#  First order $p$-adic deformations
of weight one newforms

Henri Darmon, Alan Lauder and Victor Rotger  H. D.: McGill University,
Montreal, Canada<EMAIL_ADDRESS>A. L.: University of Oxford, U. K.
<EMAIL_ADDRESS>V. R.: Universitat Politècnica de Catalunya, Barcelona,
Spain<EMAIL_ADDRESS>
###### 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

  1. A The regular setting
    1. 1 The general case
    2. 2 CM forms
    3. 3 RM forms
  2. B The irregular setting
    1. 4 Generalised eigenspaces
    2. 5 The general case
    3. 6 CM forms
    4. 7 RM forms

### Introduction

Let $g$ be a classical cuspidal newform of weight one, level $N$ and
nebentypus character
$\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\mathbb{Q}_{p}^{\times}$,
with fourier expansion

$g(q)=\sum_{n=1}^{\infty}a_{n}q^{n}.$

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

(1) $g_{\alpha}(q):=g(q)-\beta\cdot g(q^{p}),\qquad
g_{\beta}(q):=g(q)-\alpha\cdot g(q^{p}),$

where $\alpha$ and $\beta$ are the (not necessarily distinct) roots of the
Hecke polynomial

$x^{2}-a_{p}x+\chi(p)=:(x-\alpha)(x-\beta).$

The forms $g_{\alpha}$ and $g_{\beta}$ are eigenvectors for the Atkin $U_{p}$
operator, with eigenvalues $\alpha$ and $\beta$ respectively. Since $\alpha$
and $\beta$ 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 $\varrho_{g}:G_{\mathbb{Q}}{\longrightarrow}{\bf
GL}_{2}(\mathbb{C})$ 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 $g_{\alpha}$ (or to $g_{\beta}$) 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
$\alpha\neq\beta$, 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 non-classical 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 $\alpha\neq\beta$, 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 $S_{1}^{(p)}(N,\chi)$ denote the
space of $p$-adic overconvergent modular forms of weight $1$, level $N$, and
character $\chi$, and let $S_{1}^{(p)}(N,\chi)[[g]]$ denote the generalised
eigenspace attached to the system of Hecke eigenvalues of an irregular weight
one form $g\in S_{1}(N,\chi)$. The main conjecture of the second part asserts
that $S_{1}^{(p)}(N,\chi)[[g]]$ 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. The three authors would like to thank
the organizers of the Math Nisyros Conference in July 2017 for their
hospitality before and after the workshop. 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 $\Lambda=\mathbb{Z}_{p}[[1+p\mathbb{Z}_{p}]]$ denote the Iwasawa algebra,
and let

$\mathcal{W}:=\operatorname{Hom}_{\operatorname{cts}}(1+p\mathbb{Z}_{p},\mathbb{C}_{p}^{\times})=\operatorname{Hom}_{\operatorname{alg}}(\Lambda,\mathbb{C}_{p})$

denote the associated weight space. For each $k\in\mathbb{Z}_{p}$, write
$\nu_{k}\in\mathcal{W}$ for the “weight $k$” homomorphism sending the group-
like element $a\in 1+p\mathbb{Z}_{p}$ to $a^{k-1}$. The rule
$\lambda(k):=\nu_{k}(\lambda)$ realises elements of $\Lambda$ as analytic
functions on $\mathbb{Z}_{p}$. The spectrum
$\tilde{\mathcal{W}}:=\operatorname{Hom}_{\operatorname{alg}}(\tilde{\Lambda},\mathbb{C}_{p})$
of a finite flat extension $\tilde{\Lambda}$ of $\Lambda$ is equipped with a
“weight map”

$\operatorname{w}:\tilde{\mathcal{W}}{\longrightarrow}\mathcal{W}$

of finite degree. A $\mathbb{Q}_{p}$-valued point $x\in\tilde{\mathcal{W}}$ is
said to be of weight $k$ if $\operatorname{w}(x)=\nu_{k}$, and is said to be
étale over $\mathcal{W}$ if the inclusion $\Lambda\subset\tilde{\Lambda}$
induces an isomorphism between $\Lambda$ and the completion of
$\tilde{\Lambda}$ at the kernel of $x$, denoted $\tilde{\Lambda}_{x}$. An
element of this completion thus gives rise to an analytic function of
$k\in\mathbb{Z}_{p}$ in a natural way.

A Hida family is a formal $q$-series

${\mathbf{g}}:=\sum a_{n}q^{n}\in\tilde{\Lambda}[[q]]$

with coefficients in a finite flat extension $\tilde{\Lambda}$ of $\Lambda$,
specialising to a classical ordinary eigenform of weight $k$ at almost all
points $x$ of $\tilde{\mathcal{W}}$ of weight $k\in\mathbb{Z}^{\geq 2}$. Two
Hida families ${\mathbf{g}}_{1}\in\tilde{\Lambda}_{1}[[q]]$ and
${\mathbf{g}}_{2}\in\tilde{\Lambda}_{2}[[q]]$ are regarded as equal if the
$\Lambda$-algebras $\tilde{\Lambda}_{1}$ and $\tilde{\Lambda}_{2}$ can be
embedded in a common extension $\tilde{\Lambda}$ in such a way that
${\mathbf{g}}_{1}$ and ${\mathbf{g}}_{2}$ are identified. A well known theorem
of Hida and Wiles asserts the existence of a Hida family specialising to
$g_{\alpha}$ 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 $x_{\alpha}$
and $x_{\beta}$ denote the distinct points on $\tilde{\mathcal{W}}$ attached
to $g_{\alpha}$ and $g_{\beta}$ respectively. Then

  * (a)

The curve $\tilde{\mathcal{W}}$ is smooth at $x_{\alpha}$ and $x_{\beta}$, and
in particular there are unique Hida families
${\mathbf{g}}_{\alpha},{\mathbf{g}}_{\beta}\in\tilde{\Lambda}[[q]]$
specialising to $g_{\alpha}$ and $g_{\beta}$ at $x_{\alpha}$ and $x_{\beta}$
respectively.

  * (b)

The weight map
$\operatorname{w}:\tilde{\mathcal{W}}{\longrightarrow}\mathcal{W}$ is
furthermore étale at $x_{\alpha}$ if any only if

$\quad\qquad({\dagger})\quad\mbox{ $g$ 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 $\operatorname{w}$ is not étale at
$x_{\alpha}$ has been treated in [DLR2], and the remainder of Part A will
therefore focus on the scenarios where $({\dagger})$ is satisfied. In that
case, after viewing elements of the completion $\tilde{\Lambda}_{x_{\alpha}}$
of $\tilde{\Lambda}$ at $x_{\alpha}$ as analytic functions of the “weight
variable” $k$, one may consider the canonical $q$-series

$g_{\alpha}^{\prime}:=\left(\frac{d}{dk}{\mathbf{g}}_{\alpha}\right)_{k=1}$

representing the first-order infinitesimal ordinary deformation of
${\mathbf{g}}$ at the weight one point $x_{\alpha}$, along this canonical
“weight direction”. The $q$-series $g_{\alpha}^{\prime}$ 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 $g_{\alpha}^{\prime}$, which should rather be
viewed as the first order term of a “modular form of weight $1+\varepsilon$”.

  * (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 $g_{\alpha}^{\prime}$, 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 $g_{\alpha}^{\prime}$.

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

$\sigma\cdot w:=\varrho_{g}(\sigma)w\varrho_{g}(\sigma)^{-1},\qquad\sigma\in
G_{\mathbb{Q}},\quad w\in W_{g}.$

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

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 $\mathbb{Q}_{p}$, which is fixed henceforth, will
allow us, when it is convenient, to view $V_{g}$ and $W_{g}$ as
representations of $G_{\mathbb{Q}}$ with coefficients in $\mathbb{Q}_{p}$, and
the weight one form $g$ as a modular form with fourier coefficients in
$\mathbb{Q}_{p}$ rather than in $L$. The $\mathbb{Q}_{p}$-vector spaces
$V_{g}$ and $W_{g}$ are thus equipped with natural $G_{\mathbb{Q}}$-stable
$L$-rational structures, denoted $V_{g}^{L}$ and $W_{g}^{L}$ respectively.

An embedding of $\bar{\mathbb{Q}}$ into $\bar{\mathbb{Q}}_{p}$ is fixed once
and for all, determining a prime $\wp$ of $H$ and of $H_{g}$ above $p$, and an
associated frobenius element $\tau_{\wp}$ in
$\mathrm{Gal\,}(H_{g}/\mathbb{Q})$ and in $G$. Let $G_{\wp}\subset G$ be the
decomposition subgroup generated by $\tau_{\wp}$.

The representations $V_{g}$ and $W_{g}$ admit the following decompositions as
$\tau_{\wp}$-modules:

$V_{g}=V_{g}^{\alpha}\oplus V_{g}^{\beta},\qquad
W_{g}=W_{g}^{\alpha\alpha}\oplus W_{g}^{\alpha\beta}\oplus
W_{g}^{\beta\alpha}\oplus W_{g}^{\beta\beta},$

where $V_{g}^{\alpha}$ and $\mathcal{V}_{g}^{\beta}$ denote the $\alpha$ and
$\beta$-eigenspaces for the action of $\tau_{\wp}$ on $V_{g}$, and

$W_{g}^{\xi\eta}:=\hom(V_{g}^{\xi},V_{g}^{\eta}),\qquad\mbox{ for
}\xi,\eta\in\\{\alpha,\beta\\}$

is a $G_{\wp}$-stable line, on which $\tau_{\wp}$ acts with eigenvalue
$\eta/\xi$. Let

$W_{g}^{{\mathrm{ord}}}:=\hom(V_{g}/V_{g}^{\alpha},V_{g})=W_{g}^{\beta\alpha}\oplus
W_{g}^{\beta\beta}.$

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

We propose to give a general formula for the $\ell$-th fourier coefficient of
$g_{\alpha}^{\prime}$ 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 ${\mathcal{O}}_{H}^{\times}\otimes W_{g}$ diagonally on both
factors in the tensor product.

######  Lemma 1.2.

The $\mathbb{Q}_{p}$-vector space $({\mathcal{O}}_{H}^{\times}\otimes
W_{g})^{G}$ of $G$-invariant vectors is one-dimensional.

###### Proof.

Let $G_{\infty}$ 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 ${\mathcal{O}}_{H}^{\times}\otimes\mathbb{Q}_{p}$ is
isomorphic to
$\operatorname{Ind}_{G_{\infty}}^{G}(\mathbb{Q}_{p})-\mathbb{Q}_{p}$ as a
$\mathbb{Q}_{p}[G]$-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 $W_{g}=W_{g}^{0}\oplus\mathbb{Q}_{p}$, and
$W_{g}^{0}$ does not contain the trivial representation as a constituent. By
Frobenius reciprocity,

$\dim_{\mathbb{Q}_{p}}(({\mathcal{O}}_{H}^{\times}\otimes
W_{g})^{G})=\dim_{\mathbb{Q}_{p}}(({\mathcal{O}}_{H}^{\times}\otimes
W_{g}^{0})^{G})=\dim_{\mathbb{Q}_{p}}((W_{g}^{0})^{c=1})=1.$

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 ${\mathcal{O}}_{H}^{\times}\otimes L$ decomposes
as a direct sum of $V$-isotypic components,

${\mathcal{O}}_{H}^{\times}\otimes L=\oplus_{V}{\mathcal{O}}_{H}^{\times}[V],$

where $V$ runs over the irreducible representations of $G$, and
${\mathcal{O}}_{H}^{\times}[V]$ denotes the largest subrepresentation of
${\mathcal{O}}_{H}^{\times}\otimes L$ 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 ${\mathcal{O}}_{H}^{\times}[W]$ is defined as
the direct sum of the ${\mathcal{O}}_{H}^{\times}[V]$ 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 ${\mathcal{O}}_{H}^{\times}[W_{g}]$ is isomorphic to a
single irreducible subrepresentation of $W_{g}$. More precisely:

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

(2)
${\mathcal{O}}_{H}^{\times}[W_{g}]={\mathcal{O}}_{H}^{\times}[W_{g}^{0}]\simeq
W_{g}^{0},$

and hence is three-dimensional.

$\bullet$ If $\varrho_{g}$ is induced from a character $\psi_{g}$ of an
imaginary quadratic field $K$, then

$W_{g}=L\oplus L(\chi_{K})\oplus V_{\psi},$

where $\chi_{K}$ is the odd quadratic Dirichlet character associated to $K$
and $V_{\psi}$ is the two-dimensional representation of $G$ induced from the
ring class character $\psi=\psi_{g}/\psi_{g}^{\prime}$ which cuts out the
abelian extension $H$ of $K$. The representation $V_{\psi}$ is irreducible if
and only if $\psi$ is non-quadratic, and in that case,

(3)
${\mathcal{O}}_{H}^{\times}[W_{g}]={\mathcal{O}}_{H}^{\times}[V_{\psi}]\simeq
V_{\psi}.$

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

$\bullet$ If $\varrho_{g}$ is induced from a character $\psi_{g}$ of a real
quadratic field $F$, then

$W_{g}=L\oplus L(\chi_{F})\oplus V_{\psi},\qquad
V_{\psi}:=\operatorname{Ind}_{F}^{\mathbb{Q}}(\psi),\quad\psi:=\psi_{g}/\psi_{g}^{\prime},$

and one always has

(4)
${\mathcal{O}}_{H}^{\times}[W_{g}]={\mathcal{O}}_{H}^{\times}[\chi_{F}]\simeq
L(\chi_{F}),$

i.e., ${\mathcal{O}}_{H}^{\times}[W_{g}]$ is generated by a fundamental unit
of $F$.

Let $U_{g}^{\times}$ be any generator of the one-dimensional
$\mathbb{Q}_{p}$-vector space $({\mathcal{O}}_{H}^{\times}\otimes W_{g})^{G}$
and let

(5) $U_{g}:=(\log_{\wp}\otimes{\rm id})(U_{g}^{\times})\in H_{\wp}\otimes
W_{g}$

be the image of this vector under the linear map

$\log_{\wp}\otimes{\rm id}:{\mathcal{O}}_{H}^{\times}\otimes
W_{g}{\longrightarrow}H_{\wp}\otimes W_{g},$

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

######  Lemma 1.3.

There exists a non-zero endomorphism $A\in H_{\wp}\otimes W_{g}$ satisfying
the following conditions:

  1. (a)

${\rm Trace}(AU_{g})=0$.

  2. (b)

$A$ belongs to $H_{\wp}\otimes W_{g}^{\rm ord}$, i.e., $A(V_{g}^{\alpha})=0$.

This endomorphism is unique up to scaling.

