MnLargeSymbols’164 MnLargeSymbols’171 MnLargeSymbols’102 MnLargeSymbols’107

# The values of the Dedekind–Rademacher cocycle
at real multiplication points

Henri Darmon, Alice Pozzi and Jan Vonk  H. D.: Montreal, Canada
<EMAIL_ADDRESS>A. P.: London, UK<EMAIL_ADDRESS>J.V.: Leiden,
Netherlands<EMAIL_ADDRESS>
###### Abstract.

The values of the so-called Dedekind–Rademacher cocycle at certain real
quadratic arguments are shown to be global $p$-units in the narrow Hilbert
class field of the associated real quadratic field, as predicted by the
conjectures of [DD06] and [DV21]. The strategy for proving this result
combines the approach of [DPV21] with one crucial extra ingredient: the study
of infinitesimal deformations of irregular Hilbert Eistenstein series of
weight one in the anti-parallel direction, building on the techniques of
[BDP].

###### 1991 Mathematics Subject Classification:

11G18, 14G35

###### Contents

  1. 1 The Dedekind–Rademacher cocycle
  2. 2 The winding cocycle
  3. 3 Deformations of Hilbert Eisenstein series
  4. 4 Diagonal restrictions and RM values

## Introduction

Let $\mathcal{H}_{p}$ denote Drinfeld’s $p$-adic upper half plane, and let
$\operatorname{\mathcal{M}}^{\times}$ denote the multiplicative group of non-
zero rigid meromorphic functions on $\mathcal{H}_{p}$, equipped with the
translation action of the discrete group
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}[1/p])$ by Möbius transformations. A
rigid meromorphic cocycle on a congruence subgroup
$\Gamma\subset\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}[1/p])$ is a class in
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{M}}^{\times})$. If
$\tau\in\mathcal{H}_{p}$ is a real multiplication, or RM, point, i.e.,
generates a real quadratic extension of $\mathbb{Q}$, the value of $J$ at
$\tau$ is defined to be

(1) $J[\tau]:=J(\gamma_{\tau})(\tau)\in\mathbb{C}_{p}\cup\\{\infty\\},$

where $\gamma_{\tau}\in\Gamma$ is the automorph of $\tau$, a suitably
normalised generator of the stabiliser of $\tau$ in $\Gamma$. The relevance of
the RM values of rigid meromorphic cocycles to explicit class field theory for
real quadratic fields has been explored in [Da01], [DD06] [DV21], and [DV],
where it is conjectured, broadly speaking, that they behave in many key
respects just like the values of classical modular functions at CM points, and
in particular that they belong to, and often generate, narrow ring class
fields of real quadratic fields.

Theorem B below gives some theoretical evidence for this general conjecture in
the simplest case where
$\Gamma=\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}[1/p])$ and $J$ is analytic,
i.e., takes values in the subgroup
$\operatorname{\mathcal{A}}^{\times}\subset\operatorname{\mathcal{M}}^{\times}$
of rigid analytic functions. Strictly speaking, there are no interesting rigid
analytic cocycles: the group
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times})$ is generated, up
to torsion, by the class $J_{{\rm triv}}$ given by

$J_{{\rm triv}}\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)(z)=cz+d,$

whose RM values are units in the associated quadratic order – hence,
algebraic, but not in an interesting way for explicit class field theory.

There is a less trivial class in
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/p^{\mathbb{Z}})$
arising from the classical Dedekind–Rademacher homomorphism $\varphi_{\rm
DR}:\Gamma_{0}(p)\operatorname{\longrightarrow}\mathbb{Z}$ describing the
periods of the weight two Eisenstein series

(2)
$E_{2}^{(p)}(q)=\operatorname{\mathrm{dlog}}\left(\frac{\Delta(q^{p})}{\Delta(q)}\right)=\left(p-1+24\sum_{n=1}^{\infty}\sigma^{(p)}(n)q^{n}\right)\frac{dq}{q},\qquad\mbox{
where }\sigma^{(p)}(n):=\sum_{p\nmid d|n}d,$

and given by

(3) $\varphi_{\rm DR}(\gamma):=\frac{1}{2\pi i}\int_{z_{0}}^{\gamma
z_{0}}\\!\\!2E_{2}^{(p)}(z)dz.$

More precisely, the description of $\Gamma$ as an amalgamated product of two
conjugate copies of $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ intersecting
in $\Gamma_{0}(p)$ leads to an injection

$\mathrm{H}^{1}(\Gamma_{0}(p),\mathbb{Z})\hookrightarrow\mathrm{H}^{2}(\Gamma,\mathbb{Z}).$

Let $\alpha_{{}_{\rm DR}}\in Z^{2}(\Gamma,\mathbb{Z})$ be a two-cocycle whose
cohomology class is the image of $\varphi_{\rm DR}$ under this map. Refining a
construction of [DD06], Theorem A below asserts that the cocycle
$p^{\alpha_{\rm DR}}$ with values in $p^{\mathbb{Z}}$ is trivialised in the
larger group $\operatorname{\mathcal{A}}^{\times}\supset p^{\mathbb{Z}}$:

Theorem A. There is a one-cochain $J_{\rm DR}\in
C^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times})$ satisfying

$\gamma_{1}J_{\rm DR}(\gamma_{2})\div J_{\rm DR}(\gamma_{1}\gamma_{2})\times
J_{\rm DR}(\gamma_{1})=p^{\alpha_{{}_{\rm
DR}}(\gamma_{1},\gamma_{2})},\quad\mbox{ for all
}\gamma_{1},\gamma_{2}\in\Gamma.$

The essential triviality of
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times})$ shows that
$J_{\rm DR}$ is uniquely determined up to coboundaries and powers of the
cocycle $J_{\rm triv}$ above. The proof of Theorem A is given in § 1, and
constructs an explicit cochain $J_{\rm DR}$ which is well defined up to
coboundaries. The natural image of $J_{\rm DR}$ in
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/p^{\mathbb{Z}})$ is
the Dedekind–Rademacher cocycle of the title. The rigid analytic cocycles of
higher level studied in [DD06] are all multiplicative combinations of
$\mathrm{GL}_{2}(\mathbb{Q})$-translates of this basic cocycle. The proof of
Theorem A complements the approach of [DD06], producing a more canonical
object in level $1$ which can be envisaged as an avatar of the Eisenstein
series $E_{2}$ in the setting of rigid meromorphic cocycles. The RM values of
$J_{\rm DR}$ are well defined modulo $p^{\mathbb{Z}}$, and it therefore makes
sense to enquire about their algebraicity, and their factorisation away from
$p$.

An RM point $\tau\in\mathcal{H}_{p}$ is said to be of discriminant  $D$ if it
satisfies an equation of the form $Q(\tau,1)=0$, where
$Q(x,y)=Ax^{2}+Bxy+Cy^{2}$ is a primitive integral binary quadratic form of
discriminant $D$. The set $\mathcal{H}_{p}^{D}$ of $\tau$ of a fixed
discriminant $D$ is non-empty precisely when $p$ is inert or ramified in the
quadratic field $F=\mathbb{Q}(\sqrt{D})$, and is preserved by the action of
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$. The orbit set
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})\backslash\mathcal{H}_{p}^{D}$ is
in natural bijection with the class group $\operatorname{\mathrm{Cl}}(D)$, by
sending the orbit of $\tau$ to the narrow equivalence class
$\mathcal{C}_{\tau}$ of the fractional ideal generated by $\tau$ and $1$ when
$\tau-\tau^{\prime}$ is positive. The reciprocity map

$\mathrm{rec}:\operatorname{\mathrm{Cl}}(D)\operatorname{\longrightarrow}\mathrm{Gal}(H/F)$

of global class field theory identifies $\operatorname{\mathrm{Cl}}(D)$ with
the Galois group of the narrow ring class field of $H$ over $F$ attached to
$D$. If $\mathcal{H}_{p}^{D}$ is non-empty and $p\nmid D$, then the prime $p$
is inert in $F/\mathbb{Q}$ and splits completely in $H/F$. The choice of an
embedding $\bar{\mathbb{Q}}\subset\bar{\mathbb{Q}}_{p}$ hence determines a
prime $\mathfrak{p}$ of $H$ above $p$, which is fixed once and for all. Fix
also a complex embedding $\bar{\mathbb{Q}}\subset\mathbb{C}$ and write
$x\mapsto\bar{x}$ for the action of complex conjugation on $H$ (which is
independent of the choice of embedding). Let

$\mathcal{O}_{H}[1/p]^{\times}_{-}$

be the group of $p$-units of $H$ which are in the minus-eigenspace for the
action of complex conjugation. By the Dirichlet $S$-unit theorem, it is a
$\mathbb{Z}$-module of rank $[H:F]/2$ if $F$ does not possess a unit of
negative norm, and is finite otherwise. In particular, there is a unique
element $u_{\tau}\in(\mathcal{O}_{H}[1/p]^{\times}_{-})\otimes\mathbb{Q}$
satisfying

(4) ${\rm
ord}_{\mathfrak{p}^{\sigma}}(u_{\tau})=-L(F,\mathcal{C}_{\tau}^{\sigma},0),\quad\mbox{
for all }\sigma\in{\rm Gal}(H/F),$

where $L(F,\mathcal{C}_{\tau}^{\sigma},s)$ is the partial zeta function of the
narrow ideal class $\mathcal{C}_{\tau}^{\sigma}$ (cf. [Gr82, Prop. 3.8]). The
$p$-unit $u_{\tau}$ is called the Gross–Stark unit attached to $H/F$ (and the
prime $\mathfrak{p}$). The Brumer–Stark conjecture implies that
$u_{\tau}^{12}$ belongs to $\mathcal{O}_{H}[1/p]^{\times}$ rather than to the
tensor product of this group with $\mathbb{Q}$. The proof by Samit Dasgupta
and Mahesh Kakde of (the prime to $2$ part of) the Brumer–Stark conjecture
[DKa] in this setting shows that $u_{\tau}^{12}$ belongs to
$(\mathcal{O}_{H}[1/p]^{\times})\otimes\mathbb{Z}[1/2]$.

The principal conjecture of [DD06], and its refinement covering the
Dedekind–Rademacher cocycle itself, asserts that $J_{\rm DR}[\tau]$ is equal,
up to a small torsion ambiguity and powers of $p$, to an integer power of the
Gross–Stark unit $u_{\tau}$. The weaker equality

(5) ${\rm Norm}_{\mathbb{Q}_{p^{2}}/\mathbb{Q}_{p}}(J_{\rm DR}[\tau])={\rm
Norm}_{\mathbb{Q}_{p^{2}}/\mathbb{Q}_{p}}(u_{\tau}^{12})\pmod{(\mathbb{Q}_{p}^{\times})_{\rm
tors},p^{\mathbb{Z}}}$

involving the norms to $\mathbb{Q}_{p}^{\times}$ of these invariants was shown
in [DD06] to follow from Gross’s $p$-adic analogue of the Stark conjecture on
$p$-adic abelian $L$-series of totally real fields at $s=0$ – at least, after
replacing $J_{\rm DR}[\tau]$ by the closely allied quantities denoted
$u(\alpha,\tau)$ in [DD06], which depend on the choice of a suitable modular
unit $\alpha\in\mathcal{O}_{Y_{1}(N)}^{\times}$ with auxiliary level
structure. The Gross–Stark conjecture was then proved in [DDP11]. An important
recent work of Samit Dasgupta and Mahesh Kakde [DKb] has significantly refined
the approach of [DDP11] to prove Gross’s tame refinement  of the Gross–Stark
conjecture, for arbitrary totally real fields. Specialising this result to the
case of a real quadratic field leads to the refinement

(6) $J_{\rm DR}[\tau]=u_{\tau}^{12}\pmod{(\mathbb{Q}_{p^{2}}^{\times})_{\rm
tors},\ \ p^{\mathbb{Z}}}$

of (5) in which the norm is removed. The removal of this ambiguity is crucial
for a truly satisfying approach to explicit class field theory for real
quadratic fields.

The main contribution of this paper is an independent and more direct proof of
(6) for fundamental discriminants:

Theorem B. Let $D>0$ be a fundamental discriminant that is prime to $p$. If
$\tau$ is an RM point in $\mathcal{H}_{p}$ of discriminant $D$, then $J_{\rm
DR}[\tau]$ is equal to the Gross–Stark unit $u_{\tau}^{12}$, up to torsion in
$\mathbb{Q}_{p^{2}}$ and powers of $p$, and in particular belongs to
$(\mathcal{O}_{H}[1/p]^{\times})\otimes\mathbb{Z}[1/2]$.

To situate the approach of this paper in the context of previous works, note
that Dasgupta and Kakde tackle Theorem B by studying Mazur–Tate style “tame
refinements" of the techniques of [DDP11], leading to a proof of Gross’s tame
refinement of his $p$-adic Stark conjecture (known as the “tower of fields
conjecture" [Gr88]). They then show that this tame refinement implies Theorem
B. Like [DKb], the present work rests on the careful study of deformations of
Galois representations that was also exploited in [DDP11], but otherwise
differs in its approach to Theorem B by avoiding the recourse to tame
deformations. Its key idea is to package the RM values of $J_{\rm DR}$ as the
coefficients of certain modular generating series. The resulting identities
(cf. Theorem C below) are of interest in their own right and enrich the
tapestry of analogies between RM values of rigid meromorphic cocycles and CM
values of modular functions.

The Dedekind–Rademacher cocycle, taken modulo $\mathbb{C}_{p}^{\times}$ rather
than $p^{\mathbb{Z}}$, is a prototypical instance of a rigid analytic theta-
cocycle: a function
$J:\Gamma\operatorname{\longrightarrow}\operatorname{\mathcal{A}}^{\times}$
which satisfies the one-cocycle relation, but only up to multiplicative
scalars. The proof of Theorem B rests on the study of another theta-cocycle,
the so-called winding cocycle

(7)
$J_{w}\in\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times}),$

whose key properties are recalled in § 2. The notion of RM value can be
extended to theta cocycles by noting that, if the RM point $\tau$ has
discriminant prime to $p$, then its automorph $\gamma_{\tau}$ belongs to
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$. The groups
$\mathrm{H}^{1}(\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),\mathbb{C}_{p}^{\times})$
and
$\mathrm{H}^{2}(\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),\mathbb{C}_{p}^{\times})$
are finite of order dividing $12$, which implies that the restriction of
$J^{12}$ to $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ admits an essentially
unique lift
$\tilde{J}\in\mathrm{H}^{1}(\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),\operatorname{\mathcal{A}}^{\times})$,
and the value $J[\tau]$ can then be defined as in (1), with $J$ replaced by
$\tilde{J}^{1/12}$ on the right hand side. Although there is some torsion
ambiguity in the resulting RM values, the $p$-adic logarithms of these RM
values are well-defined.

The explicit nature of $J_{w}$ can be parlayed into a proof of the following
result:

Theorem C. Let $\tau$ be as in Theorem B. There is a classical modular form
$G_{\tau}$ of weight two on $\Gamma_{0}(p)$ with $p$-adic Fourier
coefficients, whose $q$-expansion is given by

$G_{\tau}(q)=\log(u_{\tau})+\sum_{n=1}^{\infty}\log((T_{n}J_{w})[\tau])q^{n},$

where $\log\ :\
\mathcal{O}_{\mathbb{C}_{p}}^{\times}\operatorname{\longrightarrow}\mathbb{C}_{p}$
is the $p$-adic logarithm. The modular form $G_{\tau}$ is non-trivial if and
only if $\mathbb{Q}(\sqrt{D})$ does not admit a unit of norm $-1$.

The modular generating series of Theorem C is constructed from the diagonal
restriction of a nearly ordinary deformation of a weight one Hilbert
Eisenstein series for $\operatorname{\mathrm{SL}}_{2}(\mathcal{O}_{F})$ in the
anti-parallel direction. The logarithm of the global $p$-unit $u_{\tau}$
enters into the proof as the eigenvalue of the Frobenius at $p$ on a quotient
of the associated $p$-adic Galois representation, via a calculation which
exploits the reciprocity law of global class field theory, thereby leveraging
class field theory for $H$ into explicit class field theory for $F$. An
essential ingredient in the proof of Theorem C is the study of $p$-adic
deformations of irregular Hilbert Eisenstein series of weight one, which is
explained in § 3 and forms the technical core of this article. This approach
is inspired by the study of the local geometry of the modular eigenvariety in
the neighbourhood of irregular Eisenstein points of weight one carried out in
[BDP], and its extension to the Hilbert setting in [BDS].

Derivatives of $p$-adic families of (classical, or Hilbert) modular forms can
be viewed as $p$-adic counterparts of incoherent Eisenstein series in the
sense of Kudla, and provide a protoypical instance of what might be envisaged
as $p$-adic mock modular forms. Deformations of weight one Hilbert modular
Eisenstein series in the parallel weight direction and their diagonal
restrictions are studied in [DPV21], where they are related to the norms to
$\mathbb{Q}_{p}$ of $J_{\rm DR}[\tau]$. Because of the loss of information
inherent in taking the norm, Theorem C represents a significant strengthening
of the main theorem of [DPV21], just as Theorem B strengthens the equality (5)
resulting from the proof of Gross–Stark conjecture in the setting of odd ring
class characters of real quadratic fields.

In § 1 the Dedekind–Rademacher cocycle is constructed, thereby proving Theorem
A. The definition and main properties of the winding cocycle appear in § 2,
where Theorem B is reduced to Theorem C. The modular generating series
$G_{\tau}$ of Theorem C is constructed in § 3–4. The pivotal § 3 studies
infinitesimal $p$-adic deformations of weight one Hilbert Eisenstein series
and their Fourier expansions. Finally, through a calculation carried out in §
4, the form $G_{\tau}$ is obtained from the ordinary projection of the
diagonal restriction of this infinitesimal deformation.

##  1\. The Dedekind–Rademacher cocycle

This section constructs a one-cochain satisfying Theorem A, which is well-
defined up to coboundaries and whose image in
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/p^{\mathbb{Z}})$ is
the Dedekind–Rademacher cocycle $J_{\rm DR}$ of the introduction.

###  1.1. Siegel units

Let $\mathcal{O}_{\mathcal{H}}^{\times}$ denote the multiplicative group of
nowhere vanishing holomorphic functions on the Poincaré upper half-plane,
endowed with the right action of $\operatorname{\mathrm{SL}}_{2}(\mathbb{R})$
given by

$h|\gamma(z)=h(\gamma z),$

where $\gamma z$ denotes the usual action of $\gamma$ by Möbius
transformations.

The construction of $J_{\rm DR}$ rests on the Siegel units
${{}_{c}}g_{\alpha,\beta}\in\mathcal{O}_{\mathcal{H}}^{\times}$ indexed by
pairs $(\alpha,\beta)\in(\mathbb{Q}/\mathbb{Z})^{2}-\\{(0,0)\\}$ of order
$N>1$, depending on an auxiliary integer $c$ which is relatively prime to
$6N$. They satisfy the transformation properties

(8) ${{}_{c}}g_{v\gamma}={{}_{c}}g_{v}|\gamma\qquad\mbox{ for all
}v=(\alpha,\beta)\in(\mathbb{Q}/\mathbb{Z})^{2},\quad\gamma\in\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}).$

(Cf. [Ka04, Lemma 1.7(1)].) In particular, ${{}_{c}}g_{\alpha,\beta}$ is a
unit on the open modular curve attached to the congruence subgroup of
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ that fixes $(\alpha,\beta)$, and
hence belongs to $\mathcal{O}^{\times}(Y_{0}(N))$. The Siegel units also
satisfy the distribution relations:

(9)
$\prod_{m\alpha^{\prime}=\alpha}{{}_{c}}g_{\alpha^{\prime},\beta}(z)={{}_{c}}g_{\alpha,\beta}(z/m),\qquad\prod_{m\beta^{\prime}=\beta}{{}_{c}}g_{\alpha,\beta^{\prime}}(z)={{}_{c}}g_{\alpha,\beta}(mz),$

which together imply that

(10)
$\prod_{m(\alpha^{\prime},\beta^{\prime})=(\alpha,\beta)}{{}_{c}}g_{\alpha^{\prime},\beta^{\prime}}(z)={{}_{c}}g_{\alpha,\beta}(z).$

(Cf. [Ka04, Lemma 1.7(2)] or [LLZ14, Prop. 2.2.1 and 2.2.2].)

