefficients of this nonclassical form were expressed as p-adic logarithms of algebraic numbers in a ring class field of F, suggesting that a closer examination of such deformations could have some relevance to explicit class field theory for real quadratic fields.

The primary purpose of this note is extend the results of [DLR2] to general weight one eigenforms.

Part A considers the regular setting where α ≠ β, in which the results exhibit a close analogy to those of [DLR2].

Part B takes up the case where g is irregular at p. Here the results are more fragmentary and less definitive. Let S_{1}^{(p)}(N, χ) denote the space of p-adic overconvergent modular forms of weight 1, level N, and character χ, and let S_{1}^{(p)}(N, χ)[[g]] denote the generalised eigenspace attached to the system of Hecke eigenvalues of an irregular weight one form g ∈ S_{1}(N, χ). The main conjecture of the second part asserts that S_{1}^{(p)}(N, χ)[[g]] is always four dimensional, with a two-dimensional subspace consisting of classical forms. Under this conjecture, an explicit description of the elements of the generalised eigenspace in terms of their q-expansions is provided. The resulting concrete description of the generalised eigenspace that emerges from Part B is an indispensible ingredient in the extension of the “elliptic Stark conjectures” of [DLR1] to the irregular setting that will be presented in [DLR3].

Acknowledgements. The first author was supported by an NSERC Discovery grant, and the third author was supported by Grant MTM2015-63829-P. The second author would like that thank Takeshi Saito of Tokyo University and Kenichi Bannai of Keio University for their hospitality. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682152).

Part A. The regular setting

Let Λ = ℤ_{p}[[1 + pℤ_{p}]] denote the Iwasawa algebra, and let

W := Hom_{cts}(1 + pℤ_{p}, ℂ_{p}^{×}) = Hom_{alg}(Λ, ℂ_{p})

denote the associated weight space. For each k ∈ ℤ_{p}, write ν_{k} ∈ W for the “weight k” homomorphism sending the group-like element a ∈ 1 + pℤ_{p} to a^{k−1}. The rule λ(k) := ν_{k}(λ) realises elements of Λ as analytic functions on ℤ_{p}. The spectrum W̃ := Hom_{alg}(Λ̃, ℂ_{p}) of a finite flat extension Λ̃ of Λ is equipped with a “weight map”

w : W̃⟶W

of finite degree. A ℚ_{p}-valued point x ∈ W̃ is said to be of weight k if w(x) = ν_{k}, and is said to be étale over W if the inclusion Λ ⊂ Λ̃ induces an isomorphism between Λ and the completion of Λ̃ at the kernel of x, denoted Λ̃_{x}. An element of this completion thus gives rise to an analytic function of k ∈ ℤ_{p} in a natural way.

A Hida family is a formal q-series

g := ∑ a_{n}q^{n} ∈ Λ̃[[q]]

with coefficients in a finite flat extension Λ̃ of Λ, specialising to a classical ordinary eigenform of weight k at almost all points x of W̃ of weight k ∈ ℤ^{≥2}. Two Hida families g_{1} ∈ Λ̃_{1}[[q]] and g_{2} ∈ Λ̃_{2}[[q]] are regarded as equal if the Λ-algebras Λ̃_{1} and Λ̃_{2} can be embedded in a common extension Λ̃ in such a way that g_{1} and g_{2} are identified. A well known theorem of Hida and Wiles asserts the existence of a Hida family specialising to g_{α} in weight one. The following uniqueness result for this Hida family plays an important role in our study.

Theorem (Bellaiche, Dimitrov). Assume that the weight one form g is regular at p, and let x_{α} and x_{β} denote the distinct points on W̃ attached to g_{α} and g_{β} respectively. Then

(a) The curve W̃ is smooth at x_{α} and x_{β}, and in particular there are unique Hida families g_{α}, g_{β} ∈ Λ̃[[q]] specialising to g_{α} and g_{β} at x_{α} and x_{β} respectively.

(b) The weight map w : W̃⟶W is furthermore étale at x_{α} if any only if

(†) g is not the theta series of a character of a real quadratic K in which p splits.

The setting where g is regular at p but w is not étale at x_{α} has been treated in [DLR2], and the remainder of Part A will therefore focus on the scenarios where (†) is satisfied. In that case, after viewing elements of the completion Λ̃_{x_{α}} of Λ̃ at x_{α} as analytic functions of the “weight variable” k, one may consider the canonical q-series

g′_{α} := (\frac{d}{dk} g_{α})_{k=1}

representing the first-order infinitesimal ordinary deformation of g at the weight one point x_{α}, along this canonical “weight direction”. The q-series g′_{α} is analogous to the overconvergent generalised eigenform considered in [DLR2], with the following differences:

(a) While the overconvergent generalised eigenform of [DLR2] is a (non-classical, but overconvergent) modular form of weight one, such an interpretation is not available for the q-series g′_{α}, which should rather be viewed as the first order term of a “modular form of weight 1 + ε”.

(b) In the non-étale setting of [DLR2], the absence of a natural local coordinate with respect to which the derivative would be computed meant that the overconvergent generalised eigenform of loc.cit. could only be meaningfully defined up to scaling by a non-zero multiplicative factor. This ambiguity is not present in the definition of g′_{α}, whose fourier coefficients are therefore entirely well-defined.

The main results of Part A give explicit formulae for these fourier coefficients: they are stated in Theorems 1.10, 2.1, 2.3, and 3.1 below.

1. The general case

The goal of this section is to describe a general formula for the fourier coefficients of g′_{α}.

The Artin representation V_{g} can be realised as a two-dimensional L-vector space, where L is a finite extension of ℚ, contained in a cyclotomic field. Let W_{g} = hom(V_{g}, V_{g}) denote the adjoint equipped with its usual conjugation action of G_{ℚ}, denoted

σ · w := ϱ_{g}(σ)wϱ_{g}(σ)^{−1}, σ ∈ G_{ℚ}, w ∈ W_{g}.

Let H ⊂ H_{g} denote the finite Galois extensions of ℚ cut out by the representations W_{g} and V_{g} respectively, and write G := Gal (H/ℚ).

For notational simplicity, the following assumption is made in the rest of this paper:

Assumption 1.1. The prime p splits completely in the field L of coefficients of the Artin representation V_{g}.

This assumption amounts to a simple congruence condition on p. The choice of an embedding of L into ℚ_{p}, which is fixed henceforth, will allow us, when it is convenient, to view V_{g} and W_{g} as representations of G_{ℚ} with coefficients in ℚ_{p}, and the weight one form g as a modular form with fourier coefficients in ℚ_{p} rather than in L. The ℚ_{p}-vector spaces V_{g} and W_{g} are thus equipped with natural G_{ℚ}-stable L-rational structures, denoted V_{g}^{L} and W_{g}^{L} respectively.

An embedding of ℚ̄ into ℚ̄_{p} is fixed once and for all, determining a prime ℘ of H and of H_{g} above p, and an associated frobenius element τ_{℘} in Gal (H_{g}/ℚ) and in G. Let G_{℘} ⊂ G be the decomposition subgroup generated by τ_{℘}.

The representations V_{g} and W_{g} admit the following decompositions as τ_{℘}-modules:

V_{g} = V_{g}^{α} ⊕ V_{g}^{β}, W_{g} = W_{g}^{αα} ⊕ W_{g}^{αβ} ⊕ W_{g}^{βα} ⊕ W_{g}^{ββ},

where V_{g}^{α} and V_{g}^{β} denote the α and β-eigenspaces for the action of τ_{℘} on V_{g}, and

W_{g}^{ξη} := hom(V_{g}^{ξ}, V_{g}^{η}), for ξ, η ∈ {α, β}

is a G_{℘}-stable line, on which τ_{℘} acts with eigenvalue η/ξ. Let

W_{g}^{ord} := hom(V_{g}/V_{g}^{α}, V_{g}) = W_{g}^{βα} ⊕ W_{g}^{ββ}.

Of course, W_{g}^{ord} is stable under the action of G_{℘} but not under the action of G.

We propose to give a general formula for the ℓ-th fourier coefficient of g′_{α} as the trace of a certain explicit endomorphism of V_{g}, which is constructed via a series of lemmas. In the lemma below, we let G act on O_{H}^{×} ⊗ W_{g} diagonally on both factors in the tensor product.

Lemma 1.2. The ℚ_{p}-vector space (O_{H}^{×} ⊗ W_{g})^{G} of G-invariant vectors is one-dimensional.

Proof. Let G_{∞} be the subgroup of G generated by a complex conjugation c, which has order two, since V_{g} is odd. By Dirichlet’s unit theorem, the global unit group O_{H}^{×} ⊗ℚ_{p} is isomorphic to Ind_{G_{∞}}^{G}(ℚ_{p}) − ℚ_{p} as a ℚ_{p}[G]-module. Let W_{g}^{0} denote the three-dimensional representation of G consisting of trace zero endomorphisms of V_{g}. As a representation of G, we have W_{g} = W_{g}^{0} ⊕ ℚ_{p}, and W_{g}^{0} does not contain the trivial representation as a constituent. By Frobenius reciprocity,

dim_{ℚ_{p}}((O_{H}^{×} ⊗ W_{g})^{G}) = dim_{ℚ_{p}}((O_{H}^{×} ⊗ W_{g}^{0})^{G}) = dim_{ℚ_{p}}((W_{g}^{0})^{c=1}) = 1.

The result follows. □

Assume that the field L of coefficients is large enough so that the semisimple ring L[G] becomes isomorphic to a direct sum of matrix algebras over L. The L[G]-module O_{H}^{×} ⊗ L decomposes as a direct sum of V -isotypic components,

O_{H}^{×} ⊗ L = ⊕_{V} O_{H}^{×}[V],

where V runs over the irreducible representations of G, and O_{H}^{×}[V] denotes the largest subrepresentation of O_{H}^{×} ⊗ L which is isomorphic to a direct sum of copies of V as an L[G]-module. For a general, not necessarily irreducible, representation W, the module O_{H}^{×}[W] is defined as the direct sum of the O_{H}^{×}[V] as V ranges over the irreducible constituents of W. Because W_{g} (viewed, for now, as a representation with coefficients in L) is self-dual, Lemma 1.2 can be recast as the assertion that O_{H}^{×}[W_{g}] is isomorphic to a single irreducible subrepresentation of W_{g}. More precisely:

• In the case of “exotic weight one forms” where ϱ_{g} has non-dihedral projective image (isomorphic to A_{4}, S_{4} or A_{5}), then

(2)  O_{H}^{×}[W_{g}] = O_{H}^{×}[W_{g}^{0}] ≃ W_{g}^{0},

and hence is three-dimensional.

• If ϱ_{g} is induced from a character ψ_{g} of an imaginary quadratic field K, then

W_{g} = L ⊕ L(χ_{K}) ⊕ V_{ψ},

where χ_{K} is the odd quadratic Dirichlet character associated to K and V_{ψ} is the twodimensional representation of G induced from the ring class character ψ = ψ_{g}/ψ′_{g} which cuts out the abelian extension H of K. The representation V_{ψ} is irreducible if and only if ψ is non-quadratic, and in that case,

(3)  O_{H}^{×}[W_{g}] = O_{H}^{×}[V_{ψ}] ≃ V_{ψ}.

In the special case where ψ is quadratic, the representation V_{ψ} further decomposes as the direct sum of one-dimensional representations attached to an even and an odd quadratic Dirichlet character, denoted χ_{1} and χ_{2} respectively. That special case, in which V_{g} is also induced from a character of the real quadratic field cut out by χ_{1}, is thus subsumed under (4) below.

• If ϱ_{g} is induced from a character ψ_{g} of a real quadratic field F, then

W_{g} = L ⊕ L(χ_{F}) ⊕ V_{ψ}, V_{ψ} := Ind_{F}^{ℚ}(ψ), ψ := ψ_{g}/ψ′_{g},

and one always has

(4)  O_{H}^{×}[W_{g}] = O_{H}^{×}[χ_{F}] ≃ L(χ_{F}),

i.e., O_{H}^{×}[W_{g}] is generated by a fundamental unit of F.

Let U_{g}^{×} be any generator of the one-dimensional ℚ_{p}-vector space (O_{H}^{×} ⊗ W_{g})^{G} and let

(5)  U_{g} := (log_{℘} ⊗id)(U_{g}^{×}) ∈ H_{℘} ⊗ W_{g}

be the image of this vector under the linear map

log_{℘} ⊗id : O_{H}^{×} ⊗ W_{g}⟶H_{℘} ⊗ W_{g},

where log_{℘} is the p-adic logarithm on the ℘-adic completion H_{℘} of H at ℘.

Lemma 1.3. There exists a non-zero endomorphism A ∈ H_{℘} ⊗ W_{g} satisfying the following conditions:

(a) Trace(AU_{g}) = 0.

(b) A belongs to H_{℘} ⊗ W_{g}^{ord}, i.e., A(V_{g}^{α}) = 0. This endomorphism is unique up to scaling.

Proof. The space H_{℘} ⊗ W_{g} is four-dimensional over H_{℘} and the conditions in Lemma 1.3 amount to three linear conditions on A. More precisely, choose a τ_{℘}-eigenbasis (v_{α}, v_{β}) for V_{g} for which

τ_{℘}v_{α} = αv_{α}, τ_{℘}v_{β} = βv_{β}.

Relative to this basis, the endomorphism U_{g} is represented by a matrix of the form

U_{g} : ( \begin{matrix}log_{℘}(u_{1}) & log_{℘}(u_{β/α}) \\ log_{℘}(u_{α/β}) & − log_{℘}(u_{1}) \end{matrix} ) ,

where u_{1}, u_{α/β}, and u_{β/α} are generators of O_{H}^{×}[W_{g}] which (when non-zero) are eigenvectors for τ_{℘}, satisfying

τ_{℘}(u_{1}) = u_{1},  τ_{℘}(u_{β/α}) = (β/α)u_{β/α},  τ_{℘}(u_{α/β}) = (α/β)u_{α/β}.

