# Monodromy of elliptic logarithms:
some topological methods and effective results

Francesco Tropeano

###### Abstract

We present some effective approaches in studying the relative monodromy group
of elliptic logarithms with respect to periods of elliptic schemes. We provide
explicit ways of constructing explicit loops which leave periods unchanged but
along which logarithms have non-trivial variations. We also get some
topological methods and effective results which allow to manage the
ramification locus of sections. The paper was inspired by a theorem of Corvaja
and Zannier which abstractly determine the relative monodromy group of non-
torsion sections.

toc

##  0 Introduction

Period functions, elliptic logarithms and the Betti map are very useful tools
in the context of elliptic and abelian schemes, for instance in studying the
torsion points of a section and many related questions in ‘unlikely
intersections’. Often, a key point in using these tools is that they are only
locally defined as holomorphic functions: some natural problems emerge when we
ask about their analytic continuation. Generally, it turns out that it’s
impossible to globally define these functions, in the sense that they turn out
to be multi-valued functions and to have a non-trivial monodromy action when
we travel along closed paths. We will deal with some questions relative to the
said monodromy action, in particular this paper is devoted to the study of the
monodromy of elliptic logarithms.

##### General notations.

We denote by $B$ a smooth affine curve and by $\pi:\mathcal{E}\rightarrow B$ a
complex elliptic scheme, where $\mathcal{E}$ is a quasi-projective surface and
$\pi:\mathcal{E}\rightarrow B$ is a surjective morphism all of whose fibers
$\pi^{-1}(b)$ are elliptic curves. We denote by $\mathcal{E}_{b}$ the fiber
$\pi^{-1}(b)$. We always suppose that there exists a zero-section
$\sigma_{0}:B\rightarrow\mathcal{E}$ which marks the origin in each fiber.
Moreover, we always suppose that the family is not isotrivial and that on any
compactification $\overline{\mathcal{E}}\rightarrow\overline{B}$ each fiber
over $\overline{B}-B$ is singular. When speaking about paths in $B$ we use the
word ‘loop’ as synonymous of closed path and we usually employ the word ‘path’
referring to paths which are not loops.

A natural problem consists in finding out the minimal (unramified) cover of
$B$ on which the relevant functions can be globally defined, which means
studying their monodromy. In this context, Corvaja and Zannier in [7] focused
on elliptic logarithms proving that the relative monodromy group of logarithm
of a non-torsion section $\sigma$ with respect to periods of
$\mathcal{E}\rightarrow B$ is non-trivial (more precisely they proved it is
isomorphic to the group $\mathbb{Z}^{2}$). In other words, this is equivalent
to the assertion that there exists a certain loop $\Gamma$ in $B$ which leaves
unchanged periods but not logarithm via analytic continuation. Anyway, they
used cohomological techniques and holomorphic sections to obtain the
conclusion but their proof does not give a method for constructing such a loop
explicitly. This paper was inspired by the work of Corvaja and Zannier with
the initial aim of providing an explicit construction of the loop $\Gamma$. As
a result, our investigation led to a different approach to the problem which
provides a new viewpoint on the issue and an alternative proof which has the
advantage of being _effective_. Further, our approach provides methods for
managing the ramification locus of a section $\sigma:B\rightarrow\mathcal{E}$
and consequently the property of being non-torsion.

To be more precise, the theorem we want to prove appear in this paper as 1.9
and can be also stated as follows:

######  Theorem.

Let $\sigma:B\rightarrow\mathcal{E}$ be a non-torsion section of the elliptic
scheme. Then the relative monodromy group of logarithm of $\sigma$ with
respect to periods of $\mathcal{E}\rightarrow B$ is non-trivial, hence
infinite with Galois group isomorphic to $\mathbb{Z}^{2}$. In particular,
there exists a loop $\Gamma$ in $B$ of the form

$\Gamma=\alpha_{1}\delta_{i}\alpha_{i}^{-1}\delta_{i}^{-1},$

where $\alpha_{1}$ is a loop whose homotopy class is one of the generators of
the fundamental group of $B$ (with some fixed base point), $\alpha_{1}$ and
$\alpha_{i}$ have the same monodromy action on periods, $\delta_{i}$ is an
open path with trivial monodromy action on periods.

Several difficulties arise in our approach. The first one comes out when
constructing the paths $\delta_{i}$: obtaining paths connecting two fixed
points of $B$ with trivial monodromy action turns out to be not obvious. We
reach this goal by getting some control on the ramification locus of the
section. In a second moment we propose an alternative way of controlling the
ramification of sections by using Abhyankar’s lemma and Belyi pairs.111When
our paper was finished, we realized that Belyi pairs also appear in connection
with other problems in number theory as pointed out for instance in [23], [24]
and [25]. However, our approach which consists in ‘separating the branch locus
of sections’ is completely new to our knowledge. The second approach has some
limitations but allows to obtain a more immediate way of constructing the
$\delta_{i}$’s. Then, a second difficulty arises when controlling variations
of logarithm along $\alpha_{1}$ and $\alpha_{i}$. This problem is fixed by
using Mordell-Weil theorem for function fields and Shioda’s theorem.
Developing our topological tools we also obtain a way of constructing all the
loops which acts trivially by monodromy on periods.

The theorem is of independent interest since it contains as particular cases
some classical theorems as: Manin’s theorem on the non-constancy of the Betti
map in the particular case of elliptic surfaces (see [15] or [3]); Shioda’s
theorem on the vanishing of the rank of elliptic modular surfaces ([20,
Theorem 5.1]); a theorem of Bertrand in [4] (generalized by André [1, Theorem
3]) which we state as 1.7. In particular, our methods and new proofs allow to
get a topological and explicit approach to some classical results in
transcendence of logarithms and periods. This kind of results are very useful
tools in some problems in ‘unlikely intersections’, in fact a result of
independence between logarithms and periods is a standard step when we are
interested in confining torsion of subvarieties of abelian schemes (for
instance see [5], [16], [17], [18] or [6], [10]). We mention the fact that the
1.9 has been conjecturally generalized in [22]; in the same paper the author
proved the instances of fibered product of elliptic schemes.

##### Structure of the paper.

For the sake of clarity, we briefly explain how the paper was organized.

In the first section we introduce the general context fixing some notations
and preliminary results needed in the following sections.

In the second part of the paper we focus on the main theorem and prove it is
invariant under some assumptions and operations. We also provide some
effective results, determining explicit paths where periods has no monodromy.

The third part regards the first effective approach to the main theorem, which
end providing an explicit loop $\Gamma$ with desired properties. The method we
use is based on some ‘control of the branch locus of the section’, which is
reached by using Legendre elliptic scheme, suitable factorizations of
morphisms and topological considerations. The rest of the proof consists in
controlling variations of elliptic logarithm by Mordell-Weil theorem and
Shioda’s theorem, concluding with an accurate analysis of analytic
continuations of the relevant functions.

In the fourth part we propose a second approach to the problem with the aim of
giving a more immediate construction of the loop $\Gamma$ in some cases. We
analyze the advantages and limitations of this way. This approach provides a
different method of controlling the branch locus, which we call ‘separation of
the branch locus of the section’: it is based on the use of Abhyankar’s Lemma
(here as 4.2) or Belyi pairs. The rest of the proof essentially follows the
idea of our previous effective proof.

The last part is devoted to some additional considerations about our methods.
In particular, we leave some questions and we give a concrete example showing
how our ideas can be applied. We obtain some stronger (effective) conclusions
in the case of Masser’s section: in fact we determine explicit generators of
the full relative monodromy group of the section. This last example also
tacitly hides how the initial idea of the paper was born.

Acknowledgements. This work was part of the author’s PhD thesis. The author is
grateful to Umberto Zannier, Pietro Corvaja and Paolo Antonio Oliverio for
helpful discussions, for their kind attention, their useful advice and
references. The author also thanks Paolo Dolce for useful opinions and
suggestions.

##  1 Elliptic schemes and monodromy problems

In this section we introduce some notations and preliminary results needed in
the following sections. At the same time, we give an (unpretentious) overview
on some monodromy problems on elliptic schemes.

Let us keep all the notations introduced above. Moreover, we introduce the
following notation, which will be repeatedly used when speaking about
monodromy: if $\alpha:I\rightarrow S$ is a path and $f:S\rightarrow\mathbb{C}$
is a function defined in a neighbourhood of $\alpha(0)$ that admits an
analytic continuation along $\alpha$, we denote by $c_{\alpha}(f)$ the
analytic continuation of $f$ in $\alpha(1)$ along $\alpha$.

Now, fix an elliptic scheme $\mathcal{E}\rightarrow B$. Any fiber
$\mathcal{E}_{b}$ is analytically isomorphic to a complex torus
$\mathbb{C}/\Lambda_{b}$; thus in corrispondence with the elliptic scheme we
have a family of lattices $\Lambda_{b}$ varying with $b\in B$. We can locally
define a basis of the period lattice: on suitable open simply connected
subsets $U\subset B$ in the complex topology which cover $B$, we can find
holomorphic functions

$\omega_{U,1},\omega_{U,2}:U\rightarrow\mathbb{C}$

such that $\omega_{U,1}(b),\omega_{U,2}(b)$ is a basis for $\Lambda_{b}$ for
each $b\in U$.

We want to be a bit more explicit on this point. To this end, define
$S:=\mathbb{P}_{1}-\\{0,1,\infty\\}$ and denote by $\mathcal{L}\rightarrow S$
the Legendre elliptic scheme, i.e. the elliptic scheme whose fiber
corresponding to $\lambda\in S$ is given by

$\mathcal{L}_{\lambda}:y^{2}=x(x-1)(x-\lambda).$

A possible choice of a basis of periods for the elliptic curve
$\mathcal{L}_{\lambda}$ is obtained by integrating the invariant differential
$y^{-1}dx$ of the curve along two paths generating the homology of the torus
corresponding to $\mathcal{L}_{\lambda}$. We saw that for instance in the
domain $\max(|\lambda|,|1-\lambda|)<1$ one obtains these periods as series
expansions of suitable hypergeometric functions (see [12]):

$\omega^{\mathcal{L}}_{1}(\lambda)=\pi\sum_{n=0}^{\infty}\binom{-\frac{1}{2}}{n}^{2}\lambda^{n},\qquad\omega^{\mathcal{L}}_{2}(\lambda)=i\pi\sum_{n=0}^{\infty}\binom{-\frac{1}{2}}{n}^{2}(1-\lambda)^{n}.$
(1)

The functions $\omega^{\mathcal{L}}_{1},\omega^{\mathcal{L}}_{2}$ representing
the periods in the above region, may be analytically continued along paths in
$\mathbb{C}-\\{0,1\\}$, obtaining functions
$\omega^{\mathcal{L}}_{U,1},\omega^{\mathcal{L}}_{U,2}$ which give locally a
basis for the lattices $\Lambda_{\lambda}$ made up of holomorphic functions on
open simply connected sets $U$ of an open covering of $\mathbb{C}-\\{0,1\\}$.

We want to emphasize the fact that the Legendre elliptic scheme is more than a
simple example. In fact, up to a finite base change, an elliptic scheme
$\mathcal{E}\rightarrow B$ can be obtained as pullback of the Legendre scheme.
Thus, in many situations if we prove a result on the Legendre elliptic scheme,
we can say something about the starting elliptic scheme
$\mathcal{E}\rightarrow B$. To be more precise about this construction, we
briefly explain that in the next remark:

######  Remark 1.1.

Let $\mathcal{E}\rightarrow B$ be an elliptic scheme as above and suppose that
the generic fibre of the family is given by a Weierstrass equation whose
affine form is

$y^{2}=f(x),$

where the coefficients of $f$ are rational functions on $B$. Let us perform a
finite cover $B^{\prime}\rightarrow B$ of the base $B$, where $B^{\prime}$ is
chosen so large so that the two-torsion points are well-defined functions on
$B^{\prime}$. We obtain an elliptic scheme $\mathcal{E}^{\prime}\rightarrow
B^{\prime}$ by pullback via $B^{\prime}\rightarrow B$:

${\mathcal{E}^{\prime}}$${\mathcal{E}}$${B^{\prime}}$${B.}$

Now, the $2$-torsion of the curve is rational, so we can reduce the
Weierstrass equation in Legendre form. This gives a morphism
$B^{\prime}\rightarrow S$ such that $\mathcal{E}^{\prime}$ is obtained as
pullback of $\mathcal{L}$ via the morphism $B^{\prime}\rightarrow S$.