The unit ${}_{c}g_{\alpha,\beta}$ is equal to $g_{\alpha,\beta}^{c^{2}}\cdot
g_{c\alpha,c\beta}^{-1}$, where the $q$-expansion of
$g_{\alpha,\beta}\in\mathcal{O}^{\times}(Y(N))\otimes\mathbb{Q}$ is given by

(11) $g_{\alpha,\beta}(q)=-q^{w}\prod_{n\geq 0}(1-q^{n+\alpha}e^{2\pi
i\beta})\prod_{n>0}(1-q^{n-\alpha}e^{-2\pi i\beta}),$

where $w=1/12-\alpha/2+(1/2)\alpha/N$, with $0\leq\alpha<1$. (Cf. [Ka04,
§1.9].)

Fix a rational prime $p$, and assume that $(\alpha,\beta)$ is of $p$-power
order in $(\mathbb{Q}/\mathbb{Z})^{2}$. To lighten notations, it will be
assumed below that $p\neq 5$, and the choice $c=5$ will be fixed. (The
constructions are readily adapted to the case $p=5$ by changing the value of
$c$.)

###  1.2. The Siegel distribution

Let $\operatorname{\mathbb{X}}_{0}:=(\mathbb{Z}_{p}^{2})^{\prime}$ be the set
of vectors $(a,b)\in\mathbb{Z}_{p}^{2}$ that are primitive, i.e., satisfy
$\gcd(a,b)=1$, and let

(12)
$\operatorname{\mathbb{X}}:=(\mathbb{Q}_{p}^{2}-\\{0,0\\})=\bigcup_{j=-\infty}^{\infty}p^{j}\operatorname{\mathbb{X}}_{0}.$

Let $A$ be an $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$-module, and let
$\operatorname{\mathrm{LC}}(\operatorname{\mathbb{X}}_{0},\mathbb{Z})$ be the
space of locally constant $\mathbb{Z}$-valued functions on
$\operatorname{\mathbb{X}}_{0}$. An $A$-valued distribution on
$\operatorname{\mathbb{X}}_{0}$ is a homomorphism from
$\operatorname{\mathrm{LC}}(\operatorname{\mathbb{X}}_{0},\mathbb{Z})$ to $A$.
Because $\operatorname{\mathbb{X}}_{0}$ is compact, a distribution $\mu$ is
determined by its values $\mu(U)$ on the characteristic functions of compact
open subsets $U\subset\operatorname{\mathbb{X}}_{0}$. Let
$\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}}_{0},A)$ denote the module
of $A$-valued distributions. It is endowed with the (right)
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$-action defined by

(13) $(\mu|\gamma)(U)=\mu(U\gamma^{-1})|\gamma,\qquad\mbox{ for
}\gamma\in\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),\ \
U\subset\operatorname{\mathbb{X}}_{0}.$

A distribution on $\operatorname{\mathbb{X}}$ is said to be $p$-invariant if
it is invariant under multiplication by $p$, i.e.,

(14) $\mu(p^{j}U)=\mu(U)\mbox{ for all $j\in\mathbb{Z}$ and all compact open
}U\subset\operatorname{\mathbb{X}}.$

Denote by $\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},A)$ the module
of $p$-invariant distributions on $\operatorname{\mathbb{X}}$. Because
$\operatorname{\mathbb{X}}_{0}$ is a fundamental domain for the action of $p$
on $\operatorname{\mathbb{X}}$ (cf. (12)), every distribution on
$\operatorname{\mathbb{X}}_{0}$ extends uniquely to a $p$-invariant
distribution, yielding an isomorphism

(15)
$\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}}_{0},A)\stackrel{{\scriptstyle\simeq}}{{\operatorname{\longrightarrow}}}\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},A).$

The target space is equipped with a natural action of the larger group
$\Gamma$ when $A$ is a $\Gamma$-module, defined by (13) with
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ replaced by $\Gamma$. For all
$\mu\in\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},A)$ and for all
locally constant, compactly supported $\mathbb{Z}$-valued functions $f$ on
$\operatorname{\mathbb{X}}$, the $\Gamma$-action is determined by

$\int_{\operatorname{\mathbb{X}}}f(x,y)d(\mu|\gamma)(x,y)=\int_{\operatorname{\mathbb{X}}}f((x,y)\gamma)d\mu(x,y).$

As was implicitly observed in the work of Kubert and Lang, the collection of
Siegel units of $p$-power level are conveniently packaged into a distribution
on $\operatorname{\mathbb{X}}_{0}$, by setting

$\mu_{\rm
Siegel}\left((a,b)+p^{n}(\mathbb{Z}_{p}^{2})\right):=_{c}g_{\frac{a}{p^{n}},\frac{b}{p^{n}}},\quad\mbox{
for all }(a,b)\in(\mathbb{Z}^{2})^{\prime}.$

Since every compact open subset of $\operatorname{\mathbb{X}}_{0}$ is a union
of sets of the form $(a,b)+p^{n}(\mathbb{Z}_{p}^{2})$, the above rule
determines $\mu_{\rm Siegel}$ on all compact open subsets of
$\operatorname{\mathbb{X}}_{0}$. The fact that it is well-defined follows from
the distribution relation (10) with $m=p$.

View $\mu_{\rm Siegel}$ as an element of
$\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathcal{O}_{\mathcal{H}}^{\times})$
via (15). A key feature of $\mu_{\rm Siegel}$ is its invariance under
$\Gamma=\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}[1/p])$, and even under the
full group $\mathrm{GL}_{2}^{+}(\mathbb{Z}[1/p])$ of invertible matrices with
coefficients in $\mathbb{Z}[1/p]$ and positive determinant.

######  Theorem 1.1.

The distribution $\mu_{\rm Siegel}$ satisfies

(16) $\mu_{\rm Siegel}(U\gamma)=\mu_{\rm Siegel}(U)|\gamma,$

for all compact open subsets $U\subset\operatorname{\mathbb{X}}$ and all
$\gamma\in\mathrm{GL}_{2}^{+}(\mathbb{Z}[1/p])$.

  * Proof.

Let $(\alpha,\beta)=(\frac{a}{p^{n}},\frac{b}{p^{n}})$ be an element of order
$p^{n}$ in $(\mathbb{Q}/\mathbb{Z})^{2}$. Since the sets
$U_{\alpha,\beta}=(a,b)+p^{n}\mathbb{Z}_{p}^{2}$ and their translates under
multiplication by $p$ form a basis for the topology on
$\operatorname{\mathbb{X}}$, it suffices to prove the theorem for the sets of
this form. The equivariance (16) for
$\gamma\in\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ follows directly from
(8). Since $\mathrm{GL}_{2}^{+}(\mathbb{Z}[1/p])$ is generated by
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ and the matrix
$T:=\bigl{(}\begin{smallmatrix}p&0\\\ 0&1\end{smallmatrix}\bigr{)}$, one is
reduced to showing the relation

$\mu_{\rm Siegel}(U_{\alpha,\beta}T)=\mu_{\rm Siegel}(U_{\alpha,\beta})|T.$

To see this, note that

$\displaystyle U_{\alpha,\beta}T$ $\displaystyle=$
$\displaystyle(pa+p^{n+1}\mathbb{Z}_{p})\times(b+p^{n}\mathbb{Z}_{p})$
$\displaystyle=$ $\displaystyle\bigcup_{b^{\prime}\equiv
b(p^{n})}(pa+p^{n+1}\mathbb{Z}_{p})\times(b^{\prime}+p^{n+1}\mathbb{Z}_{p})=\bigcup_{p\beta^{\prime}=\beta}U_{\alpha,\beta^{\prime}}.$

It then follows from (9) that

$\mu_{\rm
Siegel}(U_{\alpha,\beta}T)=\prod_{p\beta^{\prime}=\beta}{{}_{c}g_{\alpha,\beta^{\prime}}}(z)=\
_{c}g_{\alpha,\beta}(pz)=\mu_{\rm Siegel}(U_{\alpha,\beta})|T,$

as was to be shown. ∎

The invariance of $\mu_{\rm Siegel}$ under translation by the full
$p$-arithmetic group $\Gamma$, which is hinted at in [LLZ14, Rem. 2.2.3],
combines the $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$-invariance
properties (8) and norm compatibility relations (9), (10) satisfied by the
Siegel units of $p$-power level into a single unified statement.

The following Lemma evaluates the Siegel distribution at some distinguished
open subsets of $\operatorname{\mathbb{X}}$.

######  Lemma 1.2.

The distribution $\mu_{\rm Siegel}$ satisfies

(17) $\begin{array}[]{llll}\mu_{\rm
Siegel}(\operatorname{\mathbb{X}}_{0})&=&1&\pmod{\pm p^{\mathbb{Z}}},\\\
\mu_{\rm
Siegel}(p\mathbb{Z}_{p}\\!\times\\!\mathbb{Z}_{p}^{\times})&=&(\Delta(q^{p})/\Delta(q))^{2}&\pmod{\pm
p^{\mathbb{Z}}}.\end{array}$

  * Proof.

The first assertion follows from the fact that $\operatorname{\mathbb{X}}_{0}$
is stabilised by $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$, and therefore
that its associated Siegel unit is a unit on the open modular curve $Y_{0}(1)$
of level $1$, which contains no non-constant elements. More precisely,
$\mu_{\rm Siegel}(\operatorname{\mathbb{X}}_{0})$ belongs to
$\mathcal{O}^{\times}(Y_{0}(1)_{\mathbb{Z}[1/p]})=\pm p^{\mathbb{Z}}$. (Cf.
[LLZ14, Prop.2.3.2] for instance.) The second assertion follows from the
calculation

$\mu_{\rm
Siegel}(p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times})=\prod_{i=1}^{p-1}\mu_{\rm
Siegel}((0,i)+p\mathbb{Z}_{p}^{2})=\prod_{i=1}^{p-1}{{}_{c}g}_{0,i/p}=\pm
p^{c^{2}-1}(\Delta(q^{p})/\Delta(q))^{(c^{2}-1)/12},$

where the last equality can be read off from the $q$-expansions of the Siegel
units given in (11). The result now follows, since $c=5$. ∎

###  1.3. The Dedekind–Rademacher distributions

The following general Lemmas concerning $p$-invariant distributions will be
useful later.

######  Lemma 1.3.

Let $\mu$ be any element of
$\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},A)$. If $\Lambda$ is any
$\mathbb{Z}_{p}$-lattice in $\mathbb{Q}_{p}^{2}$, and $\Lambda^{\prime}$ is
its set of primitive vectors, then
$\mu(\Lambda^{\prime})=\mu(\operatorname{\mathbb{X}}_{0})$.

  * Proof.

By compactness, there is an integer $N\geq 0$ for which
$p^{-N}\mathbb{Z}_{p}^{2}\subset\Lambda\subset p^{N}\mathbb{Z}_{p}^{2}$, and
hence each $v\in\Lambda^{\prime}$ belongs to a translate
$p^{j}\operatorname{\mathbb{X}}_{0}$ for a unique $j\in[-N,N]$. Hence one may
write

$\Lambda^{\prime}=p^{m_{1}}U_{1}\sqcup\cdots\sqcup p^{m_{t}}U_{t},$

for a suitable decomposition

$\operatorname{\mathbb{X}}_{0}=U_{1}\sqcup\cdots\sqcup U_{t}$

of $\operatorname{\mathbb{X}}_{0}$ as a disjoint union of compact open
subsets. The additivity properties of $\mu$ combined with its $p$-invariance
implies that $\mu(\Lambda^{\prime})=\mu(\operatorname{\mathbb{X}}_{0})$, as
claimed. ∎

######  Lemma 1.4.

The rule which to $A$ associates
$\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},A)$ is an exact
(covariant) functor from the category of $\Gamma$-modules to itself.

  * Proof.

The issue is right exactness. If $\varphi:A\operatorname{\longrightarrow}B$ is
a surjective module homomorphism and
$\mu\in\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},B)$ is a
$B$-valued, $p$-invariant distribution on $\operatorname{\mathbb{X}}$, one can
construct a distribution
$\tilde{\mu}\in\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},A)$ that
maps to it by choosing, for each successsive $n\geq 1$ and for each primitive
vector $v=(\mathbb{Z}/p^{n}\mathbb{Z})^{\prime}$, the value
$\tilde{\mu}(v+p^{n}\mathbb{Z}_{p}^{2})\in A$ satisfying
$\varphi(\tilde{\mu}(v+p^{n}\mathbb{Z}_{p}^{2}))=\mu(v+p^{n}\mathbb{Z}_{p}^{2})$,
taking care at each stage that the additivity relations required of
distributions be satisfied. One obtains in this way an element of
$\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}}_{0},B)$, giving rise to
the desired lift in $\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},B)$
via (15). ∎

Thanks to Lemma 1.4, the exponential sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathcal{H}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{2\pi
iz}}$$\textstyle{{\mathcal{O}_{\mathcal{H}}^{\times}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

induces a short exact sequence

$1\rightarrow\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathbb{Z})\operatorname{\longrightarrow}\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathcal{O}_{\mathcal{H}})\operatorname{\longrightarrow}\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathcal{O}_{\mathcal{H}}^{\times})\rightarrow
1$

of $\Gamma$-modules. Let

$\delta:\mathrm{H}^{0}(\Gamma,\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathcal{O}_{\mathcal{H}}^{\times}))\operatorname{\longrightarrow}\mathrm{H}^{1}(\Gamma,\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathbb{Z}))$

be the connecting homomorphism arising from the resulting long exact sequence
in $\Gamma$-cohomology. The image

$\mu_{\rm DR}:=\delta(\mu_{\rm
Siegel})\in\mathrm{H}^{1}(\Gamma,\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathbb{Z}))$

is a one-cococyle on $\Gamma$, i.e., it satisfies the relation

$\mu_{\rm DR}(\gamma_{1}\gamma_{2})=\mu_{\rm DR}(\gamma_{1})+\mu_{\rm
DR}(\gamma_{2})|\gamma_{1}^{-1}.$

It is obtained by lifting $\mu_{\rm Siegel}$ to an
$\mathcal{O}_{\mathcal{H}}$-valued distribution

$\tilde{\mu}_{\rm Siegel}:=\frac{1}{2\pi i}\log(\mu_{\rm
Siegel})\in\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathcal{O}_{\mathcal{H}}),$

and setting

(18) $\mu_{\rm DR}(\gamma):=\tilde{\mu}_{\rm
Siegel}|{\gamma^{-1}}-\tilde{\mu}_{\rm Siegel}.$

Recall the Dedekind–Rademacher homomorphism $\varphi_{\rm
DR}:\Gamma_{0}(p)\operatorname{\longrightarrow}\mathbb{Z}$ evoked in the
introduction, which encodes the periods of the Eisenstein series
$E_{2}^{(p)}=d\log(\Delta(pz)/\Delta(z))$ of weight two.

######  Lemma 1.5.

The one-cocycle $\mu_{\rm DR}$ satisfies

(19) $\begin{array}[]{llll}\mu_{\rm
DR}(\gamma)(\operatorname{\mathbb{X}}_{0})&=&0&\qquad\mbox{for all }\
\gamma\in\Gamma,\\\ \mu_{\rm
DR}(\gamma)(p\mathbb{Z}_{p}\\!\times\\!\mathbb{Z}_{p}^{\times})&=&\varphi_{\rm
DR}(\gamma)&\qquad\mbox{for all }\ \gamma\in\Gamma_{0}(p).\end{array}$

  * Proof.

Observe that, for all $\gamma\in\Gamma$,

$\mu_{\rm DR}(\gamma)(\operatorname{\mathbb{X}}_{0})=\tilde{\mu}_{\rm
Siegel}|{\gamma^{-1}}(\operatorname{\mathbb{X}}_{0})-\tilde{\mu}_{\rm
Siegel}(\operatorname{\mathbb{X}}_{0})=\tilde{\mu}_{\rm
Siegel}(\operatorname{\mathbb{X}}_{0}\gamma)|\gamma^{-1}-\tilde{\mu}_{\rm
Siegel}(\operatorname{\mathbb{X}}_{0}).$

Lemma (1.3) implies that $\tilde{\mu}_{\rm
Siegel}(\operatorname{\mathbb{X}}_{0}\gamma)=\tilde{\mu}_{\rm
Siegel}(\operatorname{\mathbb{X}}_{0})$, and Lemma 1.2 shows that this common
value is a constant function on $\mathcal{H}$. The first assertion follows. As
for the second, equation (18) implies that

$\mu_{\rm
DR}(\gamma)(p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times})=(\tilde{\mu}_{\rm
Siegel}|{\gamma^{-1}}-\tilde{\mu}_{\rm
Siegel})(p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times}).$

By Lemma 1.2,

$\tilde{\mu}_{\rm
Siegel}(p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times})=\frac{2}{2\pi
i}\log\left(\Delta(pz)/\Delta(z)\right)\pmod{\mathbb{C}}.$

Since $\gamma\in\Gamma_{0}(p)$ preserves the region
$p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times}$, it follows that

$(\tilde{\mu}_{\rm Siegel}|{\gamma^{-1}}-\tilde{\mu}_{\rm
Siegel})(p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times})=\frac{2}{2\pi
i}\int_{z_{0}}^{\gamma^{-1}z_{0}}d\log(\Delta(pz)/\Delta(z))=\varphi_{\rm
DR}(\gamma),$

as was to be shown. ∎

###  1.4. The multiplicative Poisson transform

Because a distribution
$\mu\in\operatorname{\mathbb{D}}(\operatorname{\mathbb{X}},\mathbb{Z})$ is
$\mathbb{Z}$-valued, and hence $p$-adically bounded, it also gives rise to a
measure: one can extend $\mu$ to arbitrary continuous, compactly supported
functions on $\operatorname{\mathbb{X}}$. There is even a multiplicative
refinement of the integral against $\mu$, defined by

$\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\operatorname{\mathbb{X}}}f(x,y)d\mu(x,y):=\lim_{\\{U_{\alpha}\\}}\prod_{\alpha}f(x_{\alpha},y_{\alpha})^{\mu(U_{\alpha})},$

where the limit is taken over finer and finer open covers $\\{U_{\alpha}\\}$
of the support of $f$, and $(x_{\alpha},y_{\alpha})$ is a sample point in
$U_{\alpha}$. Here
$f:\operatorname{\mathbb{X}}\operatorname{\longrightarrow}\mathbb{C}_{p}^{\times}$
is a continuous, compactly supported function on $\operatorname{\mathbb{X}}$
(which means that it takes the value $1$ outside a compact subset of
$\operatorname{\mathbb{X}}$).

