# The derived Hecke algebra
for dihedral weight one forms

Henri Darmon, Michael Harris, Victor Rotger, Akshay Venkatesh  H. D.:
Department of Mathematics and Statistics, McGill University, Montreal, Canada
<EMAIL_ADDRESS>K. H.: Department of Mathematics, Columbia University,
New York, U.S<EMAIL_ADDRESS>V. R.: IMTech, UPC and Centre de
Recerca Matemàtiques, C. Jordi Girona 1-3, 08034 Barcelona, Spain
<EMAIL_ADDRESS>A. V.: School of Mathematics, Institute for Advanced
Study, Princeton, U.S<EMAIL_ADDRESS>
###### Abstract.

We study the action of the derived Hecke algebra in the setting of dihedral
weight one forms, and prove a conjecture of the second- and fourth- named
authors relating this action to certain Stark units associated to the
symmetric square $L$-function. The proof exploits the theta correspondence
between various Hecke modules as well as ideas of Merel and Lecouturier on
higher Eisenstein elements.

###### 1991 Mathematics Subject Classification:

11G18, 14G35

The first author was supported by an NSERC Discovery grant. The second and
fourth author were supported by the National Science Foundation under Grant
No. DMS-1440140 while they were in residence at the Mathematical Sciences
Research Institute in Berkeley, California, during the Spring 2019 semester.
The second author was also supported by the National Science Foundation under
Grants No. DMS-1701651 and DMS-2001369. The third author was supported by an
ERC consolidator grant and Icrea Academia. The last-named author was supported
by NSF grant DMS-1931087. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 682152).

###### Contents

  1. 1 Introduction
  2. 2 A trace identity for definite theta series
  3. 3 A trace identity for indefinite theta series
  4. 4 Higher Eisenstein elements
  5. 5 Proof of the main theorem

##  1\. Introduction

In the theory of modular forms, the case of weight one is exceptional in
several ways. The space of weight one forms, which can be interpreted as the
global sections of the Hodge line bundle $\omega$ on a modular curve $X$, does
not admit a simple dimension formula. This occurs precisely because the higher
cohomology group $H^{1}(X,\omega)$ can be nontrivial — that is to say, the
space of weight one forms manifests itself in two different cohomological
degrees.

A conjecture proposed in [PV, GV, Ve2, HV] asserts that, in situations where
spaces of automorphic forms occur across multiple cohomological degrees, the
different degrees are related by means of a hidden action of a motivic
cohomology group. The last mentioned paper [HV], in particular, formulates
this story in the context of weight $1$ forms for the modular curve, and
translates the general conjectures into a numerically testable statement. This
statement, which is summarised below, is the main topic of this paper.

###  1.1. The Shimura class

While the general definition of derived Hecke operators shall not be recalled
here, one crucial ingredient in their construction is to take the cup product
with a certain distinguished class in coherent cohomology, the so-called
Shimura class.

As explained in detail in [HV, §3.1], the Shimura class attached to a prime
$N\geq 5$ arises from the covering $X_{1}(N)\rightarrow X_{0}(N)$ of classical
modular curves, which (at least away from elliptic points) is étale with deck
group $(\mathbb{Z}/N\mathbb{Z})^{\times}$ and thus furnishes an element

$\mathfrak{S}^{\times}\in H^{1}_{\rm
et}(X_{0}(N),(\mathbb{Z}/N\mathbb{Z})^{\times}\otimes\mathbb{Z}[1/6]).$

(Here, and in what follows, modular curves will be regarded as schemes over
the ring $Z=\mathbb{Z}[\frac{1}{6N}]$ to avoid any technical issues.)

Let $p>3$ be a prime, let $p^{t}$ be the highest power of $p$ dividing $N-1$,
assume $t\geq 1$ and fix a surjective discrete logarithm

(1)
$\log:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\mathbb{Z}/p^{t}\mathbb{Z}.$

This choice determines a class $\mathfrak{S}:=\log(\mathfrak{S}^{\times})\in
H^{1}_{\mathrm{et}}(X_{0}(N),\mathbb{Z}/p^{t})$. Restricting to the fiber
product of $X_{0}(N)$ over $\operatorname{Spec}(Z)$ with
$\operatorname{Spec}(\mathbb{Z}/p^{t}\mathbb{Z})$, denoted
$\bar{X}=X_{0}(N)_{/\mathbb{Z}/p^{t}\mathbb{Z}}$, the resulting class can be
pushed into Zariski cohomology, using the inclusion of
$\mathbb{Z}/p^{t}\mathbb{Z}$ into the étale sheaf represented by
$\mathbb{G}_{a}$: in this way $\mathfrak{S}$ can be viewed as a class in
coherent cohomology. It is called the Shimura class, denoted (by a slight
abuse of notation)

$\mathfrak{S}\in
H^{1}(\bar{X},\mathcal{O}_{\bar{X}})=\operatorname{Hom}(S_{2}(N),\mathbb{Z}/p^{t}\mathbb{Z}),$

where the last identification is provided by Serre duality, and $S_{2}(N)$ is
the space of weight $N$ cusp forms (with $q$-expansions integral at $p$). Note
that $\mathfrak{S}$ depends on $N$, on $p$, and on the choice of discrete
logarithm.

###  1.2. The main result

Let $g\in H^{0}(X_{1}(d),\omega)$ be a Hecke new cusp form of weight $1$,
level $d$ and nebentype $\chi$, and let $g^{*}\in H^{0}(X_{1}(d),\omega)$ be
the dual newform, whose Fourier expansion is related to that of $g$ by complex
conjugation, and whose automorphic representation is obtained from that of $g$
by twisting by $\chi^{-1}$. Assume for simplicity that the primes $N$ and $p$
do not divide $6d$.

Let $\rho_{g}:G_{\mathbb{Q}}\longrightarrow\mathrm{GL}_{2}(L)\simeq{\rm
Aut}(V_{g})$ be the odd $2$-dimensional Artin representation attached by
Deligne and Serre to $g$, acting on a two-dimensional $L$-vector space
$V_{g}$, for a suitable finite extension $L$ of $\mathbb{Q}$ (containing the
fourier coefficients of $g$, and contained in a cyclotomic field). Let
$\operatorname{Ad}(\rho_{g})$ denote the $3$-dimensional subrepresentation of
$\mathrm{End}_{L}(V_{g})$ consisting of $L$-linear endomorphisms of $V_{g}$ of
trace zero, equipped with the natural action of $G_{\mathbb{Q}}$ by
conjugation.

Let $R$ be the ring of integers of $L$, with $6N$ inverted. The product
$g(z)g^{*}(Nz)$ is a weight $2$ cuspidal modular form of level $Nd$ with
trivial Nebentypus character and coefficients in $R$, and can thus be viewed
as an element of the space $S_{2}(Nd)=H^{0}(X_{0}(Nd),\Omega^{1})$ of global
regular differential forms. Let

$G(z):=\mathsf{Tr}^{Nd}_{N}(g(z)g^{*}(Nz))\in
S_{2}(N;R)=H^{0}(X_{0}(N)_{/R},\Omega^{1})$

denote the trace of $g(z)g^{*}(Nz)$ to the space of modular forms of weight
$2$ and level $N$.

The pairing between $G$ and the Shimura class $\mathfrak{S}$ arising from
Serre duality gives rise to a numerical invariant

$\langle G,\mathfrak{S}\rangle\in R/p^{t},$

see §1.5 for details. The conjecture of [HV] relates this quantity to the
discrete logarithm of a suitable Stark unit attached to $g$, which we now
proceed to describe.

The image of the integral group ring $R[G_{\mathbb{Q}}]$ in
$\operatorname{Ad}(\rho_{g})$ endows this space with a Galois-stable
$R$-sublattice, which is denoted $\operatorname{Ad}(\rho_{g})^{\circ}$, and
whose $R$-linear dual is denoted $\operatorname{Ad}^{*}(\rho_{g})^{\circ}$.

Let $H$ denote the finite extension of $\mathbb{Q}$ which is cut out by
$\operatorname{Ad}(\rho_{g})$. Because complex conjugation acts with
eigenvalues $1$, $-1$ and $-1$ on this representation, Dirichlet’s unit
theorem asserts that the $R$-module

$U_{g}:=({\mathcal{O}}_{H}^{\times}\otimes\operatorname{Ad}^{*}(\rho_{g})^{\circ})^{G_{\mathbb{Q}}}$

is of rank one (cf. Lemma 2.7 of [HV]). The choice of a prime $\mathcal{N}$ of
$H$ above $N$ gives rise to a frobenius element $\sigma_{N}$, whose image
under $\rho_{g}$ is a natural element of $\operatorname{Ad}(\rho_{g})^{\circ}$
which is invariant under the conjugation action of $\sigma_{N}$. Evaluation at
$\sigma_{N}$ thus gives rise to a homomorphism from
$\operatorname{Ad}^{*}(\rho_{g})^{\circ}$ to $R$ which is
$\sigma_{N}$-equivariant (for the trivial $\sigma_{N}$ action on $R$).
Combining this evaluation with the reduction modulo $\mathcal{N}$ gives a “mod
$N$ reduction map”

$\operatorname{red}_{N}:U_{g}:=({\mathcal{O}}_{H}^{\times}\otimes\operatorname{Ad}^{*}(\rho_{g})^{\circ})^{G_{\mathbb{Q}}}\longrightarrow(({\mathcal{O}}_{H}/\mathcal{N})^{\times}\otimes
R)^{\sigma_{N}=1}=(\mathbb{Z}/N\mathbb{Z})^{\times}\otimes R.$

A version of main conjecture of [HV] (Conjecture 3.1 in loc. cit.) may be
phrased as follows.

######  Conjecture 1.1.

There exists an integer $m=m_{g}\geq 1$ and $u_{g}\in U_{g}$ such that, for
all primes $N$ and $p$ as above,

$m\cdot\langle G,\mathfrak{S}\rangle=\log(\mathrm{red}_{N}(u_{g})).$

Note that both sides of this conjectured identity belong to $R/p^{t}$, and
that both depend linearly on the choice of discrete logarithm made in (1). The
validity of Conjecture 1.1 is thus independent of this choice. Similarly, both
sides of the conjecture are independent of the choice of $\mathcal{N}$. (In
loc. cit. the conjecture was formulated differently, and was slightly more
precise about the primes dividing $m$; the version above is more explicit and
is what we will prove in certain cases.)

This article presents a proof of Conjecture 1.1 when $g$ is dihedral under
certain simplifying assumptions on ramification. Recall that $g$ is said to be
dihedral if the Galois representation $\rho_{g}$ is induced from a ray class
character $\psi_{1}$ of the Galois group of an (imaginary or real) quadratic
extension $K$ of $\mathbb{Q}$. In that case $g=\theta_{\psi_{1}}$ is Hecke’s
classical theta series associated to $\psi_{1}$, i.e. the modular form whose
$L$-series is given by $L(K,\psi_{1})$. We assume throughout that
$\psi_{1}^{2}\neq 1$, as this implies that $\theta_{\psi_{1}}$ is cuspidal.

Let $D$ denote the discriminant of $K$ and $\delta=(\sqrt{D})$ its different.

######  Theorem 1.2.

If $K$ is imaginary, assume that $D$ is an odd prime and that $\psi_{1}$ is
unramified. If $K$ is real assume that $D$ is odd and that $\psi_{1}$ has
conductor dividing $\delta$. Then Conjecture 1.1 is true for
$g=\theta_{\psi_{1}}$.

######  Remark 1.3.

The proof of Theorem 1.2 described in Chapter 5 shows that the integer $m$ of
Conjecture 1.1 divides $24$ in the real case, and, in the imaginary case, that
it divides $6$ unless the order of $\psi_{1}^{2}$ is a power of a prime
$\ell$, in which case $m$ divides $6\ell$. No claim is made that these bounds
for $m$ are optimal; they are merely what comes directly out of the proofs.

The key idea in the proof of Theorem 1.2 is to express $G$ as the theta lift
of an appropriate Heegner cycle, and to compute the image of $\mathfrak{S}$
under the adjoint of the theta lift as a combination of higher Eisenstein
elements. While the latter computation is performed in full generality, the
expression for $G$ in terms of Heegner cycles has only been worked out in a
non-trivial simple scenario.

In particular, the ramification conditions force the following simplifying
feature. Let $\psi_{1}^{\prime}$ denote the
$\mathrm{Gal\,}(K/\mathbb{Q})$-conjugate of $\psi_{1}$ and set
$\psi=\psi_{1}/\psi_{1}^{\prime}$. Then

(2)
$\rho_{g}\otimes\rho_{g^{*}}=\mathrm{Ind}_{\mathbb{Q}}^{K}(1)\oplus\mathrm{Ind}_{\mathbb{Q}}^{K}(\psi)$

decomposes as the direct sum of the induced representations of two characters
of $G_{K}$, the trivial character $1$ and an unramified character $\psi$. R.
Zhang’s forthcoming Ph.D thesis [Zh] will contain a proof under less
restrictive ramification conditions for $K$ imaginary. When $K$ is real, we
envisage a method for calculating $G$ invoking Kudla-Millson theory but in
order to cover the general case one needs to solve some issues related to the
regularization of the theta lift from the split orthogonal group; cf. §1.4 for
more details.

###  1.3. Trivial cases

If $K$ is imaginary quadratic and $N=\mathfrak{N}\cdot\mathfrak{N}^{\prime}$
splits in $K$, then $u_{g}$ belongs to
$({\mathcal{O}}_{H}^{\times}\otimes\mathrm{Ind}_{\mathbb{Q}}^{K}(\psi))^{G_{\mathbb{Q}}}$,
while $\sigma_{N}$ belongs to $\mathrm{Ind}_{\mathbb{Q}}^{K}(1)$. If $K$ is
real quadratic and $N$ is inert in $K$, then $u_{g}$ belongs to the unit group
$\mathfrak{o}^{\times}$ of $K$, on which $\sigma_{N}$ acts as $-1$. In both
cases the regulator $\mathrm{red}_{N}(u_{g})$ of Conjecture 1.1 vanishes
trivially.

This is consistent with the fact that the modular form $G$ is identically zero
in these two scenarios. Indeed, the main theorem of [HK] asserts that, for all
newforms $f$ of weight two on $\Gamma_{0}(N)$,

(3) $\langle G,f\rangle^{2}=C\cdot L(f,g,g^{*},1),$

where $C$ is a product of local automorphic terms and $L(f,g,g^{*},s)$ is the
triple product $L$-series associated to $f$, $g$ and $g^{*}$. The Artin
formalism applied to (2) implies that

(4) $L(f,g,g^{*},s)=L(f/K,s)\cdot L(f/K,\psi,s),$

where the two $L$-functions appearing on the right-hand side are the ones
associated to the base change of $f$ to $K$, twisted by suitable characters.
Since $(d,N)=1$, both $L(f/K,s)$ and $L(f/K,\psi,s)$ satisfy a functional
equation with $s=1$ as center of symmetry and global sign $(\frac{-N}{K})$.
This sign is $-1$, and hence

$L(f/K,1)=L(f/K,\psi,1)=0.$

It follows that $\langle G,f\rangle=0$ for all $f$, and hence that $G=0$.

###  1.4. Outline of the paper

The interesting cases of Theorem 1.2 occur when $(\frac{-N}{K})=1$, i.e., when

  * •

$K$ is imaginary quadratic, and $N$ is inert in $K$;

  * •

$K$ is real quadratic, and $N$ is split in $K$.

The body of the article is devoted to the proof of Theorem 1.2 in these non-
trivial cases, referred to as the definite and indefinite cases respectively.
The main idea is to transfer the computation to a suitable (definite, resp.
indefinite) quaternion algebra $B$ over $\mathbb{Q}$ by means of a theta lift
$\Theta$:

$\Theta:\mbox{modular forms on $B$}\rightarrow S_{2}(N).$

This allows the identity in Conjecture 1.1 to be recast on $B$. Indeed, $G$
and $\mathfrak{S}$ are obtained via $\Theta$ from objects arising
(respectively) from

  * (i)

CM points or real quadratic closed geodesics;

  * (ii)

Siegel units.

Let us examine these two key ingredients in further detail.

Ingredient (i), in general, takes the form

(5) $G\,=_{\mathfrak{S}}\,\Theta(Z_{K,\psi}),$

where $Z_{K,\Psi}$ is a suitable Heegner cycle and the symbol
”$=_{\mathfrak{S}}$” means equality up to modular forms that pair to $0$ with
the Shimura class (see §1.5 for some details). More precisely, $Z_{K,\Psi}$ is
a formal linear combination of supersingular points in characteristic $N$,
obtained as a weighted combination of the mod $N$ reductions of CM elliptic
curves in the definite case (Theorem 2.2), and a linear combination of real
quadratic geodesics in the homology of $X_{0}(N)$ in the indefinite case
(Theorem 3.1). The equality (5) can be interpreted as coming from a certain
see-saw (Remark 1.4) although we give a rather direct proof.

The content of (ii) is the computation of the image of the Shimura class under
the adjoint (dual, in other words) of the theta lift. The outcome is an
expression

(6) $\Theta^{*}(\mathfrak{S})=\mbox{explicit {\it higher Eisenstein element}
$\mathfrak{U}_{N}$}.$

In the definite case, $\mathfrak{U}_{N}$ is obtained by restricting a suitable
Siegel unit to the supersingular locus in characteristic $N$. In the
indefinite case $\mathfrak{U}_{N}$ is built out of the modular symbol arising
from the (logarithmic derivative of the) same Siegel unit. In all cases, the
basic idea of proof is that $\Theta^{*}(\mathfrak{S})$ is uniquely
characterized by its behavior with respect to Hecke operators and so can be
proved to equal $\mathfrak{U}_{N}$, up to an irrelevant ambiguity, by a purely
Hecke-theoretic computation. Most of the work for the proof of (6) is given in
Chapter 4, building on ideas of Mazur, Merel and Lecouturier on (higher)
Eisenstein elements and the classical theory of modular units and modular
symbols.

Combining (5) and (6) leads to an identity of the form

(7) $\langle
G,\mathfrak{S}\rangle=\langle\Theta(Z_{K,\psi}),\mathfrak{S}\rangle=\langle
Z_{K,\psi},\Theta^{*}(\mathfrak{S})\rangle=\langle
Z_{K,\psi},\mathfrak{U}_{N}\rangle.$

In the definite setting, the right-hand quantity can be interpreted as the
discrete logarithm of an elliptic unit, obtained by evaluating the Siegel unit
attached to $\mathfrak{U}_{N}$ on the CM divisor attached to $Z_{K,\psi}$. In
the indefinite setting, the regulator involves only the logarithm of the
fundamental unit of $K$, and this fundamental unit emerges in $\langle
Z_{K,\psi},\mathfrak{U}_{N}\rangle$ from the eigenvalues of certain hyperbolic
matrices in $\Gamma_{0}(N)$. The details of these calculations, concluding
with the proof of Theorem 1.2, are supplied in Chapter 5.

It is worth insisting on a crucial feature of the dihedral case, namely, that
the desired units can be constructed explicitly, as CM values of modular units
in the definite case, or as eigenvalues of suitable matrices in
${\mathrm{SL}}_{2}(\mathbb{Z})$ in the indefinite case. This is what accounts
for Stark’s conjecture being known for the adjoint $L$-functions of dihedral
forms. It remains open, however, for the adjoint $L$-functions of so-called
exotic weight one forms with non-dihedral projective image. Although the
existence and essential uniqueness of the predicted unit is still guaranteed
by Dirichlet’s unit theorem, no analogue of the Kronecker limit formula
relating it to $L$-functions attached to $g$ is available. The numerical
experiments described in [HV] test Conjecture 1.1 numerically, but only in CM
dihedral cases that now fall under the purview of Theorem 1.2. The article
[Mar] provides numerical evidence for Conjecture 1.1 in several more
interesting instances where $g$ is exotic.

######  Remark 1.4.

It may be helpful to indicate the see-saw that underlies the crucial
computation (5). We emphasize, however, that the proofs in §2 and §3 do not
use this in any explicit way.111 However, the representation-theoretic
perspective appears to be indispensable to treat cases in which the Hecke
characters have more general ramification. Here we will proceed purely
formally.