Thanks to 1.1 we can focus on elliptic schemes $\mathcal{E}\rightarrow B$
endowed with a finite map $B\rightarrow S$ such that $\mathcal{E}$ is obtained
as pull-back of the Legendre scheme via $B\rightarrow S$, i.e. we have an
elliptic scheme $\mathcal{E}\rightarrow B$ with a diagram

${\mathcal{E}}$${\mathcal{L}}$${B}$${S,}$$\scriptstyle{p}$ (2)

where $p:B\rightarrow S$ is in general a ramified morphism.

We are able to construct explicitly periods of any elliptic scheme. We fix a
base point $b\in B$, not a ramification point for $p:B\rightarrow S$, and let
$s=p(b)\in S$. In a connected and simply connected neighborhood $U$ of $b$ in
$B$, we can holomorphically define a basis $\omega_{1},\omega_{2}$ of the
period lattices by the equation

$\omega_{i}=\omega^{\mathcal{L}}_{i}\circ p.$ (3)

Locally on suitable open subsets $U\subset B$ (in the complex topology), this
gives a basis for $\Lambda_{b}$ made up of holomorphic functions
$\omega_{U,1},\omega_{U,2}:U\rightarrow\mathbb{C}$.

###  1.1 Monodromy of periods

Combining Equation 1 and Equation 3 we obtain locally period functions of any
elliptic scheme $\mathcal{E}\rightarrow B$. These functions may be
analytically continued through the whole of $B$, but it’s impossible to
globally define them: they turn out to be multi-valued functions, i.e. they
have quite nontrivial monodromy when traveling along closed paths.

First of all, note that the monodromy action corresponds to the action of the
fundamental group $G:=\pi_{1}(B,b)$ on the first homology group
$H_{1}(\mathcal{E}_{b},\mathbb{Z})$: thus, we get a monodromy representation

$\rho=\rho_{B}:G\rightarrow\textnormal{Aut}(H_{1}(\mathcal{E}_{b},\mathbb{Z}))\cong\textnormal{GL}_{2}(\mathbb{Z}).$
(4)

Note that the period lattice as a whole does depend only on $b$, namely there
is no monodromy (as a whole): in fact, the lattice is uniquely determined by
the Weierstrass equation of the elliptic curve; also, the orientation of the
tangent space coming from the complex structure provides an orientation of the
set of bases for these lattices. To be more precise, since monodromy preserves
the intersection form then any action of monodromy corresponds to an element
of $\textrm{SL}_{2}(\mathbb{Z})$: therefore the above representation writes
down as $\rho:G\rightarrow\textnormal{SL}_{2}(\mathbb{Z})$.

We define the monodromy group of $\mathcal{E}\rightarrow B$ at $b$ as
$\textnormal{M}(\mathcal{E},b)=\rho(G)$. Since $B$ is path connected, all the
groups $\textrm{Mon}(\mathcal{E},b)$ are conjugate when we vary the base point
$b\in B$: in other words, the subgroups $\textrm{Mon}(\mathcal{E})$ will be
defined up to an inner automorphism of the group
$\textrm{SL}_{2}(\mathbb{Z})$. Thus, fix once and for all a base point
$b_{0}\in B$ and denote the group $\textrm{Mon}(\mathcal{E},b_{0})$ by
$\textrm{Mon}(\mathcal{E})$, without writing any dependencies on the base
point.

In view of Equation 3, in order to obtain information about
$\textrm{Mon}(\mathcal{E})$ it is enough to study the analytic continuation of
power series in Equation 1. Since $\pi_{1}(S)$ is freely generated by two
loops from the basepoint around $0,1$, define $\mathfrak{a}_{0}$ as a small
circle centred at $0$ and $\mathfrak{a}_{1}$ as a small circle centred at $1$.
The generators of $\pi_{1}(S)$ can be chosen as the homotopy classes
$g_{0},g_{1}$ of $\mathfrak{a}_{0},\mathfrak{a}_{1}$, respectively. In [13]
and [8], explicit calculations about series expansions of periods are
performed: in particular, in the second paper the authors explicitly exhibit
the monodromy action on periods. They write down the following relations:

$c_{\mathfrak{a}_{0}}(\omega^{\mathcal{L}}_{1})=\omega^{\mathcal{L}}_{1},$ |  | $c_{\mathfrak{a}_{0}}(\omega^{\mathcal{L}}_{2})=\omega^{\mathcal{L}}_{2}+2\omega^{\mathcal{L}}_{1};$
---|---|---
$c_{\mathfrak{a}_{1}}(\omega^{\mathcal{L}}_{1})=\omega^{\mathcal{L}}_{1}+2\omega^{\mathcal{L}}_{2},$ |  | $c_{\mathfrak{a}_{1}}(\omega^{\mathcal{L}}_{2})=\omega^{\mathcal{L}}_{2}.$

This determines the full monodromy group $\textnormal{Mon}(\mathcal{L})$. In
fact, given a period
$\omega:=n\omega^{\mathcal{L}}_{1}+m\omega^{\mathcal{L}}_{2}$, we obtain

$\displaystyle c_{\mathfrak{a}_{0}}(\omega)$
$\displaystyle=n\omega^{\mathcal{L}}_{1}+m(\omega^{\mathcal{L}}_{2}+2\omega^{\mathcal{L}}_{1})=(n+2m)\omega^{\mathcal{L}}_{1}+m\omega^{\mathcal{L}}_{2},$
$\displaystyle c_{\mathfrak{a}_{1}}(\omega)$
$\displaystyle=n(\omega^{\mathcal{L}}_{1}+2\omega^{\mathcal{L}}_{2})+m\omega^{\mathcal{L}}_{2}=n\omega^{\mathcal{L}}_{1}+(2n+m)\omega^{\mathcal{L}}_{2}.$

In conclusion, the monodromy group $\textnormal{Mon}(\mathcal{L})$ is
generated by the matrices

$\left(\begin{matrix}1&0\\\
2&1\end{matrix}\right),\qquad\left(\begin{matrix}1&2\\\
0&1\end{matrix}\right),$

which are known to generate freely the subgroup $\Gamma_{2}$ of
$\Gamma(2)\subset\textrm{SL}_{2}(\mathbb{Z})$ consisting of matrices congruent
to the identity modulo $2$ and with elements congruent to $1$ modulo $4$ on
the diagonal. Hence, we obtain a (surjective) representation
$\rho:\pi_{1}(S)\rightarrow\Gamma_{2}$ and the equality
$\textnormal{Mon}(\mathcal{L})=\Gamma_{2}$. Precisely, we have an action of
$\pi_{1}(S)$ on $\mathbb{Z}^{2}$ (viewed as the coordinates of periods with
respect to the basis $\omega_{1}^{\mathcal{L}},\omega_{2}^{\mathcal{L}}$) as
follows:

$g\cdot\left(\begin{matrix}n\\\
m\end{matrix}\right)=\rho(g)\left(\begin{matrix}n\\\ m\end{matrix}\right),$
(5)

where

$\rho(g_{0}):=\left(\begin{matrix}1&2\\\
0&1\end{matrix}\right),\qquad\rho(g_{1}):=\left(\begin{matrix}1&0\\\
2&1\end{matrix}\right).$

Now consider a general elliptic scheme $\mathcal{E}\rightarrow B$. By 1.1, we
can assume to have a finite map $B\rightarrow S$ which realizes
$\mathcal{E}\rightarrow B$ as pull-back of the Legendre elliptic scheme with
periods given by Equation 3. Hence, the monodromy group
$\textrm{Mon}(\mathcal{E})$ is a finite-index subgroup of
$\textrm{SL}_{2}(\mathbb{Z})$. Moreover, since $\Gamma_{2}$ is Zariski-dense
in $\textrm{SL}_{2}(\mathbb{Z})$ and $\textrm{SL}_{2}(\mathbb{Z})$ acts
irreducibly on the lattice of periods then the monodromy group
$\textrm{Mon}(\mathcal{E})$ is Zariski-dense in $\textrm{SL}_{2}(\mathbb{Z})$
and acts irreducibly on the lattice of periods.

######  Remark 1.2.

The previous considerations yield some conclusion on the well-definition of
periods on the base $B$. Let $\omega_{1},\omega_{2}$ be a basis for the period
lattice, defined locally as complex functions of $b\in U\subset B$, where $U$
is a simply connected open set of $B$. Since the group
$\textrm{Mon}(\mathcal{E})$ is non-trivial, the functions
$\omega_{1},\omega_{2}$ cannot be both defined continuously on the whole of
$B$. Moreover, since the action of $\textrm{Mon}(\mathcal{E})$ on the lattice
of periods is irreducible, neither $\omega_{1}$ nor $\omega_{2}$ nor any
single element of the period lattice can be defined on the whole of $B$.

####  1.1.1 Modular elliptic schemes

At a certain point, we shall be interested in a particular type of elliptic
schemes: the elliptic schemes obtained from the Legendre scheme
$\mathcal{L}\rightarrow S$ by pull-back under an _unramified_ cover
$p:B\rightarrow S$. These elliptic schemes (or, to be more precise, their
compactifications) are named modular elliptic surfaces in Shioda’s paper [20].
We call them _modular elliptic schemes_ to underline the fact that we only
have smooth fibres.

Let $\mathcal{E}\rightarrow B$ a modular elliptic scheme. Obviously, this is a
particular case of an elliptic scheme which satisfies Equation 2: in addition,
we have that $p$ is unramified. Thus, we can define periods as in Equation 3
and consider the associated monodromy action $\rho$ as in Equation 4. In this
case, [2, Theorem 5] ensures the representation $\rho$ to be faithful: we
obtain an embedding $\rho:G\hookrightarrow\Gamma_{2}$.

Generally, if $\mathcal{E}\rightarrow B$ is not a modular elliptic scheme, the
monodromy action on periods is not faithful. The kernel of the action
coincides with the kernel of the induced map $p_{*}:G\rightarrow\Gamma_{2}$
between fundamental groups.

In this context, we will be interested in [20, Theorem 5.1] which we will use
as stated in [7, Theorem 2.5]. We write down this last statement, since it
will be an important tool in our approaches:

######  Theorem 1.3.

(Shioda’s Theorem) Every (rational) section $\sigma:B\rightarrow\mathcal{E}$
of a modular elliptic scheme is torsion.

###  1.2 Monodromy group of a section

Now, let us consider a section $\sigma:B\rightarrow\mathcal{E}$ of an elliptic
scheme. Over every point $b\in B$, we have an elliptic exponential map
$\textrm{Lie}(\mathcal{E}_{b})\rightarrow\mathcal{E}_{b}$, whose kernel is the
period lattice. The family of Lie algebras
$\textrm{Lie}(\mathcal{E})\rightarrow B$ defines a line bundle over $B$ (it is
actually a complex Lie algebra bundle), which is trivial in the holomorphic
sense and it can then be analytically identified with the product
$B\times\mathbb{C}$. By using the fiberwise exponential maps one can define a
global map

$\exp:B\times\mathbb{C}\rightarrow\mathcal{E}.$

Observe that by restricting to $U\times\mathbb{C}$, we obtain the covering map

$U\times\mathbb{C}\rightarrow\mathcal{E}_{|U},$

where we are denoting $\pi^{-1}(U)$ by $\mathcal{E}_{|U}$. Given a simply
connected open set $U\subset B$ where periods are well-defined, we define an
_elliptic logarithm of $\sigma$ in $U$_ as a lifting of $\sigma_{|U}$ to
$U\times\mathbb{C}$: precisely, we define the logarithm as a function
$\log_{\sigma}:U\rightarrow\mathbb{C}$ which makes commutative the diagram

${U\times\mathbb{C}}$${B\supset
U}$${\pi^{-1}(U)\subset\mathcal{E}.}$$\scriptstyle{\sigma_{|U}}$$\scriptstyle{(\textnormal{id},\log_{\sigma})}$

In this context, we simply mention the fact that considering the real
coordinates of elliptic logarithms with respect to the basis of periods we
obtain the so called Betti map $\beta_{\sigma}:U\rightarrow\mathbb{R}^{2}$:
this is a real-analytic map and is very useful in many problems in ‘unlikely
intersections’.

