$\kappa_{0}^{+}$, is a distinguished Eisenstein element in ${\mathbb{H}}^{+}$,
which corresponds to the boundary symbol sending $(a,b)$ to
$f_{\infty}(b)-f_{\infty}(a)$, where $f_{\infty}$ is the unique
$\Gamma_{0}(N)$-invariant function on $\mathbb{P}_{1}(\mathbb{Q})$ which sends
$\infty$ to $1$ and $0$ to $0$. Let

$\bar{\kappa}_{1}^{+}:\Gamma_{0}(N)\longrightarrow(\mathbb{Z}/p^{t}\mathbb{Z}),\qquad\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\mapsto\log(a).$

Since it is trivial on parabolic elements, it can be viewed as an element of
$H^{1}_{\mathrm{B}}(X_{0}(N),\mathbb{Z}/p^{t}\mathbb{Z})^{+}$. Let
$\kappa_{1}^{+}\in\mathbb{H}^{+}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$ be the
class obtained by choosing a preimage of $\bar{\kappa}_{1}^{+}$ under $i^{*}$,
in the exact sequence obtained from (82) by replacing $Z$ with
$(\mathbb{Z}/p^{t}\mathbb{Z})$, and projecting it to ${\mathbb{H}}^{+}$. This
class depends on the choice of preimage, but only up to the addition of a
multiple of $\kappa_{0}^{+}$. Furthermore, it is annihilated by $I_{\rm
Eis}^{2}$, since $\bar{\kappa}_{1}^{+}$ is annihilated by $I_{\rm Eis}$, and
therefore, for all rational primes $\ell\neq N$, the class
$(T_{\ell}-(\ell+1))\kappa_{1}^{+}$ is a multiple of the boundary symbol
$\kappa_{0}^{+}$.

######  Theorem 4.8.

The class $\kappa_{1}^{+}$ is the higher Eisenstein element in
$\mathbb{H}^{+}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$ attached to
$\kappa_{0}^{+}$.

###### Proof.

The modular symbol attached to $\kappa_{1}^{+}$ admits an explicit description
when restricted to $\Gamma_{0}(N)0\times\Gamma_{0}(N)0$. Namely, if $r/s$ and
$t/u$ (viewed as fractions in lowest terms, with the convention that
$\infty=1/0$, so that, in particular, $s$ and $u$ belong to
$(\mathbb{Z}/N\mathbb{Z})^{\times}$) are elements of this
$\Gamma_{0}(N)$-orbit, we have

$\kappa_{1}^{+}(\\{r/s,t/u\\})=\log(s/u).$

This fact is proved by observing that the matrix

$\gamma:=\left(\begin{array}[]{cc}u^{\prime}&t\\\
\ast&u\end{array}\right)\left(\begin{array}[]{cc}s&-r\\\
\ast&s^{\prime}\end{array}\right)\in\Gamma_{0}(N),\qquad uu^{\prime}\equiv
ss^{\prime}\equiv 1\pmod{N}$

sends $r/s$ to $t/u$, and hence
$\kappa_{1}^{+}(\\{r/s,t/u\\})=\bar{\kappa}_{1}^{+}(\gamma)=\log(su^{\prime})$.
To calculate the constant of proportionality relating
$(T_{\ell}-(\ell+1))\kappa_{1}^{+}$ and $\kappa_{0}^{+}$, we exploit the usual
formula for the action of the Hecke operators on modular symbols (cf. [Maz77,
Prop 18.9]):

$\displaystyle(T_{\ell}-(\ell+1))\kappa_{1}^{+}(\\{0,\infty\\})$
$\displaystyle=$
$\displaystyle\kappa_{1}^{+}\left(\\{0,\infty\\}+\sum_{i=0}^{\ell-1}\\{i/\ell,\infty\\}-(\ell+1)\\{0,\infty\\}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{\ell-1}\kappa_{1}^{+}(\\{i/\ell,0\\})=(\ell-1)\log(\ell)$
$\displaystyle=$
$\displaystyle(\ell-1)\log(\ell)\cdot\kappa_{0}^{+}(\\{0,\infty\\}).$

The result follows. ∎

###  4.4. The Betti cohomology of the open modular curve

Consider now the case where

${\mathbb{X}}={\mathbb{H}}^{-}=H^{1}_{\mathrm{B}}(Y_{0}(N),Z)^{-}_{\mathfrak{m}}.$

The exact sequence

$0\longrightarrow
H^{1}_{\mathrm{B}}(X_{0}(N),Z)^{-}{\longrightarrow}H^{1}_{\mathrm{B}}(Y_{0}(N),\mathbb{Z})^{-}\longrightarrow
Z\longrightarrow 0$

produces an explicit rank one quotient of
$H^{1}_{\mathrm{B}}(Y_{0}(N),\mathbb{Z})^{-}$ which is Eisenstein. The
Eisenstein element $\kappa_{0}^{-}$ in ${\mathbb{H}}^{-}$ is described by the
Dedekind-Rademacher homomorphism on $\Gamma_{0}(N)$ described in [Maz79,
§II.2]:

$\kappa_{0}^{-}(\gamma)=\frac{1}{2\pi i}\left(\log(\Delta_{N})(\gamma
z)-\log(\Delta_{N})(z)\right),\qquad\Delta_{N}(z):=\Delta(Nz)/\Delta(z),$

which encodes the periods of the modular unit
$\Delta_{N}\in{\mathcal{O}}_{Y_{0}(N)}^{\times}$. It is given by the formula

$\kappa_{0}^{-}\left(\begin{array}[]{cc}a&b\\\
Nc&d\end{array}\right)=\begin{cases}(N-1)b/d&\mbox{ if }c=0;\\\
\frac{(N-1)(a+d)}{cN}+12{\rm sign}(c)D^{N}\left(\frac{a}{N|c|}\right)&\mbox{
if }c\neq 0,\end{cases}$

where ${\mathbf{D}}^{N}(x)={\mathbf{D}}(x)-{\mathbf{D}}(Nx)$ and
${\mathbf{D}}$ is the Dedekind sum

${\mathbf{D}}(a/m)=\sum_{j=1}^{m-1}B_{1}(j/m)B_{1}(aj/m),\qquad\mbox{ for
}m>0,\quad\gcd(a,m)=1.$

The homomorphism $\kappa_{0}^{-}$ can also be written as

(84) $\kappa_{0}^{-}\left(\begin{array}[]{cc}a&b\\\
Nc&d\end{array}\right)=\varphi\left(\begin{array}[]{cc}a&b\\\
Nc&d\end{array}\right)-\varphi\left(\begin{array}[]{cc}a&Nb\\\
c&d\end{array}\right),$

where $\varphi:{\mathrm{SL}}_{2}(\mathbb{Z})\longrightarrow\mathbb{Z}$ is the
Rademacher $\varphi$-function given by

(85) $\varphi\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)=\begin{cases}-b/d&\mbox{ if }c=0;\\\
\frac{-(a+d)}{c}+12{\rm sign}(c)\
{\mathbf{D}}\left(\frac{a}{|c|}\right)&\mbox{ if }c\neq 0.\end{cases}$

In [Lec], a formula for the higher Eisenstein element attached to
$\kappa_{0}^{-}$ is given, which we omit because it shall not be needed in
this work.

###  4.5. The dual of the modular forms

This section considers the case where ${\mathbb{X}}:={\mathbb{M}}^{*}$ is the
completion of

$M_{2}(N)^{\vee}=\hom(M_{2}(N),Z)$

at the $p$-Eisenstein ideal. It is a free ${\mathbb{T}}$-module of rank one,
and is also equipped with an Eisenstein element ${\mathfrak{S}}_{0}$ defined
by

${\mathfrak{S}}_{0}(f)=a_{0}(f),$

where $a_{0}(f)$ denotes the constant term of the modular form $f$ at the cusp
$\infty\in X_{0}(N)$. Let ${\mathfrak{S}}_{1}$ denote the higher Eisenstein
element in
$\bar{\mathbb{M}}^{*}:={\mathbb{M}}^{*}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$
attached to ${\mathfrak{S}}_{0}$. It turns out to be related to the Shimura
class ${\mathfrak{S}}$ described in the introduction.

More precisely, the inclusion $S_{2}(N)\hookrightarrow M_{2}(N)$ induces a
surjection ${\mathbb{M}}^{*}\rightarrow{\mathbb{S}}^{*}$. Fix any lift of
${\mathfrak{S}}$ to
$\bar{\mathbb{M}}^{*}={\mathbb{M}}^{*}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$ via
this surjection, denoted ${\mathfrak{S}}_{1}$. Note that ${\mathfrak{S}}_{1}$
is not completely well-defined, but that any two choices of lift differ by a
multiple of ${\mathfrak{S}}_{0}$ (mod $p^{t}$).

######  Theorem 4.9.

The class ${\mathfrak{S}}_{1}$ is the Higher Eisenstein element in
${\mathbb{M}}^{*}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$ attached to the
Eisenstein class ${\mathfrak{S}}_{0}$.

###### Proof.

The class ${\mathfrak{S}}_{1}$ arises from the $\bar{\kappa}_{1}^{+}$
described in the discussion preceeding Theorem 4.8 by means of the “étale to
coherent” morphism $H^{1}_{\rm
et}(X_{0}(N),\mathbb{Z}/p^{t}\mathbb{Z})\longrightarrow
H^{1}(X_{0}(N)_{/\mathbb{Z}/p^{t}\mathbb{Z}},{\mathbb{G}}_{a})$.

For reasons that will become clear below, instead of working with $X_{0}(N)$
over the spectrum of $\mathbb{Z}_{p}$, we will use instead an unramified
extension $W$ of $\mathbb{Z}_{p}$ containing the $N$th roots of unity. Clearly
it is enough to prove the claimed statement in
${\mathbb{M}}^{*}\otimes(W/p^{t}W)$ instead of
${\mathbb{M}}^{*}\otimes\mathbb{Z}/p^{t}$ since
$\mathbb{Z}/p^{t}\hookrightarrow W/p^{t}W$.