The endomorphism A satisfies condition (b) above if and only if the matrix representing it in the basis (v_{α}, v_{β}) is of the form

A : ( \begin{matrix}0 & x \\ 0 & y \end{matrix} ) , x, y ∈ H_{℘},

and condition (a) implies the further linear relation

(6)  log_{℘}(u_{α/β}) · x − log_{℘}(u_{1}) · y = 0.

The injectivity of log_{℘} : O_{H}^{×} ⊗ L⟶H_{℘}, which follows from the linear independence over ℚ̄ of logarithms of algebraic numbers, implies that the coefficients log_{℘}(u_{α/β}) and log_{℘}(u_{1}) in (6) vanish simultaneously if and only if

u_{α/β} = u_{1} = 0

in O_{H}^{×} ⊗ L, i.e., if and only if O_{H}^{×}[W_{g}] is one-dimensional over L and generated by u_{β/α}. This immediately rules out (2) and (3) as scenarios for the structure of O_{H}^{×}[W_{g}], leaving only (4). Hence, V_{g} is induced from a character of a real quadratic field F. In that case, the lines spanned by u_{α/β} and u_{β/α} are interchanged under the action of any reflection in G, and hence the condition u_{α/β} = 0 implies that u_{β/α} = 0 as well, thus forcing the vanishing of the full O_{H}^{×}[W_{g}]. This contradiction to Lemma 1.2 shows that (6) imposes a non-trivial linear condition on x and y, and therefore that A is unique up to scaling. □

Lemma 1.4. Let A be any element of H_{℘} ⊗W_{g} satisfying the conditions in Lemma 1.3. Then the following are equivalent:

(a) Trace(A) ≠ 0;

(b) The representation ϱ_{g} is not induced from a character of a real quadratic field in which the prime p splits.

Proof. The vanishing of Trace(A) is equivalent to the vanishing of the entry y in (6), and hence to the vanishing of log_{℘}(u_{α/β}), and therefore of u_{α/β} and u_{β/α} as well. This implies that O_{H}^{×}[W_{g}] is one-dimensional and generated by u_{1}. As in the proof of Lemma 1.3, this rules out (2) and (3), leaving only (4) as a possibility, i.e., V_{g} is necessarily induced from a character of a real quadratic field F. Furthermore, τ_{℘} fixes the group O_{H}^{×}[W_{g}] generated by the fundamental unit of F, which occurs precisely when p splits in F. The lemma follows. □

Assume from now on that the equivalent conditions of Lemma 1.4 hold. One can then define A_{g} ∈ H_{℘} ⊗ W_{g} to be the unique H_{℘}^{×}- multiple of A satisfying

Trace(A_{g}) = 1.

As in lemma 1.2, H_{℘} ⊗ W_{g} is endowed with the diagonal action of G_{℘} which acts on both H_{℘} and on W_{g} in a natural way. Given A ∈ H_{℘} ⊗ W_{g} and σ ∈ G_{℘}, let us write ^{σ}A for the image of A by the action of σ on the first factor H_{℘}, and σ · A_{g} for the image of A by the action of σ by conjugation on the second factor W_{g}.

Lemma 1.5. The endomorphism A_{g} belongs to the space (H_{℘} ⊗ W_{g})^{G_{℘}} of G_{℘}-invariants for the diagonal action of G_{℘} on H_{℘} ⊗ W_{g}, i.e.,

τ_{℘}A_{g} = τ_{℘}^{−1} · A_{g}.

Proof. Relative to the ℚ_{p}-basis for V_{g} used in the proof of Lemma 1.3, the endomorphism A_{g} is represented by a matrix of the form

( \begin{matrix}0 & \frac{log_{℘}(u_{1})}{log_{℘}(u_{α/β})} \\ 0 & 1 \end{matrix} ) .

The lemma follows immediately from this in light of the fact that conjugation by ϱ_{g}(τ_{℘}) preserves the diagonal entries in such a matrix representation while multiplying its upper right hand entry by α/β, whereas τ_{℘} acts on the upper right-hand entry of the above matrix as multiplication by β/α. □

The matrix A_{g} gives rise to a G-equivariant homomorphism Φ_{g} : H^{×}⟶H_{℘} ⊗W_{g} by setting

(7)  Φ_{g}(x) = ∑_{σ∈G} log_{℘}(^{σ}x) · (σ^{−1} · A_{g}),

where, just as above, the group G acts on H_{℘} ⊗ W_{g} trivially on the first factor and through the usual conjugation action induced by ρ_{g} on the second factor.

Lemma 1.6. The homomorphism Φ_{g} takes values in W_{g}.

Proof. For any x ∈ (H ⊗ ℚ_{p})^{×} we have

τ_{℘}Φ_{g}(x) = ∑_{σ∈G} log_{℘}(^{τ_{℘}σ}x) · (σ^{−1} ·^{τ_{℘}}A_{g})
= ∑_{σ∈G} log_{℘}(^{τ_{℘}σ}x) · (σ^{−1} · τ_{℘}^{−1} · A_{g})
= ∑_{σ∈G} log_{℘}(^{τ_{℘}σ}x) · ((τ_{℘}σ)^{−1} · A_{g})
= Φ_{g}(x),

where Lemma 1.5 has been used to derive the second equation. □

By a slight abuse of notation, we shall continue to denote with the same symbol the homomorphism

Φ_{g} : (H ⊗ ℚ_{p})^{×}⟶H_{℘} ⊗ W_{g}

obtained from (7) by extending scalars. Note that H_{℘}^{×} embeds naturally in (H ⊗ ℚ_{p})^{×}.

Lemma 1.7. The homomorphism Φ_{g} vanishes on O_{H}^{×} ⊗ ℚ_{p} and Φ_{g}(H_{℘}^{×}) ⊆ H_{℘} ⊗ W_{g}^{ord}.

Proof. Picking u ∈ O_{H}^{×} and an arbitrary B ∈ W_{g}, set

U_{g}^{×} := ∑_{σ∈G} ^{σ}u ⊗ (σ · B) ∈ (O_{H}^{×} ⊗ W_{g})^{G}, U_{g} := (log_{℘} ⊗id)(U_{g})

as in the statement of Lemma 1.3. Note that U_{g}^{×} is either trivial or a generator of the one-dimensional space (O_{H}^{×} ⊗ W_{g})^{G}. We have

Trace(Φ_{g}(u) · B) = Trace ((∑_{σ∈G} log_{℘}(^{σ}u) · (σ^{−1} · A_{g})) · B)

= Trace (A_{g} · (∑_{σ∈G} log_{℘}(^{σ}u) · (σ · B)))
= Trace (A_{g} · (log_{℘} ⊗Id)(U_{g}^{×})) = Trace (A_{g} · U_{g}) .

It follows from Property (a) satisfied by A (and hence A_{g} in particular) in Lemma 1.3 that

Trace (Φ_{g}(u) · B) = 0, for all B ∈ H_{℘} ⊗ W_{g}.

The first assertion in the lemma follows from the non-degeneracy of the H_{℘}-valued trace pairing on H_{℘} ⊗ W_{g}. The second assertion follows from Property (b) satisfied by A and by A_{g} in Lemma 1.3.

□

Let now ℓ ∤ Np be a rational prime, and let λ be a prime of H above ℓ. Let u(λ) ∈ O_{H}[1/λ]^{×} ⊗ ℚ be a λ-unit of H satisfying

(8)  Norm_{ℚ}^{H}(u(λ)) = ℓ.

This condition makes u(λ) well-defined up to the addition of elements in O_{H}^{×} ⊗ ℚ, and hence the element

A_{g}(λ) := Φ_{g}(u(λ)) = ∑_{σ∈G} log_{℘}(^{σ}u(λ)) · (σ^{−1} · A_{g})

is well-defined, by Lemma 1.7. Although A_{g}(λ) only belongs to H_{℘} ⊗ W_{g} a priori, we have:

Lemma 1.8. The trace of the endomorphism A_{g}(λ) is equal to log_{p}(ℓ).

Proof. Since the trace of A_{g} and its conjugates are all equal to 1, we have

Trace(A_{g}(λ)) = ∑_{σ∈G} log_{℘}(^{σ}u(λ)) · Trace(σ^{−1} · A_{g})
= ∑_{σ∈G} log_{℘}(^{σ}u(λ))
= log_{℘} (Norm_{ℚ}^{H}(u(λ))) .

The latter expression is equal to log_{p}(ℓ), by (8). □

Remark 1.9. Although A_{g}(λ) belongs to W_{g} by Lemma 1.6, the entries of the matrix representing A_{g}(λ) relative to an L-basis for V_{g}^{L} are L-linear combinations of products of ℘-adic logarithms of units and ℓ-units in H, and in particular A_{g}(λ) need not lie in W_{g}^{L}. (In fact, it never does, since its trace is not algebraic.)

In addition to the invariant A_{g}(λ), the choice of the prime λ of H above ℓ also determines a well-defined Frobenius element τ_{λ} in G = Gal (H/ℚ), and even in Gal (H_{g}/ℚ), since Gal (H_{g}/H) lies in the center of this group.

We are now ready to state the main theorem of this section:

Theorem 1.10. For all rational primes ℓ ∤ Np,

a_{ℓ}(g′_{α}) = Trace(ϱ_{g}(τ_{λ})A_{g}(λ)).

Remark 1.11. This invariant does not depend on the choice of a prime λ of H above ℓ, since replacing λ by another such prime has the effect of conjugating the endomorphisms ϱ_{g}(τ_{λ}) and A_{g}(λ) by the same element of Aut(V_{g}).

Proof of Theorem 1.10. Let ℚ[ε] denote the ring of dual numbers over ℚ_{p}, with ε^{2} = 0, and let

ϱ̃_{g} : G_{ℚ}⟶GL_{2}(ℚ_{p}[ε])

be the unique first order α-ordinary deformation of ϱ_{g} satisfying

det ϱ̃_{g} = χ_{g}(1 + log_{p} χ_{cyc} · ε).

This representation may be written as

(9)  ϱ̃_{g} = (1 + ε · κ_{g}) · ϱ_{g} for some κ_{g} : G_{ℚ}⟶W_{g}.

The multiplicativity of ϱ̃_{g} implies that the function κ_{g} is a 1-cocycle on G_{ℚ} with values in W_{g}, whose class in H^{1}(ℚ, W_{g}) (which shall be denoted with the same symbol, by a slight abuse of notation) depends only on the isomorphism class of ϱ̃_{g}. Furthermore,

a_{ℓ}(g_{α}) + ε · a_{ℓ}(g′_{α}) = Trace(ϱ̃_{g}(τ_{λ})) = a_{ℓ}(g) + ε · Trace(κ_{g}(τ_{λ})ϱ_{g}(τ_{λ})),

and hence

(10)  a_{ℓ}(g′_{α}) = Trace(ϱ_{g}(τ_{λ})κ_{g}(τ_{λ})).

To make κ_{g}(τ_{λ}) explicit, observe that the inflation-restriction sequence combined with global class field theory for H gives rise to a series of identifications

H^{1}(ℚ, W_{g}) ⟶^{res_{H}} hom(G_{H}, W_{g})^{G}
= hom_{G} (\frac{(O_{H} ⊗ ℚ_{p})^{×}}{O_{H}^{×} ⊗ ℚ_{p}} , W_{g}) .

Under this identification, the class κ_{g} can be viewed as an element of the space

H_{ord}^{1}(ℚ, W_{g}) = \{Φ ∈ hom_{G} (\frac{(O_{H} ⊗ ℚ_{p})^{×}}{O_{H}^{×} ⊗ ℚ_{p}} , W_{g}) such that Φ(H_{℘}^{×}) ⊂ W_{g}^{ord}\} .

But the homomorphism Φ_{g} of (7) belongs to the same one-dimensional space, by Lemma 1.6 and 1.7. By global class field theory, the endomorphism κ_{g}(τ_{λ}) is therefore a ℚ_{p}^{×}-multiple of Φ_{g}(u_{g}(λ)) = A_{g}(λ). The fact that these endomorphisms are actually equal now follows by comparing their traces and noting that

Trace(κ_{g}(τ_{λ})) = log_{p} χ_{cyc}(ℓ) = log_{p}(ℓ),

while

Trace(A_{g}(λ)) = log_{p}(ℓ),

by Lemma 1.8. Theorem 1.10 follows. □

Corollary 1.12. If the rational prime ℓ ∤ Np splits completely in H/ℚ, then

a_{ℓ}(g′_{α}) = (1/2) · a_{ℓ}(g) · log_{p}(ℓ).

Proof. The hypothesis implies that ϱ_{g}(τ_{λ}) is a scalar, and hence that ϱ_{g}(τ_{λ}) = \frac{1}{2}a_{ℓ}(g). It follows that

Trace(ϱ_{g}(τ_{λ})A_{g}(λ)) = (1/2) · a_{ℓ}(g) · Trace(A_{g}(λ)) = (1/2) · a_{ℓ}(g) · log_{p}(ℓ).

The corollary now follows from Theorem 1.10. □

Example 1.13. Let χ be a Dirichlet character of conductor 171 with order 3 at 9 and 2 at 19. Then S_{1}(171, χ) is a ℚ(χ)-vector space of dimension 2. It is spanned by an eigenform

g = q + ζq^{2} + ζ^{3}q^{3} − ζ^{2}q^{5} + (ζ^{2} − 1)q^{6} + ⋯

defined over L := ℚ(ζ), with ζ a primitive 12th root of unity, and its Galois conjugate. (See [BL] for all weight one eigenforms of level at most 1500.) The associated projective representation ϱ_{g} has A_{4}-image and factors through the field

H = ℚ(a), a^{4} + 10a^{3} + 45a^{2} + 81a + 81 = 0.

Let p = 13, which splits completely in L. The representation ϱ_{g} is regular at 13, with eigenvalues α = ζ and β = −ζ^{3}. We computed the first order deformations through each of g_{α} and g_{β} to precision 13^{10}, and q-adic precision q^{37,000}, using methods based upon the algorithms in [La].

The predictions made from Theorem 1.10 for a_{ℓ}(g′_{α}) depend upon the conjugacy class of the Frobenius at ℓ in Gal (H/ℚ). For all primes ℓ < 37, 000 which split completely in H, such as ℓ = 109, 179, 449, 467, 521, …, we verified that

a_{ℓ}(g′_{α}) = (1/2) · a_{ℓ}(g) · log_{13}(ℓ) (mod 13^{10}),

as asserted by Corollary 1.12.