Similarly to period functions, once we have locally defined the logarithm of a
section we can think about its analytic continuation through the whole of $B$.
As a first example, let us consider the zero-section $\sigma_{0}$ which
associates to each $b\in B$ the origin $O_{b}$ of the corresponding fiber
$\mathcal{E}_{b}$. A logarithm of $\sigma_{0}$ is given by the zero function

$\log_{\sigma_{0}}:B\rightarrow\mathbb{C},\qquad\log_{\sigma_{0}}(b)=0\textrm{
for each }b\in B.$

Thus, in this case we can find a well-defined logarithm on the whole of $B$,
in fact it has no monodromy. For rational sections, this is the only case in
which such a global logarithm exists. For the sake of completeness, we briefly
resume some approaches to a proof of this.

######  Remark 1.4.

Let’s give a sketch of a proof of the following fact: a non-zero rational
section $\sigma:B\rightarrow\mathcal{E}$ cannot admit a well-defined logarithm
on the whole of $B$. A way to prove it is by using Mordell-Weil theorem for
the Legendre curve over the function field of $B(\mathbb{C})$ in a way that
appeared also in [15]. If a non-zero rational section admits an elliptic
logarithm which is well defined on the whole of $B(\mathbb{C})$, then we may
divide it, and hence the section, by any prescribed positive integer and again
we have maps well defined on $B(\mathbb{C})$. Thus the section would be
infinitely divisible on $B(\mathbb{C})$ (since the submultiples of the
sections would be algebraic and well defined on $B(\mathbb{C})$, hence
rational on $B(\mathbb{C})$). But this violates the Mordell-Weil theorem for
the Legendre curve over the function field of $B(\mathbb{C})$. In 1.6 we
recall an alternative argument to prove this statement with different
technique.

In general, the fundamental group $G$ of $B$ acts by monodromy on logarithm of
$\sigma$. Clearly, two determinations of the logarithm (defined over an open
set $U\subset B$) differ by an integral combination of
$\omega_{1},\omega_{2}$. Thus, the action of $G$ on the determinations of the
logarithm is as follows: fixed a determination of $\log_{\sigma}$ and given a
loop $\alpha_{g}$ which represents the homotopy class $g\in G$ we have

$g\cdot\log_{\sigma}=c_{\alpha_{g}}(\log_{\sigma})=\log_{\sigma}+u_{g}\omega_{1}+v_{g}\omega_{2},$
(6)

where $u_{g},v_{g}\in\mathbb{Z}$ and $c_{\alpha_{g}}$ denotes the analytic
continuation along $\alpha_{g}$. Let us consider the map

$G\rightarrow\mathbb{Z}^{2},\qquad g\mapsto(u_{g},v_{g}).$

Let’s continue denoting by $\omega_{1},\omega_{2}$ a basis of periods and by
$\log_{\sigma}$ a logarithm of $\sigma$. Choose $g,h\in G$ and denote by
$\bar{g}:=\rho(g),\bar{h}:=\rho(h)\in\textrm{Mon}(\mathcal{E})$ their
monodromy representations. By looking at the action of $gh$ and recalling
Equation 5, we have

$\displaystyle\log_{\sigma}\xrightarrow{h}\log_{\sigma}+(u_{h},v_{h})\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right)\xrightarrow{g}\log_{\sigma}+(u_{g},v_{g})\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right)+(u_{h},v_{h})\bar{g}^{t}\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right).$

In other words, we obtain

$(u_{gh},v_{gh})=(u_{g},v_{g})+(u_{h},v_{h})\bar{g}^{t},$

i.e. the map $g\mapsto(u_{g},v_{g})$ _is a cocycle for the described action of
$G$ on $\mathbb{Z}^{2}$_.

The cocycle just defined describes the obstruction for a(n analytic) section
to have a well-defined logarithm. In fact, a section which has a well-defined
logarithm is characterized by the fact that the map $g\mapsto(u_{g},v_{g})$ is
a coboundary for the above mentioned action, i.e. there exists a fixed vector
$(u,v)\in\mathbb{Z}^{2}$ such that $(u_{g},v_{g})=(u,v)(\bar{g}^{t}-I)$ for
all $g\in G$. This statement is clarified in the following proposition:

######  Proposition 1.5.

Let $\sigma:B\rightarrow\mathcal{E}$ be a section and $\log_{\sigma}$ a
determination of its logarithm over an open set $U\subset B$. The section
admits a well-defined logarithm on $B$ if and only if the associated cocycle
$g\mapsto(u_{g},v_{g})$ is a coboundary.

###### Proof.

Suppose that $\sigma$ admits a well-defined logarithm
$\ell:B\rightarrow\mathbb{C}$. Then the two determinations $\ell$ and
$\log_{\sigma}$ over $U$ differ by a period $\omega:=n\omega_{1}+m\omega_{2}$,
i.e.

$\log_{\sigma}=\ell+n\omega_{1}+m\omega_{2}.$

For $g\in G$, we have

$\displaystyle g\cdot\log_{\sigma}$
$\displaystyle=g\cdot(\ell+n\omega_{1}+m\omega_{2})=\ell+(n,m)\bar{g}^{t}\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right)=$
$\displaystyle=\log_{\sigma}+\left[(n,m)\bar{g}^{t}-(n,m)\right]\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right).$

Thus the corresponding cocycle is given by

$g\mapsto(n,m)\bar{g}^{t}-(n,m)$

for $g\in G$ and a fixed pair $(n,m)\in\mathbb{Z}^{2}$, hence it is a
coboundary.

Viceversa, let us suppose to have a logarithm $\log_{\sigma}$ over $U$ and
that there exists a fixed pair $(n,m)\in\mathbb{Z}^{2}$ such that

$g\cdot\log_{\sigma}=\log_{\sigma}+\left[(n,m)\bar{g}^{t}-(n,m)\right]\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right)$

for each $g\in G$. Let us define the function

$\ell:=\log_{\sigma}-n\omega_{1}-m\omega_{2},$

which is another determination of the logarithm of $\sigma$. Looking at the
action of $G$ we obtain

$\displaystyle g\cdot\ell$
$\displaystyle=g\cdot(\log_{\sigma}-n\omega_{1}-m\omega_{2})=$
$\displaystyle=\log_{\sigma}+\left[(n,m)\bar{g}^{t}-(n,m)\right]\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right)-(n,m)\bar{g}^{t}\left(\begin{matrix}\omega_{1}\\\
\omega_{2}\end{matrix}\right)=$
$\displaystyle=\log_{\sigma}-n\omega_{1}-m\omega_{2}=\ell.$

Therefore, $\ell$ is a well-defined logarithm of $\sigma$ on the whole of $B$.
∎

The above argument leads to consider the cohomology group
$H^{1}(G,\mathbb{Z}^{2})$, which is the method used in [7].

Now, let us look at the simultaneous monodromy action of $G$ on periods and
logarithm. By Equation 5 and Equation 6, we can provide a new representation

$\theta_{\sigma}:G\rightarrow\textrm{SL}_{3}(\mathbb{Z}),$

where every matrix $\theta_{\sigma}(g)$ is of the form

$\theta_{\sigma}(g)=\left(\begin{matrix}\rho(g)&w_{g}\\\
0&1\end{matrix}\right),$ (7)

where $w_{g}=(u_{g},v_{g})^{t}\in\mathbb{Z}^{2}$. Note that the matrix
$\rho(g)$ acts on the periods as specified in Equation 5, and does not depend
on $\sigma$. Moreover, the vector $w_{g}$ encodes the action of $g$ on
determinations of the logarithm $\log_{\sigma}$ as in Equation 6. Define the
monodromy group of the section $\sigma$ as $M_{\sigma}:=\theta_{\sigma}(G)$.

######  Remark 1.6.

As mentioned in 1.4, we recap an alternative argument to prove that a non-zero
rational section $\sigma:B\rightarrow\mathcal{E}$ cannot admit a well-defined
logarithm on the whole of $B$. In this case, we use the just introduced
representation $\theta_{\sigma}$ and the mentioned theorem of Bertrand, which
we state here for the sake of completeness:

######  Theorem 1.7.

If the (rational) section $\sigma:B\rightarrow\mathcal{E}$ is non-torsion, the
kernel of the homomorphism
$\theta_{\sigma}(G)^{\textrm{Zar}}\rightarrow\textrm{SL}_{2}$ is isomorphic to
$\mathbb{G}_{a}^{2}$. In particular, the algebraic group
$\theta_{\sigma}(G)^{\textrm{Zar}}$ has dimension five.

If a rational section $\sigma$ admits a logarithm, then $\theta_{\sigma}(g)$
has $w_{g}=(0,0)^{t}$ for all $g\in G$. This means that
$\theta_{\sigma}(G)^{\textrm{Zar}}$ has not dimension five, thus $\sigma$ is a
torsion section by Bertrand’s theorem. Then a logarithm of $\sigma$ has the
form $\log_{\sigma}=q_{1}\omega_{1}+q_{2}\omega_{2}$, where
$q_{1},q_{2}\in\mathbb{Q}$. We can see, by explicit computation, that such a
function is well-defined on the whole of $B$ if and only if $q_{1}=q_{2}=0$,
i.e. $\sigma$ is the zero-section.

###  1.3 Relative monodromy group of a section

Let’s continue considering an elliptic scheme $\mathcal{E}\rightarrow B$ and a
rational non-zero section $\sigma:B\rightarrow\mathcal{E}$. By our previous
considerations, neither a basis of periods nor a logarithm $\log_{\sigma}$ can
be well-defined on the whole of $B$. Anyway, the relevant functions can be
globally defined on the universal cover $\mathbb{H}$ of $B$; note that $B$ is
hyperbolic in view of our hypothesis that the scheme is not isotrivial.
Studying the related monodromy problems corresponds to finding out the minimal
(unramified) cover of $B$ on which both a basis of the periods and a logarithm
of the section can be defined. With this in mind, we first call
$B^{*}\rightarrow B$ the minimal (unramified) cover of $B$ on which a basis
for the period lattice can be globally defined and we set
$B_{\sigma}\rightarrow B^{*}$ to be the minimal cover of $B^{*}$ where one can
define the logarithm of $\sigma$. The tower of covers is represented in the
diagram:

$\mathbb{H}\rightarrow B_{\sigma}\rightarrow B^{*}\rightarrow B.$ (8)

In particular, the group $\textnormal{Mon}(\mathcal{E})$ corresponds to the
Galois group of the covering map $B^{*}\rightarrow B$, while the group
$M_{\sigma}$ corresponds to the Galois group of the covering map
$B_{\sigma}\rightarrow B$. Our interest is in studying the relative monodromy
of logarithm of a section with respect to the monodromy of periods, i.e.
studying the covering map $B_{\sigma}\rightarrow B^{*}$. Topologically, this
is the same as looking at the variation of logarithm via analytic continuation
along loops which leave periods unchanged. Moreover, in terms of Equation 4
and Equation 7 this corresponds to studying the group
$M_{\sigma}^{\textnormal{rel}}:=\theta_{\sigma}(\ker{\rho})$, which we define
as _relative monodromy group of $\sigma$_.

######  Remark 1.8.

Note that the Zariski-closures of the discrete algebraic groups
$\textnormal{Mon}(\mathcal{E}),M_{\sigma}$ and $M_{\sigma}^{\textnormal{rel}}$
are the differential Galois groups of some Picard-Vessiot extensions of
$\mathbb{C}(B)$ obtained with $\omega_{1},\omega_{2},\log_{\sigma}$ and their
derivatives (see [9] for further details about differential Galois theory and
Picard-Vessiot extensions). In this terms, the differential Galois group
$\overline{M_{\sigma}^{\textnormal{rel}}}$ was just determined in [4]; anyway,
this result says nothing on the relative monodromy of the logarithm over
$B^{*}$ but only on its Zariski closure, and this information is not as strong
as we want. In fact, in [8] and [7] the authors pointed out that it may happen
that the Zariski closure of a group contains quite limited information on the
group itself since it may be larger than expected in comparison with the group
itself. More precisely, the Zariski closures may be the same for groups which
are of infinite index one in the other. Thus, in order to obtain information
on $M_{\sigma}^{\textnormal{rel}}$, we have to introduce considerations of
different nature with respect to Bertrand’s theorem.

In [7, Theorem 2.1] the authors proved a result which we can state as follows:

######  Theorem 1.9.

If $\sigma:B\rightarrow\mathcal{E}$ is a non-torsion (rational) section, then
$M_{\sigma}^{\textnormal{rel}}\cong\mathbb{Z}^{2}$.