Set

$G(L(2)\times
L(2))=\\{(g_{1},g_{2})\in\mathrm{GL}(2)\times\mathrm{GL}(2),\det(g_{1})=\det(g_{2})\\}.$

We examine the following see-saw:

$\textstyle{\theta_{\psi_{1}^{-1}}^{(N)}\boxtimes\theta_{\psi_{2}^{-1}}\in}$$\textstyle{[G(L(2)\times
L(2))]}$$\textstyle{[\mathrm{GO}(B)]\sim(B^{\times}\times
B^{\times})/\mathrm{GL}(1)}$$\textstyle{\ni\Theta(\mathfrak{S})}$$\textstyle{\mathfrak{S}\in}$$\textstyle{[\mathrm{GL}(2)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left[\mathrm{G}(\mathrm{O}(W_{1})\times\mathrm{O}(W_{2}))\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\ni\psi_{{}_{12}}\times\psi_{{}_{12^{\prime}}}}$

The arrow $\Theta^{*}$ from lower-left to upper-right is a realization of the
Jacquet-Langlands correspondence and we denote its formal adjoint simply by
$\Theta$. Of course $\mathfrak{S}$ is not in fact a characteristic zero
modular form, but let us proceed as if it were; in the end the proof uses
integral normalizations of the $\theta$-correspondence to get around this. The
see-saw principle and adjointness respectively give

$\langle\theta_{\psi_{1}^{-1}}^{(N)}\theta_{\psi_{2}^{-1}},\mathfrak{S}\rangle=\langle
Z_{K,\psi},\Theta^{*}(\mathfrak{S})\rangle=\langle\Theta(Z_{K,\psi}),\mathfrak{S}\rangle,$

where $Z_{K,\psi}$ arises from pushing forward
$\psi_{{}_{12}}\times\psi_{{}_{12^{\prime}}}$ under $j$. In the special case
$\psi_{2}=\psi_{1}^{-1}$ this recovers (5) from the point of view of the see-
saw formalism.

###  1.5. Notation

We will fix here some notation that is used throughout the paper. This
notation will also be introduced where we use it; we have gathered some of it
here as a convenient reference.

Throughout the paper, $K$ will denote a quadratic field, with ring of integers
$\mathfrak{o}$ and discriminant $D$. In §2 this field is imaginary, and in §3
it is real. The symbol $x\mapsto x^{\prime}$ denotes the nontrivial
automorphism of $K$, and we will allow ourselves to apply it to various
associated constructions (elements of $K$, ideals, characters, etc.)

The narrow class group of $K$ (i.e., the usual class group in the imaginary
case) is denoted by $\mathcal{C}$. In §3, $\mathcal{C}_{D}$ will denote the
ray class group of $K$ allowing level $\delta$, the different ideal of $K$.
The symbols $\psi_{1}$ and $\psi_{2}$ denote characters of $\mathcal{C}_{D}$
with inverse central characters, and we put

(8)
$\psi_{{}_{12}}:=\psi_{1}\psi_{2},\qquad\psi_{{}_{12^{\prime}}}:=\psi_{1}\psi_{2}^{\prime},$

which in all cases descend to characters of $\mathcal{C}$. In the special case
$\psi_{2}=\psi_{1}^{-1}$ which is germane to the proof of Theorem 1.2, the
above definitions simplify to

(9)
$\psi_{{}_{12}}=1,\quad\psi_{{}_{12^{\prime}}}=\psi_{1}/\psi_{1}^{\prime}:=\psi,\quad\mbox{
say}.$

We will use $L$ for a coefficient field for characters $\psi$ as above, i.e.
$L$ is a number field containing the values of $\psi:\mathcal{C}\mbox{ or
}\mathcal{C}_{D}\rightarrow L^{\times}$. In this context, $R$ will denote a
suitable ring of integers of $L$ (possibly with denominators at some primes).

We will often denote by $g$ (respectively $h$) the dihedral forms associated
to $\psi_{1}$ (respectively $\psi_{2}$) with associated Galois representation
$\rho_{g}:G_{\mathbb{Q}}\rightarrow\mathrm{GL}(V_{g})$. In this situation, we
will often denote

$G:=\mbox{trace to level $\Gamma_{0}(N)$ of
$\theta_{\psi_{1}^{-1}}(Nz)\theta_{\psi_{2}^{-1}}(z)$,}$

which in particular becomes the trace of
$\theta_{\psi_{1}}(z)\theta_{\psi_{1}^{-1}}(Nz)$ in the case
$\psi_{2}=\psi_{1}^{-1}$.

The integer $N>3$ always denotes a prime, and $Z$ denotes the ring
$\mathbb{Z}[\frac{1}{6N}]$. The modular curves $X_{0}(N)$ and $X_{1}(N)$ are
understood222 Note that the distinction between “stack” or the associated
coarse moduli scheme will make very little difference for our purposes; the
cover $X_{1}(N)\rightarrow X_{0}(N)$ is étale only when considering the
stacks, but in any case we are interested only in the
$(\mathbb{Z}/p^{t})$-subcover which is also étale over the scheme. to be
schemes over $Z$. We write $H^{1}_{\operatorname{et}}(X_{0}(N))$ and
$H^{1}_{\mathrm{B}}(X_{0}(N))$ to denote, respectively, the étale cohomology
of $X_{0}(N)$ as a scheme over $Z$, and the Betti cohomology of the Riemann
surface $X_{0}(N)(\mathbb{C})$.

We define the space of cusp forms $S_{2}(N)$ and the space of modular forms
$M_{2}(N)$ as (free) $Z$-modules accordingly and define

$S_{2}(N)^{\vee}=\operatorname{Hom}(S_{2}(N),Z)$

as their $Z$-linear duals. For $R$ a $Z$-algebra,
$S_{2}(N;R):=S_{2}(N)\otimes_{Z}R$ is similarly defined as the space of cusp
forms with coefficients in $R$, and likewise for $M_{2}(N;R)$. For an element
$f\in M_{2}(N;R)$ we will denote by $f^{(d)}\in M_{2}(Nd;R)$ the modular form
with $q$-expansion $f(q^{d})$.

Let $p$ be an odd prime $\geq 5$ and $p^{t}$ the largest power of $p$ dividing
$N-1$. Fix a surjective “discrete logarithm”

$\log:(\mathbb{Z}/N\mathbb{Z})^{\times}\longrightarrow\mathbb{Z}/p^{t}\mathbb{Z},$

where $p^{t}$ is the largest power of $p$ dividing $N-1$. Note that this
logarithm factors through the quotient $G_{N}$ of (10)

(10) $G_{N}:=(\mathbb{Z}/N\mathbb{Z})^{\times}/\langle\pm 1\rangle,$

since $p$ is assumed to be odd. This logarithm also extends uniquely to the
multiplicative group of the quadratic extension $\mathbb{F}_{N^{2}}$ and this
extension will also be denoted by $\log$. All formulas will be independent of
the choice of logarithm: both sides will scale the same way if one alters it.

######  Remark 1.5.

In the indefinite case the prime $N$ splits in $K$ and the correct definition
of the discrete logarithm entails the choice of one of the two prime divisors
of $N$. This choice is denoted $\mathfrak{N}$ and the need to pin down a
choice introduces a “breaking of symmetry” in the final formula Proposition
5.11 as well as in the intermediate calculations. The choice intervenes at the
beginning of §3.2.

Given two modular forms $F$ and $G$ of level $N$, the notation

$F=_{\mathfrak{S}}G$

means that “$F$ and $G$ have the same pairing with the Shimura class.”
(Strictly, the prime $p$ should have been included in the notation, but the
choice of $p$ is understood to be fixed.) More precisely, $F=_{\mathfrak{S}}G$
means that:

  1. (1)

$F,G$ lie inside $M_{2}(N;R)$ for $R$ the ring of $p$-integers in some
algebraic number field, and

  2. (2)

the reductions $\bar{F},\bar{G}\in
M_{2}(N;R/p^{t}R)=H^{0}(X_{0}(N)_{R},\Omega^{1})$ have the same pairing with
$\mathfrak{S}$ under the Serre duality pairing

(11) $H^{0}(X_{0}(N)_{R/p^{t}},\Omega^{1})\otimes
H^{1}(X_{0}(N)_{R/p^{t}},\mathcal{O})\rightarrow R/p^{t}$

obtained by taking the cup product to $H^{1}(X_{0}(N)_{R/p^{t}},\Omega^{1})$
and using the “trace” map on the latter. 333In the current setting, if $t>1$,
this can be defined using Grothendieck duality for the structural morphism
$X_{0}(N)_{R/p^{t}}\rightarrow\mathrm{Spec}\ R/p^{t}$, after e.g. adding
auxiliary level structure to remove any “stacky” structure. This identifies
$H^{1}(\Omega^{1})$ with
$\operatorname{Hom}(\mathrm{R}\pi_{*}\mathcal{O},R/p^{t})$, homomorphisms in
the derived category of $R/p^{t}$ modules. In particular, each element of
$H^{1}(\Omega^{1})$ induces (by passage to $H^{0}$) a map $R/p^{t}\rightarrow
R/p^{t}$, i.e. an element of $R/p^{t}$.

This notion is readily seen to be independent of $R$, i.e, compatible with
extension of scalars in the obvious sense.

###  1.6. Acknowledgements

The authors are grateful to Jan Vonk for the valuable insights which guided
their approach to proving the main theorem of Chapter 3. They also thank Frank
Calegari, Hélène Esnault, and Alice Pozzi for stimulating discussions
surrounding the topics of this paper.

The anonymous referee made a very careful reading of the paper and made
several valuable corrections and suggestions. We thank her or him for their
substantial effort, which has substantially improved the paper.

The authors are happy to express their appreciation to Gopal Prasad by
dedicating this article to him. The second- and fourth-named authors would
like to take this opportunity to add a few personal words:

M.H.: “ I vividly remember Gopal’s patient and enthusiastic explanation of his
own work on arithmetic groups, and his sincere interest in my work, when we
first met, just one year after my Ph.D. And I am grateful for Gopal’s
generosity and friendship at all our subsequent meetings, on three continents,
over the following decades.”

A.V.: “The clarity and beauty of Gopal’s work on $p$-adic groups speaks for
itself, and has influenced my work on too many occasions to readily enumerate.
And it is with much warmth that I recall the kindness that Gopal has showed
throughout my career; I still remember clearly that when I, as a graduate
student, visited the University of Michigan, Gopal took the time to speak with
me and encourage my work. It is therefore with the greatest pleasure that I
dedicate this paper to him, with admiration for a great mathematical career
and the best wishes for the future.”

##  2\. A trace identity for definite theta series

In [Gr87] and [GZ], Gross and Zagier proved a formula for the central critical
value (resp. derivative) of the $L$-function attached to the convolution of a
cusp form $f$ of weight $2$ and a theta series $g$ of weight $1$ associated to
a character of an imaginary quadratic field $K$. A substantial step in the
proof of both formulas is the computation, for a given prime $N$, of the trace
of the product $g(z)E(Nz)$ of $g$ and a suitable Eisenstein series $E$ to the
space of modular forms of level $N$.

In this note we need to carry the computation of the trace of the product
$g(z)h(Nz)$ of two cuspidal theta series attached to ray class characters of
$K$. We did not attempt to adapt the computations of [Gr87, §7,8,9] to the
present setting, but rather follow a different method invoking the Weil
representation of ${\mathrm{SL}}_{2}(\mathbb{A}_{f})$ (where $\mathbb{A}_{f}$
denotes the ring of finite adèles) on the space of Schwartz functions on the
adélic points of the underlying quadratic spaces.

###  2.1. Setup on Heegner points

The computation will be carried out in slightly greater generality than in
Theorem 1.2 of the introduction. Let $K$ be an imaginary quadratic field of
odd discriminant $D$ with maximal order $\mathfrak{o}$ and let
$\mathcal{C}={\mathrm{Pic}}(\mathfrak{o})$ denote the class group. For $I$ an
ideal, note that the image $I^{\prime}$ by conjugation defines the same class
in $\mathcal{C}$ as $I^{-1}$. Denote by $a$ the number of distinct prime
factors of $D$.

Let $N$ be an odd prime with the property that $-N$ is a square modulo $D$.
When $D$ is prime, as assumed in the introduction, this condition is
equivalent to $N$ being inert in $K$; in general it always implies that $N$
remains inert but is a stricter condition.

Fix an algebraic closure $\overline{\mathbb{F}_{N}}$, and let
$\mathbb{F}_{N^{2}}$ be the subfield of size $N^{2}$.

Choose an auxiliary odd prime $q$ such that $q\equiv-N\,(\mathrm{mod}\,D)$. An
elementary computation of quadratic symbols shows that $q$ is split in $K$.
Assume throughout that $q$ is such that the ideals $\mathfrak{q}$,
$\overline{\mathfrak{q}}$ in $K$ above $q$ are principal. The existence of
such $q$ is guaranteed by Cebotarev density theorem.

A calculation with Hilbert symbols (cf. [Vig, §2.1]) shows that

(12) $B\simeq K+Kj,\quad\mbox{ with }j^{2}=-qN\quad\mbox{and}\quad
zj=jz^{\prime},\ \ \mbox{ for all }z\in K.$

is the definite quaternion algebra over $\mathbb{Q}$ of discriminant $N$. Let
$b\mapsto b^{\prime}$ denote the canonical anti-involution on $B$; it
coincides with complex conjugation when restricted to $K$. Let
$n(b)=bb^{\prime}$ denote the reduced norm on $B$.

An orientation on a maximal order ${\mathcal{M}}$ in $B$ is a choice of
homomorphism

$\mathbf{o}:{\mathcal{M}}\rightarrow\mathbb{F}_{N^{2}}$

onto $\mathbb{F}_{N^{2}}$. Note that ${\mathcal{M}}$ admits exactly two
possible orientations. Two oriented maximal orders
$\vec{{\mathcal{M}}}_{1}=({\mathcal{M}}_{1},\mathbf{o}_{1})$,
$\vec{{\mathcal{M}}}_{2}=({\mathcal{M}}_{2},\mathbf{o}_{2})$ are equivalent if
there exists an isomorphism $i:{\mathcal{M}}_{1}\rightarrow{\mathcal{M}}_{2}$
satisfying $\mathbf{o}_{1}=\mathbf{o}_{2}\circ i$.

Write ${\mathrm{Pic}}(B)$ for the set of equivalence classes of oriented
maximal orders. By a classical result of Deuring (cf. [Vo, §42.3]),
${\mathrm{Pic}}(B)$ is in bijection with the set $\mathcal{E}$ of isomorphism
classes of supersingular elliptic curves over $\overline{\mathbb{F}}_{N}$ as
follows: we associate to an elliptic curve $E$ the order $\mathrm{End}(E)$,
which acquires an orientation by considering its action on the tangent space.

Fix a basepoint $\vec{{\mathcal{M}}}\in{\mathrm{Pic}}(B)$ containing
$\mathfrak{o}\oplus\mathfrak{o}j$. Define the map

(13) $\iota:{\mathrm{Pic}}(\mathfrak{o})\longrightarrow{\mathrm{Pic}}(B)$

that takes an ideal class $I$ to the oriented maximal order
$\iota(I)=I^{-1}\vec{{\mathcal{M}}}I$.

###  2.2. Statement of the trace identity

Define $\mathrm{Div}(\mathcal{E})$ to be the module of $\mathbb{Z}$-valued
functions on ${\mathrm{Pic}}(B)$, equipped with its natural action of the
Hecke algebra ${\mathbb{T}}$ as described e.g. in [Gr87, §4]. If
$\vec{{\mathcal{M}}}\in{\mathrm{Pic}}(B)$, let ${\mathcal{M}}$ denote the
underlying unoriented order and set
$w_{\vec{{\mathcal{M}}}}=\frac{1}{2}|{\mathcal{M}}^{\times}|$. Denote by
$e_{x}\in\mathrm{Div}(\mathcal{E})$ the characteristic function of
$x\in{\mathrm{Pic}}(B)$ and set as in the introduction
$\Sigma_{0}=\sum_{x}\frac{e_{x}}{w_{x}}\in\mathrm{Div}(\mathcal{E})\otimes\mathbb{Q}$.
The space $\mathrm{Div}(\mathcal{E})$ is endowed with a natural symmetric
bilinear form

(14) $\langle\ ,\
\rangle:\mathrm{Div}(\mathcal{E})\times\mathrm{Div}(\mathcal{E})\rightarrow\mathbb{Z},\qquad\langle
e_{x},e_{y}\rangle:=w_{x}\delta_{xy},$

relative to which the Hecke operators $T_{\ell}$ are self-adjoint for all
$\ell$, including for $\ell=N$.

The Jacquet-Langlands correspondence identifies $\mathrm{Div}(\mathcal{E})$
and $M_{2}(\Gamma_{0}(N))$ as Hecke modules. This identification can be
described explicitly by means of the $\Theta$-correspondence, which is the
Hecke-equivariant map

(15)
$\Theta:\mathrm{Div}(\mathcal{E})\otimes_{\mathbb{T}}\mathrm{Div}(\mathcal{E})\rightarrow
M_{2}(\Gamma_{0}(N))$

given by (cf. e.g. [Em] and [Gr87, Prop. 5.6])

(16)
$\Theta(\phi_{1}\otimes\phi_{2})=\frac{1}{2}\langle\phi_{1},\Sigma_{0}\rangle\langle\phi_{2},\Sigma_{0}\rangle+\sum_{m\geq
1}\langle\phi_{1},T_{m}\phi_{2}\rangle q^{m}.$

######  Remark 2.1.

Formula (16) makes it clear that

$\Theta(\phi_{1}\otimes\phi_{2})=\Theta(\phi_{2}\otimes\phi_{1}),$

because each $T_{m}$ is self-adjoint, including for $m=N$.

Given a character $\psi:{\mathrm{Pic}}(\mathfrak{o})\rightarrow L^{\times}$
with values in some finite field extension $L/\mathbb{Q}$, define

(17)
$[\psi]:=\iota_{*}(\psi)=\sum_{I\in{\mathrm{Pic}}(\mathfrak{o})}\psi(I)\iota(I)\in\mathrm{Div}(\mathcal{E})\otimes
L.$

The main result of this section is the following. Let $\theta_{\psi}$ denote
the theta series associated to $\psi$ as recalled in (22) below. Note also
that $\theta_{\psi}=\theta_{\psi^{-1}}$ because characters of
${\mathrm{Pic}}(\mathfrak{o})$ are anticyclotomic in the sense that
$\psi^{\prime}=\psi^{-1}$; this accounts for the discrepancy in phrasing
between the statement below, and the analogous Theorem 3.1 in the RM scenario.

######  Theorem 2.2.

Let $\psi_{1}$ and $\psi_{2}$ be characters of $\mathcal{C}$ and let
$\theta_{\psi_{i}}$ be the newforms associated to $\psi_{i}$ (equivalently: to
$\psi_{i}^{-1}$). Put
$\psi_{{}_{12}}=\psi_{1}\psi_{2},\psi_{{}_{12^{\prime}}}=\psi_{1}\psi_{2}^{\prime}$.
Then there exists $p_{0}$ such that, for any $N$ and any $p\geq p_{0}$ with
$p\mid N-1$:

(18)
$\mathsf{Tr}^{ND}_{N}(\theta_{\psi_{1}}(Nz)\theta_{\psi_{2}}(z))\,=_{\mathfrak{S}}\,4\cdot\Theta([\psi_{{}_{12}}]\otimes[\psi_{{}_{12^{\prime}}}]).$

where, as in (5), the notation “$f=_{\mathfrak{S}}g$” means that both modular
forms have the same pairing with the Shimura class of level $N$. If $D$ is
prime, (18) is a strict identity (not just up to $\mathfrak{S}$) for all
primes $p$.

We expect a similar trace identity to hold for general ray class characters
$\psi_{1}$, $\psi_{2}$ of $K$ with opposite central character, which amounts
to allowing the ring class characters $\psi_{{}_{12}}$,
$\psi_{{}_{12^{\prime}}}$ to have arbitrary conductor $c\geq 1$. In such
generality however we do not expect the constant to be as simple as $C=4$ and
(18) should hold up to a suitable constant $C=C(\psi_{1},\psi_{2})$ that
depends on $\psi_{1}$, $\psi_{2}$ but not on $N$. The reader is referred to R.
Zhang’s forthcoming Ph.D thesis [Zh] for the proof in greater generality in
the adelic language.

Theorem 2.2 above and Theorem 3.1 below cover the simplest non-trivial
settings in both definite and indefinite cases. The proofs are different
because we chose to be as direct as possible in each case and avoid
repetition, but the approaches in §2 and §3 are in some ways complementary.

###  2.3. Summary of the proof

Theorem 2.2 will be proved subject to three Propositions given below; these
will be proved in the remaining subsections.

Define ${\mathcal{O}}\subset B$ via

(19) ${\mathcal{O}}={\mathcal{O}}(q):=\mathfrak{o}\oplus\mathfrak{o}j.$

An elementary computation using [Vig, 1.4.7] shows that
${\mathcal{O}}={\mathcal{O}}(q)$ has square-free discriminant $DNq$, and
therefore (cf. [Vig, 3.5.3]) ${\mathcal{O}}$ is an Eichler order, that is to
say, the intersection of two maximal orders. Let us fix now and for the rest
of this section a maximal order ${\mathcal{M}}\supset{\mathcal{O}}$ as well as
an orientation on it. All other orders containing ${\mathcal{O}}$ can be
obtained from ${\mathcal{M}}$ as

(20) ${\mathcal{M}}_{d}:=\mathfrak{d}^{-1}{\mathcal{M}}\mathfrak{d},$

where $d$ ranges over positive divisors $d\mid Dq$ and $\mathfrak{d}$ is an
ideal in $\mathfrak{o}$ of norm $d$. This is because locally at every prime
$\ell$ dividing $Dq$ there are exactly two local maximal orders containing
${\mathcal{O}}\otimes\mathbb{Z}_{\ell}$: one is obtained from the other by
conjugating by any element of norm $\ell$ normalizing
${\mathcal{O}}\otimes\mathbb{Z}_{\ell}$ (cf. e.g. [Vig, §3.5]).

