# The Eisenstein cycles and Manin-Drinfeld properties

Debargha Banerjee  INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH, PUNE,
INDIA  and  Loïc Merel  Université Paris Cité and Sorbonne Université, CNRS,
IMJ-PRG, F-75013 Paris, France

###### Abstract.

Let $\Gamma$ be a subgroup of finite index of ${\mathrm{SL}}_{2}({\bf Z})$. We
give an analytic criterion for a cuspidal divisor to be torsion in the
Jacobian $J_{\Gamma}$ of the corresponding modular curve $X_{\Gamma}$. By
Belyi’s theorem, such a criterion would apply to any curve over a number
field. Our main tool is the explicit description, in terms of modular symbols,
of what we call Eisenstein cycles. The latter are representations of relative
homology classes over which integration of any holomorphic differential forms
vanishes. Our approach relies in an essential way on the specific case
$\Gamma\subset\Gamma(2)$, where we can consider convenient generalized
Jacobians instead of $J_{\Gamma}$. The Eisenstein classes are the real part of
certain homology classes with complex coefficients. The imaginary part of
those classes are related to the scattering constants attached to Eisenstein
series. Finally, we illustrate our theory by considering Fermat curves.

###### Key words and phrases:

Eisenstein series, Modular symbols, Special values of $L$-functions

###### 2010 Mathematics Subject Classification:

Primary: 11F67, Secondary: 11F11, 11F20, 11F30

##  1\. Introduction

Let $\Gamma$ be a subroup of finite index of ${\mathrm{SL}}_{2}({\bf Z})$,
acting on the upper half-plane $\mathbb{H}$. Consider the modular curve
$Y_{\Gamma}=\Gamma\backslash\mathbb{H}$. Let
$\partial_{\Gamma}=\Gamma\backslash{\mathbb{P}}^{1}({\bf Q})$ be the set of
cusps and $X_{\Gamma}$ be the compact modular curve obtained from
$Y_{\Gamma}=\Gamma\backslash\mathbb{H}$ by adding the cusps. We have a long
exact sequence of relative homology:

$0\rightarrow\mathrm{H}_{1}(X_{\Gamma};{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf
Z})\xrightarrow[]{\delta}{\bf Z}[\partial_{\Gamma}]\rightarrow{\bf
Z}\rightarrow 0,$

where the map $\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf
Z})\xrightarrow[]{\delta}{\bf Z}[\partial_{\Gamma}]$ is obtained from the
boundary map $\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf
Z})\rightarrow\mathrm{H}_{0}(\partial_{\Gamma};{\bf Z})$. Let
$\mathrm{H}^{0}(X_{\Gamma},\Omega^{1})$ denote the complex vector space of
global sections of the sheaf of holomorphic differentials on the Riemann
surface $X_{\Gamma}$. Let $D=\sum_{j\in\partial_{\Gamma}}m_{j}[j]$ be a
divisor of degree $0$ supported on $\partial_{\Gamma}$. We call Eisenstein
classes corresponding to $D$ to be the unique element
${\mathcal{E}}_{D}\in\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf R})$
such that $\int_{{\mathcal{E}}_{D}}\omega=0$ for all
$\omega\in\mathrm{H}^{0}(X_{\Gamma},\Omega^{1})$ and
$\delta({\mathcal{E}}_{D})=D$. We will determine this class in this article,
with the following application in mind: the divisor $D$ is torsion in the
Jacobian $J_{\Gamma}$ of $X_{\Gamma}$ if and only if
${\mathcal{E}}_{D}\in\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf Q})$.

It is known since Manin and Drinfeld that $D$ is torsion when $\Gamma$ is a
congruence subgroup [9, 16]. Scholl [23], Murty-Ramakrishnan [19] and recently
Burrin [5] have given criteria for $D$ being torsion in $J_{\Gamma}$, without
appealing to ${\mathcal{E}}_{D}$. Our approach to this question is different
from those authors and is a continuation of [4] and [17] (where we limited
ourselves to congruence subgroups).

We impose a more specific setting (as we will see, without losing generality),
in connection with the theory of Belyi maps and dessins d’enfants, see section
2. We assume in this article that $\Gamma$ is contained in $\Gamma(2)$ (the
principal congruence subgroup of level $2$) and that $-{\rm Id}\in\Gamma$. The
corresponding modular curve $X(2)=X_{\Gamma(2)}$ has three cusps corresponding
to the classes of $0$, $1$ and $\infty$. Denote by $\partial_{\Gamma}^{+}$
(resp. $\partial_{\Gamma}^{-}$) the cusps above either $0$ or $\infty$ (resp.
above $1$) via the canonical morphism $X_{\Gamma}\rightarrow X(2)$.

We reformulate the question of the Eisenstein cycles in the mixed homology
group
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
R})$ and its dual
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
R})$. Let $\alpha$, $\beta$ be rational numbers with odd numerators and
denominators (resp. with either even numerator or even denominator). Denote by
${\\{\alpha,\beta\\}}^{-}$ (resp. $\\{\alpha,\beta\\}^{+}$) the class in
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
R})$ (resp.
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
R})$) of the geodesic path in $\mathbb{H}$ from $\alpha$ to $\beta$. Such a
consideration is justified by the fact that the elements
$\\{g0,g\infty\\}^{+}$ and $\\{g(1),g(-1)\\}^{-}$ form bases of the free ${\bf
Z}$-modules
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ and
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})$ respectively when $g$ runs though a set of representatives of
$\Gamma\backslash\Gamma(2)$ [17], and see section 4.

For $j\in\partial_{\Gamma}$, let $\Gamma_{j}$ the stabilizer of
$\sigma_{j}(\infty)$ in $\Gamma$. For $x\in{\mathbb{P}}^{1}({\bf Q})$, such
that $m_{\Gamma x}=0$, set

$\mathcal{S}_{D}(x)=\frac{1}{2\pi i}\lim_{s\rightarrow
1^{+}}\sum_{j\in\partial_{\Gamma}}m_{j}\sum_{r=1}^{\infty}\sum_{\sigma\in\Gamma_{j}\backslash\Gamma/\Gamma_{j}}\frac{e^{2\pi
ir(x+\frac{d}{c})}}{|c|^{2s}},$

where the entries $c$ and $d$ are such that
$\sigma_{j}^{-1}\sigma\sigma_{j}=\left(\begin{smallmatrix}\star&\star\\\
c&d\\\ \end{smallmatrix}\right)$ and $c$ is required to be $>0$. The internal
sum over $\sigma$ is sometimes called a Kloosterman zeta function [13, Chapter
9, p. 121], [23, p. 14]. A priori, such Dirichlet series converge absolutely
for $\Re(s)>1$. Their values at $1$ exist since $D$ is of degree $0$.

Let $D^{+}\in\text{Div}^{0}(\partial_{\Gamma}^{+})$ (resp.
$D^{-}\in\text{Div}^{0}(\partial_{\Gamma}^{-})$). Define $F_{D^{+}}$ and
$F_{D^{+}}$: $\Gamma(2)\rightarrow{\bf C}$ by

$F_{D^{+}}(h)=\mathcal{S}_{D^{+}}(h(-1))-\mathcal{S}_{D^{+}}(h(1))\hskip
28.45274pt{\rm and}\hskip
28.45274ptF_{D^{-}}(h)=\mathcal{S}_{D^{-}}(h(\infty))-\mathcal{S}_{D^{-}}(h(0)).$

We will see that those $\Gamma$-invariant quantities can be understood as
periods of certain Eisenstein series. Set

${\mathcal{E}}_{D^{+}}=\sum_{g\in\Gamma\backslash\Gamma(2)}F_{D^{+}}(g)\\{g0,g\infty\\}^{+}\hskip
28.45274pt{\rm and}\hskip
28.45274pt{\mathcal{E}}_{D^{-}}=\sum_{g\in\Gamma\backslash\Gamma(2)}F_{D^{-}}(g)\\{g(1),g(-1)\\}^{-}$

in
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{{\bf
C}})$ and
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{{\bf
C}})$ respectively. We decompose them into their real and imaginary parts:
${\mathcal{E}}_{D^{+}}=\mathcal{R}_{D^{+}}+\frac{1}{2\pi
i}\mathcal{I}_{D^{+}}$ and
${\mathcal{E}}_{D^{-}}=\mathcal{R}_{D^{-}}+\frac{1}{2\pi
i}\mathcal{I}_{D^{-}}$.

######  Theorem 1.

For all differential forms of the third kind $\omega$ on $X_{\Gamma}$ with
poles in $\partial_{\Gamma}^{-}$ (resp. $\partial_{\Gamma}^{+}$), one has
$\int_{\mathcal{R}_{D^{+}}}\omega=0$ (resp.
$\int_{\mathcal{R}_{D^{-}}}\omega=0$) and the boundary of
$\mathcal{R}_{D^{+}}$ (resp. $\mathcal{R}_{D^{-}}$) is $-D^{+}$ (resp.
$-D^{-}$). Moreover ${\mathcal{E}}_{D^{+}}$ (resp. ${\mathcal{E}}_{D^{-}}$)
has boundary $D^{+}$ (resp. $D^{-}$).

It follows that the images of $\mathcal{R}_{D^{+}}$ and $\mathcal{R}_{D^{-}}$
in $\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf R})$ are Eisenstein
classes. They are the classes of what we call Eisenstein cycles. We will
explain in section 10 how to derive all Eisenstein classes in
$\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf R})$ from such images. We
show in section 7 that $\mathcal{I}_{D^{-}}$ and $\mathcal{I}_{D^{+}}$ are
deduced from the scattering constants, obtained from specializing scattering
matrices of Eisenstein series at $s=1$.

We have a canonical map $\lambda^{+}$ :
$\mathrm{H}_{0}(\partial_{\Gamma}^{-},{\bf
R})\rightarrow\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+},{\bf
R})$ (resp. $\lambda^{-}$ : $\mathrm{H}_{0}(\partial_{\Gamma}^{+},{\bf
R})\rightarrow\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-},{\bf
R})$) dual to the boundary map. Its image contains $\mathcal{I}_{D^{+}}$
(resp. $\mathcal{I}_{D^{-}}$), see section 8. Let $J_{\Gamma}^{-}$ (resp.
$J_{\Gamma}^{+}$) be the generalized Jacobian of $X_{\Gamma}$ with respect to
the set $\partial_{\Gamma}^{-}$ (resp. $\partial_{\Gamma}^{+}$).

######  Theorem 2.

The class of $D^{+}$ (resp. $D^{-}$) in $J_{\Gamma}^{-}$ (resp.
$J_{\Gamma}^{+}$) is torsion if and only if for every
$g\in\Gamma\backslash\Gamma(2)$, one has $F_{D^{+}}(g)\in{\bf Q}$ (resp.
$F_{D^{-}}(g)\in{\bf Q}$).

The class of $D^{+}$ (resp. $D^{-}$) in $J_{\Gamma}$ is torsion if and only if
one has $\sum_{g\in\Gamma\backslash\Gamma(2)}F_{D^{+}}(g)[g]\in{\bf
Q}[\Gamma\backslash\Gamma(2)]+\lambda^{+}({\bf R}[\partial_{\Gamma}^{-}])$
(resp. $\sum_{g\in\Gamma\backslash\Gamma(2)}F_{D^{-}}(g)[g]\in{\bf
Q}[\Gamma\backslash\Gamma(2)]+\lambda^{-}({\bf R}[\partial_{\Gamma}^{+}])$.

The first criterion is similar to the following statement: $e^{2\pi iz}$ is
torsion in ${\bf C}^{\times}$ if and only if the real part of $z$ is rational,
and the imaginary part of $z$ is $0$. Note that the first statement is
substantial even if $X_{\Gamma}$ has genus $0$. The second criterion can be
derived from the first. The maps $\lambda^{+}$ and $\lambda^{-}$ are computed
explicitly by proposition 5.