###### Proof.

The space $H_{\wp}\otimes W_{g}$ is four-dimensional over $H_{\wp}$ and the
conditions in Lemma 1.3 amount to three linear conditions on $A$. More
precisely, choose a $\tau_{\wp}$-eigenbasis $(v_{\alpha},v_{\beta})$ for
$V_{g}$ for which

$\tau_{\wp}v_{\alpha}=\alpha v_{\alpha},\qquad\tau_{\wp}v_{\beta}=\beta
v_{\beta}.$

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

$U_{g}:\left(\begin{array}[]{cc}\log_{\wp}(u_{1})&\log_{\wp}(u_{\beta/\alpha})\\\
\log_{\wp}(u_{\alpha/\beta})&-\log_{\wp}(u_{1})\end{array}\right),$

where $u_{1},u_{\alpha/\beta}$, and $u_{\beta/\alpha}$ are generators of
${\mathcal{O}}_{H}^{\times}[W_{g}]$ which (when non-zero) are eigenvectors for
$\tau_{\wp}$, satisfying

$\tau_{\wp}(u_{1})=u_{1},\qquad\tau_{\wp}(u_{\beta/\alpha})=(\beta/\alpha)u_{\beta/\alpha},\qquad\tau_{\wp}(u_{\alpha/\beta})=(\alpha/\beta)u_{\alpha/\beta}.$

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

$A:\left(\begin{array}[]{cc}0&x\\\ 0&y\end{array}\right),\qquad x,y\in
H_{\wp},$

and condition (a) implies the further linear relation

(6) $\log_{\wp}(u_{\alpha/\beta})\cdot x-\log_{\wp}(u_{1})\cdot y=0.$

The injectivity of $\log_{\wp}:{\mathcal{O}}_{H}^{\times}\otimes
L{\longrightarrow}H_{\wp}$, which follows from the linear independence over
$\bar{\mathbb{Q}}$ of logarithms of algebraic numbers, implies that the
coefficients $\log_{\wp}(u_{\alpha/\beta})$ and $\log_{\wp}(u_{1})$ in (6)
vanish simultaneously if and only if

$u_{\alpha/\beta}=u_{1}=0$

in ${\mathcal{O}}_{H}^{\times}\otimes L$, i.e., if and only if
${\mathcal{O}}_{H}^{\times}[W_{g}]$ is one-dimensional over $L$ and generated
by $u_{\beta/\alpha}$. This immediately rules out (2) and (3) as scenarios for
the structure of ${\mathcal{O}}_{H}^{\times}[W_{g}]$, leaving only (4). Hence,
$V_{g}$ is induced from a character of a real quadratic field $F$. In that
case, the lines spanned by $u_{\alpha/\beta}$ and $u_{\beta/\alpha}$ are
interchanged under the action of any reflection in $G$, and hence the
condition $u_{\alpha/\beta}=0$ implies that $u_{\beta/\alpha}=0$ as well, thus
forcing the vanishing of the full ${\mathcal{O}}_{H}^{\times}[W_{g}]$. 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 $H_{\wp}\otimes W_{g}$ satisfying the conditions in
Lemma 1.3. Then the following are equivalent:

  * (a)

${\rm Trace}(A)\neq 0$;

  * (b)

The representation $\varrho_{g}$ is not induced from a character of a real
quadratic field in which the prime $p$ splits.

###### Proof.

The vanishing of ${\rm Trace}(A)$ is equivalent to the vanishing of the entry
$y$ in (6), and hence to the vanishing of $\log_{\wp}(u_{\alpha/\beta})$, and
therefore of $u_{\alpha/\beta}$ and $u_{\beta/\alpha}$ as well. This implies
that ${\mathcal{O}}_{H}^{\times}[W_{g}]$ 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, $\tau_{\wp}$ fixes the
group ${\mathcal{O}}_{H}^{\times}[W_{g}]$ 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 $A_{g}\in H_{\wp}\otimes W_{g}$ to be the unique
$H_{\wp}^{\times}$\- multiple of $A$ satisfying

${\rm Trace}(A_{g})=1.$

As in lemma 1.2, $H_{\wp}\otimes W_{g}$ is endowed with the diagonal action of
$G_{\wp}$ which acts on both $H_{\wp}$ and on $W_{g}$ in a natural way. Given
$A\in H_{\wp}\otimes W_{g}$ and $\sigma\in G_{\wp}$, let us write
${}^{\sigma}A$ for the image of $A$ by the action of $\sigma$ on the first
factor $H_{\wp}$, and $\sigma\cdot A_{g}$ for the image of $A$ by the action
of $\sigma$ by conjugation on the second factor $W_{g}$.

######  Lemma 1.5.

The endomorphism $A_{g}$ belongs to the space $(H_{\wp}\otimes
W_{g})^{G_{\wp}}$ of $G_{\wp}$-invariants for the diagonal action of $G_{\wp}$
on $H_{\wp}\otimes W_{g}$, i.e.,

${}^{\tau_{\wp}}A_{g}=\tau_{\wp}^{-1}\cdot A_{g}.$

###### Proof.

Relative to the $\mathbb{Q}_{p}$-basis for $V_{g}$ used in the proof of Lemma
1.3, the endomorphism $A_{g}$ is represented by a matrix of the form

$\left(\begin{array}[]{cc}0&\frac{\log_{\wp}(u_{1})}{\log_{\wp}(u_{\alpha/\beta})}\\\
0&1\end{array}\right).$

The lemma follows immediately from this in light of the fact that conjugation
by $\varrho_{g}(\tau_{\wp})$ preserves the diagonal entries in such a matrix
representation while multiplying its upper right hand entry by $\alpha/\beta$,
whereas $\tau_{\wp}$ acts on the upper right-hand entry of the above matrix as
multiplication by $\beta/\alpha$. ∎

The matrix $A_{g}$ gives rise to a $G$-equivariant homomorphism
$\Phi_{g}:H^{\times}{\longrightarrow}H_{\wp}\otimes W_{g}$ by setting

(7) $\Phi_{g}(x)=\sum_{\sigma\in
G}\log_{\wp}(^{\sigma}x)\cdot(\sigma^{-1}\cdot A_{g}),$

where, just as above, the group $G$ acts on $H_{\wp}\otimes W_{g}$ trivially
on the first factor and through the usual conjugation action induced by
$\rho_{g}$ on the second factor.

######  Lemma 1.6.

The homomorphism $\Phi_{g}$ takes values in $W_{g}$.

###### Proof.

For any $x\in(H\otimes\mathbb{Q}_{p})^{\times}$ we have

${}^{\tau_{\wp}}\Phi_{g}(x)$ $\displaystyle=$ $\displaystyle\sum_{\sigma\in
G}\log_{\wp}(^{\tau_{\wp}\sigma}\\!x)\cdot(\sigma^{-1}\cdot^{\tau_{\wp}}\\!\\!A_{g})$
$\displaystyle=$ $\displaystyle\sum_{\sigma\in
G}\log_{\wp}(^{\tau_{\wp}\sigma}\\!x)\cdot(\sigma^{-1}\cdot{\tau_{\wp}}^{-1}\cdot
A_{g})$ $\displaystyle=$ $\displaystyle\sum_{\sigma\in
G}\log_{\wp}(^{\tau_{\wp}\sigma}\\!x)\cdot((\tau_{\wp}\sigma)^{-1}\cdot
A_{g})$ $\displaystyle=$ $\displaystyle\Phi_{g}(x),$

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

$\Phi_{g}:(H\otimes\mathbb{Q}_{p})^{\times}{\longrightarrow}H_{\wp}\otimes
W_{g}$

obtained from (7) by extending scalars. Note that $H_{\wp}^{\times}$ embeds
naturally in $(H\otimes\mathbb{Q}_{p})^{\times}$.

######  Lemma 1.7.

The homomorphism $\Phi_{g}$ vanishes on
${\mathcal{O}}_{H}^{\times}\otimes\mathbb{Q}_{p}$ and
$\Phi_{g}(H_{\wp}^{\times})\subseteq H_{\wp}\otimes W_{g}^{{\mathrm{ord}}}$.

###### Proof.

Picking $u\in{\mathcal{O}}_{H}^{\times}$ and an arbitrary $B\in W_{g}$, set

$U_{g}^{\times}:=\sum_{\sigma\in G}\phantom{x}^{\sigma}u\otimes(\sigma\cdot
B)\in({\mathcal{O}}_{H}^{\times}\otimes W_{g})^{G},\qquad
U_{g}:=(\log_{\wp}\otimes{\rm id})(U_{g})$

as in the statement of Lemma 1.3. Note that $U_{g}^{\times}$ is either trivial
or a generator of the one-dimensional space
$({\mathcal{O}}_{H}^{\times}\otimes W_{g})^{G}$. We have

$\displaystyle{\rm Trace}(\Phi_{g}(u)\cdot B)$ $\displaystyle=$
$\displaystyle{\rm Trace}\left(\left(\sum_{\sigma\in
G}\log_{\wp}(^{\sigma}u)\cdot(\sigma^{-1}\cdot A_{g})\right)\cdot B\right)$
$\displaystyle=$ $\displaystyle{\rm
Trace}\left(A_{g}\cdot\left(\sum_{\sigma\in
G}\log_{\wp}(^{\sigma}u)\cdot(\sigma\cdot B)\right)\right)$ $\displaystyle=$
$\displaystyle{\rm Trace}\left(A_{g}\cdot(\log_{\wp}\otimes{\rm
Id})(U_{g}^{\times})\right)={\rm Trace}\left(A_{g}\cdot U_{g}\right).$

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

${\rm Trace}\left(\Phi_{g}(u)\cdot B\right)=0,\qquad\mbox{ for all }B\in
H_{\wp}\otimes W_{g}.$

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

∎

Let now $\ell\nmid Np$ be a rational prime, and let $\lambda$ be a prime of
$H$ above $\ell$. Let
$u(\lambda)\in{\mathcal{O}}_{H}[1/\lambda]^{\times}\otimes\mathbb{Q}$ be a
$\lambda$-unit of $H$ satisfying

(8) ${\rm Norm}^{H}_{\mathbb{Q}}(u(\lambda))=\ell.$

This condition makes $u(\lambda)$ well-defined up to the addition of elements
in ${\mathcal{O}}_{H}^{\times}\otimes\mathbb{Q}$, and hence the element

$A_{g}(\lambda):=\Phi_{g}(u(\lambda))=\sum_{\sigma\in
G}\log_{\wp}(^{\sigma}u(\lambda))\cdot(\sigma^{-1}\cdot A_{g})$

is well-defined, by Lemma 1.7. Although $A_{g}(\lambda)$ only belongs to
$H_{\wp}\otimes W_{g}$ a priori, we have:

######  Lemma 1.8.

The trace of the endomorphism $A_{g}(\lambda)$ is equal to $\log_{p}(\ell)$.

###### Proof.

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

$\displaystyle{\rm Trace}(A_{g}(\lambda))$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in G}\log_{\wp}(^{\sigma}u(\lambda))\cdot{\rm
Trace}(\sigma^{-1}\cdot A_{g})$ $\displaystyle=$ $\displaystyle\sum_{\sigma\in
G}\log_{\wp}(^{\sigma}u(\lambda))$ $\displaystyle=$
$\displaystyle\log_{\wp}\left({\rm Norm}^{H}_{\mathbb{Q}}(u(\lambda))\right).$

The latter expression is equal to $\log_{p}(\ell)$, by (8). ∎

######  Remark 1.9.

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

In addition to the invariant $A_{g}(\lambda)$, the choice of the prime
$\lambda$ of $H$ above $\ell$ also determines a well-defined Frobenius element
$\tau_{\lambda}$ in $G=\mathrm{Gal\,}(H/\mathbb{Q})$, and even in
$\mathrm{Gal\,}(H_{g}/\mathbb{Q})$, since $\mathrm{Gal\,}(H_{g}/H)$ 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 $\ell\nmid Np$,

$a_{\ell}(g_{\alpha}^{\prime})={\rm
Trace}(\varrho_{g}(\tau_{\lambda})A_{g}(\lambda)).$

######  Remark 1.11.

This invariant does not depend on the choice of a prime $\lambda$ of $H$ above
$\ell$, since replacing $\lambda$ by another such prime has the effect of
conjugating the endomorphisms $\varrho_{g}(\tau_{\lambda})$ and
$A_{g}(\lambda)$ by the same element of $\mathrm{Aut}(V_{g})$.

###### Proof of Theorem 1.10.

Let $\mathbb{Q}[\varepsilon]$ denote the ring of dual numbers over
$\mathbb{Q}_{p}$, with $\varepsilon^{2}=0$, and let

$\tilde{\varrho}_{g}:G_{\mathbb{Q}}{\longrightarrow}{\bf
GL}_{2}(\mathbb{Q}_{p}[\varepsilon])$

be the unique first order $\alpha$-ordinary deformation of $\varrho_{g}$
satisfying

$\det\tilde{\varrho}_{g}=\chi_{g}(1+\log_{p}\chi_{\operatorname{cyc}}\cdot\varepsilon).$

This representation may be written as

(9)
$\tilde{\varrho}_{g}=(1+\varepsilon\cdot\kappa_{g})\cdot\varrho_{g}\quad\mbox{
for some }\quad\kappa_{g}:G_{\mathbb{Q}}{\longrightarrow}W_{g}.$

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

$a_{\ell}(g_{\alpha})+\varepsilon\cdot a_{\ell}(g_{\alpha}^{\prime})={\rm
Trace}(\tilde{\varrho}_{g}(\tau_{\lambda}))=a_{\ell}(g)+\varepsilon\cdot{\rm
Trace}(\kappa_{g}(\tau_{\lambda})\varrho_{g}(\tau_{\lambda})),$

and hence

(10) $a_{\ell}(g_{\alpha}^{\prime})={\rm
Trace}(\varrho_{g}(\tau_{\lambda})\kappa_{g}(\tau_{\lambda})).$

To make $\kappa_{g}(\tau_{\lambda})$ explicit, observe that the inflation-
restriction sequence combined with global class field theory for $H$ gives
rise to a series of identifications

$\displaystyle H^{1}(\mathbb{Q},W_{g})$
$\displaystyle\stackrel{{\scriptstyle\mathrm{res}_{H}}}{{{\longrightarrow}}}$
$\displaystyle\hom(G_{H},W_{g})^{G}$ $\displaystyle=$
$\displaystyle\hom_{G}\left(\frac{({\mathcal{O}}_{H}\otimes\mathbb{Q}_{p})^{\times}}{{\mathcal{O}}_{H}^{\times}\otimes\mathbb{Q}_{p}},W_{g}\right).$

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

$H^{1}_{\rm
ord}(\mathbb{Q},W_{g})=\left\\{\Phi\in\hom_{G}\left(\frac{({\mathcal{O}}_{H}\otimes\mathbb{Q}_{p})^{\times}}{{\mathcal{O}}_{H}^{\times}\otimes\mathbb{Q}_{p}},W_{g}\right)\mbox{
such that }\Phi(H_{\wp}^{\times})\subset W_{g}^{\rm ord}\right\\}.$

But the homomorphism $\Phi_{g}$ of (7) belongs to the same one-dimensional
space, by Lemma 1.6 and 1.7. By global class field theory, the endomorphism
$\kappa_{g}(\tau_{\lambda})$ is therefore a $\mathbb{Q}_{p}^{\times}$-multiple
of $\Phi_{g}(u_{g}(\lambda))=A_{g}(\lambda)$. The fact that these
endomorphisms are actually equal now follows by comparing their traces and
noting that

${\rm
Trace}(\kappa_{g}(\tau_{\lambda}))=\log_{p}\chi_{\operatorname{cyc}}(\ell)=\log_{p}(\ell),$

while

${\rm Trace}(A_{g}(\lambda))=\log_{p}(\ell),$

by Lemma 1.8. Theorem 1.10 follows. ∎

######  Corollary 1.12.

If the rational prime $\ell\nmid Np$ splits completely in $H/\mathbb{Q}$, then

$a_{\ell}(g_{\alpha}^{\prime})=(1/2)\cdot a_{\ell}(g)\cdot\log_{p}(\ell).$

###### Proof.

The hypothesis implies that $\varrho_{g}(\tau_{\lambda})$ is a scalar, and
hence that $\varrho_{g}(\tau_{\lambda})=\frac{1}{2}a_{\ell}(g)$. It follows
that

${\rm Trace}(\varrho_{g}(\tau_{\lambda})A_{g}(\lambda))=(1/2)\cdot
a_{\ell}(g)\cdot{\rm Trace}(A_{g}(\lambda))=(1/2)\cdot
a_{\ell}(g)\cdot\log_{p}(\ell).$

The corollary now follows from Theorem 1.10. ∎

######  Example 1.13.

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

$g=q+\zeta q^{2}+\zeta^{3}q^{3}-\zeta^{2}q^{5}+(\zeta^{2}-1)q^{6}+\cdots$

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

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

Let $p=13$, which splits completely in $L$. The representation $\varrho_{g}$
is regular at $13$, with eigenvalues $\alpha=\zeta$ and $\beta=-\zeta^{3}$. We
computed the first order deformations through each of $g_{\alpha}$ and
$g_{\beta}$ 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 $a_{\ell}(g_{\alpha}^{\prime})$
depend upon the conjugacy class of the Frobenius at $\ell$ in
$\mathrm{Gal\,}(H/\mathbb{Q})$. For all primes $\ell<37,000$ which split
completely in $H$, such as $\ell=109,179,449,467,521,\ldots$, we verified that

$a_{\ell}(g_{\alpha}^{\prime})=(1/2)\cdot
a_{\ell}(g)\cdot\log_{13}(\ell)\pmod{13^{10}},$

as asserted by Corollary 1.12.