If $I_{1},I_{2}$ are ideal classes for $\mathfrak{o}$ we can form

$I_{1}{\mathcal{O}}I_{2}=I_{1}I_{2}\oplus I_{1}I_{2}^{\prime}j\subset B.$

By definition, the left hand side means the additive subgroup of $B$ generated
by all threefold products $i_{1}\cdot o\cdot i_{2}$.

We regard $K$ and $B$ as quadratic spaces by means of the norm and reduced
norm respectively. For every ideal $I$ in either $V=K$ or $B$, let

(21) $\theta_{I}=\theta(I)=\sum_{a\in I}q^{\frac{n(a)}{n(I)}}$

denote the theta series associated to $I$; here $n(I)$ stands for the single
positive generator of the ideal of $\mathbb{Q}$ spanned by the norms of all
elements in $I$. The theta series $\theta_{I}$ is a modular form of weight
$[V:\mathbb{Q}]/2$. With this normalization, $\theta_{I}$ only depends, in the
case $V=K$, on the class of $I$ up to principal ideals, since
$\theta_{I}=\theta_{Ix}$ for any $x\in K^{\times}$. Moreover, for any
character of $\mathcal{C}$,

(22) $\theta_{\psi}=\sum_{I\in\mathcal{C}}\psi(I)^{-1}\theta_{I}$

is the new theta series associated to $\psi$, a classical modular newform of
weight $1$, level $D$ and nebentype character $\chi_{K}$, the quadratic
Dirichlet character associated to $K/\mathbb{Q}$. (As mentioned above, in the
current situation one could omit the inverse on the right hand side, but the
formula above is valid under less restrictive ramification conditions and
facilitates comparison with the RM case.)

For any $d\geq 1$, recall that $\theta^{(d)}(q):=\theta(q^{d})$. We will
include forward references to Propositions in the RM case that play a similar
role, although because of the slightly different setups the statements are not
entirely parallel.

######  Proposition 2.3.

(See §2.4, cf. also Prop. 3.10). For any pair of classes $I_{1},I_{2}$ of
$\mathcal{C}$, we have

(23)
$\mathsf{Tr}^{DN}_{N}\theta(I_{1}I_{2})\theta^{(N)}(I_{1}I_{2}^{\prime})=\frac{1}{2}\sum_{d|Dq}\theta(I_{1}{\mathcal{M}}_{d}I_{2}),$

where, on the left, $\mathsf{Tr}$ is the trace from level $\Gamma_{0}(DN)$ to
level $\Gamma_{0}(N)$, and ${\mathcal{M}}_{d}$ is as in (20).

######  Proposition 2.4.

(See §2.5, cf. also Prop. 3.4). For any pair of ideal classes $I,J$ we have

(24) $\theta(J^{\prime}{\mathcal{M}}I)=2\cdot\Theta(e_{I}\otimes e_{J}).$

Here $e_{I},e_{J}$ are as in §2.2, where we use $\iota$ of (13) to identify
$I,J$ with elements of $\mathrm{Pic}(B)$.

For every quadratic character $\chi$ of $\mathcal{C}$, set

(25)
$G_{DN}(\chi)=\theta_{\psi_{1}^{-1}\chi}^{(N)}\,\cdot\,\theta_{\psi_{2}^{-1}\chi}\quad\mbox{
and }\quad
G_{DN}:=\sum_{\chi\in(\mathcal{C}/\mathcal{C}^{2})^{*}}G_{DN}(\chi).$

In the case when $D$ is prime, the only quadratic character is trivial, so
$G_{DN}(\chi)=G_{DN}$, and the following Proposition is vacuous:

######  Proposition 2.5.

(See §2.6.) With the notation of (5)

$\mathsf{Tr}^{DN}_{N}(G_{DN}(\chi))\,\sim_{\mathfrak{S}}\,\mathsf{Tr}^{DN}_{N}(G_{DN}(\chi^{\prime})),$

for all quadratic characters $\chi,\chi^{\prime}$ of $\mathcal{C}$, so long as
the prime $p$ is sufficiently large relative to $D$.

Let us see how these three results imply the theorem. For every choice of
quadratic character $\chi$ we have

(26)
$G_{DN}(\chi)=\sum_{I,J\in\mathcal{C}}\chi(IJ)\psi_{1}(I)\psi_{2}(J)\theta_{I}^{(N)}\theta_{J}.$

But $\sum_{\chi}\chi(IJ)$ is zero unless $IJ$ is a square inside
$\mathcal{C}$; in that case, it equals $\\#\mathcal{C}[2]$. Moreover, any pair
$(I,J)$ with $IJ\in\mathcal{C}^{2}$ is of the form
$(I_{1}I_{2}^{\prime},I_{1}I_{2})$ for precisely $\\#\mathcal{C}[2]$ pairs
$(I_{1},I_{2})$ and then
$\psi_{1}(I)\psi_{2}(J)=\psi_{12}(I_{1})\psi_{12^{\prime}}(I_{2}^{\prime}).$
It follows that

(27)
$G_{DN}=\sum_{I_{1},I_{2}\in\mathcal{C}}\psi_{{}_{12}}(I_{1})\psi_{{}_{12^{\prime}}}(I_{2}^{\prime})\theta(I_{1}I_{2})\theta^{(N)}(I_{1}I_{2}^{\prime}).$

Using Proposition 2.3 we get

(28)
$\mathsf{Tr}^{DN}_{N}(G_{DN})=\frac{1}{2}\sum_{I_{1},I_{2}\in\mathcal{C}}\psi_{{}_{12}}(I_{1})\psi_{{}_{12^{\prime}}}(I_{2}^{\prime})\big{(}\sum_{d\mid
Dq}\theta(\mathfrak{d}^{-1}I_{1}{\mathcal{M}}I_{2}\mathfrak{d})\big{)}.$

Note however that all terms for every fixed $d$ in (28) are equal: this
follows after reindexing
$(I_{1},I_{2})\leftrightarrow(\mathfrak{d}^{-1}I_{1},I_{2}\mathfrak{d})$ and
recalling that the class of the ideal $\mathfrak{d}$ has order $2$ in
$\mathcal{C}$ when $d\mid D$, while the class of $\mathfrak{q}$ is trivial.
The number of such terms is equal to $2^{a+1}$, with $a$ the number of prime
factors of $D$. Hence

$\displaystyle\mathsf{Tr}^{DN}_{N}(G_{DN})$ $\displaystyle=$ $\displaystyle
2^{a}\sum_{I_{j}}\psi_{{}_{12}}(I_{1})\psi_{{}_{12^{\prime}}}(I_{2}^{\prime})\theta(I_{1}{\mathcal{M}}I_{2})$
$\displaystyle=$ $\displaystyle
2^{a}\sum_{I_{j}}\psi_{{}_{12}}(I_{1}^{\prime})\psi_{{}_{12^{\prime}}}(I_{2}^{\prime})\theta(I_{1}^{\prime}{\mathcal{M}}I_{2}).$

Proposition 2.4, as well as the symmetry of $\Theta$ in its arguments, can be
invoked to transform the right hand side, to get

$\mathsf{Tr}^{DN}_{N}(G_{DN})=2^{a+1}\sum\psi_{12}^{-1}(I_{1})\psi_{12^{\prime}}^{-1}(I_{2})\Theta(e_{I_{2}}\otimes
e_{I_{1}})=2^{a+1}\Theta([\psi_{{}_{12}}^{-1}]\otimes[\psi_{{}_{12^{\prime}}}^{-1}]).$

This directly yields Theorem 2.2 when $D$ is prime, as in that case the order
of $\mathcal{C}$ is odd and hence $G_{DN}=G$. When $D$ is composite,
Proposition 2.5 shows that we can replace $G_{DN}$ on the left by
$2^{a-1}G_{DN}(1)$ if we are only interested in pairing with the Shimura
class, since $\mathcal{C}[2]$ has rank $a-1$ by genus theory (cf. e.g. [Co,
§13]). Unwinding the notation

$\mathsf{Tr}^{ND}_{N}(\theta_{\psi_{1}^{-1}}(Nz)\theta_{\psi_{2}^{-1}}(z))\sim_{\mathfrak{S}}\,4\cdot\Theta([\psi^{-1}_{{}_{12}}]\otimes[\psi^{-1}_{{}_{12^{\prime}}}]).$

This proves Theorem 2.2 after recalling that
$\theta_{\psi_{i}}=\theta_{\psi_{i}^{-1}}$.

###  2.4. Proof of Proposition 2.3

We must show that

(30)
$\mathsf{Tr}^{DN}_{N}\theta(I_{1}I_{2})\theta^{(N)}(I_{1}I_{2}^{\prime})=\frac{1}{2}\sum_{d|Dq}\theta(I_{1}{\mathcal{M}}_{d}I_{2}).$

Let $T_{q}$ denote the Hecke operator at $q$ and $\mathsf{Tr}^{N_{2}}_{N_{1}}$
the trace map from modular forms of level $N_{2}$ to level $N_{1}$, for any
$N_{1}\mid N_{2}$. Then

$\mathsf{Tr}^{DNq}_{DN}\theta^{(Nq)}_{J}=T_{q}\cdot\theta^{(N)}_{J}=\theta^{(N)}_{J\mathfrak{q}}+\theta^{(N)}_{J\mathfrak{q}^{\prime}}=2\theta^{(N)}_{J},$

where the second equality follows from e.g. [Kan, §2], and the third since we
are supposing that $\mathfrak{q}$ is principal. Hence

$\mathsf{Tr}^{DNq}_{N}\theta_{J_{1}}\theta^{(Nq)}_{J_{2}}=\mathsf{Tr}^{DN}_{N}\theta_{J_{1}}\left(\mathsf{Tr}^{DNq}_{DN}\theta^{(Nq)}_{J_{2}}\right)=2\mathsf{Tr}^{DN}_{N}\theta_{J_{1}}\theta^{(N)}_{J_{2}}.$

Taking $J_{1}=I_{1}I_{2},J_{2}=I_{1}I_{2}^{\prime}$ and switching sides we get

(31)
$\mathsf{Tr}^{DN}_{N}\theta_{I_{1}I_{2}}\theta^{(N)}_{I_{1}I_{2}^{\prime}}=\frac{1}{2}\mathsf{Tr}^{DNq}_{N}\theta_{I_{1}{\mathcal{O}}I_{2}}.$

where we noted that $\theta_{I_{1}{\mathcal{O}}I_{2}}=\theta_{I_{1}I_{2}\oplus
I_{1}I_{2}^{\prime}j}=\theta_{I_{1}I_{2}}\theta^{(Nq)}_{I_{1}I_{2}^{\prime}}$
since $j^{2}=-qN$. So Proposition 2.3 reduces to

######  Proposition 2.6.

For any pair of classes $I_{1},I_{2}$ of ${\mathrm{Pic}}(\mathfrak{o})$,

(32)
$\mathsf{Tr}^{DNq}_{N}(\theta_{I_{1}{\mathcal{O}}I_{2}})=\sum_{d|Dq}\theta(I_{1}{\mathcal{M}}_{d}I_{2}).$

In order to prove Proposition 2.6, note that – with $V=B$ or $K$ as before –
the rule that associates to every lattice $L$ a modular form $\theta_{L}$ of
weight $\dim(V)/2$ may be extended to the space of Schwartz functions on
$V\otimes\mathbb{A}_{f}$ where $\mathbb{A}_{f}$ denotes the ring of finite
adèles of $\mathbb{Q}$ (cf. e.g. [GH] for background). Namely, such a function
may be identified with a function $\Phi$ supported on some lattice $L\subset
V$ and constant on the cosets of a sublattice of $L$. We can form the
$\theta$-function

(33) $\theta_{\Phi}:=\sum_{z\in V}\Phi(z)q^{Q(z)}.$

with $Q$ the norm form (or, as we will actually use, a rescaling of it). This
is compatible with the previous definition in the sense that
$\theta_{[L]}=\theta_{L}$, where $[L]$ is the characteristic function of the
closure of a lattice $L$ inside $V\otimes\mathbb{A}_{f}$.

The function $\Phi\mapsto\theta_{\Phi}$ is equivariant for the action of
${\mathrm{SL}}_{2}(\mathbb{A}_{f})$ on Schwartz functions arising from the
Weil representation on one side, and the natural action on the space of
modular forms on the other; this is a straightforward consequence of the
adelic interpretation of $\theta$-series; we will give a more detailed sketch
of a similar equivariance in the more complicated RM setting in §3.5. This
equivariance implies that the trace, from level $DNq$ to $N$, of
$\theta_{I_{1}{\mathcal{O}}I_{2}}$, can be computed by first computing the
corresponding trace

(34) $\mathsf{Tr}^{DNq}_{N}\mbox{ of the Schwartz function
$[I_{1}{\mathcal{O}}I_{2}]$}.$

Proposition 2.6 thus follows after computing the trace of
$[I_{1}{\mathcal{O}}I_{2}]$ with reference to the
${\mathrm{SL}}_{2}(\mathbb{A}_{f})$-action on Schwartz functions, taking into
account that we have an equality of rescaling factors
$n(I_{1}{\mathcal{M}}_{d}I_{2})=n(I_{1}{\mathcal{O}}I_{2})$ (this can be
readily deduced from the fact that $I_{1},I_{2}$ are locally principal).

This Weil representation is a tensor product of representations of
${\mathrm{SL}}_{2}(\mathbb{Q}_{\ell})$ on Schwartz functions on
$V\otimes\mathbb{Q}_{\ell}$. We review the formulas in §3.5 and will summarize
them here. Given a prime $\ell$, let
$\mu:\mathbb{Q}_{\ell}\rightarrow\mathbb{C}^{\times}$ be the restriction of
the standard character of $\mathbb{A}/\mathbb{Q}$ which is given by $x\mapsto
e^{2\pi ix}$ on $\mathbb{R}$ and is trivial on each $\mathbb{Z}_{p}$. In
particular $\mu$ is trivial on $\mathbb{Z}_{\ell}$ but not on
$\ell^{-1}\mathbb{Z}_{\ell}$. For any $t\in\mathbb{Q}_{\ell}$, denote
$m(t)=\bigl{(}\begin{smallmatrix}1&t\\\ 0&1\end{smallmatrix}\bigr{)}$ and set
$w=\bigl{(}\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\bigr{)}$. Given a
Schwartz function $\Phi_{\ell}$ on $V\otimes\mathbb{Q}_{\ell}$:

(35) $\displaystyle m(t)\cdot\Phi_{\ell}(x)=\mu(t\langle
x,x\rangle)\Phi_{\ell}(x)$ $\displaystyle\mbox{ for any
}\,t\in\mathbb{Q}_{\ell},$ $\displaystyle
w\cdot\Phi_{\ell}(y)=\gamma_{\ell}\int_{V\otimes\mathbb{Q}_{\ell}}\Phi_{\ell}(x)\mu(\langle
y,x\rangle)dx.$

where, in particular, $\gamma_{\ell}=1$ for $\ell$ not dividing $N$. Here $dx$
is taken to be the self-dual Haar measure.

The desired trace from (34) can be calculated piecewise at every prime
$\ell\mid Dq$ and then packaging together the local outputs.

######  Lemma 2.7.

Let $\ell$ be a prime divisor of $Dq$. Let $(W,\langle\,,\rangle)$ be the
quadratic space over $\mathbb{Q}_{\ell}$ given by $K\otimes\mathbb{Q}_{\ell}$
equipped with the norm form divided by $N(I_{1}I_{2})$, and let $L\subset W$
be a maximal integral lattice. Let
$(W^{\prime},L^{\prime},\langle\,,\rangle^{\prime})$ be obtained from
$(W,L,\langle\,,\rangle)$ by multiplying the form $\langle\,,\rangle$ by
$-qN$.

Then – for the Weil representation action of
${\mathrm{SL}}_{2}(\mathbb{Q}_{\ell})$ on Schwartz functions on $W\oplus
W^{\prime}$ – the characteristic function $1_{L\oplus L^{\prime}}$ of $L\oplus
L^{\prime}$ is invariant by
$\Gamma_{0}(\ell)\subset{\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})$, and

(36)
$\mathsf{Tr}^{{\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})}_{\Gamma_{0}(\ell)}1_{L\oplus
L^{\prime}}=1_{{\mathcal{M}}_{+}}+1_{{\mathcal{M}}_{-}},$

where ${\mathcal{M}}_{\pm}$ are the two self-dual integral lattices containing
$(L\oplus L^{\prime})$.

Before we prove Lemma 2.7 we explain why it implies Proposition 2.6. There is
no loss of generality in choosing $I_{1},I_{2}$ relatively prime to $Dq$. It
follows from (12) that $W\oplus W^{\prime}$ is isometric to
$B\otimes\mathbb{Q}_{\ell}$ with its reduced norm form, and this
identification carries $L\oplus L^{\prime}$ to the closure of
$I_{1}\mathcal{O}I_{2}$. Thus, combining together (37) at all primes $\ell\mid
Dq$, Proposition 2.6 follows after noticing that if we take the self-dual
lattice ${\mathcal{M}}_{+}$ to be the localization at $\ell$ of the global
maximal order ${\mathcal{M}}_{d}$ for some $d$ with $\ell\nmid d$, then
${\mathcal{M}}_{-}={\mathcal{M}}_{d\ell}\otimes\mathbb{Z}_{\ell}$ .

###### Proof.

(of Lemma 2.7) Write $\mathbf{e}$ for the characteristic function of $L\oplus
L^{\prime}$ and $\mathbf{e}^{*}$ for the characteristic function of the dual
lattice $(L\oplus L^{\prime})^{*}$. Note that we have inclusions

$(L\oplus L^{\prime})\subset{\mathcal{M}}_{+},{\mathcal{M}}_{-}\subset(L\oplus
L^{\prime})^{*},$

with both inclusions of index $\ell$, and indeed the quotient $\frac{(L\oplus
L^{\prime})^{*}}{L\oplus L^{\prime}}$ is isomorphic to
$(\mathbb{Z}/\ell\mathbb{Z})^{2}$ where the induced
$\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}$-valued quadratic form takes the form
$(x_{1},x_{2})\mapsto\ell^{-1}(x_{1}^{2}-x_{2}^{2})$; in these coordinates
${\mathcal{M}}_{\pm}$ correspond to $x_{1}=\pm x_{2}$. Invariance of
$\mathbf{e}$ by $\Gamma_{0}(\ell)$ follows readily from the definitions. Now a
set of coset representatives for $\Gamma_{0}(\ell)$ in
${\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})$ is

$\\{w\\}\cup\\{wm(t)w:t\in\mathbb{Z}/\ell\mathbb{Z}\\}.$

Note that $w\mathbf{e}=\ell^{-1}\mathbf{e}^{*}$:

  * -

for $y\not\in(L\oplus L^{\prime})^{*}$, $w\mathbf{e}(y)$ is the integral on
$L\oplus L^{\prime}$ of the character $\mu(\langle y,x\rangle)$, which
vanishes since that character is not trivial;

  * -

for $y\in(L\oplus L^{\prime})^{*}$ we have $w\mathbf{e}(y)=\
\mathrm{vol}(L)\mathrm{vol}(L^{\prime})=\ell^{-1}$ (the self-dual Haar measure
on $W\oplus W^{\prime}$ assigns mass $\ell^{-1}$ to $L\oplus L^{\prime}$).

It thus follows that

$\left(\sum_{t\in\mathbb{Z}/\ell\mathbb{Z}}m(t)\right)w\mathbf{e}=1_{S},$

where $S=\\{x\in(L\oplus L^{\prime})^{*}:\langle
x,x\rangle\in\mathbb{Z}_{\ell}\\}.$ But $S$ is just the union of
${\mathcal{M}}_{+}$ and ${\mathcal{M}}_{-}$, and also
${\mathcal{M}}_{+}\cap{\mathcal{M}}_{-}=L\oplus L^{\prime}$. Thus
$1_{S}=1_{{\mathcal{M}}_{+}}+1_{{\mathcal{M}}_{-}}-\mathbf{e}$, and we deduce
that

(37) $\mathsf{Tr}\ \mathbf{e}\ \ =\ \
w\mathbf{e}+w1_{{\mathcal{M}}_{+}}+w1_{{\mathcal{M}}_{-}}\\!\\!-w\mathbf{e}\ \
=\ \ 1_{{\mathcal{M}}_{+}}+1_{{\mathcal{M}}_{-}}.$

∎

###  2.5. Proof of Proposition 2.4

Let $E_{I}$ denote the supersingular elliptic curve associated to $\iota(I)$
and $e_{I}$ for the corresponding element in $\mathrm{Div}(\mathcal{E})$. Set
$w_{I}=w_{\iota(I)}$.