The principal weakness of our approach resides in the fact that the functions
$F_{D^{+}}$ and $F_{D^{-}}$ are not easily computable for two reasons : (i)
the convergence of Dirichlet series is notoriously difficult to understand,
and (ii) we have no convenient way to pass from a combinatorial description of
$P\Gamma(2)=\Gamma(2)/\\{\pm 1\\}$ as a subgroup of the free group on two
generators $P\Gamma(2)$ to a description of the elements of $\Gamma$ as
$2\times 2$ matrices. But it is expected that the Eisenstein classes are of a
transcendental nature in general, and we hope that our method cast a new light
on this nature.

We computed the functions $F_{D}^{+}$ and $F_{D}^{-}$ when $\Gamma$ is
congruence subgroup [2, 3, 4]. Here is a classical instance of a non-
congruence subgroup where the computation of these formulas is possible.

Let $N$ be an integer $>0$. Let $\Phi_{N}$ be the kernel of the composed maps
$\Gamma(2)\rightarrow\Gamma(2)^{\rm ab}\rightarrow\Gamma(2)^{\rm ab}/N$, where
the first map is the abelianization. It is a non-congruence subgroup, when
$N>8$. The corresponding modular curve $X_{\Phi_{N}}$ is the $N$-th Fermat
curve (i.e. it admits the model $X^{N}+Y^{N}=1$, the cusps correspond to the
points such that $X^{N}=0$, $1$ or $\infty$). Rohrlich has shown that every
cuspidal divisor of degree $0$ is torsion in $J_{\Phi_{N}}$ [22]. As an
application of our theory, we describe the cuspidal subgroups of
$J_{\Phi_{N}}^{+}$ and $J_{\Phi_{N}}^{-}$; as expected they are finite.

###  1.1. Acknowledgements

It is a pleasure to acknowledge several e-mail communication, advice and
remark of Professors Rohrlich and Professor Kumar Murty. The first author was
partially supported by the SERB grant MTR$/2017/000357$ and CRG$/2020/000223$.

##  2\. The combinatorics of Belyi maps

Let $X$ be a smooth projective curve over ${{\bf C}}$. A Belyi map is a
morphism $f\mathrel{\mathop{\ordinarycolon}}X\rightarrow{\mathbb{P}}^{1}({\bf
C})$ unramified outside $0$, $1$, $\infty$ [25].

A dessin d’enfant [12] is a connected graph, composed of a finite set of
vertices $V$ and a finite set of edges $E$, with the additional structures :

  * •

a map from $V$ to the pair $\\{0,\infty\\}$ (the graph is bicolored)

  * •

the extremities of a given edge have different colors

  * •

the set of of edges adjacent to a given vertice is endowed with a cyclic
ordering (i.e. a transitive action of ${\bf Z}$).

The dessin d’enfant associated to $f$ is the bicolored graph $G$ whose
vertices of color $0$ (resp. $\infty$) constitute $f^{-1}(0)$ (resp.
$f^{-1}(\infty)$). The edges of the graph are the components the inverse
images by $f$ of the arc form $\infty$ to $0$ in ${\mathbb{P}}^{1}({\bf C})$
(we depart here from the usual convention, where $0$ and $1$ are used instead
of $0$ and $\infty$).

Consider the function on the upper half-plane [6, p. 86 and p. 105]:

$\lambda(z)=16e^{2\pi iz}\prod_{n\geq 1}\left(\frac{1+e^{2n\pi
iz}}{1+e^{(2n-1)\pi iz}}\right)^{8}.$

It identifies the modular curve $X(2)$ to the projective line. We resist the
tentation to use $1/\lambda$; instead we follow here the customary convention
so that $\lambda(\infty)=0$, $\lambda(0)=\infty$, $\lambda(1)=\lambda(-1)=1$.
Thus the arc from $\infty$ to $0$ on the projective line is the image by
$\lambda$ of the arc from $0$ to $\infty$ in the upper half-plane.

Since $\Gamma(2)/\\{-1,1\\}$ identifes to the fundamental group of the affine
modular curve $Y(2)$, and the upper half-plane is the universal covering of
$Y(2)$, the morphism $f$ can be regarded as the morphism
$X_{\Gamma}\rightarrow X(2)$, for an appropriate subgroup
$\Gamma\subset\Gamma(2)$ of finite index, and containing $-{\rm Id}$. The
group $\Gamma(2)/\\{-1,1\\}$ is freely generated by $\\{A,B\\}$ where $A$ is
the image of $\left(\begin{smallmatrix}1&2\\\ 0&1\\\ \end{smallmatrix}\right)$
and $B$ is the image of $\left(\begin{smallmatrix}1&0\\\ 2&1\\\
\end{smallmatrix}\right)$.

The dessin d’enfant attached to $X_{\Gamma}$ is the graph whose vertices are
the cusps of $X_{\Gamma}$ above $0$ (of colour $\infty$) and $\infty$ (of
colour $0$). The edges are the translates by $\Gamma(2)$ of the image in
$X_{\Gamma}$ of the arc from $0$ to $\infty$ (in the upper half-plane). The
action of ${\bf Z}$ on the edges attached to a vertice of color $0$ (resp.
$\infty$) follows from the action of $A$ (resp. $B^{-1}$).

Conversely, $\Gamma$ can be recovered as follows from the dessin. It acts on
the edges of the dessin: consider an edge $e$ attached to a pair of vertices
$v_{e}(0)$ and $v_{e}(\infty)$ (coloured $0$ and $\infty$ respectively), the
image of $e$ by $A$ (resp. $B$) is the successor (resp. predecessor) of $e$
along the vertice $v_{e}(0)$ (resp. $v_{e}(\infty)$). The stabilizers of all
edges constitute $\Gamma$. Thus a dessin d’enfant encodes the situation
considered in this article.

Question: Is there a natural interpretation of the Kloosterman zeta function
purely in terms of the geometry of the dessin, without involving the entries
of matrices in $\Gamma(2)$?

##  3\. Preliminaries and previous work

Let $X$ be a compact, connected, non-empty Riemann surface. Let $S$ be a
finite subset of $X$. Let $D$ be a divisor of degree $0$ supported on $S$.
Recall that a differential of the third kind on $X$ is a meromorphic
differential whose poles are simple and whose residues are integers.

Let $\omega$ be a differential of the third kind on $X$, with divisor of poles
equal to $D$. It exists by the Riemann Roch theorem. There exists a unique
holomorphic differential form $\omega^{\prime}\in\Omega^{1}(X)$ such that, for
every $c\in{\mathrm{H}}_{1}(X,S;{\bf Z})$, one has ${\rm
Re}(\int_{c}\omega)={\rm Re}(\int_{c}\omega^{\prime})$. Therefore there exists
a unique differential of the third kind $\omega_{D}$ of residue divisor $D$
such that ${\rm Re}(\int_{c}\omega_{D})=0$ for every
$c\in{\mathrm{H}}_{1}(X,S;{\bf Z})$. This is the canonical differential of the
third kind associated to $D$. The notion extends obviously to the situation
where $D$ is a divisor of degree $0$ with real coefficients.

Let $f_{D}$ be a multivalued function on $X-S$ such that $df_{D}=\omega_{D}$.
Since the periods of $\omega_{D}$ are imaginary, the real part $g_{D}$ of
$f_{D}$ is single-valued and harmonic.

We turn now to the specific situation of modular curves. Let $\Gamma$ be a
subgroup of finite index of the modular group. The cases of noncongruence
subgroups has been studied since Atkin and Swinnerton-Dyer [1]. Suppose $X$ is
the modular curve $X_{\Gamma}$ associated to $\Gamma$, and $S$ is the set
$\partial_{\Gamma}$ of cusps. The pullbacks to the upper half-plane of
differential of the third kind care of the form $2\pi if(z)\,dz$, where $f$ is
a holomorphic modular form of weight $2$ for $\Gamma$. The pullback of the
canonical differential of the third kind associated to $D$ is of the form
$2\pi iG_{D}(z)\,dz$, where $G_{D}$ is by definition the Eisenstein series
associated to $D$.

The Eisenstein series $G_{D}$ has been determined explicitly see section 6. We
find convenient to follow Scholl’s account, which in turn follows Hecke,
Selberg, Kubota etc.

Let $j\in\partial_{\Gamma}$ and choose $\sigma_{j}\in{\mathrm{SL}}_{2}({\bf
Z})$ such that $\sigma_{j}\infty\in j$ and denote by $\Gamma_{j}$ the
stabilizer of $\sigma_{j}\infty$ in $\Gamma$. For $s\in{\bf C}$, with ${\rm
Re}(s)>1$ and $z=x+iy\in\mathbb{H}$, set

$E_{j}(z,s)=\sum_{\sigma\in\Gamma_{j}\backslash\Gamma}{\rm
Im}(\sigma_{j}^{-1}\sigma(z))^{s}=\sum_{\sigma\in\Gamma_{j}\backslash\Gamma}\frac{y^{s}}{|cz+d|^{s}},$

where $(c,d)$ is the lower row of $\sigma_{j}^{-1}\sigma$. As a function of
$s$, $E_{j}(z,s)$ admits a meromorphic continuation to ${\bf C}$, with a
simple pole at $s=1$.

Its study involves the corresponding Kloosterman zeta function at the pair of
cusps $(j,k)$, which has been introduced by Selberg, even though the
terminology seems due in print to Goldfeld and Sarnak [10], see also the
monographies of Kubota [14, Chapter 2, p. 16] and Iwaniec [13, Chapter 9, p.
121]. For $r\in{\bf Z}$, it is defined as

$\phi_{jk,r}(s)\mathrel{\mathop{\ordinarycolon}}=\sum_{\sigma\in\Gamma_{j}\backslash\Gamma/\Gamma_{k}}\frac{e^{\frac{2\pi
ird}{c}}}{|c|^{2s}},$

where the entries $c$ and $d$ are such that
$\sigma_{j}^{-1}\sigma\sigma_{k}=\left(\begin{smallmatrix}\star&\star\\\
c&d\\\ \end{smallmatrix}\right)$ and $c$ is required to be $>0$. The series is
absolutely convergent if ${\rm Re}(s)>1$, and can be extended to a meromorphic
function on the complex plane, which is due to Selberg [24], see for instance
[13, Chapter 9, p. 122]. It has a pole at $s=1$ when $r=0$, and extends
holomorphically to the neighbourhood of $s=1$, when $r\neq 0$. The notation
$\phi_{jk,r}$ comes from Kubota’s book [14], and has been retained by
Goldstein and Scholl [11, 23], and has been modified by Iwaniec in [13]. The
expansion of $E_{j}$ at $\sigma_{k}(\infty)$ is given by the formula :

(3.1) $E_{j}(\sigma_{k}(z),s)=\sum_{r\in{\bf Z}}a_{jk,r}(y,s)e^{2\pi irx}.$

where, for $r\geq 1$,

(3.2) $a_{jk,r}(y,s)=\phi_{jk,r}(s)F_{r}(y,s),$

and

(3.3)
$a_{jk,0}(y,s)=\delta_{j,k}y^{s}+\phi_{jk,0}(s)y^{1-s}\pi^{1/2}\frac{\Gamma(s-1/2)}{\Gamma(s)}$

here $\delta_{j,k}$ is the Kronecker symbol, and $F_{r}(y,s)$ is built out of
Bessel functions, and does not concern us, except for the value at $s=1$. Note
that Scholl’s formula for $a_{jk,0}(y,s)$ seems to contain a misprint: it
involves $\pi^{s}$ instead of $\pi^{1/2}$, contra Kubota and Iwaniec. The
coefficients $\phi_{jk,0}(s)\pi^{1/2}{\Gamma(s-1/2)}/{\Gamma(s)}$ are the
entries of the scattering matrix at the pair $(j,k)$. Such expressions have
all a simple pole at $s=1$, whose residue does not depend on $j$ and $k$. More
precisely, the following limit exists

(3.4) $C_{j,k}=\lim_{s\to
1}(\phi_{jk,0}(s)\pi^{1/2}\frac{\Gamma(s-1/2)}{\Gamma(s)}-\frac{2}{\pi|\Gamma(2)/\Gamma|(s-1)})=\lim_{s\to
1}(\pi\phi_{jk,0}(s)-\frac{2}{\pi|\Gamma(2)/\Gamma|(s-1)}).$

It is the scattering constant, see [15], at the pair $(i,j)$. As a remark on
the terminology, we note that the term seems to be of more recent use than the
calculations that justifies its existence. It does not appear in [13], for
instance. The constants could be, and have been [20], normalized differently,
for instance par using $\zeta(s)$ instead of $1/(s-1)$. Since we will only be
interested in differences between those constants, this doesn’t matter to us.