Note that $M_{\sigma}^{\textnormal{rel}}$ is clearly a subgroup of
$\mathbb{Z}^{2}$; the previous theorem asserts that this subgroup is as large
as possible. Let us illustrate what happens in the case not covered by 1.9,
i.e. when the section is torsion; moreover the example aims at introducing our
topological approach more concretely.

######  Example 1.10.

Let us consider a torsion section $\sigma:B\rightarrow\mathcal{E}$. By the
properties of Betti map, any logarithm of $\sigma$ is a rational constant
combination of periods; i.e.

$\log_{\sigma}=q_{1}\omega_{1}+q_{2}\omega_{2},$

where $q_{1},q_{2}\in\mathbb{Q}$. Therefore, a loop which leaves unchanged
periods via analytic continuation, leaves also unchanged the logarithm of such
a section. In other words, the cover $B_{\sigma}\rightarrow B^{*}$ is trivial
in this case and then $M_{\sigma}^{\textnormal{rel}}\cong\\{0\\}$.

##  2 Preliminary results and first thoughts on effectivity

###  2.1 Invariance results

Let’s start by proving some preliminary results about 1.9 and the group
$M_{\sigma}^{\textnormal{rel}}$.

######  Lemma 2.1.

Theorem 1.9 is invariant by finite base change: in other words, if
$\varphi:\widetilde{B}\rightarrow B$ is a finite morphism and
$\widetilde{\mathcal{E}}$ is the pull-back of $\mathcal{E}\rightarrow B$ via
$\varphi$, then the theorem for
$\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$ implies the theorem for
$\mathcal{E}\rightarrow B$.

###### Proof.

Let us denote by $\omega_{1},\omega_{2}$ a basis of periods of
$\mathcal{E}\rightarrow B$. We can define
$\widetilde{\omega_{1}},\widetilde{\omega_{2}}$, a basis of periods of
$\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$, by the equations

$\widetilde{\omega_{1}}=\omega_{1}\circ\varphi,\qquad\widetilde{\omega_{2}}=\omega_{2}\circ\varphi.$
(9)

Fix two base points $\widetilde{b}\in\widetilde{B}$ and $b\in B$ such that
$\varphi(\widetilde{b})=b$; we omit to explicitly write them in the
fundamental group notation. If we denote by $\bar{G},\widetilde{G}$ the
monodromy groups of periods of $\mathcal{E}\rightarrow B$,
$\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$ respectively, we obtain the
associated monodromy representations:

$\rho:\pi_{1}(B)\rightarrow\bar{G},\qquad\widetilde{\rho}:\pi_{1}(\widetilde{B})\rightarrow\widetilde{G}.$

By $(\ref{eqn3})$, we have $\widetilde{\rho}(g)=\rho(\varphi_{*}(g))$ for each
$g\in\pi_{1}(\widetilde{B})$ (here
$\varphi_{*}:\pi_{1}(\widetilde{B})\rightarrow\pi_{1}(B)$ is the induced map
between fundamental groups). In particular, we obtain

$\varphi_{*}(\ker{\widetilde{\rho}})=\ker{\rho}\cap\varphi_{*}(\pi_{1}(\widetilde{B})).$
(10)

Let $\sigma:B\rightarrow\mathcal{E}$ be a non-torsion section of
$\mathcal{E}\rightarrow B$. Since
$\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$ is obtained as pull-back of
$\mathcal{E}\rightarrow B$, then the elliptic curves
$\widetilde{\mathcal{E}}_{\widetilde{b}}$ and $\mathcal{E}_{b}$ are
canonically identified; therefore the pull-back
$\varphi^{*}(\sigma):=\sigma\circ\varphi:\widetilde{B}\rightarrow\widetilde{\mathcal{E}}$
is a non-torsion section of $\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$.
We have the associated monodromy representations

$\theta_{\sigma}:\pi_{1}(B)\rightarrow\textrm{SL}_{3}(\mathbb{Z}),\qquad\theta_{\varphi^{*}(\sigma)}:\pi_{1}(\widetilde{B})\rightarrow\textrm{SL}_{3}(\mathbb{Z}).$

Using again the fact that the elliptic curves
$\widetilde{\mathcal{E}}_{\widetilde{b}}$ and $\mathcal{E}_{b}$ are
canonically identified, we obtain that a determination of the logarithm of
$\varphi^{*}(\sigma)$ over $\widetilde{b}$ can be defined by the equation

$\log_{\varphi^{*}(\sigma)}(\widetilde{b}):=\log_{\sigma}(b).$ (11)

Hence, we have
$\theta_{\varphi^{*}(\sigma)}(g)=\theta_{\sigma}(\varphi_{*}(g))$ for each
$g\in\pi_{1}(\widetilde{B})$.

Since we are supposing that 1.9 holds for
$\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$, then we have

$M_{\varphi^{*}(\sigma)}^{\textnormal{rel}}=\theta_{\varphi^{*}(\sigma)}(\ker{\widetilde{\rho}})\cong\mathbb{Z}^{2}.$

By Equation 10 and Equation 11, this means

$\theta_{\sigma}(\ker{\rho}\cap\varphi_{*}(\pi_{1}(\widetilde{B})))=\theta_{\sigma}(\varphi_{*}(\ker{\widetilde{\rho}}))=M_{\varphi^{*}(\sigma)}^{\textnormal{rel}}\cong\mathbb{Z}^{2}.$

Since $\ker{\rho}\cap\varphi_{*}(\pi_{1}(\widetilde{B}))\subset\ker{\rho}$, we
get $M_{\varphi^{*}(\sigma)}^{\textnormal{rel}}\subseteq
M_{\sigma}^{\textnormal{rel}}$. This implies

$M_{\sigma}^{\textnormal{rel}}\cong\mathbb{Z}^{2},$

which is equivalent to saying that Theorem 1.9 holds for
$\mathcal{E}\rightarrow B$.
∎

Let us consider a non-torsion section $\sigma:B\rightarrow\mathcal{E}$. Note
that the representation $\theta_{\sigma}$ is defined in terms of Equation 6,
thus it depends on the branch of logarithm we fix. Here, we want to prove that
$M_{\sigma}^{\textnormal{rel}}$ is independent of the choice of branch of
$\log_{\sigma}$ and remains unchanged under some operations on the section.

######  Proposition 2.2.

Let $\sigma:B\rightarrow\mathcal{E}$ be a non-torsion section. Then

  * (i)

the group $M_{\sigma}^{\textnormal{rel}}$ does not depend on the choice of
branch of $\log_{\sigma}$;

  * (ii)

the groups $M_{\sigma}^{\textnormal{rel}}$ and
$M_{n\sigma}^{\textnormal{rel}}$ are isomorphic.

###### Proof.

$(i)$: Choose two branches of $\log_{\sigma}$, say $\ell_{\sigma}^{1}$ and
$\ell_{\sigma}^{2}$ and denote by $M_{\sigma,1}^{\textnormal{rel}}$ and
$M_{\sigma,2}^{\textnormal{rel}}$ the corresponding relative monodromy groups.
Since the two branches $\ell_{\sigma}^{1}$ and $\ell_{\sigma}^{2}$ differs by
a period, we get the thesis. In fact, since any loop $\alpha$ in $B$ whose
homotopy class lies in $\ker{\rho}$ leaves periods unchanged, then
$\ell_{\sigma}^{1}$ and $\ell_{\sigma}^{2}$ have the same variation by
analytic continuation along $\alpha$. Thus we get
$M_{\sigma,1}^{\textnormal{rel}}=M_{\sigma,2}^{\textnormal{rel}}$.

$(ii)$: Fixed a branch $\ell_{\sigma}$ of $\log_{\sigma}$, a determination
$\ell_{n\sigma}$ of logarithm of $n\sigma$ can be defined by the equation

$\ell_{n\sigma}=n\ell_{\sigma}.$

For any loop $\alpha$ we have the corresponding variations:

$c_{\alpha}(\ell_{n\sigma})=\ell_{n\sigma}+\omega_{\alpha}^{n\sigma},\qquad
c_{\alpha}(\ell_{\sigma})=\ell_{\sigma}+\omega_{\alpha}^{\sigma},$

where $\omega_{\alpha}^{n\sigma}=n\omega_{\alpha}^{\sigma}$. Thus
$M_{\sigma}^{\textnormal{rel}}$ and $M_{n\sigma}^{\textnormal{rel}}$ are
isomorphic.
∎

###  2.2 Some ‘effective’ considerations and results

Let $\mathcal{E}\rightarrow B$ an elliptic scheme which satisfies Equation 2
and look at the relative monodromy group $M_{\sigma}^{\textnormal{rel}}$ of a
section $\sigma$. When the morphism $p:B\rightarrow S$ is unramified we are
dealing with a modular elliptic scheme; since the representation $\rho$ is
faithful in this case, we obtain $\ker{\rho}=\\{0\\}$.222As a consequence of
faithfulness of $\rho$ and 1.9, we find again 1.3. However, note that this is
not a new proof of Shioda’s theorem, since the proof of 1.9 exploits 1.3 in an
essential way. On the other hand, when the morphism $p$ is ramified and we
have a non-torsion section $\sigma:B\rightarrow\mathcal{E}$, as a consequence
of 1.9 we get $\ker{\rho}\neq\\{0\\}$. Thus, asking about the shape of loops
whose homotopy class lies in $\ker{\rho}$ is a rather natural question. Here,
we want to say something more about the group $\ker{\rho}$ and obtain an
‘effective’ result on its shape in terms of loops.

Consider the Legendre elliptic scheme $\mathcal{L}\rightarrow S$ and denote by
$R=\\{r_{1},\ldots,r_{k}\\}$ the branch locus of the morphism $p$. Denote the
generators of the free group $\pi_{1}(S-R,s)$ by
$\mathfrak{a}_{0},\mathfrak{a}_{1},\mathfrak{d}_{i}$: to be more precise,
$\mathfrak{a}_{0},\mathfrak{a}_{1}$ are homotopy classes of small loops around
$0,1$ respectively; $\mathfrak{d}_{i}$ is the homotopy class of a small loop
around $r_{i}$. We will denote by the symbol $\langle Y\rangle$ the free group
generated by a set $Y$. Let’s put

$X^{(0)}:=\\{\mathfrak{d}_{i}:i=1,\ldots,k\\},\qquad D^{(0)}:=\langle
X^{(0)}\rangle$ (12)

and define by recursion

$X^{(n)}:=\\{\mathfrak{a}\mathfrak{d}\mathfrak{a}^{-1}\mathfrak{d}^{-1}:\mathfrak{a}\in\\{\mathfrak{a}_{0},\mathfrak{a}_{1}\\},\mathfrak{d}\in
D^{(n-1)}\\},\qquad D^{(n)}:=\langle\bigcup_{i=0}^{n}X^{(i)}\rangle.$

Define $K:=\langle\bigcup_{n\in\mathbb{N}}D^{(n)}\rangle$.

######  Theorem 2.3.

We have

$\ker{\rho_{B}}=p_{*}^{-1}(K).$

###### Proof.

Let us consider the inclusion $i:S-R\hookrightarrow S$ and the induced
homomorphism $i_{*}$ of fundamental groups. We claim that

$i_{*}^{-1}(1)=K.$

First of all note that $D^{(0)}\subseteq i_{*}^{-1}(1)$. By induction, we
easily get $D^{(n)}\subseteq i_{*}^{-1}(1)$ for any $n\in\mathbb{N}$. Hence we
have the inclusion $K\subseteq i_{*}^{-1}(1)$.

Now, we prove the reverse inclusion. Observe that each element
$g\in\pi_{1}(S-R,s)$ is a word in the alphabet

$\mathfrak{A}:=\\{1,\mathfrak{a}_{0},\mathfrak{a}_{1},\mathfrak{d}_{1},\ldots,\mathfrak{d}_{k},\mathfrak{a}^{-1}_{0},\mathfrak{a}^{-1}_{1},\mathfrak{d}^{-1}_{1},\ldots,\mathfrak{d}^{-1}_{k}\\}.$

Define the subset
$\mathfrak{A}^{\prime}:=\\{\mathfrak{a}_{0},\mathfrak{a}_{1},\mathfrak{a}^{-1}_{0},\mathfrak{a}^{-1}_{1}\\}$.
We write down words without using exponentiation, so that when writing $g$ as
a word we intend that each factor lies in $\mathfrak{A}$. Now, consider a word
$g$ and delete all elements of the type $\mathfrak{d}_{j}$; label the
remaining elements by $\mathfrak{e}_{1},\ldots,\mathfrak{e}_{m}$, where each
$\mathfrak{e}_{j}$ is an element of $\mathfrak{A}^{\prime}$. Note that the
element $i_{*}(g)$ is obtained by $g$ exactly deleting all elements of the
type $\mathfrak{d}_{j}$.