In order to prove (24), it suffices to show that both sides have the same
Fourier coefficients for all $m\geq 1$. The $m$-th Fourier coefficient of the
r.h.s. of (24) is

(38) $2a_{m}(\Theta(e_{I}\otimes e_{J}))=2\langle
T_{m}e_{I},e_{J}\rangle=2w_{J}B_{I,J}(m).$

Here $B(m)$ is the $m$-th Brandt matrix and $B_{I,J}(m)$ is the entry in
$B(m)$ associated to $E_{I}$ and $E_{J}$ (cf. [Gr87, §1, §2]). The equalities
in (38) follow from the definition of $\Theta$ in (16) and [Gr87, 4.4, 4.5,
4.6]. Since $2w_{J}=|\mathrm{Aut}(E_{J})|$, it follows from [Gr87, Prop. 2.3]
that $2w_{J}B_{I,J}(m)$ is also equal to the number of isogenies of degree $m$
from the supersingular elliptic curve $E_{I}$ to $E_{J}$; see also the proof
of [Gr87, Prop. 2.7 (6)] in p. 128 of loc. cit. The $\mathbb{Z}$-module of
such isogenies is identified with $J^{-1}{\mathcal{M}}I$, where the degree is
identified with $z\mapsto n(z)n(J)/n(I)$ (cf. [Gr87, 2.1] combined with the
definition of $M_{ij}$ in p. 118 of loc. cit.). Consequently

$2a_{m}(\Theta(e_{I}\otimes e_{J}))=|\\{z\in
J^{-1}{\mathcal{M}}I:n(z)n(J)/n(I)=m\\}|.$

This in turn is the $m$-th Fourier coefficient of the l.h.s. of (24), as
$J^{-1}{\mathcal{M}}I$ is homothetic to $J^{\prime}{\mathcal{M}}I$.

###  2.6. Proof of Proposition 2.5

Recall that this proposition is used only for the case of $D$ composite, and
thus is not strictly necessary e.g. for the statement of Theorem 1.2.

Each quadratic character $\chi$ cuts out an extension $H_{\chi}/K$ which is
the composition of $K$ and a quadratic extension
$\mathbb{Q}_{\chi}/\mathbb{Q}$ of discriminant dividing $D$. Hence $\chi$ may
be regarded as the restriction to $G_{K}$ of the Dirichlet character of
conductor dividing $D$ attached to $\mathbb{Q}_{\chi}/\mathbb{Q}$, that we
still denote with the same symbol. As it is readily seen by comparing the
associated Galois representations, $\theta_{\psi_{1}^{-1}\chi}$ is the twist
of $g=\theta_{\psi_{1}^{-1}}$ by $\chi$, and $\theta_{\psi_{2}^{-1}\chi}$ is
the twist of $h=\theta_{\psi_{2}^{-1}}$ by $\chi$.

Let $\pi_{1},\pi_{2}$ be the automorphic representations for $\mathrm{GL}_{2}$
associated to $g,h$. Let $K_{0}(D)\subset\mathrm{GL}_{2}(\mathbb{A}_{f})$ be
the standard compact open subgroup and let $K_{1}(D)$ be the kernel of the
natural “diagonal” maps
$K_{0}(D)\rightarrow\left((\mathbb{Z}/D\mathbb{Z})^{\times}\right)^{2}$. Note
that $K_{1}(D)$ in the $\mathrm{GL}_{2}$ context is sometimes defined to only
impose one constraint, but here we understand that both the diagonal entries
are congruent to $1$ modulo $D$.

Set

$X_{1}(D)=\mathrm{GL}_{2}(\mathbb{Q})\backslash\mathcal{H}^{\ast}\times\mathrm{GL}_{2}(\mathbb{A}_{f})/K_{1}(D),$

whose set of connected components identified with

$\mathbb{Q}^{\times}\backslash\mathbb{A}_{f}^{\times}/\det(K_{1}(D))=(\mathbb{Z}/D\mathbb{Z})^{\times}.$

There are embeddings

$\pi_{1,f}^{K_{1}(D)}\mbox{ and }\pi_{2,f}^{K_{1}(D)}\hookrightarrow
H^{0}(X_{1}(D),\omega_{X_{1}(D)})$

carrying the new vectors to $g$ and $h$ respectively.

The new vectors are characterized, uniquely up to scalar, by the fact that
they transform under

$k=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\subset K_{0}(D)$

by the character $k\mapsto\chi_{K}(a)$. For each $\chi$ as above, we can
consider the “pseudo-new” vector

$g^{\chi}\mbox{ or }h^{\chi}\in\pi_{1,f}^{K_{1}(D)}\mbox{ or
}\pi_{2,f}^{K_{1}(D)}$

uniquely characterized up to scalar by similarly transforming by the character
$k\mapsto\chi_{K}(a)\chi(ad)$. (The uniqueness of such a vector follows by
applying the usual new vector theory to the representation
$\pi_{1,f}\otimes\chi$, which has the same conductor as $\pi_{1,f}$.
Explicitly, we may construct a pseudo-new vector from a new vector by
multiplying the associated function in the Kirillov model by the character
$\chi$; this statement is the representation-theoretic manifestation of the
fact that twisting by $\chi$ multiplies coefficients of the $q$-expansion by
$\chi$. A nice short reference for basic properties of the Kirillov model and
new vectors is the paper [Sc] and a more encyclopaedic treatment is [JL], in
particular Theorem 2.13).

With this construction, we have the following properties:

  * (a)

the standard newforms $g_{\chi}$ and $h_{\chi}$ in the twisted automorphic
representation correspond to the cup products:

$g_{\chi}=g^{\chi}\cdot\chi,\qquad h_{\chi}=h^{\chi}\cdot\chi,$

where we pull back $\chi$ to a complex-valued function on $X_{1}(D)$ by means
of the map $X_{1}(D)\rightarrow(\mathbb{Z}/D\mathbb{Z})^{\times}$ to the group
of connected components.

  * (b)

$\langle g^{\chi},h^{\chi}\rangle=\langle g,h\rangle$, with reference to any
nontrivial $\mathrm{GL}_{2}(\mathbb{A}_{f})$-invariant pairing
$\pi_{1,f}\times\pi_{2,f}\rightarrow\mathbb{C}$.

Let $X_{10}(D,N)$ be obtained from $X_{1}(D)$ by imposing a further
$K_{0}(N)$-level structure. Let $\mathfrak{S}_{DN}\in
H^{1}(X_{10}(D,N),\omega)\otimes\mathbb{Z}/p^{t}\mathbb{Z}$ denote the pull-
back of the Shimura class. let $\pi_{1},\pi_{2}:X_{10}(D,N)\longrightarrow
X_{1}(D)$ denote the two forgetful maps intertwined by the Atkin-Lehner
involution at $N$. It follows that

$\langle\mathsf{Tr}^{DN}_{N}\ G_{DN}(\chi),\mathfrak{S}\rangle=\langle
G_{DN}(\chi),\mathfrak{S}_{DN}\rangle=\int\pi_{1}^{*}(g_{\chi})\cup\pi_{2}^{*}(h_{\chi})\cup\mathfrak{S}_{DN}$
$=\int\pi_{1}^{*}(g^{\chi})\cup\pi_{2}^{*}(h^{\chi})\cup\mathfrak{S}_{DN},$

where
$\int:H^{1}(X_{1}(D)_{\mathbb{Z}/p^{t}\mathbb{Z}},\omega)\rightarrow\mathbb{Z}/p^{t}\mathbb{Z}$
is the trace map. It remains to verify that

(39)
$\int\pi_{1}^{*}(g^{\chi})\cup\pi_{2}^{*}(h^{\chi})\cup\mathfrak{S}_{DN}=\int\pi_{1}^{*}(g)\cup\pi_{2}^{*}(h)\cup\mathfrak{S}_{DN}.$

Now (b) above implies that $g^{\chi}\otimes h^{\chi}$ and $g\otimes h$ have
the same image in the diagonal coinvariants on $\pi_{1,f}\otimes\pi_{2,f}$.
That is to say, considered inside $\pi_{1,f}\otimes\pi_{2,f}$,

(40) $g^{\chi}\otimes h^{\chi}-g\otimes h=\sum_{i\in
I}c_{i}\left[s_{i}v_{1}\otimes s_{i}v_{2}-(v_{1}\otimes v_{2})\right],$

where $c_{i}\in\mathbb{C}$ and
$s_{i}\in\prod_{v|D}\mathrm{GL}_{2}(\mathbb{Q}_{v})$. Moreover, a
straightforward argument with rational structures shows that we may even take
$c_{i}$ to belong to the field $L=\mathbb{Q}(\psi_{1},\psi_{2})$, and
similarly $v_{1}$ and $v_{2}$ to be $L$-rational modular forms, and for
sufficiently large $p$ we can suppose $c_{i}$, $v_{1}$, $v_{2}$, and $D(D-1)$
to be $p$-integral. Then

$\int(s_{i}v_{1})\cup(s_{i}v_{2})\cup\mathfrak{S}=\int v_{1}\cup
v_{2}\cup\mathfrak{S},$

where $\mathfrak{S}$ is a Shimura class at a sufficiently deep level $N\cdot
D^{r}$; this follows from the invariance of the Shimura class under the adèle
group away from $N$ after pullback to a further cover. Therefore (40) implies
the desired (39).

##  3\. A trace identity for indefinite theta series

The goal of this chapter is to prove the counterpart of Theorem 2.2 in the
case where $(g,h)=(\theta_{\psi_{1}^{-1}},\theta_{\psi_{2}^{-1}})$ is a pair
of new weight one $\theta$-series associated to ray class characters
$\psi_{1}$ and $\psi_{2}$ of a common real quadratic field $K$, whose central
characters, denoted $\chi_{1}$ and $\chi_{2}$ respectively, satisfy
$\chi_{1}=\chi_{2}^{-1}$. Let $D$ denote the discriminant of $K$, and let
$\delta=(\sqrt{D})$ be its different. We will assume that the discriminant $D$
is odd.

Since $g$ and $h$ are holomorphic, the characters $\psi_{1}$ and $\psi_{2}$,
whose induced representations are odd two-dimensional Artin representations,
are necessarily of mixed signature at $\infty$. This means that the hypotheses
of Section 2, in which $\psi_{1}$ and $\psi_{2}$ were assumed to be
unramified, are restrictive to the point of being vacuous: indeed, the
presence of the unit $-1$ precludes the existence of unramified idèle class
characters of $K$ of mixed signature. It will therefore only be assumed that
the conductors of $\psi_{1}$ and $\psi_{2}$ divide the different
$\delta:=(\sqrt{D})$ of $K$, which means that the levels of

$g=\theta_{\psi_{1}^{-1}},\qquad h=\theta_{\psi_{2}^{-1}}$

divide $D^{2}$. In particular, these forms belong to the spaces
$M_{1}(\Gamma_{1}(D^{2}),\chi^{-1})$ and $M_{1}(\Gamma_{1}(D^{2}),\chi)$
respectively.

Because the $\theta$-series for $\psi_{2}$ and its Galois conjugate
$\psi_{2}^{\prime}$ coincide, it is harmless to suppose that $\psi_{1}$ and
$\psi_{2}$ both have the same signature at $\infty$, namely the one for which
$\psi_{1}$ and $\psi_{2}$ are trivial relative to the standard real embedding
of $K$.

Because the restrictions of $\psi_{1}$ and $\psi_{2}$ (viewed as characters of
the idèles ${\mathbb{A}}_{K}^{\times}$ of $K$) to the group
${\mathbb{A}}_{\mathbb{Q}}^{\times}$ of idèles of $\mathbb{Q}$ are inverses of
each other, it follows that, for all primes $v$ of $K$ dividing $D$ where
$\psi_{1}$ and $\psi_{2}$ are possibly ramified,

$\psi_{1,v}|_{{\mathcal{O}}_{v}^{\times}}=\psi_{2,v}^{-1}|_{{\mathcal{O}}_{v}^{\times}}.$

But the Galois conjugation map $x\mapsto x^{\prime}$ induces the identity on
the residue fields of $K_{v}$ for such $v$, and hence the characters

(41)
$\psi_{{}_{12}}:=\psi_{1}\psi_{2},\qquad\psi_{{}_{12^{\prime}}}:=\psi_{1}\psi_{2}^{\prime}$

appearing in Theorem 2.2 are trivial on ${\mathcal{O}}_{v}^{\times}$ for all
primes $v$, including those dividing $D$. It follows that $\psi_{{}_{12}}$ and
$\psi_{{}_{12^{\prime}}}$ are everywhere unramified. The character
$\psi_{{}_{12}}$ is furthermore totally even, and $\psi_{{}_{12^{\prime}}}$ is
totally odd.

The existence of the odd unramified character $\psi_{{}_{12^{\prime}}}$
implies that the narrow class number of $K$ is twice its class number, and
hence, that all the units of $K$ have positive norm. The fundamental unit
$\varepsilon$ is chosen so that $\varepsilon>1$ relative to the fixed standard
real embedding $K\hookrightarrow{\mathbb{R}}$ of $K$ evoked in the
introduction

In fact, it will be shown below that any pair of unramified characters of $K$
with trivial restrictions to ${\mathbb{A}}_{\mathbb{Q}}^{\times}$ and opposite
pure signatures can be obtained from a pair $(\psi_{1},\psi_{2})$ as above, in
an essentially unique way; this fact plays a crucial role in the proof of
Theorem 3.1 below, because it eliminates the need for an analogue of
Proposition 2.5 and thus leads to a more precise result.

###  3.1. Setup on Heegner cycles

As above, $K$ is a real quadratic field of odd discriminant $D$, all of whose
units have norm $1$. Let $\mathfrak{o}$ be the maximal order of $K$. Let
$N\nmid D$ be an odd prime that splits in $K$.

Choose $\delta_{N}\in\mathbb{Z}$ satisfying

$\delta_{N}^{2}\equiv D\pmod{N}.$

This choice determines an ideal $\mathfrak{N}=(N,\delta_{N}-\sqrt{D})$ of
$\mathfrak{o}$ of norm $N$. Also, let

$M_{0}(N):=\left\\{\left(\begin{array}[]{cc}a&b\\\ Nc&d\end{array}\right)\ \
\mbox{ with }a,b,c,d\in\mathbb{Z}\right\\}\subset M_{2}(\mathbb{Z})$

be the standard Eichler order of level $N$ in the matrix ring
$M_{2}(\mathbb{Z})$. This Eichler order is equipped with the standard
orientation

$\mathbf{o}:M_{0}(N)\rightarrow\mathbb{F}_{N}=(\mathbb{Z}/N\mathbb{Z})$

onto the field of $N$ elements, sending a matrix to the mod $N$ residue class
of its upper left hand entry.

Let $I\subset\mathfrak{o}$ be an ideal. Writing $I\cap\mathbb{Z}=(a)$ with
$a>0$, we can write

$I=\left(a,\frac{-b+\sqrt{D}}{2}\right),$

with $b$ uniquely determined modulo $a$. The action of $\mathfrak{o}$ on $I$
with respect to the basis $(a,(-b+\sqrt{D})/2)$ gives a homomorphism

(42) $\alpha:\mathfrak{o}\rightarrow M_{2}(\mathbb{Z}),\ \ \
\sqrt{D}\mapsto\left[\begin{array}[]{cc}b&-2c\\\ 2a&-b\end{array}\right],$

where $c$ is defined by stipulating that the binary quadratic form
$ax^{2}+bxy+cy^{2}$ has discriminant $D$.

An eigenvector $v\in K^{2}$ for the action of $\alpha(K)$ is given by

$v=\left(\begin{array}[]{c}(b+\sqrt{D})/2\\\ a\end{array}\right).$

Write $\tau:=\frac{b+\sqrt{D}}{2a}$, and let $v^{\prime}$ and $\tau^{\prime}$
denote the algebraic conjugates of $v$ and $\tau$ respectively over $K$.

Suppose that $I$ is divisible by $\mathfrak{N}$ but not by
$\mathfrak{N}^{\prime}$. Then $a$ is divisible by $N$, $b$ is congruent to
$\delta_{N}$ modulo $N$, and $\alpha$ is an embedding of $\mathfrak{o}$ into
$M_{0}(N)$. Indeed, the basis vector $a\in I$ belongs to
$\mathfrak{N}^{\prime}I$ since it is divisible by $N$, and its image in $I/NI$
generates the index N subgroup $\mathfrak{N}^{\prime}I/NI$, which is preserved
under multiplication by $\mathfrak{o}$. Hence multiplication by any element of
$\mathfrak{o}$ is represented by a matrix in $M_{0}(N)$ relative to the basis
$(a,(b+\sqrt{D})/2)$. Moreover the composition
$\mathbf{o}\circ\alpha:\mathfrak{o}\rightarrow\mathbb{F}_{N}$ of $\alpha$ with
the orientation $\mathbf{o}:M_{0}(N)\rightarrow\mathbb{F}_{N}$ is reduction
modulo $\mathfrak{N}$.

Replacing the basis $(a,(b+\sqrt{D})/2)$ of $I$ by another positively
oriented444Here, a basis $(e_{1}e_{2})$ is said to be positively oriented if
it is in the ${\mathrm{SL}}_{2}(\mathbb{Z})$-orbit of the specified one, or,
said more intrinsically, $e_{1}\wedge e_{2}$ equals the norm of $I$ multiplied
by $1\wedge\sqrt{D}/2$.  basis of the same form conjugates the resulting
embedding by an element of $\Gamma_{0}(N)$, hence the embedding $\alpha$
attached to $I$ is independent of this choice of basis, up to conjugation in
$\Gamma_{0}(N)$.

The standard real embedding $K\hookrightarrow{\mathbb{R}}$ that was fixed
previously yields a geodesic $(\tau,\tau^{\prime})\subset{\mathcal{H}}$ in the
upper half-plane. Recall the fundamental unit
$\varepsilon\in\mathfrak{o}_{1}^{\times}$ of $K$ of norm one, and let

(43)
$\gamma_{I}=\alpha(\varepsilon)^{\mathbb{Z}}\backslash(\tau,\tau^{\prime})$

denote the closed geodesics on $\Gamma_{0}(N)\backslash{\mathcal{H}}$ attached
to $I$. We regard it as oriented from $\tau$ to $\tau^{\prime}$. This depends
only on the class of $I$ in

(44) $\mathcal{C}:=\mbox{ the narrow ideal class group of $K$},$

and correspondingly we will freely write $\gamma_{I}$ for $I\in\mathcal{C}$.

Note that $\tau^{\prime}<\tau$ and moreover the derivative of the fractional
linear transformation of $\mathbb{R}$ induced by $\alpha(\varepsilon)$ at
$\tau^{\prime}$ (resp. $\tau$) is given by $(\varepsilon^{\prime})^{-2}$
(resp. $\varepsilon^{-2}$). Since $\varepsilon>1>\varepsilon^{\prime}$, we
conclude that the action of $\alpha(\varepsilon)$ on $(\tau,\tau^{\prime})$
moves along the direction opposite to the orientation of the geodesic.

###  3.2. Statement of the trace identity

Given two narrow ideal classes, choose representatives $I_{1}$ and $I_{2}$
that are divisible by $\mathfrak{N}$ but not by $\mathfrak{N}^{\prime}$. Let
$\alpha_{i}$ for $i\in\\{1,2\\}$ denote the two embeddings attached to $I_{1}$
and $I_{2}$ as in §3.1, and let $v_{i},v_{i}^{\prime}\in K^{2}$ and
$\tau_{i},\tau_{i}^{\prime}\in K$ be the associated eigenvectors and fixed
points, respectively.

Write $\langle\gamma_{{}_{I_{1}}}\cdot
T_{m}\gamma_{{}_{I_{2}}}\rangle_{{}_{N}}$ for the topological intersection
pairing of the homology cycles $\gamma_{{}_{I_{1}}}$ and
$T_{m}\gamma_{{}_{I_{2}}}$ on the Riemann surface $X_{0}(N)(\mathbb{C})$. The
generating series

(45)
$\Theta(\gamma_{{}_{I_{1}}}\otimes\gamma_{{}_{I_{2}}}):=\sum_{m=1}^{\infty}\langle\gamma_{{}_{I_{1}}}\cdot
T_{m}\gamma_{{}_{I_{2}}}\rangle_{{}_{N}}q^{m}$

is a cusp form of weight two and level $N$. This definition can be extended by
linearity to arbitrary linear combinations of RM geodesics, notably the paths

(46)
$\gamma_{\psi_{{}_{12}}}(q)=\sum_{I\in\mathcal{C}}\psi_{{}_{12}}(I)\gamma_{{}_{I}},\qquad\gamma_{\psi_{{}_{12^{\prime}}}}(q)=\sum_{I\in\mathcal{C}}\psi_{{}_{12^{\prime}}}(I)\gamma_{{}_{I}}$

associated to the unramified characters $\psi_{{}_{12}}$ and
$\psi_{{}_{12^{\prime}}}$ respectively.