###  2\. CM forms

This section focuses on the case where $g=\theta_{\psi_{g}}$ is the CM theta
series attached to a character

${\psi_{g}}:G_{K}{\longrightarrow}L^{\times}$

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
$\mathbb{Q}_{p}$ allows us to view $\psi_{g}$ as a
$\mathbb{Q}_{p}^{\times}$-valued character, and the weight one form $g$ as a
modular form with coefficients in $\mathbb{Q}_{p}$.

For a character $\psi:G_{K}{\longrightarrow}L^{\times}$, the notation
$\psi^{\prime}$ will be used to designate the composition of $\psi$ with
conjugation by the non-trivial element in $\mathrm{Gal\,}(K/\mathbb{Q})$:

$\psi^{\prime}(\sigma)=\psi(\tau\sigma\tau^{-1}),$

where $\tau$ is any element of $G_{\mathbb{Q}}$ which acts non-trivially on
$K$.

The Artin representation $\varrho_{g}$ is induced from $\psi_{g}$ and its
restriction to $G_{K}$ is the direct sum $\psi_{g}\oplus\psi_{g}^{\prime}$ of
two characters of $K$, which are distinct by the irreducibility of
$\varrho_{g}$ 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 ${\psi}:={\psi_{g}}/{\psi_{g}}^{\prime}$. The
Galois group $G:=\mathrm{Gal\,}(H/\mathbb{Q})$ is a generalised dihedral group
containing $Z:=\mathrm{Gal\,}(H/K)$ 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
$p{\mathcal{O}}_{K}=\operatorname{\mathfrak{p}}\operatorname{\mathfrak{p}}^{\prime}$,
and fix a prime $\wp$ of $\bar{\mathbb{Q}}$ above
$\operatorname{\mathfrak{p}}$. The roots of the $p$-th Hecke polynomial of $g$
are

$\alpha=\psi_{g}(\operatorname{\mathfrak{p}}),\qquad\beta=\psi_{g}(\operatorname{\mathfrak{p}}^{\prime}).$

This case is notable in that the Hida family ${\mathbf{g}}$ passing through
$g_{\alpha}$ can be written down explicitly as a family of theta series. Its
weight $k$ specialisation ${\mathbf{g}}_{k}$ is the theta-series attached to
the character ${\psi_{g}}\Psi^{k-1}$, where $\Psi$ is a CM Hecke character of
weight $(1,0)$ which is unramified at $\operatorname{\mathfrak{p}}$. For all
rational primes $\ell\nmid Np$, the $\ell$-th fourier coefficient of
${\mathbf{g}}_{k}$ is given by

(11)
$a_{\ell}({\mathbf{g}}_{k})=\left\\{\begin{array}[]{ll}{\psi_{g}}(\lambda)\Psi^{k-1}(\lambda^{\prime})+{\psi_{g}}(\lambda^{\prime})\Psi^{k-1}(\lambda)&\mbox{
if }\ell=\lambda\lambda^{\prime}\mbox{ splits in $K$};\\\ 0&\mbox{ if $\ell$
is inert in $K$}.\end{array}\right.$

Letting $h$ be the class number of $K$ and $t$ the cardinality of the unit
group ${\mathcal{O}}_{K}^{\times}$, the character $\Psi$ satisfies

$\Psi(\lambda)^{ht}=u_{\lambda}^{t},\qquad\mbox{ where
}(u_{\lambda}):=\lambda^{h},$

for any prime ideal $\lambda$ of ${\mathcal{O}}_{K}$ whose norm is the
rational prime $\ell=\lambda\lambda^{\prime}$. Let $u_{\lambda}^{\prime}$
denote the conjugate of $u_{\lambda}$ in $K/\mathbb{Q}$. It follows that

$\displaystyle\frac{d}{dk}\Psi^{k-1}(\lambda)_{k=1}$ $\displaystyle=$
$\displaystyle\log_{\operatorname{\mathfrak{p}}}(u(\lambda)),\qquad\mbox{
where
}u(\lambda):=u_{\lambda}\otimes\frac{1}{h}\in{\mathcal{O}}_{H}[1/\ell]^{\times}\otimes\mathbb{Q},$

and likewise that

$\displaystyle\frac{d}{dk}\Psi^{k-1}(\lambda^{\prime})_{k=1}$ $\displaystyle=$
$\displaystyle\log_{\operatorname{\mathfrak{p}}}(u(\lambda)^{\prime}),\qquad\mbox{
where
}u(\lambda)^{\prime}:=u_{\lambda}^{\prime}\otimes\frac{1}{h}\in{\mathcal{O}}_{H}[1/\ell]^{\times}\otimes\mathbb{Q}.$

In light of (11), we have obtained:

######  Theorem 2.1.

For all rational primes $\ell$ that do not divide $Np$,

(12)
$a_{\ell}(g_{\alpha}^{\prime})=\left\\{\begin{array}[]{ll}\left({\psi_{g}}(\lambda)\log_{\operatorname{\mathfrak{p}}}(u(\lambda^{\prime}))+{\psi_{g}}(\lambda^{\prime})\log_{\operatorname{\mathfrak{p}}}(u(\lambda))\right)&\mbox{
if }\ell=\lambda\lambda^{\prime}\mbox{ splits in $K$};\\\ 0&\mbox{ if $\ell$
is inert in $K$}.\end{array}\right.$

Thus, the prime fourier coefficients of $g_{\alpha}^{\prime}$ are supported at
the primes $\ell$ which are split in $K$, where they are (algebraic multiples
of) the $\operatorname{\mathfrak{p}}$-adic logarithms of $\ell$-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
$g_{\alpha}^{\prime}$ will be more complicated expressions involving, in
general, the $p$-adic logarithms of units and $\ell$-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
$\tau_{\wp}$) which are interchanged by some element $\tau\in
G_{\mathbb{Q}}-G_{K}$. Relative to such a basis, the endomorphisms $U_{g}$ and
$A_{g}$ are represented by the following matrices, in which $u_{\psi}$ and
$\tau u_{\psi}$ are generators of the spaces of $\psi$ and
$\psi^{-1}$-isotypic vectors in the group of elliptic units in
${\mathcal{O}}_{H}^{\times}\otimes L$:

$U_{g}:\left(\begin{array}[]{cc}0&\log_{\wp}(u_{\psi})\\\
\log_{\wp}(u_{\psi}^{\prime})&0\end{array}\right),\qquad
A_{g}:\left(\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right).$

It follows that, if $\ell=\lambda\lambda^{\prime}$ is split in $K$, then
$A_{g}(\lambda)$ is represented by the matrix

$A_{g}(\lambda):\left(\begin{array}[]{cc}\log_{\wp}(u(\lambda^{\prime}))&0\\\
0&\log_{\wp}(u(\lambda))\end{array}\right),$

while $A_{g}(\lambda)=\frac{1}{2}\log_{p}(\ell)$ is the scalar matrix with
trace equal to $\log_{p}(\ell)$ if $\ell$ 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
$\sigma_{\wp}:=\tau_{\wp}^{2}$ denote the frobenius element in $G_{K}$
attached to the prime $\wp$ of $H$ (which is well-defined modulo the inertia
subgroup at $\wp$). Note that the prime $p$ splits completely in $H/K$, since
the image of $\tau_{\wp}$ in $G$ is a reflection in this generalised dihedral
group. The image of $\sigma_{\wp}$ in $\mathrm{Gal\,}(H_{g}/K)$ therefore
belongs to the subgroup $\mathrm{Gal\,}(H_{g}/H)$ whose image under
$\varrho_{g}$ consists of scalar matrices. Similar notations and remarks apply
to any rational prime $\ell$ which is inert in $K/\mathbb{Q}$.

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

(13) $\varrho_{g}(\sigma)=\left(\begin{array}[]{cc}\psi_{g}(\sigma)&0\\\
0&\psi_{g}^{\prime}(\sigma)\end{array}\right)\ \ \mbox{for }\sigma\in G_{K}.$

The homomorphisms
$\psi_{g},\psi_{g}^{\prime}:G_{K}{\longrightarrow}\mathbb{Q}_{p}^{\times}$
factor through $\mathrm{Gal\,}(H_{g}/K)$ and satisfy

$\psi_{g}(\tau\sigma\tau^{-1})=\psi_{g}^{\prime}(\sigma),\quad\mbox{ for all
}\tau\in G_{\mathbb{Q}}-G_{K},\ \ \ \sigma\in G_{K}.$

It follows that $\varrho_{g}(\tau)$ interchanges the lines spanned by $e_{1}$
and $e_{2}$, for any element $\tau\in G_{\mathbb{Q}}-G_{K}$. The restriction
of $\varrho_{g}$ to $G_{\mathbb{Q}}-G_{K}$ can therefore be described in
matrix form by

(14) $\varrho_{g}(\tau)=\left(\begin{array}[]{cc}0&\eta_{g}(\tau)\\\
\eta_{g}^{\prime}(\tau)&0\end{array}\right)\ \ \mbox{for }\tau\in
G_{\mathbb{Q}}-G_{K},$

where $\eta_{g}$ and $\eta_{g}^{\prime}$ are $L$-valued functions on
$G_{\mathbb{Q}}-G_{K}$ that satisfy

(15)
$\eta_{g}(\tau_{1})\eta_{g}^{\prime}(\tau_{2})=\psi_{g}(\tau_{1}\tau_{2})=\psi_{g}^{\prime}(\tau_{2}\tau_{1}),\qquad\mbox{
for all }\tau_{1},\tau_{2}\in G_{\mathbb{Q}}-G_{K},$

as well as the relations

(16)
$\begin{array}[]{ll}\eta_{g}(\sigma\tau)=\psi_{g}(\sigma)\eta_{g}(\tau),&\quad\eta_{g}(\tau\sigma)=\psi_{g}^{\prime}(\sigma)\eta_{g}(\tau),\\\
\eta_{g}^{\prime}(\sigma\tau)=\psi_{g}^{\prime}(\sigma)\eta_{g}^{\prime}(\tau),&\quad\eta_{g}^{\prime}(\tau\sigma)=\psi_{g}(\sigma)\eta_{g}^{\prime}(\tau),\end{array}\qquad\mbox{
for all }\sigma\in G_{K},\ \ \tau\in G_{\mathbb{Q}}-G_{K}.$

After re-scaling $e_{1}$ and $e_{2}$ if necessary, we may assume that
$\tau_{\wp}\in G_{\mathbb{Q}}-G_{K}$ is sent to the matrix

(17) $\varrho_{g}(\tau_{\wp})=\left(\begin{array}[]{cc}0&\zeta\\\
\zeta&0\end{array}\right),\qquad\mbox{ with }\zeta^{2}=-\chi_{g}(p).$

The eigenvalues of $\varrho_{g}(\tau_{\wp})$ are equal to $\alpha:=\zeta$ and
$\beta:=-\zeta$, and hence $g$ is always regular in this setting.

Let

$\tilde{\varrho}_{g}:G_{\mathbb{Q}}{\longrightarrow}{\bf GL}(\tilde{V}_{g})$

denote the first-order infinitesimal deformation of $\varrho_{g}$ attached to
the Hida family ${\mathbf{g}}$ passing through a choice of $p$-stabilization
$g_{\alpha}$ of $g$, where $\alpha\in\\{\zeta,-\zeta\\}$. The module
$\tilde{V}_{g}$ is free of rank two over the ring
$\mathbb{Q}_{p}[\varepsilon]/(\varepsilon^{2})=\mathbb{Q}_{p}[[T]]/(T^{2})$
arising from the mod $T^{2}$ reduction of the representation
$\varrho_{{\mathbf{g}}}$ attached to ${\mathbf{g}}$. Choose any
$\mathbb{Q}_{p}[\varepsilon]$-basis $(\tilde{e}_{1},\tilde{e}_{2})$ of
$\tilde{V}_{g}$ lifting $(e_{1},e_{2})$, and note that the restriction of
$\tilde{\varrho}_{g}$ to $G_{K}$ is given by:

(18)
$\tilde{\varrho}_{g}(\sigma)=\left(\begin{array}[]{cc}{\psi_{g}}(\sigma)\cdot(1+\kappa(\sigma)\cdot\varepsilon)&\psi_{g}^{\prime}(\sigma)\cdot\kappa_{\psi}(\sigma)\cdot\varepsilon\\\
{\psi_{g}}(\sigma)\cdot\kappa_{\psi}^{\prime}(\sigma)\cdot\varepsilon&\psi_{g}^{\prime}(\sigma)\cdot(1+\kappa^{\prime}(\sigma)\cdot\varepsilon)\end{array}\right),\quad\mbox{
for all }\sigma\in G_{K}.$

In this expression,

  * (a)

The functions $\kappa$ and $\kappa^{\prime}$ are continuous homomorphisms from
$G_{K}$ to $\mathbb{Q}_{p}$, i.e., elements of $H^{1}(K,\mathbb{Q}_{p})$,
which are interchanged by conjugation by the involution in
$\mathrm{Gal\,}(K/\mathbb{Q})$:

$\kappa(\tau\sigma\tau^{-1})=\kappa^{\prime}(\sigma),\qquad\tau\in
G_{\mathbb{Q}}-G_{K},\ \ \ \sigma\in G_{K}.$

  * (b)

The functions
$\kappa_{\psi},\kappa_{\psi}^{\prime}:G_{K}{\longrightarrow}\mathbb{Q}_{p}$
are one-cocycles with values in $\mathbb{Q}_{p}({\psi})$, and give rise to
well defined classes

$\kappa_{\psi}\in
H^{1}(K,\mathbb{Q}_{p}({\psi})),\qquad\kappa_{\psi}^{\prime}\in
H^{1}(K,\mathbb{Q}_{p}({\psi}^{-1})),$

which also satisfy

$\kappa_{\psi}(\tau\sigma\tau^{-1})=\kappa_{\psi}^{\prime}(\sigma),\qquad\tau\in
G_{\mathbb{Q}}-G_{K},\ \ \ \sigma\in G_{K}.$

For each rational prime $\ell\nmid Np$, the $\ell$-th fourier coefficient
$a_{\ell}(g_{\alpha}^{\prime})$ is given by

(19)
$a_{\ell}(g_{\alpha}^{\prime})=\frac{d}{dk}\mbox{Trace}(\varrho_{{\mathbf{g}}}(\tau_{\lambda}))_{k=1}$

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

(20)
$H^{1}(K,\mathbb{Q}_{p})=\hom({\mathcal{O}}_{K_{p}}^{\times},\mathbb{Q}_{p})=\mathbb{Q}_{p}\log_{p}(z)\oplus\mathbb{Q}_{p}\log_{p}(z^{\prime}).$

Let ${\mathcal{O}}_{H}^{\times,\psi}$ denote the (one-dimensional)
$\psi$-isotypic component of ${\mathcal{O}}_{H}^{\times}\otimes\mathbb{Q}_{p}$
on which $\mathrm{Gal\,}(H/K)$ acts through the character $\psi$, and denote
by $\wp$ the prime of $H$ above $p$ arising from our chosen embedding of
$\bar{\mathbb{Q}}$ into $\bar{\mathbb{Q}}_{p}$. Restriction to the inertia
group at $\wp$ in $G_{H}$ likewise gives rise to an identification

(21)
$H^{1}(K,\mathbb{Q}_{p}(\psi))=\hom({\mathcal{O}}_{H_{\wp}}^{\times}/{\mathcal{O}}_{H}^{\times,\psi},\mathbb{Q}_{p})=\mathbb{Q}_{p}\cdot(\log_{\wp}(u_{\psi}^{\prime})\log_{\wp}(z)-\log_{\wp}(u_{\psi})\log_{\wp}(z^{\prime})).$

In the above equation, $u_{\psi}$ is to be understood as the natural image in
${\mathcal{O}}_{H_{\wp}}^{\times}={\mathcal{O}}_{K_{p}}^{\times}$ of an
element of the form

$\sum_{\sigma\in
Z}\psi^{-1}(\sigma)u^{\sigma}\in({\mathcal{O}}_{H}^{\times}\otimes L)^{\psi},$

where $u$ is an $L[G]$-module generator of ${\mathcal{O}}_{H}^{\times}\otimes
L$, and $u_{\psi}^{\prime}$ is the image of $u_{\psi}$ under the conjugation
action $K_{p}{\longrightarrow}K_{p}$. Note that replacing $u$ by $\lambda u$
for some $\lambda\in L[G]$ has the effect of multiplying both $u_{\psi}$ and
$u_{\psi}^{\prime}$ by $\psi(\lambda)\in\mathbb{Q}_{p}$, so that the
$\mathbb{Q}_{p}$-line spanned by the right-hand side of (21) is independent of
the choice of $u\in{\mathcal{O}}_{H}^{\times}$.