From now on, we only consider elements $g,h\in i_{*}^{-1}(1)$. Since
$\pi_{1}(S,s)$ is a free group, the condition $g\in i_{*}^{-1}(1)$ implies
that $m$ is even and moreover $\mathfrak{e}_{m-j+1}$ is the inverse of
$\mathfrak{e}_{j}$. We want to prove that $i_{*}^{-1}(1)\subseteq K$. To this
end, define the integer $n:=m/2$ and proceed by induction over $n$; the
integer $n$ will be called _relative length of $g$_.

If $n=0$, then $g\in D^{(0)}$ and we have done. Now, suppose that if $h$ is a
word of relative length $n-1$ then $h\in D^{(n-1)}$ and consider a word $g$ of
relative length $n$. Let $\mathfrak{a}$ be the first element of
$\mathfrak{A}^{\prime}$ which appear in the word $g$ in left-right order.
Then, $g$ is of the form

$\mathfrak{d}\mathfrak{a}\mathfrak{h}\mathfrak{a}^{-1}\mathfrak{d}^{\prime},$

where $\mathfrak{a}\in\mathfrak{A}^{\prime}$,
$\mathfrak{d},\mathfrak{d}^{\prime}\in D^{(0)}$ and $\mathfrak{h}$ is a word
of relative length $n-1$. By inductive hypothesis, we get $\mathfrak{h}\in
D^{(n-1)}$. With simple manipulations we get

$g=\mathfrak{d}(\mathfrak{a}\mathfrak{h}\mathfrak{a}^{-1}\mathfrak{h}^{-1})\mathfrak{h}\mathfrak{d}^{\prime},$

which proves $g\in D^{(n)}$ and consequently the claim. By Equation 3 we
finally obtain

$\ker{\rho_{B}}=p_{*}^{-1}(K).$

∎

2.3 gives a way of finding explicit loops in $B$ which leaves periods
unchanged: we obtain them as liftings of some loops in $S$ whose homotopy
classes lie in $K$. Clearly, not all the elements of $K$ provide loops in $B$:
in general such a lifting is an open path. However, making use of
considerations about ramification of the morphism $p$ we can always find some
of them. We exhibit some of them in the next sections.

Before closing this section, we prove an invariance result in the spirit of
Lemma 2.1. In other terms, the following lemma states that the effectivity of
1.9 is invariant under finite base change. However, we will be more precise
about the meaning of ‘effectivity of 1.9’ in the next sections, which entirely
concern this topic.

######  Lemma 2.4.

Let $\varphi:\widetilde{B}\rightarrow B$ be a finite morphism and let
$\widetilde{\mathcal{E}}$ be the pull-back of $\mathcal{E}\rightarrow B$ via
$\varphi$. Let $\widetilde{\alpha}$ be a loop in $\widetilde{B}$ whose
homotopy class $g$ is such that $g\in\ker{\widetilde{\rho}}$ and
$\theta_{\varphi^{*}(\sigma)}(g)\neq 0$. Then
$\alpha:=\varphi\circ\widetilde{\alpha}$ is a loop in $B$ whose homotopy class
$h$ satisfies $h\in\ker{\rho}$ and $\theta_{\sigma}(h)\neq 0$.

###### Proof.

This is straightforward by Equation 10 and Equation 11.
∎

##  3 Effective proof

In this section we prove the main (effective) result of the paper. To this
end, we focus on 1.9. Once we have proved that $M_{\sigma}^{\textnormal{rel}}$
is non-trivial we get the complete theorem by a purely algebraic argument.
Thus, here we focus on the crucial step of the proof, that is proving
$M_{\sigma}^{\textnormal{rel}}$ is non-trivial. In other words, the statement
$M_{\sigma}^{\textnormal{rel}}\neq\\{0\\}$ means that there exists a loop in
$B$ along which periods can be continuously defined but logarithm cannot be
defined.

Our strategy is _to explicitly find such a loop $\Gamma$ in $B$ with a certain
base point $b_{1}$, along which the logarithm changes value in $b_{1}$ but
periods don’t_, which gives an effective answer to the problem.

###  3.1 Setup of the proof

We keep all the notations introduced above. Morever, we shall always use the
symbol $\mathcal{B}$ to denote the compactification of an affine curve $B$. We
will do a little abuse of notation by denoting with the same letter a morphism
$p:B\rightarrow S$ between affine curves and its extension
$p:\mathcal{B}\rightarrow\mathbb{P}_{1}$. We denote by $\alpha,\beta,\ldots$
paths and loops in $B$, while we denote by $\mathfrak{a},\mathfrak{b},\ldots$
their projection in $S$ via the morphism $p$. In particular, we denote by
$\mathfrak{a}_{0}$ and $\mathfrak{a}_{1}$ two fixed and small enough circles
centered at $0$ and $1$, respectively. Note that we can choose all paths in
$B$ and $S$ such that they avoid all the ramification points and the branch
points of $p:B\rightarrow S$, respectively.

Let’s start with some preliminary considerations. Fix a non-torsion section
$\sigma$ of an elliptic scheme $\mathcal{E}\rightarrow B$. By 1.1, Lemma 2.1
and Lemma 2.4, in what follows we can always assume that
$\mathcal{E}\rightarrow B$ is obtained as pull-back of $\mathcal{L}\rightarrow
S$ via a finite Galois morphism $p:B\rightarrow S$. In this case, the section
$\sigma$ is also called an algebraic section of $\mathcal{L}\rightarrow S$,
i.e. we have the following diagram

${\mathcal{E}}$${\mathcal{L}}$${\mathbb{H}}$${B_{\sigma}}$${B^{*}}$${B}$${S,}$$\scriptstyle{p}$$\scriptstyle{\sigma}$
(13)

where we are referring to the tower of covers in Equation 8. By 1.3 the
section $\sigma$ is ramified, in the sense that the morphism $p:B\rightarrow
S$ introduced above is ramified. However, we look at the branch points of
$\sigma$ in $\mathbb{P}_{1}$ admitting also $0,1,\infty$ as possible branch
points, i.e. we are interested in the branch points of
$p:\mathcal{B}\rightarrow\mathbb{P}_{1}$. We will denote the branch locus of
$p:\mathcal{B}\rightarrow\mathbb{P}_{1}$ by $\mathcal{R}$ and the branch locus
of $p:B\rightarrow S$ by $R:=\\{r_{1},\ldots,r_{k}\\}$; thus we have
$R=\mathcal{R}\cap S$ and $R\neq\emptyset$. Fix a base point $s\in S-R$ and a
base point $b=b_{1}\in p^{-1}(s)$; denote by $b_{i}$ the elements of
$p^{-1}(s)$. Since the section is algebraic, then each branch point has finite
branching order.

######  Remark 3.1.

This remark simply to briefly clarify why $p$ can be assumed to be Galois. We
call $p$ a _Galois (branched) cover_ if the morphism
$p:\mathcal{B}-p^{-1}(\mathcal{R})\rightarrow\mathbb{P}_{1}-\mathcal{R}$ is a
Galois unramified cover. By [21, Theorem 3.3.7] this corresponds to having a
Galois extension of function fields $\mathbb{C}(S)\subseteq\mathbb{C}(B)$.
Given an elliptic scheme, by 1.1, Lemma 2.1 and Lemma 2.4, in a first moment
we can simply assume to have a finite morphism $B\rightarrow S$ which realizes
our elliptic scheme as pullback of the Legendre scheme. Generally, the
morphism $B\rightarrow S$ is ramified and is not Galois. Anyway, the Galois
closure of the field extension $\mathbb{C}(S)\subseteq\mathbb{C}(B)$ is a
finite extension of $\mathbb{C}(S)$. Thus, appealing to Lemma 2.1 again, we
can always assume $\mathcal{E}\rightarrow B$ to be obtained as pull-back of
$\mathcal{L}\rightarrow S$ via a finite Galois morphism $p:B\rightarrow S$.

##### Plan of the proof.

We explain how the proof is organized in order to make the reader aware of
what we are doing in the different steps:

  1. 1.

Control on ramification locus of sections: this is the fundamental step of our
construction. A method for controlling the ramification of sections is
provided and it allows to lift loops between covering maps getting control on
the monodromy action on periods and logarithms.

  2. 2.

Explicit loop: this step regards explicit construction of $\Gamma$. To this
end we need some auxiliary paths $\delta_{i}$ and loops $\alpha_{i}$ which
induce a controlled monodromy action on periods and logarithm. The auxiliary
paths will be obtained by using the control on ramification deriving from the
previous step, the monodromy action of a covering map, Mordell-Weil theorem
for elliptic schemes and Shioda’s theorem.

  3. 3.

Monodromy action and analytic continuation: a careful analysis of analytic
continuation of the relevant functions concludes the proof.

###  3.2 Control on ramification locus of sections

The first step of our proof consists in finding a way to “control the branch
locus $\mathcal{R}$” (the formal meaning of this sentence is clarified in the
next lemma). The right way of carrying out this step has been found in a
factorization of the morphism $p:\mathcal{B}\rightarrow\mathbb{P}_{1}$ which
appears in [7].333The authors use this factorization in [7] in a different way
with respect to us. They use the cohomological interpretation of variations of
logarithm and they need using holomorphic (not merely algebraic) sections. Our
approach involves analysis on the ramification locus of the section.

######  Lemma 3.2.

Given the ramified Galois morphism $p:\mathcal{B}\rightarrow\mathbb{P}_{1}$
associated to the elliptic scheme $\mathcal{E}\rightarrow B$ as in Equation
13, there exist a smooth curve $\mathcal{B}^{\prime}$ and finite morphisms
$p_{1}:\mathcal{B}\rightarrow\mathcal{B}^{\prime}$,
$p_{2}:\mathcal{B}^{\prime}\rightarrow\mathbb{P}_{1}$ such that the map $p$
factorizes in the following way

${\mathcal{B}}$${\mathcal{B}^{\prime}}$${\mathbb{P}_{1},}$$\scriptstyle{p_{1}}$$\scriptstyle{p}$$\scriptstyle{p_{2}}$

where

  1. $(i)$

$p_{*}(\pi_{1}(B,b))=p_{2*}(\pi_{1}(B^{\prime},b^{\prime}))$ for base points
$b\in B$ and $b^{\prime}:=p_{1}(b)\in B^{\prime}$;

  2. $(ii)$

$p_{1}$ is a ramified Galois morphism with branch locus $A$ satisfying
$R\subseteq p_{2}(A)\subseteq\mathcal{R}$;

  3. $(iii)$

$p_{2}$ is a ramified morphism with branch locus contained into the set
$\\{0,1,\infty\\}$.

###### Proof.

Fix base points $b\in B-p^{-1}(R)$ and $s:=p(b)\in S-R$. The continuous map
$p:B\rightarrow S$ induces a homomorphism of the corresponding fundamental
groups, i.e.

$p_{*}:\pi_{1}(B,b)\rightarrow\pi_{1}(S,s)=\Gamma_{2}.$

Define $\overline{G}:=p_{*}(\pi_{1}(B,b))$. The inclusion
$\overline{G}\subseteq\Gamma_{2}$ induces an unramified covering map
$p_{2}:B^{\prime}\rightarrow S$ between Riemann surfaces such that
$p_{2*}(\pi_{1}(B^{\prime},b^{\prime}))=\overline{G}$, where $b^{\prime}$ is a
fixed base point in $B^{\prime}-A$ which satisfies $p_{2}(b^{\prime})=s$.
Since $p_{*}(\pi_{1}(B,b))=p_{2*}(\pi_{1}(B^{\prime},b^{\prime}))$, there
exists a morphism $p_{1}:B\rightarrow B^{\prime}$ such that
$p_{1}(b)=b^{\prime}$. Considering the corresponding elliptic schemes we
obtain the diagram

${\mathcal{E}}$${\mathcal{L}^{\prime}}$${\mathcal{L}}$${B}$${B^{\prime}}$${S.}$$\scriptstyle{p_{1}}$$\scriptstyle{p}$$\scriptstyle{p_{2}}$
(14)

We extend each morphism to the compactification of the curves. Part $(i)$ is
straightforward by construction. Observe that part $(iii)$ is also true by
construction, since $p_{2}:B^{\prime}\rightarrow S$ is unramified. Moreover,
note that $p_{1}$ is Galois since $p$ is. Now, call $A$ the branch locus of
$p_{1}$ and observe that

$\textnormal{Branch}(p)=\textnormal{Branch}(p_{2})\cup
p_{2}(\textnormal{Branch}(p_{1})),$

where the meaning of the just introduced notation ‘Branch’ is clear from the
context. Hence, we get $R\subseteq p_{2}(A)\subseteq\mathcal{R}$.
∎

From now on, we fix base points $b\in B-p^{-1}(R)$, $b^{\prime}:=p_{1}(b)\in
B^{\prime}-A$ and $s:=p_{2}(b^{\prime})\in S-R$. We will write

$p_{1}^{-1}(b^{\prime})=\\{b_{1},\ldots,b_{N}\\},$

where $b$ is identified with $b_{1}$.