Let $\operatorname{\mathbb{D}}_{0}(\operatorname{\mathbb{X}}_{0},\mathbb{Z})$
be the $\mathbb{Z}$-module of distributions on $\operatorname{\mathbb{X}}_{0}$
satisfying

$\mu(\operatorname{\mathbb{X}}_{0})=0.$

The multiplicative Poisson transform of
$\mu\in\operatorname{\mathbb{D}}_{0}(\operatorname{\mathbb{X}}_{0},\mathbb{Z})$
is the rigid analytic function $J(\mu)$ on $\mathcal{H}_{p}$ defined by
setting

$J(\mu)(\tau)=\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\operatorname{\mathbb{X}}_{0}}(x\tau+y)d\mu(x,y).$

This assignment gives rise to an
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$-equivariant map

$J:\operatorname{\mathbb{D}}_{0}(\operatorname{\mathbb{X}}_{0},\mathbb{Z})\operatorname{\longrightarrow}\operatorname{\mathcal{A}}^{\times},$

i.e.,

$J({\mu|\gamma})(\tau)=J(\mu)|\gamma(\tau)=J(\mu)(\gamma\tau),\qquad\mbox{ for
all }\gamma\in\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}).$

Identifying
$\operatorname{\mathbb{D}}_{0}(\operatorname{\mathbb{X}}_{0},\mathbb{Z})$ with
the module
$\operatorname{\mathbb{D}}_{0}(\operatorname{\mathbb{X}},\mathbb{Z})$ of
distributions on $\operatorname{\mathbb{X}}$ satisfying

$\mu(\operatorname{\mathbb{X}}_{0})=0,\qquad\mu(pU)=\mu(U),$

the same rule $J$ (where one continues to integrate over the compact subset
$\operatorname{\mathbb{X}}_{0}\subset\operatorname{\mathbb{X}}$) determines a
$\Gamma$-equivariant map

(20)
$J:\operatorname{\mathbb{D}}_{0}(\operatorname{\mathbb{X}},\mathbb{Z})\operatorname{\longrightarrow}\operatorname{\mathcal{A}}^{\times}/p^{\mathbb{Z}}.$

The reason for this somewhat weaker invariance property is that while
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ preserves the region
$\operatorname{\mathbb{X}}_{0}$ of integration defining $J(\mu)$, the full
$p$-arithmetic group $\Gamma$ does not. Nonetheless, if $\gamma\in\Gamma$, one
still can write (following the reasoning in the proof of 1.3)

$\operatorname{\mathbb{X}}_{0}\gamma=p^{m_{1}}U_{1}\sqcup\cdots\sqcup
p^{m_{t}}U_{t},\qquad\mbox{ with
}\quad\operatorname{\mathbb{X}}_{0}=U_{1}\sqcup\cdots\sqcup U_{t},$

and the integrand $(x-\tau y)$ arising in the definition of $J$ obeys a simple
transformation property under multiplication by $p$. It follows that
$J(\mu|\gamma)=J(\mu)|\gamma\pmod{p^{\mathbb{Z}}}$, for all $\gamma\in\Gamma$.

Let

$J_{\rm DR}:=J(\mu_{\rm
DR})\in\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/p^{\mathbb{Z}})$

be the image of the measure-valued cocycle $\mu_{\rm DR}$ under the
multiplicative Poisson transform of (20). It is represented by the one-cochain
$J_{\rm
DR}:\Gamma\operatorname{\longrightarrow}\operatorname{\mathcal{A}}^{\times}$
(denoted by the same symbol, by an abuse of notation) defined by

$J_{\rm DR}(\gamma)(\tau)=J(\mu_{\rm DR}(\gamma))(\tau),$

which satisfies the cocycle relation modulo $p^{\mathbb{Z}}$,

$J_{\rm DR}(\gamma_{1}\gamma_{2})=J_{\rm DR}(\gamma_{1})\times J_{\rm
DR}(\gamma_{2})|\gamma_{1}^{-1}\pmod{p^{\mathbb{Z}}}.$

Its restriction to $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ also satisfies
the full cocycle relation, with no $p^{\mathbb{Z}}$-ambiguity, because of the
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$-equivariance of $J$.

In order to prove Theorem A of the introduction, it now suffices to calculate
the image of $J_{\rm DR}$ under the sequence of maps

$\eta:\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/p^{\mathbb{Z}})\operatorname{\longrightarrow}\mathrm{H}^{2}(\Gamma,p^{\mathbb{Z}})=\mathrm{H}^{1}(\Gamma_{0}(p),p^{\mathbb{Z}}).$

######  Theorem 1.6.

The image of $J_{\rm DR}$ under $\eta$ is

$\eta(J_{\rm DR})=p^{\varphi_{\rm DR}}.$

  * Proof.

The action of $\Gamma$ on the Bruhat-Tits tree of
${\mathrm{PGL}}_{2}(\mathbb{Q}_{p})$ leads to an expression for $\Gamma$ as an
amalgamated product of the groups

$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),\qquad\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})^{\prime}=\left(\begin{array}[]{cc}p&0\\\
0&1\end{array}\right)^{-1}\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})\left(\begin{array}[]{cc}p&0\\\
0&1\end{array}\right),$

whose intersection is $\Gamma_{0}(p)$. The fact that
$\mathrm{H}^{1}(\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),p^{\mathbb{Z}})=0$
and that
$\mathrm{H}^{2}(\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),p^{\mathbb{Z}})$ is
of order $12$ ensures the existence of unique lifts to
$\operatorname{\mathcal{A}}^{\times}$ of the restrictions of $J_{\rm DR}^{12}$
to $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})$ and
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})^{\prime}$:

$\mathcal{J}_{\rm
DR}\in\mathrm{H}^{1}(\operatorname{\mathrm{SL}}_{2}(\mathbb{Z}),\operatorname{\mathcal{A}}^{\times}),\qquad\mathcal{J}_{\rm
DR}^{\prime}\in\mathrm{H}^{1}(\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})^{\prime},\operatorname{\mathcal{A}}^{\times}).$

One then has, for all $\gamma\in\Gamma_{0}(p)$,

(21) $\eta(J_{\rm DR}^{12})(\gamma)=\mathcal{J}_{\rm
DR}(\gamma)\div\mathcal{J}_{\rm DR}^{\prime}(\gamma).$

Concretely, $\mathcal{J}_{\rm DR}$ and $\mathcal{J}_{\rm DR}^{\prime}$ may be
expressed as multiplicative Poisson transforms of $\mu_{\rm DR}$, by setting

$\mathcal{J}_{\rm DR}(\gamma)(\tau):=\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\operatorname{\mathbb{X}}_{0}}(x\tau+y)^{12}d\mu_{\rm
DR}(\gamma)(x,y),\qquad\mathcal{J}_{\rm
DR}^{\prime}(\gamma)(\tau):=\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\operatorname{\mathbb{X}}_{0}^{\prime}}(x\tau+y)^{12}d\mu_{\rm
DR}(\gamma)(x,y),$

where
$\operatorname{\mathbb{X}}_{0}^{\prime}:=(p\mathbb{Z}_{p}\times\mathbb{Z}_{p})^{\prime}$
is the translate of $\operatorname{\mathbb{X}}_{0}$ under the matrix
$\left(\begin{array}[]{cc}p&0\\\ 0&1\end{array}\right)$, a region whose
stabiliser in $\Gamma$ is the group
$\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})^{\prime}$. Observe that

(22)
$\operatorname{\mathbb{X}}_{0}\cap\operatorname{\mathbb{X}}_{0}^{\prime}=p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times},\qquad\operatorname{\mathbb{X}}_{0}-\operatorname{\mathbb{X}}_{0}^{\prime}=\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p},\qquad\operatorname{\mathbb{X}}_{0}^{\prime}-\operatorname{\mathbb{X}}_{0}=p(\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p}).$

Hence, for all $\gamma\in\Gamma_{0}(p)$,

$\displaystyle\mathcal{J}_{\rm DR}(\gamma)\div\mathcal{J}_{\rm
DR}^{\prime}(\gamma)$ $\displaystyle=$
$\displaystyle\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\operatorname{\mathbb{X}}_{0}}(x\tau+y)^{12}d\mu_{\rm
DR}(\gamma)(x,y)\div\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\operatorname{\mathbb{X}}_{0}^{\prime}}(x\tau+y)^{12}d\mu_{\rm
DR}(\gamma)(x,y)$ $\displaystyle=$
$\displaystyle\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p}}(x\tau+y)^{12}d\mu_{\rm
DR}(\gamma)(x,y)\div\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{p(\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p})}(x\tau+y)^{12}d\mu_{\rm
DR}(\gamma)(x,y)$ $\displaystyle=$
$\displaystyle\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p}}p^{-12}d\mu_{\rm
DR}(\gamma)(x,y),$

where the penultimate equality follows from (22) and the last from the
invariance of $\mu_{\rm DR}(\gamma)$ under multiplication by $p$. Because
$(\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p})$ is the complement of
$(p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times})$ in
$\operatorname{\mathbb{X}}_{0}$, and $\mu_{\rm
DR}(\gamma)(\operatorname{\mathbb{X}}_{0})=0$, this implies that

$\mathcal{J}_{\rm DR}(\gamma)\div\mathcal{J}_{\rm
DR}^{\prime}(\gamma)=\times\\!\\!\\!\\!\\!\\!\\!\hskip
0.97pt\int_{p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times}}p^{12}d\mu_{\rm
DR}(\gamma)(x,y)=p^{12\mu_{\rm
DR}(\gamma)(p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times})}=p^{12\varphi_{\rm
DR}(\gamma)},$

where the last equality follows from Lemma 1.5. Combining this with (21) shows
that $\eta(J_{\rm DR})$ and $p^{\varphi_{\rm DR}}$ agree, since the group they
belong to is torsion-free. This completes the proof of Theorem A. ∎

##  2\. The winding cocycle

The goal of this section is to recall the definition and key properties of the
winding cocycle introduced in [DPV21, § 2.3] and to reduce Theorem B of the
introduction to Theorem C.

###  2.1. The residue map

The group
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times})$
of rigid analytic theta cocycles is finitely generated and closely related to
the space of modular forms of weight two on the Hecke congruence group
$\Gamma_{0}(p)$. More precisely, it is a module over the Hecke algebra
${\mathbb{T}}_{0}(p)$ of Hecke operators acting faithfully on the weight two
modular forms on $\Gamma_{0}(p)$. To see this, let

$U:=\\{z\in\mathbb{P}_{1}(\mathbb{C}_{p})\mbox{ with
}1<|z|<p\\}\subset\mathcal{H}_{p}$

be the standard annulus whose stabiliser in $\Gamma$ is $\Gamma_{0}(p)$. The
logarithmic annular residue map

(23)
$\partial_{U}:\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times}\operatorname{\longrightarrow}\mathbb{Z}_{p},\qquad\partial_{U}(f):={\rm
Res}_{U}(\operatorname{\mathrm{dlog}}f)$

is equivariant for the action of $\Gamma_{0}(p)$, and hence composing it with
the restriction to $\Gamma_{0}(p)$ yields a map on cohomology

(24)
$\partial_{U}:\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}\\!/\mathbb{C}_{p}^{\times})\operatorname{\longrightarrow}\mathrm{H}^{1}(\Gamma_{0}(p),\mathbb{Z}_{p}),$

which is denoted by the same symbol by abuse of notation. This map is
compatible with the action of the Hecke operators, and with the involution
$W_{\infty}$ determined by the matrix $\bigl{(}\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\bigr{)}$, which lies in the normaliser of both $\Gamma$
and $\Gamma_{0}(p)$. Let
$\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}\\!/\mathbb{C}_{p}^{\times})^{\pm}$
denote the plus and minus eigenspaces for this involution in the space of
rigid analytic theta cocycles, and denote by
$\mathrm{H}^{1}(\Gamma_{0}(p),\mathbb{Z}_{p})^{\pm}$ the corresponding
eigenspaces in the cohomology of $\Gamma_{0}(p)$.

While the map in (23) has an infinite rank kernel, it is notable that the
induced map on rigid analytic theta cocycles is essentially an isomorphism:

######  Lemma 2.1.

Up to torsion kernels and cokernels, the map $\partial_{U}$ of (24) is
surjective, and its kernel is generated by the “trivial" theta-cocycle

$J_{\rm
triv}\in\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}),\qquad
J_{\rm triv}\left(\bigl{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\right)(z)=cz+d.$

In particular, the induced map

(25) $\partial_{U}^{-}\ :\
\mathbb{Q}\otimes\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times})^{-}\
\operatorname{\longrightarrow}\ \mathrm{H}^{1}(\Gamma_{0}(p),\mathbb{Q})^{-}$

is an isomorphism.

  * Proof.

The first assertion is a reformulation of [DPV21, Theorem 3.1]. The last
follows from the fact that $J_{\rm triv}$ is fixed by $W_{\infty}$, as can be
checked directly from the definition of $J_{\rm triv}$. ∎

###  2.2. The winding cocycle

In [DPV21, §2.3], the so-called winding cocycle

$J_{w}\in\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times})^{-}$

is introduced. Unlike the Dedekind–Rademacher cocycle, it is not an eigenclass
for the Hecke operators, although it belongs to the $-1$ eigenspace for the
involution $W_{\infty}$. The greater complexity of $J_{w}$ on the spectral
side is offset by a gain in simplicity on the geometric side, evidenced by the
fact that the rigid analytic functions $J_{w}(\gamma)$ admit explicit infinite
product expansions.

Let $Y_{0}(p)=\Gamma_{0}(p)\backslash\mathcal{H}$ be the open modular curve,
and let $X_{0}(p)$ be its standard compactification, obtained by adding the
two cusps $0$ and $\infty$. The intersection pairing on homology (Poincaré
duality) defines isomorphisms

(26)
$\mathrm{H}_{1}(X_{0}(p);\\{0,\infty\\},\mathbb{Q})^{\pm}=\mathrm{H}^{1}(Y_{0}(p),\mathbb{Q})^{\mp}=\mathrm{H}^{1}(\Gamma_{0}(p),\mathbb{Q})^{\mp}.$

Mazur’s winding element

$\varphi_{w}\in\mathrm{H}^{1}(\Gamma_{0}(p),\mathbb{Z})^{-}$

is defined to be the class of the path from $0$ to $\infty$ in the homology of
the modular curve $X_{0}(p)$ relative to the cusps, viewed as an element of
$\mathrm{H}^{1}(\Gamma_{0}(p),\mathbb{Z})$ via (26). By [DPV21, Prop. 3.3] and
its proof, the winding cocycle is characterised by the identity

(27) $\partial_{U}^{-}(J_{w})=2\varphi_{w}.$

###  2.3. Theorem C implies Theorem B

Theorem B of the introduction is reduced to Theorem C by writing the modular
form $G_{\tau}$ of this theorem as a linear combination of eigenforms.

To this end, observe that $\mathrm{H}^{1}(\Gamma_{0}(p),\bar{\mathbb{Q}})^{-}$
is generated as a $\bar{\mathbb{Q}}$-vector space by the Dedekind–Rademacher
morphism $\varphi_{\rm DR}$ of (3) encoding the periods of the weight two
Eisenstein series $E_{2}^{(p)}$ defined in the introduction, and the
homomorphisms $\varphi_{f}^{-}$ attached to the minus modular symbol for $f$,
where $f$ runs through a basis of cuspidal Hecke eigenforms in
$S_{2}(\Gamma_{0}(p))$. A direct calculation of integration pairings in
[DPV21, Lemma 3.4] then yields the spectral decomposition of the winding
element,

(28) $\varphi_{w}\ =\ \frac{1}{p-1}\cdot\varphi_{\rm DR}\ +\
\sum_{f}\lambda_{f}\cdot\varphi_{f}^{-},$

where the coefficient $\lambda_{f}\in\bar{\mathbb{Q}}$ is a suitable non-zero
multiple of $L(f,1)$ whose exact nature is not germane to the proof of Theorem
B. (But see [DPV21, § 3] for more details.)

Now consider the rigid analytic theta-cocycle

$J_{f}^{-}\ \in\
\bar{\mathbb{Q}}\otimes\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times})^{-}$

characterised by $\partial_{U}^{-}(J_{f}^{-})=\varphi_{f}^{-}$. By Lemma 2.1
and (27),

(29) $J_{w}\ =\ \frac{2}{p-1}\cdot J_{\rm DR}\ \ +\ \
\sum_{f}2\lambda_{f}\cdot J_{f}^{-}\qquad\mbox{in }\
\bar{\mathbb{Q}}\otimes\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times})^{-},$

where additive notation has been adopted to describe the operations in this
group in spite of its multiplicative nature. For each $n\geq 1$, applying the
Hecke operator $T_{n}$ to this identity then gives

$\displaystyle T_{n}J_{w}$ $\displaystyle=$ $\displaystyle\frac{2}{p-1}\cdot
T_{n}J_{\rm DR}\ \ +\ \ \sum_{f}2\lambda_{f}\cdot T_{n}J_{f}^{-}$
$\displaystyle=$ $\displaystyle\frac{2}{p-1}\cdot J_{\rm
DR}\cdot\sigma_{1}^{(p)}(n)\ \ +\ \ \sum_{f}2\lambda_{f}\cdot J_{f}^{-}\cdot
a_{n}(f)$

in
$\bar{\mathbb{Q}}\otimes\mathrm{H}^{1}(\Gamma,\operatorname{\mathcal{A}}^{\times}/\mathbb{C}_{p}^{\times})^{-}$.
After evaluating at the RM point $\tau$ and taking $p$-adic logarithms, it
follows that

(31) $\log_{p}(T_{n}J_{w}[\tau])\ \ \ =\ \ \ \frac{2\log_{p}(J_{\rm
DR}[\tau])}{p-1}\cdot\sigma_{1}^{(p)}(n)\ \ +\ \
\sum_{f}2\lambda_{f}\log_{p}(J_{f}^{-}[\tau])\cdot a_{n}(f).$

Substituting this identity into Theorem C of the introduction yields the
spectral expansion

(32) $G_{\tau}(q)\ \ =\ \ \frac{\log J_{\rm DR}[\tau]}{12(p-1)}\cdot
E_{2}^{(p)}(q)\ +\ \sum_{f}\beta_{f}\cdot f(q),$

where $E_{2}^{(p)}$ is the Eisenstein series of (2), and

(33) $\beta_{f}=2\lambda_{f}\log_{p}(J_{f}^{-}[\tau]).$

Comparing the zero-th Fourier coefficient of $G_{\tau}$ in (32) with the one
in Theorem C shows that

$\frac{\log J_{\rm DR}[\tau]}{12}=\log(u_{\tau}),$

thereby reducing Theorem B of the introduction to Theorem C.

The remainder of the paper is devoted to the construction of the modular
generating series required for the proof of Theorem C.

Remark. The coefficients $\beta_{f}$ in (33) are immaterial to the proof of
Theorem B but are of independent interest, insofar as they involve the RM
values of the elliptic rigid analytic theta-cocycles $J_{f}^{-}$: these values
are the formal group logarithms of certain Stark–Heegner points in the modular
Jacobian $J_{0}(p)$. Although poorly understood theoretically, these
Stark–Heegner points are conjectured to be defined over the narrow ring class
field $H_{\tau}$. The approach to the algebraicity of $J_{\rm DR}[\tau]$ based
on deformations of Galois representation does not seem to shed any immediate
light on the algebraicity of these more mysterious invariants.

##  3\. Deformations of Hilbert Eisenstein series

This section studies the derivatives of certain $p$-adic analytic families of
Hilbert modular forms for $F$ parametrised by the weight and specialising to a
certain Hilbert Eisenstein series of parallel weight one.

This Eisenstein series has several notable features. Firstly, it is cuspidal
when viewed as a $p$-adic modular form, and admits cuspidal $p$-adic
deformations. Secondly, it vanishes upon diagonal restriction. This implies
that the derivatives of both cuspidal and Eisenstein families specialising to
$f$, in spite of not displaying any simple modularity properties themselves,
yield $p$-adic modular forms after taking diagonal restriction. A suitable
linear combination of these derivatives is considered in § 4, and the Fourier
coefficients of its ordinary projection are related to the RM values of the
winding cocycle.