Following Scholl [23, p. 15], define

$G_{j}(z,s)=2i\frac{\partial}{\partial z}E_{j}(z,s).$

Furthermore, write $G_{j}(z)=\lim_{s\rightarrow 1}G_{j}(z,s)$. The Kloosterman
zeta function is involved in the $q$-expansion of $G_{D}$ by the formula [23]
(based on [14])

$G_{j}(\sigma_{k}(z))J(\sigma_{j},z)^{-2}=\delta_{j,k}-\frac{\pi
C}{y}-4\pi^{2}\sum_{r=1}^{\infty}r\phi_{jk,r}(1)q^{r}$

where $z=x+iy$, $q=e^{2\pi iz}$, $J(\gamma,z)=cz+d$ when
$\gamma=\left(\begin{smallmatrix}\star&\star\\\ c&d\\\
\end{smallmatrix}\right)$, $C$ is a constant independent of $j$, and
$\delta_{j,k}$ is the Kronecker symbol. Note that if we write
$G_{j|\sigma_{j}}(z)=G_{j}(\sigma_{j}(z))J(\sigma_{j},z)^{-2}$, one has
$G_{j|\sigma_{j}}=G_{\sigma_{j}^{-1}\Gamma
j}=G_{\sigma_{j}^{-1}\Gamma\sigma_{j}\infty}$.

Write $D=\sum_{j\in\partial_{\Gamma}}m_{j}[j]$. The limit $\lim_{s\to
1}E_{j}(z,s)$ makes no sense, because of the pole at $s=1$. But
$E_{D}(z)=\lim_{s\to 1}\sum_{j\in\partial_{\Gamma}}m_{j}E_{j}(z,s)$ exists, as
the residue at $s=1$ disappears. The modular form $G_{D}$ is given by

$G_{D}(z)=\sum_{j=1}^{m}m_{j}G_{j}(z).$

It is holomorphic since $D$ is of degree $0$. Write

$G_{D}(z)=\sum_{r=0}^{\infty}a_{r}e^{2\pi irz}.$

Scholl, using Waldschmidt’s work in transcendental number theory [27],
established the following criterion.

######  Theorem 3 (Scholl, [23]).

The divisor $D$ is torsion in $J_{\Gamma}$ if and only if, for every integer
$r\geq 1$, the coefficient

(3.5)
$a_{r}=-4\pi^{2}r\sum_{j\in\partial_{\Gamma}}m_{j}\phi_{j\Gamma\infty,r}(1)$

is an algebraic number.

Note that the differential form $G_{D}(z)\,dz$ is determined by finitely many
(in terms of the genus of $X_{\Gamma}$) coefficients $a_{r}$. Therefore the
criterion can be reduced to verifying the algebricity of finitely many
numbers. The coefficients $a_{r}$ have been computed slightly more explicitly
by K. Murty and Ramakrishnan, in terms of what they call generalized Ramanujan
sums [19].

In [15], Kühn expressed the Néron-Tate pairing of two divisors of degree zero
supported on $\partial_{\Gamma}$ as a rational linear combination of (1)
logarithms of integers and (2) products of $\pi$ by scattering constants, at
least when $X_{\Gamma}$ is defined over ${{\bf Q}}$. He derived as a
consequence a formula for the scattering constants when all such divisors are
torsion in $J_{\Gamma}$.

##  4\. Mixed homology groups

We retain the notations of the introduction. We follow [17] and study the
mixed homology groups
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ and
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})$. The intersection pairing provides a perfect bilinear pairing

$\bullet\mathrel{\mathop{\ordinarycolon}}\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})\times\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})\rightarrow{\bf Z}.$

For $g\in\Gamma(2)$, set $\xi^{+}(\Gamma g)=\\{g0,g\infty\\}^{+}$ and
$\xi^{-}(\Gamma g)=\\{g1,g(-1)\\}^{-}$. By linearity, we extend the maps
$\xi^{+}$ and $\xi^{-}$ to ${\bf
Z}[\Gamma\backslash\Gamma(2)]\rightarrow\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ and ${\bf
Z}[\Gamma\backslash\Gamma(2)]\rightarrow\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})$ respectively.

######  Theorem 4 ( [17]).

The map $\xi^{+}$ and $\xi^{-}$ thus obtained are group isomorphisms.
Furthermore, for $g$, $h\in\Gamma(2)$ the intersection pairing
$\xi^{+}(g)\bullet\xi^{-}(h)$ is equal to $1$ if $\Gamma g=\Gamma h$ and to
$0$ otherwise.

Consider the maps:
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma}^{+};{\bf Z})$ and
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma}^{-};{\bf Z})$. The
kernel of those maps are the image of the maps $\lambda^{+}$: ${\bf
Z}[\partial_{\Gamma}^{-}]\simeq\mathrm{H}_{0}(\partial_{\Gamma}^{-},{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})$ and $\lambda^{-}$: ${\bf
Z}[\partial_{\Gamma}^{+}]\simeq\mathrm{H}_{0}(\partial_{\Gamma}^{+};{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ respectively. To be precise, $\lambda^{-}(x)$ (resp. $\lambda^{+}(x)$) is
homologous to a counterclockwise loop around the cusp $x$. The width of the
cusp is the ramification index of $\pi_{0}$ at that cusp. Recall that $A$
(resp. $B^{-1}$, resp. $BA^{-1}$) is the generator of the stabiliser of
$\infty$ (resp. $0$, resp. $1$) in $P\Gamma(2)$ such that, for $z$ in the
upper half-plane, the image in $Y(2)$ of a path from $z$ to $Az$ (resp.
$B^{-1}z$, resp. $BA^{-1}z$) is a counterclockwise loop around
$\Gamma(2)\infty$ (resp. $\Gamma(2)0$, resp. $\Gamma(2)1$).

######  Proposition 5.

Let $g\in\Gamma(2)$ and denote by $w_{\infty}$ the width of the cusp $j=\Gamma
g\infty$. One has

$\lambda^{-}(\Gamma
g\infty)=-\sum_{k=0}^{w_{\infty}-1}\xi^{-}(gA^{k})=\sum_{h\in\Gamma\backslash\Gamma(2),h\infty=j}\xi^{-}(h).$

Denote by $w_{0}$ the width of the cusp $\Gamma g0$. One has

$\lambda^{-}(\Gamma
g0)=\sum_{k=0}^{w_{0}-1}\xi^{-}(gB^{k})=\sum_{h\in\Gamma\backslash\Gamma(2),h0=j}\xi^{-}(h).$

Denote by $w_{1}$ the width of the cusp $\Gamma g1$. One has

$\lambda^{+}(\Gamma
g1)=\sum_{k=0}^{w_{1}-1}\xi^{+}(g(AB^{-1})^{k}B)-\xi^{+}(g(AB^{-1})^{k})=\sum_{h\in\Gamma\backslash\Gamma(2),h(-1)=j}\xi^{-}(h)-\sum_{h\in\Gamma\backslash\Gamma(2),h1=j}\xi^{-}(h).$

###### Proof.

Note that $A\infty=\infty$, $B0=0$, $A(-1)=B(-1)=1$. The class of a loop
around the cusp $\Gamma g\infty$ is given by

$\displaystyle\\{g(-1),(gAg^{-1})^{w_{\infty}}g(-1)\\}^{-}$ $\displaystyle=$
$\displaystyle\\{g(-1),gA^{w_{\infty}}(-1)\\}^{-}$ $\displaystyle=$
$\displaystyle\\{g(-1),gA^{w_{\infty}-1}1\\}^{-}$ $\displaystyle=$
$\displaystyle\\{g(-1),gA^{w_{\infty}-1}(-1)\\}^{-}+\\{gA^{w_{\infty}-1}(-1),gA^{w_{\infty}-1}1\\}^{-}$
$\displaystyle=$
$\displaystyle\\{g(-1),gA^{w_{\infty}-1}(-1)\\}^{-}+\\{gA^{w_{\infty}-1}(-1),gA^{w_{\infty}-1}1\\}^{-}$
$\displaystyle=$
$\displaystyle\\{g(-1),gA^{w_{\infty}-1}(-1)\\}^{-}-\xi^{-}(gA^{w_{\infty}-1}).$

Which gives the first formula by iteration. The second formula is proved by
the same method (replace $A$ by $B^{-1}$).

The third formula is obtained similarly. The class of a loop around the cusp
$\Gamma g1$ is given by

$\displaystyle\\{g\infty,(gBA^{-1}g^{-1})^{w_{1}}g\infty\\}^{+}$
$\displaystyle=$ $\displaystyle\\{g\infty,g(BA^{-1})^{w_{1}-1}B\infty\\}^{+}$
$\displaystyle=$
$\displaystyle\\{g\infty,g(BA^{-1})^{w_{1}-1}B0\\}^{+}+\\{g(BA^{-1})^{w_{1}-1}B0,g(BA^{-1})^{w_{1}-1}B\infty\\}^{+}$
$\displaystyle=$
$\displaystyle\\{g\infty,g(BA^{-1})^{w_{1}-1}0\\}^{+}+\xi^{+}(g(BA^{-1})^{w_{1}-1}B)$
$\displaystyle=$
$\displaystyle\\{g\infty,g(BA^{-1})^{w_{1}-1}\infty\\}^{+}+\\{g(BA^{-1})^{w_{1}-1}\infty,g(BA^{-1})^{w_{1}-1}0\\}^{+}$
$\displaystyle+\xi^{+}(g(AB^{-1})^{w_{1}-1}B)$ $\displaystyle=$
$\displaystyle\\{g\infty,g(BA^{-1})^{w_{1}-1}\infty\\}^{+}-\xi^{+}(g(BA^{-1})^{w_{1}-1})+\xi^{+}(g(AB^{-1})^{w_{1}-1}B),$

which leads to the third formula, by iterating again.

∎

The boundary maps
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})\rightarrow{\bf Z}[\partial_{\Gamma}^{+}]$ (resp.
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})\rightarrow{\bf Z}[\partial_{\Gamma}^{-}]$) associates to $\xi^{-}(g)$
(resp. $\xi^{+}(g)$) the divisor $(\Gamma g1)-(\Gamma g(-1))$ (resp. $(\Gamma
g\infty)-(\Gamma g0)$). They are dual to the maps $\lambda^{-}$ and
$\lambda^{+}$ respectively.

There is a notion of Eisenstein class in
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
R})$ and
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
R})$. Let $D^{+}$ (resp. $D^{-}$) be a divisor of degree $0$ supported on
$\partial_{\Gamma}^{+}$ (resp. $\partial_{\Gamma}^{-}$). The corresponding
Eisenstein class belongs to
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
R})$ (resp.
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
R})$).

It is the unique element $c$ of boundary $D^{+}$ (resp. $D^{-}$) such that
$\Re(\int_{c}\omega)=0$ for all $\omega$ differential form of the third kind
whose poles are supported on $\partial_{\Gamma}^{-}$ (resp.
$\partial_{\Gamma}^{+}$) and whose residues are real.

##  5\. Inner product formula

Let $M_{2}(\Gamma)$ be the space of holomorphic modular forms of weight $2$
for $\Gamma$. The following formula is akin to the formula [18, p. 21, Theorem
2].

######  Theorem 6.

Let $f_{+}$ and $f_{-}$ be elements $M_{2}(\Gamma)$ such that the poles of
$f_{+}(z)dz$ and $f_{-}(z)dz$ belong to $\partial_{\Gamma}^{+}$ and
$\partial_{\Gamma}^{-}$ respectively. Thus the Petersson inner product
$<f_{+},f_{-}>$ is well defined. We have the formula:

$<f_{+},f_{-}>=\frac{1}{12i[\Gamma(2)\mathrel{\mathop{\ordinarycolon}}\Gamma]}\sum_{g\in\Gamma\backslash\Gamma(2)}\int_{g(1)}^{g(-1)}f_{+}(z)\,dz\int_{g0}^{g\infty}\overline{f_{-}(z)dz}.$

###### Proof.

We prove first a more abstract statement which seems a basic statement in the
theory of Riemann surfaces. However we did not know any reference for this.