The following theorem, which is the main result of this chapter, relates the
trace of products of binary theta series to modular generating series of real
quadratic geodesic cycles as in (45).

######  Theorem 3.1.

For all theta series $g=\theta_{\psi_{1}^{-1}}$ and $h=\theta_{\psi_{2}^{-1}}$
of $K$ as above,

$\mathsf{Tr}^{ND^{2}}_{N}(\theta_{\psi_{1}^{-1}}(Nz)\theta_{\psi_{2}^{-1}}(z))=C\cdot\psi_{1}({\mathfrak{N}}^{\prime})\cdot\Theta(\gamma_{\psi_{{}_{12}}}\otimes\gamma_{\psi_{{}_{12^{\prime}}}}),$

where

(47) $C=D\sum_{D=D_{1}D_{2}}\mu(D_{1})\cdot
D_{2}\cdot\psi_{{}_{12}}\psi_{{}_{12^{\prime}}}(j_{{}_{D_{1}}})\ =\
D\prod_{p|D}(p-\psi_{{}_{12}}\psi_{{}_{12^{\prime}}}(j_{p})),$

and $j_{{}_{D_{1}}}$ is the order two element represented by the ideal
$(D_{1},\sqrt{D})$ in the narrow class group of $K$.

The reader should compare this theorem to Theorem 2.2, which is less precise.
It turns out that allowing the ray class characters $\psi_{i}$ to be ramified
at primes dividing the discriminant simplifies rather than complicates the
situation. Transposing the proof of Theorem 3.1 to the setting of Chapter 2
would presumably lead to a refined and slightly more general variant of
Theorem 2.2.

######  Remark 3.2.

In the extension above of the generating series (45) to linear combinations of
geodesics we are always taking representatives of $I\in\mathcal{C}$ that are
divisible by $\mathfrak{N}$ but not by $\mathfrak{N}^{\prime}$. This choice
introduces an asymmetry that reappears throughout this section, and explains
the appearance of the factor $\psi_{1}({\mathfrak{N}}^{\prime})$ on the right-
hand side of the identity in the Theorem. Since the right-hand side is
invariant under exchange of ${\mathfrak{N}}$ and ${\mathfrak{N}}^{\prime}$,
the second factor
$\Theta(\gamma_{\psi_{{}_{12}}}\otimes\gamma_{\psi_{{}_{12^{\prime}}}})$ must
also depend on the choice of $\mathfrak{N}$.

The proof of Theorem 3.1 is summarized in §3.3 below, and the details of this
sketch are fleshed out in the remainder of the chapter.

###  3.3. Summary of the proof

Let

$\mathcal{C}:={{\mathscr{I}}}(\mathfrak{o})/{\rm
P}_{+}(\mathfrak{o}),\qquad\mathcal{C}_{D}:={{\mathscr{I}}}_{\delta}(\mathfrak{o})/{\rm
P}_{\delta,+}(\mathfrak{o}),$

be the narrow class group and generalised class group of conductor $\delta$,
defined by letting

  * •

${{\mathscr{I}}}(\mathfrak{o})$, resp. ${\mathscr{I}}_{\delta}(\mathfrak{o})$,
be the semi-group of ideals of $\mathfrak{o}$, resp. the ideals that are prime
to $\delta$;

  * •

${\rm P}_{+}(\mathfrak{o})$ be the semi-group of principal ideals with a
totally positive generator;

  * •

${\rm P}_{\delta,+}(\mathfrak{o})$ be the semi-group of principal ideals with
a totally positive generator that is congruent to $1$ modulo $\delta$.

Given ideals $I_{1}$ and $I_{2}$, let

(48)
$\mathcal{A}:=\\{(x,y)\in({\mathfrak{N}}^{\prime}I_{1})_{+}\times(I_{2})_{-}\mbox{
satisfying }x\equiv y\pmod{\delta}\\},$

and the $+$ and $-$ subscripts mean, respectively, positive and negative norm.
The group

(49) $\mathcal{U}:=\\{\pm(\varepsilon^{a},\varepsilon^{b})\mbox{ satisfying
}a\equiv b\pmod{2}\\}$

operates naturally on $\mathcal{A}$. Let

$\displaystyle\Theta^{\sharp}(I_{1},I_{2})$ $\displaystyle:=$
$\displaystyle\sum_{(x,y)\in\mathcal{A}/(\varepsilon^{2\mathbb{Z}}\times\varepsilon^{2\mathbb{Z}})}{\rm
sign}(x)\cdot{\rm sign}(y)\cdot q^{\frac{xx^{\prime}}{D{\rm
N}(I_{1})}-\frac{yy^{\prime}}{D{\rm N}(I_{2})}}$ $\displaystyle=$
$\displaystyle 4\sum_{(x,y)\in\mathcal{A}/\mathcal{U}}{\rm sign}(x)\cdot{\rm
sign}(y)\cdot q^{\frac{xx^{\prime}}{D{\rm N}(I_{1})}-\frac{yy^{\prime}}{D{\rm
N}(I_{2})}}.$

The function $\Theta^{\sharp}(I_{1},I_{2})(e^{2\pi i\tau})$ is a finite sum of
suitable pairs of indefinite binary theta series attached to certain cosets in
$I_{1}\oplus I_{2}$, and is a modular form of weight two. It is readily
verified that it depends only on the classes of $I_{1}$ and $I_{2}$ in
$\mathcal{C}_{D}$.

The proof of Theorem 3.1 follows from two key Propositions. The first will be
proved in §3.6 and the second in §3.7.

######  Proposition 3.3.

(See §3.6, also cf. (2.3)). There is an equality of modular forms on
$\Gamma_{0}(N)$:

$\displaystyle{\rm
Tr}^{ND^{2}}_{N}\left(\theta_{\psi_{1}^{-1}}(q^{N})\cdot\theta_{\psi_{2}^{-1}}(q)\right)$
$\displaystyle=$
$\displaystyle\psi_{1}({\mathfrak{N}}^{\prime})\cdot\frac{C}{4}\cdot\sum_{\mathcal{C}\times\mathcal{C}}\psi_{{}_{12}}(I_{1})\psi_{{}_{12^{\prime}}}(I_{2})\cdot\Theta^{\sharp}(I_{1}I_{2},I_{1}I_{2}^{\prime}),$

where $C$ is as in (47).

######  Proposition 3.4.

(See §3.7, also cf. Prop. 2.4). The generating series of (45) is equal to

$\Theta({\gamma_{{}_{I_{1}}}\otimes\gamma_{{}_{I_{2}}}})(q)=\frac{1}{4}\cdot\Theta^{\sharp}(I_{1}I_{2},I_{1}I_{2}^{\prime})(q).$

Taken together, these two propositions imply that the trace appearing in
Proposition 3.3 is equal to

$\psi_{1}({\mathfrak{N}}^{\prime})\cdot
C\cdot\sum_{\mathcal{C}\times\mathcal{C}}\psi_{{}_{12}}(I_{1})\psi_{{}_{12^{\prime}}}(I_{2})\cdot\Theta(\gamma_{{}_{I_{1}}}\otimes\gamma_{{}_{I_{2}}}),$

and the sum appearing here, by definition, equals
$\Theta(\gamma_{\psi_{{}_{12}}}\otimes\gamma_{\psi_{{}_{21}}})$. That is
precisely the statement of Theorem 3.1.

###  3.4. Setup on class groups

The running assumption that all units of $K$ have norm one implies that
equivalence of ideals in the narrow sense is strictly finer than equivalence
in the wide sense, i.e., that the narrow class number of $K$ is twice its
class number. It also implies, by genus theory, that the odd discriminant $D$
is a product of two negative fundamental discriminants, and hence is not
prime. Let $a\geq 2$ be the number of prime divisors of $D$.

Although $K$ possesses no unramified idèle class characters of mixed
signature, such characters always appear in conductor dividing the different
$\delta$ of $K$, since the units of $\mathfrak{o}$ which are $1$ modulo
$\delta$ are all totally positive.

Let

$\iota:=\mbox{ the class of $\varepsilon$ modulo $\delta$}.$

It is one of the $2^{a}-2$ possible non-trivial ($\neq\pm 1$) square roots of
$1$ in $\mathfrak{o}/\delta=\mathbb{Z}/D\mathbb{Z}$. For if $\iota=\pm 1$, the
fundamental unit $\pm\varepsilon$ gives rise to a solution
$(x,y)\in\mathbb{Z}^{2}$ of the Pell’s equation

(51) $\displaystyle x^{2}-Dy^{2}=1,$ $\displaystyle\qquad x\equiv
1\pmod{D},\qquad x\mbox{ odd},\quad y\mbox{ even},\qquad\mbox{or }$ (52)
$\displaystyle x^{2}-Dy^{2}=4,$ $\displaystyle\qquad x\equiv 2\pmod{D},\qquad
x,y\mbox{ odd}.$

In the second case, the factorisation of $Dy^{2}=(x-2)(x+2)$ into relatively
prime integers implies that

$x+2=\pm u^{2}\ \ \mbox{ and }\ \ x-2=\pm Dv^{2},$

for some $(u,v)\in\mathbb{Z}^{2}$, and hence $(u,v)$ is a solution of the
equation $u^{2}-Dv^{2}=\pm 4$ of height strictly smaller than that of $(x,y)$.
Likewise, a solution to (51) leads to a pair $(u,v)$ satisfying

$x+1=\pm 2u^{2}\mbox{ and }x-1=\pm 2Dv^{2},\quad$

and hence to a unit of $\mathfrak{o}$ of smaller height, contradicting in both
cases the assumption that $\varepsilon$ is a fundamental unit.

There is a natural exact sequence

$0\longrightarrow\langle\iota\rangle\longrightarrow(\mathbb{Z}/D\mathbb{Z})^{\times}\longrightarrow\mathcal{C}_{D}\longrightarrow\mathcal{C}\longrightarrow
0.$

where the first inclusion sends $t\in(\mathbb{Z}/D\mathbb{Z})^{\times}$ to the
principal ideal generated by any totally positive integer congruent to $t$
modulo $\delta$. Let

$Z:=\ker(\mathcal{C}_{D}\longrightarrow\mathcal{C})\simeq(\mathbb{Z}/D\mathbb{Z})^{\times}/\iota$

be the kernel of the natural projection. Next, let
$W\subset(\mathbb{Z}/D\mathbb{Z})^{\times}$ be the index $2$ subgroup which is
the kernel of the quadratic Dirichlet character associated to $K$, and

(53) $\mathrm{N}:\mathcal{C}_{D}\rightarrow
W\subset(\mathbb{Z}/D\mathbb{Z})^{\times}$

be the norm map sending the class of an ideal to the mod $D$ residue class of
its norm. The triviality of the Herbrand quotient of the finite group
$\mathcal{C}_{D}$ as a ${\rm Gal}(K/\mathbb{Q})$-module implies that

$\mathcal{C}_{D}^{-}:=\mbox{kernel of $\mathrm{N}$}=\left\\{g/g^{\prime}\mbox{
with }g\in\mathcal{C}_{D}\right\\},$

where $g\mapsto g^{\prime}$ is induced by the Galois automorphism of $K$ over
$\mathbb{Q}$; thus $W$ is now identified with
$\mathcal{C}_{D}/\mathcal{C}_{D}^{-}$.

The groups $Z$ and $W$ have the same cardinality $\varphi(D)/2$, but the
natural homomorphism $Z\longrightarrow W$ obtained by composing the inclusion
$Z\hookrightarrow\mathcal{C}_{D}$ with the surjection
$\mathcal{C}_{D}\rightarrow W$ is not an isomorphism; its kernel is the two-
torsion subgroup of $Z$, of cardinality $2^{a-1}$.

Global class field theory identifies $\mathcal{C}$ with the Galois group of
the Hilbert class field $H$ over $K$, and $\mathcal{C}_{D}$ with the Galois
group of $H_{D}$ over $K$, where $H_{D}$ is the ray class field of $K$ of
conductor $\delta$, an extension of $H$ of degree $\varphi(D)/2$. The subgroup
$\mathcal{C}_{D}^{-}$ is identified with the Galois group of $H_{D}$ over the
maximal subfield of $H_{D}$ which is Galois and abelian over $\mathbb{Q}$,
namely, the cyclotomic field $\mathbb{Q}(\zeta_{D})$. The group $W$ is
identified with the Galois group of $\mathbb{Q}(\zeta_{D})$ over $K$, an index
two subgroup of $(\mathbb{Z}/D\mathbb{Z})^{\times}$. The situation is
summarised in the field diagram below.

$\textstyle{H_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2^{a-1}}$$\scriptstyle{Z}$$\scriptstyle{\mathcal{C}_{D}^{-}}$$\scriptstyle{{\mathcal{C}}_{D}}$$\textstyle{H(\zeta_{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{C}}$$\textstyle{\mathbb{Q}(\zeta_{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W}$$\textstyle{H\cap\mathbb{Q}(\zeta_{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2^{a-1}}$$\textstyle{K}$

We can now state and prove the crucial

######  Lemma 3.5.

There is an isomorphism

(54)
$\xi:\mathcal{C}^{2}\longrightarrow({\mathcal{C}}_{D}\times_{W}{\mathcal{C}}_{D})/Z,\qquad(I_{1},I_{2})\mapsto(I_{1}I_{2},I_{1}I_{2}^{\prime}),$

where the target is defined after choosing lifts $I_{1}$ and $I_{2}$ of the
eponymous ideal classes $I_{1},I_{2}\in\mathcal{C}$ to the ray class group
${\mathcal{C}}_{D}$.

The validity of this Lemma is the main reason that the current (RM) section
obtains a more precise result than the CM section.

###### Proof.

Observe, first, that the map is well-defined, since the kernel of
$\mathcal{C}_{D}\rightarrow\mathcal{C}$ is the image of
$(\mathbb{Z}/D\mathbb{Z})^{\times}$, represented by principal ideals $(t)$ for
$t\in\mathbb{Z}$, and multiplying $I_{1}$ or $I_{2}$ by such a principal ideal
of norm prime to $D$ only changes $(I_{1}I_{2},I_{1}I_{2}^{\prime})$ by an
element of the diagonally embedded $Z$. The two groups have the same
cardinality, by the discussion above; so it is enough to prove that $\xi$ is
surjective. But clearly a pair $(J_{1},J_{2})$ lies in the image if and only
if $J_{2}J_{1}^{-1}$ has the form $I_{2}/I_{2}^{\prime}$, i.e., belongs to
$\mathcal{C}_{D}^{-}$. ∎

###  3.5. Setup on binary $\theta$ series

For lack of a reference, let us briefly sketch the general situation, before
specializing to the case of a quadratic space arising from the quadratic field
$K$.

Consider a $2n$-dimensional anisotropic quadratic space $(V,q)$ over
$\mathbb{Q}$. The space of Schwartz functions on $V\otimes\mathbb{A}_{f}$ is
endowed with an action of ${\mathrm{SL}}_{2}(\mathbb{A}_{f})$ via the Weil
representation which at any finite prime of $\mathbb{Q}$ is given by the
following formulas:

(55) $\begin{gathered}r_{\mu}\left(\begin{array}[]{cc}1&a\\\
0&1\end{array}\right)f(x)=\mu(aq(x))f(x)\\\
r_{\mu}\left(\begin{array}[]{cc}a&0\\\
0&a^{-1}\end{array}\right)f(x)=|a|\omega(a)f(ax)\\\
r_{\mu}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)f(x)=\gamma\widehat{f}(x)\end{gathered}$

Here $\mu$ is a chosen additive character, $\omega$ is the quadratic
discriminant character corresponding to the space $V$, the function
$\widehat{f}$ is Fourier transform of $f$ relative to $\mu$ and a self-dual
Haar measure on $V$, and $\gamma$ is an eighth root of unity. We apply this
only in the case when $V\otimes\mathbb{Q}_{\ell}$ is a split $4$-dimensional
quadratic space; in this case $\omega=1$ and also $\gamma=1$ (for the latter,
see [We, p. 176]).

Now suppose $\dim V=2$, that $(V,q)$ has signature $(1,1)$, and suppose that
$\Psi_{f}$ is a Schwartz function on $V\otimes\mathbb{A}_{f}$ with stabilizer
$\Gamma\leqslant\mathrm{SO}_{q}(\mathbb{Q})$.

######  Proposition 3.6.

Let

(56) $\theta_{\Psi_{f}}(z):=\sum_{\begin{subarray}{c}v\in\Gamma\backslash
V,\\\ q(v)>0\end{subarray}}\mathrm{sign}(v)e^{2\pi iq(v)z}\Psi_{f}(v),$

where $\mathrm{sign}(v)$ is positive on one connected component of $q(v)>0$
and negative on the other. Then $\theta_{\Psi_{f}}(z)$ is a modular form on
${\mathrm{SL}}_{2}$ and the association $\Psi_{f}\mapsto\theta_{\Psi_{f}}$ is
equivariant for the action of ${\mathrm{SL}}_{2}(\mathbb{A}_{f})$ via the Weil
representation. The same conclusion applies replacing the condition $q(v)>0$
by $q(v)<0$ and $e^{2\pi iq(v)z}$ by $e^{-2\pi iq(v)z}$.

###### Sketch of proof.

To check this we use the dual pair $\mathrm{SO}_{q}\times{\mathrm{SL}}_{2}$.
Fix an isomorphism $(V\otimes{\mathbb{R}},q)\simeq({\mathbb{R}}^{2},xy)$, let
$\Psi_{\infty}(x,y)=(x+y)e^{-\pi(x^{2}+y^{2})}$, and let
$\Psi=\Psi_{\infty}\otimes\Psi_{f}$ be the associated Schwartz function on
$V\otimes\mathbb{A}$. The function $\Psi_{\infty}$ is chosen so that its
average $\overline{\Psi}_{\infty}$ over the connected component of
$\mathrm{SO}_{q}({\mathbb{R}})$ is explicitly computable:

$\overline{\Psi}_{\infty}(x,y)=\int_{\lambda\in\mathbb{R}_{+}^{\times}}(\lambda
x+\lambda^{-1}y)e^{-\pi(\lambda^{2}x^{2}+\lambda^{-2}y^{2})}\frac{d\lambda}{\lambda}\\\
=\begin{cases}\mathrm{sign}(x)e^{-2\pi xy},\ \ \mbox{ if }xy>0,\\\ 0,\ \ \ \ \
\mbox{ otherwise}.\end{cases}$

In particular, fixing $h\in\mathrm{SO}_{q}(\mathbb{A}),$ the rule

$g\mapsto\tilde{\theta}_{\Psi}(g,h):=\sum_{x\in V}(g,h)\cdot\Psi(x),$

where $(g,h)\cdot\Psi$ refers to the actions of
$g\in{\mathrm{SL}}_{2}(\mathbb{A})$ on $\Psi$ via the Weil representation, and
of $h\in\mathrm{SO}_{q}(\mathbb{A})$ via translation on the arguments, defines
an automorphic form on ${\mathrm{SL}}_{2}(\mathbb{A})$. The rule
$\Psi\mapsto\tilde{\theta}_{\Psi}$ is equivariant for the
${\mathrm{SL}}_{2}(\mathbb{A})$-actions on both sides. We now integrate over
$h\in\Gamma\backslash\mathrm{SO}_{q}({\mathbb{R}})$ to check that

$(g_{\infty},g_{f})\in{\mathrm{SL}}_{2}(\mathbb{A})\mapsto\theta:=\sum_{x\in\Gamma\backslash
V}\overline{\Psi}_{\infty}^{g_{\infty}}(x)\cdot\Psi_{f}^{g_{f}}(x)$

is again a modular form for ${\mathrm{SL}}_{2}(\mathbb{A})$. This gives the
claimed statement. ∎

Now, we will explicitly take $V$ to be $K$ together with a suitable rescaling
of the norm as quadratic form, and explicate the above construction when
$\Psi_{f}$ is given by suitable characteristic functions.