While $p$-adic Eisenstein families only occur in parallel weight, cuspidal
families vary over a larger weight space. The main result of this section is
Theorem 3.13, which describes the Fourier coefficients of the derivatives of a
cuspidal family in the “anti-parallel” direction of the weight space. Much
like in the archimedean settings, the Fourier expansions of $p$-adic
Eisenstein families are entirely explicit; however, no general expression is
available for cuspidal families. Our approach to studying cuspidal
deformations of a Hilbert Eisenstein series rests on the analysis of the
associated Galois deformation problems. Roughly speaking, first order
deformations of the Artin representation attached to a Hilbert Eisenstein
series of parallel weight one are described in terms of the Galois cohomology
of the adjoint representation, which cuts out a finite abelian extension $H$
of $F$. A class in the Galois cohomology of the adjoint cuts out an abelian
$p$-adic Lie extension of $H$, and the Frobenius traces on the associated
Galois deformation involve $p$-adic logarithms of global $p$-units in $H$, via
the reciprocity law of global class field theory for $H$. This translates into
the appearance of the logarithms of Gross–Stark units in the Fourier
coefficients of first order deformations of Hilbert Eisenstein series, and
accounts for the presence of the same quantities in the constant term of the
generating series $G_{\tau}$ of Theorem C.

The Galois deformation arguments are clarified and not substantially
lengthened by working in the setting where $F$ is an arbitrary totally real
field of degree $d$ in which $p$ is inert. This will be assumed until § 3.5,
when the main results will be specialised to the case where $F$ is real
quadratic.

###  3.1. Hilbert modular forms and Hecke algebras.

Fix an embedding $\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$. Let
$F$ be a totally real field in which $p$ is inert, and denote by
$\mathfrak{d}$ the different of its ring of integers $\mathcal{O}_{F}$. Write
$\alpha_{1},\ldots,\alpha_{d}$ for the distinct embeddings of $F$ into
$\bar{\mathbb{Q}}_{p}$, so that $\alpha_{1}$ is the embedding given by the
restriction to $F$ of the chosen embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$. Via the choice of an
isomorphism $\mathbb{C}\simeq\mathbb{C}_{p}$, one obtains a corresponding
indexing of embeddings $\bar{\mathbb{Q}}\hookrightarrow\mathbb{C}.$ For $x\in
F$, let $(x_{1},\dots,x_{d})$ denote the image of $x$ under the embeddings
$\alpha_{1},\ldots,\alpha_{d}$, viewed as a $d$-tuple of either complex or
$p$-adic numbers, depending on the context.
_It is assumed throughout that the Leopoldt Conjecture holds for $F$_. (When
$F$ is quadratic, this assumption is known to be satisfied.)

Fix a totally odd character $\psi$ of the narrow class group of $F$. Let $E$
be a finite extension of $\mathbb{Q}_{p}$ containing the images of $F$ under
all embeddings $\alpha_{1},\ldots,\alpha_{d}$ and the values of the character
$\psi$.

We now recall some definitions and conventions related to Hilbert modular
forms and their associated Hecke algebras, following the treatment that is
given in [Shi78], [Hi88, § 2] and [Hi91, § 3].

Let ${\bf k}=(k_{j})\in\mathbb{Z}^{d}_{\geq 2}$ be a $d$-tuple of integers.
Denote ${\bf t}=(t_{j})$ the vector with $t_{j}=1$ for every $1\leq j\leq d$.
Choose a vector ${\bf v}\in\mathbb{Z}^{d}$ of non-negative integers such that
${\bf k}+2{\bf v}=m{\bf t}$ for some $m\in\mathbb{Z}$, and define
$\bf{w}=\bf{k}+\bf{v}-\bf{t}$. The space of Hilbert modular forms of weight
$({\bf k},{\bf w})$ and full level ${\mathfrak{N}}$, defined as in [Hi88, §
2], is a finite-dimensional complex vector space. Let $\mathscr{H}_{{\bf
k},{\bf w}}({\mathfrak{N}})$ be the algebra of Hecke operators acting
faithfully on the subspace of cuspforms. It is free of finite rank as a
$\mathbb{Z}$-module.

Fix $({\bf k,\bf w})$ as above. The $p$-adic Hecke algebra is defined to be

(34) $\mathcal{T}\ :=\
\varprojlim_{\alpha}\,\mathscr{H}_{\bf{k},\bf{w}}(p^{\alpha})\otimes\mathcal{O}_{E},$

where the inverse limit is taken with respect to restriction of increasing
full level structure at $p$. It contains in particular diamond operators
$\langle\mathfrak{l}\rangle$ for every integral ideal $\mathfrak{l}$ coprime
to $p$, as well as Hecke operators

(35) $\mathbf{T}(y)\ =\ \varprojlim_{\alpha}T(y)y_{p}^{-{\bf v}}$

for any idèle
$y\in\widehat{\mathcal{O}_{F}}\cap\operatorname{\mathbf{A}}_{F}^{\times}$,
whose component at $p$ is denoted $y_{p}$. When $y_{p}$ is a unit, the
operator $T(y)$ depends only on the integral ideal $\mathfrak{m}$ defined by
$y$, and we write $T_{\mathfrak{m}}$ and $\mathbf{T}_{\mathfrak{m}}$ for
$T(y)$ and $\mathbf{T}(y)$. The compact ring $\mathcal{T}$ has a unique
decomposition
$\mathcal{T}=\mathcal{T}^{\mathrm{n.ord}}\oplus\mathcal{T}^{\scriptstyle\mathrm{ss}}$
such that $\mathbf{T}(p)$ is a unit in $\mathcal{T}^{\mathrm{n.ord}}$ and is
topologically nilpotent in $\mathcal{T}^{\scriptstyle\mathrm{ss}}$. The ring
$\mathcal{T}^{\mathrm{n.ord}}$ is called the nearly ordinary cuspidal Hecke
algebra. It is independent of the choice of $({\bf k},{\bf w})$.

Write
$\operatorname{{\mathsf{U}}}=(\mathcal{O}_{F}\otimes\mathbb{Z}_{p})^{\times}$
and let $\operatorname{{\mathsf{Z}}}$ be the Galois group of the maximal
abelian extension of $F$ unramified outside $p$ and $\infty$. The Iwasawa
algebra

$\Lambda:=\mathcal{O}_{E}\lsem\operatorname{{\mathsf{U}}}\times\operatorname{{\mathsf{Z}}}\rsem$

is abstractly isomorphic to a ring of power series in several variables with
coefficients in a finite group ring over $\mathcal{O}_{E}$. Denote

$\kappa^{\mathrm{univ}}\colon\operatorname{{\mathsf{U}}}\times\operatorname{{\mathsf{Z}}}\operatorname{\longrightarrow}\Lambda^{\times}$

its universal character. Since $p$ is inert in $F$, the group
$\operatorname{{\mathsf{U}}}$ is identified with the units of $F_{p}$. Denote
$\operatorname{{\mathsf{U}}^{\circ}}$ and
$\operatorname{{\mathsf{Z}}^{\circ}}$ the torsion free parts of
$\operatorname{{\mathsf{U}}}$ and $\operatorname{{\mathsf{Z}}}$ respectively
and let
$\Lambda^{\circ}=\mathcal{O}_{E}\lsem\operatorname{{\mathsf{U}}^{\circ}}\times\operatorname{{\mathsf{Z}}^{\circ}}\rsem$.
Let $\chi_{p}$ and $\omega_{p}$ be the cyclotomic and the Teichmüller
characters of $G_{F}$ respectively. They factor through the quotient
$\operatorname{{\mathsf{Z}}}$. Define $\mathsf{q}=4$ when $p=2$, and
$\mathsf{q}=p$ otherwise. Then the homomorphism

$\chi_{p}\cdot\omega_{p}^{-1}\colon\operatorname{{\mathsf{Z}}}\to
1+\mathsf{q}\mathbb{Z}_{p}$

induces an isomorphism when restricted to
$\operatorname{{\mathsf{Z}}^{\circ}}$ if the Leopoldt conjecture holds for
$F$. For any weight $({\bf k},{\bf w})$ as above, consider the character
$\kappa_{{\bf k},{\bf
w}}\colon\operatorname{{\mathsf{G}}}=\operatorname{{\mathsf{U}}}\times\operatorname{{\mathsf{Z}}}\operatorname{\longrightarrow}\bar{\mathbb{Q}}_{p}^{\times}$
defined by

(36) $(a,z)\ \longmapsto\ a^{\bf v}\cdot\chi_{p}^{m-1}(z).$

With a slight abuse of notation, the corresponding ring homomorphism will also
be denoted by $\kappa_{{\bf k},{\bf w}}$ and referred to as the weight
$({\mathbf{k}},{\mathbf{w}})$-specialisation. If ${{\mathbf{v}}}=0$, so that
${\mathbf{k}}=m\cdot{\mathbf{t}}$, the pair $({\mathbf{k}},{\mathbf{w}})$ will
be called parallel weight $m$. Parallel weight specialisations are
parametrised by the Iwasawa algebra
$\mathcal{O}_{E}\lsem\operatorname{{\mathsf{Z}}}\rsem$, which shall be
regarded as a quotient of $\Lambda$.

The nearly ordinary Hecke algebra $\mathcal{T}^{\mathrm{n.ord}}$ is a
$\Lambda$-algebra via the action of the diamond operators. The main theorem of
[Hi89a] asserts that $\mathcal{T}^{\mathrm{n.ord}}$ is finitely generated and
torsion-free as a $\Lambda^{\circ}$-module. In addition, the quotient of
$\mathcal{T}^{\mathrm{n.ord}}$ by the ideal generated by the kernel of
$\kappa_{{\mathbf{k}},{\mathbf{w}}}$, for $({\mathbf{k}},{\mathbf{w}})$ as
above, is isomorphic to the ordinary part of the classical Hecke algebra of
weight $({\mathbf{k}},{\mathbf{w}})$ and Iwahori level at $p$. This result is
often referred to as Hida’s Control Theorem.

$p$-adic families of Hilbert modular forms and weight one Eisenstein series.

The space of Hilbert modular forms of weight $({\mathbf{k}},{\mathbf{w}})$ and
any level is automatically cuspidal unless $({\mathbf{k}},{\mathbf{w}})$ is
parallel [Shi78, ($1.8_{a}$)]. However, for parallel weights, non-trivial
Eisenstein forms exist and can be interpolated in explicit $p$-adic families
parametrised by $\mathcal{O}_{E}\lsem\operatorname{{\mathsf{Z}}}\rsem$. The
study of congruences between cuspidal and Eisenstein families of Hilbert
modular forms is at the heart of Wiles’ proof of Iwasawa Main Conjecture over
totally real fields [Wi90]. In a similar spirit, we consider certain cuspidal
and Eisenstein families sharing the same specialisation at parallel weight
one, i.e. ${\mathbf{k}}={\mathbf{t}}$ and ${\mathbf{w}}=0$.

For any pair $(\varphi,\eta)$ of unramified characters of $F$ with
$\varphi\eta$ totally odd, there exists a family $\mathcal{E}(\varphi,\eta)$
with Fourier expansions as described in [DDP11, § 3]. The cases where
$(\varphi,\eta)=(1,\psi)$ or $(\psi,1)$ are of particular relevance in the
calculations leading to the proof of Theorem C.

Let $E_{1}(1,\psi)$ be the classical Eisenstein series of weight 1 and trivial
level with Fourier expansion:

$E_{1}(1,\psi)(z_{1},\dots,z_{d})=L(F,\psi,0)+\psi^{-1}(\mathfrak{d})L(F,\psi^{-1},0)+2^{d}\sum_{\nu\in{\mathfrak{d}}_{+}^{-1}}\sigma_{0,\psi}(\nu\mathfrak{d})e^{2\pi
i(\nu_{1}z_{1}+\dots+\nu_{d}z_{d})},$

where $z=(z_{1},\dots z_{d})\in\mathcal{H}^{d}$ and

$\sigma_{k,\psi}(\alpha)=\sum_{I\lhd\mathcal{O}_{F},I\mid(\alpha)}\psi(I)\mathrm{Nm}(I)^{k}\qquad\mbox{and}\qquad
L(F,\psi,s)=\sum_{I\lhd\mathcal{O}_{F}}\psi(I)\mathrm{Nm}(I)^{-s},$

the latter converging for $\mathrm{Re}(s)$ large enough, analytically
continued to all $s\in\mathbb{C}$.

In the case where $p$ is inert in $F$, one has $\psi(p)=1$, and the $p$-adic
$L$-functions $L_{p}(\psi,s)$ and $L_{p}(\psi^{-1},s)$ have exceptional zeros
at $s=0$. The Eisenstein series $E_{1}(1,\psi)$ then admits a unique
$p$-stabilisation

(37) $f(z):=E_{1}^{(p)}(1,\psi)(z):=E_{1}(1,\psi)(z)-E_{1}(1,\psi)(pz)$

which is the weight one specialisation of the Eisenstein families
$\mathcal{E}(1,\psi)$ and $\mathcal{E}(\psi,1)$. The derivatives of the
Fourier coefficients of $\mathcal{E}(1,\psi)$ and $\mathcal{E}(\psi,1)$ at
weight 1 will be exploited in §4. Let

$\kappa_{1+\varepsilon}\colon\mathcal{O}_{E}\lsem\operatorname{{\mathsf{Z}}}\rsem\to
E[\varepsilon]/(\varepsilon^{2})$

be the algebra homomorphism whose restriction to $\operatorname{{\mathsf{Z}}}$
is given by

$\kappa_{1+\varepsilon}(u,z)=\chi_{p}(z)^{-1}(1+\log_{p}(\chi_{p}(z))\varepsilon),$

and let $E_{1+\varepsilon}^{(p)}(\eta,\varphi)$ be the image of
$\mathcal{E}(\eta,\varphi)$ under $\kappa_{1+\varepsilon}$. The Fourier
expansion of $E_{1+\varepsilon}^{(p)}(\eta,\varphi)$ can be written as

(38) $E_{1+\varepsilon}^{(p)}(\eta,\varphi)\ =\ a_{0}(\eta,\varphi)\ +\
\sum_{\nu}a_{\nu}(\eta,\varphi)q^{\nu}.$

where the coefficients can be read off from [DDP11, Prop. 2.1, 3.2], as
summarised in the following lemma:

######  Lemma 3.1.

The Fourier coefficients of $E_{1+\varepsilon}^{(p)}(\eta,\varphi)$ are given
by

(39) $\begin{array}[]{lll}a_{0}(\eta,\varphi)&=&\
\displaystyle\frac{L_{p}^{\prime}(\eta^{-1}\varphi,0)}{4\eta(\mathfrak{d})}\cdot\varepsilon\\\\[7.0pt]
a_{\nu}(\eta,\varphi)&=&\displaystyle\sum_{p\nmid
I\mid(\nu)\mathfrak{d}}\eta\left(\frac{(\nu)\mathfrak{d}}{I}\right)\varphi(I)\left(1+\varepsilon\log_{p}\operatorname{\mathrm{Nm}}(I)\right).\\\
\end{array}$

The article [DDP11] constructs an explicit cuspidal family parametrised by
$\mathcal{O}_{E}\lsem\operatorname{{\mathsf{Z}}}\rsem$ specialising to $f$ at
weight 1. The family is not an eigenform over
$\mathcal{O}_{E}\lsem\operatorname{{\mathsf{Z}}}\rsem$. Nevertheless, since
$f$ itself is an eigenform, one can deduce the existence of a morphism

(40) $\pi_{f}\colon\mathcal{T}^{\mathrm{n.ord}}\to\mathcal{O}_{E}$

encoding the eigenvalues of Hecke operators acting on $f$. The composition
with the morphism $\Lambda\to\mathcal{T}^{\mathrm{n.ord}}$ will be denoted by

(41) $\pi_{1}\colon\Lambda\to\mathcal{O}_{E}$

and corresponds to the character $\kappa_{{\mathbf{t}},\mathbf{0}}\cdot\psi$.

The remainder of § 3 will be dedicated to studying lifts of the morphism
$\pi_{f}$ to $E[\varepsilon]/(\varepsilon^{2})$. Geometrically, this
corresponds to studying the geometry of
$\mathrm{Spec}(\mathcal{T}^{\mathrm{n.ord}})$ in an infinitesimal neighborhood
of the prime ideal defined by $\pi_{f}$.

######  Remark 1.

The cuspidal family appearing in [DDP11] was used to obtain the explicit
formula for the derivative of the $p$-adic $L$-function $L_{p}(\psi,s)$ at
$s=0$ conjectured by Gross, asserting that

(42) $L_{p}^{\prime}(\psi,0)=\mathscrbf{L}(\psi)L(\psi,0),$

where $\mathscrbf{L}(\psi)$ is the $\mathscrbf{L}$-invariant described in §
3.2. In recent work of Betina, Dimitrov and Shih [BDS] Gross’ formula is
linked to the study of the geometry of eigenvarieties from a Galois theoretic
perspective. The approach of [BDS] informs the present work, and is carried
out in a broader setting.

###  3.2. Galois cohomology and $\mathscr{L}$-invariants

This section develops some results on Galois cohomology, which will be used
later, notably in § 3.4, to describe the tangent space of certain Galois
deformation functors. These preliminary results are well known to experts.

Although it will not be used, it is worth noting that most of the arguments
below are quite general and also work for general number fields $F$. Let $H$
be a Galois extension of $F$ with Galois group $G$. The $E[G]$-module
$\mathrm{Hom}(G_{H}^{\mathrm{ab}},E)$ can be described explicitly via class
field theory. This will be used to show that certain global Galois cohomology
classes for the totally odd character $\psi$ are determined by their images in
local cohomology.
A finite place $v$ of $H$ determines a prime ideal of $H$ whose decomposition
group is isomorphic to the absolute Galois group of $H_{v}$, denoted by
$G_{v}$. For each $v$, there is an isomorphism between the completion of
$H_{v}^{\times}$ and $G_{v}^{\mathrm{ab}}$ induced by the local Artin
reciprocity map

$\mathrm{rec}_{v}\colon H_{v}^{\times}\ \operatorname{\longrightarrow}\
G_{v}^{\mathrm{ab}}$

for which the geometric normalisation is adopted. Note that the image of
$G_{v}$ in $G_{H}^{\mathrm{ab}}$ is canonical. Thus, there is a restriction
map $\mathrm{res}_{v}$ defined by

(43)
$\begin{array}[]{lcll}\mathrm{res}_{v}\colon&\mathrm{Hom}(G_{H}^{\mathrm{ab}},E)&\operatorname{\longrightarrow}&\mathrm{Hom}(H_{v}^{\times},E)\\\
&f&\longmapsto&\mathrm{rec}_{v}\circ f|_{G_{v}^{\mathrm{ab}}}.\end{array}$

The following lemma is well-known to experts, but its proof is sketched below
for completeness.

######  Lemma 3.2.

There is an exact sequence of $E[G]$-modules

(44)
$0\to\mathrm{Hom}(G_{H}^{\mathrm{ab}},E)\xrightarrow{(\mathrm{res}_{v})_{v\mid
p}}\prod_{v\mid
p}\mathrm{Hom}(H_{v}^{\times},E)\,\operatorname{\longrightarrow}\,\mathrm{Hom}(\mathcal{O}_{H}[1/p]^{\times},E).$

In addition, the rightmost map is surjective if and only if Leopoldt’s
Conjecture holds for $H$.

  * Proof.