Let $\iota:\mathrm{cusps}\hookrightarrow X_{0}(N)$, be the inclusion of the
cuspidal divisor, a relative divisor over $Z$. Let $j:Y_{0}(N)\rightarrow
X_{0}(N)$ be the complementary open immersion. Now, there are compatible short
exact sequences of étale sheaves on $X_{0}(N)_{W/p^{t}}$, the base change of
$X_{0}(N)$ along $Z\rightarrow W/p^{t}$:

(86)
$\textstyle{j_{!}(\mathbb{Z}/p^{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(-\mathrm{cusps})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{Z}/p^{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{*}(\mathbb{Z}/p^{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathrm{cusps}}}$

Note that we are dealing here with étale sheaves whose order is not prime to
the residual degrees, but all we are using is the existence of this diagram.
Taking cohomology now gives the following commutative diagram which is
compatible with Hecke operators:

(87) $\textstyle{H^{0}_{\rm et}({\rm
cusps}_{W};\mathbb{Z}/p^{t}\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathrm{cusps}_{W/p^{t}},\mathcal{O})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\rm
et}(X_{0}(N)_{W},{\rm
cusps}_{W};\mathbb{Z}/p^{t}\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(X_{0}(N)_{W/p^{t}},\mathcal{O}(-\mathrm{cusps}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}(M_{2}(N),W/p^{t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\rm
et}(X_{0}(N)_{W},\mathbb{Z}/p^{t}\mathbb{Z})^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(X_{0}(N)_{W/p^{t}},{\mathcal{O}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}(S_{2}(N),W/p^{t}).}$

Here the groups in the middle column are Zariski cohomology groups; the map
from left to middle column arises from, first of all, restricting to
$W/p^{t}$, then using (86) and the fact that coherent sheaves have the same
cohomology in Zariski and étale topology. The zero superscript in the bottom
left of (87) refers to classes that are trivial when pulled back to the cusps.
The maps from middle to right are induced by the Serre duality pairings as in
(11).

The Shimura class $\mathfrak{S}\in
H^{1}_{\operatorname{et}}(X_{0}(N)_{\mathbb{Z}_{p}},\mathbb{Z}/p^{t})$ gives
rise to a class in the group
$H^{1}_{\operatorname{et}}(X_{0}(N)_{W},\mathbb{Z}/p^{t})^{(0)}$ in the lower
left of (87) (which we also denote by $\mathfrak{S}$), i.e. $\mathfrak{S}$
becomes trivial when pulled back to the cusps – because the cusps are defined
over $W$. Fix a lift

$\tilde{\mathfrak{S}}\in H^{1}_{\rm et}(X_{0}(N)_{W},{\rm
cusps}_{W};\mathbb{Z}/p^{t}\mathbb{Z})$

to the middle left group in (87). Now this left hand term can be compared with
(82) via restriction to the geometric generic fiber, i.e., the fiber over
$\overline{\mathbb{Q}}_{p}$, and it follows from Theorem 4.8 that

$(T_{\ell}-\ell-1)\tilde{\mathfrak{S}}=(\ell-1)\log(\ell)\mathfrak{S}$

holds after restriction to this geometric generic fiber.

We claim that “restriction to the geometric generic fiber” is injective on the
group
$H^{1}_{\operatorname{et}}(X_{0}(N)_{W},\mbox{cusps}_{W};\mathbb{Z}/p^{t})$.
To see this, let $E=W\otimes\mathbb{Q}_{p}$ be the quotient field of $W$. In
view of the diagram (87), it is enough to check that the kernel of the map

$q:H^{1}_{\operatorname{et}}(X_{0}(N)_{W},\mathbb{Z}/p^{t})\rightarrow
H^{1}_{\operatorname{et}}(X_{0}(N)_{\overline{\mathbb{Q}}_{p}},\mathbb{Z}/p^{t})$

is precisely the image of
$H^{1}_{\operatorname{et}}(\operatorname{Spec}W,\mathbb{Z}/p^{t})$ on the
left.

A class in the kernel of $q$ amounts to an étale $\mathbb{Z}/p^{t}$-cover of
$X_{0}(N)_{W}$ which becomes trivial on the geometric generic fiber. This
cover is uniquely determined by its restriction to $X_{0}(N)_{E}$ (see [SGA1,
Théorème 3.8, Exposé X]) where it becomes trivial on passage to a finite field
extension of $E$, i.e. the cover on $X_{0}(N)_{E}$ necessarily arises from a
character
$\mathrm{Gal\,}(\overline{\mathbb{Q}}_{p}/E)\rightarrow\mathbb{Z}/p^{t}$. For
such a cover to extend over $X_{0}(N)_{W}$ the character $\chi$ must be
unramified. (For instance, this can be seen by restricting to the cuspidal
sections.) This implies the claim regarding $\mathrm{ker}(q)$ and concludes
the proof. ∎

######  Remark 4.10.

Theorem 4.9 implies Merel’s theorem that
$\langle{\mathfrak{S}}_{1},E_{2}^{(N)}\rangle={\mathcal{M}}$, where
${\mathcal{M}}$ is the Merel constant of (79), since, letting $E^{\prime}$ be
the mod $p^{t}$ modular form defined in Proposition 4.1,
$\langle{\mathfrak{S}}_{1},E_{2}^{(N)}\rangle=\langle{\mathfrak{S}}_{0},E^{\prime}\rangle=a_{0}(E^{\prime})={\mathcal{M}}.$

###  4.6. Supersingular divisors and modular units

Recall from §2.2 the module $\mathrm{Div}(\mathcal{E})$ of $\mathbb{Z}$-linear
combinations of isomorphism classes of supersingular elliptic curves over
$\overline{\mathbb{F}}_{N}$.

The Jacquet-Langlands correspondence shows that
$\mathrm{Div}(\mathcal{E})\otimes\mathbb{C}$ is abstractly isomorphic to
$M_{2}(N;\mathbb{C})$ as a module over the ring of Hecke operators, and in
particular the Hecke ring for $\mathrm{Div}(\mathcal{E})$ can be identified
with $\mathbb{T}(N)$. In this section we consider the case where
${\mathbb{X}}:={\mathbb{D}}$ is the $p$-Eisenstein completion of
$\mathrm{Div}(\mathcal{E})$.

The vector (in the notation of §2.2)

(88) $\Sigma_{0}:=\sum_{i=1}^{n}\frac{e_{i}}{w_{i}}\in{\mathbb{D}}$

satisfies $T_{\ell}\Sigma_{0}=(\ell+1)\Sigma_{0}$ for all $\ell\neq N$, and is
thus an Eisenstein element in ${\mathbb{D}}$.

Let $\Sigma_{1}\in{\mathbb{D}}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$ denote the
Higher Eisenstein element associated to $\Sigma_{0}$, as specified in
Definition 4.6. The main goal of this section is to give an explicit
construction of $\Sigma_{1}$ in terms of the restrictions of certain modular
units to the supersingular locus. This construction is inspired from [Lec] and
involves the Eisenstein series $E_{N+1}$ of weight $N+1$, and the cusp form
$\Delta$ of weight $12$, viewed as modular forms mod $N$ of level $1$.

Let ${\mathcal{O}}_{N}$ denote the ring of (meromorphic) modular functions on
the modular curve of level one over
$\operatorname{Spec}(\mathbb{Z}/N\mathbb{Z})$ that are regular at its
supersingular points. Since $p$ is odd and $p\nmid N+1$, the discrete
logarithm $\log$ extends uniquely to the multiplicative group
$\mathbb{F}_{N^{2}}^{\times}$, and can therefore be used to define a
homomorphism

(89)
$\operatorname{Log}:{\mathcal{O}}_{N}^{\times}\longrightarrow\mathrm{Div}(\mathcal{E})\otimes(\mathbb{Z}/p^{t}\mathbb{Z}),\qquad\operatorname{Log}(U):=\sum_{i=1}^{n}\log(U(e_{i}))\cdot\frac{e_{i}}{w_{i}}.$

It shall be useful to introduce multiplicative Hecke operators acting on the
multiplicative monoid in the graded ring of modular forms mod $N$. To describe
these operators, we shall adopt Katz’s point of view to describe modular forms
over a ring. Recall that a Katz test object over
$\mathbb{F}_{N}=\mathbb{Z}/N\mathbb{Z}$ is a pair $(A,\omega)_{/R}$, where

  1. (i)

$A$ is an elliptic curve over a $\mathbb{F}_{N}$-algebra $R$;

  2. (ii)

$\omega$ in an $R$-module generator of $H^{0}(A,\Omega^{1}_{A})$.

A weakly holomorphic modular form of weight $k$ and level $1$ over
$\mathbb{F}_{N}$ is a rule $f$ which to any such test object associates an
invariant $f(A,\omega)\in R$, satisfying

  1. (1)

$f(A,\omega)$ depends only on the $R$-isomorphism class of $(A,\omega)$;

  2. (2)

$f$ commutes with base change with respect to any homomorphism $R\rightarrow
R^{\prime}$ of $\mathbb{F}_{N}$-algebras, in the obvious sense;

  3. (3)

$f(A,u\omega)=u^{-k}f(A,\omega)$, for any $u\in R^{\times}$.

Let $(A_{q},\omega_{\rm can})$ denote the “Tate test object” over
$\mathbb{F}_{N}((q))$, whose points over this local field are identified with
$\mathbb{F}_{N}((q))^{\times}/q^{\mathbb{Z}}$, equipped with its canonical
differential $\omega_{\rm can}=dt/t$. If $f(A_{q},\omega_{\rm can})$ lies in
$\mathbb{F}_{N}[[q]]$ (resp. $q\mathbb{F}_{N}[[q]]$), then $f$ is called a
modular form (resp. a cusp form). The space of modular forms and cusp forms of
weight $k$ and level $1$ over $\mathbb{F}_{N}$ shall simply be denoted $M_{k}$
and $S_{k}$ respectively.

Let $\ell\neq N$ be a prime. The multiplicative Hecke operator

$T_{\ell}^{\times}:M_{k}\longrightarrow M_{k(\ell+1)}$

is defined by setting

(90)
$(T_{\ell}^{\times}f)(A,\omega)=\prod_{\varphi}f(A^{\prime},\omega^{\prime}),$

where the product is taken over the distinct isogenies $\varphi:A\rightarrow
A^{\prime}$ of degree $\ell$, with $\omega^{\prime}$ determined by
$\omega:=\varphi^{*}\omega^{\prime}$. Up to language this is already in [Hur].
One readily checks that $T_{\ell}^{\times}$ maps $M_{k}$ to $M_{(\ell+1)k}$,
as claimed. Of course, $T_{\ell}^{\times}$ is not additive but it is
compatible with multiplication on the graded ring of modular forms over
$\mathbb{F}_{N}$:

$T_{\ell}^{\times}(fg)=T_{\ell}^{\times}(f)T_{\ell}^{\times}(g).$

In particular, it induces homomorphisms
$T_{\ell}^{\times}:{\mathcal{O}}_{N}^{\times}\longrightarrow{\mathcal{O}}_{N}^{\times}$
for which the diagram

(91)
$\textstyle{{\mathcal{O}}_{N}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T_{\ell}^{\times}}$$\scriptstyle{\operatorname{Log}}$$\textstyle{{\mathcal{O}}_{N}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Log}}$$\textstyle{\mathrm{Div}(\mathcal{E})\otimes(\mathbb{Z}/p^{t}\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T_{\ell}}$$\textstyle{\mathrm{Div}(\mathcal{E})\otimes(\mathbb{Z}/p^{t}\mathbb{Z})}$

commutes.