Now, we want to determine some paths $\delta_{i}$ which join the points
$b_{i}$ between them and such that their projections $p(\delta_{i})$ in $S$
are homotopically trivial. This last condition is essential and is equivalent
to saying that periods does not have monodromy along the paths, in the sense
that $c_{\delta_{i}}(\omega)=\omega(\delta_{i}(0))$ for any period function
$\omega$. To make this possible, we use the monodromy action of
$\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})$ on the fiber
$p_{1}^{-1}(b^{\prime})$, but we need enough control on the branch locus
$\mathcal{R}$ which is provided by Lemma 3.2.

######  Lemma 3.3.

For each $i,j\in\\{1,\ldots,N\\}$ there exists a path
$\delta_{i,j}:I\rightarrow B-p_{1}^{-1}(A)$ such that

  1. 1.

$\delta_{i,j}(0)=b_{i}$ and $\delta_{i,j}(1)=b_{j}$;

  2. 2.

$p(\delta_{i,j})$ is a homotopically trivial loop in $S$.

###### Proof.

Let us consider the unramified cover

$p_{1}:\mathcal{B}-p_{1}^{-1}(A)\rightarrow\mathcal{B}^{\prime}-A$

which is a Galois cover. Therefore, if we denote by $\textrm{Bij}(*)$ the set
of all bijections on a set, we have a monodromy representation

$\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})\rightarrow\textrm{Bij}(p_{1}^{-1}(b^{\prime}))=\textrm{Bij}(\\{b_{1},\ldots,b_{N}\\}),$

where the image is a transitive subgroup of $S_{N}$. In other words this
amounts to saying that for each $i,j$ there exists a loop $\mu^{\prime}_{i,j}$
in $\mathcal{B}^{\prime}-A$ with base point $b^{\prime}$ which lifts to a path
$\mu_{i,j}:I\rightarrow\mathcal{B}-p_{1}^{-1}(A)$ such that
$\mu_{i,j}(0)=b_{i},\mu_{i,j}(1)=b_{j}$.

Denote by $[*]$ the homotopy class of a loop. By part $(i)$ of Lemma 3.2,
there exists a loop $\xi$ in $B$ such that
$p_{*}([\xi])=p_{2*}([\mu^{\prime}_{i,j}])$. Further, $\xi$ defines a loop
$\xi^{\prime}:=p_{1}(\xi)$ in $B^{\prime}$ such that
$p_{2*}([\xi^{\prime}])=p_{2*}([\mu^{\prime}_{i,j}])$. Since the monodromy
representation $\rho_{B^{\prime}}$ is faithful, we get that
$\mu^{\prime}_{i,j}$ and $\xi^{\prime}$ are homotopically equivalent in
$B^{\prime}$. Define $\delta_{i,j}:=\xi^{-1}*\mu_{i,j}$. By construction, the
loop $p(\delta_{i,j})$ is homotopically trivial in $S$ and satisfies
$\delta_{i,j}(0)=b_{i}$ and $\delta_{i,j}(1)=b_{j}$.

Observe that all paths and loops can be chosen avoiding all the ramification
and branch points of the maps. Therefore, $\delta_{i,j}$ can be viewed as a
loop in $B-p_{1}^{-1}(A)$.
∎

###  3.3 Explicit loop

From now on, we fix the paths $\delta_{i-1,i}$ for each $i=2,\ldots,N$ coming
from Lemma 3.3, and define the paths $\delta_{i}$ as follows:

$\delta_{i}:=\delta_{1,2}*\cdots*\delta_{i-1,i}.$ (15)

In other words, $\delta_{i}$ is a path joining $b_{1}$ to $b_{i}$ such that
$p(\delta_{i})$ is a homotopically trivial loop in $S$ with base point $s$.
Let’s consider a branch $\log_{\sigma}(b_{1})$ of logarithm of $\sigma$ at
$b_{1}$ and define $\log_{\sigma}$ at the points $b_{i}$ by analytic
continuation along the path considered in Equation 15, i.e.

$\log_{\sigma}(b_{i}):=c_{\delta_{i}}(\log_{\sigma}).$ (16)

Consider the map $p_{1}$ defined in Equation 14 and denote by $\mathcal{E}(B)$
the group of rational sections of the elliptic scheme. Now, we refer to the
trace operator
$\textrm{Tr}_{p_{1}}:\mathcal{E}(B)\rightarrow\mathcal{L}^{\prime}(B^{\prime})$
defined as follows: for each $t\in B$ and $t^{\prime}=p_{1}(t)\in B^{\prime}$,
the elliptic curves $\mathcal{E}_{t}$ and $\mathcal{L}^{\prime}_{t^{\prime}}$
are canonically identified; then, for each section
$\sigma:B\rightarrow\mathcal{E}$ define
$\textrm{Tr}(\sigma)=\textrm{Tr}_{p_{1}}(\sigma):B^{\prime}\rightarrow\mathcal{L}^{\prime}$
by setting

$\textrm{Tr}(\sigma)(t^{\prime})=\sum_{t\in
p_{1}^{-1}(t^{\prime})}\sigma(t)\in\mathcal{L}^{\prime}_{t^{\prime}},$ (17)

where the sum is taken counting multiplicities. Note that
$\textrm{Tr}(\sigma)$ is a rational section of the modular elliptic scheme
$\mathcal{L}^{\prime}\rightarrow B^{\prime}$, thus by 1.3 it is a torsion
section. Therefore, by 2.2 we can multiply the section by a non-zero integer
and assume $\textrm{Tr}_{p_{1}}(\sigma)=0$; we continue denoting by $\sigma$
the new section. In other words, this means that

$\sigma(b_{1})+\cdots+\sigma(b_{N})=0,$

which is equivalent to saying

$\log_{\sigma}(b_{1})+\cdots+\log_{\sigma}(b_{N})=\omega,$

where $\omega$ is a period.

######  Lemma 3.4.

Up to multiplying the section by a non-zero integer and to changing
determination of $\log_{\sigma}$, we can suppose

$\log_{\sigma}(b_{1})+\cdots+\log_{\sigma}(b_{N})=0.$

###### Proof.

Let us consider

$\ell(b_{1}):=N\cdot\log_{\sigma}(b_{1})-\omega,$

where $N$ is the cardinality of $\\{b_{1},\ldots,b_{N}\\}$; observe that
$\ell(b_{1})$ is a logarithm of the section $N\cdot\sigma$ over the point
$b_{1}$ and that this section still satisfies the condition
$\textrm{Tr}_{p_{1}}(N\sigma)=0$. Since periods have no monodromy along the
paths $\delta_{i}$, by analytic continuation as above we obtain

$\ell(b_{i}):=c_{\delta_{i}}(\ell)=N\cdot\log_{\sigma}(b_{i})-\omega,$

Therefore, we get

$\ell(b_{1})+\cdots+\ell(b_{N})=N\cdot[\log_{\sigma}(b_{1})+\cdots+\log_{\sigma}(b_{N})-\omega]=0.$

∎

In view of 2.2, it is enough to prove 1.9 for a multiple of the section with a
suitable fixed branch of its elliptic logarithm. Thus, from now on we replace
our section by a suitable multiple but we keep the notation $\sigma$.
Moreover, we fix the branch $\ell$ at $b_{1}$ which satisfies Lemma 3.4. In
other words, we can assume that the section $\sigma$ admits a branch $\ell$ of
logarithm which satisfies

$\ell(b_{1})+\cdots+\ell(b_{N})=0.$ (18)

Generally, analytic continuation of $\ell$ along a loop $\zeta$ gives us a new
branch of logarithm, which differs from the first one by a period. In other
words we have

$c_{\zeta}(\ell)=\ell(b_{1})+\omega_{\zeta},$

where $\omega_{\zeta}$ is a period which may also be zero. In order to obtain
our explicit loop we need some control on the variation $\omega_{\zeta}$ for
some loop $\zeta$. By 1.4, for some loop $\zeta$ (with base point $b_{1}$) we
can assume $\omega_{\zeta}\neq 0$. Notice that if this loop leaves unchanged
the periods, we just have the conclusion of our theorem. Anyway, this would
not be enough to provide effectivity nor to allow the proof to work. In order
to really obtain effectivity, observe that the loop $\zeta$ can be chosen into
a set of finitely many loops, i.e. the generators of the fundamental group
$\pi_{1}(B,b_{1})$ where $B$ is a genus $g$ Riemann surface minus the finite
set of points $p^{-1}(\mathcal{R})$. From now on, we denote by $\alpha$ a
generator of $\pi_{1}(B,b_{1})$ such that

$c_{\alpha}(\ell)=\ell(b_{1})+\omega_{\alpha},$ (19)

where $\omega_{\alpha}$ is a non-zero period. Now we introduce some auxiliary
loops:

  1. 1.

let us denote by $\alpha^{\prime}$ the loop in $B^{\prime}-A$ with base point
$b^{\prime}$ defined by $\alpha^{\prime}:=p_{1}(\alpha)$ and by $\mathfrak{a}$
the loop in $S-R$ with base point $s$ obtained as the projection $p(\alpha)$;

  2. 2.

since $p_{1}:B\rightarrow B^{\prime}$ is a Galois morphism and since one of
the liftings of $\alpha^{\prime}$ via $p_{1}$ is the closed path $\alpha$,
then all the liftings of $\alpha^{\prime}$ via $p_{1}$ are closed paths. In
other words, $\alpha^{\prime}$ can be lifted by $p_{1}$ to a closed path
$\alpha_{i}$ in $B-p_{1}^{-1}(A)$ with base point $b_{i}$ for each
$i=1,\ldots,N$; with this notation we have $\alpha_{1}=\alpha$. Observe that
$p(\alpha_{i})=\mathfrak{a}$ for each $i$.

By considering analytic continuation along $\alpha_{i}$ we obtain

$c_{\alpha_{i}}(\ell)=\ell(b_{i})+\omega_{\alpha_{i}},$

where $\omega_{\alpha_{i}}$ is a period which may also be zero. Thanks to our
previous considerations we have some control on the variation
$\omega_{\alpha_{1}}$, but we need something more. To this end, the next lemma
provides enough control on the variations $\omega_{\alpha_{i}}$ too.