2. CM forms

This section focuses on the case where g = θ_{ψ_{g}} is the CM theta series attached to a character

ψ_{g} : G_{K}⟶L^{×}

of a quadratic imaginary field K. The main theorems are Theorems 2.1 and 2.3 below, which will be derived in two independent ways, both “from first principles” and by specialising Theorem 1.10.

As in the previous section, the choice of an embedding of L into ℚ_{p} allows us to view ψ_{g} as a ℚ_{p}^{×}-valued character, and the weight one form g as a modular form with coefficients in ℚ_{p}.

For a character ψ : G_{K}⟶L^{×}, the notation ψ′ will be used to designate the composition of ψ with conjugation by the non-trivial element in Gal (K/ℚ):

ψ′(σ) = ψ(τστ^{−1}),

where τ is any element of G_{ℚ} which acts non-trivially on K.

The Artin representation ϱ_{g} is induced from ψ_{g} and its restriction to G_{K} is the direct sum ψ_{g} ⊕ ψ′_{g} of two characters of K, which are distinct by the irreducibility of ϱ_{g} resulting from the fact that g is a cusp form. In this case, the field H is the ring class field of K which is cut out by the non-trivial ring class character ψ := ψ_{g}/ψ_{g}′. The Galois group G := Gal (H/ℚ) is a generalised dihedral group containing Z := Gal (H/K) as its abelian normal subgroup of index two.

The case of CM forms can be further subdivided into two sub-cases, depending on whether p is split or inert in K.

2.1. The case where p splits in K. Write pO_{K} = 𝔭 𝔭′, and fix a prime ℘ of ℚ̄ above 𝔭. The roots of the p-th Hecke polynomial of g are

α = ψ_{g}(𝔭), β = ψ_{g}(𝔭′).

This case is notable in that the Hida family g passing through g_{α} can be written down explicitly as a family of theta series. Its weight k specialisation g_{k} is the theta-series attached to the character ψ_{g}Ψ^{k−1}, where Ψ is a CM Hecke character of weight (1, 0) which is unramified at 𝔭. For all rational primes ℓ ∤ Np, the ℓ-th fourier coefficient of g_{k} is given by

(11)  a_{ℓ}(g_{k}) = \{ \begin{matrix}ψ_{g}(λ)Ψ^{k−1}(λ′) + ψ_{g}(λ′)Ψ^{k−1}(λ) & if ℓ = λλ′ splits in K; \\ 0 & if ℓ is inert in K. \end{matrix}

Letting h be the class number of K and t the cardinality of the unit group O_{K}^{×}, the character Ψ satisfies

Ψ(λ)^{ht} = u_{λ}^{t}, where (u_{λ}) := λ^{h},

for any prime ideal λ of O_{K} whose norm is the rational prime ℓ = λλ′. Let u′_{λ} denote the conjugate of u_{λ} in K/ℚ. It follows that

\frac{d}{dk} Ψ^{k−1}(λ)_{k=1} = log_{𝔭}(u(λ)),  where u(λ) := u_{λ} ⊗ \frac{1}{h} ∈ O_{H}[1/ℓ]^{×} ⊗ ℚ,

and likewise that

\frac{d}{dk} Ψ^{k−1}(λ′)_{k=1} = log_{𝔭}(u(λ)′),  where u(λ)′ := u′_{λ} ⊗ \frac{1}{h} ∈ O_{H}[1/ℓ]^{×} ⊗ ℚ.

In light of (11), we have obtained:

Theorem 2.1. For all rational primes ℓ that do not divide Np,

(12) a_{ℓ}(g′_{α}) = \{ \begin{matrix}(ψ_{g}(λ) log_{𝔭}(u(λ′)) + ψ_{g}(λ′) log_{𝔭}(u(λ))) & if ℓ = λλ′ splits in K; \\ 0 & if ℓ is inert in K. \end{matrix}

Thus, the prime fourier coefficients of g′_{α} are supported at the primes ℓ which are split in K, where they are (algebraic multiples of) the 𝔭-adic logarithms of ℓ-units in this quadratic field. This general pattern will persist in the other settings to be described below, with the notable feature that the fourier coefficients of g′_{α} will be more complicated expressions involving, in general, the p-adic logarithms of units and ℓ-units in the full ring class field H.

The reader will note Theorem 2.1 is consistent with Theorem 1.10, and could also have been deduced from it. More precisely, choose a basis of V_{g} consisting of eigenvectors for the action of G_{K} (and hence also, of τ_{℘}) which are interchanged by some element τ ∈ G_{ℚ}−G_{K}. Relative to such a basis, the endomorphisms U_{g} and A_{g} are represented by the following matrices, in which u_{ψ} and τu_{ψ} are generators of the spaces of ψ and ψ^{−1}-isotypic vectors in the group of elliptic units in O_{H}^{×} ⊗ L:

U_{g} : ( \begin{matrix}0 & log_{℘}(u_{ψ}) \\ log_{℘}(u′_{ψ}) & 0 \end{matrix} ) , A_{g} : ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) .

It follows that, if ℓ = λλ′ is split in K, then A_{g}(λ) is represented by the matrix

A_{g}(λ) : ( \begin{matrix}log_{℘}(u(λ′)) & 0 \\ 0 & log_{℘}(u(λ)) \end{matrix} ) ,

while A_{g}(λ) = \frac{1}{2} log_{p}(ℓ) is the scalar matrix with trace equal to log_{p}(ℓ) if ℓ is inert in K.

2.2. The case where p is inert in K. We now turn to the more interesting case where p is inert in K. Let σ_{℘} := τ_{℘}^{2} denote the frobenius element in G_{K} attached to the prime ℘ of H (which is well-defined modulo the inertia subgroup at ℘). Note that the prime p splits completely in H/K, since the image of τ_{℘} in G is a reflection in this generalised dihedral group. The image of σ_{℘} in Gal (H_{g}/K) therefore belongs to the subgroup Gal (H_{g}/H) whose image under ϱ_{g} consists of scalar matrices. Similar notations and remarks apply to any rational prime ℓ which is inert in K/ℚ.

Relative to an eigenbasis (e_{1}, e_{2}) for the action of G_{K} on V_{g}, the Galois representation ϱ_{g} takes the form

(13)  ϱ_{g}(σ) = ( \begin{matrix}ψ_{g}(σ) & 0 \\ 0 & ψ′_{g}(σ) \end{matrix} ) for σ ∈ G_{K}.

The homomorphisms ψ_{g}, ψ′_{g} : G_{K}⟶ℚ_{p}^{×} factor through Gal (H_{g}/K) and satisfy

ψ_{g}(τστ^{−1}) = ψ′_{g}(σ), for all τ ∈ G_{ℚ} − G_{K}, σ ∈ G_{K}.

It follows that ϱ_{g}(τ) interchanges the lines spanned by e_{1} and e_{2}, for any element τ ∈ G_{ℚ}−G_{K}. The restriction of ϱ_{g} to G_{ℚ} − G_{K} can therefore be described in matrix form by

(14)  ϱ_{g}(τ) = ( \begin{matrix}0 & η_{g}(τ) \\ η′_{g}(τ) & 0 \end{matrix} ) for τ ∈ G_{ℚ} − G_{K},

where η_{g} and η′_{g} are L-valued functions on G_{ℚ} − G_{K} that satisfy

(15)  η_{g}(τ_{1})η′_{g}(τ_{2}) = ψ_{g}(τ_{1}τ_{2}) = ψ′_{g}(τ_{2}τ_{1}),  for all τ_{1}, τ_{2} ∈ G_{ℚ} − G_{K},

as well as the relations

(16) \begin{matrix}η_{g}(στ) = ψ_{g}(σ)η_{g}(τ), & η_{g}(τσ) = ψ′_{g}(σ)η_{g}(τ), \\ η′_{g}(στ) = ψ′_{g}(σ)η′_{g}(τ), & η′_{g}(τσ) = ψ_{g}(σ)η′_{g}(τ), \end{matrix}  for all σ ∈ G_{K}, τ ∈ G_{ℚ} − G_{K}.

After re-scaling e_{1} and e_{2} if necessary, we may assume that τ_{℘} ∈ G_{ℚ} − G_{K} is sent to the matrix

(17)  ϱ_{g}(τ_{℘}) = ( \begin{matrix}0 & ζ \\ ζ & 0 \end{matrix} ) ,  with ζ^{2} = −χ_{g}(p).

The eigenvalues of ϱ_{g}(τ_{℘}) are equal to α := ζ and β := −ζ, and hence g is always regular in this setting.

Let

ϱ̃_{g} : G_{ℚ}⟶GL(Ṽ_{g})

denote the first-order infinitesimal deformation of ϱ_{g} attached to the Hida family g passing through a choice of p-stabilization g_{α} of g, where α ∈ {ζ, −ζ}. The module Ṽ_{g} is free of rank two over the ring ℚ_{p}[ε]/(ε^{2}) = ℚ_{p}[[T]]/(T^{2}) arising from the mod T^{2} reduction of the representation ϱ_{g} attached to g. Choose any ℚ_{p}[ε]-basis (ẽ_{1}, ẽ_{2}) of Ṽ_{g} lifting (e_{1}, e_{2}), and note that the restriction of ϱ̃_{g} to G_{K} is given by:

(18)  ϱ̃_{g}(σ) = ( \begin{matrix}ψ_{g}(σ) · (1 + κ(σ) · ε) & ψ′_{g}(σ) · κ_{ψ}(σ) · ε \\ ψ_{g}(σ) · κ′_{ψ}(σ) · ε & ψ′_{g}(σ) · (1 + κ′(σ) · ε) \end{matrix} ) , for all σ ∈ G_{K}.

In this expression,

(a) The functions κ and κ′ are continuous homomorphisms from G_{K} to ℚ_{p}, i.e., elements of H^{1}(K, ℚ_{p}), which are interchanged by conjugation by the involution in Gal (K/ℚ):

κ(τστ^{−1}) = κ′(σ), τ ∈ G_{ℚ} − G_{K}, σ ∈ G_{K}.

(b) The functions κ_{ψ}, κ′_{ψ} : G_{K}⟶ℚ_{p} are one-cocycles with values in ℚ_{p}(ψ), and give rise to well defined classes

κ_{ψ} ∈ H^{1}(K, ℚ_{p}(ψ)), κ′_{ψ} ∈ H^{1}(K, ℚ_{p}(ψ^{−1})),

which also satisfy

κ_{ψ}(τστ^{−1}) = κ′_{ψ}(σ), τ ∈ G_{ℚ} − G_{K}, σ ∈ G_{K}.

For each rational prime ℓ ∤ Np, the ℓ-th fourier coefficient a_{ℓ}(g′_{α}) is given by

(19)  a_{ℓ}(g′_{α}) = \frac{d}{dk} Trace(ϱ_{g}(τ_{λ}))_{k=1}

Observe that the spaces H^{1}(K, ℚ_{p}) and H^{1}(K, ℚ_{p}(ψ)) are of dimensions two and one respectively over ℚ_{p}, since ψ ≠ 1. More precisely, restriction to the inertia group at p combined with local class field theory induces an isomorphism

(20)  H^{1}(K, ℚ_{p}) = hom(O_{K_{p}}^{×}, ℚ_{p}) = ℚ_{p} log_{p}(z) ⊕ ℚ_{p} log_{p}(z′).

Let O_{H}^{×,ψ} denote the (one-dimensional) ψ-isotypic component of O_{H}^{×}⊗ℚ_{p} on which Gal (H/K) acts through the character ψ, and denote by ℘ the prime of H above p arising from our chosen embedding of ℚ̄ into ℚ̄_{p}. Restriction to the inertia group at ℘ in G_{H} likewise gives rise to an identification

(21) H^{1}(K, ℚ_{p}(ψ)) = hom(O_{H_{℘}}^{×}/O_{H}^{×,ψ}, ℚ_{p}) = ℚ_{p} · (log_{℘}(u′_{ψ}) log_{℘}(z) − log_{℘}(u_{ψ}) log_{℘}(z′)).

In the above equation, u_{ψ} is to be understood as the natural image in O_{H_{℘}}^{×} = O_{K_{p}}^{×} of an element of the form

∑_{σ∈Z} ψ^{−1}(σ)u^{σ} ∈ (O_{H}^{×} ⊗ L)^{ψ},

where u is an L[G]-module generator of O_{H}^{×} ⊗ L, and u′_{ψ} is the image of u_{ψ} under the conjugation action K_{p}⟶K_{p}. Note that replacing u by λu for some λ ∈ L[G] has the effect of multiplying both u_{ψ} and u′_{ψ} by ψ(λ) ∈ ℚ_{p}, so that the ℚ_{p}-line spanned by the right-hand side of (21) is independent of the choice of u ∈ O_{H}^{×}.

It follows from (20) and (21) that the total deformation space of ϱ_{g} (before imposing any ordinarity hypotheses, or restrictions on the determinant) is three dimensional.

Let v_{g}^{+} := e_{1} +e_{2} and v_{g}^{−} := e_{1} −e_{2} be the eigenvectors for τ_{℘} acting on V_{g}, with eigenvalues ζ and −ζ respectively. Let κ_{p} and κ_{ψ,℘} denote the restrictions κ and κ_{ψ} to the inertia groups at p and ℘ in G_{H} and G_{K}respectively. Both can be viewed as characters of K_{p}^{×} = H_{℘}^{×} after identifiying the abelianisations of G_{K_{p}} and G_{H_{℘}} with a quotient of K_{p}^{×} via local class field theory.

Lemma 2.2. The following are equivalent:

(a) The inertia group at ℘ acts as the identity on some lift ṽ_{g}^{+} of v_{g}^{+} to Ṽ_{g};

(b) The inertia group at ℘ acts as the identity on all lifts ṽ_{g}^{+} of v_{g}^{+} to Ṽ_{g};

(c) The restrictions κ_{p} and κ_{ψ,℘} satisfy

κ_{p}(x) = −κ_{ψ,℘}(x), for all x ∈ O_{K_{p}}^{×}.