Let $\mathbb{A}_{H}^{\times}/H^{\times}$ be the idèle class group of $H$. The
global Artin reciprocity map
$\mathrm{rec}_{H}\colon\mathbb{A}_{H}^{\times}/H^{\times}\to
G_{H}^{\mathrm{ab}}$ is compatible with its local versions via the restriction
maps, and gives a sequence

(45)
$0\to\mathrm{Hom}(G_{H}^{\mathrm{ab}},E)\to\mathrm{Hom}(\mathbb{A}_{H}^{\times},E)\to\mathrm{Hom}(H^{\times},E)$

of continuous group homomorphisms and the topology on $H^{\times}$ is
discrete. This sequence must be exact for topological reasons, and the two
terms on the right are understood via their restrictions to the places above
$p$, by the commutative diagram

${\mathrm{Hom}(\mathbb{A}_{H}^{\times},E)}$${\mathrm{Hom}(H^{\times},E)}$${\displaystyle\prod_{v\mid
p}\mathrm{Hom}(H_{v}^{\times},E)}$${\mathrm{Hom}(\mathcal{O}_{H}[1/p]^{\times},E)}$$\scriptstyle{\Delta^{\vee}}$$\scriptstyle{\Delta_{p}^{\vee}}$

where $\Delta^{\vee}$, $\Delta_{p}^{\vee}$ are induced by the diagonal
embeddings. Note that any element in $\mathrm{Hom}(\mathbb{A}_{H}^{\times},E)$
must be trivial on the units $\mathcal{O}_{v}^{\times}$ of $H_{v}$ for any
$v\nmid p\infty$, and standard continuity arguments then show that the
resulting map $\ker\Delta^{\vee}\to\ker\Delta_{p}^{\vee}$ is an isomorphism.
Finally, by Dirichlet’s Unit Theorem,

$\mathrm{rk}_{\mathbb{Z}}\,\mathcal{O}_{H}[1/p]^{\times}=\mathrm{rk}_{\mathbb{Z}}\,\mathcal{O}_{H}^{\times}+|\\{v\mid
p\\}|,$

so the rightmost map of (44) is surjective if and only if the
$\mathbb{Z}_{p}$-rank of the closure of the image of
$\mathcal{O}_{H}^{\times}$ in $\prod_{v\mid p}\mathcal{O}_{v}^{\times}$ is
equal to the $\mathbb{Z}$-rank of $\mathcal{O}_{H}^{\times}$, that is, if
Leopoldt’s Conjecture holds for $H$. ∎

Let $\varphi\colon G\to E^{\times}$ be any character, viewed as a character of
$G_{F}$, and consider the global Galois cohomology group
$\mathrm{H}^{1}(F,E(\varphi)).$ The inflation-restriction sequence for
continuous group cohomology of $E(\varphi)$ leads to the identification

(46)
$\mathrm{H}^{1}(F,E(\varphi))\simeq\mathrm{H}^{1}(H,E)^{\varphi^{-1}}=\mathrm{Hom}(G_{H}^{\mathrm{ab}},E)^{\varphi^{-1}}.$

The cohomology group $H^{1}(G,E(\varphi))$ is related to certain units in $H$.
Let $S$ be a finite set of places of $F$, containing all infinite places and
let $\mathcal{O}_{H,S}^{\times}$ be the $S$-units of $H$, and write

$U_{\varphi}:=(\mathcal{O}_{H,S}^{\times}\otimes E)^{\varphi^{-1}}.$

Then, the Galois-equivariant version of Dirichlet’s Unit Theorem yields

(47) $\dim_{E}U_{\varphi}=|\\{w\in
S\mid\varphi(w)=1\\}|-\dim_{E}E(\varphi)^{G}.$

The above discussion will now be specialised to the case where $H$ is the
narrow Hilbert class field of $F$. In what follows, the character $\varphi$
will be taken to be either the trivial character or the unramified totally odd
character $\psi$, viewed as a character of $G=\mathrm{Gal}(H/F)$. Let $S$ be
the set of places containing the place corresponding to the prime $(p)$ of $F$
and all infinite places of $F$. It follows from (47) that $\dim_{E}U_{\mathbbm
1}=d$, since $F$ is totally real and $\dim_{E}U_{\psi}=1$, because $(p)$
splits completely in $H/F$ and $\psi$ is totally odd.

Recall that we wrote

$\alpha_{1},\ldots,\alpha_{d}:F\hookrightarrow\bar{\mathbb{Q}}_{p}$

for the distinct $p$-adic embeddings of $F$. The prime ideal
$p\mathcal{O}_{F}$ splits completely in $H/F$, and the choice of a prime
$\mathfrak{p}$ of $H$ above $p$ determines an identification
$H_{\mathfrak{p}}=F_{p}$. Fix the choice of $\mathfrak{p}$ corresponding to
the chosen embedding $\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$
once and for all, and write

$\tilde{\alpha}_{1},\ldots,\tilde{\alpha}_{d}:H\hookrightarrow\bar{\mathbb{Q}}_{p}$

for the $p$-adic embeddings of $H$ extending $\alpha_{1},\ldots,\alpha_{d}$
respectively, i.e.,

$\tilde{\alpha}_{j}|_{F}=\alpha_{j}\qquad\text{and}\qquad\tilde{\alpha}_{j}^{-1}(\mathfrak{m}_{\bar{\mathbb{Z}}_{p}})=\mathfrak{p}.$

######  Lemma 3.3.

If the Leopoldt Conjecture holds for $H$, then:

(48)
$\dim_{E}\mathrm{H}^{1}(F,E)=1\qquad\text{and}\qquad\dim_{E}\mathrm{H}^{1}(F,E(\psi))=d.$

  * Proof.

Combining (46) with Lemma 3.2, for any character $\varphi$ of $G$, the
cohomology group $\mathrm{H}^{1}(F,E(\varphi))$ is equal to the
$\varphi^{-1}$-eigenspace of the kernel of

(49) $\prod_{\mathfrak{p}_{i}\mid
p}\mathrm{Hom}(H_{\mathfrak{p}_{i}}^{\times},E)\operatorname{\longrightarrow}\mathrm{Hom}(\mathcal{O}_{H}[1/p]^{\times},E).$

Let $\mathfrak{p}_{1}=\mathfrak{p}$ be the chosen prime. Then
$\mathrm{Hom}(H_{\mathfrak{p}_{1}}^{\times},E)$ is spanned by
$(\log_{p}\circ\tilde{\alpha}_{j})_{j}$ and $\mathrm{ord}_{\mathfrak{p}_{1}}$.
Since $p$ splits completely in $H/F$, the $E$-vector space
$\mathrm{Hom}(H_{\mathfrak{p}_{1}}^{\times},E)$ generates the source of (49)
as an $E[G]$-module. The $E[G]$-span of each basis element of
$\mathrm{Hom}(H_{\mathfrak{p}_{1}}^{\times},E)$ is isomorphic to the right
regular representation of $G$. In particular, the multiplicity of every one-
dimensional representation of $G$ in the source of (49) is equal to
$\dim_{E}(\mathrm{Hom}(H_{\mathfrak{p}_{1}}^{\times},E))=(d+1)$.

Thus, (47) leads to the inequality (which is an equality if Leopoldt’s
Conjecture holds for $H$)

$\dim_{E}\mathrm{H}^{1}(F,E(\varphi))\geq(d+1)-\dim_{E}U_{\varphi^{-1}}=\begin{cases}1,\quad\text{if
$\varphi=\mathbbm 1,$}\\\ d,\quad\text{if $\varphi$ is totally
odd}.\end{cases}$

∎

When $\varphi$ is either totally odd or trivial, we wish to describe the
restriction to the decomposition group at the prime $p$ of a basis of
$\mathrm{H}^{1}(F,E(\varphi))$. Denote

$\mathrm{res}_{p}\colon\mathrm{H}^{1}(F,E(\varphi))\to\mathrm{H}^{1}(F_{p},E(\varphi))$

for the restriction to the decomposition group at $p$, which is characterised
by the choice of embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$. Since
$\varphi|_{G_{F_{p}}}=1$, by local class field theory, we can choose a basis
of the target given by the $p$-adic valuation in $F_{p}^{\times}$, denoted by
$o_{p}$, together with the homomorphisms

$\ell_{p,j}=\log_{p}\circ\alpha_{j},\qquad\qquad 1\leq j\leq d.$

Choose an auxiliary prime $\mathfrak{p}_{i}$, for some $1\leq i\leq n$. Then,
under the running assumptions, $U_{\psi}$ is a one-dimensional $E$-vector
space. Choose a generator $u_{\psi}$ of $U_{\psi}$, and note that
$\mathrm{ord}_{\mathfrak{p}_{i}}(u_{\psi})\neq 0$, since the
$\psi^{-1}$-eigenspace of $\mathcal{O}_{H}^{\times}\otimes E$ is trivial by
(47).

######  Definition 3.4.

The quantity

(50)
$\mathscr{L}_{j}(\psi)=-\frac{(\log_{p}\circ\,\tilde{\alpha}_{j})(u_{\psi})}{(\mathrm{ord}_{p}\circ\tilde{\alpha}_{j})(u_{\psi})},\qquad(1\leq
j\leq d)$

is called the partial $\mathscr{L}$-invariant of $\psi$ with respect to the
$j$-th embedding of $F$ into $\bar{\mathbb{Q}}_{p}$.

Note that this definition is independent of the choice of the generator
$u_{\psi}$ of $U_{\psi}$ and of the auxiliary choice of the prime
$\mathfrak{p}$ of $H$ above $p$. However, it depends on the choice of the
$p$-adic embedding $\alpha_{j}$ of $F$, thus justifying the notation.

The following lemma related the partial $\mathscr{L}$-invariants of Definition
3.4 with the Gross–Stark unit $u_{\tau}$ of the introduction. In order to
state it precisely, fix a choice of the unit $u_{\psi}$ by fixing an RM point
$\tau$ of discriminant $D$ and setting

(51) $u_{\psi}=\prod_{\sigma\in{\rm Gal}(H/F)}(\sigma
u_{\tau})^{\psi(\sigma^{-1})}.$

######  Lemma 3.5.

For all odd characters $\psi$ of ${\rm Gal}(H/F)$, and all $1\leq j\leq d$,

$\mathscr{L}_{j}(\psi)L(\psi,0)=\log_{p}(\tilde{\alpha}_{j}(u_{\psi})).$

  * Proof.

This follows after noting that, by definition of the Gross–Stark unit
$u_{\psi}$,

$\mathrm{ord}_{p}(\tilde{\alpha}_{j}(u_{\psi}))=-L(\psi,0).$

∎

The full $\mathscrbf{L}$-invariant of $\psi$ is the quantity

$\mathscrbf{L}(\psi)=\sum_{j=1}^{d}\mathscr{L}_{j}(u_{\psi}).$

It can alternatively be defined as

$\mathscrbf{L}(\psi)=-\frac{(\log_{p}\circ\operatorname{\mathrm{Nm}}^{F_{p}}_{\mathbb{Q}_{p}}\circ\,\tilde{\alpha}_{j})(u_{\psi})}{(\mathrm{ord}_{p}\circ\tilde{\alpha}_{j})(u_{\psi})}.$

######  Remark 2.

Of primary interest is the case where $F$ is quadratic and $\psi$ is an odd
narrow class character. Under theses assumptions, the character $\psi$
satisfies $\psi^{-1}(\sigma)=\psi(\tau\sigma\tau^{-1})$ for any $\tau\in
G_{\mathbb{Q}}\smallsetminus G_{F}$. This implies that the partial
$\mathscr{L}$-invariants satisfy the relations

$\mathscr{L}_{1}(\psi)=\mathscr{L}_{2}(\psi^{-1}),\qquad\mathscr{L}_{2}(\psi)=\mathscr{L}_{1}(\psi^{-1}).$

######  Proposition 3.6.

Let $\varphi$ be any character of $G$. Denote $\chi_{p}\colon
G_{F}\to\mathbb{Z}_{p}^{\times}$ the $p$-adic cyclotomic character. If
Leopoldt’s Conjecture holds for $H$, then:

  1. (1)

If $\varphi$ is trivial, $\eta_{\mathbbm 1}:=-\log_{p}\circ\chi_{p}$ generates
$\mathrm{H}^{1}(F,E(\varphi))$. Its restriction to the decomposition group at
$p$ satisfies

$\mathrm{res}_{p}(\eta_{\mathbbm 1})=\sum_{j=1}^{d}\ell_{p,j};$

  2. (2)

If $\varphi$ is totally odd, the cohomology group
$\mathrm{H}^{1}(F,E(\varphi))$ has a basis $\\{\eta_{\varphi,j}\\}_{1\leq
j\leq d}$ such that

$\mathrm{res}_{p}(\eta_{\varphi,j})=\ell_{p,j}+\mathscr{L}_{j}(\varphi^{-1})o_{p}.$

  * Proof.

By Proposition 3.6, $\mathrm{H}^{1}(F,E)$ is one dimensional; thus, it is
generated by the (non-zero element) class of $\eta_{\mathbbm
1}=-\log_{p}(\chi_{p})$. The restriction to the decomposition group at $p$ can
be calculated by observing that, since $\chi_{p}$ is obtained by restriction
to $G_{F}$ of a character of $G_{\mathbb{Q}}$, the same applies to the local
characters at $p$. This implies that $\mathrm{res}_{p}(\eta_{\mathbbm 1})$
factors through the norm map from $F_{p}$ to $\mathbb{Q}_{p}$.

Since $\mathfrak{p}_{1}$ is the prime of $H$ determined by the fixed embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$, the diagram

${\mathrm{H}^{1}(F,E(\varphi))}$${\mathrm{H}^{1}(F_{p},E(\varphi))}$${\mathrm{H}^{1}(H,E(\varphi))}$${\mathrm{H}^{1}(H_{\mathfrak{p}_{1}},E(\varphi))}$$\scriptstyle{\mathrm{res}_{p}}$$\scriptstyle{\mathrm{res}_{H}}$$\scriptstyle{\mathrm{res}_{H_{\mathfrak{p}_{1}}}}$$\scriptstyle{\mathrm{res}_{\mathfrak{p}_{1}}}$

commutes, where all the maps are given by restriction. In addition,
$\mathrm{res}_{H_{\mathfrak{p}_{1}}}$ is an isomorphism, because $(p)$ splits
completely in $H/F$; more precisely $\mathrm{res}_{H_{\mathfrak{p}_{1}}}$
satisfies

(52)
$\mathrm{res}_{H_{\mathfrak{p}_{1}}}(o_{p})=\mathrm{ord}_{p}\circ\tilde{\alpha}_{j}\quad\text{and}\quad\mathrm{res}_{H_{\mathfrak{p}_{1}}}(\ell_{j,p})=\log_{p}\circ\tilde{\alpha}_{j}$

for every $1\leq j\leq d$. It is worth noting at this stage that
$\mathrm{ord}_{p}\circ\tilde{\alpha}_{j}$ is independent of $j$, while
$\log_{p}\circ\tilde{\alpha}_{j}$ depends very much on $j$. After identifying
$\mathrm{H}^{1}(F,E(\varphi))$ with the $\varphi^{-1}$ eigenspace of the
kernel of (49), the image of $\mathrm{res}_{p}$ is isomorphic to the image of
this subspace via the projection

$\bigoplus_{i=1}^{n}\mathrm{Hom}(H^{\times}_{\mathfrak{p}_{i}},E)\operatorname{\longrightarrow}\mathrm{Hom}(H^{\times}_{\mathfrak{p}_{1}},E)$

on the first component (which is, of course, not Galois equivariant). Let
$\\{1,2,\dots,n\\}$ be the $G$-set characterised by
$\sigma\mathfrak{p}_{i}=\mathfrak{p}_{\sigma i}$ for every $\sigma\in G$. Let
$(f_{i})_{i\in I}$ an element of
$\bigoplus_{i=1}^{n}\mathrm{Hom}(H_{\mathfrak{p}_{i}}^{\times},E)$. The action
of $\sigma\in G$ on $(f_{i})_{i}$ is given by

$\sigma(f_{i})_{i}=(f_{\sigma^{-1}i}\circ\sigma^{-1})_{i}.$

In particular, let $(f_{i})_{i}$ belong to the $\varphi^{-1}$-eigenspace. The
action of $G$ on primes above $p$ is simply transitive. Let $i=\sigma^{-1}1$
for $\sigma\in G$. Then

$f_{i}=\varphi(\sigma)^{-1}(f_{1}\circ\sigma)$

Let $u_{\varphi^{-1}}$ be a generator of the $\varphi$-eigenspace of
$\mathcal{O}_{H}^{\times}\otimes E$. Then

$(f_{i})_{i\in
I}(\Delta_{p}(u_{\varphi^{-1}}))=\sum_{i=1}^{n}f_{i}(u_{\varphi})=\sum_{i=1}^{n}\varphi(\sigma)^{-1}(f_{1}(\sigma
u_{\varphi}))=nf_{1}(u_{\varphi}).$

Thus, $(f_{i})_{i}$ belongs to the $\varphi^{-1}$-component of the kernel of
(49) if and only if $f_{1}(u_{\varphi^{-1}})=0$. Let

$f_{1}=\sum_{j=1}^{d}x_{j}\log_{p}\circ\tilde{\alpha}_{j}+y\mathrm{ord}_{p}\circ\tilde{\alpha}_{j}.$

The condition $f_{1}(u_{\varphi^{-1}})=0$ cuts out a $d$-dimensional subspace,
since $\mathrm{ord}_{p}\circ\tilde{\alpha}_{j}(u_{\varphi^{-1}})\neq 0$. After
re-writing this condition in terms of the $\mathscr{L}$-invariants
$\\{\mathscr{L}_{j}(\varphi^{-1})\\}$ and comparing with (52), the proposition
follows. ∎

###  3.3. $\Lambda$-adic Galois representations

A general result of Hida establishes the connection between the nearly
ordinary Hecke algebra introduced in §3.1 and Galois representations. More
precisely, in [Hi89b], certain Galois representations are constructed which
interpolate the representation corresponding to classical specialisations of
Hida families for the Hecke algebra $\mathcal{T}^{\mathrm{n.ord}}$. Exploiting
the properties of these Galois representations, the study of the Hecke algebra
$\mathcal{T}^{\mathrm{n.ord}}$ infinitesimally at the prime ideal
corresponding to the system of eigenvalues of $f$ can be reduced to that of a
deformation ring that will be introduced in §3.4.

The ultimate goal is to leverage the properties of the Galois representation
to extract explicit formulae for the derivatives of the cuspidal family
specialising to $f$, in the spirit of [DLR15]. For this purpose, it suffices
to consider the completed local ring $\mathcal{T}_{f}$ obtained as the
nilreduction of the completion of the localisation of
$\mathcal{T}^{\mathrm{n.ord}}$ at the prime ideal $\mathfrak{q}_{f}$ given by
the kernel of the morphism $\pi_{f}$ defined in (40). (Although this is not
crucial for this application, it can be showed as in [C05, Prop. 6.4] that the
completion of the localisation of $\mathcal{T}^{\mathrm{n.ord}}$ at the point
corresponding to $\mathfrak{q}_{f}$ is automatically reduced.)

It is natural to view $\mathcal{T}_{f}$ as an algebra over $\Lambda_{1}$, the
completion of the localisation of $\Lambda$ at the prime ideal
$\mathfrak{p}_{1}=\ker\pi_{1}$; the latter is isomorphic to a ring of power
series in $d+1$ variables over $E$.

In this section, Hida’s results are slightly refined in order to obtain a two-
dimensional representation with coefficients in $\mathcal{T}_{f}$ (Prop. 3.8),
satisfying certain additional properties. The proof follows Mazur and Wiles’
approach to the (somewhat delicate) study of deformations of residually
reducible representations. The treatment of Bellaiche and Chenevier [BC06],
which is well-suited to working over reduced henselian local rings such as
$\mathcal{T}_{f}$, will be followed.

Write $K_{f}$ for the total ring of fractions of $\mathcal{T}_{f}$; it is
isomorphic to a product of fields $\prod_{i}K_{f,i}$, each corresponding to a
minimal local component at of $\mathrm{Spec}(\mathcal{T}^{\mathrm{n.ord}})$ at
the point corresponding to $\mathfrak{q}_{f}$.