It follows from (20) and (21) that the total deformation space of
$\varrho_{g}$ (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 $\tau_{\wp}$ acting on $V_{g}$, with eigenvalues $\zeta$ and $-\zeta$
respectively. Let $\kappa_{p}$ and $\kappa_{\psi,\wp}$ denote the restrictions
$\kappa$ and $\kappa_{\psi}$ to the inertia groups at $p$ and $\wp$ in $G_{H}$
and $G_{K}$respectively. Both can be viewed as characters of
$K_{p}^{\times}=H_{\wp}^{\times}$ after identifiying the abelianisations of
$G_{K_{p}}$ and $G_{H_{\wp}}$ with a quotient of $K_{p}^{\times}$ via local
class field theory.

######  Lemma 2.2.

The following are equivalent:

  * (a)

The inertia group at $\wp$ acts as the identity on some lift
$\tilde{v}_{g}^{+}$ of $v_{g}^{+}$ to $\tilde{V}_{g}$;

  * (b)

The inertia group at $\wp$ acts as the identity on all lifts
$\tilde{v}_{g}^{+}$ of $v_{g}^{+}$ to $\tilde{V}_{g}$;

  * (c)

The restrictions $\kappa_{p}$ and $\kappa_{\psi,\wp}$ satisfy

$\kappa_{p}(x)=-\kappa_{\psi,\wp}(x),\qquad\mbox{ for all
}x\in{\mathcal{O}}_{K_{p}}^{\times}.$

Similar statements hold when $v_{g}^{+}$ is replaced by $v_{g}^{-}$, where the
conclusion is that $\kappa_{p}=\kappa_{\psi,\wp}$.

###### Proof.

The equivalence of the first two conditions follows from the fact that
$\varepsilon\tilde{V}_{g}\simeq V_{g}$ is unramified at $p$ and hence that
inertia acts as the identity on the kernel of the natural map
$\tilde{V}_{g}{\longrightarrow}V_{g}$. 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 $\sigma\in I_{p}$ sends
$\tilde{e}_{1}+\tilde{e}_{2}$ to

$\displaystyle\tilde{\varrho}_{g}(\sigma)(\tilde{e}_{1}+\tilde{e}_{2})$
$\displaystyle=$
$\displaystyle\tilde{e}_{1}+\tilde{e}_{2}+\varepsilon\cdot(\kappa(\sigma)\tilde{e}_{1}+\kappa_{\psi}^{\prime}(\sigma)\tilde{e}_{2}+\kappa_{\psi}(\sigma)\tilde{e}_{1}+\kappa^{\prime}(\sigma)\tilde{e}_{2})$
$\displaystyle=$
$\displaystyle\tilde{e}_{1}+\tilde{e}_{2}+\varepsilon\cdot((\kappa(\sigma)+\kappa_{\psi}(\sigma))\tilde{e}_{1}+(\kappa^{\prime}(\sigma)+\kappa_{\psi}^{\prime}(\sigma))\tilde{e}_{2}).$

The lemma follows. ∎

A lift $\tilde{\varrho}_{g}$ of $\varrho_{g}$ 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 $g_{\alpha}$. More precisely, the
general ordinary first-order deformation of $\varrho_{g}$ is completely
determined by the pair $(\kappa_{p},\kappa_{\psi,\wp})$, which depends on a
single linear parameter $\mu\in\bar{\mathbb{Q}}_{p}$ and is given by the rule

(22) $\displaystyle\kappa_{p}(z)$ $\displaystyle=$
$\displaystyle\mu(\log_{\wp}(u_{\psi}^{\prime})\cdot\log_{\wp}(z)-\log_{\wp}u_{\psi}\cdot\log_{\wp}(z^{\prime})),$
(23) $\displaystyle\kappa_{\psi,p}(z)$ $\displaystyle=$
$\displaystyle\pm\mu(\log_{\wp}(u_{\psi}^{\prime})\cdot\log_{\wp}(z)-\log_{\wp}u_{\psi}\cdot\log_{\wp}(z^{\prime})),$

where the sign in the second formula depends on whether one is working with
the ordinary deformation of $g_{\alpha}$ or $g_{\beta}$.

Let us now make use of the fact that

$\det(\tilde{\varrho}_{g})=1+\varepsilon\log_{p}\chi_{\operatorname{cyc}}=1+\varepsilon\log_{p}(zz^{\prime}).$

Since $\det(\tilde{\varrho}_{g})=1+\varepsilon(\kappa+\kappa^{\prime})$, this
condition implies that

$\mu=\frac{1}{\log_{\wp}(u_{\psi}^{\prime})-\log_{\wp}(u_{\psi})},$

and hence that $\kappa_{p}$ and $\kappa_{\psi,\wp}$ are given by

(24) $\displaystyle\kappa_{p}(z)$ $\displaystyle=$
$\displaystyle\frac{\log_{\wp}(u_{\psi}^{\prime})\cdot\log_{\wp}(z)-\log_{\wp}(u_{\psi})\cdot\log_{\wp}(z^{\prime})}{\log_{\wp}(u_{\psi}^{\prime})-\log_{\wp}(u_{\psi})},$
(25) $\displaystyle\kappa_{\psi,\wp}(z)$ $\displaystyle=$
$\displaystyle\pm\frac{\log_{\wp}(u_{\psi}^{\prime})\cdot\log_{\wp}(z)-\log_{\wp}(u_{\psi})\cdot\log_{\wp}(z^{\prime})}{\log_{\wp}(u_{\psi}^{\prime})-\log_{\wp}(u_{\psi})}.$

Equations (24) and (25) give a completely explicit description of the first
order deformation $\tilde{\varrho}_{g_{\alpha}}$ and
$\tilde{\varrho}_{g_{\beta}}$, from which the fourier coefficients of
$g_{\alpha}^{\prime}$ and $g_{\beta}^{\prime}$ shall be readily calculated.

The formula for the $\ell$-th fourier coefficient of $g_{\alpha}^{\prime}$
involves the unit $u_{\psi}$ above as well as certain $\ell$-units in
${\mathcal{O}}_{H}[1/\ell]^{\times}\otimes L$ whose definition depends on
whether or not the prime $\ell$ is split or inert in $K/\mathbb{Q}$.

If $\ell=\lambda\lambda^{\prime}$ splits in $K/\mathbb{Q}$, let $u(\lambda)$
and $u(\lambda^{\prime})$ denote, as before, the $\ell$-units in
${\mathcal{O}}_{K}[1/\ell]^{\times}\otimes\mathbb{Q}$ of norm $\ell$ with
prime factorisation $\lambda$ and $\lambda^{\prime}$ respectively. Set

$u_{g}(\lambda):=u(\lambda)\otimes{{\psi_{g}}(\lambda)}+u(\lambda^{\prime})\otimes{{\psi_{g}}(\lambda^{\prime})},\qquad
u_{g}(\lambda^{\prime}):=u(\lambda^{\prime})\otimes{{\psi_{g}}(\lambda)}+u(\lambda)\otimes{{\psi_{g}}(\lambda^{\prime})}.$

In other words, $u_{g}(\lambda)$ is the unique element of
${\mathcal{O}}_{K}[1/\ell]^{\times}\otimes L$ whose prime factorisation is
equal to
${\psi_{g}}(\lambda)\cdot\lambda+{\psi_{g}}(\lambda^{\prime})\cdot\lambda^{\prime}$.
Note that, if $\ell$ splits completely in $H/\mathbb{Q}$, i.e., if
$\varrho_{g}(\tau_{\lambda})$ is equal to a scalar $\zeta$, then
$u_{g}(\lambda)=u_{g}(\lambda^{\prime})=\ell\otimes\zeta$, but that otherwise
$u_{g}(\lambda)$ and $\ell$ generate the $L$-vector space
${\mathcal{O}}_{K}[1/\ell]^{\times}\otimes L$ of $\ell$-units of $K$ (tensored
with $L$).

If $\ell$ is inert in $K/\mathbb{Q}$, choose a prime $\lambda$ of $H$ lying
above $\ell$, and let
$u(\lambda)\in{\mathcal{O}}_{H}[1/\lambda]^{\times}\otimes\mathbb{Q}$ be any
$\lambda$-unit of $H$ satisfying ${\mathrm{ord}}_{\lambda}(u(\lambda))=1$,
which is well defined up to units in ${\mathcal{O}}_{H}^{\times}$. Define the
elements

$\displaystyle u_{\psi}(\lambda)$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in Z}{\psi}^{-1}(\sigma)\otimes\
^{\sigma}\\!u(\lambda)\quad\in\quad
L\otimes{\mathcal{O}}_{H}[1/\ell]^{\times},$ $\displaystyle
u_{\psi}^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\tau_{\wp}u_{\psi}(\lambda)\quad\in\quad
L\otimes{\mathcal{O}}_{H}[1/\ell]^{\times}.$

Thus $u_{\psi}(\lambda)$ lies in the $\psi$-component
${\mathcal{O}}_{H}[1/\ell]^{\times}[\psi]$ and is well-defined up to the
addition of multiples of $u_{\psi}$, where

$u_{\psi}:=\sum_{\sigma\in Z}{\psi}^{-1}(\sigma)\otimes\
^{\sigma}\\!u\quad\in\quad L\otimes{\mathcal{O}}_{H}[1/\ell]^{\times},$

for any unit $u\in{\mathcal{O}}_{H}^{\times}$, while
$u_{\psi}^{\prime}(\lambda)$ lies in the $\psi^{-1}$ component and is well-
defined up to the addition of multiples of $u_{\psi}^{\prime}$, where

$u_{\psi}^{\prime}=\tau_{\wp}u_{\psi}.$

Recall the function $\eta_{g}^{\prime}:G_{\mathbb{Q}}\setminus G_{K}$
introduced in (14), with values in the roots of unity of $L^{\times}$. The
main result of this section is:

######  Theorem 2.3.

Let $\ell\nmid Np$ be a rational prime.

  * (a)

If $\ell=\lambda\lambda^{\prime}$ splits in $K/\mathbb{Q}$, then

(26)
$a_{\ell}(g_{\alpha}^{\prime})=a_{\ell}(g_{\beta}^{\prime})=\frac{\log_{\wp}(u_{\psi}^{\prime})\cdot\log_{\wp}(u_{g}(\lambda))-\log_{\wp}(u_{\psi})\cdot\log_{\wp}(u_{g}(\lambda^{\prime}))}{\log_{\wp}(u_{\psi}^{\prime})-\log_{\wp}(u_{\psi})}.$

  * (b)

If $\ell$ remains inert in $K/\mathbb{Q}$, then

$a_{\ell}(g_{\alpha}^{\prime})=\eta_{g}^{\prime}(\tau_{\lambda})\frac{\log_{\wp}({u}_{\psi}^{\prime})\log_{\wp}(u_{\psi}(\lambda))-\log_{\wp}(u_{\psi})\log_{\wp}({u}_{\psi}^{\prime}(\lambda))}{\log_{\wp}(u_{\psi}^{\prime})-\log_{\wp}(u_{\psi}))}.$

###### Proof.

Let us first compute first the fourier coefficients at primes $\ell\nmid Np$
that split as $\ell=\lambda\lambda^{\prime}$ in $K$. Let $\sigma_{\lambda}$
and $\sigma_{\lambda^{\prime}}$ be the frobenius elements associated to
$\lambda$ and $\lambda^{\prime}$ respectively. They are well-defined elements
in the Galois group of any abelian extension of $K$ in which $\ell$ is
unramified.

It follows from (19) and the matrix expression for $\tilde{\varrho}_{g|G_{K}}$
given in (18) that

$\displaystyle a_{\ell}(g_{\alpha}^{\prime})$ $\displaystyle=$
$\displaystyle{\psi_{g}}(\lambda)\kappa(\lambda)+{\psi_{g}}(\lambda^{\prime})\kappa(\lambda^{\prime})$
$\displaystyle=$
$\displaystyle{\psi_{g}}(\lambda)\kappa_{p}(u(\lambda))+{\psi_{g}}(\lambda^{\prime})\kappa_{p}(u^{\prime}(\lambda)).$

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

We now turn now to the computation of the fourier coefficients of
$g^{\prime}_{\alpha}$ at primes $\ell\nmid Np$ that remain inert in $K$. Let
$\tau_{\lambda}$ denote the Frobenius element in
$\mathrm{Gal\,}(M/\mathbb{Q})$ associated to the choice of a prime ideal
$\lambda$ above $\ell$ in $\bar{\mathbb{Q}}$, and let
$\sigma_{\lambda}:=\tau_{\lambda}^{2}$ denote the associated frobenius element
in $\mathrm{Gal\,}(M/K)$.