######  Proposition 7.

Let $X$ be a connected, compact, non-empty Riemann surface endowed with a
quadrangulation. In such a situation, the set of vertices of the
quadrangulation is partitioned in two subsets $E^{+}$ and $E^{-}$ such that no
two vertices of $E^{+}$ (resp. $E^{-}$) are connected by an edge. Let $Q$ be
the set of faces. For every $q\in Q$, we fix an oriented path $\delta_{q}^{+}$
(resp. $\delta_{q}^{-}$) whose extremities are the vertices of $q$ in $E^{+}$
(resp. in $E^{-}$). We impose furthermore that the intersection product is
given by $\delta_{q}^{+}\bullet\delta_{q}^{-}=1$ (here, our convention is that
$\delta_{q}^{+}$ and $\delta_{q}^{-}$ cross counterclockwise). Let
$\omega_{+}$ and $\omega_{-}$ be meromorphic differential forms on $X$ whose
poles are at most simple and reside in $E^{+}$ and $E^{-}$ respectively. Then
one has

$\int_{X}\omega_{+}\wedge\bar{\omega}_{-}=\sum_{q\in
Q}\int_{\delta_{q}^{-}}\omega_{+}\int_{\delta_{q}^{+}}\bar{\omega}_{-},$

where $q$ runs through the faces of the quadrangulation.

###### Proof.

Note that the formula makes sense. Set $V=E^{-}$. Let $Q$ be the set of faces
of the quadrangulation. We can suppose that for every $q\in Q$, the paths
$\delta_{q}^{+}$ and $\delta_{q}^{-}$ are supported on a set that divides $q$
in two connected components. Denote by $V_{q}$ the subset of vertices in $V$
that are adjacent to $q$. For $v\in V$, denote by $\tau(q,v)$ the connected
component containing $v$ of $q$ deprived of the support of $\delta_{q}^{+}$.
Denote by $Q_{v}$ the set of faces of the quadrangulation which are adjacent
to $v$. One has a disjoint union (up to a negligible subset)

$X=\cup_{v\in V}\cup_{q\in Q_{v}}\tau(q,v).$

We turn now to the computation. One has

$\int_{X}\omega_{+}\wedge\bar{\omega}_{-}=\sum_{v\in V}\sum_{q\in
Q_{v}}\int_{\tau(v,q)}\omega_{+}\wedge\bar{\omega}_{-}=\sum_{v\in
V}\int_{\cup_{q\in Q_{v}}\tau(v,q)}\omega_{+}\wedge\bar{\omega}_{-}.$

The boundary of $\cup_{q\in Q_{v}}\tau(v,q)$ is $\sum_{q\in
Q_{v}}\alpha(q,v)\delta_{q}^{+}$, where $\alpha(q,v)=1$ (resp. $-1$) if
$\delta_{q}^{+}$ goes left (resp. right) from $v$’s point of view.

For $v\in V$, and $z\in\cup_{q\in Q_{v}}\tau(v,q)$ denote by
$F_{v}(z)=\int_{v}^{z}\omega_{+}$, which is well defined since $\cup_{q\in
Q_{v}}\tau(v,q)$ is simply connected. One has
$dF_{v}\omega_{-}=\omega_{+}\wedge\omega_{-}$. By Stokes’ theorem, one has

$\int_{X}\omega_{+}\wedge\bar{\omega}_{-}=\sum_{v\in V}\sum_{q\in
Q_{v}}\alpha(q,v)\int_{\delta_{q}^{+}}F_{v}(z)\omega_{-}.$

Note that $\sum_{q\in Q_{v}}\alpha(q,v)F_{v}=\int_{\delta_{q}^{-}}\omega_{+}$,
because of the condition $\delta_{q}^{+}\bullet\delta_{q}^{-}=1$. Thus we get
the desired formula. ∎

The theorem follows. Indeed, the translates by $\Gamma$ of the fundamental
domain $D_{0}$ of $X(2)$ given by the hyperbolic quadrangle with vertices $0$,
$1$, $-1$ and $\infty$ provide a quadrangulation of $X_{\Gamma}$. The faces
are given by the $\Gamma gD_{0}$, for $g\in\Gamma\backslash\Gamma(2)$. The
$\delta_{\Gamma gD_{0}}^{+}$ and $\delta_{\Gamma gD_{0}}^{-}$ can be chosen as
the image of the geodesic paths from $g0$ to $g\infty$ and $g1$ to $g(-1)$
respectively. The Petersson inner product [8, p. 182]:

$<f_{+},f_{-}>=\frac{1}{[{\mathrm{SL}}_{2}({\bf
Z})\mathrel{\mathop{\ordinarycolon}}\Gamma]}\int_{D_{\Gamma}}f_{+}(z)\overline{f_{-}}(z)dxdy=\frac{1}{12i[\Gamma(2)\mathrel{\mathop{\ordinarycolon}}\Gamma]}\int_{X_{\Gamma}}\omega_{f_{+}}\wedge\overline{\omega_{f_{-}}},$

where $D_{\Gamma}$ is a fundamental domain for $\Gamma$ in the upper half-
plane. The last term can be expressed using proposition 7. ∎

Let $D^{+}$ and $D^{-}$ be divisors of degree $0$ supported on
$\partial_{\Gamma}^{+}$ and $\partial_{\Gamma}^{-}$ respectively.

######  Proposition 8.

With the notations of the preceeding theorem, suppose furthermore that
$f_{+}(z)\,dz$ (resp. $f_{-}(z)\,dz$) is the pullback of a canonical
differential form of the third kind. One has

$<f_{+},f_{-}>=0.$

###### Proof.

Call $\omega_{+}$ the differential form whose pullback is $f_{+}(z)\,dz$. The
proposition is true whenever $\omega_{+}$ is holomorphic on $X_{\Gamma}$. Thus
we can suppose that $\omega_{+}$ is a canonical differential form of the third
kind. It can be written as $\partial h$, where $h$ is a harmonic function on
$X_{\Gamma}$. Hence $\int_{X_{\Gamma}}\partial h\wedge{\bar{\omega}}=0$, for
all meromorphic differential forms $\omega$ whose poles are concentrated in
$\partial_{\Gamma}^{-}$. ∎

##  6\. Periods of Eisenstein series

We use the setup of section 3. Let $m=2\pi|\Gamma/\Gamma(2)|$ be the
hyperbolic volume of the modular curve $X_{\Gamma}$ (it will only play a
transitory role below). Set $q=e^{2i\pi z}$. Consider the function on the
upper half-plane:

$\log(\eta_{\Gamma,j}(z))=-\frac{mz}{4i}-\pi
m[\sum_{r=1}^{\infty}\phi_{jjr}(1)q^{r}].$

It is connected to the Eisenstein series $E_{j}(z,s)$ via a version of the
Kronecker limit theorem established by Goldstein [11, Theorem 3-1; 3-3]

(6.1) $\lim_{s\rightarrow 1}[\frac{1}{2\pi}E_{j}(z,s)-\frac{1}{2\pi
m(s-1)}]=\frac{\beta_{j}}{2}-\frac{1}{\pi m}\log(2)-\frac{1}{\pi
m}\log|\sqrt{y}\eta_{\Gamma,j}(z)^{2}|,$

where $z=x+iy$ and $\beta_{j}$ is a complex number independent of $z$ (the
scattering constants $C_{j,j}$ up to a scalar multiple independent of $j$).
Goldstein has explored the properties of the function $\eta_{\Gamma,j}$, which
is a modular form of weight $1/2$ for the group
$\sigma_{j}\Gamma\sigma_{j}^{-1}$, and whose periods he sees as analogues of
Dedekind sums [11].

The multivalued function on the upper half-plane associated to the divisor $D$
is then given by

$\log(\eta_{\Gamma,D}(z))=\sum_{j}m_{j}\log(\eta_{\Gamma,j})(z).$

Note that

$E_{D}(z)\mathrel{\mathop{\ordinarycolon}}=\frac{2}{m}\log|\eta_{\Gamma,D}(z)^{2}|.$

######  Proposition 9.

We have an equality of differential forms

$2\pi iG_{D}(z)\,dz=\frac{4\pi}{m}\,d\eta_{\Gamma,D}.$

###### Proof.

We compare the $q$-expansions. One has, taking into account that $D$ is of
degree $0$,

$d\eta_{\Gamma,D}(z)=2\pi i\sum_{j}m_{j}\,d\log(\eta_{\Gamma,j})(z)=2\pi
i\sum_{j}-m_{j}\pi m[\sum_{r=1}^{\infty}r\phi_{jjr}(1)q^{r}\,dz].$

Similarly, constant terms disappear in $G_{D}$. So we just need to check that
the $q$-expansions corresponding to each cusp agree. When $j=\Gamma\infty$, it
is obvious by examining the $q$-expansions.

The other terms agree as well. Indeed, one has
$\log(\eta_{\Gamma,j})(\sigma_{j}z)=\log(\eta_{\sigma_{j}^{-1}\Gamma\sigma_{j},\Gamma\infty(z)})$.
On the other side we use the identity
$G_{j|\sigma_{j}}=G_{\sigma_{j}^{-1}\Gamma\sigma_{j}\infty}$, and we are left
with the case of the cusp $\sigma_{j}^{-1}\Gamma\sigma_{j}\infty$.

∎

Proposition 9 can be reformulated as

$\displaystyle G_{D}(z)\,dz=\frac{2}{mi}\,d\eta_{\Gamma,D}(z)=2\pi
i\sum_{j\in\partial_{\Gamma}}m_{j}\,d(\sum_{r=1}^{\infty}\phi_{jj,r}(1)q^{r}).$

Recall that

$\displaystyle\mathcal{S}_{D}(x)=\frac{1}{2\pi
i}\sum_{j\in\partial_{\Gamma}}m_{j}\sum_{r=1}^{\infty}\phi_{jj,r}(1)e^{2ri\pi
x}.$

Equipped with these formulas, we relate the periods of Eisenstein series to
the functions defined in the introduction.

######  Proposition 10.

One has, for $g\in\Gamma(2)$,

$\int_{g(1)}^{g(-1)}G_{D^{+}}(z)\,dz=F_{D^{+}}(g)\hskip
28.45274pt\text{and}\hskip
28.45274pt\int_{g(0)}^{g(\infty)}G_{D^{-}}(z)\,dz=F_{D^{-}}(g).$

###### Proof.

We apply Proposition 9 and its reformulation Proposition 6 to $D=D^{+}$. Thus
we have

$\displaystyle\int_{g(1)}^{g(-1)}G_{D^{+}}(z)\,dz=2\pi
i\int_{g(1)}^{g(-1)}\sum_{j\in\partial_{\Gamma}}m_{j}\sum_{r=1}^{\infty}\phi_{jj,r}(1)q^{r}\,dz=\mathcal{S}_{D^{+}}(g(-1))-\mathcal{S}_{D^{+}}(g(1)).$

The other statement is proved similarly. ∎

The integrals occuring in Proposition 10 play a key role in our work. It is
tempting to ask whether they have a $p$-adic counterpart, using for instance
Coleman’s integration. A priori, the notion of canonical differential of the
third kind does not make sense without making choice in the $p$-adic world.
However, Pierre Colmez suggests that it still does using the notion of
Wintenberger splitting [7].

##  7\. Scattering constants

We provide another expression for the functions ${\mathcal{I}}_{D^{+}}$ and
${\mathcal{I}}_{D^{-}}$ of the introduction, in terms of scattering constants.
Recall that, for $j$, $k\in\partial_{\Gamma}$, one has

$C_{j,k}=\lim_{s\to
1}\pi(\sum_{\sigma\in\Gamma_{j}\backslash\Gamma/\Gamma_{k}}\frac{1}{|c|^{2s}})-\frac{2}{\pi|\Gamma(2)/\Gamma|(s-1)}$

where the entries $c$ and $d$ are such that
$\sigma_{j}^{-1}\sigma\sigma_{k}=\left(\begin{smallmatrix}\star&\star\\\
c&d\\\ \end{smallmatrix}\right)$ and $c$ is required to be $>0$.