Thus $\mathcal{T}_{f}$ can be viewed as a subring of $K_{f}$. In this context,
the main result of [Hi89b] can be phrased as follows.

######  Theorem 3.7 (Hida).

There exists a totally odd, continuous Galois representation

$\rho_{K_{f}}\colon G_{F}\to\mathrm{GL}_{2}(K_{f})$

satisfying the following properties:

  1. (1)

the pushforward $\rho_{K_{f_{i}}}$ of $\rho_{K_{f}}$ to $K_{f,i}$ is
absolutely irreducible, for every i;

  2. (2)

$\rho_{K_{f}}$ is unramified outside $p$;

  3. (3)

For every $\mathfrak{l}$ prime ideal of $F$ such that $\mathfrak{l}\nmid p$,
let $\mathrm{Frob}_{\mathfrak{l}}$ be a Frobenius element. Then

$\det(1-X\rho(\mathrm{Frob}_{\mathfrak{l}}))=1-\mathbf{T}_{\mathfrak{l}}X+\langle\mathfrak{l}\rangle\mathrm{Nm}(\mathfrak{l})X^{2};$

  4. (4)

The restriction of $\rho_{K_{f}}$ to $G_{F_{p}}$ is nearly ordinary, i.e., it
satisfies

$\rho_{K_{f}}|_{G_{F_{p}}}\simeq\left[\begin{matrix}\epsilon&*\\\
0&\delta\end{matrix}\right],$

where

(53) $\displaystyle\delta\circ\mathrm{rec}_{p}(p)$
$\displaystyle=\mathbf{T}(p),$ (54)
$\displaystyle\delta\circ\mathrm{rec}_{p}(u)$
$\displaystyle=(\kappa^{\mathrm{univ}})((u,1))\quad\mbox{for all
}u\in\operatorname{{\mathsf{U}}},$

where $\mathrm{rec}_{p}\colon F_{p}^{\times}\to G_{F_{p}}$ is the local Artin
reciprocity map.

In order to relate the ring $\mathcal{T}_{f}$ to a deformation ring, it is
important to refine the Galois representation $\rho_{K_{f}}$ to an integral
version with coefficients in $\mathcal{T}_{f}$. By the Čebotarev density
theorem, the trace of $\rho_{K_{f}}$, as well as the characters $\delta$ and
$\epsilon$, take values in $\mathcal{T}_{f}\subset K_{f}$. Following
Bellaïche–Chenevier [BC06], the existence of a free rank two
$\mathcal{T}_{f}$-module stable under the action of $G_{F}$ can be related to
a certain global cohomology group. In addition, the condition that the Galois
representation $\rho_{K_{f}}$ is nearly ordinary imposes some local conditions
on the global cohomology classes, that allow to show that $\rho_{K_{f}}$ is
conjugate to a representation with coefficients in $\mathcal{T}_{f}$,
following an argument which will now be described.

Let $M_{K_{f}}\simeq K_{f}^{2}$ be the two-dimensional Galois representation
of $G_{F}$ provided by Theorem 3.7. Relative to a basis $(e^{+},e^{-})$ of
$M_{K_{f}}$ on which a choice of complex conjugation for $F$ acts diagonally
as $\left[\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right]$, the
representation $\rho_{K_{f}}$ is given by

$\rho_{K_{f}}=\left[\begin{matrix}a_{f}&b_{f}\\\
c_{f}&d_{f}\end{matrix}\right]\colon G_{F}\to\mathrm{GL}_{2}(K_{f}).$

The fact that the traces of $\rho_{K_{f}}$ lie in $\mathcal{T}_{f}$ implies
that $a_{f}(\sigma)\pm d_{f}(\sigma)$ belong to $\mathcal{T}_{f}$, and hence,
that

$a_{f}(\sigma),d_{f}(\sigma)\in\mathcal{T}_{f},\quad\mbox{ for all }\sigma\in
G_{F}.$

It follows that

$\quad b_{f}(\sigma)\cdot
c_{f}(\tau)=a_{f}(\sigma\tau)-a_{f}(\sigma)a_{f}(\tau)\in\mathcal{T}_{f},\quad\mbox{
for all }\sigma,\tau\in G_{F}.$

Let $B_{f}$ and $C_{f}$ be the $\mathcal{T}_{f}$-submodules of $K_{f}$
generated by the values of the functions $b_{f}$ and $c_{f}$ respectively. The
reducibility ideal $I^{\mathrm{red}}_{f}$ is the (proper) integral ideal of
$\mathcal{T}_{f}$ generated by the products $b_{f}(\sigma)c_{f}(\tau)$ for all
$\sigma,\tau\in G_{F}$.

Fix a $G_{F_{p}}$-stable free one-dimensional submodule $L_{K_{f}}$ of
$M_{K_{f}}=K_{f}^{2}$, and denote by $\epsilon_{f}$ and $\delta_{f}$ the local
characters of $G_{F_{p}}$ acting on $L_{K_{f}}$ and $M_{K_{f}}/L_{K_{f}}$
respectively.

######  Proposition 3.8.

There exists a free $\mathcal{T}_{f}[G_{F}]$-submodule
$M_{\mathcal{T}_{f}}\subset M_{K_{f}}$ of rank two over $\mathcal{T}_{f}$,
whose associated Galois representation $\rho_{\mathcal{T}_{f}}\colon
G_{F}\to\mathrm{GL}(M_{\mathcal{T}_{f}})$ satisfies the following properties:

  1. (1)

The residual representation $M_{E}:=M_{\mathcal{T}_{f}}\otimes E$ is
semisimple;

  2. (2)

There exists a free rank one summand $L_{\mathcal{T}_{f}}$ of
$M_{\mathcal{T}_{f}}$ such that

     * •

$L_{\mathcal{T}_{f}}$ is $G_{F_{p}}$-stable and $G_{F_{p}}$ acts on
$M_{\mathcal{T}_{f}}/L_{\mathcal{T}_{f}}$ via $\delta_{f}$;

     * •

The subspace $L_{E}:=L_{\mathcal{T}_{f}}\otimes E$ of $M_{E}$ is _not_
$G_{F}$-stable.

  * Proof.

Let $B_{p,f}\subset B_{f}$ be the $\mathcal{T}_{f}$-module generated by
$b(G_{F_{p}})$, and likewise for $C_{f,p}\subset C_{f}$. We claim that the
natural inclusion $B_{p,f}\hookrightarrow B_{f}$ is surjective. The
$\mathcal{T}_{f}$-module $B_{f}$ is finitely generated by continuity of the
representation $\rho_{K_{f}}$, and hence, by Nakayama’s lemma, it suffices to
show that the induced map

$i_{B}:B_{f,p}/\mathfrak{m}_{\mathcal{T}_{f}}B_{f,p}\rightarrow
B_{f}/\mathfrak{m}_{\mathcal{T}_{f}}B_{f}$

is surjective. Consider the commutative diagram

(55) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
48.95573pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Hom}(B_{f}/\mathfrak{m}_{\mathcal{T}_{f}}B_{f},E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-21.1422pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.18056pt\hbox{$\scriptstyle{i_{B}^{\vee}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-31.7844pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.66049pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\qquad\Gamma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 72.95573pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.95573pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}^{1}(F,E(\psi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
101.69603pt\raise-21.1422pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.6875pt\hbox{$\scriptstyle{\mathrm{res}_{p}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
101.69603pt\raise-30.64664pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Hom}(B_{p,f}/\mathfrak{m}_{\mathcal{T}_{f}}B_{p.f},E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
34.15424pt\raise-36.07329pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.57222pt\hbox{$\scriptstyle{\quad\qquad\Gamma_{p}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
78.13728pt\raise-42.2844pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
78.13728pt\raise-42.2844pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}^{1}(F_{p},E)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where the top horizontal map $\Gamma$ maps
$\theta\in\mathrm{Hom}(B_{f}/\mathfrak{m}_{\mathcal{T}_{f}}B_{f},E)$ to the
class of the cocycle

$\sigma\ \longmapsto\ \theta(b(\sigma))$

for every $\sigma\in G_{F}$, and $\Gamma_{p}$ is the corresponding map on
local cohomology. By [BC06, Lemma 3], the map $\Gamma$ is injective, and
Proposition 3.6 implies that $\mathrm{res}_{p}$ is also injective. The
commutativity of diagram (55) implies that $i_{B}^{\vee}$ is injective, and
therefore $i_{B}$ is surjective, as claimed, so that $B_{f}=B_{p,f}$. The same
argument shows that $C_{f}=C_{p,f}$.

From the fact that $\rho_{K_{f_{i}}}$ is absolutely irreducible for every $i$,
it follows that the the annihilator of the module $B_{f}$ (respectively
$C_{f}$) is 0. One can deduce that, without loss of generality, the vectors
$(e^{+},e^{-})$ can be rescaled by a pair of elements of $K_{f}^{\times}$ so
that

$L_{K_{f}}=\langle e^{+}+e^{-}\rangle.$

Note that this basis is unique up to scaling, and hence, the resulting matrix
representation of $\rho_{K_{f}}$ is uniquely determined.

Changing the basis $(e^{+},e^{-})$ to $(e^{-}+e^{+},e^{+})$, the
representation $\rho_{K_{f}}$ is given in matrix form by

$\left[\begin{matrix}1&0\\\
-1&1\end{matrix}\right]\left[\begin{matrix}a_{f}&b_{f}\\\
c_{f}&d_{f}\end{matrix}\right]\left[\begin{matrix}1&0\\\
1&1\end{matrix}\right]=\left[\begin{matrix}a_{f}+b_{f}&b_{f}\\\
-b_{f}+(d_{f}-a_{f})+c_{f}&-b_{f}+d_{f}\end{matrix}\right].$

In particular, for every $\sigma\in G_{F_{p}}$,

$a_{f}(\sigma)+b_{f}(\sigma)=\epsilon_{f}(\sigma)\qquad\text{and}\qquad-
b_{f}(\sigma)+d_{f}(\sigma)-a_{f}(\sigma)+c_{f}(\sigma)=0.$

Since $a_{f}$ and $d_{f}$ are valued in $\mathcal{T}_{f}$ and
$a_{f}(\sigma)=d_{f}(\sigma)=1\pmod{\mathfrak{m}_{\mathcal{T}_{f}}}$, the
first equation implies that $b_{f}$ takes values in
$\mathfrak{m}_{\mathcal{T}_{f}}$. Similarly, because
$d_{f}(\sigma)-1\in\mathfrak{m}_{\mathcal{T}_{f}}$, from the second equation
it also follows that $c_{f}(\sigma)\in\mathfrak{m}_{\mathcal{T}_{f}}$. Thus,
$B_{f}=B_{p,f}\subset\mathfrak{m}_{\mathcal{T}_{f}}$ and
$C_{p}=C_{p,f}\subset\mathfrak{m}_{\mathcal{T}_{f}}$. It follows that
$\rho_{K_{f}}$ has coefficients in $\mathcal{T}_{f}$ relative to the basis
$(e^{+},e^{-})$, giving rise to a Galois stable $\mathcal{T}_{f}$-lattice
$M_{\mathcal{T}_{f}}:=\mathcal{T}_{f}e^{+}+\mathcal{T}_{f}e^{-}$ which
satisfies all the claims of Proposition 3.8. ∎

###  3.4. A deformation ring for residually reducible representations

This section describes an abstract deformation ring
$\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}$ relevant to the Eisenstein series $f$
defined in (37). This deformation ring is equipped with a natural
$\Lambda_{1}$-algebra structure

(56) $\Phi_{\mathcal{R}}\colon\Lambda_{1}\ \operatorname{\longrightarrow}\
\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}.$

The construction of $\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}$ is complicated by
the residual reducibility, and we follow the approach of Calegari–Emerton
[CE05] to overcome this. The main result of this section is Proposition 3.10,
which computes the map induced by $\Phi_{\mathcal{R}}$ on tangent spaces and
shows it is an isomorphism.

####  3.4.1. The deformation functor.

Let $\mathfrak{C}_{E}$ be the category of local complete noetherian rings with
residue field $E$. Consider the functor

$\mathcal{D}^{\det}\times\mathcal{D}^{\mathrm{loc}}\colon\mathfrak{C}_{E}\,\operatorname{\longrightarrow}\,\mathbf{Sets}$

which takes any $(A,\mathfrak{m}_{A})\in\mathrm{Ob}(\mathfrak{C}_{E})$ to the
pairs of continuous characters

$(\upsilon_{A},\vartheta_{A})\colon\operatorname{{\mathsf{U}}}\times\operatorname{{\mathsf{Z}}}\to
A^{\times},\quad\text{ with
}\quad(\upsilon_{A},\vartheta_{A})=(1,1)\pmod{\mathfrak{m}_{A}}.$

This functor is represented by the completed local ring $\Lambda_{1}$ of the
Iwasawa algebra introduced in § 3.3. The ring $\Lambda_{1}$ is isomorphic to a
ring of formal power series in $(d+1)$ variables over $E$.

We first define a deformation ring $\mathcal{R}_{\rho,L}$. Let
$\rho=\psi\oplus 1$ with the standard basis $(v_{1},v_{2})$ of $V=E^{2}$. This
is a semisimple reducible representation, and as such it admits non-scalar
endomorphisms. To obtain a representable functor, the deformation problem
needs to be suitably rigidified. This is done, following Calegari–Emerton
[CE05], by setting $L=\langle v_{1}+v_{2}\rangle$, which is a line that is not
stable under the action of $G_{F}$. A strict deformation of $(V,L,\rho)$ over
an $E$-algebra $A$ is a quadruple $(V_{A},L_{A},\rho_{A},g_{A})$ where

  * •

$V_{A}$ is a free $A$-module of rank $2$;

  * •

$L_{A}$ is a free rank $1$ summand of $V_{A}$;

  * •

$\rho_{A}\colon G_{F}\to\mathrm{GL}(V_{A})$ is a continuous representation;

  * •

$g_{A}\colon V_{A}\otimes_{A}E\simeq V$ is an isomorphism of
$E[G_{\mathbb{Q}}]$-modules sending $L_{A}\otimes E$ to $L$.

Two strict deformations $(V_{A},L_{A},\rho_{A},g_{A})$ and
$(V_{A}^{\prime},L_{A}^{\prime},\rho_{A}^{\prime},g_{A}^{\prime})$ are said to
be equivalent if there is an $A[G_{\mathbb{Q}}]$-module isomorphism
$h:V_{A}\operatorname{\longrightarrow}V_{A}^{\prime}$ sending $L_{A}$ to
$L_{A}^{\prime}$ and for which the diagram

$\textstyle{V_{A}\otimes_{A}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
g_{A}}$$\scriptstyle{h\otimes
E}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{A}^{\prime}\otimes_{A}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
g_{A}^{\prime}}$$\textstyle{V}$

commutes. A deformation of $(V,L,\rho)$ over an $E$-algebra $A$ is an
equivalence class of strict deformations over $A$.

Consider the functor

$\mathcal{D}_{\rho,L}\colon\mathfrak{C}_{E}\,\operatorname{\longrightarrow}\,\mathbf{Sets}$

which associates to an object $A$ in $\mathfrak{C}_{E}$ the set of
deformations of $(V,L,\rho)$ over $A$. The functor $\mathcal{D}_{\rho,L}$ is
representable by a complete local Noetherian ring $\mathcal{R}_{\rho,L}$ with
residue field $E$. The representability can be verified as in [Ma89, Prop. 1].
It is ensured by the additional datum of a line lifting $L$, which can be
viewed a “partial framing" of the functor parametrising deformations, forcing
automorphisms of a deformation to consist only of scalars. Thus rigidified,
the deformation functor presents the advantage of being fine enough to be
representable, while still being conceivably comparable with a Hecke algebra,
as in [CE05].

Finally, consider the functor $\mathcal{D}^{\mathrm{n.ord}}_{\rho,L}$
classifying quintuples $(V_{A},L_{A},\rho_{A},g_{A},\iota_{A})$ such that:

  * •

the equivalence class of the quadruple $(V_{A},L_{A},\rho_{A},g_{A})$ belongs
to $\mathcal{D}_{\rho,L}(A)$;

  * •

the free rank one summand $L_{A}$ is $G_{F_{p}}$-stable;

  * •

$\iota_{A}\colon F_{p}^{\times}\to A^{\times}$ is the character satisfying

$((\rho_{A}\circ\mathrm{rec}_{p})y)v=\iota_{A}(y)v\mod L_{A}$

for every $y\in F_{p}^{\times}$ and $v\in V_{A}$, where $\mathrm{rec}_{p}$
denotes the local Artin reciprocity map.

######  Remark 3.

Of course the datum of a character $\iota_{A}\colon F_{p}^{\times}\to
A^{\times}$ is redundant in the previous definition; it is nonetheless useful
to keep track of it, since it plays a key role in the calculations.

The deformation functor $\mathcal{D}^{\mathrm{n.ord}}_{\rho,L}$ is
representable by a quotient of $\mathcal{R}_{\rho,L}$, denoted
$\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}$. Indeed, choose an
$\mathcal{R}_{\rho,L}$-basis $(\tilde{v}_{1},\tilde{v}_{2})$ for the universal
representation, lifting $(v_{1},v_{2})$, in such a way that the universal free
rank one summand is given by

$L_{\mathcal{R}_{\rho,L}}=\langle\tilde{v}_{1}+\tilde{v}_{2}\rangle.$

Then the ring $\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}$ is the quotient of
$\mathcal{R}_{\rho,L}$ by the ideal

$J^{\mathrm{n.ord}}=\langle\alpha(\sigma)+\beta(\sigma)-\gamma(\sigma)-\delta(\sigma)\mid\forall\sigma\in
G_{F_{p}}\rangle,$

where $\alpha,\beta,\gamma,\delta$ are the entries of
$\rho_{\mathcal{R}_{\rho,L}}=\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]$ with respect to the chosen basis.

We now describe the $\Lambda_{1}$-algebra structure of
$\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}$. Given a quintuple
$(V_{A},L_{A},\rho_{A},g_{A},\iota_{A})\in\mathcal{D}^{\mathrm{n.ord}}_{\rho,L}(A)$,
the pair $(\det\rho_{A}\cdot\psi^{-1},\iota|_{\mathcal{O}_{F_{p}}^{\times}})$
belongs to $\mathcal{D}^{\det}\times\mathcal{D}^{\mathrm{loc}}(A)$. In
particular, taking (a representative of) the universal object of
$\mathcal{D}^{\mathrm{n.ord}}_{\rho,L}$ yields a morphism
$\Phi_{\mathcal{R}}$. As a consequence of Proposition 3.8, the universal
property of $\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}$ yields a morphism
$\Upsilon$ from the deformation ring to the Hecke algebra, which makes the
following diagram commute:

(57) $\textstyle{\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Upsilon}$$\textstyle{\quad{\mathcal{T}_{f}}}$$\textstyle{\Lambda_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{\mathcal{R}}}$$\scriptstyle{\Phi_{\mathcal{T}}}$

####  3.4.2. Tangent spaces.

We now come to the main results of this subsection, and describe the map on
tangent spaces induced by $\Phi_{R}$. Let $E[\varepsilon]$ be the ring of dual
numbers over $E$. Let

$t^{\det,\mathrm{loc}}=(\mathcal{D}^{\det}\times\mathcal{D}^{\mathrm{loc}})(E[\varepsilon]),\qquad
t_{\rho,L}=\mathcal{D}_{\rho,L}(E[\varepsilon])\qquad\text{and}\qquad
t^{\mathrm{n.ord}}_{\rho,L}=\mathcal{D}^{\mathrm{n.ord}}_{\rho,L}(E[\varepsilon])$

be the tangent spaces of the three functors introduced above. The following
lemma describes them explicitly.