Consider the meromorphic modular function

(92) $\Sigma^{\times}:=\frac{E_{N+1}^{12}}{\Delta^{N+1}}$

of level one. By a result of Katz ([Kat, Theorem 3.1]), $E_{N+1}$ has no
common zero with the Hasse invariant. Since the Hasse invariant has simple
zeroes at the supersingular points, it follows that $\Sigma^{\times}$ belongs
to ${\mathcal{O}}_{N}^{\times}$ and therefore that the vector

(93)
$\Sigma_{1}:=\frac{1}{12}\operatorname{Log}(\Sigma^{\times})\in(\mathbb{Z}/p^{t}\mathbb{Z})\otimes\mathrm{Div}(\mathcal{E})$

is well-defined. Note that the class of $\Sigma_{1}$ mod
$\mathbb{Z}/p^{t}\mathbb{Z}\cdot\Sigma_{0}$ does not depend on the way one
normalizes the constant term of $E_{N+1}$.

######  Theorem 4.11.

For all primes $\ell\neq N$,

$(T_{\ell}-(\ell+1))\Sigma_{1}=(\ell-1)\log(\ell)\Sigma_{0},$

and $\Sigma_{1}$ is therefore equal to the higher Eisenstein element attached
to $\Sigma_{0}\in{\mathbb{D}}$.

###### Proof.

While the Eisenstein series $E_{N+1}$ presumably exhibits a complicated
behavior under the multiplicative Hecke operators, a result of G. Robert
([Rob, Théorème B]) asserts that if $(A,\omega)$ and
$(A^{\prime},\omega^{\prime})$ are marked supersingular elliptic curves and
$\varphi:A\longrightarrow A^{\prime}$ is an isogeny of degree $\ell$
satisfying $\varphi^{*}(\omega^{\prime})=\omega$, then

(94) $E_{N+1}(A^{\prime},\omega^{\prime})=\ell E_{N+1}(A,\omega),\qquad\mbox{
for all }A\in{\mathcal{E}}.$

It follows that

(95) $T^{\times}_{\ell}E_{N+1}=\ell^{\ell+1}E^{\ell+1}_{N+1}.$

In addition, for every prime $\ell\neq N$,

(96) $T_{\ell}^{\times}\Delta={\ell}^{12}\Delta^{\ell+1}.$

This follows by noting that

$T_{\ell}^{\times}(\Delta)(E_{q},\omega_{\rm
can})=\Delta(E_{q^{\ell}},\ell^{-1}\omega_{\rm
can})\times\prod_{\zeta\in\mu_{\ell}}\Delta(E_{\zeta q^{1/\ell}},\omega_{\rm
can})=\ell^{12}\Delta(q)^{\ell+1}.$

Combining (95) and (96), we obtain

$T_{\ell}^{\times}(\Sigma^{\times})=\ell^{12(\ell-1)}(\Sigma^{\times})^{\ell+1}.$

It follows that

$T_{\ell}(\operatorname{Log}(\Sigma^{\times}))=12(\ell-1)\log(\ell)\Sigma_{0}+(\ell+1)\operatorname{Log}(\Sigma^{\times}),$

as claimed. ∎

######  Example 4.12.

Take $N=23$ and $p=11$. The supersingular $j$-invariants mod $N$ are
$\\{1728,19,0\\}$ and we have $\Sigma_{0}=(6,1,4)$ with respect to this basis.
Normalize
$\log:(\mathbb{Z}/N\mathbb{Z})^{\times}\longrightarrow\mathbb{Z}/p\mathbb{Z}$
by setting $\log(5)=1$. Vector
$\Sigma_{1}=\frac{1}{12}\operatorname{Log}(\Sigma^{*})=\operatorname{Log}(\frac{E_{24}}{\Delta^{2}})\in\mathbb{F}_{p}e_{1728}\oplus\mathbb{F}_{p}e_{19}\oplus\mathbb{F}_{p}e_{0}$
is then computed to be

$\Sigma_{1}=(-1,-1,-3).$

This can readily be checked for instance by means of the identity

$\frac{E_{24}}{\Delta^{2}}=(aj^{2}+b(j^{2}-1728j)+c(j-1728)^{2})/d,$

where

$a=49679091,\ \ b=176400000,\ \ c=10285000,\ \ d=236364091,$

which follows by comparing the $q$-expansions of $E_{4}$, $E_{6}$ and
$E_{24}$. A computation with Brandt matrices allows to verify numerically the
identity of Theorem 4.11.

The description of $\Sigma_{1}$ given in Theorem 4.11 makes it possible to
relate some of its pullbacks to modular units. More precisely, let $q\neq N$
be an auxiliary prime, let $\mathcal{E}^{(q)}$ denote the set of of
supersingular points of the modular curve $X_{0}(q)$ in characteristic $N$
(i.e. over $\overline{\mathbb{F}_{N}}$) and let
$\mathrm{Div}(\mathcal{E}^{(q)})$ and ${\mathbb{D}}^{(q)}$ denote
(respectively) the space of $Z$\- and $(\mathbb{Z}/p^{t}\mathbb{Z})$-linear
combinations of elements of $\mathcal{E}^{(q)}$. Note that in carrying over
constructions from $\mathcal{E}$ to $\mathcal{E}^{(q)}$ we must take account
of the fact that the weights $w_{x}$ for $x\in\mathcal{E}^{(q)}$ take into
account the level structure and thus will not in general coincide with the
weight $w_{\bar{x}}$ of the image $\bar{x}\in\mathcal{E}$.

The two degeneracy maps

(97) $\pi_{1},\pi_{2}:X_{0}(q)\longrightarrow
X(1),\qquad\pi_{1}(A,C)=A,\quad\pi_{2}(A)=A/C$

induce maps $\pi_{1},\pi_{2}:\mathcal{E}^{(q)}\longrightarrow\mathcal{E}$, and
correspondingly push-forward maps

$\pi_{1*},\pi_{2*}:\mathrm{Div}(\mathcal{E}^{(q)})\longrightarrow\mathrm{Div}(\mathcal{E}).$

The dual of these maps are pullback maps $\pi_{1}^{*},\pi_{2}^{*}$, defined so
as to satisfy

$\langle\pi_{j}^{*}a,b\rangle_{q}=\langle a,\pi_{j\ast}b\rangle,\qquad\mbox{
for all }a\in\mathrm{Div}(\mathcal{E}),\ b\in\mathrm{Div}(\mathcal{E}^{(q)}).$

where $\langle-,-\rangle_{q}$ and $\langle-,-\rangle$ are the natural pairings
(cf. (14)). In particular we get

$\pi_{1}^{\ast},\pi_{2}^{\ast}:{\mathbb{D}}\longrightarrow{\mathbb{D}}^{(q)},$

which is, now, compatible with the corresponding pullback of functions on the
ambient modular curves by means of the map (89).

Just as in (89) we have a homomorphism

(98)
$\operatorname{Log}:{\mathcal{O}}_{q,N}^{\times}\longrightarrow\mathrm{Div}(\mathcal{E}^{(q)})\otimes(\mathbb{Z}/p^{t}\mathbb{Z}),$

where now ${\mathcal{O}}_{q,N}^{\times}$ denotes the multiplicative group of
(meromorphic) modular functions on $X_{0}(q)_{\mathbb{F}_{N}}$ regular at the
supersingular points. This applies to the case where
$f=\pi_{1}^{*}(\Delta)/\pi_{2}^{*}(\Delta)=\Delta(z)/\Delta(qz)$, which is a
modular unit of level $q$.

######  Theorem 4.13.

For any auxiliary prime $q\neq N$, denote by

(99) $u_{q}:=\Delta(z)/\Delta(qz)$

the modular unit of level $q$, considered as an element of
$\mathcal{O}_{q,N}^{\times}$ (see (98)). Then

$\pi_{1}^{*}(\Sigma_{1})-\pi_{2}^{*}(\Sigma_{1})=-\frac{1}{6}\operatorname{Log}(u_{q})\pmod{\Sigma_{0}^{(q)}},$

where $\Sigma_{0}^{(q)}=\pi_{1}^{*}(\Sigma_{0})=\pi_{2}^{*}(\Sigma_{0})$ is an
Eisenstein eigenvector on ${\mathbb{D}}^{(q)}$.

The use of the auxiliary prime $q$ simplifies the situation: the map
$(\pi_{1}^{*}-\pi_{2}^{*})$ kills $\Sigma_{0}$; thus
$(\pi_{1}^{*}-\pi_{2}^{*})\Sigma_{1}$ is independent of the choice of
$\Sigma_{1}$ and is strictly Eisenstein, rather than higher Eisenstein. In
fact, in the case $q=2$, this general idea appears in the work of Lecouturier;
the role of the modular unit (99) is replaced in his work by the
$\lambda$-invariant, cf. [Lec, Prop 3.25].

###### Proof.

Equation (94) shows that $\pi_{1}^{*}(E_{N+1})/\pi_{2}^{*}(E_{N+1})$ is
constant on $\mathcal{E}^{(q)}$, and hence

$\operatorname{Log}(\pi_{1}^{*}(E_{N+1})/\pi_{2}^{*}(E_{N+1}))\sim\Sigma_{0}^{(q)},$

where $\sim$ indicates that the two vectors are proportional to each other. It
follows from the definition (92) of $\Sigma^{\times}$ and $N\equiv 1$ modulo
$p^{t}$ that

$\operatorname{Log}(\pi_{1}^{*}(\Sigma^{\times})/\pi_{2}^{*}(\Sigma^{\times}))=2\operatorname{Log}(\Delta(qz)/\Delta(z))\pmod{\Sigma_{0}^{(q)}},$

and the claim follows from the definition (93) of $\Sigma_{1}$. ∎

###  4.7. Tensor products

Let $M$ and $N$ be any two free modules of rank one over ${\mathbb{T}}$. The
tensor product $M\otimes_{{\mathbb{T}}}N$ is still free of rank one. If
$m_{0}$ and $m_{1}$ (resp. $n_{0}$ and $n_{1}$) are Eisenstein and higher
Eisenstein elements in $M$ (resp. $N$), there seems to be no simple expression
for the higher Eisenstein element in $M\otimes_{{\mathbb{T}}}N$ in terms of
these elements. (For instance, the vector $m_{0}\otimes n_{0}$ fails to
generate the Eisenstein subspace in $M\otimes_{{\mathbb{T}}}N$ in general.)