Similar statements hold when v_{g}^{+} is replaced by v_{g}^{−}, where the conclusion is that κ_{p} = κ_{ψ,℘}.

Proof. The equivalence of the first two conditions follows from the fact that εṼ_{g} ≃ V_{g} is unramified at p and hence that inertia acts as the identity on the kernel of the natural map Ṽ_{g}⟶V_{g}. To check the third, note that the inertia group I_{p} at p is contained in G_{K}, since K is unramified at p, and observe that any σ ∈ I_{p} sends ẽ_{1} + ẽ_{2} to

ϱ̃_{g}(σ)(ẽ_{1} + ẽ_{2}) = ẽ_{1} + ẽ_{2} + ε · (κ(σ)ẽ_{1} + κ′_{ψ}(σ)ẽ_{2} + κ_{ψ}(σ)ẽ_{1} + κ′(σ)ẽ_{2})
= ẽ_{1} + ẽ_{2} + ε · ((κ(σ) + κ_{ψ}(σ))ẽ_{1} + (κ′(σ) + κ′_{ψ}(σ))ẽ_{2}).

The lemma follows. □

A lift ϱ̃_{g} of ϱ_{g} is ordinary relative to the space spanned by v_{g}^{+} if and only if it satisfies the equivalent conditions of Lemma 2.2. This lemma merely spells out the proof of the Bellaiche-Dimitrov theorem on the one-dimensionality of the tangent space of the eigencurve at the point associated to g_{α}. More precisely, the general ordinary first-order deformation of ϱ_{g} is completely determined by the pair (κ_{p}, κ_{ψ,℘}), which depends on a single linear parameter μ ∈ ℚ̄_{p} and is given by the rule

(22)  κ_{p}(z) = μ(log_{℘}(u′_{ψ}) · log_{℘}(z) − log_{℘} u_{ψ} · log_{℘}(z′)),
(23)  κ_{ψ,p}(z) = ±μ(log_{℘}(u′_{ψ}) · log_{℘}(z) − log_{℘} u_{ψ} · log_{℘}(z′)),

where the sign in the second formula depends on whether one is working with the ordinary deformation of g_{α} or g_{β}.

Let us now make use of the fact that

det(ϱ̃_{g}) = 1 + ε log_{p} χ_{cyc} = 1 + ε log_{p}(zz′).

Since det(ϱ̃_{g}) = 1 + ε(κ + κ′), this condition implies that

μ = \frac{1}{log_{℘}(u′_{ψ}) − log_{℘}(u_{ψ})},

and hence that κ_{p} and κ_{ψ,℘} are given by

(24)  κ_{p}(z) = \frac{log_{℘}(u′_{ψ}) · log_{℘}(z) − log_{℘}(u_{ψ}) · log_{℘}(z′)}{log_{℘}(u′_{ψ}) − log_{℘}(u_{ψ})},

(25)  κ_{ψ,℘}(z) = ±\frac{log_{℘}(u′_{ψ}) · log_{℘}(z) − log_{℘}(u_{ψ}) · log_{℘}(z′)}{log_{℘}(u′_{ψ}) − log_{℘}(u_{ψ})}.

Equations (24) and (25) give a completely explicit description of the first order deformation ϱ̃_{g_{α}} and ϱ̃_{g_{β}}, from which the fourier coefficients of g′_{α} and g′_{β} shall be readily calculated.

The formula for the ℓ-th fourier coefficient of g′_{α} involves the unit u_{ψ} above as well as certain ℓ-units in O_{H}[1/ℓ]^{×} ⊗ L whose definition depends on whether or not the prime ℓ is split or inert in K/ℚ.

If ℓ = λλ′ splits in K/ℚ, let u(λ) and u(λ′) denote, as before, the ℓ-units in O_{K}[1/ℓ]^{×} ⊗ ℚ of norm ℓ with prime factorisation λ and λ′ respectively. Set

u_{g}(λ) := u(λ) ⊗ ψ_{g}(λ) + u(λ′) ⊗ ψ_{g}(λ′), u_{g}(λ′) := u(λ′) ⊗ ψ_{g}(λ) + u(λ) ⊗ ψ_{g}(λ′).

In other words, u_{g}(λ) is the unique element of O_{K}[1/ℓ]^{×}⊗L whose prime factorisation is equal to ψ_{g}(λ) · λ + ψ_{g}(λ′) · λ′. Note that, if ℓ splits completely in H/ℚ, i.e., if ϱ_{g}(τ_{λ}) is equal to a scalar ζ, then u_{g}(λ) = u_{g}(λ′) = ℓ ⊗ ζ, but that otherwise u_{g}(λ) and ℓ generate the L-vector space O_{K}[1/ℓ]^{×} ⊗ L of ℓ-units of K (tensored with L).

If ℓ is inert in K/ℚ, choose a prime λ of H lying above ℓ, and let u(λ) ∈ O_{H}[1/λ]^{×} ⊗ ℚ be any λ-unit of H satisfying ord_{λ}(u(λ)) = 1, which is well defined up to units in O_{H}^{×}. Define the elements

u_{ψ}(λ) = ∑_{σ∈Z} ψ^{−1}(σ) ⊗ ^{σ}u(λ) ∈ L ⊗ O_{H}[1/ℓ]^{×},
u′_{ψ}(λ) = τ_{℘}u_{ψ}(λ) ∈ L ⊗ O_{H}[1/ℓ]^{×}.

Thus u_{ψ}(λ) lies in the ψ-component O_{H}[1/ℓ]^{×}[ψ] and is well-defined up to the addition of multiples of u_{ψ}, where

u_{ψ} := ∑_{σ∈Z} ψ^{−1}(σ) ⊗ ^{σ}u ∈ L ⊗ O_{H}[1/ℓ]^{×},

for any unit u ∈ O_{H}^{×}, while u′_{ψ}(λ) lies in the ψ^{−1} component and is well-defined up to the addition of multiples of u′_{ψ}, where

u′_{ψ} = τ_{℘}u_{ψ}.

Recall the function η′_{g} : G_{ℚ} \ G_{K} introduced in (14), with values in the roots of unity of L^{×}. The main result of this section is:

Theorem 2.3. Let ℓ ∤ Np be a rational prime.

(a) If ℓ = λλ′ splits in K/ℚ, then

(26)  a_{ℓ}(g′_{α}) = a_{ℓ}(g′_{β}) = \frac{log_{℘}(u′_{ψ}) · log_{℘}(u_{g}(λ)) − log_{℘}(u_{ψ}) · log_{℘}(u_{g}(λ′))}{log_{℘}(u′_{ψ}) − log_{℘}(u_{ψ})}.

(b) If ℓ remains inert in K/ℚ, then

a_{ℓ}(g′_{α}) = η′_{g}(τ_{λ})\frac{log_{℘}(u′_{ψ}) log_{℘}(u_{ψ}(λ)) − log_{℘}(u_{ψ}) log_{℘}(u′_{ψ}(λ))}{log_{℘}(u′_{ψ}) − log_{℘}(u_{ψ}))}.

Proof. Let us first compute first the fourier coefficients at primes ℓ ∤ Np that split as ℓ = λλ′ in K. Let σ_{λ} and σ_{λ′} be the frobenius elements associated to λ and λ′ respectively. They are well-defined elements in the Galois group of any abelian extension of K in which ℓ is unramified.

It follows from (19) and the matrix expression for ϱ̃_{g|G_{K}} given in (18) that

a_{ℓ}(g′_{α}) = ψ_{g}(λ)κ(λ) + ψ_{g}(λ′)κ(λ′)
= ψ_{g}(λ)κ_{p}(u(λ)) + ψ_{g}(λ′)κ_{p}(u′(λ)).

Equation (26) then follows from the formula for κ_{p}(z) given in (24).

We now turn now to the computation of the fourier coefficients of g′_{α} at primes ℓ ∤ Np that remain inert in K. Let τ_{λ} denote the Frobenius element in Gal (M/ℚ) associated to the choice of a prime ideal λ above ℓ in ℚ̄, and let σ_{λ} := τ_{λ}^{2} denote the associated frobenius element in Gal (M/K).

Since τ_{λ} belongs to G_{ℚ} − G_{K}, it follows from (14) that the matrix ϱ̃_{g}(τ_{λ}) is of the form

ϱ̃_{g}(τ_{λ}) = ( \begin{matrix}r_{ℓ} · ε & η_{g}(τ_{λ})(1 + s_{ℓ} · ε) \\ η′_{g}(τ_{λ})(1 + t_{ℓ} · ε) & u_{ℓ} · ε \end{matrix} ) ,

for suitable scalars r_{ℓ}, s_{ℓ}, t_{ℓ}, and u_{ℓ} ∈ ℚ̄_{p}. Since

a_{ℓ}(g_{α}) + a_{ℓ}(g′_{α})ε = Trace(ϱ̃_{g}(τ_{λ})) = (r_{ℓ} + u_{ℓ}) · ε,

it follows that

(27)  a_{ℓ}(g′_{α}) = r_{ℓ} + u_{ℓ}.

In order to compute this trace, we observe that it arises in the upper right-hand and lower left-hand entries of the matrix

(28)  ϱ̃_{g}(σ_{λ}) = ϱ̃_{g}(τ_{λ})^{2} = ( \begin{matrix}ψ_{g}(σ_{λ})(1 + (s_{ℓ} + t_{ℓ}) · ε) & η_{g}(τ_{λ})(r_{ℓ} + u_{ℓ}) · ε \\ η′_{g}(τ_{λ})(r_{ℓ} + u_{ℓ}) · ε & ψ_{g}(σ_{λ})(1 + (s_{ℓ} + t_{ℓ}) · ε) \end{matrix} ) .

On the other hand, since σ_{λ} belongs to G_{K} it follows from (18) that

(29)  ϱ̃_{g}(σ_{λ}) = ( \begin{matrix}ψ_{g}(σ_{λ}) · (1 + κ(σ_{λ}) · ε) & ψ′_{g}(σ_{λ}) · κ_{ψ}(σ_{λ}) · ε \\ ψ_{g}(σ_{λ}) · κ′_{ψ}(σ_{λ}) · ε & ψ′_{g}(σ_{λ}) · (1 + κ′(σ_{λ}) · ε). \end{matrix} )

By comparing upper-right entries in the matrices in (28) and (29) and invoking (27) together with the relation ψ′_{g}(σ_{λ})η_{g}(τ_{λ})^{−1} = η′_{g}(τ_{λ}) arising from (15), we deduce that

a_{ℓ}(g′_{α}) = η′_{g}(τ_{λ})κ_{ψ}(σ_{λ}).

It is worth noting that each of the expressions η′_{g}(τ_{λ}) and κ_{ψ}(σ_{λ}) depend on the choice of a prime λ of H above ℓ that was made to define τ_{λ} and σ_{λ}, since changing this prime replaces τ_{λ} and σ_{λ} by their conjugates στ_{λ}σ^{−1} and σσ_{λ}σ^{−1} by some σ ∈ G_{K}. More precisely, by (16) and the cocycle property of κ_{ψ},

η′_{g}(στ_{λ}σ^{−1}) = ψ^{−1}(σ)η′_{g}(τ_{λ}), κ_{ψ}(σσ_{λ}σ^{−1}) = ψ(σ)κ_{ψ}(σ_{λ}).

In particular, the product η′_{g}(τ_{λ})κ_{ψ}(σ_{λ}) is independent of the choice of a prime above ℓ, as it should be. Note that η′_{g}(τ_{λ}) is a simple root of unity belonging to the image of ψ_{g}, while κ_{ψ}(σ_{λ}) represents the interesting “transcendental” contribution to the fourier coefficient a_{ℓ}(g′_{α}).

By the description of κ_{ψ}(σ_{λ}) arising from local and global class field theory, we conclude from (25) that

(30)  a_{ℓ}(g′_{α}) = η′_{g}(τ_{λ})\frac{log_{℘}(u′_{ψ}) log_{℘}(u_{ψ}(λ)) − log_{℘}(u_{ψ}) log_{℘}(u′_{ψ}(λ))}{log_{℘}(u′_{ψ}) − log_{℘}(u_{ψ})},

as was to be shown. □

A more efficient (but somewhat less transparent) route to the proof of Theorem 2.3 is to specialise Theorem 1.10 to this setting. Relative to a basis of the form (v, τ_{℘}v) for V_{g}, where v spans a G_{K}-stable subspace of V_{g} on which G_{K} acts via ψ_{g}, the matrix for U_{g} is proportional to one of the form

U_{g} : ( \begin{matrix}0 & log_{℘}(u_{ψ}) \\ log_{℘}(τ_{℘}u_{ψ}) & 0 \end{matrix} ) ,

and the ordinarity condition implies that the matrix representing A_{g} is proportional to a matrix of the form

A : ( \begin{matrix}x & −x \\ y & −y \end{matrix} ) .

The relations Trace(A_{g}U_{g}) = 0 and Trace(A_{g}) = 1 show that A_{g} is represented by the matrix

A_{g} : \frac{1}{log_{℘}(u_{ψ}) − log_{℘}(τ_{℘}u_{ψ})} · ( \begin{matrix}log_{℘}(u_{ψ}) & − log_{℘}(u_{ψ}) \\ log_{℘}(τ_{℘}u_{ψ}) & − log_{℘}(τ_{℘}u_{ψ}) \end{matrix} ) ,

and Theorem 2.3 is readily deduced from the general formula for the fourier coefficients of g′_{α} given in Theorem 1.10. The details are left to the reader.

2.3. Numerical examples. We begin with an illustration of Theorem 2.3 in which the image of ϱ_{g} is isomorphic to the symmetric group S_{3}.

Example 2.4. Let χ be the quadratic character of conductor 23 and

g = q − q^{2} − q^{3} + q^{6} + q^{8} + ⋯ ∈ S_{1}(23, χ)

be the theta series attached to the imaginary quadratic field K = ℚ(\sqrt{−23}). The Hilbert class field H of K is

H = ℚ(α) where α^{6} − 6α^{4} + 9α^{2} + 23 = 0.

Write Gal (H/K) = ⟨σ⟩. The smallest prime which is inert in K is p = 5. The deformations g′_{1} and g′_{−1} were computed to a 5-adic precision of 5^{40} (and q-adic precision q^{600}).