######  Lemma 3.9.

  1. (1)

There is an isomorphism $G:\mathrm{H}^{1}(F,\mathrm{ad}(\rho))\to t_{\rho,L}$
sending the cohomology class of $\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]$ in
$\mathrm{H}^{1}(F,\mathrm{ad}(\rho))$ to the equivalence class of

(58)
$\left(E[\varepsilon]^{2},\left(1+\varepsilon\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]\right)\rho,\left\langle\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]\right\rangle,g_{\varepsilon}\right),$

where $g_{\varepsilon}$ sends the standard basis of $E[\varepsilon]^{2}$ to
$(v_{1},v_{2})$.

  2. (2)

Let $\mathrm{H}^{1}(F,\mathrm{ad}(\rho))^{\mathrm{n.ord}}$ be the subspace of
$\mathrm{H}^{1}(F,\mathrm{ad}(\rho))$ consisting of cocycles for which

$\alpha(\sigma)+\beta(\sigma)-\gamma(\sigma)-\delta(\sigma)=0,\qquad\forall\sigma\in
G_{F_{p}}.$

There is an isomorphism
$G^{\prime}\colon\mathrm{H}^{1}(F,\mathrm{ad}(\rho))^{\mathrm{n.ord}}\to
t^{\mathrm{n.ord}}_{\rho,L}$ given by

$G^{\prime}\left(\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]\right)=\left(G\left(\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]\right),1+\varepsilon(\delta-\beta)\circ\mathrm{rec}_{p})\right).$

  * Proof.

    1. (1)

For any equivalence class in
$t_{\rho,L}=\mathcal{D}_{\rho,L}(E[\varepsilon])$, we can choose a
representative of the form (58) for some cocycle in
$\mathrm{Z}^{1}(F,\mathrm{ad}(\rho))$. It suffices to verify that $G$ is well-
defined; in other words, it is enough to show that two lifts if
$\rho_{\varepsilon},\rho^{\prime}_{\varepsilon}$ of $\rho$ are conjugate by a
matrix

$g\in\ker(\mathrm{GL}_{2}(E[\varepsilon])\to\mathrm{GL}_{2}(E)),$

then they are conjugate by a matrix stabilizing the line
$\left\langle\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]\right\rangle$. This follows from the fact that the
space of coboundaries for the adjoint representation of the form

$\sigma\mapsto\left[\begin{matrix}\psi(\sigma)&0\\\
0&1\end{matrix}\right]\left[\begin{matrix}r&s\\\
t&u\end{matrix}\right]\left[\begin{matrix}\psi(\sigma)^{-1}&0\\\
0&1\end{matrix}\right]-\left[\begin{matrix}r&s\\\
t&u\end{matrix}\right]=\left[\begin{matrix}0&(\psi(\sigma)-1)s\\\
(\psi(\sigma^{-1})-1)t&0\end{matrix}\right]$

for $\sigma\in G_{F}$ and $\left[\begin{smallmatrix}r&s\\\
t&u\end{smallmatrix}\right]\in\mathrm{M}_{2}(E)$ is spanned by coboundaries of
matrices fixing the line $\left\langle\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]\right\rangle$.

    2. (2)

Let $X$ be the cohomology class of $\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]\in\mathrm{Z}^{1}(F,\mathrm{ad}(\rho))$.
Denote
$\rho_{\varepsilon}=(1+\varepsilon\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right])\rho$. The image of $X$ under $G$
belongs to the $t_{\rho,L}^{\mathrm{n.ord}}$ if the line
$\left\langle\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]\right\rangle$ is stable for the action of
$G_{F_{p}}$. In the basis $\left(\left\langle\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]\right\rangle,\left\langle\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]\right\rangle\right)$ the matrix of
$\rho_{\varepsilon}$ is given by

$\left[\begin{matrix}1&0\\\
1&1\end{matrix}\right]^{-1}\cdot\left[\begin{matrix}\alpha&\beta\\\
\gamma&\delta\end{matrix}\right]\cdot\left[\begin{matrix}1&0\\\
1&1\end{matrix}\right]=\left[\begin{matrix}\psi+\varepsilon(\psi\alpha+\beta)&\beta\varepsilon\\\
(1-\psi)+\varepsilon(-\alpha\psi-\beta+\gamma\psi+\delta)&1+\delta\varepsilon-\beta\varepsilon\end{matrix}\right].$

The claim follows by observing that the restriction of $\psi$ to $G_{F_{p}}$
is trivial.

∎

With these identifications, explicit bases for the relevant tangent spaces can
be obtained by using the results in § 3.2. Identify $t^{\det,\mathrm{loc}}$
with
$\mathrm{Hom}(\operatorname{{\mathsf{Z}}},E)\oplus\mathrm{Hom}(\operatorname{{\mathsf{U}}},E)$,
and choose the basis

(59) $e_{0}:=(\eta_{\mathbbm 1},0),\qquad e_{j}=(0,\ell_{p,j}),\qquad 1\leq
j\leq d.$

Given this choice of basis of the tangent space, an identification between
$\Lambda_{1}=E\lsem X_{0},X_{1},\dots,X_{d}\rsem$ can be chosen so that the
universal characters are given by the pair
$(\upsilon^{\mathrm{univ}},\vartheta^{\mathrm{univ}})$ satisfying

$\displaystyle\vartheta^{\mathrm{univ}}=1+X_{0}e_{0}\pmod{\mathfrak{m}^{2}_{\Lambda_{1}}}$
$\displaystyle\upsilon^{\mathrm{univ}}=1+\sum_{j=1}^{d}X_{j}e_{j}\pmod{\mathfrak{m}^{2}_{\Lambda_{1}}}.$

For the tangent space $t_{\rho,L}$, note that since $\rho=\psi\oplus\mathbbm
1$, it follows that

$\mathrm{ad}(\rho)=\mathbbm 1^{2}\oplus\psi\oplus\psi^{-1}.$

i.e. the adjoint representation of $\rho$ splits completely. Hence, by Lemma
3.3, the cohomology group $\mathrm{H}^{1}(F,\mathrm{ad}(\rho))$ has dimension
$2d+2$, and we may choose the $E$-basis consisting of

(60) $A=\left[\begin{matrix}\eta_{\mathbbm 1}&0\\\
0&0\end{matrix}\right],\quad D=\left[\begin{matrix}0&0\\\ 0&\eta_{\mathbbm
1}\end{matrix}\right],\quad B_{j}=\left[\begin{matrix}0&\eta_{\psi,j}\\\
0&0\end{matrix}\right],\quad C_{j}=\left[\begin{matrix}0&0\\\
\eta_{\psi^{-1},j}&0\end{matrix}\right],\qquad 1\leq j\leq d.$

where the entries are described by Proposition 3.6. With respect to these
choices of bases, we now explicitly describe the map on tangent spaces induced
by $\Phi_{\mathcal{R}}$, denoted

(61) $\Theta\colon
t^{\mathrm{n.ord}}_{\rho,L}\,\operatorname{\longrightarrow}\,t^{\det,\mathrm{loc}}.$

######  Proposition 3.10.

If $\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1})\neq 0$, the map $\Theta$ is
an isomorphism and its inverse satisfies

(62) $\displaystyle\Theta^{-1}(e_{0})$ $\displaystyle=\frac{\mathscrbf
L(\psi^{-1})}{\mathscrbf L(\psi)+\mathscrbf
L(\psi^{-1})}\left(A+\textstyle\sum_{k=1}^{d}C_{k}\right)+\frac{\mathscrbf
L(\psi)}{\mathscrbf L(\psi)+\mathscrbf
L(\psi^{-1})}\left(D+\textstyle\sum_{k=1}^{d}B_{k}\right)$ (63)
$\displaystyle\Theta^{-1}(e_{j})$
$\displaystyle=\frac{\mathscr{L}_{j}(\psi)-\mathscr{L}_{j}(\psi^{-1})}{\mathscrbf
L(\psi)+\mathscrbf
L(\psi^{-1})}(A-D+\textstyle\sum_{k=1}^{d}C_{k}-\sum_{k=1}^{d}B_{k})-B_{j}-C_{j}$

for $1\leq j\leq d$.

  * Proof.

By Lemma 3.9, up to composing with $G^{\prime}$, the tangent space
$t^{\mathrm{n.ord}}_{\rho,L}$ is identified with a subspace of
$\mathrm{H}^{1}(F,\mathrm{ad}(\rho))$ given by the kernel of

$\displaystyle
P_{1}\colon\mathrm{H}^{1}(F,\mathrm{ad}(\rho))\to\mathrm{H}^{1}(F_{p},E),$
$\displaystyle\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]\mapsto\mathrm{res}_{p}(\alpha+\beta-\gamma-\delta).$

where $\mathrm{res}_{p}$ denotes the restriction to $G_{F_{p}}$ (note that
this is well defined, because $\psi(G_{F_{p}})=1$). Similarly, the map
$\Theta\colon t^{\mathrm{n.ord}}_{\rho,L}\to t^{\det,\mathrm{loc}}$ sends a
quintuple
$(V_{\varepsilon},L_{\varepsilon},\rho_{\varepsilon},g_{\varepsilon},\iota_{\varepsilon})$
to the pair
$(\det\rho_{\varepsilon}\cdot\psi^{-1},\iota_{\varepsilon}|_{\mathcal{O}_{F_{p}}^{\times}})$.
Again, by Lemma 3.9, it can be interpreted in terms of Galois cohomology as
the restriction to the kernel of $P_{1}$ of

$\displaystyle
P_{2}\colon\mathrm{H}^{1}(F,\mathrm{ad}(\rho))\to\mathrm{Hom}(\operatorname{{\mathsf{Z}}^{\circ}},E)\oplus\mathrm{Hom}(\operatorname{{\mathsf{U}}^{\circ}},E),$
$\displaystyle\left[\begin{smallmatrix}\alpha&\beta\\\
\gamma&\delta\end{smallmatrix}\right]\mapsto\left(\alpha+\delta,(\delta-\beta)\circ\mathrm{rec}_{p}|_{\mathcal{O}_{F_{p}}^{\times}}\right).$

Thus, in order to show that $\Theta$ is an isomorphism, it suffices to show
that $P:=(P_{1},P_{2})$ is an isomorphism. Choose the bases
$(A,D,B_{1},\dots,B_{d},C_{1},\dots,C_{d})$ for
$\mathrm{H}^{1}(F,\mathrm{ad}(\rho))$ and

$\displaystyle(o_{p},0),\ (\ell_{p,1},0),\dots,(\ell_{p,d},0),\
(0,e_{0}),\dots,(0,e_{d})$

for the target of $P$. Let
$(x_{A},x_{D},x_{B_{1}},\dots,x_{B_{d}},x_{C_{1}},\dots,x_{C_{d}})$ be the
coordinates of a class in $\mathrm{H}^{1}(F,\mathrm{ad}(\rho))$ with respect
to the basis above. The map $P$ yields a system of $(2d+2)$ linear equations
in $(2d+2)$-variables:

$\displaystyle\textstyle\sum_{j=1}^{d}(\mathscr{L}_{j}(\psi)x_{C_{j}}-\mathscr{L}_{j}(\psi^{-1})x_{B_{j}})=0$
$\displaystyle x_{A}-x_{D}+x_{B_{j}}-x_{C_{j}}=0$ $\displaystyle
x_{A}+x_{D}=0$ $\displaystyle x_{D}-x_{B_{j}}=0$

where $1\leq j\leq d$. The corresponding matrix has determinant
$\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1})$. The expressions for the
inverse of $\Theta$ can be obtained by the inverting the matrix of $P$ with
respect to the above bases. ∎

######  Remark 4.

In the case of interest to this paper where $[F:\mathbb{Q}]=2$, the non-
vanishing of $\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1})$ is clear since
$\mathscrbf{L}(\psi)=\mathscrbf{L}(\psi^{-1})\neq 0$. In fact, this non-
vanishing holds in general. For details, compare with [BDS].

Recall the commutative diagram (57) arising from Proposition 3.8.

######  Theorem 3.11.

The maps $\Phi_{\mathcal{R}},\Phi_{\mathcal{T}}$ and $\Upsilon$ are
isomorphisms.

  * Proof.

Since $\Phi_{\mathcal{R}}$ is a morphism of complete local noetherian rings
with residue field $E$, the injectivity of (61) implies that
$\Phi_{\mathcal{R}}$ is surjective. The top row is surjective because all
Hecke operators are in the image; thus it follows that $\Phi_{\mathcal{T}}$ is
surjective. But $\mathcal{T}_{f}$ is a torsion free $\Lambda_{1}$-algebra; in
particular $\Phi_{\mathcal{T}}$ is an injective, hence an isomorphism. It
follows that $\Phi_{\mathcal{R}}$ and $\Upsilon$ are isomorphisms as well. ∎

######  Proposition 3.12.

The inverse of $\Phi_{\mathcal{T}}$ satisfies

(64)
$\begin{array}[]{llll}\Phi_{\mathcal{T}}^{-1}(\mathbf{T}_{\mathfrak{l}})&=&1+\psi(\mathfrak{l})+\log_{p}(\mathrm{Nm}(\mathfrak{l}))\cdot(\lambda+\mu\psi(\mathfrak{l}))&\pmod{\mathfrak{m}_{\Lambda_{1}}^{2}}\\\
\Phi_{\mathcal{T}}^{-1}(\langle\mathfrak{l}\rangle)\operatorname{\mathrm{Nm}}(\mathfrak{l})&=&\psi(\mathfrak{l})(1+(\lambda+\mu)\log_{p}(\mathrm{Nm}(\mathfrak{l})))&\pmod{\mathfrak{m}_{\Lambda_{1}}^{2}}\\\
\Phi_{\mathcal{T}}^{-1}(\mathbf{T}(p))&=&1+\xi&\pmod{\mathfrak{m}_{\Lambda_{1}}^{2}},\end{array}$

where $\mathfrak{l}$ is a prime ideal of $F$ such that
$p\nmid\rm{Nm}(\mathfrak{l})$ and
$\lambda,\mu,\xi\in\mathfrak{m}_{\Lambda_{1}}$ are given by

$\begin{array}[]{lll}\lambda&=&(\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1}))^{-1}\left(\mathscrbf{L}(\psi^{-1})X_{0}+\sum_{j=1}^{d}(\mathscr{L}_{j}(\psi)-\mathscr{L}(\psi^{-1}))X_{j}\right)\\\
\mu&=&(\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1}))^{-1}\left(\mathscrbf{L}(\psi)X_{0}-\sum_{j=1}^{d}(\mathscr{L}_{j}(\psi)-\mathscr{L}(\psi^{-1}))X_{j}\right)\\\
\xi&=&(\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1}))^{-1}\left(-\mathscrbf{L}(\psi)\mathscrbf{L}(\psi^{-1})X_{0}+\sum_{j=1}^{d}(\mathscr{L}_{j}(\psi)\mathscrbf{L}(\psi^{-1})+\mathscr{L}_{j}(\psi^{-1})\mathscrbf{L}(\psi))X_{j}\right).\end{array}$

  * Proof.

Let
$\left(V^{\mathrm{univ}},L^{\mathrm{univ}},\rho^{\mathrm{univ}},g^{\mathrm{univ}},\iota^{\mathrm{univ}}\right)$
be a representative of the universal object of the functor
$\mathcal{D}^{\mathrm{n.ord}}_{\rho,L}$ over the deformation ring
$\mathcal{R}^{\mathrm{n.ord}}_{\rho,L}$. Then

$\Upsilon^{-1}(\mathbf{T}_{\mathfrak{l}})=\mathrm{Tr}(\rho^{\mathrm{univ}})(\mathrm{Frob}_{\mathfrak{l}})\qquad\text{and}\qquad\Upsilon^{-1}(\mathbf{T}(p))=\iota^{\mathrm{univ}}(\mathrm{Frob}_{p}).$

Denoting

$\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)=\Phi^{-1}_{\mathcal{R}}\circ\rho^{\mathrm{univ}}\colon
G_{F}\ \operatorname{\longrightarrow}\ \mathrm{GL}_{2}(\Lambda_{1}),$

it follows from Proposition 3.10 that modulo $\mathfrak{m}_{\Lambda_{1}}^{2}$,

$\begin{array}[]{lll}a&=&\psi+\psi\eta_{\mathbbm
1}(\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1}))^{-1}\left(\mathscrbf{L}(\psi^{-1})X_{0}+\sum_{j=1}^{d}(\mathscr{L}_{j}(\psi)-\mathscr{L}(\psi^{-1}))X_{j}\right)\\\
d&=&1+\eta_{\mathbbm{1}}(\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1}))^{-1}\left(\mathscrbf{L}(\psi)X_{0}-\sum_{j=1}^{d}(\mathscr{L}_{j}(\psi)-\mathscr{L}(\psi^{-1}))X_{j}\right)\end{array}$

from which the expression for $\Upsilon^{-1}(\mathbf{T}_{\mathfrak{l}})$ is
obtained. Similarly, $\Phi_{\mathcal{R}}^{-1}\circ\Phi^{\mathrm{univ}}=d-b$.
Since $\eta_{\mathbbm 1}(\mathrm{Frob}_{p})=0$, it can be seen that
$(d-b)(\mathrm{Frob}_{p})=1-b(\mathrm{Frob}_{p})$, which is equal to

$1+\left(\frac{\mathscrbf{L}(\psi)(\sum_{k=1}^{d}\eta_{\psi,k}(\mathrm{Frob}_{p}))}{\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1})}X_{0}+\sum_{j=1}^{d}{\frac{(\mathscr{L}_{j}(\psi^{-1})-\mathscr{L}_{j}(\psi))\sum_{k=1}^{d}\eta_{\psi,k}(\mathrm{Frob}_{p})}{\mathscrbf{L}(\psi)+\mathscrbf{L}(\psi^{-1})}}-\eta_{\psi,j}(\mathrm{Frob}_{p})\right)X_{j}$

modulo $\mathfrak{m}^{2}_{\Lambda_{1}}$ by Proposition 3.10. The equality (64)
then follows from Lemma 3.6. ∎

###  3.5. Fourier coefficients

The above results will now be specialised to the case where $F$ is a real
quadratic field and $\psi$ is an unramified totally odd character of $F$. In
this case, we compute the Fourier coefficients of the anti-parallel family
through the Eisenstein series $f$ of parallel weight 1 discussed above.

The anti-parallel weight direction in the tangent space of the Iwasawa algebra
is the direction corresponding to the morphism
$\Lambda_{1}\operatorname{\longrightarrow}E[\varepsilon]/(\varepsilon^{2})$
given in terms of generators as follows:

(65) $\left\\{\begin{array}[]{cll}X_{0},X_{1}&\longmapsto&\varepsilon,\\\
X_{2}&\longmapsto&0.\end{array}\right.$

Since, by Theorem 3.11, the structural map
$\Phi_{\mathcal{T}}\colon\Lambda_{1}\to\mathcal{T}_{f}$ is an isomorphism, the
map (65) gives rise to a morphism from the nearly ordinary Hecke algebra to
the ring of dual numbers

(66)
$\tilde{\pi}_{f}:\mathcal{T}^{\mathrm{n.ord}}\operatorname{\longrightarrow}E[\varepsilon]/(\varepsilon^{2}).$

lifting the morphism $\pi_{f}$ defined in (40). This corresponds to the system
of Hecke eigenvalues of a first order eigenfamily of Hilbert modular forms
$\mathcal{F}$, whose Fourier coefficients, which can be recovered from
$\tilde{\pi}_{f}$, play a central role in what follows.