######  Proposition 11.

Let $g\in\Gamma(2)$. One has

${\mathcal{I}}_{D^{+}}=\pi\sum_{g\in\Gamma\backslash\Gamma(2)}\sum_{j\in\partial_{\Gamma}}m_{j}(C_{j,\Gamma
g(-1)}-C_{j,\Gamma g(1)})\\{g0,g\infty\\}^{+}$

and

${\mathcal{I}}_{D^{-}}=\pi\sum_{g\in\Gamma\backslash\Gamma(2)}\sum_{j\in\partial_{\Gamma}}m_{j}(C_{j,\Gamma
g(\infty)}-C_{j,\Gamma g(0)})\\{g(1),g(-1)\\}^{-}.$

###### Proof.

We prove the first equality. Let $g\in\Gamma$. We use the formula
$\int_{g(1)}^{g(-1)}G_{D^{+}}(z)\,dz=F_{D^{+}}(g)$. Thus we get

$F_{D^{+}}(g)=\frac{2}{mi}[\log(\eta_{\Gamma,D})]_{g(1)}^{g(-1)}.$

Recall that $\mathcal{I}_{D^{+}}$ is defined as the real part of $2\pi
i{\mathcal{E}}_{D^{+}}=2\pi
i\sum_{g\in\Gamma\backslash\Gamma(2)}F_{D^{+}}(g)\\{g0,g\infty\\}^{+}$. We get

$\Re(2\pi
iF_{D^{+}}(g))=\Re(\frac{4\pi}{m}[\log(\eta_{\Gamma,D})]_{g(1)}^{g(-1)})=\pi[E_{D}]_{g(1)}^{g(-1)}.$

We use the formulas 3.1 and 3.3 for the development 3.1 of $E_{j}(z,s)$ at the
cusps $g(1)$ and $g(-1)$. Note that , when $m_{j}\neq 0$, the cusp $j$ is
distinct from both cusps $\Gamma g(1)$ and $\Gamma g(-1)$, since
$\Gamma(2)j=\Gamma(2)0$ or $\Gamma(2)\infty$. The constant terms given by
equation 3.3 are $\phi_{j\Gamma
g(1),0}(1)\pi^{1/2}\Gamma(1/2)/\Gamma(1)=\pi\phi_{j\Gamma g(1),0}(1)$ and
$\pi\phi_{j\Gamma g(-1),0}(1)$ respectively. We get

$\Re(2\pi
if_{D^{+}}(g))=\sum_{j\in\partial_{\Gamma}}\pi^{2}m_{j}(\phi_{j\Gamma
g(-1),0}(1)-\phi_{j\Gamma
g(1),0}(1))=\pi\sum_{j\in\partial_{\Gamma}}m_{j}(C_{j,\Gamma
g(-1)}-C_{j,\Gamma g(1)})$

(because of the cancellation of the poles, the middle term makes sense). The
formula follows. The second formula is proved similarly.

∎

It is interesting to compare to the work of Kühn [15]. For instance, when
$D^{+}$ (resp. $D^{-}$) is torsion, and the Belyi map is defined over ${\bf
Q}$, Kühn shows that

$e^{2\pi\sum_{j\in\partial_{\Gamma}}m_{j}(C_{j,\Gamma g(-1)}-C_{j,\Gamma
g(1)})}$

(resp. $e^{2\pi\sum_{j\in\partial_{\Gamma}}m_{j}(C_{j,\Gamma
g\infty}-C_{j,\Gamma g0})}$) is a rational number for every $g\in\Gamma$.

More precisely, still in the case where the Belyi map is defined over ${\bf
Q}$, Kühn proves that the Néron-Tate pairing of two divisors
$D=\sum_{j}m_{j}[j]$ and
$D^{\prime}=\sum_{j^{\prime}}m^{\prime}_{j^{\prime}}$, both of degree $0$,
satisfies

(7.1) $[D,D^{\prime}]_{\rm NT}\in{\rm log}({\bf
Q}^{\times}_{+})+2\pi\sum_{j,j^{\prime}\in\partial_{\Gamma}}m_{j}m^{\prime}_{j^{\prime}}C_{j,j^{\prime}}.$

A similar formula holds even if the Belyi map is not defined over ${\bf Q}$;
but one needs to consider the conjugates of the Belyi map. The second term in
fomula 7.1 can be expressed as follows, when $D=D^{+}$ and $D^{\prime}=D^{-}$,

$2\pi\sum_{j,j^{\prime}\in\partial_{\Gamma}}m_{j}m^{\prime}_{j^{\prime}}C_{j,j^{\prime}}=-2\mathcal{I}_{D^{+}}\bullet\mathcal{R}_{D^{-}}.$

Indeed, we just have to note that the boundary of $\mathcal{R}_{D^{-}}$ is
$-D^{-}$ (that will be proved in section 8). Note that Néron-Tate pairing
considered by Kühn is obtained by summing contribution coming from all places
of ${\bf Q}$ and is relative to the jacobian $J_{\Gamma}$, whereas our
intersection product, of complex analytic nature, takes place relatively to
the 1-motives we are considering. One wonders whether an adjustment of the
Néron-Tate pairing for the mixed situation would not provide a clearer
connection between Kühn’s formula and ours.

##  8\. Proof of theorem 1

Let $D$ a divisor supported on $\partial_{\Gamma}^{-}$ (resp.
$\partial_{\Gamma}^{+}$). Let $c_{D}$ be a cycle on
$X_{\Gamma}-\partial_{\Gamma}^{+}$ (resp. $X_{\Gamma}-\partial_{\Gamma}^{-}$)
of boundary $D$. Then the real number
$\Im(\frac{1}{2i\pi}\int_{c_{D}}\omega_{D^{+}})$ (resp.
$\Im(\frac{1}{2i\pi}\int_{c_{D}}\omega_{D^{-}})$) depends only on $D$. Let
$j$, $j_{0}$ in $\partial_{\Gamma}^{-}$ (resp $\partial_{\Gamma}^{+}$ ). When
$D=[j]-[j_{0}]$, we abuse notations and write
$\Im(\frac{1}{2i\pi}\int_{j_{0}}^{j}\omega_{D^{-}})$ (resp.
$\Im(\frac{1}{2i\pi}\int_{j_{0}}^{j}\omega_{D^{+}})$) instead.

Since $\sum_{j\in\partial_{\Gamma}^{-}}\lambda^{+}(j)=0$ (resp.
$\sum_{j\in\partial_{\Gamma}^{+}}\lambda^{-}(j)=0$), the class
$\sum_{j\in\partial_{\Gamma}^{-}}\Im(\frac{1}{2i\pi}\int_{j_{0}}^{j}\omega_{D^{+}})\lambda^{+}(j)$
in the real vector space
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
R})$ (resp.
$\sum_{j\in\partial_{\Gamma}^{+}}\Im(\frac{1}{2i\pi}\int_{j_{0}}^{j}\omega_{D^{-}})\lambda^{+}(j)$
in $\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
R})$) is independent of the choice of $j_{0}$. In other words,
$\sum_{j\in\partial_{\Gamma}^{-}}\Re(\int_{j_{0}}^{j}\omega_{D^{+}})\lambda^{+}(j)$
(resp.
$\sum_{j\in\partial_{\Gamma}^{+}}\Re(\int_{j_{0}}^{j}\omega_{D^{-}})\lambda^{-}(j)$)
makes sense and is independent of $j_{0}$.

Recall that $\mathcal{I}_{D^{+}}$ (resp. $\mathcal{I}_{D^{-}}$) is defined as
the real part of $2\pi i{\mathcal{E}}_{D^{+}}$ (resp. $2\pi
i{\mathcal{E}}_{D^{-}}$).

######  Proposition 12.

One has

$\mathcal{I}_{D^{+}}=\sum_{j\in\partial_{\Gamma}^{-}}\Re(\int_{j_{0}}^{j}\omega_{D^{+}})\lambda^{+}(j)\hskip
28.45274pt{\rm and}\hskip
28.45274pt\mathcal{I}_{D^{-}}=-\sum_{j\in\partial_{\Gamma}^{+}}\Re(\int_{j_{0}}^{j}\omega_{D^{-}})\lambda^{-}(j).$

In particular, those classes belong to the real vector spaces spanned by the
images of $\lambda^{+}$ and $\lambda^{-}$ respectively.

###### Proof.

By proposition 10, one has, by using proposition 5

$\displaystyle\mathcal{I}_{D^{+}}$ $\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash\Gamma(2)}\Re(\int_{g(1)}^{g(-1)}\omega_{D^{+}})\xi^{+}(g)$
$\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash\Gamma(2)}\Re(\int_{j_{0}}^{g(-1)}\omega_{D^{+}}-\int_{j_{0}}^{g(1)})\xi^{+}(g)$
$\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash\Gamma(2)}\Re(\int_{j_{0}}^{g(1)}\omega_{D^{+}})(\xi^{+}(gB)-\xi^{+}(g))$
$\displaystyle=$
$\displaystyle\sum_{j\in\partial_{\Gamma}^{-}}\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g(1)=j}\Re(\int_{j_{0}}^{g(1)}\omega_{D^{+}})(\xi^{+}(gB)-\xi^{+}(g))$
$\displaystyle=$
$\displaystyle\sum_{j\in\partial_{\Gamma}^{-}}\Re(\int_{j_{0}}^{j}\omega_{D^{+}})\lambda^{+}(j).$

Similarly, one has

$\displaystyle\mathcal{I}_{D^{-}}$ $\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash\Gamma(2)}\Re(\int_{g0}^{g\infty}\omega_{D^{-}})\xi^{-}(g)$
$\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash\Gamma(2)}\Re(\int_{j_{0}}^{g\infty}\omega_{D^{-}}-\int_{j_{0}}^{g0}\omega_{D^{-}})\xi^{-}(g)$
$\displaystyle=$
$\displaystyle\sum_{j\in\partial_{\Gamma}^{+}}\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g\infty=j)\Re(\int_{j_{0}}^{g\infty}\omega_{D^{-}})\xi^{-}(g)-\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g0=j}\Re}\int_{j_{0}}^{g0}\omega_{D^{-}})\xi^{-}(g)$ $\displaystyle=$
$\displaystyle-\sum_{j\in\partial_{\Gamma}^{-},\Gamma(2)j=\Gamma(2)\infty}\Re(\int_{j_{0}}^{j}\omega_{D^{+}})\lambda^{-}(j)-\sum_{j\in\partial_{\Gamma}^{-},\Gamma(2)j=\Gamma(2)0}\Re(\int_{j_{0}}^{j}\omega_{D^{+}})\lambda^{-}(j)$
$\displaystyle=$
$\displaystyle-\sum_{j\in\partial_{\Gamma}^{+}}\Re(\int_{j_{0}}^{j}\omega_{D^{-}})\lambda^{-}(j).$

∎

######  Corollary 13.

One has $\mathcal{I}_{D^{+}}\bullet\mathcal{I}_{D^{-}}=0$.

###### Proof.

Indeed, one has $\lambda^{+}(j)\bullet\lambda^{-}(j^{\prime})$, for
$j\in\partial_{\Gamma}^{-}=0$, $j^{\prime}\in\partial_{\Gamma}^{+}$ (small
loops do not intersect). ∎

The following consequence is certainly well-known. However it follows nicely
from our proposition.

######  Corollary 14.

One has $\mathcal{I}_{D^{+}}=0$ (resp. $\mathcal{I}_{D^{-}}=0$) if and only
if, for all $j$, $j^{\prime}\in\partial_{\Gamma}^{-}$ (resp. $j$,
$j^{\prime}\in\partial_{\Gamma}^{-}$) one has
$\Re(\int_{j^{\prime}}^{j}\omega_{D^{+}})=0$ (resp.
$\Re(\int_{j^{\prime}}^{j}\omega_{D^{-}})=0$).