Since ${\mathbb{T}}$ is Gorenstein, the $\mathbb{Z}_{p}$-dual
$M^{*}=\operatorname{Hom}(M,\mathbb{Z}_{p})$ is again a free
${\mathbb{T}}$-module of rank $1$ and hence it makes sense to consider
(higher) Eisenstein elements on it.

######  Proposition 4.14.

If $m_{0}^{*}$ and $m_{1}^{*}$ (resp. $n_{0}^{*}$ and $n_{1}^{*}$) are the
Eisenstein and higher Eisenstein elements of $M^{*}$ and $N^{*}$ respectively,
then

  1. (1)

The element $m_{0}^{*}\otimes n_{0}^{*}$ is an Eisenstein element of
$(M\otimes_{{\mathbb{T}}}N)^{*}$.

  2. (2)

The element $m_{0}^{*}\otimes n_{1}^{*}+m_{1}^{*}\otimes n_{0}^{*}$ is the
higher Eisenstein element of $(M\otimes_{{\mathbb{T}}}N)^{*}/p^{t}$ associated
to $m_{0}^{*}\otimes n_{0}^{*}$.

Note that there is a natural module homomorphism
$M^{*}\otimes_{\mathbb{Z}_{p}}N^{*}\longrightarrow(M\otimes_{\mathbb{Z}_{p}}N)^{*}$
sending $m^{*}\otimes n^{*}$ to the functional defined by $(m^{*}\otimes
n^{*})(a\otimes b)=m^{*}(a)n^{*}(b)$. The meaning of the first statement above
is, then, that the displayed expressions in fact belong to
$(M\otimes_{{\mathbb{T}}}N)^{*}\subset(M\otimes_{\mathbb{Z}_{p}}N)^{*}$, and
moreover are Eisenstein/higher Eisenstein considered in the former group.
Similarly for the second statement (see below for details).

###### Proof.

As for (1), we first check that $m_{0}^{*}\otimes n_{0}^{*}$ belongs to the
submodule $(M\otimes_{{\mathbb{T}}}N)^{*}$ of
$(M\otimes_{\mathbb{Z}_{p}}N)^{*}$. The kernel of the surjection
$M\otimes_{\mathbb{Z}_{p}}N\longrightarrow M\otimes_{{\mathbb{T}}}N$ is
generated by $(T\otimes 1-1\otimes T)(M\otimes N)$ for $T\in{\mathbb{T}}$.
Hence it suffices to verify that $(T\otimes 1-1\otimes T)(m_{0}^{*}\otimes
n_{0}^{*})=0$ for all $T\in{\mathbb{T}}$, and this follows because
${\mathbb{T}}$ is a simple algebra over $\mathbb{Z}_{p}$, generated by an
element of $I_{\rm Eis}$. Now (1) follows, as it is obvious that
$m_{0}^{*}\otimes n_{0}^{*}$ is a generator of the $\mathbb{Z}_{p}$-module
$(M\otimes_{{\mathbb{T}}}N)^{*}[I_{\rm Eis}]$.

As for (2), write $\bar{M}:=M/p^{t}M$ and $\bar{N}:=N/p^{t}N$. Note that
$M^{*}/p^{t}\simeq\bar{M}^{*}$ where, on the right, $*$ denotes
$\operatorname{Hom}(-,\mathbb{Z}/p^{t})$. The expression $m_{0}^{*}\otimes
n_{1}^{*}+m_{1}^{*}\otimes n_{0}^{*}$ lies in

$(M^{*}\otimes_{\mathbb{Z}_{p}}N^{*})/p^{t}=\bar{M}^{*}\otimes_{\mathbb{Z}/p^{t}}\bar{N}^{*}.$

We argue as before that $m_{0}^{*}\otimes n_{1}^{*}+m_{1}^{*}\otimes
n_{0}^{*}$ lies in $(\bar{M}\otimes_{{\mathbb{T}}}\bar{N})^{*}$. The
${\mathbb{T}}$-module structure is given by applying $T\in\mathbb{T}$ to
either the first or second argument. Applying $T_{\ell}-\ell-1$ to the first
argument gives:

$\displaystyle(T_{\ell}-\ell-1)\left[m_{0}^{*}\otimes
n_{1}^{*}+m_{1}^{*}\otimes n_{0}^{*}\right]$ $\displaystyle=$
$\displaystyle(T_{\ell}-\ell-1)m_{0}^{*}\otimes
n_{1}^{*}+(T_{\ell}-\ell-1)m_{1}^{*}\otimes n_{0}^{*}$ $\displaystyle=$
$\displaystyle(\ell-1)\log(\ell)m_{0}^{*}\otimes n_{0}^{*},$

as desired.

∎

##  5\. Proof of the main theorem

This chapter proves Conjecture 1.1 for dihedral modular forms.

###  5.1. Elliptic units

We put ourselves in the situation of §2.1 and §2.2 with
$\psi_{2}=\psi_{1}^{-1}$ and $\psi_{2}\neq\psi_{1}$. In particular: $K$ is an
imaginary quadratic field of odd discriminant $D<0$ and ring of integers
$\mathfrak{o}$; the level $N$ is prime, $p>3$ a prime dividing $N-1$, and
$\psi_{1}:\mathcal{C}\longrightarrow L^{\times}$ a class group character into
some cyclotomic field $L$. Let $R$ be the ring of integers of $L$.

Finally put

(100) $\psi=\psi_{1}/\psi_{1}^{\prime}=\psi_{1}^{2}:\mathcal{C}\rightarrow
L^{\times}.$

When $N$ splits in $K$, Conjecture 1.1 reduces to the equality $0=0$, as
explained in §1.3 of the introduction. Hence it shall be assumed throughout
that $N$ is inert in $K$.

Let $\mathbb{F}_{N^{2}}$ be the quotient $\mathfrak{o}/N$, a finite field of
size $N^{2}$, and fix an algebraic closure $\overline{\mathbb{F}}_{N}$ of
$\mathbb{F}_{N^{2}}$.

In the current section only $\psi$ will be relevant (and the discussion would
be valid for an arbitrary character $\psi$, not just one of the form (100)).
We will construct an elliptic unit $u_{\psi}$ associated to $\psi$ and explain
how its discrete logarithm at various primes is related to the geometry of
supersingular points. We will use the setup of §2.1 regarding double coset
spaces attached to definite quaternion algebras, but will now use the
incarnation of these spaces in terms of supersingular elliptic curves.

More precisely, global class field theory identifies $\mathcal{C}$ with the
Galois group of an abelian extension $H$ of $K$: the Hilbert class field of
$K$, generated over $K$ by the $j$-invariants of elliptic curves over
$\bar{K}$ with endomorphism ring equal to $\mathfrak{o}$. The set of all such
elliptic curves up to ${\bar{K}}$-isomorphism, denoted
$\mathcal{E}_{\mathfrak{o}}$, is a principal transitive $\mathcal{C}$-set and
the choice of a base point $A\in\mathcal{E}_{\mathfrak{o}}$ identifies the two
sets

$\mathfrak{a}\in\mathcal{C}\mapsto
A_{\mathfrak{a}}\in\mathcal{E}_{\mathfrak{o}}.$

via tensoring with the inverse of $\mathfrak{a}$.

The prime $N$, which is inert in $K/\mathbb{Q}$, splits completely in $H/K$,
and the choice of a prime $\mathfrak{N}$ of $H$ above $N$ determines reduction
maps

$\iota:\mathcal{E}_{\mathfrak{o}}\longrightarrow\mathcal{E},\qquad\iota:{\mathrm{Pic}}(\mathfrak{o})\longrightarrow\mathcal{E},$

where $\mathcal{E}$ is the set of isomorphism classes of supersingular curves
over $\overline{\mathbb{F}}_{N}$. Since the end result we are proving is
independent of the choice of $\mathfrak{N}$, we can and will choose
$\mathfrak{N}$ in such a way that the reduction $\iota(A)\in\mathcal{E}$
matches with one of the basepoints for $\mathcal{E}$ chosen before (13), i.e.,
to reprise, the endomorphism ring of the reduction of $A$ at $\mathfrak{N}$
should contain an order of the form $\mathfrak{o}\oplus\mathfrak{o}j$.

The map $\iota$ coincides with the map (13) after identifying $\mathcal{E}$
with maximal orders in the associated quaternion algebras, as specified prior
to (13). As in (17), the image of $\psi$ under the pushforward map
$\iota_{\ast}:R[{\mathrm{Pic}}(\mathfrak{o})]\longrightarrow\mathrm{Div}(\mathcal{E})\otimes
R$ is denoted by $[\psi]:=\iota_{*}(\psi)\in\mathrm{Div}(\mathcal{E})\otimes
R.$

Let $q$ be an auxiliary rational prime which does not divide $DN$. A Heegner
point on $X_{0}(q)(\bar{K})$ attached to $\mathfrak{o}$ is a pair $(A,C)$
where $A$ an elliptic curve over $\bar{K}$ equipped with a cyclic subgroup
$C\subset A$ of order $q$, for which both $A$ and $A/C$ belong to
$\mathcal{E}_{\mathfrak{o}}$. The set $\mathcal{E}_{\mathfrak{o}}^{(q)}$ of
Heegner points on $X_{0}(q)(\bar{K})$ is non-empty precisely when the prime
$q\nmid D$ is split in $K/\mathbb{Q}$, i.e., when
$q=\mathfrak{q}\bar{\mathfrak{q}}$. It is then contained in $X_{0}(q)(H)$.
Just as above, the choice of a prime $\mathfrak{N}$ of ${\mathcal{O}}_{H}$
induces reduction maps
$\mathcal{E}_{\mathfrak{o}}^{(q)}\rightarrow\mathcal{E}^{(q)}$.

The set $\mathcal{E}_{\mathfrak{o}}^{(q)}$ is equipped with the two degeneracy
maps

$\pi_{1},\pi_{2}:\mathcal{E}_{\mathfrak{o}}^{(q)}\longrightarrow\mathcal{E}_{\mathfrak{o}};\qquad\pi_{1}(A,C)=A,\quad\pi_{2}(A,C)=A/C,$

obtained by restricting the corresponding degeneracy maps $X_{0}(q)\rightarrow
X(1)$. The choice of a prime divisor $\mathfrak{q}$ of $q$ determines a
section
$\eta_{\mathfrak{q}}:\mathcal{E}_{\mathfrak{o}}\longrightarrow\mathcal{E}_{\mathfrak{o}}^{(q)}$
of $\pi_{1}$ by setting

$\eta_{\mathfrak{q}}(A)=\tilde{A}:=(A,A[\mathfrak{q}]).$

Observe that the action of ${\mathrm{Pic}}(\mathfrak{o})$ on
$\mathcal{E}_{\mathfrak{o}}$ satisfies

(101)
$A_{\mathfrak{a}\mathfrak{q}}=\pi_{2}(\eta_{\mathfrak{q}}(A_{\mathfrak{a}})).$

######  Definition 5.1.

The elliptic unit attached to $\psi$ and $\mathfrak{q}$ is the element

(102) $u_{\psi,q}=\sum_{\mathfrak{a}\in{\rm
Pic}(\mathfrak{o})}u_{q}(\eta_{\mathfrak{q}}(A_{\mathfrak{a}}))\otimes\psi(\mathfrak{a})\in
H^{\times}\otimes R,$

with $u_{q}$ the modular unit defined in (99).