Since $\tau_{\lambda}$ belongs to $G_{\mathbb{Q}}-G_{K}$, it follows from (14)
that the matrix $\tilde{\varrho}_{g}(\tau_{\lambda})$ is of the form

$\tilde{\varrho}_{g}(\tau_{\lambda})=\left(\begin{array}[]{cc}r_{\ell}\cdot\varepsilon&\eta_{g}(\tau_{\lambda})(1+s_{\ell}\cdot\varepsilon)\\\
\eta_{g}^{\prime}(\tau_{\lambda})(1+t_{\ell}\cdot\varepsilon)&u_{\ell}\cdot\varepsilon\end{array}\right),$

for suitable scalars $r_{\ell},s_{\ell},t_{\ell}$, and
$u_{\ell}\in\bar{\mathbb{Q}}_{p}$. Since

$a_{\ell}(g_{\alpha})+a_{\ell}(g_{\alpha}^{\prime})\varepsilon=\mbox{Trace}(\tilde{\varrho}_{g}(\tau_{\lambda}))=(r_{\ell}+u_{\ell})\cdot\varepsilon,$

it follows that

(27) $a_{\ell}(g_{\alpha}^{\prime})=r_{\ell}+u_{\ell}.$

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)
$\tilde{\varrho}_{g}(\sigma_{\lambda})=\tilde{\varrho}_{g}(\tau_{\lambda})^{2}=\left(\begin{array}[]{cc}\psi_{g}(\sigma_{\lambda})(1+(s_{\ell}+t_{\ell})\cdot\varepsilon)&\eta_{g}(\tau_{\lambda})(r_{\ell}+u_{\ell})\cdot\varepsilon\\\
\eta_{g}^{\prime}(\tau_{\lambda})(r_{\ell}+u_{\ell})\cdot\varepsilon&\psi_{g}(\sigma_{\lambda})(1+(s_{\ell}+t_{\ell})\cdot\varepsilon)\end{array}\right).$

On the other hand, since $\sigma_{\lambda}$ belongs to $G_{K}$ it follows from
(18) that

(29)
$\tilde{\varrho}_{g}(\sigma_{\lambda})=\left(\begin{array}[]{cc}{\psi_{g}}(\sigma_{\lambda})\cdot(1+\kappa(\sigma_{\lambda})\cdot\varepsilon)&\psi_{g}^{\prime}(\sigma_{\lambda})\cdot\kappa_{\psi}(\sigma_{\lambda})\cdot\varepsilon\\\
{\psi_{g}}(\sigma_{\lambda})\cdot\kappa_{\psi}^{\prime}(\sigma_{\lambda})\cdot\varepsilon&\psi_{g}^{\prime}(\sigma_{\lambda})\cdot(1+\kappa^{\prime}(\sigma_{\lambda})\cdot\varepsilon).\end{array}\right)$

By comparing upper-right entries in the matrices in (28) and (29) and invoking
(27) together with the relation
$\psi_{g}^{\prime}(\sigma_{\lambda})\eta_{g}(\tau_{\lambda})^{-1}=\eta_{g}^{\prime}(\tau_{\lambda})$
arising from (15), we deduce that

$a_{\ell}(g_{\alpha}^{\prime})=\eta_{g}^{\prime}(\tau_{\lambda})\kappa_{\psi}(\sigma_{\lambda}).$

It is worth noting that each of the expressions
$\eta_{g}^{\prime}(\tau_{\lambda})$ and $\kappa_{\psi}(\sigma_{\lambda})$
depend on the choice of a prime $\lambda$ of $H$ above $\ell$ that was made to
define $\tau_{\lambda}$ and $\sigma_{\lambda}$, since changing this prime
replaces $\tau_{\lambda}$ and $\sigma_{\lambda}$ by their conjugates
$\sigma\tau_{\lambda}\sigma^{-1}$ and $\sigma\sigma_{\lambda}\sigma^{-1}$ by
some $\sigma\in G_{K}$. More precisely, by (16) and the cocycle property of
$\kappa_{\psi}$,

$\eta_{g}^{\prime}(\sigma\tau_{\lambda}\sigma^{-1})=\psi^{-1}(\sigma)\eta_{g}^{\prime}(\tau_{\lambda}),\qquad\kappa_{\psi}(\sigma\sigma_{\lambda}\sigma^{-1})=\psi(\sigma)\kappa_{\psi}(\sigma_{\lambda}).$

In particular, the product
$\eta_{g}^{\prime}(\tau_{\lambda})\kappa_{\psi}(\sigma_{\lambda})$ is
independent of the choice of a prime above $\ell$, as it should be. Note that
$\eta_{g}^{\prime}(\tau_{\lambda})$ is a simple root of unity belonging to the
image of $\psi_{g}$, while $\kappa_{\psi}(\sigma_{\lambda})$ represents the
interesting “transcendental” contribution to the fourier coefficient
$a_{\ell}(g_{\alpha}^{\prime})$.

By the description of $\kappa_{\psi}(\sigma_{\lambda})$ arising from local and
global class field theory, we conclude from (25) that

(30)
$a_{\ell}(g_{\alpha}^{\prime})=\eta_{g}^{\prime}(\tau_{\lambda})\frac{\log_{\wp}({u}_{\psi}^{\prime})\log_{\wp}(u_{\psi}(\lambda))-\log_{\wp}(u_{\psi})\log_{\wp}({u}_{\psi}^{\prime}(\lambda))}{{\log_{\wp}(u_{\psi}^{\prime})-\log_{\wp}(u_{\psi})}},$

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 $(v,\tau_{\wp}v)$ for $V_{g}$, where $v$ spans a $G_{K}$-stable subspace
of $V_{g}$ on which $G_{K}$ acts via ${\psi_{g}}$, the matrix for $U_{g}$ is
proportional to one of the form

$U_{g}:\left(\begin{array}[]{cc}0&\log_{\wp}(u_{\psi})\\\
\log_{\wp}(\tau_{\wp}u_{\psi})&0\end{array}\right),$

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

$A:\left(\begin{array}[]{cc}x&-x\\\ y&-y\end{array}\right).$

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

$A_{g}:\frac{1}{\log_{\wp}(u_{\psi})-\log_{\wp}(\tau_{\wp}u_{\psi})}\cdot\left(\begin{array}[]{cc}\log_{\wp}(u_{\psi})&-\log_{\wp}(u_{\psi})\\\
\log_{\wp}(\tau_{\wp}u_{\psi})&-\log_{\wp}(\tau_{\wp}u_{\psi})\end{array}\right),$

and Theorem 2.3 is readily deduced from the general formula for the fourier
coefficients of $g_{\alpha}^{\prime}$ 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
$\varrho_{g}$ is isomorphic to the symmetric group $S_{3}$.

######  Example 2.4.

Let $\chi$ be the quadratic character of conductor $23$ and

$g=q-q^{2}-q^{3}+q^{6}+q^{8}+\cdots\in S_{1}(23,\chi)$

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

$H=\mathbb{Q}(\alpha)\ \ \ \ \mbox{ where }\ \ \ \
\alpha^{6}-6\alpha^{4}+9\alpha^{2}+23=0.$

Write $\mathrm{Gal\,}(H/K)=\langle\sigma\rangle$. The smallest prime which is
inert in $K$ is $p=5$. The deformations $g^{\prime}_{1}$ and $g^{\prime}_{-1}$
were computed to a $5$-adic precision of $5^{40}$ (and $q$-adic precision
$q^{600}$).

Consider the inert prime $\ell=7$ in $K$. Let
$u(7)=(2\alpha^{4}-7\alpha^{2}+5)/9$, a root of $x^{3}-x^{2}+2x-7=0$. Taking
$\omega$ a primitive cube root of unity we have

$\begin{array}[]{rcl}\log_{5}(u_{\psi}(7))&=&\log_{5}(u(7))+\omega\log_{5}(u(7)^{\sigma})+\omega^{2}\log_{5}(u(7)^{\sigma^{2}})\\\
\log_{5}({u}^{\prime}_{\psi}(7))&=&\log_{5}(u(7))+\omega^{2}\log_{5}(u(7)^{\sigma})+\omega\log_{5}(u(7)^{\sigma^{2}}).\end{array}$

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

$\begin{array}[]{rcl}\log_{5}(u_{\psi})&=&\log_{5}(u)+\omega\log_{5}(u^{\sigma})+\omega^{2}\log_{5}(u^{\sigma^{2}})\\\
\log_{5}({u}^{\prime}_{\psi})&=&\log_{5}(u)+\omega^{2}\log_{5}(u^{\sigma})+\omega\log_{5}(u^{\sigma^{2}}).\end{array}$

Now

$a_{7}(g^{\prime}_{1})=-a_{7}(g^{\prime}_{-1})=4083079847610157092272537548\cdot
5\bmod{5^{40}}$

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

$a_{7}(g^{\prime}_{1})=\frac{\log_{5}(u_{\psi}(7))\log_{5}({u}^{\prime}_{\psi})-\log_{5}({u}^{\prime}_{\psi}(7))\log_{5}(u_{\psi})}{\log_{5}(u_{\psi})-\log_{5}({u}^{\prime}_{\psi})}$

as predicted by part (b) of Theorem 2.3.

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

$\begin{array}[]{rcl}\log_{5}(u_{g}(\lambda))&=&\left(\omega\log_{5}(u(\lambda))+\omega^{2}\log_{5}({u}^{\prime}(\lambda))\right)\\\
\log_{5}({u}_{g}^{\prime}(\lambda))&=&\frac{1}{3}\left(\omega^{2}\log_{5}(u(\lambda))+\omega\log_{5}({u}^{\prime}(\lambda))\right).\end{array}$

We have

$a_{13}(g^{\prime}_{1})=a_{13}(g^{\prime}_{-1})=-638894131680830198852008592\cdot
5\bmod{5^{40}}$

and one sees that

$a_{13}(g^{\prime}_{\pm
1})=\frac{\log_{5}({u}^{\prime}(\psi))\log_{5}(u_{g}(\lambda))-\log_{5}(u(\psi))\log_{5}(u_{g}^{\prime}(\lambda))}{\log_{5}({u}_{\psi}^{\prime})-\log_{5}(u_{\psi})}$

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

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

$a_{\ell}(g_{\alpha}^{\prime})=\frac{-\log_{p}u_{\psi}\cdot(\log_{p}(u_{g}(\ell))+\log_{p}(u_{g}^{\prime}(\ell)))}{-2\log_{p}(u_{\psi})}=\frac{1}{2}\cdot(\log_{p}(u_{g}(\ell))+\log_{p}(u_{g}^{\prime}(\ell))).$

It follows from the definition of $u_{g}(\ell)$ that

$a_{\ell}(g_{\alpha}^{\prime})={\rm
trace}(\varrho_{g}(\lambda))\cdot\log_{p}(\ell).$

In particular, we obtain

(31)
$a_{\ell}(g_{\alpha}^{\prime})=\left\\{\begin{array}[]{ll}\log_{p}(\ell)&\mbox{
if }{\psi_{g}}(\lambda)=1,\\\ 0&\mbox{ if }{\psi_{g}}(\lambda)=\pm i,\\\
-\log_{p}(\ell)&\mbox{ if }{\psi_{g}}(\lambda)=-1,\end{array}\right.$

in perfect agreement with the experiments below.

######  Example 2.5.

Let $\chi=\chi_{3}\chi_{13}$ where $\chi_{3}$ and $\chi_{13}$ are the
quadratic characters of conductors $3$ and $13$, respectively. The space
$S_{1}(39,\chi)$ is one dimensional and spanned by the form
$g=q-q^{3}-q^{4}+q^{9}+\cdots$. The representation $\rho_{g}$ 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 $\psi_{g}$ of the Hilbert class field

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

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

First consider the case of $\ell=\lambda\lambda^{\prime}$ split in
$\mathbb{Q}(\sqrt{-39})$. Then one observes to $20$-digits of $7$-adic
precision and all such $\ell<900$ that both $a_{\ell}(g_{1}^{\prime})$ and
$a_{\ell}(g_{-1}^{\prime})$ satisfy (31). Next we consider the case that
$\ell$ is inert in $\mathbb{Q}(\sqrt{-39})$. Here one observes numerically
that the Fourier coefficients are zero when $\ell$ is inert in
$\mathbb{Q}(\sqrt{-3})$. When $\ell$ is split in $\mathbb{Q}(\sqrt{-3})$ the
Fourier coefficients of the two stabilisations are opposite in sign and both
equal to the $p$-adic logarithm of a fundamental $\ell$ unit of norm $1$ in
$\mathbb{Q}(\sqrt{-3})$. (Observe that $p$ is split in $\mathbb{Q}(\sqrt{-3})$
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

${\psi_{g}}:G_{K}{\longrightarrow}L^{\times}$

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

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 $K/\mathbb{Q}$. The prime $p$ then splits completely in $H/K$ and
we fix a prime $\wp$ of $H$ above $p$. This choice determines an embedding
$H{\longrightarrow}H_{\wp}=K_{p}$ and we write $z\mapsto{z^{\prime}}$ for the
conjugation action of $\mathrm{Gal\,}(K_{p}/\mathbb{Q}_{p})$. Let
$u_{K}\in{\mathcal{O}}_{K}^{\times}$ denote the fundamental unit of
${\mathcal{O}}_{K}$ of norm $1$, which we regard as an element of
$K_{p}^{\times}=H_{\wp}^{\times}$ through the above embedding, and let
$u_{K}^{\prime}=u_{K}^{-1}$ 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
$\tau_{\wp}$. Just as in the previous section, relative to this basis the
Galois representation $\varrho_{g}$ takes the form

(32) $\varrho_{g}(\sigma)=\left(\begin{array}[]{cc}\psi_{g}(\sigma)&0\\\
0&\psi_{g}^{\prime}(\sigma)\end{array}\right)\ \mbox{for }\sigma\in
G_{K},\qquad\varrho_{g}(\tau)=\left(\begin{array}[]{cc}0&\eta_{g}(\tau)\\\
\eta_{g}^{\prime}(\tau)&0\end{array}\right)\ \mbox{for }\tau\in
G_{\mathbb{Q}}-G_{K},$

where $\eta_{g}$ and $\eta_{g}^{\prime}$ are functions taking values in the
group of roots of unity in $L^{\times}$. The element $U_{g}\in(H_{\wp}\otimes
W_{g})$ of (5) is thus represented the matrix

$U_{g}:\left(\begin{array}[]{cc}\log_{\wp}(u_{K})&0\\\
0&\log_{\wp}(u_{K}^{\prime})\end{array}\right),$

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

$A_{g}:\frac{1}{2}\left(\begin{array}[]{rr}1&-1\\\ -1&1\end{array}\right).$

It follows that, if $\ell=\lambda\lambda^{\prime}$ is split in $K/\mathbb{Q}$,
we have

(33) $A_{g}(\lambda):\frac{1}{2}\left(\begin{array}[]{cc}\log_{p}(\ell)&0\\\
0&\log_{p}(\ell)\end{array}\right),$

while if $\ell$ is inert in $K/\mathbb{Q}$ and $\lambda$ is a prime of $H$
lying above $\ell$,

(34)
$A_{g}(\lambda):\frac{1}{2}\left(\begin{array}[]{cc}\log_{p}(\ell)&-\log_{\wp}(u_{\psi}(\lambda))\\\
-\log_{\wp}(u_{\psi}^{\prime}(\lambda))&\log_{p}(\ell)\end{array}\right).$

######  Theorem 3.1.

For all rational primes $\ell\nmid Np$,

  1. (a)

If $\ell$ is split in $K/\mathbb{Q}$, then

$a_{\ell}(g_{\alpha}^{\prime})=\frac{1}{2}a_{\ell}(g)\cdot\log_{p}(\ell).$

  2. (b)

If $\ell$ is inert in $K/\mathbb{Q}$, then

$a_{\ell}(g_{\alpha}^{\prime})=-\left(\eta_{g}(\lambda)\log_{\wp}(u_{\psi}^{\prime}(\lambda))+\eta_{g}(\lambda^{\prime})\log_{\wp}(u_{\psi}(\lambda)\right)).$

###### 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 $\infty$ 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 $\chi=\chi_{5}\chi_{29}$ where $\chi_{5}$ and $\chi_{29}$ are quadratic
and quartic characters of conductor $5$ and $29$, respectively. Then
$S_{1}(145,\chi)$ is one dimensional and spanned by the eigenform

$g=q+iq^{4}-iq^{5}+iq^{9}+(-i-1)q^{11}-q^{16}+(-i-1)q^{19}+\cdots.$

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

$H=\mathbb{Q}(\alpha)\mbox{ where
}\alpha^{8}-2\alpha^{7}+4\alpha^{6}-26\alpha^{5}+94\alpha^{4}-212\alpha^{3}+761\alpha^{2}-700\alpha+980=0.$

Write $\mathrm{Gal\,}(H/K)=\langle\psi\rangle$. Take $p=13$, and note that
$\chi(p)=1$ and so $\alpha=i$ and $\beta=-i$. We compute $g^{\prime}_{\pm i}$
to $10$ digits of $13$-adic precision (and $q$-adic precision $q^{28,000}$).

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

$\log_{13}(u(7,\pm i)):=\log_{13}(u(7))\mp
i\log_{13}(u(7)^{\psi})\mp\log_{13}(u(7)^{\psi^{2}})\pm
i\log_{13}(u(7)^{\psi^{3}})$

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

$a_{7}(g^{\prime}_{i})=-\frac{1}{6}\cdot\log_{13}(v)$

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

$a_{11}(g^{\prime}_{i})=-\frac{i+1}{2}\cdot\log_{13}(11)$

exactly as predicted by Theorem 3.1.