###### Proof.

Indeed, the kernel of the map $\lambda^{+}$ : ${\bf
Z}[\partial_{\Gamma}^{-}]\rightarrow\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ is spanned by $\sum_{j\in\partial_{\Gamma}^{-}}[j]$. It follows that all
the coefficients in the expression of $\mathcal{I}_{D^{+}}$ need to be equal
for $\mathcal{I}_{D^{+}}$ to vanish. The other statement is proved similarly.
∎

###### Proof.

We prove theorem 1.

By proposition 8 and theorem 5, one has
$\int_{{\mathcal{E}}_{D^{+}}}\omega=0$. We write
$\omega=\omega_{D^{-}}+\omega_{0}$, where $\omega_{0}$ is holomorphic on
$X_{\Gamma}$ and $\omega_{D^{-}}$ is the canonical differential form of the
third kind associated to $D^{-}$, where $D^{-}$ is the divisor of $\omega$.
Since $\omega_{0}$ has no pole on $X_{\Gamma}$, proposition 12 implies that
$\int_{\mathcal{I}_{D^{+}}}\omega_{0}=0$, and therefore
$\int_{\mathcal{R}_{D^{+}}}\omega_{0}=0$. Thus one gets
$\int_{{\mathcal{E}}_{D^{+}}}\omega=\int_{{\mathcal{E}}_{D^{+}}}\omega_{D^{-}}=0$.

One has in ${\bf C}$, with the notations of the introduction, and using the
formula for the intersection products and proposition 8,

${\mathcal{E}}_{D^{+}}\bullet{\mathcal{E}}_{D^{-}}=0.$

Since $\mathcal{I}_{D^{+}}\bullet\mathcal{I}_{D^{-}}=0$, if follows that we
have the following identities in ${\bf R}$

$\mathcal{R}_{D^{+}}\bullet\mathcal{R}_{D^{-}}=0,$

and

$\mathcal{R}_{D^{+}}\bullet\mathcal{I}_{D^{-}}+\mathcal{I}_{D^{+}}\bullet\mathcal{R}_{D^{-}}=0.$

Thus we get

$\Re(\int_{\mathcal{R}_{D^{+}}}\omega)=\Re(\int_{\mathcal{R}_{D^{+}}}\omega_{D^{-}})=\Re(\mathcal{R}_{D^{+}}\bullet\mathcal{R}_{D^{-}}+\frac{1}{2\pi
i}\mathcal{R}_{D^{+}}\bullet\mathcal{I}_{D^{-}})=\mathcal{R}_{D^{+}}\bullet\mathcal{R}_{D^{-}}=0.$

It remains to prove that the boundary of ${\mathcal{E}}_{D^{+}}$ (resp.
${\mathcal{E}}_{D^{-}}$) is ${-D^{+}}$ (resp. ${-D^{-}}$). Indeed,

$\displaystyle\delta({\mathcal{E}}_{D^{+}})$ $\displaystyle=$
$\displaystyle\sum_{g\in\Gamma\backslash\Gamma(2)}\int_{g(1)}^{g(-1)}G_{D^{+}}(z)\,dz((\Gamma
g\infty)-(\Gamma g0))$ $\displaystyle=$
$\displaystyle\sum_{j\in\partial_{\Gamma}^{+}}\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g\infty=j}\int_{g(1)}^{g(-1)}G_{D^{+}}(z)\,dz(j)-\sum_{j\in\partial_{\Gamma}^{+}}\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g0=j}\int_{g(1)}^{g(-1)}G_{D^{+}}(z)\,dz(j)$ $\displaystyle=$
$\displaystyle\sum_{j\in\partial_{\Gamma}^{+}}\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g\infty=j}\frac{1}{2\pi
i}\int_{\xi^{-}(g)}\omega_{D^{+}}(j)-\sum_{j\in\partial_{\Gamma}^{+}}\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g0=j}\frac{1}{2\pi i}\int_{\xi^{-}(g)}\omega_{D^{+}}(j).$

Since, by proposition 5, for every $j\in\partial_{\Gamma}^{+}$, one has
$\frac{1}{2\pi
i}\int_{\lambda^{+}(j)}\omega_{D^{+}}=-\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g\infty=j}\frac{1}{2\pi i}\int_{\xi^{-}(g)}\omega_{D^{+}}$ and, similarly,
$\frac{1}{2\pi
i}\int_{\lambda^{+}(j)}\omega_{D^{+}}=\sum_{j\in\partial_{\Gamma}^{+}}\sum_{g\in\Gamma\backslash\Gamma(2),\Gamma
g0=j}\frac{1}{2\pi i}\int_{\xi^{-}(g)}\omega_{D^{+}}$, one gets

$\delta({\mathcal{E}}_{D^{+}})=-\sum_{j\in\partial_{\Gamma}^{+}}{\rm
Res}_{j}(\omega_{D^{+}})=-D^{+}.$

The boundary of ${\mathcal{E}}_{D^{-}}$ is computed similarly. ∎

##  9\. Manin-Drinfeld properties and generalized Jacobians

Let $D$ be an effective divisor on $X_{\Gamma}$ supported on $S$. Recall that
the generalized Jacobian of $X_{\Gamma}$ with respect to $D$ admits a complex
analytic description as follows. Denote by $\Omega(-D)$ the space of
meromorphic differential forms on $X_{\Gamma}$ whose divisor is $\geq-D$.
Denote by $E={\rm Hom}_{\bf C}(\Omega(-D),{\bf C})$. Denote by $L$ the image
of the map $\mathrm{H}_{1}(X_{\Gamma}-S;{\bf Z})\rightarrow E$ which
associates to $c$ the map $\omega\mapsto\int_{c}\omega$. The complex Lie group
formed by the complex points of the generalized Jacobian of $X_{\Gamma}$ with
respect to $D$ is isomorphic to $E/L$.

Suppose $D=\sum_{s\in S}[s]$, then the group of complex point of the
generalized Jacobian can be identified to a subgroup of
$\mathrm{H}_{1}(X_{\Gamma}-S;{\bf C})/\mathrm{H}_{1}(X_{\Gamma}-S;{\bf Z})$.
This subgroup is $(\mathrm{H}_{1}(X_{\Gamma}-S;{\bf
R})+i\lambda(\mathrm{H}_{0}(S;{\bf R})))/\mathrm{H}_{1}(X_{\Gamma}-S;{\bf
Z})$, where $\lambda$ is the canonical map $\mathrm{H}_{0}(S;{\bf
R})\rightarrow\mathrm{H}_{1}(X_{\Gamma}-S;{\bf R})$.

When $D^{\prime}$ is a divisor of degree $0$ of $X_{\Gamma}$ supported outside
$D$, its image in the generalized Jacobian is the class of the map
$\omega\mapsto\int_{c_{D}^{\prime}}\omega$, where
$c_{D^{\prime}}\in\mathrm{H}_{1}(X_{\Gamma}-S,D^{\prime};{\bf Z})$ has
boundary $D^{\prime}$.

In the case of interest to us $S=\partial_{\Gamma}^{-}$ (resp. $\partial^{+}$)
and $D^{\prime}=D^{+}$ (resp. $D^{-}$) is supported on $\partial_{\Gamma}^{+}$
(resp. $\partial_{\Gamma}^{-}$).

We define the cuspidal subgroup of $J_{\Gamma}^{-}$ (resp. $J_{\Gamma}^{+}$)
as the subgroup spanned by the divisors of degree $0$ supported on
$\partial_{\Gamma}^{+}$ (resp. $\partial_{\Gamma}^{-}$). To simplify notation,
we see $\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-};{\bf Z})$ (resp.
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+};{\bf Z})$) as a subgroup of
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ (resp.
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})$).

######  Proposition 15.

The converse of the map $c\mapsto(\omega\mapsto\int_{c}\omega)$ provides the
following group isomorphisms :

$J_{\Gamma}^{-}({\bf
C})\simeq(\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-};{\bf
R})+i\lambda^{+}(\mathrm{H}_{0}(\partial_{\Gamma}^{-};{\bf
R})))/\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-};{\bf Z})$

and

$J_{\Gamma}^{+}({\bf
C})\simeq(\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+};{\bf
R})+i\lambda^{-}(\mathrm{H}_{0}(\partial_{\Gamma}^{+};{\bf
R})))/\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+};{\bf Z}).$

The groups on the right-hand sides identify canonically to subgroups of

$(\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
R})+i\lambda^{+}(\mathrm{H}_{0}(\partial_{\Gamma}^{-};{\bf
R})))/\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$

and

$(\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
R})+i\lambda^{-}(\mathrm{H}_{0}(\partial_{\Gamma}^{+},{\bf
R})))/\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})$

respectively. In the latter groups, the class of $D^{+}$ (resp. $D^{-}$) in
$J_{\Gamma}^{-}({\bf C})$ (resp. $J_{\Gamma}^{+}({\bf C})$) is the class of
$-{\mathcal{E}}_{D^{+}}$ (resp. $-{\mathcal{E}}_{D^{-}}$).

###### Proof.

The first statement follows from what we have just recalled on generalized
Jacobians. We prove now the second statement regarding $D^{+}$. Let
$c\in\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ such that the boundary of $c$ is $D^{+}$. By Abel’s theorem, the image of
$D^{+}$ in $J_{\Gamma}^{-}({\bf C})$ is the map $\omega\mapsto\int_{c}\omega$.
Let $\omega$ be a meromorphic differential form of the third kind with divisor
supported on $\partial_{\Gamma}^{-}$. One has
$\int_{{\mathcal{E}}_{D^{+}}}\omega=0$. Thus
$\int_{c}\omega=\int_{c-{\mathcal{E}}_{D^{+}}}\omega$. The boundary of
$c-{\mathcal{E}}_{D^{+}}\in\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
R})+i\lambda^{-}(\mathrm{H}_{0}(\partial_{\Gamma}^{+};{\bf R}))$ is $0$. The
desired result follows. The assertion concerning $D^{-}$ is proved similarly.

∎

######  Corollary 16.

The cuspidal subgroup of the generalized Jacobian $J_{\Gamma}^{-}$ (resp.
$J_{\Gamma}^{+}$) is isomorphic to the subgroup of
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
C})/\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})$ (resp.
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
C})/\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})$) generated by the classes ${\mathcal{E}}_{D^{+}}$ (resp.
${\mathcal{E}}_{D^{-}}$) when $D^{+}$ runs through the divisors of degree $0$
supported on $\partial_{\Gamma}^{+}$ (resp. $\partial_{\Gamma}^{-}$).

We now turn to theorem 2. The first part follows directly from the following
proposition.

######  Corollary 17.

The divisor $D^{+}$ (resp. $D^{-}$) is torsion in $J_{\Gamma}^{-}$ (resp.
$J_{\Gamma}^{+}$) if and only if one has
$\mathcal{R}_{D^{+}}\in\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Q})$ (resp.
$\mathcal{R}_{D^{-}}\in\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Q})$) and $\mathcal{I}_{D^{+}}=0$ (resp. $\mathcal{I}_{D^{-}}=0$).

###### Proof.

It follows from proposition generalizedjacobian that $D^{+}$ is torsion if and
only if
${\mathcal{E}}_{D^{+}}\in\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Q})$ which translates immediately into
$\mathcal{R}_{D^{+}}\in\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Q})$ and $\mathcal{I}_{D^{+}}=0$. The alternate statement is proved similarly.
∎

The second part of theorem 2 can be deduced from the first part. Indeed, it
follows from the fact that the image of $\lambda^{+}$ (resp. $\lambda^{-}$) is
the kernel of the map
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+};{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma}^{+};{\bf Z})$ (resp.
$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{+},\partial_{\Gamma}^{-};{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma}^{-};{\bf Z})$).

##  10\. Passage to any modular curve

Our purpose in this section is to derive from Theorem 1 all Eisenstein classes
in $\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf R})$, where $\Gamma$ is
still a subgroup of $\Gamma(2)$. In fact we will need to apply Theorem 1 for
several conjugates of $\Gamma$ in the full modular group.