In the anti-parallel weight direction, one immediately checks that the
quantities $\lambda,\mu$, and $\xi$ appearing in the description of the
tangent space of $\mathcal{T}^{\mathrm{n.ord}}$ in Proposition 3.12 specialise
to

(67)
$\lambda=\frac{\mathscr{L}_{1}(\psi)}{\mathscrbf{L}(\psi)}\qquad\qquad\mu=\frac{\mathscr{L}_{2}(\psi)}{\mathscrbf{L}(\psi)}\qquad\qquad\xi=0.$

Using the results in § 3.4, the image under the anti-parallel weight morphism
$\tilde{\pi}_{f}$ of the operators $\langle l\rangle$ and
$\mathbf{T}_{\mathfrak{l}^{n}}$ for $\mathfrak{l}\neq(p)$, as well as
$\mathbf{T}(p^{n})$, can now be computed in terms of these quantities:

  * •

Let $\mathfrak{l}\neq(p)$ be a prime ideal. Proposition 3.12 immediately
implies that

$\begin{array}[]{llcll}\tilde{\pi}_{f}(\mathbf{T}_{\mathfrak{l}})&=&1+\psi(\mathfrak{l})&+&\varepsilon\log_{p}\left(\operatorname{\mathrm{Nm}}(\mathfrak{l})\right)\cdot\left(\lambda\psi(\mathfrak{l})+\mu\right)\\\
\tilde{\pi}_{f}(\langle\mathfrak{l}\rangle)\operatorname{\mathrm{Nm}}(\mathfrak{l})&=&\psi(\mathfrak{l})&+&\varepsilon\log_{p}\left(\operatorname{\mathrm{Nm}}(\mathfrak{l})\right)\cdot\psi(\mathfrak{l}).\end{array}$

The recursion relation proved in [Hi88, Corollary 4.2], which states

$\mathbf{T}_{\mathfrak{l}^{n}}\mathbf{T}_{\mathfrak{l}}\ =\
\mathbf{T}_{\mathfrak{l}^{n+1}}\ +\
\langle\mathfrak{l}\rangle\operatorname{\mathrm{Nm}}(\mathfrak{l})\mathbf{T}_{\mathfrak{l}^{n-1}}$

can be used to determine the image of the Hecke operators attached to powers
of $\mathfrak{l}$. A straightforward inductive argument now shows that

$\begin{array}[]{lll}\tilde{\pi}_{f}(\mathbf{T}_{\mathfrak{l}^{n}})&=&\displaystyle\sum_{j=0}^{n}\psi(\mathfrak{l})^{j}\left(1+\varepsilon\log_{p}\operatorname{\mathrm{Nm}}(\mathfrak{l})\left(j\cdot\lambda+(n-j)\cdot\mu\right)\right)\\\
&=&\displaystyle\sum_{I\mid\mathfrak{l}^{n}}\psi(I)\left(1+\varepsilon\left(\lambda\log_{p}\operatorname{\mathrm{Nm}}(I)+\mu\log_{p}\operatorname{\mathrm{Nm}}\left(\frac{\mathfrak{l}^{n}}{I}\right)\\!\\!\right)\\!\\!\right)\end{array}$

  * •

For the nearly ordinary Hecke operators at $p$, it follows from [Hi89b,
Proposition 2.3] that

$\tilde{\pi}_{f}(\mathbf{T}(p^{n}))=\tilde{\pi}_{f}(\mathbf{T}(p))^{n}=1.$

We are now ready to compute the Fourier coefficients of the anti-parallel
deformation $\mathcal{F}$. Recall that the algebraic notion of $q$-expansions
(cf. Hida [Hi04, Chapter 4]) gives a tuple of power series
$\mathcal{F}_{i}(q)$ indexed by a set $\operatorname{\mathfrak{t}}_{i}$ of
integral ideals representing the classes in the narrow ideal class group of
$F$:

(68) $\mathcal{F}_{i}(q)\ =\ a_{0}(\operatorname{\mathfrak{t}}_{i})\
+\sum_{\nu\,\in\,(\operatorname{\mathfrak{t}}_{i})_{+}}a_{\nu}\,q^{\nu},\qquad\qquad
a_{\nu}\in E[\varepsilon]/(\varepsilon^{2}),$

where as usual $(\operatorname{\mathfrak{t}}_{i})_{+}$ denotes the subset of
totally positive elements of $\operatorname{\mathfrak{t}}_{i}$. Abbreviate the
$q$-expansion corresponding to the inverse different $\mathfrak{d}^{-1}$ by
$\mathcal{F}(q)$. The main result of this section is:

######  Theorem 3.13.

The anti-parallel family

(69) $\mathcal{F}(q)\ =\
\sum_{\nu\,\in\,\mathfrak{d}^{-1}_{+}}a_{\nu}\,q^{\nu},\qquad a_{\nu}\in
E[\varepsilon]/(\varepsilon^{2}),$

has Fourier coefficients given, to first order, by

(70)
$a_{\nu}=\displaystyle\sum_{I\mid(\nu)\mathfrak{d}}\psi(I)\left(1+\varepsilon\left(-\log_{p}(\nu)+\frac{\mathscr{L}_{1}(\psi)}{\mathscrbf{L}(\psi)}\log_{p}\operatorname{\mathrm{Nm}}(I)+\frac{\mathscr{L}_{2}(\psi)}{\mathscrbf{L}(\psi)}\log_{p}\operatorname{\mathrm{Nm}}\left(\frac{(\nu)\mathfrak{d}}{I}\right)\\!\\!\right)\\!\\!\right),$

for all $\nu$ that are relatively prime to $p$. Furthermore
$a_{p^{m}\nu}=a_{\nu}$ for all $m\geq 1$.

  * Proof.

The family $\mathcal{F}$ is $p$-adically cuspidal, so the constant term
vanishes. To compute the higher Fourier coefficients, we compute that, for any
$\nu\in\mathfrak{d}_{+}^{-1}$ such that $p\nmid\nu$,

$\displaystyle\tilde{\pi}_{f}(\mathbf{T}_{(\nu)\mathfrak{d}})$
$\displaystyle=$
$\displaystyle\prod_{\mathfrak{l}^{n}\mid\\!\mid(\nu)\mathfrak{d}}\tilde{\pi}_{f}\left(\mathbf{T}_{\mathfrak{l}^{n}}\right)$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{I\mid(\nu)\mathfrak{d}}\psi(I)\left(1+\varepsilon\left(\frac{\mathscr{L}_{1}(\psi)}{\mathscrbf{L}(\psi)}\log_{p}\operatorname{\mathrm{Nm}}(I)+\frac{\mathscr{L}_{2}(\psi)}{\mathscrbf{L}(\psi)}\log_{p}\operatorname{\mathrm{Nm}}\left(\frac{(\nu)\mathfrak{d}}{I}\right)\\!\\!\right)\\!\\!\right)$

to first order. To determine the Fourier coefficients from this value, the
$p$-adic interpolation properties of the coefficients $a_{\nu}$, and the
density of classical forms, are used to reduce to the relations between
Fourier coefficients and the Hecke algebra proved for classical forms in
[Hi91].

Consider the rigid analytic fiber of the formal scheme attached to
$\mathcal{T}^{\mathrm{n.ord}}$. For a sufficiently small affinoid
neighbourhood $V=\mathrm{Spm}(A_{V})$ of the point corresponding to the
morphism $\pi_{f}$, there is a rigid analytic family
$\mathcal{F}_{V}=\sum_{\nu}a_{V,\nu}q^{\nu}$ with normalised Fourier
coefficients in $a_{V,\nu}\in A_{V}$, specialising to $\mathcal{F}$ in the
anti-parallel direction. By Hida’s Control Theorem, there is a Zariski-dense
set of points in $V$ corresponding to systems of Hecke eigenvalues
$\pi_{g}\colon A_{V}\to\bar{\mathbb{Q}}_{p}$ of classical modular forms $g$ of
weight $({\mathbf{k}}_{g},{\mathbf{w}}_{g})$ and with Fourier coefficients
$a_{g,\nu}$ given by the image of $a_{V,\nu}$ under $\pi_{g}$. Combining the
relations for classical forms proved in [Hi91, Eqn. (2.3) et seq./Eqn. (1.5)],
one obtains

$\begin{array}[]{lll}\pi_{g}(\mathbf{T}_{(\nu)\mathfrak{d}})&=&a_{g,\nu}\cdot\nu^{{\mathbf{v}}_{g}}\\\
\pi_{g}(\mathbf{T}_{p^{m}})&=&a_{g,p^{m}}\end{array}$

where as before
${\mathbf{v}}_{g}={\mathbf{w}}_{g}-{\mathbf{k}}_{g}+{\mathbf{t}}$. The
quantity $\nu^{{\mathbf{v}}_{g}}$ may be identified with the weight
$({\mathbf{k}}_{g},{\mathbf{w}}_{g})$-specialisation of the universal
character $\kappa^{\mathrm{univ}}$ evaluated at $(\nu,1)$. The image of
$\kappa^{\mathrm{univ}}(\nu,1)$ under the morphism (65) defining the anti-
parallel direction is given by

$1+\varepsilon\log_{p}(\nu)$

so that the density of classical points in $V$ implies that

$a_{\nu}=(1-\varepsilon\log_{p}(\nu))\tilde{\pi}_{f}(\mathbf{T}_{(\nu)\mathfrak{d}}).$

The result follows. ∎

##  4\. Diagonal restrictions and RM values

This section explains how to parlay Theorem 3.13 of § 3 into a proof of
Theorem C. In a nutshell, the generating series of Theorem C is obtained from
the ordinary projection of the diagonal restriction of a modification of the
anti-parallel cuspidal deformation $\mathcal{F}$ described in Theorem 3.13.

Retain the setup of § 3.5. Namely, $F$ denotes a real quadratic field, and
$\psi$ is a totally odd unramified character. Let $D$ be the discriminant of
$F$, with ring of integers $\mathcal{O}_{F}$, set of integral ideals
$\mathscr{I}_{F}$, and different ideal $\mathfrak{d}$. The notation
$\mathfrak{d}^{-1}_{+}$ is used for the subset of totally positive elements of
the inverse different $\mathfrak{d}^{-1}$. Write $\operatorname{\mathrm{Nm}}$
and $\mathrm{Tr}$ for the norm and trace functions from $F$ to $\mathbb{Q}$.
If $\tau\in\mathcal{H}_{p}^{D}$ is an RM point of discriminant $D$, denote by
$\mathfrak{a}_{\tau}\in{\rm Cl}(D)$ the narrow ideal class attached to $\tau$.
If $J$ is a rigid cocycle, then

$J[\psi]:=\prod_{\tau\in\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})\backslash\mathcal{H}_{p}^{\circ,D}}J[\tau]^{\psi(\mathfrak{a}_{\tau})}\in\mathbb{C}_{p}^{\times}\otimes\mathbb{Q}(\psi).$

######  Remark 5.

The character $\psi$ was assumed to be unramified for simplicity, and it would
be interesting to generalise the arguments to the case of an arbitrary totally
odd ring class character

(71) $\psi:\operatorname{\mathrm{Cl}}(D)\ \operatorname{\longrightarrow}\
\mathbb{C}_{p}^{\times}$

of discriminant $D=f^{2}D_{0}$ with $D_{0}$ fundamental, and $(p,f)=1$. The
deformations studied in § 3 are not sensitive in an essential way to this
additional ramification. Moreover, a version of Lemma 4.2 for non-trivial
conductors can be found in [LV], and the explicit formula (74) continues to
hold. One may therefore expect that Proposition 4.7 is amenable to this
generalisation via the strategy of this paper, provided that the left hand
side of the equality is replaced by the series obtained by taking the trace to
level $\Gamma_{0}(p)$:

(72) $\mathrm{Tr}^{\Gamma_{0}(fp)}_{\Gamma_{0}(p)}\left(e^{\rm
ord}(\partial_{\varepsilon}f^{+})\right)\quad\in M_{2}(\Gamma_{0}(p)).$

###  4.1. The RM values of the winding cocycle

In contrast the approach of [DD06], the calculations below build on the
viewpoint of rigid (theta) cocycles introduced in [DV, §3], by making
essential use of the winding cocycle $J_{w}$ of the prequel [DPV21], some of
whose properties were already recalled in § 2.2. This section describes some
further results from [DPV21] concerning its RM values. The first key result is
an explicit formula for $T_{n}J_{w}[\tau]$, which was established in [DPV21,
Theorem 2.9].

In order to state it, choose, for any integer $n\geq 1$ and any RM point
$\tau$ in $\mathcal{H}_{p}$, a finite set $M_{n}(\tau)$ of representatives for
the double coset space $\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})\backslash
M_{2}(\mathbb{Z})_{n}/\Gamma_{\tau}$, where

$M_{2}(\mathbb{Z})_{n}:=\\{\alpha\in M_{2}(\mathbb{Z})\mbox{ with
}\det(\alpha)=n\\},\qquad\quad\Gamma_{\tau}=\operatorname{\mathrm{Stab}}_{\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})}(\tau).$

In other words

(73) $M_{2}(\mathbb{Z})_{n}=\bigsqcup_{\delta\in
M_{n}(\tau)}\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})\cdot\delta\cdot\Gamma_{\tau}.$

Let $\tilde{\Gamma}:=\mathrm{GL}_{2}^{+}(\mathbb{Z}[1/p])$ be the group of
invertible matrices over $\mathbb{Z}[1/p]$ with positive determinant.

######  Theorem 4.1.

Let $n\geq 1$ be an integer coprime to $p$. Then

(74) $T_{n}J_{w}[\tau]=\prod_{\delta\in
M_{n}(\tau)}\prod_{\begin{subarray}{c}w\,\in\,\tilde{\Gamma}\delta\tau\\\
v_{p}(w)\,=\,0\end{subarray}}w^{[0,\infty]\cdot(w^{\prime},w)}.$

  * Proof.

See [DPV21, Theorem 2.9]. ∎

Lemma 4.2 below recalls the existence of a bijection between “level $n$” sets
of RM points and ideals that was constructed in [DPV21, § 1]. Define the
(multi)set

(75) $\operatorname{\mathrm{RM}}_{n}^{+}(\tau):=\bigsqcup_{\delta\in
M_{n}(\tau)}\left\\{w\in\tilde{\Gamma}\delta\tau\ :\
\begin{array}[]{c}w>0>w^{\prime}\\\ v_{p}(w)=0,\,v_{p}(\mathrm{disc}(w))\leq
v_{p}(n)\end{array}\right\\}$

where $\mathrm{disc}(w)$ is defined to be the discriminant of a primitive
integral quadratic form that has $w$ as a root. Similarly, define
$\operatorname{\mathrm{RM}}_{n}^{-}(\tau)$ as above, with the condition
$w>0>w^{\prime}$ replaced by $w^{\prime}>0>w$.

######  Remark 6.

Note that an RM point $w$ may appear several times in the set
$\operatorname{\mathrm{RM}}_{n}^{+}(\tau)$, and the multiplicity with which it
does is a subtle actor in the bijections discussed below. It is therefore
important to use a disjoint union in this definition. The nature of the
matrices $\delta$, which index the multiplicity with which an RM point $w$
arises, was made clearer in the proof of [DPV21, Lemma 1.9].

The sets $\operatorname{\mathrm{RM}}_{n}^{\pm}(\tau)$ play a crucial role in
the explicit formulae for the Fourier coefficients of the diagonal
restrictions of the Eisenstein family $\mathcal{E}$ investigated in [DPV21].
It will be observed below that they appear again in the analysis of the
diagonal restriction of the anti-parallel family $\mathcal{F}$ studied in § 3.

######  Lemma 4.2.

There exist two bijections

$\begin{array}[]{ccll}\varphi_{1}\
:&\operatorname{\mathrm{RM}}_{n}^{-}(-\tau)&\operatorname{\longrightarrow}&\operatorname{\mathrm{RM}}_{n}^{+}(\tau)\\\\[4.0pt]
\varphi_{2}\
:&\left\\{(I,\nu):\begin{array}[]{cc}\nu\in\mathfrak{d}^{-1}_{+},&p\nmid
I\,|\,(\nu)\mathfrak{d}\\\
\mathrm{Tr}(\nu)=n,&I\sim(1,\tau)\end{array}\right\\}&\operatorname{\longrightarrow}&\operatorname{\mathrm{RM}}_{n}^{+}(\tau)\\\
\end{array}$

such that, after writing $\nu=p^{m}\nu_{0}$, we have

$\begin{array}[]{lll}\varphi_{1}(w)&=&-w,\\\
\varphi_{2}(I,\nu)&=&\nu_{0}\sqrt{\Delta}/\operatorname{\mathrm{Nm}}(I).\end{array}$

  * Proof.

A bijection $\varphi_{1}$ as required may be constructed by letting
$W_{\infty}$ be a diagonal matrix with eigenvalues $1$ and $-1$. If
$w\in\tilde{\Gamma}\delta\tau$, then

$-w=W_{\infty}w\ \in\ \tilde{\Gamma}\delta^{\prime}(-\tau)$

where $\delta^{\prime}\in M_{n}(\tau)$ is the double coset representative of
$W_{\infty}\delta W_{\infty}$. To obtain a bijection $\varphi_{2}$ as above,
one first uses a bijection

$\Phi\ :\ \
\left\\{(I,\nu):\begin{array}[]{cc}\nu\in\mathfrak{d}^{-1}_{+},&I\,|\,(\nu)\mathfrak{d}\\\
\mathrm{Tr}(\nu)=n,&I\sim(1,\tau)\end{array}\right\\}\ \
\operatorname{\longrightarrow}\ \bigsqcup_{\delta\in
M_{n}(\tau)}\left\\{w\in\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})\delta\tau\
:\ w>0>w^{\prime}\right\\}$

which satisfies $\Phi(I,\nu)=\nu\sqrt{\Delta}/\operatorname{\mathrm{Nm}}(I)$.
Such a bijection was constructed in [DPV21, Lemma 1.9]. Note that the source
of $\Phi$ is almost equal to the source of the desired bijection, minus the
condition $p\nmid I$. Under the bijection $\Phi$, the condition $p\nmid I$ is
equivalent to the condition that $w=w_{0}p^{m}$ for some $m\geq 0$ and $p\nmid
w_{0}$. The map $w\mapsto w_{0}$ then defines a bijection between

$\left\\{w\in\operatorname{\mathrm{SL}}_{2}(\mathbb{Z})\delta\tau\ :\
\begin{array}[]{c}w>0>w^{\prime}\\\ w=w_{0}p^{m},\,p\nmid w_{0},\,m\geq
0\end{array}\right\\},$

and the set

$\left\\{w\in\tilde{\Gamma}\delta\tau\ :\ \begin{array}[]{c}w>0>w^{\prime}\\\
v_{p}(w)=0,\,v_{p}(\mathrm{disc}(w))\leq v_{p}(n)\end{array}\right\\},$

so that the result follows by definition of
$\operatorname{\mathrm{RM}}_{n}^{+}(\tau)$. ∎

###  4.2. Derivatives of diagonal restrictions.

The modular generating series for the RM values of the winding cocycle that is
the subject of Theorem C will be constructed from three different analytic
families that specialise to the Eisenstein series of parallel weight one. More
specifically, the anti-parallel cuspidal family from § 3.5, and the two
Eisenstein families of Lemma 3.1:

$E^{(p)}_{1+\varepsilon}(1,\psi)\quad\mbox{and }\quad
E^{(p)}_{1+\varepsilon}(\psi,1)$

The modularity of the generating series of Theorem C will follow from two
simple results:

  1. (1)

The vanishing of the diagonal restriction of $E^{(p)}_{1}(1,\psi)$, the
$p$-stabilisation of the parallel weight one Eisenstein series (for which the
shorthand $f$ was used in § 3),

  2. (2)

For any analytic family of $p$-adic modular forms whose specialisation
vanishes, the specialisation of its derivative is also a $p$-adic modular
form.

These results were also used in [DPV21], where full proofs may be found. Since
they play an important role in the argument, they will be briefly reviewed
here.