##  Part B The irregular setting

Denote by $S_{k}(Np,\chi)$ (resp. by $S_{k}^{(p)}(N,\chi)$) the space of
classical (resp. $p$-adic overconvergent) modular forms of weight $k$, level
$Np$ (resp. tame level $N$) and character $\chi$, with coefficients in
$\mathbb{Q}_{p}$. The Hecke algebra $\mathbb{T}$ of level $Np$ generated over
$\mathbb{Q}$ by the operators $T_{\ell}$ with $\ell\nmid Np$ and $U_{\ell}$
with $\ell\mid Np$ acts naturally on the spaces $S_{k}(Np,\chi)$ and
$S_{k}^{(p)}(N,\chi)$.

As in the introduction, let $g\in S_{1}(N,\chi)$ be a newform and let
$g_{\alpha}\in S_{1}(Np,\chi)$ be a $p$-stabilisation of $g$. The eigenform
$g_{\alpha}$ gives rise to a ring homomorphism
$\varphi_{g_{\alpha}}:\mathbb{T}{\longrightarrow}L$ to the field $L$ generated
by the fourier coefficients of $g_{\alpha}$, satisfying

(35) $\varphi_{g_{\alpha}}(T_{\ell})=a_{\ell}(g_{\alpha})\mbox{ if }\ell\nmid
Np,\qquad\varphi_{g_{\alpha}}(U_{\ell})=\left\\{\begin{array}[]{ll}a_{\ell}(g_{\alpha})&\mbox{
if }\ell\mid N;\\\ \alpha&\mbox{ if }\ell=p.\end{array}\right.$

For any ideal $I$ of a ring $R$ and any $R$-module $M$, denote by $M[I]$ the
$I$-torsion in $M$. Let $I_{g_{\alpha}}\\!\\!\triangleleft\ \mathbb{T}$ be the
kernel of $\varphi_{g_{\alpha}}$, and set

$S_{1}(Np,\chi)[g_{\alpha}]:=S_{1}(Np,\chi)[I_{g_{\alpha}}],\qquad
S_{1}(Np,\chi)[[g_{\alpha}]]:=S_{1}(Np,\chi)[I_{g_{\alpha}}^{2}].$

Our main object of study is the subspace

$S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]:=S_{1}^{(p)}(N,\chi)[I_{g_{\alpha}}^{2}].$

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

$S_{1}(Np,\chi)[[g_{\alpha}]]\hookrightarrow
S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]$

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., $\alpha=\beta$.

The study of $S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]$ in scenario (a) was carried
out in [DLR2] when $\alpha\neq\beta$. The main result of loc.cit. is the
description of a basis $(g_{\alpha},g_{\alpha}^{\flat})$ for
$S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]$ which is canonical up to scaling, and an
expression for the fourier coefficients of the non-classical
$g_{\alpha}^{\flat}$ (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 $\alpha=\beta$.
In that case, the form $g$ admits a unique $p$-stabilisation
$g_{\alpha}=g_{\beta}$. The Hecke operators $T_{\ell}$ for $\ell\nmid Np$ and
$U_{\ell}$ for $\ell\mid N$ act semisimply (i.e., as scalars) on the two-
dimensional vector space

$S_{1}(Np,\chi)[[g_{\alpha}]]=\mathbb{Q}_{p}g_{\alpha}\oplus\mathbb{Q}_{p}g^{\prime},\qquad
g^{\prime}(q):=g(q^{p}),$

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

$U_{p}g_{\alpha}=\alpha g_{\alpha},\qquad U_{p}g^{\prime}=g_{\alpha}+\alpha
g^{\prime}.$

Because

$(a_{1}(g_{\alpha}),a_{p}(g_{\alpha}))=(1,\alpha),\qquad(a_{1}(g^{\prime}),a_{p}(g^{\prime}))=(0,1),$

the classical subspace $S_{1}(Np,\chi)[[g_{\alpha}]]$ has a natural linear
complement in $S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]$, consisting of the
generalised eigenforms $\tilde{g}$ whose $q$-expansions satisfy

(36) $a_{1}(\tilde{g})=a_{p}(\tilde{g})=0.$

A modular form satisfying (36) is said to be normalised, and the space of
normalised generalised eigenforms is denoted
$S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$. 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 $2\times 2$ 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

$\varrho_{g}:G_{\mathbb{Q}}{\longrightarrow}\mathrm{Aut}_{\mathbb{Q}_{p}}(V_{g})\simeq{\bf
GL}_{2}(\mathbb{Q}_{p})$

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 $\varrho_{g}$ can be
embedded in ${\bf GL}_{2}(\mathbb{Q}_{p})$ and not just in ${\bf
GL}_{2}(\bar{\mathbb{Q}}_{p})$).

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

$\sigma\cdot M:=\varrho_{g}(\sigma)\circ
M\circ\varrho_{g}(\sigma)^{-1},\qquad\mbox{ for any }\sigma\in
G_{\mathbb{Q}},\quad M\in{W_{g}}.$

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

$0{\longrightarrow}W_{g}^{\circ}{\longrightarrow}W_{g}{\longrightarrow}\mathbb{Q}_{p}{\longrightarrow}0$

of $G$-modules admits a canonical $G$-equivariant splitting

$p:{W_{g}}{\longrightarrow}W_{g}^{\circ},\qquad p(A):=A-1/2\cdot{\rm Tr}(A).$

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

${W}_{g}^{L}:=\operatorname{Ad}(V_{g}^{L}),\qquad W_{g}^{\circ
L}:=\operatorname{Ad}^{0}(V_{g}^{L})$

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

$\iota:{W}_{g}^{L}\otimes_{L}\mathbb{Q}_{p}{\longrightarrow}{W}_{g},\qquad\iota:W_{g}^{\circ
L}\otimes_{L}\mathbb{Q}_{p}{\longrightarrow}W_{g}^{\circ}.$

The spaces ${W}_{g}$ and $W_{g}^{\circ}$ (as well as ${W}_{g}^{L}$ and
$W_{g}^{\circ L}$) are equipped with the Lie bracket $[\ ,\ ]$ and with a
symetric non-degenerate pairing $\langle\ ,\ \rangle$ defined by the usual
rules

$[A,B]:=AB-BA,\qquad\langle A,B\rangle:={\rm Tr}(AB),$

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

$[\sigma\cdot A,\sigma\cdot B]=\sigma\cdot[A,B],\qquad\langle\sigma\cdot
A,\sigma\cdot B\rangle=\langle A,B\rangle,\qquad\mbox{ for all }\sigma\in G.$

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

$\det(A,B,C):=\langle[A,B],C\rangle.$

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

$S_{1}(Np,\chi)[g_{\alpha}]\simeq\operatorname{Hom}(\mathbb{T}/I_{g_{\alpha}},\mathbb{Q}_{p}),\quad
S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]\simeq\operatorname{Hom}(\mathbb{T}/I_{g_{\alpha}}^{2},\mathbb{Q}_{p}),$

and hence the dual of the short exact sequence

$0\rightarrow
I_{g_{\alpha}}/I_{g_{\alpha}}^{2}\rightarrow\mathbb{T}/I_{g_{\alpha}}^{2}\rightarrow\mathbb{T}/I_{g_{\alpha}}\rightarrow
0$

can be identified with

$0{\longrightarrow}S_{1}(Np,\chi)[g_{\alpha}]{\longrightarrow}S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]{\longrightarrow}S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}{\longrightarrow}0.$

In particular, one has the isomorphism

(37)
$S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}\simeq\operatorname{Hom}(I_{g_{\alpha}}/I_{g_{\alpha}}^{2},\mathbb{Q}_{p}).$

Let $\mathbb{Q}_{p}[\varepsilon]=\mathbb{Q}_{p}[x]/(x^{2})$ denote the ring of
dual numbers. Given $g^{\flat}\in S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$, the
modular form ${\tilde{g}}:=g_{\alpha}+\varepsilon\cdot g^{\flat}$ is an
eigenform for $\mathbb{T}$ with coefficients in $\mathbb{Q}_{p}[\varepsilon]$.
Its associated Galois representation

$\varrho_{\tilde{g}}:G_{\mathbb{Q}}{\longrightarrow}{\bf
GL}_{2}(\mathbb{Q}_{p}[\varepsilon])$

satisfies

  1. (i)

$\varrho_{\tilde{g}}=\varrho_{g}\pmod{\varepsilon}$ and
$\det(\varrho_{\tilde{g}})=\chi$,

  2. (ii)

for every prime number $\ell\nmid Np$, the trace of an arithmetic Frobenius
$\tau_{\ell}$ at $\ell$ is

(38)
$\mathrm{Tr}(\varrho_{\tilde{g}}(\tau_{\ell}))=a_{\ell}(g_{\alpha})+\varepsilon\cdot
a_{\ell}(g^{\flat}).$

######  Conjecture 4.1.

Assume that $g$ is irregular at $p$. Then the assignment
$g^{\flat}\mapsto\varrho_{\tilde{g}}$ gives rise to a canonical isomorphism
between $S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$ and the space
$\mathrm{Def}^{0}(\varrho_{g})$ of isomorphism classes of deformations of
$\varrho_{g}$ 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
$S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$ is two-dimensional over
$\mathbb{Q}_{p}$.

###### Proof.

Since any $\tilde{\varrho}\in\mathrm{Def}^{0}(\varrho_{g})$ has constant
determinant, it may be written as

(39) $\tilde{\varrho}=(1+\varepsilon\cdot c)\cdot\varrho_{g}\quad\mbox{ for
some }\quad
c=c(\tilde{\varrho}):G_{\mathbb{Q}}{\longrightarrow}W_{g}^{\circ}.$

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

$\mathrm{Def}^{0}(\varrho_{g}){\longrightarrow}H^{1}(\mathbb{Q},W_{g}^{\circ}).$

Under Conjecture 4.1, this yields an isomorphism

(40)
$S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}\stackrel{{\scriptstyle\sim}}{{{\longrightarrow}}}H^{1}(\mathbb{Q},W_{g}^{\circ}),\quad
g^{\flat}\mapsto c_{g^{\flat}}.$

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

(41) $\displaystyle H^{1}(\mathbb{Q},W_{g}^{\circ})$
$\displaystyle\stackrel{{\scriptstyle\mathrm{res}_{H}}}{{{\longrightarrow}}}$
$\displaystyle\hom(G_{H},W_{g}^{\circ})^{G}$ $\displaystyle=$
$\displaystyle\hom_{G}\left(\frac{({\mathcal{O}}_{H}\otimes\mathbb{Z}_{p})^{\times}}{{\mathcal{O}}_{H}^{\times}\otimes\mathbb{Z}_{p}},W_{g}^{\circ}\right)$
$\displaystyle=$
$\displaystyle\hom_{G}\left(\frac{H_{p}}{U},W_{g}^{\circ}\right)$
$\displaystyle=$
$\displaystyle\ker\left(\hom_{G}(H_{p},W_{g}^{\circ})\stackrel{{\scriptstyle\mathrm{res}_{U}}}{{{\longrightarrow}}}\hom_{G}(U,W_{g}^{\circ})\right),$

where $U$ denotes the natural image of
${\mathcal{O}}_{H}^{\times}\otimes\mathbb{Z}_{p}$ in
$H_{p}:=H\otimes\mathbb{Q}_{p}$ under the $p$-adic logarithm map

$\log_{p}:H_{p}^{\times}{\longrightarrow}H_{p}.$

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

$H_{p}\simeq\operatorname{Ind}_{1}^{G}\mathbb{Q}_{p},$

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

$U\simeq\operatorname{Ind}_{G_{\infty}}^{G}\mathbb{Q}_{p}.$

Complex conjugation acts on $W_{g}^{\circ}$ with eigenvalues $1$, $-1$
and$-1$, and hence by Frobenius reciprocity,

(42)
$\dim_{\mathbb{Q}_{p}}\hom_{G}(H_{p},W_{g}^{\circ})=3,\qquad\dim_{\mathbb{Q}_{p}}\hom_{G}(U,W_{g}^{\circ})=1.$

It follows from (41) that $H^{1}(\mathbb{Q},W_{g}^{\circ})$ is two-dimensional
over $\mathbb{Q}_{p}$. Proposition 4.2 follows. ∎

For any $\ell\nmid Np$, the $\ell$-th fourier coefficient of $g^{\flat}$ is
given in terms of the associated cocycle $c_{g^{\flat}}$ by the rule

(43) $a_{\ell}(g^{\flat})={\rm
Tr}(c_{g^{\flat}}(\sigma_{\lambda})\varrho_{g}(\sigma_{\lambda}))$

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

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 $S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$ 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 $W_{g}^{\circ}$ is reducible, i.e.,

  1. (i)

the CM case where $V_{g}$ is induced from a character of an imaginary
quadratic field;

  2. (ii)

the RM case where $V_{g}$ is induced from a character of a real quadratic
field.

###  5\. The general case

The Galois representation $W_{g}^{\circ}$ is irreducible if and only if
$G:=\mathrm{Gal\,}(H/\mathbb{Q})$ is isomorphic to $A_{4}$, $S_{4}$, or
$A_{5}$. Otherwise, the representation $\varrho_{g}$ 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 $\mathbb{Q}_{p}$ after
fixing an embedding $H\hookrightarrow\mathbb{Q}_{p}$ once and for all. This
amounts to choosing a prime $\wp$ of $H$ above $p$. Let
$\log_{\wp}:H_{p}^{\times}{\longrightarrow}\mathbb{Q}_{p}$ denote the
associated ${\wp}$-adic logarithm map, which factors through $\log_{p}$.

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

$\dim_{L}({\mathcal{O}}_{H}^{\times}\otimes W_{g}^{\circ L})^{G}=1.$

In particular, for all ${\mathfrak{u}}\in{\mathcal{O}}_{H}^{\times}$ and all
$w\in W_{g}^{\circ L}$, the element

(44) $\xi({\mathfrak{u}},w):=\frac{1}{\\#G}\times\sum_{\sigma\in
G}(\sigma{\mathfrak{u}})\otimes(\sigma\cdot
w)\in({\mathcal{O}}_{H}^{\times}\otimes W_{g}^{\circ L})^{G}$

only depends on the choices of ${\mathfrak{u}}$ and $w$ up to scaling by a
(possibly zero) factor in $L$. As ${\mathfrak{u}}$ varies over
${\mathcal{O}}_{H}^{\times}$ and $w$ over $W_{g}^{\circ L}$, the elements

(45) $\xi_{\wp}({\mathfrak{u}},w):=(\log_{\wp}\otimes{\rm
id})\xi({\mathfrak{u}},w)=\frac{1}{\\#G}\times\sum_{\sigma\in
G}\log_{\wp}(\sigma{\mathfrak{u}})\cdot(\sigma\cdot w)\in W_{g}^{\circ}$

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

(46)
$w(1)=\log_{\wp}({\mathfrak{u}}_{1})e_{1}+\log_{\wp}({\mathfrak{u}}_{2})e_{2}+\log_{\wp}({\mathfrak{u}}_{3})e_{3},$

for appropriate
${\mathfrak{u}}_{i}\in({\mathcal{O}}_{H}^{\times})\otimes_{\mathbb{Z}}L$.