Consider the inert prime ℓ = 7 in K. Let u(7) = (2α^{4}−7α^{2}+5)/9, a root of x^{3}−x^{2}+2x−7 = 0. Taking ω a primitive cube root of unity we have

log_{5}(u_{ψ}(7)) = log_{5}(u(7)) + ω log_{5}(u(7)^{σ}) + ω^{2} log_{5}(u(7)^{σ^{2}})
log_{5}(u′_{ψ}(7)) = log_{5}(u(7)) + ω^{2} log_{5}(u(7)^{σ}) + ω log_{5}(u(7)^{σ^{2}}).

Let u = (α^{2} − 1)/3 be the elliptic unit in H, a root of x^{3} − x^{2} + 1 = 0. Then likewise we have

log_{5}(u_{ψ}) = log_{5}(u) + ω log_{5}(u^{σ}) + ω^{2} log_{5}(u^{σ^{2}})
log_{5}(u′_{ψ}) = log_{5}(u) + ω^{2} log_{5}(u^{σ}) + ω log_{5}(u^{σ^{2}}).

Now

a_{7}(g′_{1}) = −a_{7}(g′_{−1}) = 4083079847610157092272537548 · 5 mod 5^{40}

and one checks to 40 digits of 5-adic precision that

a_{7}(g′_{1}) = \frac{log_{5}(u_{ψ}(7)) log_{5}(u′_{ψ}) − log_{5}(u′_{ψ}(7)) log_{5}(u_{ψ})}{log_{5}(u_{ψ}) − log_{5}(u′_{ψ})}

as predicted by part (b) of Theorem 2.3.

Consider next the prime ℓ = 13, which splits in K and factors as (l) = λλ′, where λ^{3} = (u_{λ}) is a principal idead generated by u_{λ} = −6α^{3} + 18α − 37, a root of x^{2} + 74x + 2197. After setting u(λ) = u_{λ} ⊗ \frac{1}{3}, we let

log_{5}(u_{g}(λ)) = (ω log_{5}(u(λ)) + ω^{2} log_{5}(u′(λ)))
log_{5}(u′_{g}(λ)) = \frac{1}{3} (ω^{2} log_{5}(u(λ)) + ω log_{5}(u′(λ))) .

We have

a_{13}(g′_{1}) = a_{13}(g′_{−1}) = −638894131680830198852008592 · 5 mod 5^{40}

and one sees that

a_{13}(g′_{±1}) = \frac{log_{5}(u′(ψ)) log_{5}(u_{g}(λ)) − log_{5}(u(ψ)) log_{5}(u′_{g}(λ))}{log_{5}(u′_{ψ}) − log_{5}(u_{ψ})}

to 40 digits of 5-adic precision, confirming Part (a) of Theorem 2.3.

The experiment below focuses on the case where ψ_{g} is a quartic ring class character, so that ϱ_{g} has image isomorphic to the dihedral group of order 8. The associated ring class character ψ = ψ_{g}/ψ′_{g} = ψ_{g}^{2} of K is quadratic, i.e., a genus character which cuts out a biquadratic exension H of ℚ containing K. Let F denote the unique real quadratic subfield of H, and let K′ the unique imaginary quadratic subfield of H which is distinct from K. The unit u_{ψ} is a power of the fundamental unit of F. Observe that the prime p is necessarily inert in K′/ℚ, since otherwise ϱ_{g} would be induced from a character of the real quadratic field F in which p splits. It follows that u′_{ψ} = u_{ψ}^{−1}, so that by Theorem 2.3,

a_{ℓ}(g′_{α}) = \frac{− log_{p} u_{ψ} · (log_{p}(u_{g}(ℓ)) + log_{p}(u′_{g}(ℓ)))}{−2 log_{p}(u_{ψ})} = \frac{1}{2} · (log_{p}(u_{g}(ℓ)) + log_{p}(u′_{g}(ℓ))).

It follows from the definition of u_{g}(ℓ) that

a_{ℓ}(g′_{α}) = trace(ϱ_{g}(λ)) · log_{p}(ℓ).

In particular, we obtain

(31)  a_{ℓ}(g′_{α}) = \{ \begin{matrix}log_{p}(ℓ) & if ψ_{g}(λ) = 1, \\ 0 & if ψ_{g}(λ) = ±i, \\ − log_{p}(ℓ) & if ψ_{g}(λ) = −1, \end{matrix}

in perfect agreement with the experiments below.

Example 2.5. Let χ = χ_{3}χ_{13} where χ_{3} and χ_{13} are the quadratic characters of conductors 3 and 13, respectively. The space S_{1}(39, χ) is one dimensional and spanned by the form g = q − q^{3} − q^{4} + q^{9} + · · ·. The representation ρ_{g} has projective image D_{4} and is induced from characters of two imaginary quadratic fields and one real quadratic field. In particular, it is induced from the quadratic character ψ_{g} of the Hilbert class field

H = ℚ(\sqrt{−39}, a), a^{4} + 4a^{2} − 48 = 0

of ℚ(\sqrt{−39}) (and also ramified characters of ray class fields of ℚ(\sqrt{−3}) and ℚ(\sqrt{13})). Let p = 7, which is inert in ℚ(\sqrt{−39}). We computed g′_{±1} to 20-digits of 7-adic precision (and to q-adic precision q^{900}).

First consider the case of ℓ = λλ′ split in ℚ(\sqrt{−39}). Then one observes to 20-digits of 7-adic precision and all such ℓ < 900 that both a_{ℓ}(g′_{1}) and a_{ℓ}(g′_{−1}) satisfy (31). Next we consider the case that ℓ is inert in ℚ(\sqrt{−39}). Here one observes numerically that the Fourier coefficients are zero when ℓ is inert in ℚ(\sqrt{−3}). When ℓ is split in ℚ(\sqrt{−3}) the Fourier coefficients of the two stabilisations are opposite in sign and both equal to the p-adic logarithm of a fundamental ℓ unit of norm 1 in ℚ(\sqrt{−3}). (Observe that p is split in ℚ(\sqrt{−3}) and our numerical observations are consistent in this example with both our theorems for p split and p inert in the CM case.)

3. RM forms

Consider now the case where g is the theta series attached to a character

ψ_{g} : G_{K}⟶L^{×}

of mixed signature of a real quadratic field K. As before, assume for simplicity that the field L may be embedded into ℚ_{p} and fix one such embedding. We also continue to denote H_{g} the abelian extensions of K which is cut out by ϱ_{g}, and let H be the ring class field of K cut out by the non-trivial ring class character ψ := ψ_{g}/ψ_{g}′. Since ψ_{g} has mixed signature, it follows that ψ is totally odd and thus H is totally imaginary. As before, write G := Gal (H/ℚ) and Z := Gal (H/K).

As explained in the introduction, the case where p splits in K was already dealt with in [DLR2], so in this section we only consider the case where p is inert in K/ℚ. The prime p then splits completely in H/K and we fix a prime ℘ of H above p. This choice determines an embedding H⟶H_{℘} = K_{p} and we write z ↦ z′ for the conjugation action of Gal (K_{p}/ℚ_{p}). Let u_{K} ∈ O_{K}^{×} denote the fundamental unit of O_{K} of norm 1, which we regard as an element of K_{p}^{×} = H_{℘}^{×} through the above embedding, and let u′_{K} = u_{K}^{−1} denote its algebraic conjugate.

Let (v_{1}, v_{2}) be a basis for V_{g} consisting of eigenvectors for the action of G_{K}, and which are interchanged by the frobenius element τ_{℘}. Just as in the previous section, relative to this basis the Galois representation ϱ_{g} takes the form (32)

ϱ_{g}(σ) = ( \begin{matrix}ψ_{g}(σ) & 0 \\ 0 & ψ′_{g}(σ) \end{matrix} ) for σ ∈ G_{K}, ϱ_{g}(τ) = ( \begin{matrix}0 & η_{g}(τ) \\ η′_{g}(τ) & 0 \end{matrix} ) for τ ∈ G_{ℚ} − G_{K},

where η_{g} and η′_{g} are functions taking values in the group of roots of unity in L^{×}. The element U_{g} ∈ (H_{℘} ⊗ W_{g}) of (5) is thus represented the matrix

U_{g} : ( \begin{matrix}log_{℘}(u_{K}) & 0 \\ 0 & log_{℘}(u′_{K}) \end{matrix} ) ,

and hence the endomorphism A_{g} of Lemma 1.5 is represented by the particularly simple matrix

A_{g} : \frac{1}{2} ( \begin{matrix}1 & −1 \\ −1 & 1 \end{matrix} ) .

It follows that, if ℓ = λλ′ is split in K/ℚ, we have

(33)  A_{g}(λ) : \frac{1}{2} ( \begin{matrix}log_{p}(ℓ) & 0 \\ 0 & log_{p}(ℓ) \end{matrix} ) ,

while if ℓ is inert in K/ℚ and λ is a prime of H lying above ℓ,

(34)  A_{g}(λ) : \frac{1}{2} ( \begin{matrix}log_{p}(ℓ) & − log_{℘}(u_{ψ}(λ)) \\ − log_{℘}(u′_{ψ}(λ)) & log_{p}(ℓ) \end{matrix} ) .

Theorem 3.1. For all rational primes ℓ ∤ Np, (a) If ℓ is split in K/ℚ, then

a_{ℓ}(g′_{α}) = \frac{1}{2}a_{ℓ}(g) · log_{p}(ℓ).

(b) If ℓ is inert in K/ℚ, then

a_{ℓ}(g′_{α}) = − (η_{g}(λ) log_{℘}(u′_{ψ}(λ)) + η_{g}(λ′) log_{℘}(u_{ψ}(λ))).

Proof. This follows directly from Theorem 1.10 in light of (33) and (34). □

Remark 3.2. As already remarked in [DLR2], Theorem 3.1 above (and also Theorem 7.1 of Part B) display a striking analogy with Theorem 1.1. of [DLi] concerning the fourier expansions of mock modular forms whose shadows are weight one theta series attached to characters of imaginary quadratic fields. The underlying philosophy is that the p-adic deformations considered in this paper behave somewhat like mock modular forms of weight one, “with ∞ replaced by p”. This explains why the analogy remains compelling when the quadratic imaginary fields of [DLi] are replaced by real quadratic fields in which p is inert (these fields being “imaginary” from a p-adic perspective).

We illustrate Theorem 3.1 on the form of smallest level whose associated Artin representation is induced from a character of a real quadratic field, but of no imaginary quadratic field. The projective image in this example is the dihedral group D_{8} of order 8:

Example 3.3. Let χ = χ_{5}χ_{29} where χ_{5} and χ_{29} are quadratic and quartic characters of conductor 5 and 29, respectively. Then S_{1}(145, χ) is one dimensional and spanned by the eigenform

g = q + iq^{4} − iq^{5} + iq^{9} + (−i − 1)q^{11} − q^{16} + (−i − 1)q^{19} + · · · .

The form g is induced from a quartic character of a ray class group of K = ℚ(\sqrt{5}) (see [DLR1, Example 4.1] for a further discussion on this form). The relevant ring class field H is

H = ℚ(α) where α^{8} − 2α^{7} + 4α^{6} − 26α^{5} + 94α^{4} − 212α^{3} + 761α^{2} − 700α + 980 = 0.

Write Gal (H/K) = ⟨ψ⟩. Take p = 13, and note that χ(p) = 1 and so α = i and β = −i. We compute g′_{±i} to 10 digits of 13-adic precision (and q-adic precision q^{28,000}).

Consider first the prime ℓ = 7 which is inert in K. We take the 7-unit u(7) ∈ H to satisfy x^{4} + 13x^{3} + 38x^{2} + 5x + 343 = 0 and define

log_{13}(u(7, ±i)) := log_{13}(u(7)) ∓ i log_{13}(u(7)^{ψ}) ∓ log_{13}(u(7)^{ψ^{2}}) ± i log_{13}(u(7)^{ψ^{3}})

and so log_{13}(u(7, i))+log_{13}(u(7, −i)) = log_{13}(v) where v = u(7)/u(7)^{ψ^{2}} ∈ H. Then one checks that to 10 digits of 13-adic precision

a_{7}(g′_{i}) = −\frac{1}{6} · log_{13}(v)

which is in line with Theorem 3.1. Next we take the prime ℓ = 11 which is split in K. Then to 10-digits of 13-adic precision

a_{11}(g′_{i}) = −\frac{i + 1}{2} · log_{13}(11)

exactly as predicted by Theorem 3.1.

Part B. The irregular setting

Denote by S_{k}(Np, χ) (resp. by S_{k}^{(p)}(N, χ)) the space of classical (resp. p-adic overconvergent) modular forms of weight k, level Np (resp. tame level N) and character χ, with coefficients in ℚ_{p}. The Hecke algebra 𝕋 of level Np generated over ℚ by the operators T_{ℓ} with ℓ ∤ Np and U_{ℓ} with ℓ | Np acts naturally on the spaces S_{k}(Np, χ) and S_{k}^{(p)}(N, χ).

As in the introduction, let g ∈ S_{1}(N, χ) be a newform and let g_{α} ∈ S_{1}(Np, χ) be a pstabilisation of g. The eigenform g_{α} gives rise to a ring homomorphism φ_{g_{α}} : 𝕋⟶L to the field L generated by the fourier coefficients of g_{α}, satisfying

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

For any ideal I of a ring R and any R-module M, denote by M[I] the I-torsion in M. Let I_{g_{α}}◁ 𝕋 be the kernel of φ_{g_{α}}, and set

S_{1}(Np, χ)[g_{α}] := S_{1}(Np, χ)[I_{g_{α}}], S_{1}(Np, χ)[[g_{α}]] := S_{1}(Np, χ)[I_{g_{α}}^{2}].