######  Lemma 3.5.

There exists an index $i\in\\{1,\ldots,N\\}$ such that

$\omega_{\alpha_{i}}\neq\omega_{\alpha_{1}}.$

###### Proof.

Recall that $\alpha_{i}$ are liftings of the same loop $\alpha^{\prime}$.
Since $\textrm{Tr}(\sigma)(\alpha^{\prime}(t))\equiv 0$ for each $t$, by
evaluating $\ell$ along the loops $\alpha_{i}$, we have

$\sum_{i}\ell(\alpha_{i}(t))=\omega(t)\textrm{ for each }t\in I,$

where $\omega(t):=n(t)\omega_{1}(t)+m(t)\omega_{2}(t)$ is a period function
with $n(t),m(t)\in\mathbb{Z}$; $\omega_{1}(t),\omega_{2}(t)$ is as usual a
holomorphic basis for period lattices in the corresponding points. Since the
map $t\mapsto(n(t),m(t))$ is continuous and $\mathbb{Z}^{2}$ is discrete we
obtain that $(n(t),m(t))$ is constant with respect to $t$. Since
$(n(0),m(0))=(0,0)$ by $(\ref{conditionLog})$, we deduce that

$\omega(t)\equiv 0\textrm{ for each }t\in I.$

Since $\ell(\alpha_{i}(1))=\ell(\alpha_{i}(0))+\omega_{\alpha_{i}}$, we obtain

$\sum_{i}\ell(b_{i})+\sum_{i}\omega_{\alpha_{i}}=\sum_{i}\omega_{\alpha_{i}}=0.$

The conclusion follows by the conditions $\omega_{\alpha_{1}}\neq 0$ and
$\sum_{i}\omega_{\alpha_{i}}=0$.
∎

###  3.4 Monodromy action and analytic continuation

Appealing to Equation 3, we define period functions $\omega_{1},\omega_{2}$ in
a connected and simply connected neighborhood $U$ of $b\in B-p^{-1}(R)$ by

$\omega_{1}=\omega_{1}^{\mathcal{L}}\circ
p,\qquad\omega_{2}=\omega_{2}^{\mathcal{L}}\circ p.$

This argument can be retraced on an open covering of any path in
$B-p^{-1}(R)$: this means that the monodromy of the periods of
$\mathcal{E}\rightarrow B$ is non trivial only along the paths whose
projection via $p:B\rightarrow S$ is a loop in $S$ which is not homotopically
trivial. Hence, by Equation 3 and Lemma 3.3 we get

$\omega(b_{i}):=c_{\delta_{i}}(\omega)=\omega(b_{1})$

for each period function $\omega$ locally defined on $B$, and

$c_{\alpha_{i}}(\omega_{1})=c_{\mathfrak{a}}(\omega_{1}^{\mathcal{L}}),\qquad
c_{\alpha_{i}}(\omega_{2})=c_{\mathfrak{a}}(\omega_{2}^{\mathcal{L}}).$

In other words, the monodromy of $\omega_{1},\omega_{2}$ along a path in $B$
is the same as the monodromy of
$\omega_{1}^{\mathcal{L}},\omega_{2}^{\mathcal{L}}$ along the projection of
the path via the morphism $p:B\rightarrow S$.

Looking at Equation 15 and Lemma 3.5, let us fix an index $i$ which satisfies
$\omega_{\alpha_{i}}\neq\omega_{\alpha_{1}}$ and consider the path
$\delta_{i}$. Define the loop

$\Gamma:=\alpha_{1}\delta_{i}\alpha_{i}^{-1}\delta_{i}^{-1}.$ (20)

The monodromy of any period along $\Gamma$ is determined by our previous
considerations. In fact, for any period function $\omega$ we have

$c_{\Gamma}(\omega)=c_{\delta_{i}^{-1}}c_{\alpha_{i}^{-1}}c_{\delta_{i}}c_{\alpha_{1}}(\omega)=c_{\alpha_{1}^{-1}}c_{\alpha_{1}}(\omega)=\omega(b_{1}).$

So periods don’t change value in $b_{1}$ along $\Gamma$. Now, we want to study
the analytic continuation of logarithm along $\Gamma$.

######  Lemma 3.6.

We have the following relation:

$c_{\alpha_{i}^{-1}}(\log_{\sigma})=\log_{\sigma}(b_{i})-c_{\alpha_{i}^{-1}}(\omega_{\alpha_{i}}).$

###### Proof.

By linearity of $c_{\alpha}$ we have

$\displaystyle\log_{\sigma}(b_{i})$
$\displaystyle=c_{\alpha_{i}^{-1}}c_{\alpha_{i}}(\log_{\sigma})=c_{\alpha_{i}^{-1}}(\log_{\sigma}+\omega_{\alpha_{i}})=$
$\displaystyle=c_{\alpha_{i}^{-1}}(\log_{\sigma})+c_{\alpha_{i}^{-1}}(\omega_{\alpha_{i}}).$

∎

Therefore, we finally get

$\displaystyle\log_{\sigma}(b_{1})$
$\displaystyle\xrightarrow{\alpha_{1}}\log_{\sigma}(b_{1})+\omega_{\alpha_{1}}\xrightarrow{\delta_{i}}\log_{\sigma}(b_{i})+\omega_{\alpha_{1}}$
(21)
$\displaystyle\xrightarrow{\alpha_{i}^{-1}}\log_{\sigma}(b_{i})-c_{\alpha_{i}^{-1}}(\omega_{\alpha_{i}})+c_{\alpha_{i}^{-1}}(\omega_{\alpha_{1}})$
$\displaystyle\xrightarrow{\delta_{i}^{-1}}\log_{\sigma}(b_{1})+c_{\alpha_{i}^{-1}}(\omega_{\alpha_{1}}-\omega_{\alpha_{i}}).$

Thus, the value of the logarithm in $b_{1}$ has changed by a period
$c_{\alpha_{i}^{-1}}(\omega_{\alpha_{1}}-\omega_{\alpha_{i}})$. Since we are
supposing that $\omega_{\alpha_{1}}-\omega_{\alpha_{i}}\neq 0$, then the above
period is not zero: recall that $c_{\alpha_{i}}$, and so
$c_{\alpha_{i}^{-1}}$, is injective. In conclusion, we have explicitly
determined a loop $\Gamma$ which establishes

$M_{\sigma}^{\textnormal{rel}}\neq\\{0\\}.$

It’s a purely algebraic matter to prove that
$M_{\sigma}^{\textnormal{rel}}\cong\mathbb{Z}^{2}$: it follows by the
irreducibility of the action of $\rho(G)$ on the lattice of periods (see [7]
for details).

######  Remark 3.7.

Note that the proof is really effective: we obtained a way of determining
explicitly in finite time a loop $\Gamma$ with the properties we need. In
fact, each path appearing in Equation 20 can be found in a finite number of
attempts: $\alpha_{1}$ is a representative of one of the generators of
$\pi_{1}(B,b_{1})$; $\alpha_{i}$ is one of the finitely many liftings of
$\alpha^{\prime}$; $\delta_{i}$ is determined by the monodromy action of
$\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})$ on the fibre
$p_{1}^{-1}(b^{\prime})$ and by the ramification of the section.

##  4 A variation in the approach: Abhyankar’s Lemma and Belyi pairs

We propose a second approach to the problem with the aim of simplifying the
shape of the loop $\Gamma$. This approach works by adding a hypothesis on the
ramification locus of the section but it has the advantage to give a more
immediate way of constructing the paths $\delta_{i}$ considered above. Thus,
the theorem we want to prove is the following:

######  Theorem 4.1.

Let $\sigma:B\rightarrow\mathcal{E}$ be a non-torsion algebraic section of
$\mathcal{L}\rightarrow S$ and suppose that the ramification locus
$\mathcal{R}$ of $\sigma$ does not contain the whole set $\\{0,1,\infty\\}$.
Then the loop $\Gamma$ constructed in Equation 20 can be chosen of the type

$\Gamma:=\alpha_{1}\delta_{i}\alpha_{i}^{-1}\delta_{i}^{-1},$

where the homotopy class of $\alpha_{1}$ is one of the generators of
$\pi_{1}(B,b)$ and $\delta_{i}\in p_{*}^{-1}(D^{(0)})$; the group $D^{(0)}$ is
defined in Equation 12.

The hypothesis on the ramification locus $\mathcal{R}$ means that we are again
in the situation of Equation 13 but this time the intersection between the
branch locus of $p$ and the set $\\{0,1,\infty\\}$ contains at most two
points. Up to an isomorphism, we can suppose that these (at most) two points
are chosen between $\\{0,\infty\\}$. Thus, the hypothesis can be reformulated
by saying that the section is not ramified at $1$.

Thus, we assume that the section is not ramified at $1$ for the rest of the
proof and keep all the notations introduced above.

##### Plan of the proof.

The steps of the proof follow the same philosophy as before but the control on
the ramification locus of the section is reached by different techniques: in
this case we get a complete ‘separation of the branch locus’. We are exposing
the proof in two steps:

  1. 1.

Separation of the branch locus of sections: we reduce the proof of the theorem
to the case of an elliptic scheme whose morphism $p:B\rightarrow S$ admits a
factorization which allows us to have a powerful control on the ramification
of the section. Precisely, Lemma 4.3 takes the place of Lemma 3.2: this is the
essential step that allows us to carry out the construction of the loop
$\Gamma$. We will make use of the Abhyankar’s lemma, which we state as in [19,
Theorem 12.4.4]:

######  Theorem 4.2.

(Abhyankar’s Lemma) Let $L/K$ be a finite separable extension of function
fields. Suppose that $L=K_{1}K_{2}$ with $K\subseteq K_{i}\subseteq L$. Let
$\mathfrak{p}$ be a prime divisor of $K$ and $\mathfrak{P}$ a prime divisor in
$L$ above $\mathfrak{p}$. Let $\mathfrak{P}_{i}=\mathfrak{P}\cap K_{i}$ for
$i=1,2$. If at least one of the extensions $K_{i}/K$, $i=1,2$, is tamely
ramified at $\mathfrak{p}$, then

$e_{L/K}(\mathfrak{P}|\mathfrak{p})=\textrm{lcm}\\{e_{K_{1}/K}(\mathfrak{P}_{1}|\mathfrak{p}),e_{K_{2}/K}(\mathfrak{P}_{2}|\mathfrak{p})\\},$

where $e$ denotes the ramification index.

  2. 2.

Explicit paths: the remaining steps are analogous to the previous proof. We
reach the construction of $\Gamma$ by using the previous step, Mordell-Weil
theorem for elliptic schemes and Shioda’s theorem. Studying the analytic
continuation of periods and logarithm along $\Gamma$ we get the proof.

###  4.1 Separation of the branch locus of sections

Recall we are considering a section which is not ramified at $1$. We are
denoting the branch locus of $p:\mathcal{B}\rightarrow\mathbb{P}_{1}$ by
$\mathcal{R}$ and the branch locus of $p:B\rightarrow S$ by
$R:=\\{r_{1},\ldots,r_{k}\\}$. Moreover, by 1.3 we have $R\neq\emptyset$. The
first step of our proof consists in reducing the problem to a case in which
there exists a factorization of the morphism
$p:\mathcal{B}\rightarrow\mathbb{P}_{1}$ which “separates the branch locus
$\mathcal{R}$”. The formal meaning of this sentence is expressed by the
following lemma:

######  Lemma 4.3.

Given the ramified Galois morphism $p:\mathcal{B}\rightarrow\mathbb{P}_{1}$,
there exist smooth curves $\mathcal{B}^{\prime},\widetilde{\mathcal{B}}$ and a
finite morphism $\psi:\widetilde{\mathcal{B}}\rightarrow\mathcal{B}$ such that
the map $\widetilde{p}:=p\circ\psi$ factorizes in the following way

${\widetilde{\mathcal{B}}}$${\mathcal{B}^{\prime}}$${\mathbb{P}_{1},}$$\scriptstyle{\phi}$$\scriptstyle{\widetilde{p}:=p\circ\psi}$$\scriptstyle{q}$

where

  1. $(i)$

$p$ and $\widetilde{p}$ are ramified Galois morphisms and have the same branch
locus $\mathcal{R}$;

  2. $(ii)$

$q$ is a ramified Galois morphism with branch locus contained into the set
$\\{0,\infty\\}$;

  3. $(iii)$

$\phi$ is a ramified Galois morphism with branch locus $A$ satisfying
$q(A)=R$.

###### Proof.

(1) Construction of $\widetilde{\mathcal{B}}$: Since we are in characteristic
$0$, by the primitive element theorem the extension
$\mathbb{C}(\mathbb{P}_{1})\subset\mathbb{C}(\mathcal{B})$ is simple. So there
exists an algebraic function $f$ such that

$\mathbb{C}(\mathcal{B})=\mathbb{C}(\lambda)(f),$

where $\mathbb{C}(\mathbb{P}_{1})=\mathbb{C}(\lambda)$. Thus there exists an
irreducible polynomial $P(\lambda,w)\in\mathbb{C}(\lambda)[w]$ such that
$P(\lambda,f)=0$.