Let $\ell\nmid Np$ be a rational prime. For any prime $\lambda$ of $H$ above
$\ell$, let $\tilde{\mathfrak{u}}_{\lambda}$ be a generator of the principal
ideal $\lambda^{h}$, where $h$ is the class number of $H$, and set

${\mathfrak{u}}_{\lambda}:={\tilde{\mathfrak{u}}}_{\lambda}\otimes
h^{-1}\in({\mathcal{O}}_{H}[1/\ell]^{\times})\otimes_{\mathbb{Z}}L.$

Let

(47) $\tilde{w}_{\lambda}:=\varrho_{g}(\sigma_{\lambda})\in{W_{g}^{L}},\qquad
w_{\lambda}:=p({\tilde{w}_{\lambda}})\in W_{g}^{\circ L}$

be the endomorphisms of $V_{g}$ arising from the image of $\sigma_{\lambda}$
under $\varrho_{g}$. The element ${\mathfrak{u}}_{\lambda}$ is well-defined up
to multiplication by elements of ${\mathcal{O}}_{H}^{\times}$, and hence the
elements

(48) $\displaystyle\xi({\mathfrak{u}}_{\lambda},w_{\lambda})$
$\displaystyle:=$ $\displaystyle\frac{1}{\\#G}\times\sum_{\sigma\in
G}(\sigma{\mathfrak{u}}_{\lambda})\otimes(\sigma\cdot
w_{\lambda})\in({\mathcal{O}}_{H}[1/\ell]^{\times}\otimes W_{g}^{\circ
L})^{G},$ $\displaystyle
w(\ell)=\xi_{\wp}({\mathfrak{u}}_{\lambda},w_{\lambda})$ $\displaystyle:=$
$\displaystyle\frac{1}{\\#G}\times\sum_{\sigma\in
G}\log_{\wp}(\sigma{\mathfrak{u}}_{\lambda})\cdot(\sigma\cdot w_{\lambda})\in
W_{g}^{\circ}$

are defined up to translation by elements of the one-dimensional $L$-vector
spaces $({\mathcal{O}}_{H}^{\times}\otimes W_{g}^{\circ L})^{G}$ and $L\cdot
w(1)$ respectively. Furthermore, the image of $w(\ell)$ in the quotient
$W_{g}^{\circ}/(L\cdot w(1))$ does not depend on the choice of the prime
$\lambda$ of $H$ above $\ell$ that was made to define it. The Lie bracket

${\mathfrak{W}}(\ell):=[w(1),w(\ell)]\in W_{g}^{\circ}$

is thus independent of the choices that were made in defining $w(\ell)$.

######  Remark 5.1.

The coordinates of $w(\ell)$ relative to a basis $(e_{1},e_{2},e_{3})$ for
$W_{g}^{\circ L}$ are $\wp$-adic logarithms of $\ell$-units in $H$, i.e., one
can write

(49)
$w(\ell)=\log_{\wp}({\mathfrak{v}}_{1})e_{1}+\log_{\wp}({\mathfrak{v}}_{2})e_{2}+\log_{\wp}({\mathfrak{v}}_{3})e_{3},$

with ${\mathfrak{v}}_{i}\in({\mathcal{O}}_{H}[1/\ell]^{\times})_{L}$ for
$i=1,2,3$. A direct computation shows that

$\dim_{\mathbb{Q}_{p}}(W_{g}^{\circ})^{\sigma_{\lambda}=1}=\left\\{\begin{array}[]{cl}1&\mbox{
if $g$ is regular at $\ell$};\\\ 3&\mbox{ if $g$ is irregular at
$\ell$}.\end{array}\right.$

It follows that for all regular primes $\ell$,

$\dim_{L}({\mathcal{O}}_{H}[1/\ell]^{\times}\otimes W_{g}^{\circ L})^{G}=2,$

and therefore that the element $\xi({\mathfrak{v}},w)$ attached to any pair
$({\mathfrak{v}},w)\in{\mathcal{O}}[1/\ell]^{\times}\times W_{g}^{\circ L}$ as
in (48) is well-defined up to scaling by $L$ and up to translation by elements
of the one-dimensional space $({\mathcal{O}}_{H}^{\times}\otimes W_{g}^{\circ
L})^{G}$. In particular, the associated vector ${\mathfrak{W}}(\ell)$ lies in
a canonical one-dimensional subspace of $W_{g}^{\circ}$, namely, the
orthogonal complement in $W_{g}^{\circ}$ of

$(\log_{\wp}\otimes{\rm Id})({\mathcal{O}}_{H}[1/\ell]^{\times}\otimes
W_{g}^{\circ})^{G}\subset W_{g}^{\circ}.$

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

$e_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\qquad
e_{2}=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right),\qquad
e_{3}=\left(\begin{array}[]{cc}0&0\\\ 1&0\end{array}\right),$

then

${\mathfrak{W}}(\ell)\ \ =\ \
\det\left(\begin{array}[]{cc}{\mathfrak{u}}_{2}&{\mathfrak{u}}_{3}\\\
{\mathfrak{v}}_{2}&{\mathfrak{v}}_{3}\end{array}\right)\cdot e_{1}\ +\
2\det\left(\begin{array}[]{cc}{\mathfrak{u}}_{1}&{\mathfrak{u}}_{2}\\\
{\mathfrak{v}}_{1}&{\mathfrak{v}}_{2}\end{array}\right)\cdot e_{2}\ -\
2\det\left(\begin{array}[]{cc}{\mathfrak{u}}_{1}&{\mathfrak{u}}_{3}\\\
{\mathfrak{v}}_{1}&{\mathfrak{v}}_{3}\end{array}\right)\cdot e_{3}.$

######  Remark 5.2.

Observe that if the prime $\ell$ is irregular for $g$, the vector
$\tilde{w}_{\lambda}$ is a scalar endomorphism in ${W_{g}}$ and hence
$w_{\lambda}=w(\ell)={\mathfrak{W}}(\ell)=0$.

Our main result is

######  Theorem 5.3.

Assume Conjecture 4.1. For all $w\in W_{g}^{\circ}$, there exists an
overconvergent generalised eigenform $g_{w}^{\flat}\in
S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$ satisfying

$a_{\ell}(g_{w}^{\flat})=\langle
w,{\mathfrak{W}}(\ell)\rangle=\det(w,w(1),w(\ell)),$

for all primes $\ell\nmid Np$. The assignment $w\mapsto g_{w}^{\flat}$ induces
an isomorphism between $W_{g}^{\circ}/U$ and
$S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$.

###### Proof.

The semi-local field
$H_{p}=H\otimes_{\mathbb{Q}}\mathbb{Q}_{p}=\oplus_{\wp|p}\mathbb{Q}_{p}$ is
naturally identified with the set of vectors $h=(h_{\wp})_{\wp|p}$ with
entries $h_{\wp}\in\mathbb{Q}_{p}$, indexed by the primes of $H$ above $p$.
The function which to $w\in W_{g}^{\circ}$ associates the linear
transformation

$\tilde{\varphi}_{w}:H_{p}{\longrightarrow}W_{g}^{\circ},\qquad\tilde{\varphi}_{w}(h)=\frac{1}{\\#G}\times\sum_{\sigma\in
G}(\sigma^{-1}h)_{\wp}\cdot(\sigma\cdot{w})$

identifies $W_{g}^{\circ}$ with $\hom_{G}(H_{p},W_{g}^{\circ})$. The linear
function $\tilde{\varphi}_{w}$ is trivial on
$U:=\log_{p}({\mathcal{O}}_{H}^{\times})\subset H_{p}$ if and only if, for all
${\mathfrak{u}}\in{\mathcal{O}}_{H}^{\times}$ and all $w^{\prime}\in
W_{g}^{\circ}$,

$\langle\tilde{\varphi}_{w}(\log_{p}({\mathfrak{u}})),w^{\prime}\rangle=0.$

But

$\displaystyle\langle\tilde{\varphi}_{w}(\log_{p}({\mathfrak{u}})),w^{\prime}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\\#G}\times\left\langle\sum_{\sigma\in
G}\log_{\wp}(\sigma^{-1}({\mathfrak{u}}))\cdot(\sigma\cdot{w}),\
w^{\prime}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\\#G}\times\sum_{\sigma\in
G}\log_{\wp}(\sigma^{-1}({\mathfrak{u}}))\cdot\langle\sigma\cdot{w},w^{\prime}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\\#G}\times\sum_{\sigma\in
G}\log_{\wp}(\sigma^{-1}({\mathfrak{u}}))\cdot\langle{w},\sigma^{-1}\cdot
w^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\langle{w},\xi_{\wp}({\mathfrak{u}},w^{\prime})\rangle,$

and hence $\tilde{\varphi}_{w}$ is trivial on
$U=\log_{p}({\mathcal{O}}_{H}^{\times})$ if and only if $w$ is orthogonal in
$W_{g}^{\circ}$ to the line spanned by $w(1)$. It follows that the
$G$-equivariant linear function

$\varphi_{w}:=\tilde{\varphi}_{[w,w(1)]}:H_{p}{\longrightarrow}W_{g}^{\circ}$

factors through $H_{p}/U$. The assignment $w\mapsto\varphi_{w}$ identifies
$W_{g}^{\circ}/(L\cdot w(1))$ with $\hom_{G}(H_{p}/U,W_{g}^{\circ})$, and
gives an explicit description of the latter space.

Let $\tilde{g}_{w}=g+\varepsilon g_{w}^{\flat}$ be the eigenform with
coefficients in $\mathbb{Q}_{p}[\varepsilon]$ which is attached to the cocycle
$\varphi_{w}\in\hom_{G}(H_{p}/U,W_{g}^{\circ})=H^{1}(\mathbb{Q},W_{g}^{\circ})$.
Equation (43) with $g^{\flat}=g_{w}^{\flat}$ (and hence
$c_{g^{\flat}}=\varphi_{w}$) combined with (47) shows that the $\ell$-th the
fourier coefficient of $g_{w}^{\flat}$ at a prime $\ell\nmid Np$ is equal to

(50) $a_{\ell}(g_{w}^{\flat})={\rm
Tr}(\varphi_{w}(\sigma_{\lambda})\cdot\varrho_{g}(\sigma_{\lambda}))=\langle\varphi_{w}(\sigma_{\lambda}),\tilde{w}_{\lambda}\rangle=\langle\varphi_{w}(\sigma_{\lambda}),w_{\lambda}\rangle.$

Class field theory for $H$ implies that

$\varphi_{w}(\sigma_{\lambda})=\frac{1}{\\#G}\times\sum_{\sigma\in
G}\log_{\wp}(\sigma^{-1}{\mathfrak{u}}_{\lambda})\cdot\sigma\cdot[w,w(1)].$

Hence

$\displaystyle a_{\ell}(g_{w}^{\flat})$ $\displaystyle=$
$\displaystyle\frac{1}{\\#G}\times\left\langle\sum_{\sigma\in
G}\log_{\wp}(\sigma^{-1}{\mathfrak{u}}_{\lambda})\cdot\sigma\cdot[w,w(1)],\
w_{\lambda}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\\#G}\times\sum_{\sigma\in
G}\log_{\wp}(\sigma^{-1}{\mathfrak{u}}_{\lambda})\cdot\langle\sigma\cdot[w,w(1)],\
w_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\\#G}\times\sum_{\sigma\in
G}\log_{\wp}(\sigma^{-1}{\mathfrak{u}}_{\lambda})\cdot\langle[w,w(1)],\
\sigma^{-1}\cdot w_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\langle[w,w(1)],w(\ell)\rangle=\det(w,w(1),w(\ell))=\langle
w,{\mathfrak{W}}(\ell)\rangle.$

The theorem follows. ∎

If $w$ in a vector in $W_{g}^{\circ L}$, Theorem 5.3 shows that the associated
overconvergent generalised eigenform $g_{w}^{\flat}$ has fourier coefficients
which are $L$-rational linear combinations of determinants of $2\times 2$
matrices whose entries are the ${\wp}$-adic logarithms of algebraic numbers in
$H$. In the CM and RM cases to be discussed below, the representation
$W_{g}^{\circ}$ is reducible and decomposes further into non-trivial
irreducible representations. In that case the choice of an $L$-basis for
$W_{g}^{\circ L}$ which is compatible with this decomposition leads to
canonical elements of $S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$ 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

$V_{g}^{L}=\operatorname{Ind}_{K}^{\mathbb{Q}}\psi_{g},$

where $\psi_{g}:\mathrm{Gal\,}(\bar{K}/K){\longrightarrow}L^{\times}$ is a
finite order character. Let $\psi_{g}^{\prime}$ denote the character deduced
from $\psi_{g}$ by composing it with the involution in
$\mathrm{Gal\,}(K/\mathbb{Q})$. The irreducibility assumption on $V_{g}^{L}$
implies that the characters $\psi_{g}$ and $\psi_{g}^{\prime}$ 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

$V_{g}^{L}={\mathcal{L}}_{\psi_{g}}^{L}\oplus{\mathcal{L}}_{\psi_{g}^{\prime}}^{L},\qquad
V_{g}={\mathcal{L}}_{\psi_{g}}\oplus{\mathcal{L}}_{\psi_{g}^{\prime}}$

on which $G_{K}$ acts via the characters $\psi_{g}$ and $\psi_{g}^{\prime}$
respectively. The representations ${W}_{g}^{L}$ and ${W_{g}}$ also decompose
as direct sums of four $G_{K}$-stable lines

$\displaystyle{W_{g}^{L}}$ $\displaystyle=$
$\displaystyle\left(\hom({\mathcal{L}}_{\psi_{g}}^{L},{\mathcal{L}}_{\psi_{g}}^{L})\oplus\hom({\mathcal{L}}_{\psi_{g}^{\prime}}^{L},{\mathcal{L}}_{\psi_{g}^{\prime}}^{L})\right)\
\oplus\
\left(\hom({\mathcal{L}}_{\psi_{g}^{\prime}}^{L},{\mathcal{L}}_{\psi_{g}}^{L})\oplus\hom({\mathcal{L}}_{\psi_{g}}^{L},{\mathcal{L}}_{\psi_{g}^{\prime}}^{L})\right),$
$\displaystyle{W_{g}}$ $\displaystyle=$
$\displaystyle\left(\hom({\mathcal{L}}_{\psi_{g}},{\mathcal{L}}_{\psi_{g}})\oplus\hom({\mathcal{L}}_{\psi_{g}^{\prime}},{\mathcal{L}}_{\psi_{g}^{\prime}})\right)\
\oplus\
\left(\hom({\mathcal{L}}_{\psi_{g}^{\prime}},{\mathcal{L}}_{\psi_{g}})\oplus\hom({\mathcal{L}}_{\psi_{g}},{\mathcal{L}}_{\psi_{g}^{\prime}})\right).$

The direct summands in parentheses are also stable under $G_{\mathbb{Q}}$ and
are isomorphic to the induced representations
$\operatorname{Ind}_{K}^{\mathbb{Q}}1$ and
$\operatorname{Ind}_{K}^{\mathbb{Q}}\psi$ respectively, where
$\psi:=\psi_{g}/\psi_{g}^{\prime}$, is the ring class character of $K$
associated to $\psi_{g}$. It follows that

$W_{g}^{\circ L}=L(\chi_{K})\oplus Y_{g}^{L},\qquad
W_{g}^{\circ}=\mathbb{Q}_{p}(\chi_{K})\oplus Y_{g},\qquad
Y_{g}^{L}:=\operatorname{Ind}_{K}^{\mathbb{Q}}\psi,\quad
Y_{g}:=Y_{g}^{L}\otimes_{L}\mathbb{Q}_{p}.$

It will be convenient to choose a basis
$(e_{1},e_{2})\in{\mathcal{L}}_{\psi_{g}}^{L}\times{\mathcal{L}}_{\psi_{g}^{\prime}}^{L}$
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
$(i^{\prime},j^{\prime})$-entry is
$\delta{i=i^{\prime}}\delta_{j=j^{\prime}}$. Relative to the identification of
$W_{g}^{\circ L}$ with the space of $2\times 2$ matrices of trace zero with
entries in $L$ via this basis, the representation
$L(\chi_{K})=L\cdot(e_{11}-e_{22})$ is identified with the space of diagonal
matrices of trace $0$, while $Y_{g}^{L}=L\cdot e_{12}\oplus L\cdot e_{21}$ is
identified with the space of off-diagonal matrices in $M_{2}(L)$. Fix an
element $\tau\in G_{\mathbb{Q}}=G_{K}$ once and for all. By eventually re-
scaling $e_{1}$ and $e_{2}$, it can (and shall, henceforth) be assumed that
$\varrho_{g}(\tau)$ is represented by the matrix
$\left(\begin{array}[]{cc}0&t\\\ t&0\end{array}\right)$ in this basis, where
$-t^{2}:=\chi(\tau)$.