If $\psi$ is non-trivial, then $u_{\psi,q}$ belongs to
${\mathcal{O}}_{H}^{\times}\otimes R$ and more precisely to its
$\psi$-isotypical component, that is to say:

(103) $g\cdot u_{\psi,q}=\psi^{-1}(g)u_{\psi,q},\qquad\mbox{ for all
}g\in\mathrm{Gal}(H/K).$

(Cf. [KL, §11, Thms. 1.1. and 1.2].) Note that on the left hand side of (103),
$g$ acts on $H$ in the natural way. On the right-hand side, $\psi$ is
understood as a character of $\mathrm{Gal}(H/K)$ through the isomorphism
$\mathrm{Gal}(H/K)\simeq\mathcal{C}$ through which this Galois group acts on
$\mathcal{E}_{\mathfrak{o}}$, and $\psi^{-1}(g)\in R^{\times}$ acts by
multiplication on the second factor in the tensor product
${\mathcal{O}}_{H}^{\times}\otimes R$. If $\psi=1$ then $u_{\psi,q}$ may fail
to be a unit at the primes above $q$ but this case will not arise.

The following proposition plays a key role in the proof of Conjecture 1.1 for
CM forms described in the next section, since it is via this result that the
relevant Stark unit makes its appearance.

######  Proposition 5.2.

For all characters $\psi:\mathcal{C}\rightarrow R^{\times}$, and all split
primes $q=\mathfrak{q}\bar{\mathfrak{q}}$ as above, we have an equality in
$R/p^{t}$:

$(1-\psi(\bar{\mathfrak{q}}))\times\langle\Sigma_{1},[\psi]\rangle=-\frac{1}{6}\log(u_{\psi,\mathfrak{q}}),$

where
$\Sigma_{1}\in\mathrm{Div}(\mathcal{E})\otimes\mathbb{Z}/p^{t}\mathbb{Z}$ is
the higher Eisenstein element of Theorem 4.11, and we wrote
$\log:{\mathcal{O}}_{H}^{\times}\otimes R\,\longrightarrow\,R/p^{t}$ for the
composition of the reduction map
${\mathcal{O}}_{H}^{\times}\rightarrow({\mathcal{O}}_{H}/\mathfrak{N})^{\times}\simeq\mathbb{F}_{N^{2}}^{\times}$
with discrete logarithm fixed at the outset.555This discrete logarithm was
defined on $(\mathbb{Z}/N\mathbb{Z})^{\times}$ but uniquely extends to
$\mathbb{F}_{N^{2}}^{\times}$.

###### Proof.

Recall that $A$ is a fixed basepoint for $\mathcal{E}_{\mathfrak{o}}$ and
$[\psi]=\sum_{I\in\mathcal{C}}\psi(I)A_{I}$. We may write:

$\displaystyle(1-\psi(\bar{\mathfrak{q}}))\langle\Sigma_{1},[\psi]\rangle$
$\displaystyle=$
$\displaystyle\sum_{I\in{\mathrm{Pic}}(\mathfrak{o})}(\psi(I)-\psi(I\bar{\mathfrak{q}}))\langle\Sigma_{1},A_{I}\rangle$
$\displaystyle=$
$\displaystyle\sum_{I\in{\mathrm{Pic}}(\mathfrak{o})}\psi(I)\langle\Sigma_{1},A_{I}-A_{I\mathfrak{q}}\rangle.$

Letting ${\tilde{A}_{I}}:=\eta_{\mathfrak{q}}(A_{I})$, we have, by (101),

$A_{I}-A_{I\mathfrak{q}}=(\pi_{1}-\pi_{2})_{\ast}(\tilde{A}_{I}),$

and hence, by invoking Theorem 4.13,

$\displaystyle(1-\psi(\bar{\mathfrak{q}}))\langle\Sigma_{1},[\psi]\rangle$
$\displaystyle=$
$\displaystyle\sum_{I\in{\mathrm{Pic}}(\mathfrak{o})}\psi(I)\langle(\pi_{1}^{\ast}-\pi_{2}^{\ast})\Sigma_{1},{\tilde{A}}_{I}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{6}\cdot\sum_{I\in{\mathrm{Pic}}(\mathfrak{o})}\psi(I)\langle\operatorname{Log}(u_{q}),{\tilde{A}}_{I}\rangle,$

with the pairings the natural ones on $\mathrm{Div}(\mathcal{E}^{(q)})$. The
latter expression is equal to
$-\frac{1}{6}\langle\operatorname{Log}(u_{q}),[\psi]\rangle=-\frac{1}{6}\log(u_{\psi,\mathfrak{q}})$,
the equality taking place in $R/p^{t}$:

$\displaystyle\langle\operatorname{Log}(u_{q}),[\psi]\rangle$
$\displaystyle\stackrel{{\scriptstyle\eqref{Log-def2}}}{{=}}$
$\displaystyle\sum_{I\in\mathcal{C}}\log
u_{q}(\iota\circ\eta_{\mathfrak{q}}(A_{I}))\psi(I)$ $\displaystyle=$
$\displaystyle\log\left(\sum_{I\in\mathcal{C}}u_{q}(\iota\circ\eta_{\mathfrak{q}}(A_{I}))\otimes\psi(I)\right)$
$\displaystyle=$
$\displaystyle\log\mathrm{red}_{\mathfrak{N}}\sum_{I\in\mathcal{C}}u_{q}(\eta_{\mathfrak{q}}(A_{I}))\otimes\psi(I)=\log(u_{\psi,\mathfrak{q}}).$

∎

######  Remark 5.3.

One can replace the algebra $M_{2}(\mathbb{Q})$ in the above considerations by
a non-split, indefinite quaternion algebra $D_{M}$ over $\mathbb{Q}$, of
discriminant $M>1$ say, which is associated to a Shimura curve $X_{M}$ arising
from a co-compact subgroup of ${\mathrm{SL}}_{2}({\mathbb{R}})$. Given a prime
$N\nmid M$, the module $\mathcal{E}_{M,N}$ of supersingular points of $X_{M}$
in characteristic $N$ is identified with the space of functions on a finite
double coset space attached to the definite quaternion algebra $D_{MN}$ of
discriminant $MN$. If $\psi$ is a character of the class group of a quadratic
imaginary field $K$ in which all the primes dividing $MN$ are inert, one can
define an associated vector $[\psi]\in\mathcal{X}_{M,N}$ much as in the case
where $M=1$. The space $\mathcal{X}_{M,N}$ contains an Eisenstein eigenvector
$\Sigma_{0}$, whose value on a double coset is equal to the cardinality of its
stabiliser subgroup. Theorems 1.2 and 1.3 of [Y] show that the Hecke algebra
${\mathbb{T}}_{MN}$ acting on $\mathcal{X}_{M,N}$ is equipped with an
Eisenstein homomorphism $\tilde{\varphi}_{\rm Eis}$ as in (80) with
${\mathbb{T}}$ replaced by ${\mathbb{T}}_{MN}$, and suggest that, if $p>3$ is
a prime with $p^{t}||N-1$, the module
$\mathcal{X}_{M,N}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$ contains a generalised
Eisenstein eigenvector $\Sigma_{1}$ attached to a choice of discrete logarithm
$\log:\mathbb{F}_{N}^{\times}\longrightarrow\mathbb{Z}/p^{t}\mathbb{Z}$,
satisfying

$(T_{\ell}-(\ell+1))\Sigma_{1}=(\ell-1)\log(\ell)\Sigma_{0}.$

Does such a $\Sigma_{1}$, when it exists, satisfy an analogue of Proposition
5.2 relating $\langle\Sigma_{1},[\psi]\rangle$ to the discrete logarithm of
the elliptic unit $u_{\psi}$, which does not depend on $N$? Such a
relationship would be intriguing in light of the fact that the arithmetic
subgroup of ${\mathrm{SL}}_{2}({\mathbb{R}})$ defining $X_{M}$ has no
parabolic elements and hence there are no modular units on $X_{M}$ that could
be parlayed into a direct construction of $\Sigma_{1}$.

###  5.2. Proof of Conjecture 1.1 for definite theta series

We now restrict to the case $D$ prime; however, as we comment in the
statements, the proofs verbatim give results for $D$ odd under further
restrictions on $N$.

We let $g=\theta_{\psi_{1}}$ be the associated $\theta$ series. It is a cusp
form by virtue of the assumption that $\psi_{1}\neq\psi_{1}^{-1}$. The Galois
representation $\rho_{g}$ is the induction to $G_{\mathbb{Q}}$ of the finite
order character $\psi_{1}$. Let

$G\in M_{2}(\Gamma_{0}(N))=\mathsf{Tr}^{ND}_{N}g(z)g^{*}(Nz)$

denote the modular form defined as the trace to the space of modular forms of
weight $2$ and level $N$ of the product
$g(z)g^{*}(Nz)=\theta_{\psi_{1}}(z)\theta_{\psi_{1}^{-1}}(Nz)$.

Recall from (15) the $\Theta$-correspondence

$\Theta:\mathrm{Div}(\mathcal{E})\otimes_{\mathbb{T}(N)}\mathrm{Div}(\mathcal{E})\rightarrow
M_{2}(\Gamma_{0}(N)).$

In particular, this induces a map on localizations at the Eisenstein ideal
$\mathfrak{m}$, and it follows from [Em, Theorem 0.5] that the resulting map
is an isomorphism of free $\mathbb{T}=\mathbb{T}(N)_{\mathfrak{m}}$-modules of
rank one. Write

$\overline{\mathbb{T}}=\mathbb{T}/p^{t},\quad\bar{\mathbb{D}}=\mathrm{Div}(\mathcal{E})_{\mathfrak{m}}/p^{t},\quad\bar{\mathbb{M}}=M_{2}(\Gamma_{0}(N)_{\mathfrak{m}}/p^{t},\quad\bar{\mathbb{S}}=S_{2}(\Gamma_{0}(N))_{\mathfrak{m}}/p^{t}.$

Then, reducing $\Theta$ modulo $p^{t}$, we obtain an isomorphism

(104)
$\Theta:\bar{\mathbb{D}}\otimes_{\overline{\mathbb{T}}}\bar{\mathbb{D}}\simeq\bar{{\mathbb{M}}},$

with associated adjoint

(105)
$\Theta^{*}:\bar{{\mathbb{M}}}^{*}\simeq(\bar{\mathbb{D}}\otimes_{\overline{\mathbb{T}}}\bar{\mathbb{D}})^{*}.$

Here $*$ denotes $\operatorname{Hom}(-,\mathbb{Z}/p^{t})$.