Given any fractional ideal $I$ of $\mathfrak{o}$ of norm ${\rm
N}(I)\in\mathbb{Q}^{>0}$ which is relatively prime to $\delta$, the group
$\varepsilon^{2\mathbb{Z}}$ preserves the intersection $I^{+}$ (resp. $I^{-}$)
of $I$ with the cone of elements of positive (resp. negative) norm in
$K\otimes{\mathbb{R}}$, as well as the subsets

$\displaystyle I_{1}^{+}$ $\displaystyle:=$ $\displaystyle\\{x\in I^{+}\mbox{
with }x\equiv 1\pmod{\delta}\\},$ $\displaystyle I_{1}^{-}$ $\displaystyle:=$
$\displaystyle\\{x\in I^{-}\mbox{ with }x\equiv 1\pmod{\delta}\\}.$

Taking $\Psi_{f}$ to be the characteristic function of $\\{x\in
I\otimes\hat{\mathbb{Z}}:x\equiv 1(\delta)\\}$, we recover Hecke’s partial
theta series

$\displaystyle\vartheta^{+}(I)(q)$ $\displaystyle:=$ $\displaystyle\sum_{x\in
I_{1}^{+}/\varepsilon^{2\mathbb{Z}}}{\rm sign}(x)\cdot q^{xx^{\prime}/D{\rm
N}(I)},$ $\displaystyle\vartheta^{-}(I)(q)$ $\displaystyle:=$
$\displaystyle\sum_{x\in I_{1}^{-}/\varepsilon^{2\mathbb{Z}}}{\rm
sign}(x)\cdot q^{-xx^{\prime}/D{\rm N}(I)}.$

These theta-series depend only on the image of $I$ in the ray class group
$\mathcal{C}_{D}$, and are modular forms of weight one on a suitable
congruence subgroup. More precisely, by (55) or [HIM, §1] we have:

######  Lemma 3.7.

For all $\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\Gamma_{0}(D),$

$\displaystyle\vartheta^{+}(I)\left(\frac{a\tau+b}{c\tau+d}\right)$
$\displaystyle=$ $\displaystyle\left(\frac{D}{|d|}\right)e^{\frac{-2\pi
iab}{D{\rm N}(I)}}(c\tau+d)\cdot\vartheta^{+}(aI)(\tau),$
$\displaystyle\vartheta^{-}(I)\left(\frac{a\tau+b}{c\tau+d}\right)$
$\displaystyle=$ $\displaystyle\left(\frac{D}{|d|}\right)e^{\frac{2\pi
iab}{D{\rm N}(I)}}(c\tau+d)\cdot\vartheta^{-}(aI)(\tau).$

######  Lemma 3.8.

We have

(57) $\displaystyle
2\theta_{\psi_{1}^{-1}}=\sum_{I\in\mathcal{C}_{D}}\psi_{1}(I)\vartheta^{+}(I)(q^{D}),$
(58) $\displaystyle
2\theta_{\psi_{2}^{-1}}=\sum_{I\in\mathcal{C}_{D}}\psi_{2}(I)\vartheta^{-}(I)(q^{D}).$

Here, by convention, $\psi_{1}(J)$ simply means the value of $\psi_{1}$
applied to the image of $J$ in the ray class group $\mathcal{C}_{D}$.

###### Proof.

Rewrite the right hand side of (57) as

(59)
$\theta_{\psi_{1}^{-1}}^{+}:=\sum_{\begin{subarray}{c}I\in\mathcal{C}_{D},\\\
x\in I_{1}^{+}/\varepsilon^{2\mathbb{Z}}\end{subarray}}\psi_{1}(I)\cdot{\rm
sign}(x)\cdot q^{xx^{\prime}/{\rm N}(I)},$

where we have made the slight abuse of notation of choosing a representative
$I$ for each class in $\mathcal{C}_{D}$, and $I_{1}^{+}$ consists of elements
in $I$ of positive norm and congruent to $1$ modulo $\delta$. The set
$I_{1}^{+}$ is the union of its totally positive and totally negative
elements. Sending a pair $(I,x)$ in the range of summation of the right-hand
side of (59), where $x$ is totally positive (resp. totally negative) to the
integral ideal $I^{-1}x$ determines two bijections

(60) $(I,x\in I_{1}^{+}\mbox{ totally positive})\mapsto I^{-1}x,$ (61)
$(I,x\in I_{1}^{+}\mbox{ totally negative})\mapsto I^{-1}x,$

to the set of integral prime-to-$\delta$ ideals. These two bijections are
interchanged by precomposing with the involution $(I,x)\mapsto((z)I,z\cdot
x)$, where $z$ is any totally negative element congruent to $1$ modulo
$\delta$. Therefore, for a given integral prime-to-$\delta$ ideal $J$, the
preimages $(I,x)$ and $(I^{\prime},x^{\prime})$ under these two bijections do
not coincide; rather, the classes of $I$ and $I^{\prime}$ in $\mathcal{C}_{D}$
differ by the image of $(-1)\in(\mathbb{Z}/D\mathbb{Z})^{\times}$ in
$\mathcal{C}_{D}$. Being of mixed signature, the character $\psi_{1}$ sends
this element to $-1$, and reindexing via $J=I^{-1}x$ allows us to rewrite (59)
as $2\sum_{J}\psi_{1}^{-1}(J)q^{\mathrm{N}(J)},$ which is (up to the factor of
$2$) the standard expression for the $\theta$-series
$\theta_{\psi_{1}^{-1}}(q)$ attached to $\psi_{1}^{-1}$. This proves (57), and
the proof of (58) is essentially the same.

∎

###  3.6. Proof of Proposition 3.3

With preliminaries on $\theta$-series in hand, we proceed the proof of the
first key step, Proposition 3.3.

Recall that $N$ is a prime that splits in $K$ as a product
${\mathfrak{N}}{\mathfrak{N}}^{\prime}$ of two prime ideals of norm $N$. If
$I_{1}$ and $I_{2}$ are (representatives of) elements of $\mathcal{C}_{D}$,
thus, fractional ideals of $K$, the modular form

(62)
$\Theta(I_{1},I_{2})=\vartheta^{+}({\mathfrak{N}}^{\prime}I_{1})(q^{N})\cdot\vartheta^{-}(I_{2})(q)$

is of weight two on $\Gamma(D)\cap\Gamma_{0}(N)$. Define

$\displaystyle\Theta^{(1)}(I_{1},I_{2})$ $\displaystyle=$
$\displaystyle\mbox{trace of $\Theta(I_{1},I_{2})$ to level
$\Gamma_{0}(N)\cap\Gamma_{1}(D)$},$ $\displaystyle\Theta^{(0)}(I_{1},I_{2})$
$\displaystyle=$ $\displaystyle\mbox{trace of $\Theta(I_{1},I_{2})$ to level
$\Gamma_{0}(N)\cap\Gamma_{0}(D)$},$
$\displaystyle\Theta^{(\emptyset)}(I_{1},I_{2})$ $\displaystyle=$ trace of
$\Theta(I_{1},I_{2})$ to level $\Gamma_{0}(N).$

The superscripts here are intended to remind the reader of the level structure
at $D$.

######  Lemma 3.9.

For all ideals $I_{1}$ and $I_{2}$ of $\mathcal{C}_{D}$,

(63)
$\Theta^{(1)}(I_{1},I_{2})=\left\\{\begin{array}[]{cl}D\cdot\vartheta^{+}({\mathfrak{N}}^{\prime}I_{1})(q^{N})\cdot\vartheta^{-}(I_{2})(q),&\mbox{
if }{\rm N}(I_{1})={\rm N}(I_{2}),\\\ 0&\mbox{ otherwise}.\end{array}\right.$

Here $\rm N$ is the norm of (53). Moreover $\Theta^{(0)}(I_{1},I_{2})$, which
therefore vanishes unless $(I_{1},I_{2})$ belongs to the fiber product

$\mathcal{C}_{D}\times_{W}\mathcal{C}_{D}:=\\{(I_{1},I_{2})\in\mathcal{C}_{D}\times\mathcal{C}_{D}\mbox{
satisfying }{\rm N}(I_{1})={\rm N}(I_{2})\\},$

depends only on the image of $(I_{1},I_{2})$ in the quotient
$(\mathcal{C}_{D}\times_{W}\mathcal{C}_{D})/Z$.

Proof. The non-zero terms in the Fourier expansion of
$\vartheta^{+}({\mathfrak{N}}^{\prime}I_{1})(q^{N})\cdot\vartheta^{-}(I_{2})(q)$
are concentrated at powers of the form $q^{m/D}$, where

$m\equiv 1/{\rm N}(I_{1})-1/{\rm N}(I_{2})\pmod{D},$

and the result follows, since the trace from $\Gamma(D)$ to $\Gamma_{1}(D)$
annihilates any term of the form $q^{m/D}$ with $D$ not dividing $m$, and
multiplies the others by a factor of $D$. The final assertion follows from the
explicit formula

(64)
$\Theta^{(0)}(I_{1},I_{2})=\sum_{a\in(\mathbb{Z}/D\mathbb{Z})^{\times}}\Theta^{(1)}(aI_{1},aI_{2})$

which is an immediate consequence of Lemma 3.7 ∎

Note that if $(I_{1},I_{2})$ belongs to
$(\mathcal{C}_{D}\times_{W}\mathcal{C}_{D})$, then the same is true of
$(I_{1},eI_{2})$, where $e$ is any element of $K^{\times}$ whose associated
fractional ideal is prime to $\delta$, and satisfies $e^{2}=1\pmod{\delta}$.

######  Proposition 3.10.

(cf. Prop. 2.3). For all
$(I_{1},I_{2})\in(\mathcal{C}_{D}\times_{W}\mathcal{C}_{D})/Z$ we have

(65)
$\Theta^{(\emptyset)}(I_{1},I_{2})=D\cdot\sum_{D=D_{1}D_{2}}\mu(D_{1})\cdot
D_{2}\cdot\Theta^{\sharp}(I_{1},\varepsilon_{D_{1}}I_{2}),$

where $\mu$ is the Möbius function, and

  1. (1)

$\Theta^{\sharp}(I_{1},I_{2})$ is the modular form defined in (3.3);

  2. (2)

the sum on the right is taken over all factorisations of $D$ into (relatively
prime) fundamental discriminants $D_{1},D_{2}$;

  3. (3)

$\varepsilon_{D_{1}}$ is a totally positive element which is congruent to $-1$
(resp $1$) modulo the primes dividing $D_{1}$ (resp. $D_{2}$).

Proof. By Lemma 3.9, it may be assumed that $I_{1}$ and $I_{2}$ have the same
norm, and are represented by ideals that are relatively prime to $\delta$. We
must prove an equality of the form

$\mbox{Trace of $\Theta(I_{1},I_{2})$ from $\Gamma(D)\cap\Gamma_{0}(N)$ to
$\Gamma_{0}(N)$}=\mbox{sum of $\Theta^{\prime}$s}.$

We will do this in a fashion very similar to the proof of Proposition 2.6,
i.e. by reducing it to a local question about Weil representations. Both
$\Theta(I_{1},I_{2})$ and $\Theta^{\sharp}(I_{1},I_{2})$ have the general form

(66) $\Theta_{\Psi}(q):=\sum_{(x,y)\in V_{\pm}}\Psi(x,y){\rm sign}(x)\cdot{\rm
sign}(y)\cdot q^{Q(x,y)},$

where:

  * •

$V=K\oplus K$ considered as a quadratic space over $\mathbb{Q}$: we consider
it as a $\mathbb{Q}$-vector space and endow it with the quadratic form

$Q(x,y)=\frac{xx^{\prime}}{D{\rm N}(I_{1})}-\frac{yy^{\prime}}{D{\rm
N}(I_{2})}.$

  * •

$V_{\pm}$ are elements $(x,y)$ with $xx^{\prime}>0$ and $yy^{\prime}<0$.

  * •

$\Psi$ is a Schwartz function on $V\otimes{\mathbb{A}}_{f}$ (with
${\mathbb{A}}_{f}$ the ring of finite adeles), invariant by the action of the
subgroup $\mathcal{U}$ of the of the unit group
$\mathfrak{o}_{1}^{\times}\mathfrak{o}_{1}^{\times}$.

In the situation of (66) the map $\Psi\mapsto\Theta_{\Psi}$ is equivariant for
the Weil representation action of ${\mathrm{SL}}_{2}({\mathbb{A}}_{f})$ on
Schwartz functions on $V\otimes{\mathbb{A}}_{f}$; this action preserves the
invariance condition on $\Psi$. Indeed this is a a product of two copies of
the situation already discussed in §3.5 and the Weil representation for a
direct sum of quadratic spaces is simply the tensor product of the individual
factors.

The action of ${\mathrm{SL}}_{2}({\mathbb{A}}_{f})$ on Schwartz functions just
mentioned factors as a (restricted) tensor product of actions of
${\mathrm{SL}}_{2}(\mathbb{Q}_{p})$ on the space of Schwartz functions on
$V\otimes\mathbb{Q}_{p}$. The factor at $p$ is the Weil representation of
${\mathrm{SL}}_{2}(\mathbb{Q}_{p})$ on the Schwartz functions on the quadratic
space $(V_{p},Q_{p})$, where:

$V_{p}=(K\oplus K)\otimes\mathbb{Q}_{p},\qquad
Q_{p}(x,y)=\frac{xx^{\prime}}{D{\rm N}(I_{1})}-\frac{yy^{\prime}}{D{\rm
N}(I_{2})}.$

In this way, we are reduced to a problem in explicitly computing with this
Weil representation: the question of computing the trace of
$\Theta^{\sharp}(I_{1},I_{2})$ from $\Gamma_{0}(N)\cap\Gamma(D)$ to
$\Gamma_{0}(N)$ reduces, thereby, to a product of local computations over $p$
dividing $D$, which which we will spell out below.

######  Lemma 3.11.

(cf. Lemma 2.7). Let $\ell$ divide $D$.

Let $(W,L,\langle\,,\rangle)$ be the quadratic space over $\mathbb{Q}_{\ell}$
given by $K\otimes\mathbb{Q}_{\ell}$ equipped with the norm form, multiplied
by $(DN(I_{1}))^{-1}$ and $L$ be the ring of integers. Let
$(W^{\prime},L^{\prime},\langle\,,\rangle^{\prime})$ be similarly defined but
multiplying the form by $-(DN(I_{2}))^{-1}$ and taking $L^{\prime}$ to be the
ring of integers.

Call $\mathbf{e}_{1}$ the characteristic function of

$\\{(x\in L,x^{\prime}\in L^{\prime}):x\equiv x^{\prime}\equiv
1\in(\mathbb{Z}/\ell)\\},$

considered as a Schwartz function on $W\oplus W^{\prime}$. (Here the map from
$L$ to $\mathbb{Z}/\ell$ is given by reduction at the maximal ideal.)

Then, for the Weil representation action of
${\mathrm{SL}}_{2}(\mathbb{Q}_{\ell})$ on Schwartz functions on $W\oplus
W^{\prime}$, the trace

$\mathsf{Tr}^{{\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})}_{\Gamma(\ell)}\mathbf{e}_{1}=\ell\left(\ell
1_{{\mathcal{M}}_{+}}-1_{{\mathcal{M}}_{-}}\right),$

where ${\mathcal{M}}_{\pm}$ are the two self-dual integral lattices contained
in $(L\oplus L^{\prime})$, defined in (68).

Proposition 3.10 follows readily from this Lemma. Indeed, from (62) we can
write $\Theta(I_{1},I_{2})$ in the notation of (66) as the series
$\Theta_{\Psi}$ with $\Psi=\bigotimes\Psi_{\ell}$ and $\Psi_{\ell}$ simply the
characteristic function of $\mathfrak{N}^{\prime}I_{1}\oplus I_{2}$ for $\ell$
not dividing $D$, and $\Psi_{\ell}=\mathbf{e}_{1}$ for $\ell$ dividing $D$. We
must only observe that, given a factorization $D=D_{1}D_{2}$, the value of the
corresponding $\Theta$ series where we replace the role of $\mathbf{e}_{1}$ by
${\mathcal{M}}_{+}$ for $\ell|D_{1}$ and by ${\mathcal{M}}_{-}$ for
$\ell|D_{2}$ is exactly $\Theta^{\sharp}(I_{1},I_{2})$ but replacing $x\equiv
y(\delta)$ by $x\equiv\varepsilon_{D_{1}}y\ (\delta)$, and this in turn
coincides with $\Theta^{\sharp}(I_{1},\varepsilon_{D_{1}}I_{2})$ by means of
the substitution $y\leftarrow\varepsilon_{D_{1}}y$.

###### Proof.

(of Lemma 3.11). First we define ${\mathcal{M}}_{\pm}$. Let $(L\oplus
L^{\prime})^{*}$ be the dual lattice with respect to the quadratic form $Q$ on
$W\oplus W^{\prime}$ and similarly define $L^{*},(L^{\prime})^{*}$. Then
$L^{*}$ corresponds simply to the maximal ideal inside $L$, and similarly for
$L^{\prime}$, so there are canonical identifications

(67) $(L/L^{*})\simeq(\mathbb{Z}/\ell)\simeq
L^{\prime}/(L^{\prime})^{*},\qquad\frac{(L\oplus L^{\prime})}{(L\oplus
L^{\prime})^{*}}\simeq(\mathbb{Z}/\ell)^{2}.$

We let

(68) ${\mathcal{M}}_{\pm}=\mbox{ preimages of the lines $x_{1}\equiv\pm x_{2}$
in $(\mathbb{Z}/\ell\mathbb{Z})^{2}$}.$

The function $\mathbf{e}_{1}$ is readily verified, using the formulas in (55),
to be invariant by the principal congruence subgroup $\Gamma(\ell)$ of level
$\ell$ inside ${\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})$. Indeed, using the
Iwahori factorization of $\Gamma(\ell)$ it suffices to prove this for upper
triangular unipotent elements, diagonal elements, and lower triangular
unipotent elements congruent to the identity modulo $\ell$. For the first two
this is obvious from the first two lines of (55); to conclude we write the
lower triangular unipotent subgroup with the conjugate of the upper triangular
subgroup by the element

$w=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$

that appears on the last line of (55). Since the Weil constant $\gamma=1$, it
suffices to observe that $w^{-1}$ acts as the inverse of the Fourier
transform.

We must compute its trace to
${\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})$-invariants. Clearly, this projection is
the same as if we first average over the diagonal subgroup, which has the
effect of replacing $\mathbf{e}_{1}$ by $\frac{\sum_{j\neq
0}\mathbf{e}_{j}}{\ell-1}$ with where $\mathbf{e}_{j}$ be the Schwartz
function defined similarly to $\mathbf{e}_{j}$ but now considering
$x_{1}\equiv x_{2}\equiv j\mbox{ modulo $\ell$}$. Now
$1_{{\mathcal{M}}_{+}}=\sum_{j}\mathbf{e}_{j}$ and so

$\sum_{j\neq 0}\mathbf{e}_{j}=1_{{\mathcal{M}}_{+}}-\mathbf{e}_{0}.$

Now this is in fact invariant by
$K_{0}(\ell)\subset{\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})$. Indeed
$1_{{\mathcal{M}}_{+}}$ is already invariant by
${\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})$, since it is self-dual and integral for
the quadratic form, and $\mathbf{e}_{0}$ is the characteristic function of
$(L\oplus L^{\prime})^{*}$, on which the quadratic form is integral. We will
prove that (cf. (36))

(69)
${\mathrm{trace}}^{{\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})}_{K_{0}(\ell)}\mathbf{e}_{0}=1_{{\mathcal{M}}_{+}}+1_{{\mathcal{M}}_{-}}.$

From this it follows that the corresponding trace of $\sum_{j\neq
0}\mathbf{e}_{j}$ equals $\ell 1_{{\mathcal{M}}_{+}}-1_{{\mathcal{M}}_{-}}$
and the Lemma follows from this, taking into account the index
$[K_{0}(\ell):K(\ell)]=(\ell-1)\ell$.

The proof of of (69) is very similar to the computation carried out in
Proposition 2.6 of the previous chapter, and, more specifically, to (37). The
role of ${\mathcal{M}}_{\pm}$ arises from the fact that

$\\{x\in L\oplus
L^{\prime}:Q(x)\in\mathbb{Z}_{\ell}\\}={\mathcal{M}}_{+}\cup{\mathcal{M}}_{-},$

and indeed the function induced by the quadratic form upon the right-hand
group of (67) is proportional to
$(x_{1},x_{2})\in(\mathbb{Z}/\ell\mathbb{Z})^{2}\mapsto\ell^{-1}(x_{1}^{2}-x_{2}^{2})\in\ell^{-1}\mathbb{Z}/\mathbb{Z}$.
Let notation be as in (35); as discussed there, a system of coset
representatives for ${\mathrm{SL}}_{2}(\mathbb{Z}_{\ell})/K_{0}(\ell)$ is
given by $w$ together with $wm(t)w$, where $1\leq t\leq\ell$. We get
$w\mathbf{e}_{0}=\ell^{-1}1_{L\oplus L^{\prime}}$ and thus

$\sum_{t}m(t)w\mathbf{e}_{0}=1_{{\mathcal{M}}_{+}}+1_{{\mathcal{M}}_{-}}-\mathbf{e}_{0}.$

Therefore,
$\left(w\sum_{t}m(t)w\right)\mathbf{e}_{0}=1_{{\mathcal{M}}_{+}}+1_{{\mathcal{M}}_{-}}-w\mathbf{e}_{0}$,
and so the trace of $\mathbf{e}_{0}$ is
$1_{{\mathcal{M}}_{+}}+1_{{\mathcal{M}}_{-}}$ as desired. ∎

We will now parlay Prop. 3.10 into an expression for the trace of the product
$\theta_{\psi_{1}^{-1}}(q^{N})\theta_{\psi_{2}^{-1}}(q)$ of weight one theta
series. The following result immediately implies the desired Proposition 3.3,
after performing a change of variables via the isomorphism
$\mathcal{C}^{2}\longrightarrow({\mathcal{C}}_{D}\times_{W}{\mathcal{C}}_{D})/Z$
of Lemma 3.5, given explicitly by
$(I_{1},I_{2})\mapsto(I_{1}I_{2},I_{1}I_{2}^{\prime})$.