Recall that Manin [16] has defined map $\xi$ : ${\bf
Z}[\Gamma\backslash{\mathrm{SL}}_{2}({\bf
Z})]\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf Z})$, which to
$[\Gamma g]$ associates the class $\\{g0,g\infty\\}$ of the image in
$X_{\Gamma}$ of the path from $g0$ to $g\infty$ in the upper half-plane. Such
a map is surjective and its kernel is fully described by Manin.

Denote by $U=\left(\begin{smallmatrix}0&1\\\ -1&1\\\ \end{smallmatrix}\right)$
and $S=\left(\begin{smallmatrix}0&1\\\ -1&0\\\ \end{smallmatrix}\right)$. The
element $U$ permutes cyclicly $0$, $1$ and $\infty$. Denote by $\partial^{0}$,
$\partial^{1}$ and $\partial^{\infty}$ the cusps of $X_{\Gamma}$ above the
cusps $\Gamma(2)0$, $\Gamma(2)1$ and $\Gamma(2)\infty$ respectively. Let $D$
be a divisor of degree $0$ supported on $\partial_{\Gamma}$. Write

$D=-E_{0}+E_{\infty}$

where $E_{0}$ and $E_{\infty}$ are divisors of degree $0$ supported on
$\partial^{\infty}\cup\partial^{1}$ and $\partial^{0}\cup\partial^{1}$
respectively. This decomposition is well-defined up to a divisor of degree $0$
supported on $\partial^{1}$.

When $D$ is supported on $\partial_{\Gamma}^{+}$ (resp.
$\partial_{\Gamma}^{0}$), we extend the function $F_{D}$ to a map still
denoted $F_{D}$ : ${\mathrm{SL}}_{2}({\bf Z})\rightarrow{\bf C}$ by $F(g)=0$
whenever $g\notin\Gamma$.

######  Proposition 18.

The element

${\mathcal{E}}^{\prime}_{D}=\sum_{g\in\Gamma\backslash{\mathrm{SL}}_{2}({\bf
Z})}(F_{U^{-1}E_{\infty}}(U^{-1}g)-F_{UE_{0}}(Ug))\xi(g)$

belongs to $\mathrm{H}_{1}(X_{\Gamma},\partial(X_{\Gamma});{\bf R})$ and is
the Eisenstein class of boundary $D$.

###### Proof.

Note first that $UE_{0}$ and $U^{-1}E_{\infty}$ are divisors of degree $0$
supported on $\partial_{U\Gamma U^{-1}}^{+}\subset X_{U\Gamma U^{-1}}$ and
$\partial_{U^{-1}\Gamma U}^{+}\subset X_{U^{-1}\Gamma U}$ respectively.
Consider the corresponding Eisenstein elements ${\mathcal{E}}_{UE_{0}}$ and
${\mathcal{E}}_{U^{-1}E_{\infty}}$ and their respective images
${\mathcal{E}}^{\prime}_{UE_{0}}$ and
${\mathcal{E}}^{\prime}_{U^{-1}E_{\infty}}$ in $\mathrm{H}_{1}(X_{U\Gamma
U^{-1}},\partial_{U\Gamma U^{-1}},{\bf C})$ and
$\mathrm{H}_{1}(X_{U^{-1}\Gamma U},\partial_{U^{-1}\Gamma U},{\bf C})$. Those
images have in fact real coefficients, see section 1. Then
$U^{-1}{\mathcal{E}}^{\prime}_{UE_{0}}$ and
$U{\mathcal{E}}_{U^{-1}E^{\prime}_{\infty}}$ are Eisenstein elements in
$\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma};{\bf R})$ of boundary $E_{0}$ and
$E_{\infty}$ respectively.

For $g\in\Gamma(2)$, the image of $\xi^{+}(g)$ via the composition of maps

$\mathrm{H}_{1}(X_{\Gamma}-\partial_{\Gamma}^{-},\partial_{\Gamma}^{+},{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma}^{+},{\bf
Z})\rightarrow\mathrm{H}_{1}(X_{\Gamma},\partial_{\Gamma},{\bf Z})$

is $\xi(g)$. Thus, one has

${\mathcal{E}}^{\prime}_{U^{-1}E_{\infty}}=\sum_{g\in U^{-1}\Gamma
U\backslash\Gamma(2)}F_{U^{-1}E_{\infty}}(g)\xi(g)\hskip
28.45274pt\text{and}\hskip
28.45274pt{\mathcal{E}}^{\prime}_{UE_{0}}=\sum_{g\in U\Gamma
U^{-1}\backslash\Gamma(2)}F_{UE_{0}}(g)\xi(g).$

Those Eisenstein elements have boundary $U^{-1}E_{\infty}$ and $UE_{0}$
respectively. Thus
$U{\mathcal{E}}_{U^{-1}E_{\infty}}-U^{-1}{\mathcal{E}}_{UE_{0}}$ is an
Eisenstein elements of boundary $E_{\infty}-E_{0}=D$. It is precisely
${\mathcal{E}}^{\prime}_{D}$, as can be seen by changing variable in the sums.

∎

When $\Gamma^{\prime}$ is a subgroup of ${\rm SL}_{2}({\bf Z})$ containing
$\Gamma$, consider the degeneracy map $\pi$ : $X_{\Gamma}\rightarrow
X_{\Gamma^{\prime}}$. It defines a map $\pi_{*}$ :
$\mathrm{H}_{1}(X_{\Gamma},\partial(X_{\Gamma});{\bf
R})\rightarrow\mathrm{H}_{1}(X_{\Gamma^{\prime}},\partial(X_{\Gamma});{\bf
R})$, which is compatible with the boundary maps and respects Eisenstein
elements. Thus proposition 18 provides a formula for all Eisenstein elements
for modular curves associated to any subgroup of finite index of ${\rm
SL}_{2}({\bf Z})$. The formula is unsatisfactory insofar as the decomposition
$D=E_{\infty}-E_{0}$ is not unique.

Note a dual statement using $\xi^{-}$ instead of $\xi^{+}$ is missing. We make
no use of $\xi^{-}$ in proposition 18.

######  Corollary 19.

The class of the divisor $D$ is torsion in $J_{\Gamma}$ if and only if there
exists $F^{\prime}$, $F^{\prime\prime}$ : ${\bf
Z}[\Gamma\backslash{\mathrm{SL}}_{2}({\bf Z})]\rightarrow{\bf R}$ which are
right $S$-invariant and right $U$-invariant respectively such that for all
$g\in{\rm SL}_{2}({\bf Z})$

$(F_{U^{-1}E_{\infty}}(U^{-1}g)-F_{UE_{0}}(Ug))-F^{\prime}(g)+F^{\prime\prime}(g)\in{\bf
Q}.$

###### Proof.

Indeed, the class of the divisor $D$ is torsion if and only if the Eisenstein
element ${\mathcal{E}}_{D}$ belongs to
$\mathrm{H}_{1}(X_{\Gamma},\partial(X_{\Gamma});{\bf Q})$. The kernel of
Manin’s map $\xi$ is spanned by the sum of elements invariant by $U$ and
elements invariant by $S$ [16]. The corollary follows.

∎

##  11\. Fermat curves

Let $N$ be an integer $>0$. The $N$-th Fermat curve $F_{N}$ is given by the
projective equation:

$X^{N}+Y^{N}=Z^{N}.$

Fermat curves and their points at infinity (cusps) are studied extensively by
Rohrlich [21], [22], Vélu [26] and Posingies [20]. We use the notations of the
latter author. In particular, we have the Belyi map

$\beta_{N}\mathrel{\mathop{\ordinarycolon}}F_{N}\rightarrow{\mathbb{P}}^{1}$

given by
$(X\mathrel{\mathop{\ordinarycolon}}Y\mathrel{\mathop{\ordinarycolon}}Z)\rightarrow(X^{N}\mathrel{\mathop{\ordinarycolon}}Z^{N})$.
The map $\beta_{N}$ satisfy the following properties:

  * •

It is of degree $N^{2}$.

  * •

It is ramified only above the points $0,1,\infty$.

  * •

The corresponding ramification points are given by
$a_{j}=(0\mathrel{\mathop{\ordinarycolon}}\zeta^{j}\mathrel{\mathop{\ordinarycolon}}1)$,
$b_{j}=(\zeta^{j}\mathrel{\mathop{\ordinarycolon}}0\mathrel{\mathop{\ordinarycolon}}1)$,
$c_{j}=(\epsilon\zeta^{j}\mathrel{\mathop{\ordinarycolon}}1\mathrel{\mathop{\ordinarycolon}}0)$,
for $j\in{\bf Z}/N{\bf Z}$.

Here, $\zeta=e^{2\pi i/N}$is the primitive $N$ th root of unity and
$\epsilon=e^{\pi i/N}$. Each of the above points has ramification index $N$
over ${\mathbb{P}}^{1}$.

Since the $\lambda$ function identifies ${\mathbb{P}}^{1}-\\{0,1,\infty\\}$ to
$Y(2)$. A covering of ${\mathbb{P}}^{1}-\\{0,1,\infty\\}$ can be understood as
a covering of $Y(2)$, i.e. a modular curve. Consider the group morphism
$P\Gamma(2)\rightarrow({\bf Z}/N{\bf Z})^{2}$ which sends $A$ to $(1,0)$ and
sends $B$ to $(0,1)$. An explicit formula for this morphism in terms of the
entries of the matrix (and not in terms of its decomposition as product of
generators) is due to Murty and Ramakrishnan [19]. Denote by $\Phi_{N}$ its
kernel, which is a subgroup of index $N^{2}$ of $P\Gamma(2)$. Denote by
$X_{\Phi(N)}$ the corresponding modular curve.

A system of representatives for the cosets $\Phi_{N}\backslash\Gamma(2)$ is
given by $A^{a}B^{b}$ with $a,b\in\\{0,1...(N-1)\\}$.

The dessin d’enfant of the Fermat curve enjoys a special combinatorial
property: its edges are in a one-to-one correspondence with the pairs of
opposite color vertices.

######  Proposition 20.

The map
$\Phi_{N}\backslash\Gamma(2)\rightarrow\Phi_{N}\backslash\Gamma(2)0\times\Phi_{N}\backslash\Gamma(2)\infty$
which to $\Phi_{N}g$ associates $(\Phi_{N}g0,\Phi_{N}g\infty)$ is bijective.

###### Proof.

All sets involved have cardinality $N^{2}$. It is sufficient to show that the
map is injective. For the first statement, let $g,g^{\prime}\in\Gamma(2)$ such
that
$(\Phi_{N}g0,\Phi_{N}g\infty)=(\Phi_{N}g^{\prime}0,\Phi_{N}g^{\prime}\infty)$.
There exists $a$, $b\in{\bf Z}$ such that $\Phi_{N}g=\Phi_{N}g^{\prime}A^{a}$
and $\Phi_{N}g=\Phi_{N}g^{\prime}B^{b}$. Thus one has
$\Phi_{N}gA^{-a}=\Phi_{N}gB^{-b}$. Hence $a$ and $b$ belong to $N{\bf Z}$.
Thus $\Phi_{N}g=\Phi_{N}g^{\prime}$.

∎

Thus the corresponding dessin d’enfant admits as $0$-vertices the cusps
$\Phi_{N}A^{a}0$, for $a\in\\{0,1,...,N-1\\}$, as $\infty$-vertices the cusps
$\Phi_{N}B^{b}\infty$, for $b\in\\{0,1,...,N-1\\}$, as edges the pairs
$\\{\Phi_{N}A^{a}0,\Phi_{N}B^{b}\infty\\}$ for $a$, $b\in\\{0,1,...,N-1\\}$.
The cyclic ordering (i.e. action of ${\bf Z}$) on the edges attached to
$\Phi_{N}A^{a}0$ (resp. $\Phi_{N}B^{b}\infty$) is such that the successor of
$\\{\Phi_{N}A^{a}0,\Phi_{N}B^{b}\infty\\}$ is
$\\{\Phi_{N}A^{a+1}0,\Phi_{N}B^{b}\infty\\}$ (resp.
$\\{\Phi_{N}A^{a}0,\Phi_{N}B^{b-1}\infty\\}$).