The strategy of the proof of Conjecture 1.1, as outlined in §1.4, is to
express the inner product $\langle G,{\mathfrak{S}}\rangle$ as an inner
product on $\bar{\mathbb{D}}\otimes\bar{\mathbb{D}}$ via $\Theta$. It follows
from Theorem 2.2 that

(106) $\langle
G,\mathfrak{S}\rangle=4\cdot\langle\Theta([1]\otimes[\psi]),\mathfrak{S}\rangle=4\langle[1]\otimes[\psi],\Theta^{*}(\mathfrak{S})\rangle.$

Here we regard the equality as occurring inside $R/p^{t}$, and we regard
$\mathfrak{S}\in\bar{{\mathbb{M}}}^{*}$ and
$[1]\otimes[\psi]\in\bar{{\mathbb{D}}}\otimes_{\overline{\mathbb{T}}}\bar{{\mathbb{D}}}$.
We now need:

######  Theorem 5.4.

Let ${\mathfrak{S}}_{0}$ and ${\mathfrak{S}}_{1}\in\bar{\mathbb{M}}^{*}$
denote the Eisenstein and higher classes described in Section 4.5 and let
$\Sigma_{0}$ and $\Sigma_{1}$ denote the analogous classes in
$\bar{\mathbb{D}}$ described in Section 4.6. Then

  1. (1)

$\Theta^{*}({\mathfrak{S}}_{0})=\frac{1}{2}\Sigma_{0}\otimes\Sigma_{0}$;

  2. (2)

$\Theta^{*}({\mathfrak{S}}_{1})\equiv\frac{1}{2}(\Sigma_{1}\otimes\Sigma_{0}+\Sigma_{0}\otimes\Sigma_{1})\mbox{
modulo }\Sigma_{0}\otimes\Sigma_{0}.$

Here we used the pairing $\langle\ ,\ \rangle$ given in (14) to identify
$\bar{\mathbb{D}}\simeq(\bar{\mathbb{D}})^{*}$; we also used the inclusion
$\left(\bar{{\mathbb{D}}}\otimes_{\overline{\mathbb{T}}}{\bar{\mathbb{D}}}\right)^{*}\subset\left(\bar{{\mathbb{D}}}^{*}\otimes_{\mathbb{Z}/p^{t}}{\bar{\mathbb{D}}}^{*}\right)$
to describe elements of the left hand group, just as was done in Proposition
5.2.

###### Proof.

The first part of the theorem follows directly from the definition of $\Theta$
given in (15). The second follows from the Hecke equivariance of $\Theta^{*}$,
in light of the fact that
$\Sigma_{1}\otimes\Sigma_{0}+\Sigma_{1}\otimes\Sigma_{0}$ is the higher
Eisenstein element in
$(\bar{\mathbb{D}}\otimes_{{\mathbb{T}}}\bar{\mathbb{D}})^{*}$ attached to
$\Sigma_{0}\otimes\Sigma_{0}$, by Proposition 4.14. ∎

We now choose an auxiliary prime ideal $\mathfrak{q}$ so that
$\psi(\mathfrak{q})$ is a primitive root of unity of order equal to the order
of $\psi$.

######  Proposition 5.5.

There is an equality inside $R/p^{t}$

(107) $(1-\psi(\bar{\mathfrak{q}}))\langle
G,\mathfrak{S}\rangle=\frac{-h(\mathfrak{o})}{3}\log(u_{\psi,\mathfrak{q}}),$

where $u_{\psi,\mathfrak{q}}$ is the elliptic unit defined in (5.1), and
$h(\mathfrak{o})$ is the order of the class group $\mathcal{C}$.

Note that, for $D$ odd but not assumed prime, the same conclusion holds true
with the following caveats: we suppose not merely that $N$ is inert in
$\mathbb{Q}(\sqrt{-D})$, but that $-N$ is a square modulo $D$; and, owing to
the denominators potentially introduced in §2.6, it is only valid for $p$
sufficiently large, in the sense of Theorem 2.2.

###### Proof.

By (106) and part (2) of Theorem 5.4,

$\langle
G,\mathfrak{S}\rangle=2\langle[1]\otimes[\psi],\Sigma_{0}\otimes\Sigma_{1}+\Sigma_{1}\otimes\Sigma_{0}\rangle=2\langle\Sigma_{0},[1]\rangle\langle\Sigma_{1},[\psi]\rangle$

where we have used the fact that $\langle\Sigma_{0},[\psi]\rangle=0$ since
$\psi$ is non-trivial. Since $\langle\Sigma_{0},[1]\rangle=h(\mathfrak{o})$ by
definition, the theorem now follows from Proposition 5.2 ∎

To prove Theorem 1.2 of the introduction for CM weight one forms, it remains
to relate the right-hand side of (107) to the expression
$\operatorname{red}_{N}(u_{g})$ occuring in this theorem; this is done by the
following Lemma.

######  Lemma 5.6.

Let
$U_{g}:=({\mathcal{O}}_{H}^{\times}\otimes\operatorname{Ad}^{*}(\rho_{g})^{\circ})^{G_{\mathbb{Q}}}$.
There exists $u_{g}\in U_{g}$ with the property that, for all $N$ as above,

$\log(\operatorname{red}_{N}(u_{g}))=2\log(u_{\psi,\mathfrak{q}}).$

This lemma concludes the proof of Theorem 1.2, after multiplying equality
(107) by $\frac{-6n}{1-\psi(\bar{\mathfrak{q}})}\in R$ with $n$ the norm of
$1-\psi(\bar{\mathfrak{q}})$:

$(-6n)\langle G,\mathfrak{S}\rangle=\log(u_{g}^{\prime}),\qquad
u_{g}^{\prime}:=\frac{-h(\mathfrak{o})n}{(1-\psi(\mathfrak{q}))}\cdot u_{g}.$

where, in the last equality, we are implicitly using the $R$-module structure
on $U_{g}$ to form the product.

###### Proof.

For typographical simplicity we write just $u_{\psi}$ instead of
$u_{\psi,\mathfrak{q}}$.

Let $e_{1}$ be an eigenvector in $V_{g}$ for the action of $G_{K}$, on which
$G_{K}$ acts via the character $\psi_{1}$. Since $N$ is inert in $K$, the
associated Frobenius automorphism $\sigma_{N}\in G_{\mathbb{Q}}$ sends $e_{1}$
to a complementary vector $e_{2}=\sigma_{N}(e_{1})$, on which $G_{K}$ acts via
the character $\psi_{1}^{\prime}$. Since $\sigma_{N}$ has determinant $-1$, it
then sends $e_{2}$ to $e_{1}$. Representing elements of ${\rm Ad}(V_{g})$ as
matrices relative to the basis $(e_{1},e_{2})$, so that

$\rho_{g}(x)=\left(\begin{array}[]{cc}\psi_{1}(x)&0\\\
0&\psi_{1}(x)^{-1}\end{array}\right),\ \ \mbox{ for }x\in G_{K},\qquad\mbox{
and }\ \rho_{g}(\sigma_{N})=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),$

and using the trace form to identify $\operatorname{Ad}(V_{g})$ with its dual
$\operatorname{Ad}^{*}(V_{g})$, we define $u_{g}\in U_{g}$ via

(108) $u_{g}:=u_{\psi}\otimes\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)+\sigma_{N}(u_{\psi})\otimes\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right).$

Note that the matrices above do indeed define functionals on
$\operatorname{Ad}(V_{g})$ that send the image of $R[G_{\mathbb{Q}}]$ to $R$.
We readily see that $u_{g}$ is in fact $G_{\mathbb{Q}}$-invariant, e.g.
$\rho_{g}(G_{K})$ acts on the $u_{\psi}$ through $\psi^{-1}$ and on the upper
nilpotent matrix through $\psi=\psi_{1}^{2}$. We then compute an equality
inside $\mathcal{O}_{H}^{\times}\otimes R$:

$\operatorname{red}_{N}(u_{g})=\left\langle
u_{g},\rho_{g}(\sigma_{N})\right\rangle={\rm
Trace}\left(\begin{array}[]{cc}u_{\psi}&0\\\
0&\sigma_{N}(u_{\psi})\end{array}\right)=u_{\psi}+\sigma_{N}(u_{\psi}).$

(The reader is cautioned that additive notation for the group law in
${\mathcal{O}}_{H}^{\times}\otimes R/p^{t}$ has been used in this last
equation.) Since the discrete logarithm mod $N$ is equivariant for the action
of $\sigma_{N}$, which acts trivially on $(\mathbb{Z}/N\mathbb{Z})^{\times}$,
we obtain the desired equality

$\log(\operatorname{red}_{N}(u_{g}))=\log(u_{\psi}+\sigma_{N}u_{\psi})=2\log(u_{\psi}).$

∎

###  5.3. Proof of Conjecture 1.1 for indefinite theta series

We now turn to prove Conjecture 1.1 when $g$ is an RM form. We will be in the
situation of §3 with $\psi_{2}=\psi_{1}^{-1}$. More precisely, let
$\psi_{1}:G_{K}\,\longrightarrow\,R^{\times}$ be the finite order character of
mixed signature as in the beginning of §3, with values in the ring of integers
of a finite extension $L$ of $\mathbb{Q}$, such that $g=\theta_{\psi_{1}}$ is
the theta series associated to $\psi_{1}$ as described in (22). Let $N$ be an
odd prime and define $G\in S_{2}(\Gamma_{0}(N))$ as the trace to the space of
modular forms of level $N$ of
$\theta_{\psi_{1}}(z)\theta_{\psi_{1}^{-1}}(Nz)$. As explained in the
introduction, the conjecture we address in this note becomes trivial when $N$
remains inert, and hence we assume throughout that it splits in $K$ as
$N=\mathfrak{N}\cdot\bar{\mathfrak{N}}$.