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

The field $H$ through which $W_{g}^{\circ}$ factors is the ring class field of
$K$ attached to the character $\psi$. The group
${\mathcal{O}}_{H}^{\times}\otimes\mathbb{Q}$ of units of $H$ is isomorphic to
the regular representation of $Z$ minus the trivial representation, and a
finite index subgroup of ${\mathcal{O}}_{H}^{\times}$ can be constructed
explicity from the elliptic units arising in the theory of complex
multiplication. Let

$e_{\psi}:=\frac{1}{\\#Z}\sum_{\sigma\in Z}\psi^{-1}(\sigma)\sigma$

be the idempotent in the group ring of $Z$ giving rise to the projection onto
the $\psi$-isotypic component for the action of $Z$. Choose a unit
${\mathfrak{u}}\in{\mathcal{O}}_{H}^{\times}$ and let

${\mathfrak{u}}_{\psi}:=e_{\psi}{\mathfrak{u}},\qquad\tau{\mathfrak{u}}_{\psi}=e_{\psi^{\prime}}(\tau{\mathfrak{u}})$

be elements of ${\mathcal{O}}_{H}^{\times}\otimes L$ on which $Z$ acts via the
characters $\psi$ and $\psi^{\prime}=\psi^{-1}$ respectively. With these
choices, we can let

(51) $w(1)=\left(\begin{array}[]{cc}0&\log_{\wp}({\mathfrak{u}}_{\psi})\\\
\log_{\wp}(\tau{\mathfrak{u}}_{\psi})&0\end{array}\right).$

The description of the canonical vectors $w(\ell),{\mathfrak{W}}(\ell)\in
W_{g}^{\circ}$ attached to a rational prime $\ell\nmid Np$ depends in an
essential way on whether $\ell$ is split or inert in $K/\mathbb{Q}$.

If $\ell=\lambda\lambda^{\prime}$ is split in $K$ and $\ell$ is regular for
$g$, i.e., $\psi_{g}(\sigma_{\lambda})\neq\psi_{g}({\lambda^{\prime}})$, then
the natural map

$({\mathcal{O}}_{K}[1/\ell]^{\times}\otimes W_{g}^{\circ
L})^{G}\subset\left(\frac{{\mathcal{O}}_{H}[1/\ell]^{\times}}{{\mathcal{O}}_{H}^{\times}}\otimes
W_{g}^{\circ L}\right)^{G}$

is an isomorphism of $L$-vector spaces.

Let ${\tilde{\mathfrak{u}}}_{\lambda}$ be a generator of $\lambda^{h}$ where
$h$ is the class number of $K$, and set

${\mathfrak{u}}_{\lambda}:={\tilde{\mathfrak{u}}}_{\lambda}\otimes h^{-1}.$

Since

${\tilde{w}}_{\lambda}=\left(\begin{array}[]{cc}\psi_{g}(\lambda)&0\\\
0&\psi_{g}({\lambda^{\prime}})\end{array}\right),\qquad{w}_{\lambda}=\frac{\psi_{g}(\lambda)-\psi_{g}({\lambda^{\prime}})}{2}\times\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$

a direct calculation shows that

$w(\ell)=\log_{\wp}({\mathfrak{u}}_{\lambda}/{\mathfrak{u}}_{\lambda}^{\prime})\times\frac{(\psi_{g}(\lambda)-\psi_{g}({\lambda^{\prime}}))}{2}\times\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$

It follows that

(52)
${\mathfrak{W}}(\ell)=\log_{\wp}({\mathfrak{u}}_{\lambda}/{\mathfrak{u}}_{\lambda}^{\prime})\times(\psi_{g}(\lambda)-\psi_{g}(\lambda^{\prime}))\times\left(\begin{array}[]{cc}0&-\log_{\wp}({\mathfrak{u}}_{\psi})\\\
\log_{\wp}({\mathfrak{u}}_{\psi}^{\prime})&0\end{array}\right).$

If $\ell$ is inert in $K$ then $\ell$ is always regular for $g$ since
$\varrho_{g}(\tau_{\ell})$ has trace $0$ and hence has distinct eigenvalues.
The prime $\ell$ splits completely in $H/K$, and hence the group
$({\mathcal{O}}_{H}[1/\ell]^{\times})\otimes L$ is isomorphic to two copies of
the regular representation of $Z$ minus a trivial representation. The choice
of a prime $\lambda$ of $H_{g}$ above $\ell$ determines a matrix (and not just
a conjugacy class)

${\tilde{w}}_{\lambda}=w_{\lambda}=\varrho(\sigma_{\lambda})=\left(\begin{array}[]{cc}0&b_{\lambda}\\\
c_{\lambda}&0\end{array}\right)$

with entries in $L$. Let ${{\mathfrak{u}}}_{\lambda}$ be an element of
$({\mathcal{O}}_{H}[1/\ell]^{\times}/{\mathcal{O}}_{H}^{\times})\otimes L$
whose prime factorisation is given by

(53) $({\mathfrak{u}}_{\lambda})=b_{\lambda}\lambda+c_{\lambda}(\tau\lambda).$

This $\ell$-unit is only well defined by (53) up to translation by
${\mathcal{O}}_{H}^{\times}\otimes L$, and the defining equation (53) of
course depends crucially on the choice of the prime $\lambda$ above $\ell$.
However, the $\psi$-isotypic projection

(54) ${\mathfrak{u}}_{\psi}(\ell):=e_{\psi}{\mathfrak{u}}_{\lambda}$

is independent of this choice. A direct calculation shows that

$w(\ell)=\frac{1}{2}\times\left(\begin{array}[]{cc}0&\log_{\wp}({\mathfrak{u}}_{\psi}(\ell))\\\
\log_{\wp}(\tau{\mathfrak{u}}_{\psi}(\ell))&0\end{array}\right).$

It follows that

(55) ${\mathfrak{W}}(\ell)=\left(\begin{array}[]{cc}R_{\psi}(\ell)&0\\\
0&-R_{\psi}(\ell)\end{array}\right),$

where

$R_{\psi}(\ell)=\det\left(\begin{array}[]{cc}\log_{\wp}({\mathfrak{u}}_{\psi})&\log_{\wp}(\tau{\mathfrak{u}}_{\psi})\\\
\log_{\wp}({\mathfrak{u}}_{\psi}(\ell))&\log_{\wp}(\tau{\mathfrak{u}}_{\psi}(\ell))\end{array}\right)$

is an $\ell$-unit regulator attached to $\psi$, which is independent of the
choice of prime $\lambda$ of $H$ above $\ell$. The function $\ell\mapsto
R_{\psi}(\ell)$ does depend on the choice of the unit ${\mathfrak{u}}$, but
only up to scaling by $L^{\times}$.

######  Theorem 6.1.

The space $S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$ has a canonical basis
$(g_{1}^{\flat},g_{2}^{\flat})$ which is characterised by the properties:

  1. (i)

The fourier coefficients $a_{\ell}(g_{1}^{\flat})$ are $0$ for all primes
$\ell\nmid Np$ that are inert in $K$. If $\ell=\lambda\lambda^{\prime}$ is
split in $K$, then

$a_{\ell}(g_{1}^{\flat})=(\psi_{g}(\lambda)-\psi_{g}(\lambda^{\prime}))\times\log_{\wp}({\mathfrak{u}}_{\lambda}/{\mathfrak{u}}_{\lambda}^{\prime})$

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

  2. (ii)

The fourier coefficients of $g_{2}^{\flat}$ are $0$ at all the primes
$\ell\nmid Np$ that are split in $K$. If $\ell$ is inert in $K$ , then

$a_{\ell}(g_{s}^{\flat})=R_{\psi}(\ell).$

###### Proof.

This follows directly from the calculation of the matrices
${\mathfrak{W}}(\ell)$ in (52) and (55) in light of Theorem 5.3. ∎

######  Example 6.2.

Let $\chi$ be the quadratic character of conductor $59$. The space
$S(59,\chi)$ is one dimensional and spanned by the theta series

$g=q-q^{3}+q^{4}-q^{5}-q^{7}-q^{12}+q^{15}+q^{16}+2q^{17}-\cdots.$

Here $K=\mathbb{Q}(\sqrt{-59})$ and the ring class field attached to ${\psi}$
is

$H=K(\alpha)\mbox{ where }\alpha^{3}-3\alpha+46\sqrt{-59}=0.$

The inert primes $\ell$ in $K$ are $2,3,13,23,\cdots$ and the unit and first
few $\ell$-units are

$\begin{array}[]{rclrcl}u&=&\frac{1}{612}\left(13\alpha^{2}-7\sqrt{-59}\alpha-26\right),&u_{2}&=&\frac{1}{612}\left(-5\alpha^{2}-13\sqrt{-59}\alpha-194\right)\\\
\\\
u_{11}&=&\frac{1}{306}\left(5\alpha^{2}+13\sqrt{-59}\alpha-112\right),&u_{13}&=&\frac{1}{612}\left(13\alpha^{2}-7\sqrt{-59}\alpha-1250\right)\\\
\\\
u_{23}&=&\frac{1}{204}\left(-\alpha^{2}+11\sqrt{-59}\alpha+138\right).&&&\\\
\end{array}$

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 $g_{\alpha}(q)$ and
$g(q^{p})$ and in addition a complementary space of dimension two. This space
is canonically spanned by two normalised generalised eigenforms

$\tilde{g}_{1}^{\flat}=q^{3}+\cdots+0\cdot
q^{p}+\cdots\quad\mbox{and}\quad\tilde{g}_{2}^{\flat}=q^{2}+0\cdot
q^{3}+\cdots+\cdots+0\cdot q^{p}+\cdots.$

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
$\ell$ inert in $K$, or $\ell$ split in $K$ but irregular, we have
$a_{\ell}(\tilde{g}_{1}^{\flat})=0$; and for $\ell$ split in $K$ we have that

$a_{\ell}(\tilde{g}_{1}^{\flat})=\frac{\log_{p}(u_{\ell})}{\log_{p}(u_{3})}$

where $u_{\ell}$ is a fundamental $\ell$-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 $\ell<1000$. Further, one expects that

$a_{\ell}(\tilde{g}_{2}^{\flat})=\frac{R_{\psi}(\ell)}{R_{\psi}(2)}\mbox{ for
$\ell$ inert in $K$, and }a_{\ell}(\tilde{g}_{2}^{\flat})=0\mbox{ for $\ell$
split in $K$.}$

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

###  7\. RM forms

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

$V_{g}=\operatorname{Ind}_{F}^{\mathbb{Q}}\psi_{g},$

where $\psi_{g}:\mathrm{Gal\,}(\bar{F}/F){\longrightarrow}L^{\times}$ is a
finite order character of mixed signature. Letting $\psi_{g}^{\prime}$ denote
the character deduced from $\psi_{g}$ by composing it with the involution in
$\mathrm{Gal\,}(F/\mathbb{Q})$, the ratio $\psi:=\psi_{g}/\psi_{g}^{\prime}$
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 $\psi$, and set $Z:=\mathrm{Gal\,}(H/F)$ and
$G:=\mathrm{Gal\,}(H/\mathbb{Q})$. Just as in the previous section,

$W_{g}^{\circ}=\chi_{K}\oplus Y_{g},\qquad
Y_{g}:=\operatorname{Ind}_{K}^{\mathbb{Q}}\psi,$

and we can set

$w(1)=\left(\begin{array}[]{cc}\log_{\wp}({\mathfrak{u}}_{F})&0\\\
0&-\log_{\wp}({\mathfrak{u}}_{F})\end{array}\right),$

where ${\mathfrak{u}}_{F}$ is a fundamental unit of $F$.

If $\ell$ is split in $K/\mathbb{Q}$, it is easy to see that the vector
$w(\ell)$ is proportional to $w(1)$, and hence that

(56) ${\mathfrak{W}}(\ell)=0.$

If $\ell$ is inert in $K$, let $U_{g}$ and $U_{g}(\ell)$ denote the subspaces
$({\mathcal{O}}_{H}^{\times}\otimes Y_{g})^{G_{\mathbb{Q}}}$ and
$({\mathcal{O}}_{H}[1/\ell]^{\times}\otimes Y_{g})^{G_{\mathbb{Q}}}$. The the
dimensions of these spaces are $0$ and $1$ respectively. Choose a prime
$\lambda$ of $H$ above $\ell$, and let ${\mathfrak{u}}_{\lambda}$ and
${\mathfrak{u}}_{\psi}(\ell)$ be the elements of
${\mathcal{O}}_{H}[1/\ell]^{\times}$ determined by the relations

$({\mathfrak{u}}_{\lambda})=b_{\lambda}\lambda+c_{\lambda}\tau\lambda,\qquad{\mathfrak{u}}_{\psi}(\ell)=e_{\psi}({\mathfrak{u}}_{\lambda}),\qquad{\mathfrak{u}}_{\psi}^{\prime}(\ell)=\tau{\mathfrak{u}}_{\psi}(\ell),$

where

$\varrho_{g}(\sigma_{\lambda})=\left(\begin{array}[]{cc}0&b_{\lambda}\\\
c_{\lambda}&0\end{array}\right).$

The $\wp$-adic logarithms

$\log_{\wp}({\mathfrak{u}}_{\psi}(\ell)),\qquad\log_{\wp}({\mathfrak{u}}_{\psi}^{\prime}(\ell))$

are well-defined invariants of $\ell$ and $\varrho$ which do not depend on the
choice of a prime $\lambda$ lying above $\ell$, and

$w(\ell)=\frac{1}{2}\times\left(\begin{array}[]{cc}0&\log_{\wp}({\mathfrak{u}}_{\psi}(\ell))\\\
\log_{\wp}({\mathfrak{u}}_{\psi}^{\prime}(\ell))&0\end{array}\right).$

It follows that

(57)
${\mathfrak{W}}(\ell)=\log_{\wp}({\mathfrak{u}}_{F})\times\left(\begin{array}[]{cc}0&\log_{\wp}({\mathfrak{u}}_{\psi}(\ell))\\\
-\log_{\wp}({\mathfrak{u}}_{\psi}^{\prime}(\ell))&0\end{array}\right).$

######  Theorem 7.1.

The space $S_{1}^{(p)}(N,\chi)[[g_{\alpha}]]_{0}$ has a canonical basis
$(g_{1}^{\flat},g_{2}^{\flat})$ which is characterised by the properties:

  1. (i)

The fourier coefficients of $g_{1}^{\flat}$ and $g_{2}^{\flat}$ are $0$ at all
primes $\ell\nmid Np$ that are split in $F$.

  2. (ii)

If $\ell$ is inert in $F$, then

$a_{\ell}(g_{1}^{\flat})=\log_{\wp}({\mathfrak{u}}_{\psi}(\ell)),\qquad
a_{\ell}(g_{2}^{\flat})=\log_{\wp}({\mathfrak{u}}_{\psi}^{\prime}(\ell)).$

###### Proof.

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

######  Example 7.2.

Let $\chi_{8}$ and $\chi_{7}$ denote the quadratic characters of conductors
$8$ and $7$, respectively, and define $\chi:=\chi_{8}\chi_{7}$. Then
$S_{1}(56,\chi)$ is one-dimensional and spanned by the form

$g=q-q^{2}+q^{4}-q^{7}-q^{8}-q^{9}+q^{14}+q^{16}+q^{18}+2q^{23}-\cdots.$

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

$\tilde{g}_{1}^{\flat}=q^{3}+\cdots+0\cdot
q^{p}+\cdots\quad\mbox{and}\quad\tilde{g}_{2}^{\flat}=q^{2}+0\cdot
q^{3}+\cdots+\cdots+0\cdot q^{p}+\cdots.$

Take

$g_{1}^{\flat}:=\frac{1}{2}\cdot\log_{p}(u_{2})\cdot\tilde{g}_{1}^{\flat}\quad\mbox{and}\quad
g_{2}^{\flat}:=\log_{p}(u_{3})\cdot\tilde{g}_{2}^{\flat}.$

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

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