The $N$-th roots

$x\mathrel{\mathop{\ordinarycolon}}=\sqrt[N]{\lambda},y\mathrel{\mathop{\ordinarycolon}}=\sqrt[N]{1-\lambda}$

define modular units for $\Phi_{N}$. We recover thus the familiar model of the
Fermat curve.

The cusps $a_{j},b_{j}$ and $c_{j}$ have been introduced above. The divisors
of the following modular functions are given by:

${\rm div}(x-\zeta^{j})=Nb_{j}-\sum_{j}c_{j},{\rm
div}(y-\zeta^{j})=Na_{j}-\sum_{j}c_{j},{\rm
div}(x-\epsilon\xi^{j}y)=Nc_{j}-\sum_{j}c_{j}.$

It follows that the cuspidal subgroup of $J_{\Phi_{N}}$ is annihilated by $N$.
Suppose from now that $N$ is an odd integer. Rohrlich [22] has determined the
structure of the cuspidal group of $J_{\Phi_{N}}$ (we use Vélu’s alternative
proof and description [26] ).

######  Theorem 21.

(Rohrlich, [22], [26]) The group of principal divisors of $X_{\Phi(N)}$ is
spanned by by $N[\partial_{\Phi_{N}}]^{0}$ together with the following set
(the Rohrlich relations)

$\\{\sum_{i=0}^{N-1}[a_{i}]-[P],\sum_{i=0}^{N-1}[b_{i}]-[P],\sum_{i=0}^{N-1}[c_{i}]-[P],$
$\sum_{i=0}^{N-1}i([a_{i}]-[b_{i}]),\sum_{i=0}^{N-1}i([a_{i}]-[c_{i}]),\sum_{i=0}^{N-1}i^{2}([a_{i}]+[b_{i}]+[c_{i}]-3[P])\\},$

where $P$ is any cusp of $X_{\Phi_{N}}$. Consequently the cuspidal subgroup of
$J_{\Phi_{N}}$ is isomorphic to $({\bf Z}/N{\bf Z})^{3N-7}$.

We turn now to the analogue of this result for the generalized Jacobian. We
start with the Eisenstein cycles.

######  Theorem 22.

Let $j$, $k\in{{\bf Z}}/N{{\bf Z}}$, one has

${\mathcal{E}}_{(a_{j})-(c_{k})}=\frac{1}{N}(\sum_{g\in\Phi_{N}\backslash\Gamma(2),g0=a_{j}}\\{g0,g\infty\\}^{+}-\sum_{g\in\Phi_{N}\backslash\Gamma(2),g\infty=c_{k}}\\{g0,g\infty\\}^{+})$

and

$\displaystyle{\mathcal{E}}_{(b_{j})-(b_{k})}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}(\sum_{g\in\Phi_{N}\backslash\Gamma(2),g1=b_{j}}\\{g(1),g(-1)\\}^{-}-\sum_{g\in\Phi_{N}\backslash\Gamma(2),g(-1)=b_{j}}\\{g(1),g(-1)\\}^{-})$
$\displaystyle-\frac{1}{2N}(\sum_{g\in\Phi_{N}\backslash\Gamma(2),g1=b_{k}}\\{g(1),g(-1)\\}^{-}-\sum_{g\in\Phi_{N}\backslash\Gamma(2),g(-1)=b_{k}}\\{g(1),g(-1)\\}^{-}).$

###### Proof.

Set $f^{+}_{j,k}=\frac{y-\zeta^{j}}{x-\epsilon\zeta^{k}y}$. It is a modular
unit of divisor $N(a_{j}-c_{k})$. One has

$\omega_{(a_{j})-(c_{k})}=d(\log
f_{j,k})=\frac{dy}{y-\zeta^{j}}-\frac{dy}{x-\epsilon\zeta^{k}y}.$

Recall that $2\pi iG_{(a_{j})-(c_{k})}\,dz$ is the pullback on the upper half-
plane of $\omega_{(a_{j})-(c_{k})}$. Thus one has

$\displaystyle F_{(a_{j})-(c_{k})}(g)$ $\displaystyle=$
$\displaystyle\frac{-1}{2\pi
iN}[\int_{g(-1)}^{g(1)}\frac{dy}{y-\zeta^{j}}-\int_{g(-1)}^{g(1)}\frac{dy}{x-\epsilon\zeta^{k}y}]$
$\displaystyle=$ $\displaystyle\frac{-1}{2\pi iN}[\int_{\text{loop around
}g(0)}\frac{dy}{y-\zeta^{j}}-\int_{\text{loop
around}g(\infty)}\frac{dy}{x-\epsilon\zeta^{k}y}]$ $\displaystyle=$
$\displaystyle\frac{-1}{N}[\delta_{g0,a_{j}}-\delta_{g\infty,c_{k}}]$

where $\delta$ is the Kronecker symbol. The result follows from the formula

${\mathcal{E}}_{(a_{j})-(c_{k})}=\sum_{g\in\Phi_{N}\backslash\Gamma}\int_{g(1)}^{g(-1)}G_{(a_{j})-(c_{k})}(z)\,dz\\{g0,g\infty\\}^{+}.$

The second statement is proved similarly, using the function
$f^{-}_{j,k}=(x-\zeta^{j})/(x-\zeta^{k})$, a modular unite of divisor
$N((b_{j})-(b_{k}))$. Thus

$\displaystyle F_{(b_{j})-(b_{k})}(g)$ $\displaystyle=$
$\displaystyle\frac{-1}{2\pi
iN}[\int_{g0}^{g\infty}\frac{dy}{y-\zeta^{j}}-\int_{g0}^{g\infty}\frac{dy}{x-\epsilon\zeta^{k}y}]$
$\displaystyle=$ $\displaystyle\frac{-1}{2\pi iN}[\int_{\text{half-loop
around}g(1)}\frac{dy}{x-\zeta^{j}}-\int_{\text{half-loop
around}g(-1)}\frac{dy}{x-\zeta^{j}}]$ $\displaystyle+\frac{1}{2\pi
iN}[\int_{\text{half-loop around}g(1)}\frac{dy}{x-\zeta^{k}}-\int_{\text{half-
loop around}g(-1)}\frac{dy}{x-\zeta^{k}}]$ $\displaystyle=$
$\displaystyle\frac{-1}{2N}[\delta_{g(1),b_{j}}-\delta_{g(-1),b_{k}}-\delta_{g(1),b_{j}}+\delta_{g(-1),b_{k}}].$

∎

An evident consequence of theorem is that the imaginary parts of
${\mathcal{E}}_{(a_{j})-(c_{k})}$ and ${\mathcal{E}}_{(b_{j})-(b_{k})}$
vanish. This is in fact a consequence of proposition 11 and of theorem 8.1. of
, where Posingies determines the scattering constants for the Fermat curves.

######  Corollary 23.

The kernel of the map ${\bf Z}[\partial_{\Phi_{N}}^{+}]^{0}\rightarrow
J_{\Phi_{N}}^{-}$ is generated by $N{\bf
Z}[\partial_{\Phi_{N}}^{+}]^{0}\cup\\{\sum_{j=0}^{N-1}[a_{j}]-[c_{j}]\\}$
Therefore the cuspidal subgroup of $J_{\Phi_{N}}^{-}$ is isomorphic to $({\bf
Z}/N{\bf Z})^{2N-2}$.

The kernel of the map ${\bf Z}[\partial_{\Phi_{N}}^{-}]^{0}\rightarrow
J_{\Phi_{N}}^{+}$ is generated by $2N{\bf Z}[\partial_{\Phi_{N}}^{-}]^{0}$.
Therefore the cuspidal subgroup of $J_{\Phi_{N}}^{+}$ is isomorphic to $({\bf
Z}/2N{\bf Z})^{N-1}$.

###### Proof.

We use corollary 16 to understand the cuspidal subgroup of $J_{\Phi_{N}}^{-}$
as a subgroup of
$\mathrm{H}_{1}(X_{\Phi_{N}}-\partial_{\Phi_{N}}^{-},\partial_{\Phi_{N}}^{+};{\bf
Q})/\mathrm{H}_{1}(X_{\Phi_{N}}-\partial_{\Phi_{N}}^{-},\partial_{\Phi_{N}}^{+};{\bf
Z})$. Since the denominator of ${\mathcal{E}}_{(a_{j})-(c_{k})}$ is $N$, by
theorem 22, the cuspidal subgroup is contained in
$\frac{1}{N}\mathrm{H}_{1}(X_{\Phi_{N}}-\partial_{\Phi_{N}}^{-},\partial_{\Phi_{N}}^{+};{\bf
Z})/\mathrm{H}_{1}(X_{\Phi_{N}}-\partial_{\Phi_{N}}^{-},\partial_{\Phi_{N}}^{+};{\bf
Z})$. Recall that the group
$\mathrm{H}_{1}(X_{\Phi_{N}}-\partial_{\Phi_{N}}^{-},\partial_{\Phi_{N}}^{+};{\bf
Z})$ is freely generated by the $\\{g0,g\infty\\}^{+}$ when $g$ runs through
$\Phi_{N}\backslash\Gamma(2)$ by theorem 4. Thus we are left to determine the
kernel of the map :

$\theta^{+}\mathrel{\mathop{\ordinarycolon}}({\bf Z}/N{\bf
Z})[\partial_{\Phi_{N}}^{+}]^{0}\rightarrow({\bf Z}/N{\bf
Z})[\Phi_{N}\backslash\Gamma(2)]$

which to $(a_{j})-(c_{k})$ associates
$\sum_{g\in\Phi_{N}\backslash\Gamma(2),g0=a_{j}}[g]-\sum_{g\in\Phi_{N}\backslash\Gamma(2),g\infty=c_{k}}[g]$.
We use now proposition 20. It follows that the kernel of $\theta^{+}$ is
spanned by $\sum_{j=0}^{N-1}[a_{j}]-[c_{j}]$.

The other case can be treated in a similar manner. The group
$\mathrm{H}_{1}(X_{\Phi_{N}}-\partial_{\Phi_{N}}^{+},\partial_{\Phi_{N}}^{-};{\bf
Z})$ is freely generated by the $\\{g0,g\infty\\}^{-}$ when $g$ runs through
$\Phi_{N}\backslash\Gamma(2)$. Therefore we have to examine the map

$\theta^{-}\mathrel{\mathop{\ordinarycolon}}({\bf Z}/2N{\bf
Z})[\partial_{\Phi_{N}}^{+}]^{0}\rightarrow({\bf Z}/2N{\bf
Z})[\Phi_{N}\backslash\Gamma(2)]$

which to $(b_{j})-(b_{k})$ associates
$\sum_{g\in\Phi_{N}\backslash\Gamma(2),g1=b_{j}}[g]-\sum_{g\in\Phi_{N}\backslash\Gamma(2),g1=b_{k}}[g]-\sum_{g\in\Phi_{N}\backslash\Gamma(2),g(-1)=b_{j}}[g]+\sum_{g\in\Phi_{N}\backslash\Gamma(2),g(-1)=b_{k}}[g]$.

It is induced by the map

$\theta^{\prime}\mathrel{\mathop{\ordinarycolon}}({\bf Z}/2N{\bf
Z})[\partial_{\Phi_{N}}^{+}]\rightarrow({\bf Z}/2N{\bf
Z})[\Phi_{N}\backslash\Gamma(2)]$

which to $(b_{j})$ associates
$\sum_{g\in\Phi_{N}\backslash\Gamma(2),g1=b_{j}}[g]-\sum_{g\in\Phi_{N}\backslash\Gamma(2),g(-1)=b_{j}}[g]$.
The kernel of $\theta^{\prime}$ is spanned by the element $\sum_{j}(b_{j})$.
Thus $\theta^{-}$ is injective.

∎

######  Remark 24.

We did not use the sums coming from the Kloosterman zeta function to determine
the Eisenstein classes for the Fermat curves. A priori, it seems difficult to
understand a sum over the elements of $\Phi_{N}$. Indeed, how to characterize
the elements of $\Phi_{N}$ in terms of the entries of the corresponding
matrices? Murty and Ramakrishnan [19] provide an answer in terms of Dedekind
sums, but it is unclear how this would be useful.

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