The proof of Conjecture 1.1, which computes the pairing of $G$ with the
Shimura class, again relies crucially on the $\Theta$-correspondence, namely
the Hecke-equivariant map

$\Theta:H_{1,\mathrm{B}}(X_{0}(N),{\rm
cusps};Z)^{+}\otimes_{\mathbb{T}(N)}H_{1,\mathrm{B}}(Y_{0}(N),Z)^{-}\rightarrow
M_{2}(N)$

given by

(109)
$\Theta(\gamma^{+}\otimes\gamma^{-})=\frac{-1}{24}\kappa_{0}^{+}(\gamma^{+})\cdot\kappa_{0}^{-}(\gamma^{-})+\sum_{m\geq
1}\langle T_{m}\gamma^{+},\gamma^{-}\rangle q^{m}.$

Here $\kappa_{0}^{\pm}$ are as defined in §4.3 and §4.4. Note that the sign of
$\frac{-1}{24}$ depends on orientation conventions implicit in the definition
of the intersection pairing.

For lack of a suitable reference we sketch a proof. We identify the relative
homology group $H_{1,\mathrm{B}}(X_{0}(N),{\rm cusps};Z)^{+}$ with
$H^{1}_{\rm{B}}(Y_{0}(N),Z)^{-}$, a free $\mathbb{T}(N)$-module of rank one,
and thus with $\mathbb{T}(N)$ itself. We can similarly identify
$H_{1,\mathrm{B}}(Y_{0}(N),Z)^{-}$ with its dual $M_{2}(N;Z)$. Adjusting these
identifications if necessary, we can suppose that the Poincaré pairing
$\langle-,-\rangle$ corresponds to the pairing on $\mathbb{T}(N)\times
M_{2}(N;Z)$ given by $(T,f)\mapsto a_{1}(Tf)$, and $\Theta$ corresponds to
$(T,f)\mapsto Tf$. The formula (109) follows from this, up to the
identification of the constant $\frac{-1}{24}$. To compute the constant we
take $\gamma^{+}$ the element represented by the geodesic from $0$ to
$\infty$, and $\gamma^{-}$ a small loop around $\infty$ and we fix
orientations so that $\langle\gamma^{+},\gamma^{-}\rangle=1$. In particular,

$\langle
T_{m}\gamma^{+},\gamma^{-}\rangle=\sum_{d|m,(d,N)=1}d,\qquad\kappa_{0}^{+}(\gamma^{+})=1,\qquad\kappa^{-}_{0}(\gamma^{-})=N-1.$

The expansion on the right-hand side of (109) must represent $E_{2}^{(N)}$,
and therefore this fixes the constant as $\frac{-1}{24}$.

We note in particular that $\kappa^{+}_{0}$ vanishes on the image of
$H_{1,\mathrm{B}}(X_{0}(N))$ and so the formula above in fact matches with
(45) used in an earlier section.

As in the CM setting, we can observe that – with $\mathfrak{m}$ the Eisenstein
ideal as before –

  * •

the modules ${\mathbb{H}}_{+}:=H_{1,\mathrm{B}}(X_{0}(N),{\rm
cusps};\mathbb{Z})^{+}_{\mathfrak{m}}$ and
${\mathbb{H}}_{-}=H_{1,\mathrm{B}}(Y_{0}(N),\mathbb{Z})^{-}_{\mathfrak{m}}$,
obtained from completing the singular homology of the complex modular curves,
are again free ${\mathbb{T}}$-modules of rank $1$.666Note that we get, by
duality, an isomorphism of these with the (sign-altered) cohomological
analogues: ${\mathbb{H}}_{+}\simeq{\mathbb{H}}^{-}$ and
${\mathbb{H}}_{-}\simeq{\mathbb{H}}^{+}$, so this result follows from its
cohomological analogue.

  * •

the map $\Theta_{\mathfrak{m}}$ is an isomorphism:
$({\mathbb{H}}_{+}\otimes_{\mathbb{T}}{\mathbb{H}}_{-})\longrightarrow\mathbb{M}$
and so (cf. (104), (105)) we have adjoint maps

(110)
$\Theta:\bar{\mathbb{H}}_{+}\otimes_{\overline{\mathbb{T}}}\bar{\mathbb{H}}_{-}\simeq\bar{{\mathbb{M}}},\
\
\Theta^{*}:\bar{{\mathbb{M}}}^{*}\simeq(\bar{\mathbb{H}}_{+}\otimes_{\overline{\mathbb{T}}}\bar{\mathbb{H}}_{-})^{*}.$

Here bars denote tensoring with $\mathbb{Z}/p^{t}$ and $*$ denotes
$\operatorname{Hom}(-,\mathbb{Z}/p^{t})$.

The strategy of the proof of Conjecture 1.1 is, much as in the case of CM
theta series, to express the inner product $\langle G,\mathfrak{S}\rangle$ as
an inner product on ${\mathbb{H}}_{+}\otimes_{{\mathbb{T}}}{\mathbb{H}}_{-}$
via $\Theta$.

We will follow the notation of §3.2; in particular $\mathcal{C}$ is the narrow
class group of $K$, and we have introduced Heegner cycles $\gamma_{I}$
attached to $I\in\mathcal{C}$, as well as weighted combinations
$\gamma_{\psi}$ in (46). The following proposition plays a key role in the
proof of Conjecture 1.1 for RM forms, since it is via this result that the
relevant Stark unit – in this case, a fundamental unit of the real quadratic
field – makes its appearance.

######  Proposition 5.7.

For all even characters $\psi$ of the narrow Picard group $\mathcal{C}$,

$\kappa_{0}^{+}(\gamma_{\psi})=0,\qquad\mbox{and
}\quad\kappa_{1}^{+}(\gamma_{\psi})=\begin{cases}-h\log(u_{K})&\mbox{ if
}\psi=1,\\\ 0&\mbox{ if }\psi\neq 1,\end{cases}$

where $\kappa_{0}^{+}$ and $\kappa_{1}^{+}\in H^{1}_{\mathrm{B}}(X_{0}(N),{\rm
cusps};\mathbb{Z})^{+}$ are the Eisenstein and higher Eisenstein elements
described in §4.3, $h$ is the order of the narrow class group $\mathcal{C}$,
and $\log(u_{K})$ refers to the logarithm of the reduction of $u_{K}$ at the
chosen divisor $\mathfrak{N}$ of $N$.777 The definition of $\gamma_{\psi}$
also depends on the choice of divisor of $N$, although this is not indicated
in the notation. One checks that the identity remains valid upon replacing
$\mathfrak{N}$ by $\mathfrak{N}^{\prime}$ on both sides.

###### Proof.

The assertion about $\kappa_{0}^{+}$ follows from the fact that the Heegner
cycles $\gamma_{I}$, viewed as cycles in the integral homology of $X_{0}(N)$
relative to the cusps, are in the kernel of the boundary map $\partial$ of
(83), and hence are orthogonal to $\kappa_{0}$.

To show the second assertion, recall that the class $\kappa_{1}^{+}$ was
defined modulo $p^{t}$ by choosing a discrete logarithm
$\log:(\mathbb{Z}/N\mathbb{Z})^{\times}\longrightarrow\mathbb{Z}/p^{t}\mathbb{Z}$,
and setting

$\kappa_{1}^{+}\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)=\log(a).$

With this choice we have

(111) $\kappa_{1}^{+}(\gamma_{I})=-\log(u_{K}),$

where $u_{K}$ is a fundamental unit of norm $1$ of the real quadratic field
$K$, $\log(u_{K})$ refers to the logarithm of the reduction of $u_{K}$ at
$\mathfrak{N}$. . This is because (notation of §3.2) the cycle $\gamma_{I}$
arises from an embedding $\mathfrak{o}\rightarrow M_{0}(N)$, with respect to
which the ring homomorphism sending a matrix in $M_{0}(N)$ to the mod $N$
reduction of upper left hand entry restricts to reduction modulo
$\mathfrak{N}$ on $\mathfrak{o}$ (see discussion above (43)); the sign arises
for the orientation reason noted below (44). Equation (111) therefore implies
that
$\kappa_{1}^{+}(\gamma_{\psi})=-(\sum_{\mathfrak{a}}\psi(\mathfrak{a}))\log(u_{K})$,
and the result follows. ∎

######  Proposition 5.8.

For all totally odd ring class characters $\psi$,

$\kappa_{0}^{-}(\gamma_{\psi})=(1-\psi({\mathfrak{N}}))L_{\mathrm{alg}}(\psi),$

where $\kappa_{0}^{-}\in H^{1}_{\mathrm{B}}(Y_{0}(N),\mathbb{Z})^{-}$ is as
defined in Section 4.4, and $L_{\mathrm{alg}}(\psi)\in R$ will be defined in
(112) and is in particular independent of $\mathfrak{N}$.

Recall that $\kappa_{0}^{-}$ arises from the Dedekind-Rademacher function
$\varphi$ of (85) which encodes the periods of the (complex!) logarithm of the
modular unit $\Delta(Nz)/\Delta(z)$. The proposition shows that
$\kappa_{0}^{-}(\gamma_{\psi})$ exhibits a mild dependence on $N$ through the
factor $(1-\psi({\mathfrak{N}}))$.

###### Proof.

The issue to be dealt with here is, essentially, passage from level $1$ to
level $N$. Let $I\in\mathcal{C}$. Choose a representative that is relatively
prime to $N$ and an oriented basis $(e_{1},e_{2})$. The element

$\eta_{I}=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in{\mathrm{SL}}_{2}(\mathbb{Z}),\qquad\mbox{ where
}\quad\begin{array}[]{l}u_{K}e_{1}=ae_{1}+ce_{2}\\\
u_{K}e_{2}=be_{1}+de_{2}\end{array}$

has conjugacy class in ${\mathrm{SL}}_{2}(\mathbb{Z})$ that does not depend on
the choice of oriented basis, and in particular $\varphi(\eta_{\mathfrak{a}})$
is well-defined. Choose $(e_{1},e_{2})$ so that $e_{2}$ belongs to
$I\cap{\mathfrak{N}}$, and observe then that
$(e_{1}^{\prime},e_{2}^{\prime}):=(Ne_{1},e_{2})$ is an oriented basis for
$I{\mathfrak{N}}$ and that $u_{K}$ acts on this basis according to the rule
$u_{K}e_{1}^{\prime}=ae_{1}^{\prime}+(cN)e_{2}^{\prime}$ and
$u_{k}e_{2}^{\prime}=(b/N)e_{1}^{\prime}+de_{2}^{\prime}$. By (84) as well as
the definition (42) of cycles $\gamma_{J}$, we get
$\kappa_{0}^{-}(\gamma_{I{\mathfrak{N}}})=\varphi(\eta_{I{\mathfrak{N}}})-\varphi(\eta_{I})$
and it follows that