Our main object of study is the subspace

S_{1}^{(p)}(N, χ)[[g_{α}]] := S_{1}^{(p)}(N, χ)[I_{g_{α}}^{2}].

of the space of overconvergent p-adic modular forms of weight one, which is contained in the generalised eigenspace attached to I_{g_{α}}. An element of S_{1}^{(p)}(N, χ)[[g_{α}]] is called an overconvergent generalised eigenform attached to g_{α}, and it is said to be classical if it belongs to S_{1}(Np, χ)[[g_{α}]]. The theorem of Bellaiche and Dimitrov stated in the opening paragraphs of Part A implies that the natural inclusion

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

is an isomorphism, i.e., every overconvergent generalised eigenform is classical, except possibly in the following cases:

(a) g is the theta series attached to a finite order character of a real quadratic field in which the prime p splits, or

(b) g is irregular at p, i.e., α = β.

The study of S_{1}^{(p)}(N, χ)[[g_{α}]] in scenario (a) was carried out in [DLR2] when α ≠ β. The main result of loc.cit. is the description of a basis (g_{α}, g_{α}^{♭}) for S_{1}^{(p)}(N, χ)[[g_{α}]] which is canonical up to scaling, and an expression for the fourier coefficients of the non-classical g_{α}^{♭} (or rather, of their ratios) in terms of p-adic logarithms of certain algebraic numbers.

Assume henceforth that g is not regular at p, i.e., that α = β. In that case, the form g admits a unique p-stabilisation g_{α} = g_{β}. The Hecke operators T_{ℓ} for ℓ ∤ Np and U_{ℓ} for ℓ | N act semisimply (i.e., as scalars) on the two-dimensional vector space

S_{1}(Np, χ)[[g_{α}]] = ℚ_{p}g_{α} ⊕ ℚ_{p}g′, g′(q) := g(q^{p}),

but the Hecke operator U_{p} acts non-semisimply via the formulae

U_{p}g_{α} = αg_{α}, U_{p}g′ = g_{α} + αg′.

Because

(a_{1}(g_{α}), a_{p}(g_{α})) = (1, α), (a_{1}(g′), a_{p}(g′)) = (0, 1),

the classical subspace S_{1}(Np, χ)[[g_{α}]] has a natural linear complement in S_{1}^{(p)}(N, χ)[[g_{α}]], consisting of the generalised eigenforms g̃ whose q-expansions satisfy

(36)  a_{1}(g̃) = a_{p}(g̃) = 0.

A modular form satisfying (36) is said to be normalised, and the space of normalised generalised eigenforms is denoted S_{1}^{(p)}(N, χ)[[g_{α}]]_{0}. The main goal of Part B is to study this space and give an explicit description of its elements in terms of their fourier expansions. The idoneous fourier coefficients will be expressed as determinants of 2 × 2 matrices whose entries are p-adic logarithms of algebraic numbers the number field H cut out by the projective Galois representation attached to g (cf. Theorems 5.3, 6.1 and 7.1).

4. Generalised eigenspaces

We begin by recalling some of the notations that were already introduced in Part A. Let

ϱ_{g} : G_{ℚ}⟶Aut_{ℚ_{p}}(V_{g}) ≃ GL_{2}(ℚ_{p})

be the odd, two-dimensional Artin representation associated to g by Deligne and Serre (but viewed as having p-adic rather than complex coefficients; as in Part A, we assume for simplicity that the image of ϱ_{g} can be embedded in GL_{2}(ℚ_{p}) and not just in GL_{2}(ℚ̄_{p})).

The four-dimensional ℚ_{p}-vector space W_{g} := Ad(V_{g}) := End(V_{g}) of endomorphisms of V_{g} is endowed with the conjugation action of G_{ℚ},

σ · M := ϱ_{g}(σ) ∘ M ∘ ϱ_{g}(σ)^{−1}, for any σ ∈ G_{ℚ}, M ∈ W_{g}.

Let H be the field cut out by this Artin representation. The action of G_{ℚ} on W_{g} factors through a faithful action of the finite quotient G := Gal (H/ℚ). Let W_{g}^{∘} := Ad^{0}(V_{g}) denote the three-dimensional G_{ℚ}-submodule of W_{g} consisting of trace zero endomorphisms. The exact sequence

0⟶W_{g}^{∘}⟶W_{g}⟶ℚ_{p}⟶0

of G-modules admits a canonical G-equivariant splitting

p : W_{g}⟶W_{g}^{∘}, p(A) := A − 1/2 · Tr(A).

Because the action of G_{ℚ} on V_{g} also factors through a finite quotient, the field L ⊂ ℚ_{p} generated by the traces of ϱ_{g} is a finite extension of ℚ, and ϱ_{g} maps the semisimple algebra L[G_{ℚ}] to a central simple algebra of rank 4 over L. By eventually enlarging L, it can be assumed that ϱ_{g}(L[G_{ℚ}]) ≃ M_{2}(L), and therefore that ϱ_{g} is realised on a two-dimensional L-vector space V_{g}^{L} equipped with an identification ι : V_{g}^{L} ⊗_{L} ℚ_{p}⟶V_{g}. The spaces

W_{g}^{L} := Ad(V_{g}^{L}), W_{g}^{∘L} := Ad^{0}(V_{g}^{L})

likewise correspond to G-stable L-rational structures on W_{g} and W_{g}^{∘} respectively, equipped with identifications

ι : W_{g}^{L} ⊗_{L} ℚ_{p}⟶W_{g}, ι : W_{g}^{∘L} ⊗_{L} ℚ_{p}⟶W_{g}^{∘}.

The spaces W_{g} and W_{g}^{∘} (as well as W_{g}^{L} and W_{g}^{∘L}) are equipped with the Lie bracket [ , ] and with a symetric non-degenerate pairing ⟨ , ⟩ defined by the usual rules

[A, B] := AB − BA, ⟨A, B⟩ := Tr(AB),

which are compatible with the G-action in the sense that

[σ · A, σ · B] = σ · [A, B],  ⟨σ · A, σ · B⟩ = ⟨A, B⟩,  for all σ ∈ G.

These operations can be combined to define a G-invariant determinant function—i.e., a nonzero, alternating trilinear form—on W_{g}^{∘} and on W_{g}^{∘L} by setting

det(A, B, C) := ⟨[A, B], C⟩.

The rule described in (35) gives rise to natural identifications

S_{1}(Np, χ)[g_{α}] ≃ Hom(𝕋/I_{g_{α}}, ℚ_{p}), S_{1}^{(p)}(N, χ)[[g_{α}]] ≃ Hom(𝕋/I_{g_{α}}^{2}, ℚ_{p}),

and hence the dual of the short exact sequence

0 → I_{g_{α}}/I_{g_{α}}^{2} → 𝕋/I_{g_{α}}^{2} → 𝕋/I_{g_{α}} → 0

can be identified with

0⟶S_{1}(Np, χ)[g_{α}]⟶S_{1}^{(p)}(N, χ)[[g_{α}]]⟶S_{1}^{(p)}(N, χ)[[g_{α}]]_{0}⟶0.

In particular, one has the isomorphism

(37)  S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} ≃ Hom(I_{g_{α}}/I_{g_{α}}^{2}, ℚ_{p}).

Let ℚ_{p}[ε] = ℚ_{p}[x]/(x^{2}) denote the ring of dual numbers. Given g^{♭} ∈ S_{1}^{(p)}(N, χ)[[g_{α}]]_{0}, the modular form g̃ := g_{α} + ε · g^{♭} is an eigenform for 𝕋 with coefficients in ℚ_{p}[ε]. Its associated Galois representation

ϱg̃ : G_{ℚ}⟶GL_{2}(ℚ_{p}[ε])

satisfies (i) ϱg̃ = ϱ_{g} (mod ε) and det(ϱg̃) = χ, (ii) for every prime number ℓ ∤ Np, the trace of an arithmetic Frobenius τ_{ℓ} at ℓ is

(38)  Tr(ϱg̃(τ_{ℓ})) = a_{ℓ}(g_{α}) + ε · a_{ℓ}(g^{♭}).

Conjecture 4.1. Assume that g is irregular at p. Then the assignment g^{♭} ↦ ϱg̃ gives rise to a canonical isomorphism between S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} and the space Def^{0}(ϱ_{g}) of isomorphism classes of deformations of ϱ_{g} to the ring of dual numbers, with constant determinant.

We now derive some consequences of this conjecture.

Proposition 4.2. Assume Conjecture 4.1. If g is irregular at p, then the space S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} is two-dimensional over ℚ_{p}.

Proof. Since any ϱ̃ ∈ Def^{0}(ϱ_{g}) has constant determinant, it may be written as

(39)  ϱ̃ = (1 + ε · c) · ϱ_{g} for some c = c(ϱ̃) : G_{ℚ}⟶W_{g}^{∘}.

The multiplicativity of ϱ̃ implies that the function c is a 1-cocycle of G_{ℚ} with values in W_{g}^{∘}, whose class in H^{1}(ℚ, W_{g}^{∘}) (which shall be denoted with the same symbol, by a slight abuse of notation) depends only on the isomorphism class of ϱ̃. The assignment ϱ̃ ↦ c(ϱ̃) realises an isomorphism (cf. for instance [Ma, §1.2])

Def^{0}(ϱ_{g})⟶H^{1}(ℚ, W_{g}^{∘}).

Under Conjecture 4.1, this yields an isomorphism

(40)  S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} ⟶^{∼} H^{1}(ℚ, W_{g}^{∘}), g^{♭} ↦ c_{g^{♭}}.

The inflation-restriction sequence combined with global class field theory for H now gives rise to a series of identifications

H^{1}(ℚ, W_{g}^{∘}) ⟶^{res_{H}} hom(G_{H}, W_{g}^{∘})^{G}
= hom_{G} (\frac{(O_{H} ⊗ ℤ_{p})^{×}}{O_{H}^{×} ⊗ ℤ_{p}} , W_{g}^{∘})
= hom_{G} (\frac{H_{p}}{U} , W_{g}^{∘})
(41)  = ker (hom_{G}(H_{p}, W_{g}^{∘}) ⟶^{res_{U}} hom_{G}(U, W_{g}^{∘})) ,

where U denotes the natural image of O_{H}^{×} ⊗ ℤ_{p} in H_{p} := H ⊗ ℚ_{p} under the p-adic logarithm map

log_{p} : H_{p}^{×}⟶H_{p}.

As representations for G, the space H_{p} is isomorphic to the regular representation

H_{p} ≃ Ind_{1}^{G} ℚ_{p},

while U, by the Dirichlet unit theorem, is induced from the trivial representation of the subroup G_{∞} ⊂ G generated by a complex conjugation:

U ≃ Ind_{G_{∞}}^{G} ℚ_{p}.

Complex conjugation acts on W_{g}^{∘} with eigenvalues 1, −1 and−1, and hence by Frobenius reciprocity,

(42)  dim_{ℚ_{p}} hom_{G}(H_{p}, W_{g}^{∘}) = 3,  dim_{ℚ_{p}} hom_{G}(U, W_{g}^{∘}) = 1.

It follows from (41) that H^{1}(ℚ, W_{g}^{∘}) is two-dimensional over ℚ_{p}. Proposition 4.2 follows. □

For any ℓ ∤ Np, the ℓ-th fourier coefficient of g^{♭} is given in terms of the associated cocycle c_{g^{♭}} by the rule

(43)  a_{ℓ}(g^{♭}) = Tr(c_{g^{♭}}(σ_{λ})ϱ_{g}(σ_{λ}))

where λ|ℓ is any prime above ℓ and σ_{λ} denotes the arithmetic Frobenius associated to it. Note that the right-hand side of (43) does not depend on the choice of λ.

Our next goal is to parametrise the elements of (41) explicitly, and then to derive concrete formulae for the fourier expansions of the associated modular forms in S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} via (40) and (43). After treating the general case in Section 5, Sections 6 and 7 focus on the special features of the scenarios where W_{g}^{∘} is reducible, i.e.,

(i) the CM case where V_{g} is induced from a character of an imaginary quadratic field; (ii) the RM case where V_{g} is induced from a character of a real quadratic field.

5. The general case

The Galois representation W_{g}^{∘} is irreducible if and only if G := Gal (H/ℚ) is isomorphic to A_{4}, S_{4}, or A_{5}. Otherwise, the representation ϱ_{g} has dihedral projective image and G is isomorphic to a dihedral group.

The irregularity assumption implies that the prime p splits completely in H, and H can therefore be viewed as a subfield of ℚ_{p} after fixing an embedding H ↪ ℚ_{p} once and for all. This amounts to choosing a prime ℘ of H above p. Let log_{℘} : H_{p}^{×}⟶ℚ_{p} denote the associated ℘-adic logarithm map, which factors through log_{p}.

The Dirichlet unit theorem implies (via the second equation in (42)) that

dim_{L}(O_{H}^{×} ⊗ W_{g}^{∘L})^{G} = 1.

In particular, for all 𝔲 ∈ O_{H}^{×} and all w ∈ W_{g}^{∘L}, the element

(44)  ξ(𝔲, w) := \frac{1}{#G} × ∑_{σ∈G}(σ𝔲) ⊗ (σ · w) ∈ (O_{H}^{×} ⊗ W_{g}^{∘L})^{G}

only depends on the choices of 𝔲 and w up to scaling by a (possibly zero) factor in L. As 𝔲 varies over O_{H}^{×} and w over W_{g}^{∘L}, the elements

(45)  ξ_{℘}(𝔲, w) := (log_{℘} ⊗id)ξ(𝔲, w) = \frac{1}{#G} × ∑_{σ∈G} log_{℘}(σ𝔲) · (σ · w) ∈ W_{g}^{∘}

therefore lie in a one-dimensional L-vector subspace of W_{g}^{∘}. Choose a generator w(1) for this space. The coordinates of w(1) relative to a basis (e_{1}, e_{2}, e_{3}) for W_{g}^{∘L} are ℘-adic logarithms of units in O_{H}, namely, we can write

(46)  w(1) = log_{℘}(𝔲_{1})e_{1} + log_{℘}(𝔲_{2})e_{2} + log_{℘}(𝔲_{3})e_{3},

for appropriate 𝔲_{i} ∈ (O_{H}^{×}) ⊗_{ℤ} L.

Let ℓ ∤ Np be a rational prime. For any prime λ of H above ℓ, let 𝔲̃_{λ} be a generator of the principal ideal λ^{h}, where h is the class number of H, and set

𝔲_{λ} := 𝔲̃_{λ} ⊗ h^{−1} ∈ (O_{H}[1/ℓ]^{×}) ⊗_{ℤ} L.