######  Proposition 3.12.

Let

$G_{ND^{2}}(q):=\theta_{\psi_{1}^{-1}}(q^{N})\cdot\theta_{\psi_{2}^{-1}}(q),$

which belongs to the space $M_{2}(\Gamma_{0}(ND^{2}))$ of modular forms of
level $ND^{2}$ with trivial nebentypus character. Then

$\displaystyle{\rm Tr}^{ND^{2}}_{ND}\left(G_{ND^{2}}\right)$ $\displaystyle=$
$\displaystyle\psi_{1}({\mathfrak{N}}^{\prime})\frac{1}{4}\sum_{(\mathcal{C}_{D}\times_{W}\mathcal{C}_{D})/Z}\psi_{1}(I_{1})\psi_{2}(I_{2})\cdot\Theta^{(0)}(I_{1},I_{2}),$
$\displaystyle{\rm Tr}^{ND^{2}}_{N}\left(G_{ND^{2}}\right)$ $\displaystyle=$
$\displaystyle\psi_{1}({\mathfrak{N}}^{\prime})\cdot\frac{C}{4}\cdot\\!\\!\\!\\!\\!\\!\\!\sum_{(\mathcal{C}_{D}\times_{W}\mathcal{C}_{D})/Z}\psi_{1}(I_{1})\psi_{2}(I_{2})\cdot\Theta^{\sharp}(I_{1},I_{2}),$

where

$C:=D\sum_{D=D_{1}D_{2}}\mu(D_{1})\cdot
D_{2}\cdot\psi_{1}(\varepsilon_{{}_{D_{1}}})=D\prod_{p|D}(p-\psi_{1}(\varepsilon_{p}))$

is a constant that depends on $(\psi_{1},\psi_{2})$ and $D$ but not on $N$.

Proof. By Lemma 3.8

$G_{ND^{2}}(q)=\frac{1}{4}\sum_{(I_{1},I_{2})\in\mathcal{C}_{D}^{2}}\psi_{1}({\mathfrak{N}}^{\prime}I_{1})\vartheta^{+}({\mathfrak{N}}^{\prime}I_{1})(q^{ND})\cdot\psi_{2}(I_{2})\vartheta^{-}(I_{2})(q^{D}),$

where we re-indexed the sum for $\theta_{\psi_{1}^{-1}}$ via
$I\leftarrow\mathfrak{N}^{\prime}I$.

Because the restrictions to $Z$ of the characters $\psi_{1}$ and $\psi_{2}$
are inverses of each other, the right hand side can be rewritten as

$\frac{1}{4}\sum_{\mathcal{C}_{D}^{2}/Z}\psi_{1}({\mathfrak{N}}^{\prime}I_{1})\psi_{2}(I_{2})\sum_{j\in
Z}\vartheta^{+}(j{\mathfrak{N}}^{\prime}I_{1})(q^{ND})\vartheta^{-}(jI_{2})(q^{D}),$

where $Z\subset\mathcal{C}_{D}^{2}$ is embedded diagonally. It follows from
(64) and (63) that

(70)
$G_{ND^{2}}(q)=\frac{1}{4D}\sum_{\mathcal{C}_{D}^{2}/Z}\psi_{1}({\mathfrak{N}}^{\prime}I_{1})\psi_{2}(I_{2})\cdot\Theta^{(0)}(I_{1},I_{2})(q^{D}).$

Both the left and right hand sides in this identity are modular forms on
$\Gamma_{0}(ND^{2})$. Let $U_{D}$ be the Hecke operator which on
$q$-expansions is given by

$U_{D}(\sum a_{n}q^{n})=\sum_{n\in\mathbb{Z}}a_{{}_{nD}}q^{n}.$

The trace from level $D^{2}$ to level $D$ amounts to an application of $D\cdot
U_{D}$, and by the same reasoning as in Lemma 3.9, we have

$U_{D}(\Theta^{(0)}(I_{1},I_{2})(q^{D}))=\left\\{\begin{array}[]{cl}\Theta^{(0)}(I_{1},I_{2})(q)&\mbox{
if }{\rm N}(I_{1})\equiv{\rm N}(I_{2}),\\\ 0&\mbox{
otherwise}.\end{array}\right.$

Applying the trace to level $ND$ to both sides of (70) therefore gives

$\displaystyle{\rm Tr}^{ND^{2}}_{ND}(G_{ND^{2}})$ $\displaystyle=$
$\displaystyle\psi_{1}({\mathfrak{N}}^{\prime})\cdot\frac{1}{4}\cdot\sum_{(\mathcal{C}_{D}\times_{W}\mathcal{C}_{D})/Z}\psi_{1}(I_{1})\psi_{2}(I_{2})\Theta^{(0)}(I_{1},I_{2})(q),$

and the first equation in Proposition 3.12 follows directly. The second
follows from this and (65), taking into account that $\psi_{2}$ and $\psi_{1}$
agree on $\varepsilon_{p}$. ∎

###  3.7. Proof of Proposition 3.4

Recall now the setup of §3.2. We choose narrow ideal classes $I_{1}$ and
$I_{2}$, and, by choosing representatives by ideals that are divisible by
$\mathfrak{N}$ but not $\mathfrak{N}^{\prime}$, obtain a pair of real
quadratic geodesics $\gamma_{1}:=\gamma_{{}_{I_{1}}}$ and
$\gamma_{2}:=\gamma_{{}_{I_{2}}}$ in $\Gamma_{0}(N)\backslash{\mathcal{H}}$
with the same discriminant $D$. We also obtain embeddings $\alpha_{i}$ for
$i\in\\{1,2\\}$ attached to $I_{1}$ and $I_{2}$; similarly we get eigenvectors
$v_{i},v_{i}^{\prime}\in K^{2}$ and fixed points
$\tau_{i},\tau_{i}^{\prime}\in K$ for the action of $\alpha_{i}(K^{\times})$.

Proposition 3.4 asserts that the generating series of (45) is equal to

$\Theta({\gamma_{1}\otimes\gamma_{2}})(q)=\frac{1}{4}\cdot\Theta^{\sharp}(I_{1}I_{2},I_{1}I_{2}^{\prime})(q).$

The proof proceeds, much as in the proof of the Gross-Zagier formula, by the
most powerful technique known to number theory – compute and compare.

Examining the definition of $\Theta^{\sharp}(I_{1},I_{2})$ from (3.3), we see
that

(71) $\mbox{$m$th Fourier coefficient of $\Theta^{\sharp}$}=\
4\\!\\!\\!\sum_{(x,y)\in\mathcal{A}_{m}/\mathcal{U}}{\rm sign}(xy),$

where $\mathcal{A}_{m}$ consists of the pairs
$(x,y)\in\mathfrak{N}^{\prime}I_{1}I_{2}\times I_{1}I_{2}^{\prime}$ satisfying

(72) $xx^{\prime}>0,\ \ yy^{\prime}<0,\
\qquad\frac{xx^{\prime}-yy^{\prime}}{a_{1}a_{2}}=Dm,$

where $a_{1}=N(I_{1})$, $a_{2}=N(I_{2})$ and $\mathcal{U}$ is the subgroup of
$\mathfrak{o}_{1}^{\times}{\times}\mathfrak{o}_{1}^{\times}$ introduced in
(49).

Now we turn to the left hand side, which is more involved and will take up the
remainder of the subsection. We must compute the $m$-th Fourier coefficient

$a_{m}:=\langle\gamma_{1}\cdot T_{m}\gamma_{2}\rangle_{{}_{N}}.$

Letting $M_{0}(N)_{m}$ be the set of elements of $M_{0}(N)$ of determinant
$m$, and letting

$\Gamma_{1}:=\alpha_{1}(\mathfrak{o}_{1}^{\times}),\quad\Gamma_{2}:=\alpha_{2}(\mathfrak{o}_{1}^{\times}),$

this intersection number can be rewritten as

(73) $a_{m}=\sum_{A\in\Gamma_{1}\backslash
M_{0}(N)_{m}/\Gamma_{2}}\langle(\tau_{1},\tau_{1}^{\prime})\cdot(A\tau_{2},A\tau_{2}^{\prime})\rangle.$

(Note that in (73) the intersection numbers are now being computed on the
upper half-plane and not on the modular curve.) The calculation proceeds by
rewriting the coefficient $a_{m}$ of (73) as a sum over certain ideals of $K$,
by exploiting the map

$\eta:M_{2}(\mathbb{Q})\hookrightarrow K\oplus
K,\qquad\eta(A):=(\det(v_{1},Av_{2}),\det(v_{1},Av_{2}^{\prime})).$

The map $\eta$ sets up a $K\otimes K$-module isomorphism from
$M_{2}(\mathbb{Q})$ to $K\oplus K$, the module structures being given by

(74) $(a\otimes b)M:=\alpha_{1}(a^{\prime})M\alpha_{2}(b)\mbox{ and }(a\otimes
b)(x,y)=(abx,ab^{\prime}y)$

respectively.

It is also an isomorphism of quadratic spaces, after equipping $K\oplus K$
with the quadratic form $Q(x,y)=\frac{xx^{\prime}-yy^{\prime}}{Da_{1}a_{2}}$:

######  Lemma 3.13.

If $\eta(A)=(x,y)$, then

$\det(A)=\frac{xx^{\prime}-yy^{\prime}}{Da_{1}a_{2}}.$

###### Proof.

The source and the target of $\eta$ are both cyclic $(K\otimes K)$-modules, as
in (74), and both sides transform the same way, which reduces us to verifying
the assertion for a single generator; taking $A$ to be the identity and using
$Da_{1}a_{2}=\det(v_{1},v_{1}^{\prime})\det(v_{2},v_{2}^{\prime})$ this
follows from the identity

$\det(v_{1},v_{1}^{\prime})\det(v_{2},v_{2}^{\prime})-\det(v_{1},v_{2})\det(v_{1}^{\prime},v_{2}^{\prime})+\det(v_{1},v_{2}^{\prime})\det(v_{1}^{\prime},v_{2})=0.$

which can be derived by considering the determinant of the $4\times 4$ matrix
whose rows are two copies of $[v_{1},v_{1}^{\prime},v_{2},v_{2}^{\prime}]$. ∎

######  Proposition 3.14.

The image of $M_{0}(N)$ under $\eta$ is equal to

$\eta(M_{0}(N))=\left\\{(x,y)\in{\mathfrak{N}}^{\prime}I_{1}I_{2}\times
I_{1}I_{2}^{\prime}\ \ \mbox{ with }x\equiv y\pmod{\delta}\right\\},$

and $\eta$ induces a bijection between $\Gamma_{1}\backslash
M_{0}(N)_{m}/\Gamma_{2}$ and $\mathcal{A}_{m}/\mathcal{U}$.

Proof. Note that $K\otimes K$ is naturally identified with $K\oplus K$ via the
map $\varrho$ sending $a\otimes b$ to

$\varrho(a\otimes b)=(ab,ab^{\prime}).$

For $1\leq i,j\leq 2$, let $E_{ij}$ be the elementary matrix having a $1$ in
the $ij$ entry and $0$’s elsewhere, and set $\beta_{j}=(-b_{j}+\sqrt{D})/2$.
By the definition of $\eta$,

$\displaystyle\eta(E_{11})=\varrho(-a_{1}\otimes\beta_{2})$
$\displaystyle\eta(E_{12})=\varrho(-a_{1}\otimes a_{2}),$
$\displaystyle\eta(E_{21})=\varrho(\beta_{1}\otimes\beta_{2}),$
$\displaystyle\eta(E_{22})=\varrho(\beta_{1}\otimes a_{2}).$

It follows that $\eta(M_{2}(\mathbb{Z}))$ is contained in the index $D$
subgroup of $I_{1}I_{2}\times I_{1}I_{2}^{\prime}$ consisting of pairs that
are congruent modulo $\delta$. The fact that this containment is an equality
follows by comparing the determinants of the pairing matrices for the two
lattices, relative to the quadratic forms $\det(A)$ and
$\frac{xx^{\prime}-yy^{\prime}}{Da_{1}a_{2}}$ respectively. Furthermore, the
lattice $\eta(M_{0}(N))$ is obtained by replacing the $\mathbb{Z}$-module
generator $\eta(E_{21})$ by $N\eta(E_{21})$. A local analysis at $N$ shows
that

$\eta(M_{0}(N))\subset{\mathfrak{N}}^{\prime}I_{1}I_{2}\times
I_{1}I_{2}^{\prime}.$

Since it is of index at most $N$ in $\eta(M_{2}(\mathbb{Z}))$, it must be
equal to

$\\{(x,y)\in{\mathfrak{N}}^{\prime}I_{1}I_{2}\times I_{1}I_{2}^{\prime}\
\mbox{ with }x\equiv y\pmod{\delta}\\},$

as claimed. In particular, the map $\eta$ identifies $M_{0}(N)_{m}$ with
$\mathcal{A}_{m}$, and the last assertion follows from the fact that $\eta$
tranforms the left action of $\varepsilon\in\Gamma_{1}$ (resp. the right
action of $\varepsilon\in\Gamma_{2}$) into mutliplication by
$(\varepsilon,\varepsilon)$ (resp. by $(\varepsilon,\varepsilon^{-1}))$, which
together generate $\mathcal{U}$. $\square$

It is also crucial to interpret the intersection pairing
$\langle\gamma_{1}\cdot A\gamma_{2}\rangle\in\\{-1,0,1\\}$ in terms of
$\eta(A)$.

######  Lemma 3.15.

If $\det(A)=m>0$, then the intersection $\langle\gamma_{1}\cdot
A\gamma_{2}\rangle$ is non-zero if and only if $xx^{\prime}>0$ and
$yy^{\prime}<0$, where $\eta(A)=(x,y)$. In that case, it is given (after
suitable choice of orientation conventions for the intersection) by ${\rm
sign}(xy)$.

Proof. Given any four distinct elements
$t_{1},t_{1}^{\prime},t_{2},t_{2}^{\prime}$ of $\mathbb{P}_{1}({\mathbb{R}})$,
the hyperbolic geodesics $(t_{1},t_{1}^{\prime})$ and $(t_{2},t_{2}^{\prime})$
intersect non trivially if and only if the cross-ratios
$[t_{1},t_{2}^{\prime};t_{2},t_{1}^{\prime}]$ and
$[t_{1},t_{2};t_{2}^{\prime},t_{1}^{\prime}]$ belong to the open interval
$(0,1)\subset{\mathbb{R}}$. This can be seen by exploiting the invariance of
the cross ratio under Möbius transformations to reduce this statement to the
special case in which
$(t_{1},t_{1}^{\prime},t_{2},t_{2}^{\prime})=(0,\infty,1,t)$, where it can be
verified directly. In particular, the geodesics $(\gamma_{1},A\gamma_{2})$
intersect precisely when the following cross ratios belong to
$(0,1)\subset{\mathbb{R}}$:

(75)
$\frac{xx^{\prime}}{mDa_{1}a_{2}}=\frac{\det(v_{1},Av_{2})\det(v_{1}^{\prime},Av_{2}^{\prime})}{\det(v_{1},v_{1}^{\prime})\det(Av_{2},Av_{2}^{\prime})}=[\tau_{1},A\tau_{2}^{\prime};A\tau_{2},\tau_{1}^{\prime}],$
(76)
$\frac{-yy^{\prime}}{Da_{1}a_{2}}=\det(A)\frac{\det(\tau_{1},A\tau_{2}^{\prime})\det(\tau_{1}^{\prime},A\tau_{2})}{\det(\tau_{1},\tau_{1}^{\prime})\det(A\tau_{2}^{\prime},A\tau_{2})}=\det(A)[\tau_{1},A\tau_{2};A\tau_{2}^{\prime},\tau_{1}^{\prime}].$

The first assertion follows. As to the second, the sign of

$xy=\det(v_{1},Av_{2})\det(v_{1}^{\prime},Av_{2})$

determines whether or not $\tau_{1}$ lands inside or outside of the geodesic
from $A\tau_{2}$ to $A\tau_{2}^{\prime}$, and hence determines the sign of the
non-zero intersection, given a suitable choice of orientation on
${\mathcal{H}}$. $\square$

Recall that $\mathcal{U}$ acts naturally on the set $\mathcal{A}_{m}$ from
(72). By combining Lemmas 3.13, 3.14, and 3.15, we obtain:

######  Proposition 3.16.

For all $m\geq 1$,

$a_{m}=\sum_{(x,y)\in\mathcal{A}_{m}/\mathcal{U}}{\rm sign}(xy).$

Comparing this proposition with (71) shows that

$\Theta(\gamma_{{}_{I_{1}}},\gamma_{{}_{I_{2}}})=\frac{1}{4}\Theta^{\sharp}(I_{1}I_{2},I_{1}I_{2}^{\prime}),$

and Proposition 3.4 follows.

##  4\. Higher Eisenstein elements

This chapter is devoted to a review of “higher Eisenstein elements” in the
sense of Merel and Lecouturier [Me], [Lec], i.e. elements in suitable spaces
of modular forms that are not killed by the Eisenstein ideal but by its
square, see Definition 4.6. We will provide explicit formulas for Eisenstein
and higher Eisenstein elements in

  * •

the space $\mathbb{M}$ of modular forms (Proposition 4.1);

  * •

the dual space $\mathbb{M}^{*}$ to modular forms (Theorem 4.9);

  * •

the positive part of cohomology $\mathbb{H}^{+}$ of the modular curve (Theorem
4.8);

  * •

the negative part of cohomology $\mathbb{H}^{-}$ of the modular curve (§4.4;
here we don’t need higher elements), and finally

  * •

the supersingular module $\mathbb{D}$ (Theorem 4.11).

Each of these spaces $\mathbb{M},\mathbb{H}^{+},\mathbb{H}^{-},\mathbb{D}$ is
the completion of a suitable Hecke module at Mazur’s Eisenstein ideal in the
Hecke algebra.

###  4.1. Higher Eisenstein series

As in §1.5, let $N>3$ be a prime, let $M_{2}(N)$ be the module of weight two
modular forms with Fourier coefficients in $Z=\mathbb{Z}[\frac{1}{6N}]$ for
the Hecke congruence group $\Gamma_{0}(N)$, and let $S_{2}(N)\subset M_{2}(N)$
denote the submodule of cusp forms. Denote by ${\mathbb{T}}(N)$ the ring
generated by the Hecke operators $T_{n}$ (with $N\nmid n$) together with
$T_{N}:=U_{N}$, acting faithfully on $M_{2}(N)$.

The vector space $M_{2}(N)\otimes\mathbb{Q}$ is generated by
$S_{2}(N)\otimes\mathbb{Q}$ along with the weight two Eisenstein series whose
$q$-expansion is given by

(77)
$E_{2}^{{}^{{}_{(N)}}}\\!(q)=\frac{N-1}{24}+\sum_{n=1}^{\infty}\sigma_{1}^{{}^{{}_{{}_{{}_{{}_{\\!(N)}}}}}}\\!\\!(n)q^{n},\qquad\mbox{
where
}\sigma_{1}^{{}^{{}_{{}_{{}_{{}_{\\!(N)}}}}}}\\!\\!(n)=\sum_{\begin{subarray}{c}d|n,\\\
N\nmid d\end{subarray}}d.$

The homomorphism

$\varphi_{\rm Eis}:{\mathbb{T}}(N)\longrightarrow\mathbb{Z},\qquad\varphi_{\rm
Eis}(T_{n}):=\sigma_{1}^{{}^{{}_{{}_{{}_{{}_{\\!(N)}}}}}}\\!\\!(n)$

by which ${\mathbb{T}}(N)$ acts on $E_{2}^{{}^{{}_{(N)}}}\\!$ is called the
Eisenstein homomorphism, and its kernel $I_{\rm Eis}$ is called the Eisenstein
ideal.