$\displaystyle\kappa_{0}^{-}(\gamma_{\psi})$ $\displaystyle=$
$\displaystyle\sum\psi(I{\mathfrak{N}})\left(\varphi(\eta_{I{\mathfrak{N}}})-\varphi(\eta_{I})\right)$
$\displaystyle=$
$\displaystyle(1-\psi({\mathfrak{N}}))\sum_{I}\psi(I)\varphi(\eta_{I}),$

and we obtain the result upon defining

(112) $L_{\mathrm{alg}}(\psi):=\sum_{I}\psi(I)^{-1}\varphi(\eta_{I}).$

∎

######  Remark 5.9.

As is implicit in the notation, $L_{\rm alg}(\psi)$ is closely related to the
“algebraic part” of the $L$-series
$L(\psi,s)=\sum_{\mathfrak{a}\lhd\mathfrak{o}_{K}}\psi(\mathfrak{a})(N\mathfrak{a})^{-s}$
attached to $\psi$, at $s=1$. The justification for this is given by Meyer’s
analogue of the Kronecker limit formula for real quadratic fields (cf. [Za,
§4]) which asserts that, at least for all unramified, totally odd characters
$\psi$ of the narrow Hilbert class field of $K$, $L_{\rm
alg}(\psi)=\frac{12\sqrt{D}}{\pi^{2}}L(\psi^{-1},1).$

Note that if $x^{2}-a_{N}(g)+\chi_{K}(N)=(x-\alpha_{N})(x-\beta_{N})$ is the
$N$-th Hecke polynomial attached to $g$, then we may order $\alpha_{N}$ and
$\beta_{N}$ in such a way that

$\alpha_{N}=\psi_{1}({\mathfrak{N}}),\quad\beta_{N}=\psi_{1}({\mathfrak{N}}^{\prime}),\mbox{
and so
}\psi({\mathfrak{N}})=\psi_{1}({\mathfrak{N}})/\psi_{1}^{\prime}({\mathfrak{N}})=\alpha_{N}/\beta_{N},$

where we use the definition (9). Proposition 5.8 can then be rewritten as

(113) $\kappa_{0}^{-}(\gamma_{\psi})=\left(1-\alpha_{N}/\beta_{N}\right)\times
L_{\rm alg}(\psi).$

Let ${\mathfrak{S}}_{0}$ and
${\mathfrak{S}}={\mathfrak{S}}_{1}\in\bar{\mathbb{M}}^{*}$ denote the
Eisenstein and higher classes described in §4.5. It follows from Theorem 3.1
applied to the pair $(\psi_{1},\psi_{1}^{-1})$ – so by (41) $\psi_{12}=1$ and
$\psi_{12^{\prime}}=\psi_{1}/\psi_{1}^{\prime}=\psi$ – that there exists
$C_{g}\in R$ independent of $N$ such that

$\displaystyle\langle G,\mathfrak{S}\rangle$ $\displaystyle=$
$\displaystyle\beta_{N}\,C_{g}\langle\Theta([\gamma_{1}]\otimes[\gamma_{\psi}]),\mathfrak{S}\rangle$
$\displaystyle=$
$\displaystyle\beta_{N}C_{g}\cdot\langle[\gamma_{1}]\otimes[\gamma_{\psi}],\Theta^{*}(\mathfrak{S})\rangle,$

where we understand $[\gamma_{1}]\otimes[\gamma_{\psi}]$ as an element of
$(\bar{\mathbb{H}}_{+}\otimes_{\overline{\mathbb{T}}}\bar{\mathbb{H}}_{-})$,
and $\Theta^{*}(\mathfrak{S})$ as an element of the $\mathbb{Z}/p^{t}$-dual,
see (110). Here we regard $\Theta$ as normalized as in (109); the $C_{g}$ that
appears in the above equation only agrees with that constant appearing in
Theorem 3.1 up to sign, arising from the fact that the choice of orientation
convention for (109) was not compared with the choice of orientation
convention used in Theorem 3.1. This sign may be computed by the enthusiastic
reader.

The next theorem below, which determines the image of $\mathfrak{S}$ under
$\Theta^{*}$, plays exactly the same role in the RM proof as Theorem 5.4 in
the CM setting.

######  Theorem 5.10.

We have

  1. (1)

$\Theta^{*}({\mathfrak{S}}_{0})=\frac{-1}{24}\kappa_{0}^{+}\otimes\kappa_{0}^{-}$;

  2. (2)

$\Theta^{*}(\mathfrak{S})\equiv\frac{-1}{24}(\kappa_{1}^{+}\otimes\kappa_{0}^{-}+\kappa_{0}^{+}\otimes\kappa_{1}^{-})$
modulo $\kappa_{0}^{+}\otimes\kappa_{0}^{-}$.

where $\kappa^{+}$ are the Eisenstein classes of §4.3, or rather their image
in $(\mathbb{H}_{+})^{*}$ or $(\overline{\mathbb{H}}_{+})^{*}$, and similarly
$\kappa^{-}$ are similarly defined from the Eisenstein classes of §4.4.

The statements should be interpreted just as in Theorem 5.4: we use

$\left(\overline{\mathbb{H}}_{+}\otimes_{\overline{\mathbb{T}}}\overline{\mathbb{H}}_{-}\right)^{*}\subset\left(\overline{\mathbb{H}}_{+}\otimes_{\mathbb{Z}/p^{t}}\overline{\mathbb{H}}_{-}\right)^{*}=(\overline{\mathbb{H}}_{+})^{*}\otimes_{\mathbb{Z}/p^{t}}(\overline{\mathbb{H}}_{-})^{*},$

where $*$ means $\operatorname{Hom}(-,\mathbb{Z}/p^{t})$.

###### Proof.

The first part of the theorem follows directly from the definition of $\Theta$
given in (109). The second follows from the Hecke equivariance of
$\Theta^{*}$, in light of the fact that
$\kappa_{1}^{+}\otimes\kappa_{0}^{-}+\kappa_{1}^{+}\otimes\kappa_{0}^{-}$ is
the higher Eisenstein element in
$({\mathbb{H}}^{+}\otimes_{{\mathbb{T}}}{\mathbb{H}}^{-})^{\vee}$ attached to
$\kappa_{0}^{+}\otimes\kappa_{0}^{-}$, by Proposition 4.14. ∎

We can now prove Conjecture 1.1 in the RM setting.

######  Proposition 5.11.

We have

(115) $\langle G,\mathfrak{S}\rangle=\frac{1}{24}h(\mathfrak{o})C_{g}\cdot
L_{\rm alg}(\psi)\cdot(\beta_{N}-\alpha_{N})\cdot\log(u_{K}).$

###### Proof.

Applying (5.3) and part (2) of Theorem 5.10,

$\langle
G,\mathfrak{S}\rangle=\frac{-\beta_{N}C_{g}}{24}\cdot\langle\gamma_{1}\otimes\gamma_{\psi},\kappa_{0}^{+}\otimes\kappa_{1}^{-}+\kappa_{1}^{+}\otimes\kappa_{0}^{-}\rangle=\frac{-\beta_{N}C_{g}}{24}\cdot\kappa_{1}^{+}(\gamma_{1})\cdot\kappa_{0}^{-}(\gamma_{\psi}),$

where we have used the fact that $\kappa_{0}^{+}(\gamma_{1})=0$ to ignore the
term arising from
$\langle\gamma_{1}\otimes\gamma_{\psi},\kappa_{0}^{+}\otimes\kappa_{1}^{-}\rangle$.
The theorem now follows from Proposition 5.7 and (113), which imply that

$\kappa_{1}^{+}(\gamma_{1})=-h\log(u_{K}),\qquad\kappa_{0}^{-}(\gamma_{\psi})=(1-\alpha_{N}/\beta_{N})\cdot
L_{\rm alg}(\psi).$

∎

To prove Theorem 1.2 of the introduction when $K$ is a real quadratic field,
it remains, as before, to relate the right-hand side of (115) to the
expression $\operatorname{red}_{N}(u_{g})$ occuring in this theorem.

######  Lemma 5.12.

(cf. Lemma 5.6). Let
$U_{g}:=({\mathcal{O}}_{K}^{\times}\otimes\operatorname{Ad}^{*}(\rho_{g})^{\circ})^{G_{\mathbb{Q}}}$.
There exists $u_{g}\in U_{g}$ with the property that, for all $N$ as above,

$\log(\operatorname{red}_{N}(u_{g}))=(\alpha_{N}-\beta_{N})\log(u_{K}).$

As before, Theorem 1.2 will follow from this: we have

$24\langle G,\mathfrak{S}\rangle=\log(\mathrm{red}_{N}(u_{g}^{\prime})),$

with $u_{g}^{\prime}=-h(\mathfrak{o})L_{\mathrm{alg}}(\psi)C_{g}\cdot u_{g}$.

###### Proof.

Let $e_{1}$ and $e_{2}$ be eigenvectors in $V_{g}$ for the action of $G_{K}$,
on which $G_{K}$ acts via the characters $\psi_{1}$ and $\psi_{1}^{\prime}$
respectively. Since $N$ is split in $K$, the associated Frobenius automorphism
$\sigma_{N}\in G_{\mathbb{Q}}$ is a diagonal matrix, with entries $\alpha_{N}$
and $\beta_{N}$. Representing elements of ${\rm Ad}(V_{g})$ as matrices
relative to the basis $(e_{1},e_{2})$, so that
$\rho_{g}(\sigma_{N})=\left(\begin{array}[]{cc}\alpha_{N}&0\\\
0&\beta_{N}\end{array}\right)$, and using the trace form to identify
$\operatorname{Ad}(V_{g})$ with its dual $\operatorname{Ad}^{*}(V_{g})$, we
define

$u_{g}:=u_{K}\otimes\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),$

which is clearly $G_{\mathbb{Q}}$-invariant: it is fixed by $G_{K}$, and the
nontrivial automorphism of $K$ negates both factors. As after (108) this
indeed defines an element of $U_{g}$. One then finds

$\displaystyle\operatorname{red}_{N}(u_{g})$ $\displaystyle=$
$\displaystyle\left\langle u_{g},\rho_{g}(\sigma_{N})\right\rangle\ \ =\ \
{\rm Trace}\left(\begin{array}[]{cc}u_{K}\otimes\alpha_{N}&0\\\
0&u_{K}\otimes(-\beta_{N})\end{array}\right)$ $\displaystyle=$ $\displaystyle
u_{K}\otimes(\alpha_{N}-\beta_{N}).$

Therefore,

$\log(\operatorname{red}_{N}(u_{g}))=(\alpha_{N}-\beta_{N})\log(u_{K}).$

The lemma follows. ∎

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