Let

(47)  w̃_{λ} := ϱ_{g}(σ_{λ}) ∈ W_{g}^{L},  w_{λ} := p(w̃_{λ}) ∈ W_{g}^{∘L}

be the endomorphisms of V_{g} arising from the image of σ_{λ} under ϱ_{g}. The element 𝔲_{λ} is welldefined up to multiplication by elements of O_{H}^{×}, and hence the elements

(48)  ξ(𝔲_{λ}, w_{λ}) := \frac{1}{#G} × ∑_{σ∈G}(σ𝔲_{λ}) ⊗ (σ · w_{λ}) ∈ (O_{H}[1/ℓ]^{×} ⊗ W_{g}^{∘L})^{G},

w(ℓ) = ξ_{℘}(𝔲_{λ}, w_{λ}) := \frac{1}{#G} × ∑_{σ∈G} log_{℘}(σ𝔲_{λ}) · (σ · w_{λ}) ∈ W_{g}^{∘}

are defined up to translation by elements of the one-dimensional L-vector spaces (O_{H}^{×} ⊗W_{g}^{∘L})^{G} and L · w(1) respectively. Furthermore, the image of w(ℓ) in the quotient W_{g}^{∘}/(L · w(1)) does not depend on the choice of the prime λ of H above ℓ that was made to define it. The Lie bracket

𝔚(ℓ) := [w(1), w(ℓ)] ∈ W_{g}^{∘}

is thus independent of the choices that were made in defining w(ℓ).

Remark 5.1. The coordinates of w(ℓ) relative to a basis (e_{1}, e_{2}, e_{3}) for W_{g}^{∘L} are ℘-adic logarithms of ℓ-units in H, i.e., one can write

(49)  w(ℓ) = log_{℘}(𝔳_{1})e_{1} + log_{℘}(𝔳_{2})e_{2} + log_{℘}(𝔳_{3})e_{3},

with 𝔳_{i} ∈ (O_{H}[1/ℓ]^{×})_{L} for i = 1, 2, 3. A direct computation shows that

dim_{ℚ_{p}}(W_{g}^{∘})^{σ_{λ}=1} = \{ \begin{matrix}1 & if g is regular at ℓ; \\ 3 & if g is irregular at ℓ. \end{matrix}

It follows that for all regular primes ℓ,

dim_{L}(O_{H}[1/ℓ]^{×} ⊗ W_{g}^{∘L})^{G} = 2,

and therefore that the element ξ(𝔳, w) attached to any pair (𝔳, w) ∈ O[1/ℓ]^{×}×W_{g}^{∘L} as in (48) is well-defined up to scaling by L and up to translation by elements of the one-dimensional space (O_{H}^{×} ⊗ W_{g}^{∘L})^{G}. In particular, the associated vector 𝔚(ℓ) lies in a canonical one-dimensional subspace of W_{g}^{∘}, namely, the orthogonal complement in W_{g}^{∘} of

(log_{℘} ⊗Id)(O_{H}[1/ℓ]^{×} ⊗ W_{g}^{∘})^{G} ⊂ W_{g}^{∘}.

If the basis (e_{1}e_{2}, e_{3}) for W_{g}^{∘L} in (46) and (49) is taken to be the standard basis

e_{1} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,  e_{2} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ,  e_{3} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ,

then

𝔚(ℓ) = det ( \begin{matrix}𝔲_{2} & 𝔲_{3} \\ 𝔳_{2} & 𝔳_{3} \end{matrix} ) · e_{1} + 2 det ( \begin{matrix}𝔲_{1} & 𝔲_{2} \\ 𝔳_{1} & 𝔳_{2} \end{matrix} ) · e_{2} − 2 det ( \begin{matrix}𝔲_{1} & 𝔲_{3} \\ 𝔳_{1} & 𝔳_{3} \end{matrix} ) · e_{3}.

Remark 5.2. Observe that if the prime ℓ is irregular for g, the vector w̃_{λ} is a scalar endomorphism in W_{g} and hence w_{λ} = w(ℓ) = 𝔚(ℓ) = 0.

Our main result is

Theorem 5.3. Assume Conjecture 4.1. For all w ∈ W_{g}^{∘}, there exists an overconvergent generalised eigenform g_{w}^{♭} ∈ S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} satisfying

a_{ℓ}(g_{w}^{♭}) = ⟨w, 𝔚(ℓ)⟩ = det(w, w(1), w(ℓ)),

for all primes ℓ ∤ Np. The assignment w ↦ g_{w}^{♭} induces an isomorphism between W_{g}^{∘}/U and S_{1}^{(p)}(N, χ)[[g_{α}]]_{0}.

Proof. The semi-local field H_{p} = H ⊗_{ℚ} ℚ_{p} = ⊕_{℘|p}ℚ_{p} is naturally identified with the set of vectors h = (h_{℘})_{℘|p} with entries h_{℘} ∈ ℚ_{p}, indexed by the primes of H above p. The function which to w ∈ W_{g}^{∘} associates the linear transformation

φ̃_{w} : H_{p}⟶W_{g}^{∘},  φ̃_{w}(h) = \frac{1}{#G} × ∑_{σ∈G}(σ^{−1}h)_{℘} · (σ · w)

identifies W_{g}^{∘} with hom_{G}(H_{p}, W_{g}^{∘}). The linear function φ̃_{w} is trivial on U := log_{p}(O_{H}^{×}) ⊂ H_{p} if and only if, for all 𝔲 ∈ O_{H}^{×} and all w′ ∈ W_{g}^{∘},

⟨φ̃_{w}(log_{p}(𝔲)), w′⟩ = 0.

But

⟨φ̃_{w}(log_{p}(𝔲)), w′⟩ = \frac{1}{#G} × ⟨∑_{σ∈G} log_{℘}(σ^{−1}(𝔲)) · (σ · w), w′⟩

= \frac{1}{#G} × ∑_{σ∈G} log_{℘}(σ^{−1}(𝔲)) · ⟨σ · w, w′⟩

= \frac{1}{#G} × ∑_{σ∈G} log_{℘}(σ^{−1}(𝔲)) · ⟨w, σ^{−1} · w′⟩
= ⟨w, ξ_{℘}(𝔲, w′)⟩,

and hence φ̃_{w} is trivial on U = log_{p}(O_{H}^{×}) if and only if w is orthogonal in W_{g}^{∘} to the line spanned by w(1). It follows that the G-equivariant linear function

φ_{w} := φ̃_{[w,w(1)]} : H_{p}⟶W_{g}^{∘}

factors through H_{p}/U. The assignment w ↦ φ_{w} identifies W_{g}^{∘}/(L·w(1)) with hom_{G}(H_{p}/U, W_{g}^{∘}), and gives an explicit description of the latter space.

Let g̃_{w} = g+εg_{w}^{♭} be the eigenform with coefficients in ℚ_{p}[ε] which is attached to the cocycle φ_{w} ∈ hom_{G}(H_{p}/U, W_{g}^{∘}) = H^{1}(ℚ, W_{g}^{∘}). Equation (43) with g^{♭} = g_{w}^{♭} (and hence c_{g^{♭}} = φ_{w}) combined with (47) shows that the ℓ-th the fourier coefficient of g_{w}^{♭} at a prime ℓ ∤ Np is equal to

(50)  a_{ℓ}(g_{w}^{♭}) = Tr(φ_{w}(σ_{λ}) · ϱ_{g}(σ_{λ})) = ⟨φ_{w}(σ_{λ}), w̃_{λ}⟩ = ⟨φ_{w}(σ_{λ}), w_{λ}⟩.

Class field theory for H implies that

φ_{w}(σ_{λ}) = \frac{1}{#G} × ∑_{σ∈G} log_{℘}(σ^{−1}𝔲_{λ}) · σ · [w, w(1)].

Hence

a_{ℓ}(g_{w}^{♭}) = \frac{1}{#G} × ⟨∑_{σ∈G} log_{℘}(σ^{−1}𝔲_{λ}) · σ · [w, w(1)], w_{λ}⟩

= \frac{1}{#G} × ∑_{σ∈G} log_{℘}(σ^{−1}𝔲_{λ}) · ⟨σ · [w, w(1)], w_{λ}⟩

= \frac{1}{#G} × ∑_{σ∈G} log_{℘}(σ^{−1}𝔲_{λ}) · ⟨[w, w(1)], σ^{−1} · w_{λ}⟩
= ⟨[w, w(1)], w(ℓ)⟩ = det(w, w(1), w(ℓ)) = ⟨w, 𝔚(ℓ)⟩.

The theorem follows. □

If w in a vector in W_{g}^{∘L}, Theorem 5.3 shows that the associated overconvergent generalised eigenform g_{w}^{♭} has fourier coefficients which are L-rational linear combinations of determinants of 2 × 2 matrices whose entries are the ℘-adic logarithms of algebraic numbers in H. In the CM and RM cases to be discussed below, the representation W_{g}^{∘} is reducible and decomposes further into non-trivial irreducible representations. In that case the choice of an L-basis for W_{g}^{∘L} which is compatible with this decomposition leads to canonical elements of S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} which can sometimes be re-scaled so that their fourier expansions admit even simpler expressions, as will be described in the next two sections.

6. CM forms

Assume that g is the theta series attached to a character of a quadratic imaginary field K, i.e., that

V_{g}^{L} = Ind_{K}^{ℚ} ψ_{g},

where ψ_{g} : Gal (K̄/K)⟶L^{×} is a finite order character. Let ψ′_{g} denote the character deduced from ψ_{g} by composing it with the involution in Gal (K/ℚ). The irreducibility assumption on V_{g}^{L} implies that the characters ψ_{g} and ψ′_{g} are distinct, and therefore the representations V_{g}^{L} and V_{g} decompose canonically as a direct sum of two G_{K}-stable one-dimensional subspaces

V_{g}^{L} = L_{ψ_{g}}^{L} ⊕ L_{ψ′_{g}}^{L}, V_{g} = L_{ψ_{g}} ⊕ L_{ψ′_{g}}

on which G_{K} acts via the characters ψ_{g} and ψ′_{g} respectively. The representations W_{g}^{L} and W_{g} also decompose as direct sums of four G_{K}-stable lines

W_{g}^{L} = (hom(L_{ψ_{g}}^{L}, L_{ψ_{g}}^{L}) ⊕ hom(L_{ψ′_{g}}^{L}, L_{ψ′_{g}}^{L})) ⊕ (hom(L_{ψ′_{g}}^{L}, L_{ψ_{g}}^{L}) ⊕ hom(L_{ψ_{g}}^{L}, L_{ψ′_{g}}^{L})) ,
W_{g} = (hom(L_{ψ_{g}}, L_{ψ_{g}}) ⊕ hom(L_{ψ′_{g}}, L_{ψ′_{g}})) ⊕ (hom(L_{ψ′_{g}}, L_{ψ_{g}}) ⊕ hom(L_{ψ_{g}}, L_{ψ′_{g}})) .

The direct summands in parentheses are also stable under G_{ℚ} and are isomorphic to the induced representations Ind_{K}^{ℚ} 1 and Ind_{K}^{ℚ} ψ respectively, where ψ := ψ_{g}/ψ′_{g}, is the ring class character of K associated to ψ_{g}. It follows that

W_{g}^{∘L} = L(χ_{K}) ⊕ Y_{g}^{L},  W_{g}^{∘} = ℚ_{p}(χ_{K}) ⊕ Y_{g},  Y_{g}^{L} := Ind_{K}^{ℚ} ψ, Y_{g} := Y_{g}^{L} ⊗_{L} ℚ_{p}.

It will be convenient to choose a basis (e_{1}, e_{2}) ∈ L_{ψ_{g}}^{L} × L_{ψ′_{g}}^{L} for V_{g}^{L}, and to denote by e_{11}, e_{12}, e_{21}, e_{22} the resulting basis of W_{g}^{L}, where e_{ij} is the elementary matrix whose (i′, j′)- entry is δi = i′δ_{j=j′}. Relative to the identification of W_{g}^{∘L} with the space of 2 × 2 matrices of trace zero with entries in L via this basis, the representation L(χ_{K}) = L · (e_{11} − e_{22}) is identified with the space of diagonal matrices of trace 0, while Y_{g}^{L} = L·e_{12} ⊕L·e_{21} is identified with the space of off-diagonal matrices in M_{2}(L). Fix an element τ ∈ G_{ℚ} = G_{K} once and for all. By eventually re-scaling e_{1} and e_{2}, it can (and shall, henceforth) be assumed that ϱ_{g}(τ)

is represented by the matrix ( \begin{matrix}0 & t \\ t & 0 \end{matrix} ) in this basis, where −t^{2} := χ(τ).

Let Z := Gal (H/K) be the maximal abelian normal subgroup of the dihedral group G = Gal (H/ℚ). Note that every element in G − Z (such as the image of τ in G) is an involution, and that Z operates transitively on G − Z by either left or right multiplication.

The field H through which W_{g}^{∘} factors is the ring class field of K attached to the character ψ. The group O_{H}^{×} ⊗ℚ of units of H is isomorphic to the regular representation of Z minus the trivial representation, and a finite index subgroup of O_{H}^{×} can be constructed explicity from the elliptic units arising in the theory of complex multiplication. Let

e_{ψ} := \frac{1}{#Z} ∑_{σ∈Z} ψ^{−1}(σ)σ

be the idempotent in the group ring of Z giving rise to the projection onto the ψ-isotypic component for the action of Z. Choose a unit 𝔲 ∈ O_{H}^{×} and let

𝔲_{ψ} := e_{ψ}𝔲, τ𝔲_{ψ} = e_{ψ′}(τ𝔲)

be elements of O_{H}^{×} ⊗ L on which Z acts via the characters ψ and ψ′ = ψ^{−1} respectively. With these choices, we can let

(51)  w(1) = ( \begin{matrix}0 & log_{℘}(𝔲_{ψ}) \\ log_{℘}(τ𝔲_{ψ}) & 0 \end{matrix} ) .

The description of the canonical vectors w(ℓ), 𝔚(ℓ) ∈ W_{g}^{∘} attached to a rational prime ℓ ∤ Np depends in an essential way on whether ℓ is split or inert in K/ℚ.