For any maximal ideal $\mathfrak{m}$ of $\mathbb{T}(N)$ and any
$\mathbb{T}(N)$-module $M$, let $M_{\mathfrak{m}}$ denote the completion of
$M$ at $\mathfrak{m}$. The maximal ideal $\mathfrak{m}$ is said to be
Gorenstein if ${\mathbb{T}}:=\mathbb{T}(N)_{\mathfrak{m}}$ is a Gorenstein
ring. It is known that all maximal ideals of ${\mathbb{T}}(N)$ containing a
prime $p>3$ are Gorenstein, by a result of Mazur [Maz77, cor. II.16.3].

Let $p>3$ be a prime divisor of $N-1$. The maximal ideal
${\mathfrak{m}}:=(p,I_{\rm Eis})$ of ${\mathbb{T}}(N)$ is called the
$p$-Eisenstein ideal. Let

${\mathbb{T}}:={\mathbb{T}}(N)_{\mathfrak{m}},\qquad{\mathbb{M}}:=M_{2}(N)_{{\mathfrak{m}}}$

denote the completions of ${\mathbb{T}}(N)$ and $M_{2}(N)$ relative to this
ideal. The ring ${\mathbb{T}}$ is a complete local ring which is free of
finite rank as a $\mathbb{Z}_{p}$-module. The module ${\mathbb{M}}$ is
canonically dual to ${\mathbb{T}}$ via the pairing
${\mathbb{M}}\times{\mathbb{T}}\longrightarrow\mathbb{Z}_{p}$ given by
$\langle f,T\rangle=a_{1}(Tf)$, and hence ${\mathbb{M}}$ is free of rank one
as a ${\mathbb{T}}$-module, since ${\mathbb{T}}$ is Gorenstein. The
$\mathbb{Z}_{p}$-rank of ${\mathbb{T}}$ is strictly greater than one because
$p$ divides $N-1$. We fix a discrete log
$(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\mathbb{Z}/p^{t}\mathbb{Z}$ as in
§1.5.

The following proposition is due to Lecouturier [Lec], but the details of the
proof have been provided for the sake of being self-contained.

######  Proposition 4.1.

There is a modular form $E^{\prime}\in
M_{2}(N)\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$ having Fourier expansion of the
form

$E^{\prime}={\mathcal{M}}-\sum_{n=1}^{\infty}\left(\sum_{d|n^{\prime}}\log(d^{2}/n^{\prime})d\right)q^{n}$

for some ${\mathcal{M}}\in\mathbb{Z}/p^{t}\mathbb{Z}$, where $n^{\prime}$
denotes the prime-to-$N$ part of $n$. It satisfies $(U_{N}-1)E^{\prime}=0$,
and, for all primes $\ell\neq N$,

$(T_{\ell}-(\ell+1))E^{\prime}=(\ell-1)\log(\ell)E_{2}^{{}^{{}_{(N)}}}\\!.$

The modular form $E^{\prime}$ mod $p^{t}$ is called the higher Eisenstein
series of weight $2$ and level $N$. We will discuss abstractly such elements
in other Hecke modules in §4.2.

###### Proof.

Recall that $Z:=\mathbb{Z}[1/6N]$ and let $I$ denote the augmentation ideal in
the group ring $Z[G_{N}]$, where $G_{N}$ is as in (10). For $d$ an integer
prime to $N$ we shall denote by $\sigma_{d}$ the corresponding element of
$G_{N}$, arising from $d$ by means of the homomorphism
$\mathbb{Z}\rightarrow(\mathbb{Z}/N)^{\times}\rightarrow G_{N}$. Let
${\mathbb{E}}$ and ${\mathbb{F}}$ be the formal $q$-expansions with
coefficients in $Z[G_{N}]$ given by

(78)
${\mathbb{E}}:={\mathfrak{M}}-\sum_{n=1}^{\infty}\left(\sum_{\begin{subarray}{c}d|n,\\\
N\nmid
d\end{subarray}}d\sigma_{d}\right)q^{n},\qquad\qquad{\mathbb{F}}:=-\sum_{n=1}^{\infty}\left(\sum_{\begin{subarray}{c}d|n,\\\
N\nmid\ n/d\end{subarray}}d\sigma_{n/d}\right)q^{n},$

where

${\mathfrak{M}}:=\frac{1}{2}\sum_{j=1}^{N-1}\theta_{j}\cdot\sigma_{j},\qquad\mbox{
with }\theta_{j}:=\frac{N}{2}B_{2}(j/N),\quad B_{2}(x):=x^{2}-x+1/6.$

These formal $q$-expansions satisfy, for every Dirichlet character $\chi$ of
modulus $N$,

$\chi({\mathbb{E}})=\left\\{\begin{array}[]{ll}E_{2}^{{}^{{}_{(N)}}}\\!=E_{2}(1,1_{N})&\mbox{
if }\chi=1;\\\ E_{2}(1,\chi)&\mbox{
otherwise},\end{array}\right.\qquad\chi({\mathbb{F}})=\left\\{\begin{array}[]{ll}E_{2}(1_{N},1)&\mbox{
if }\chi=1;\\\ E_{2}(\chi,1)&\mbox{ otherwise},\end{array}\right.$

where $1_{N}$ denotes the trivial character, but viewed as having modulus $N$,
and $E_{2}(1,\chi)$ and $E_{2}(\chi,1)$ are the usual Eisenstein series
associated to the Galois representations $\chi\omega\oplus 1$ and
$\omega\oplus\chi$ respectively with $\omega$ the cyclotomic character, whose
Fourier expansions are given by

$E_{2}(1,\chi)(q)=-L(-1,\chi)/2-\sum_{n=1}^{\infty}(\sum_{d|n}\chi(d)d)q^{n},$
$E_{2}(\chi,1)(q)=-\sum_{n=1}^{\infty}(\sum_{d|n}\chi(n/d)d)q^{n}.$

These Eisenstein series are classical modular forms of weight two on the
congruence group $\Gamma_{1}(N)$, with the exception of $E_{2}(1_{N},1)$. The
latter is (the holomorphic part of) a nearly holomorphic form in the sense of
Shimura, as we see via

$E_{2}(1_{N},1)=E_{2}(q)-E_{2}(Nq)+\frac{\mathrm{const}}{y},\quad\mbox{ with
}E_{2}=(8\pi y)^{-1}-\frac{1}{24}+\sum_{n}(\sum_{d|n}d)q^{n}.$

Denote by $M_{2}^{\rm nh}(\Gamma_{1}(N);Z)$ the abelian group of
$q$-expansions of such nearly holomorphic forms, so that $E_{2}(1_{N},1)\in
M_{2}^{\rm nh}$.

It follows that ${\mathbb{E}}$ is a classical modular form with coefficients
in $Z[G_{N}]$. As for ${\mathbb{F}}:=\sum_{\sigma\in
G_{N}}F_{\sigma}\cdot\sigma$, although the individual coefficients
$F_{\sigma}\in M_{2}^{\rm nh}(\Gamma_{1}(N);Z)$ are merely nearly holomorphic,
their pairwise differences $F_{\sigma_{1}}-F_{\sigma_{2}}$ are in fact
holomorphic, since they lie in the linear span of the $E_{2}(\chi,1)$ with
$\chi$ non-trivial. It follows that one can write

${\mathbb{F}}={\mathbb{F}}_{0}+\eta\cdot{\bf N},$

where

${\mathbb{F}}_{0}\in M_{2}(\Gamma_{1}(N);Z[G_{N}]),\qquad\eta\in M_{2}^{\rm
nh}(\Gamma_{1}(N);Z),\qquad{\bf N}=\sum_{\sigma\in G_{N}}\sigma.$

Since the $q$-series $E_{2}(1,1_{N})$ and $E_{2}(1_{N},1)$ agree modulo
$p^{t}$, and the image of the norm element ${\bf N}$ in
$\mathbb{Z}/p^{t}\mathbb{Z}[G_{N}]$ belongs to $I^{2}$, the mod $p^{t}$
reduction of the difference ${\mathbb{E}}-{\mathbb{F}}$ belongs to $M_{2}^{\rm
nh}(\Gamma_{1}(N);\mathbb{Z}/p^{t}\mathbb{Z})\otimes I$. It follows that its
natural image, denoted $\overline{{\mathbb{E}}-{\mathbb{F}}}$, in $M^{\rm
nh}_{2}(\Gamma_{1}(N);\mathbb{Z}/p^{t}\mathbb{Z})\otimes(I/I^{2})$ gives rise
to an element

$\overline{{\mathbb{E}}-{\mathbb{F}}}\in
M_{2}(\Gamma_{1}(N);\mathbb{Z}/p^{t}\mathbb{Z})\otimes(I/I^{2})=M_{2}(\Gamma_{1}(N);\mathbb{Z}/p^{t}\mathbb{Z})\otimes
G_{N},$

which is invariant under the diamond operators. At the last stage we have used
the isomorphism $(I/I^{2})\simeq G_{N}\otimes Z$ uniquely characterized by the
fact that $\sum a_{j}\sigma_{j}\mapsto\prod j^{a_{j}}\otimes 1$ when
$a_{j}\in\mathbb{Z}$. Consequently, $\overline{{\mathbb{E}}-{\mathbb{F}}}$
arises from a unique element of
$M_{2}(\Gamma_{0}(N);\mathbb{Z}/p^{t}\mathbb{Z})\otimes G_{N}$, to be denoted
by the same letter. One then readily checks that the modular form $E^{\prime}$
given by

$E^{\prime}:=\log(\overline{{\mathbb{E}}-{\mathbb{F}}})$

has all the properties claimed in the proposition. For instance, since
${\mathbb{E}}$ and ${\mathbb{F}}$ are eigenvectors for $T_{\ell}$ with
eigenvalue $(1+\ell\sigma_{\ell})$ and $(\sigma_{\ell}+\ell)$ respectively,

$(T_{\ell}-(\ell+1))({\mathbb{E}}-{\mathbb{F}})=(\ell\sigma_{\ell}-\ell){\mathbb{E}}-(\sigma_{\ell}-1){\mathbb{F}}=(\ell-1)(\sigma_{\ell}-1)E_{2}^{{}^{{}_{(N)}}}\\!\pmod{I^{2}[[q]]},$

and therefore, after reducing modulo $I^{2}$ and taking the discrete
logarithms on both sides,

$(T_{\ell}-(\ell+1))E^{\prime}=(\ell-1)\log(\ell)E_{2}^{{}^{{}_{(N)}}}\\!,$

as claimed. ∎

######  Remark 4.2.

The proof of Proposition 4.1 yields an explicit formula for the constant term
$\mathcal{M}$ of $E^{\prime}$. It is attached to the Mazur-Tate, or
Stickelberger element ${\mathfrak{M}}$, which is characterised as the unique
element of $Z[G_{N}]$ satisfying

$\chi({\mathfrak{M}})=\left\\{\begin{array}[]{ll}(1-N)/24&\mbox{ if
}\chi=1;\\\ -L(-1,\chi)/2&\mbox{ otherwise},\end{array}\right.\qquad\mbox{ for
all }\chi:G_{N}\longrightarrow\mathbb{C}^{\times}.$

This Mazur-Tate element belongs to the augmentation ideal $I$ of the group
ring $(\mathbb{Z}/p^{t}\mathbb{Z})[G_{N}]$, and its natural image in
$I/I^{2}=G_{N}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$, denoted
${\mathfrak{M}}^{\prime}$, is called the “Mazur-Tate derivative” of
${\mathfrak{M}}$. The constant term ${\mathcal{M}}$ is the discrete logarithm
of this Mazur-Tate derivative:

(79) ${\mathcal{M}}=\log({\mathfrak{M}}^{\prime})$

This explicit formula for ${\mathcal{M}}$, which was first obtained (under a
slightly different guise) by Loic Merel [Me], will play no role in the
argument.

###  4.2. General Higher Eisenstein elements

From now on, the symbol $I_{\rm Eis}$ shall also be used to denote the
Eisenstein ideal in the completed Hecke algebra ${\mathbb{T}}$, whose
associated quotient ${\mathbb{T}}/I_{\rm Eis}$ is isomorphic to
$\mathbb{Z}_{p}$.

Mazur has proved that ${\mathbb{T}}$ is generated by a single element as a
$\mathbb{Z}_{p}$-algebra, i.e. ${\mathbb{T}}=\mathbb{Z}_{p}[x]$ for suitable
$x\in{\mathbb{T}}$. Indeed, one may take $x=T_{\ell}-\ell-1$ for suitable
$\ell$ and $x$ may be taken to generate $I_{\rm Eis}$. See [Maz77, §II, Prop.
18.10], as well as the discussion at the start of §19 therein. The following
result is also proved by Mazur (loc. cit. Proposition 18.8); we sketch a
direct proof.

######  Corollary 4.3.

There is an isomorphism

$\eta:I_{\rm Eis}/I_{\rm
Eis}^{2}=\mathbb{Z}_{p}\otimes(\mathbb{Z}/N\mathbb{Z})^{\times}\simeq(\mathbb{Z}/p^{t}\mathbb{Z})$

sending the element $(T_{\ell}-(\ell+1))$ to $(\ell-1)\otimes\ell$, for all
primes $\ell\neq N$, and sending $U_{N}$ to $1$.

###### Sketch of proof.

The modular form $E_{2}^{{}^{{}_{(N)}}}\\!+\varepsilon E^{\prime}$ with
coefficients in the ring $\mathbb{Z}/p^{t}\mathbb{Z}[\varepsilon]$ of dual
numbers is a Hecke eigenform on $\Gamma_{0}(N)$, and gives rise to a
surjective homomorphism with kernel $I_{\rm Eis}^{2}$

(80) $\tilde{\varphi}_{\rm
Eis}:{\mathbb{T}}\longrightarrow\mathbb{Z}/p^{t}\mathbb{Z}[\varepsilon],\qquad\tilde{\varphi}(U_{N})=1,\qquad\tilde{\varphi}(T_{\ell})=(\ell+1)+(\ell-1)\log(\ell)\varepsilon.$

The quantity $\tilde{\varphi}(T_{\ell}-(\ell+1))$ is equal to
$\log\circ\eta(T_{\ell}-(\ell+1))$, and the corollary follows. ∎

Let ${\mathbb{X}}$ be a free ${\mathbb{T}}$-module of rank one.

######  Lemma 4.4.

The module ${\mathbb{X}}[I_{\rm Eis}]$ of elements $m\in{\mathbb{X}}$
satisfying

$(T_{\ell}-(\ell+1))m=0\quad\mbox{ for all primes }\ell\neq N,\qquad
U_{N}m=m,$

is free of rank one over $\mathbb{Z}_{p}$.

###### Proof.

Since the localisation of ${\mathbb{T}}$ at $I_{\rm Eis}$ is Gorenstein, the
$I_{\rm Eis}$-torsion submodule of $X$ is isomorphic to ${\mathbb{X}}/I_{\rm
Eis}{\mathbb{X}}$, and the result therefore follows from the fact that
${\mathbb{T}}/I_{\rm Eis}$ is isomorphic to $\mathbb{Z}_{p}$. ∎

A generator of the $\mathbb{Z}_{p}$-module ${\mathbb{X}}[I_{\rm Eis}]$ is
called an Eisenstein element in ${\mathbb{X}}$. Although such generators are
only well defined up to scaling by $\mathbb{Z}_{p}^{\times}$, the concrete
Hecke modules that arise in practice are frequently equipped with a
distinguished choice of Eisenstein element $m_{0}$. Corollary 4.3 implies the
following lemma:

######  Lemma 4.5.

There is an element $m_{1}\in{\mathbb{X}}/p^{t}{\mathbb{X}}$ satisfying
$U_{N}m_{1}=m_{1}$ and

(81)
$(T_{\ell}-(\ell+1))m_{1}=(\ell-1)\log(\ell)m_{0}\pmod{p^{t}},\qquad\mbox{ for
every prime }\ell\neq N,$

and the choice of $m_{0}$ uniquely specifies $m_{1}$ up to the addition of a
multiple of $m_{0}$.

The element $m_{1}$ depends linearly on the choice of discrete logarithm,
namely, replacing $\log$ by $a\cdot\log$ with
$a\in(\mathbb{Z}/p^{t}\mathbb{Z})^{\times}$ has the effect of replacing
$m_{1}$ by $am_{1}$.

######  Definition 4.6.

The element $m_{1}$ is called the higher Eisenstein element  in
${\mathbb{X}}/p^{t}$ (associated to $m_{0}$ and to the choice of discrete
logarithm).

For example, (the Eisenstein completion)
${\mathbb{M}}=M_{2}(N)_{\mathfrak{m}}$ of the module of modular forms has a
distinguished Eisenstein element $m_{0}=E_{2}^{{}^{{}_{(N)}}}\\!$. Proposition
4.1 supplies an explicit description of the higher Eisenstein element
$m_{1}=E^{\prime}$ in ${\mathbb{M}}\otimes(\mathbb{Z}/p^{t}\mathbb{Z})$. The
proof of Conjecture 1.1 for dihedral forms rests crucially on similar explicit
expressions of the higher Eisenstein element in various other Hecke modules,
which will be described in the forthcoming sections.

######  Remark 4.7.

When ${\mathcal{M}}\equiv 0\pmod{p^{u}}$ with $u\leq t$, there is also a
second higher Eisenstein element
$m_{2}\in{\mathbb{X}}\otimes(\mathbb{Z}/p^{u}\mathbb{Z})$ satisfying, for all
primes $\ell\neq N$,

$(T_{\ell}-(\ell+1))m_{2}=(\ell-1)\log(\ell)m_{1}\pmod{m_{0}{\mathbb{X}}}.$

In fact, in ${\mathbb{X}}\otimes(\mathbb{Z}/p\mathbb{Z})$ there is an entire
sequence
$m_{0},m_{1},\ldots,m_{r}\in{\mathbb{X}}\otimes(\mathbb{Z}/p\mathbb{Z})$ of
higher Eisenstein elements obeying similar inductive relations, where $r+1$ is
the $\mathbb{Z}_{p}$-rank of ${\mathbb{T}}$. These higher Eisenstein elements
have been studied systematically in [Lec], but only the first higher
Eisenstein elements will play a role in this work. Henceforth, the terminology
“higher Eisenstein series” or “higher Eisenstein element” shall always refer
to what might be called the “first higher Eisenstein element” in [Lec].

###  4.3. The Betti cohomology relative to the cusps

One of the settings which turns out to be relevant to the proof of Conjecture
1.1 for RM dihedral forms occurs when ${\mathbb{X}}:={\mathbb{H}}^{+}$ is the
$p$-Eisenstein completion of the relative cohomology
$H^{1}_{\mathrm{B}}(X_{0}(N);\\{0,\infty\\};Z)^{+}$ with coefficients in the
ring $Z:=\mathbb{Z}[1/6N]$, where the superscript $+$ denotes the subspace
which is fixed by complex conjugation. As discussed in §1.5, the subscript
$\mathrm{B}$ means that we take the singular cohomology of the complex points
of $X_{0}(N)$. This relative cohomology is dual to
$H^{1}_{\mathrm{B}}(Y_{0}(N),Z)^{-}$, which is isomorphic, after tensoring
with $\mathbb{C}$, to the space of weight two modular forms on
$\Gamma_{0}(N)$, via integration. In particular, the ring generated by the
Hecke operators acting on $H^{1}_{\mathrm{B}}(X_{0}(N);\\{0,\infty\\};Z)^{+}$
is naturally identified with ${\mathbb{T}}(N)$.

The module $H^{1}_{\mathrm{B}}(X_{0}(N);\\{0,\infty\\};Z)^{+}$ fits into the
short exact sequence

(82) $0\longrightarrow
Z\stackrel{{\scriptstyle\partial^{*}}}{{\longrightarrow}}H^{1}_{\mathrm{B}}(X_{0}(N);\\{0,\infty\\};Z)^{+}\stackrel{{\scriptstyle
i^{*}}}{{\longrightarrow}}H^{1}_{\mathrm{B}}(X_{0}(N),Z)^{+}\longrightarrow 0$

of ${\mathbb{T}}(N)$-modules, where $\partial^{*}$ is dual to the boundary
homomorphism

(83) $\partial:H_{1,\mathrm{B}}(X_{0}(N);\\{0,\infty\\};Z)\longrightarrow
Z\cdot(0-\infty)=Z.$

The relative cohomology group $H^{1}_{\mathrm{B}}(X_{0}(N);\\{0,\infty\\};Z)$
can be described concretely in terms of $Z$-valued modular symbols:
$\Gamma_{0}(N)$-invariant functions $m$ from
$\mathbb{P}_{1}(\mathbb{Q})\times\mathbb{P}_{1}(\mathbb{Q})$ to $Z$ which are
additive  in the sense that they satisfy

$m\\{a,b\\}+m\\{b,c\\}=m\\{a,c\\}\qquad\mbox{ for all
}a,b,c\in\mathbb{P}_{1}(\mathbb{Q}).$

The image of the class $\partial^{*}(1)$ in ${\mathbb{H}}^{+}$, denoted