If ℓ = λλ′ is split in K and ℓ is regular for g, i.e., ψ_{g}(σ_{λ}) ≠ ψ_{g}(λ′), then the natural map

(O_{K}[1/ℓ]^{×} ⊗ W_{g}^{∘L})^{G} ⊂ (\frac{O_{H}[1/ℓ]^{×}}{O_{H}^{×}} ⊗ W_{g}^{∘L})^{G}

is an isomorphism of L-vector spaces.

Let 𝔲̃_{λ} be a generator of λ^{h} where h is the class number of K, and set

𝔲_{λ} := 𝔲̃_{λ} ⊗ h^{−1}.

Since

w̃_{λ} = ( \begin{matrix}ψ_{g}(λ) & 0 \\ 0 & ψ_{g}(λ′) \end{matrix} ) , w_{λ} = \frac{ψ_{g}(λ) − ψ_{g}(λ′)}{2} × ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,

a direct calculation shows that

w(ℓ) = log_{℘}(𝔲_{λ}/𝔲′_{λ}) × \frac{(ψ_{g}(λ) − ψ_{g}(λ′))}{2} × ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

It follows that

(52)  𝔚(ℓ) = log_{℘}(𝔲_{λ}/𝔲′_{λ}) × (ψ_{g}(λ) − ψ_{g}(λ′)) × ( \begin{matrix}0 & − log_{℘}(𝔲_{ψ}) \\ log_{℘}(𝔲′_{ψ}) & 0 \end{matrix} ) .

If ℓ is inert in K then ℓ is always regular for g since ϱ_{g}(τ_{ℓ}) has trace 0 and hence has distinct eigenvalues. The prime ℓ splits completely in H/K, and hence the group (O_{H}[1/ℓ]^{×}) ⊗ L is isomorphic to two copies of the regular representation of Z minus a trivial representation. The choice of a prime λ of H_{g} above ℓ determines a matrix (and not just a conjugacy class)

w̃_{λ} = w_{λ} = ϱ(σ_{λ}) = ( \begin{matrix}0 & b_{λ} \\ c_{λ} & 0 \end{matrix} )

with entries in L. Let 𝔲_{λ} be an element of (O_{H}[1/ℓ]^{×}/O_{H}^{×}) ⊗ L whose prime factorisation is given by

(53)  (𝔲_{λ}) = b_{λ}λ + c_{λ}(τλ).

This ℓ-unit is only well defined by (53) up to translation by O_{H}^{×} ⊗L, and the defining equation (53) of course depends crucially on the choice of the prime λ above ℓ. However, the ψ-isotypic projection

(54)  𝔲_{ψ}(ℓ) := e_{ψ}𝔲_{λ}

is independent of this choice. A direct calculation shows that

w(ℓ) = \frac{1}{2} × ( \begin{matrix}0 & log_{℘}(𝔲_{ψ}(ℓ)) \\ log_{℘}(τ𝔲_{ψ}(ℓ)) & 0 \end{matrix} ) .

It follows that

(55)  𝔚(ℓ) = ( \begin{matrix}R_{ψ}(ℓ) & 0 \\ 0 & −R_{ψ}(ℓ) \end{matrix} ) ,

where

R_{ψ}(ℓ) = det ( \begin{matrix}log_{℘}(𝔲_{ψ}) & log_{℘}(τ𝔲_{ψ}) \\ log_{℘}(𝔲_{ψ}(ℓ)) & log_{℘}(τ𝔲_{ψ}(ℓ)) \end{matrix} )

is an ℓ-unit regulator attached to ψ, which is independent of the choice of prime λ of H above ℓ. The function ℓ ↦ R_{ψ}(ℓ) does depend on the choice of the unit 𝔲, but only up to scaling by L^{×}.

Theorem 6.1. The space S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} has a canonical basis (g_{1}^{♭}, g_{2}^{♭}) which is characterised by the properties:

(i) The fourier coefficients a_{ℓ}(g_{1}^{♭}) are 0 for all primes ℓ ∤ Np that are inert in K. If ℓ = λλ′ is split in K, then

a_{ℓ}(g_{1}^{♭}) = (ψ_{g}(λ) − ψ_{g}(λ′)) × log_{℘}(𝔲_{λ}/𝔲′_{λ})

is a simple algebraic multiple of the p-adic logarithm of the fundamental ℓ-unit of norm 1 in K.

(ii) The fourier coefficients of g_{2}^{♭} are 0 at all the primes ℓ ∤ Np that are split in K. If ℓ is inert in K , then

a_{ℓ}(g_{s}^{♭}) = R_{ψ}(ℓ).

Proof. This follows directly from the calculation of the matrices 𝔚(ℓ) in (52) and (55) in light of Theorem 5.3. □

Example 6.2. Let χ be the quadratic character of conductor 59. The space S(59, χ) is one dimensional and spanned by the theta series

g = q − q^{3} + q^{4} − q^{5} − q^{7} − q^{12} + q^{15} + q^{16} + 2q^{17} − · · · .

Here K = ℚ(\sqrt{−59}) and the ring class field attached to ψ is

H = K(α) where α^{3} − 3α + 46\sqrt{−59} = 0.

The inert primes ℓ in K are 2, 3, 13, 23, ⋯ and the unit and first few ℓ-units are

u = \frac{1}{612} (13α^{2} − 7\sqrt{−59}α − 26) ,  u_{2} = \frac{1}{612} (−5α^{2} − 13\sqrt{−59}α − 194)
u_{11} = \frac{1}{306} (5α^{2} + 13\sqrt{−59}α − 112) , u_{13} = \frac{1}{612} (13α^{2} − 7\sqrt{−59}α − 1250)
u_{23} = \frac{1}{204} (−α^{2} + 11\sqrt{−59}α + 138) .

Let p = 17, an irregular prime for g. We computed a basis of q-expansions for the generalised eigenspace modulo p^{20} and q^{30,000}. One observes that it contains the classical space spanned by the forms g_{α}(q) and g(q^{p}) and in addition a complementary space of dimension two. This space is canonically spanned by two normalised generalised eigenforms

g̃_{1}^{♭} = q^{3} + ⋯ + 0 · q^{p} + ⋯ and g̃_{2}^{♭} = q^{2} + 0 · q^{3} + ⋯ + ⋯ + 0 · q^{p} + · · · .

Note that the natural scaling of the forms output by our algorithm is with leading Fourier coefficients equal to 1. By Theorem 6.1 one expects that for ℓ inert in K, or ℓ split in K but irregular, we have a_{ℓ}(g̃_{1}^{♭}) = 0; and for ℓ split in K we have that

a_{ℓ}(g̃_{1}^{♭}) = \frac{log_{p}(u_{ℓ})}{log_{p}(u_{3})}

where u_{ℓ} is a fundamental ℓ-unit in K (the logarithm of this is well-defined up to sign). We checked this to 20-digits of 17-adic precision for primes ℓ < 1000. Further, one expects that

a_{ℓ}(g̃_{2}^{♭}) = \frac{R_{ψ}(ℓ)}{R_{ψ}(2)} for ℓ inert in K, and a_{ℓ}(g̃_{2}^{♭}) = 0 for ℓ split in K.

We checked this for all split primes ℓ < 30, 000 and for the inert primes ℓ = 2, 3, 11 and 23, constructing R_{ψ}(ℓ) using the unit u and ℓ-unit u_{ℓ} above.

7. RM forms

We now turn to the RM setting where F is a real quadratic field and

V_{g} = Ind_{F}^{ℚ} ψ_{g},

where ψ_{g} : Gal (F̄/F)⟶L^{×} is a finite order character of mixed signature. Letting ψ′_{g} denote the character deduced from ψ_{g} by composing it with the involution in Gal (F/ℚ), the ratio ψ := ψ_{g}/ψ′_{g} is a totally odd L-valued ring class character of F.

As before, let H denote the ring class field of F which is fixed by the kernel of ψ, and set Z := Gal (H/F) and G := Gal (H/ℚ). Just as in the previous section,

W_{g}^{∘} = χ_{K} ⊕ Y_{g}, Y_{g} := Ind_{K}^{ℚ} ψ,

and we can set

w(1) = ( \begin{matrix}log_{℘}(𝔲_{F}) & 0 \\ 0 & − log_{℘}(𝔲_{F}) \end{matrix} ) ,

where 𝔲_{F} is a fundamental unit of F.

If ℓ is split in K/ℚ, it is easy to see that the vector w(ℓ) is proportional to w(1), and hence that

(56)  𝔚(ℓ) = 0.

If ℓ is inert in K, let U_{g} and U_{g}(ℓ) denote the subspaces (O_{H}^{×} ⊗ Y_{g})^{G_{ℚ}} and (O_{H}[1/ℓ]^{×} ⊗ Y_{g})^{G_{ℚ}}. The the dimensions of these spaces are 0 and 1 respectively. Choose a prime λ of H above ℓ, and let 𝔲_{λ} and 𝔲_{ψ}(ℓ) be the elements of O_{H}[1/ℓ]^{×} determined by the relations

(𝔲_{λ}) = b_{λ}λ + c_{λ}τλ,  𝔲_{ψ}(ℓ) = e_{ψ}(𝔲_{λ}),  𝔲′_{ψ}(ℓ) = τ𝔲_{ψ}(ℓ),

where

ϱ_{g}(σ_{λ}) = ( \begin{matrix}0 & b_{λ} \\ c_{λ} & 0 \end{matrix} ) .

The ℘-adic logarithms

log_{℘}(𝔲_{ψ}(ℓ)), log_{℘}(𝔲′_{ψ}(ℓ))

are well-defined invariants of ℓ and ϱ which do not depend on the choice of a prime λ lying above ℓ, and

w(ℓ) = \frac{1}{2} × ( \begin{matrix}0 & log_{℘}(𝔲_{ψ}(ℓ)) \\ log_{℘}(𝔲′_{ψ}(ℓ)) & 0 \end{matrix} ) .

It follows that

(57)  𝔚(ℓ) = log_{℘}(𝔲_{F}) × ( \begin{matrix}0 & log_{℘}(𝔲_{ψ}(ℓ)) \\ − log_{℘}(𝔲′_{ψ}(ℓ)) & 0 \end{matrix} ) .

Theorem 7.1. The space S_{1}^{(p)}(N, χ)[[g_{α}]]_{0} has a canonical basis (g_{1}^{♭}, g_{2}^{♭}) which is characterised by the properties:

(i) The fourier coefficients of g_{1}^{♭} and g_{2}^{♭} are 0 at all primes ℓ ∤ Np that are split in F. (ii) If ℓ is inert in F, then

a_{ℓ}(g_{1}^{♭}) = log_{℘}(𝔲_{ψ}(ℓ)), a_{ℓ}(g_{2}^{♭}) = log_{℘}(𝔲′_{ψ}(ℓ)).

Proof. This follows directly from Theorem 5.3 in light of equations (56) and (57). □

Example 7.2. Let χ_{8} and χ_{7} denote the quadratic characters of conductors 8 and 7, respectively, and define χ := χ_{8}χ_{7}. Then S_{1}(56, χ) is one-dimensional and spanned by the form

g = q − q^{2} + q^{4} − q^{7} − q^{8} − q^{9} + q^{14} + q^{16} + q^{18} + 2q^{23} − · · · .

We take p = 23, an irregular prime for g, and compute a basis for the generalised eigenspace modulo (p^{15}, q^{3000}). The two dimensional space complementary to the classical space has a natural basis

g̃_{1}^{♭} = q^{3} + ⋯ + 0 · q^{p} + ⋯ and g̃_{2}^{♭} = q^{2} + 0 · q^{3} + ⋯ + ⋯ + 0 · q^{p} + · · · .

Take

g_{1}^{♭} := \frac{1}{2} · log_{p}(u_{2}) · g̃_{1}^{♭} and g_{2}^{♭} := log_{p}(u_{3}) · g̃_{2}^{♭}.

Here u_{ℓ}, ℓ = 2 and 3, denotes a fundamental ℓ-unit of norm 1 in ℚ(\sqrt{−7}) and ℚ(\sqrt{−56}), respectively. One finds that the coefficients at primes ℓ which are split in ℚ(\sqrt{8}) of both forms g_{1}^{♭} and g_{2}^{♭} are zero. At inert primes the coefficients of g_{1}^{♭} are the logarithms of fundamental ℓ-units of norm 1 in ℚ(\sqrt{−7}), and those of g_{2}^{♭} are the logarithms of fundamental ℓ-units of norm 1 in ℚ(\sqrt{−56}) (such logarithms are well-defined up to sign; interestingly, the forms g_{j}^{♭} single out a consistent choice of signs).

References

[BD] J. Bellaïche, M. Dimitrov, On the eigencurve at classical weight one points, Duke Math Journal 165, Issue 2 (2016), 245–266.

[Bet] Adel Betina, Ramification of the eigencurve at classical RM points, submitted.

[BL] K. Buzzard and A. Lauder, A computation of modular forms of weight one and small level, Annales mathématique du Québec, to appear. Appendix http://people.maths.ox.ac.uk/lauder/weight1

[DLi] W. Duke and Y. Li. Harmonic Maass forms of weight 1. Duke Math. J. 164 (2015), no. 1, 39–113.

[DLR1] H. Darmon, A. Lauder and V. Rotger, Stark points and p-adic iterated integrals attached to modular forms of weight one. Forum of Mathematics, Pi (2015), Vol. 3, e8, 95 pages.

[DLR2] H. Darmon, A. Lauder, and V. Rotger, Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields. Advances in Mathematics 283 (2015) 130-142.

[DLR3] H. Darmon, A. Lauder, and V. Rotger, Elliptic Stark conjectures and irregular weight one forms, in progress.

[La] A. Lauder, Computations with classical and p-adic modular forms, LMS J. Comput. Math. 14 (2011) 214-231.

[Ma] B. Mazur, Deforming Galois representations, in Galois groups over ℚ (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York-Berlin, pp. 395–437, 1989.

[St] H. M. Stark, L-functions at s = 1. II. Artin L-functions with Rational Characters, Advances in Mathematics 17 (1975), 60–92.

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