Since $p:\mathcal{B}\rightarrow\mathbb{P}_{1}$ is a ramified cover between
Riemann surfaces, we can consider a neighbourhood $U$ of $0$ such that
$p^{-1}(U)=\cup_{i}V_{i}$ where the open sets $V_{i}^{\prime}s$ are disjoint
and biholomorphic to a disk $D$. For each $i$ we have a biholomorphic map
$\varphi_{i}:D\rightarrow V_{i}$ such that

$p\circ\varphi_{i}(\zeta_{i})=\zeta_{i}^{n_{i}}.$

We now focus on the local ring $\mathcal{O}_{0}$ of $0$. Embed
$\mathcal{O}_{0}$ in its completion $\hat{\mathcal{O}}_{0}$, which can be
identified with the ring of formal power series $\mathbb{C}[[\lambda]]$ since
$\mathbb{P}_{1}$ is a smooth curve. By looking for the solutions of
$P(\lambda,f)=0$ in an algebraic extension of $\hat{\mathcal{O}}_{0}$, for any
$i$ we obtain a Puiseux series of the form

$\sum_{\nu=k_{i}}^{\infty}c_{\nu,i}\lambda^{\frac{\nu}{n_{i}}}.$

that solves the polynomial equation. Let us consider the integer

$n:=\textrm{lcm}\\{n_{i}:i=1,\ldots,r\\}.$

Repeating the same reasoning in a neighbourhood of $\infty$, we obtain
biholomorphic maps which can be expressed as $\xi\mapsto\xi^{m_{i}}$. By
looking at the local ring $\mathcal{O}_{\infty}$, we obtain the corresponding
Puiseux series of the form

$\sum_{\nu}^{\infty}d_{\nu,i}\left(\frac{1}{\lambda}\right)^{\frac{\nu}{m_{i}}}.$

Let us consider

$m:=\textrm{lcm}\\{m_{i}:i=1,\ldots,s\\}.$

Now, let us consider the field extension

$\mathbb{C}(\lambda)\subset\mathbb{C}(\lambda)(\psi),\qquad\textrm{where
}\psi^{nm}=\lambda.$

This field extension gives us a morphism
$q:\mathcal{B}^{\prime}\rightarrow\mathbb{P}_{1}$ between smooth curves. We
can distinguish two cases:

  1. $\bullet$

if $n=m=1$, then we have
$q=\textrm{id}:\mathbb{P}_{1}\rightarrow\mathbb{P}_{1}$;

  2. $\bullet$

if $nm>1$, then $q:\mathcal{B}^{\prime}\rightarrow\mathbb{P}_{1}$ is a Galois
morphism with branch locus $\\{0,\infty\\}$.

In every case, passing to the compositum
$\mathbb{C}(\mathcal{B})\mathbb{C}(\mathcal{B}^{\prime})$ we obtain a smooth
curve $\widetilde{\mathcal{B}}$ endowed with finite Galois morphisms

$\psi:\widetilde{\mathcal{B}}\rightarrow\mathcal{B},\quad\phi:\widetilde{\mathcal{B}}\rightarrow\mathcal{B}^{\prime},\quad\widetilde{p}:\widetilde{\mathcal{B}}\rightarrow\mathbb{P}_{1}$

which make the following diagram commutative:

${\widetilde{\mathcal{B}}}$${\mathcal{B}^{\prime}}$${\mathcal{B}}$${\mathbb{P}_{1}.}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\scriptstyle{\widetilde{p}}$$\scriptstyle{q}$$\scriptstyle{p}$

(2) Distribution of ramification points: It remains to be shown that parts
$(i)$ and $(iii)$ of the lemma are true. Let’s start by showing that the
branch locus $A$ of $\phi$ satisfies the condition
$q(A)\cap\\{0,\infty\\}=\emptyset$. In other words, this means that all the
preimages $q^{-1}(0)$ and $q^{-1}(\infty)$ are not contained in $A$.

In what follows we denote by $\widetilde{x},\widetilde{y}$ any preimage of
$0,\infty$ via $\widetilde{p}$, respectively. Let us observe that by the
previous construction, we have the following divisibility relations between
ramification indexes:

$e(\psi(\widetilde{x})|0)\,|\,e(\phi(\widetilde{x})|0),\qquad
e(\psi(\widetilde{y})|\infty)\,|\,e(\phi(\widetilde{y})|\infty).$

We now use Abhyankar’s lemma (4.2) with respect to the fields
$L=\mathbb{C}(\widetilde{\mathcal{B}})$, $K_{1}=\mathbb{C}(\mathcal{B})$,
$K_{2}=\mathbb{C}(\mathcal{B}^{\prime})$, $K=\mathbb{C}(\mathbb{P}_{1})$. If
we take $\mathfrak{P}$ as the maximal ideal of the local ring
$\mathcal{O}_{\widetilde{x}}$, then $\mathfrak{P}_{1},\mathfrak{P}_{2}$ are
identified with the maximal ideals of the local rings
$\mathcal{O}_{\psi\left(\widetilde{x}\right)}$,
$\mathcal{O}_{\phi\left(\widetilde{x}\right)}$ and $\mathfrak{p}$ with the
maximal ideal of the local ring $\mathcal{O}_{0}$. Thus we obtain

$e(\widetilde{x}|0)=\textrm{lcm}\\{e(\psi(\widetilde{x})|0),e(\phi(\widetilde{x})|0)\\}=e(\phi(\widetilde{x})|0).$

If we consider the local ring in $\widetilde{y}$, we analougously obtain

$e(\widetilde{y}|\infty)=\textrm{lcm}\\{e(\psi(\widetilde{y})|\infty),e(\phi(\widetilde{y})|\infty)\\}=e(\phi(\widetilde{y})|\infty).$

Since the ramification index is multiplicative, we obtain:

$e(\widetilde{x}|\phi(\widetilde{x}))=1,\qquad
e(\widetilde{y}|\phi(\widetilde{y}))=1.$

Thus, $\phi$ is not ramified into the points of $\widetilde{p}^{-1}(0)$ and
$\widetilde{p}^{-1}(\infty)$.

Finally, we want to determine the branch locus of $\phi$ in the affine part
$B^{\prime}$ of $\mathcal{B}^{\prime}$. By using the same reasoning as above,
starting with the local rings $\mathcal{O}_{r_{i}}$ where $r_{i}$ are the
branch points in $R$, we obtain that the branch locus $A$ of $\phi$ is such
that $q(A)\cap S=R$. Thus, we just proved that $q(A)=R$. Part $(i)$ follows
directly by Abhyankar’s lemma.

∎

######  Remark 4.4.

Let us consider the morphism
$q:\mathcal{B}^{\prime}\rightarrow\mathbb{P}_{1}$. During the proof, we saw
that when the section is unramified over the set $\\{0,\infty\\}$, then
$\mathcal{B}^{\prime}=\mathbb{P}_{1}$ and
$q=\textrm{id}:\mathbb{P}_{1}\rightarrow\mathbb{P}_{1}$. In the case where the
section is ramified over $0$ or $\infty$, by using Riemann-Hurwitz formula we
obtain again that $\mathcal{B}^{\prime}$ has genus $0$, therefore
$\mathcal{B}^{\prime}\cong\mathbb{P}_{1}$. In particular, in the next sections
we will use the fact that $\mathcal{B}^{\prime}-A\cong\mathbb{P}_{1}-A$, where
$A$ is a finite set of points.

By restricting to $S$, the previous lemma gives us a chain of morphisms
between affine curves

${\widetilde{B}}$${B^{\prime}}$${S,}$$\scriptstyle{\widetilde{p}}$$\scriptstyle{\phi}$$\scriptstyle{q}$
(22)

where

  1. $\bullet$

$B^{\prime}=\mathcal{B}^{\prime}-q^{-1}(\\{0,1,\infty\\})$ and
$\widetilde{B}=\widetilde{\mathcal{B}}-\phi^{-1}(q^{-1}(\\{0,1,\infty\\}))$;

  2. $\bullet$

$q:B^{\prime}\rightarrow S$ is an unramified cover whose extension to
compactifications $q:\mathcal{B}^{\prime}\rightarrow\mathbb{P}_{1}$ has branch
locus either empty or equals to $\\{0,\infty\\}$;

  3. $\bullet$

$\phi:\widetilde{\mathcal{B}}\rightarrow\mathcal{B}^{\prime}$ has branch locus
$A$ such that $q(A)=R$;

  4. $\bullet$

$\widetilde{p}:\widetilde{\mathcal{B}}\rightarrow\mathbb{P}_{1}$ has branch
locus $\mathcal{R}$.

###  4.2 Explicit paths

By Lemma 4.3 we can put in the situation described in Equation 22 with the
just listed properties. We can pull-back the Legendre scheme via these
morphisms and obtain the diagram

${\widetilde{\mathcal{E}}}$${\mathcal{L}^{\prime}}$${\mathcal{L}}$${\widetilde{B}}$${B^{\prime}}$${S.}$

By Lemma 2.1, Lemma 2.4 and Lemma 4.3, we just need to prove 4.1 for
$\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$. From now on we rename
$\widetilde{\mathcal{E}}\rightarrow\widetilde{B}$ with $\mathcal{E}\rightarrow
B$ and we will put ourselves in the situation expressed by the following
diagram

${\mathcal{E}}$${\mathcal{L}^{\prime}}$${\mathcal{L}}$${B}$${B^{\prime}}$${S.}$$\scriptstyle{\phi}$$\scriptstyle{p}$$\scriptstyle{q}$
(23)

and by the properties in Equation 22. Let us fix a base point $s$ in $S-R$ and
let us choose a preimage $b^{\prime}\in q^{-1}(s)$. We will write

$\phi^{-1}(b^{\prime})=\\{b_{1},\ldots,b_{N}\\}.$

Now, we want to determine some paths $\delta_{i}$ which join the points
$b_{i}$ between them and such that their projections $p(\delta_{i})$ are
homotopically trivial loops in $S$. To make this possible, we use the
monodromy action of $\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})$ on the fiber
$\phi^{-1}(b^{\prime})$, where $A$ denotes the branch locus of $\phi$.

######  Lemma 4.5.

For each $i,j\in\\{1,\ldots,N\\}$ there exists a path
$\delta_{i,j}:I\rightarrow B-\phi^{-1}(A)$ such that

  1. 1.

$\delta_{i,j}(0)=b_{i}$ and $\delta_{i,j}(1)=b_{j}$;

  2. 2.

$p(\delta_{i,j})$ is a homotopically trivial loop in $S$.

###### Proof.

Let us consider the unramified cover

$\phi:\mathcal{B}-\phi^{-1}(A)\rightarrow\mathcal{B}^{\prime}-A$

which is a Galois cover. Therefore, if we denote by $\textrm{Bij}(*)$ the set
of all bijections on a set, we have a monodromy representation

$\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})\rightarrow\textrm{Bij}(\phi^{-1}(b^{\prime}))=\textrm{Bij}(\\{b_{1},\ldots,b_{N}\\}),$

where the image is a transitive subgroup of $S_{N}$. In other words this
amounts to saying that for each $i,j$ there exists a loop
$\delta^{\prime}_{i,j}$ in $\mathcal{B}^{\prime}-A$ with base point
$b^{\prime}$ which lifts to a path
$\delta_{i,j}:I\rightarrow\mathcal{B}-\phi^{-1}(A)$ such that
$\delta_{i,j}(0)=b_{i},\delta_{i,j}(1)=b_{j}$.

By 4.4, we have that $\mathcal{B}^{\prime}\cong\mathbb{P}_{1}$. Now we want to
choose some particular loops in $\mathcal{B}^{\prime}-A$ whose homotopy
classes are generators of $\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})$. These
loops are constructed in the following steps:

  1. $\bullet$

fix $r^{\prime}_{i}\in A$ and let
$\mathbb{D}_{i}\subset\mathcal{B}^{\prime}-A$ be an open neighbourhood of
$r^{\prime}_{i}$ isomorphic to a punctured disc; moreover, suppose that
$\mathbb{D}_{i}$ is sufficiently small such that $\mathbb{D}_{i}\cap
q^{-1}\\{0,1,\infty\\}=\emptyset$ and $q(\mathbb{D}_{i})$ is a punctured disc
in $S-R$ centered at a point of $R$. Choose a loop $\xi$ in $\mathbb{D}_{i}$
with winding number one around $r^{\prime}_{i}$ and with a base point which we
denote by $z_{i}\in\mathbb{D}_{i}$. Observe that $q(\xi)$ is a loop contained
in $q(\mathbb{D}_{i})$, thus it is homotopically trivial in
$S=\mathbb{P}_{1}-\\{0,1,\infty\\}$.

  2. $\bullet$

Choose a path $\mu$ from the point $b^{\prime}$ to the point
$z_{i}\in\mathbb{D}_{i}$; this path can be chosen in such a way that it does
not pass through any point of $q^{-1}(\\{0,1,\infty\\})$.

  3. $\bullet$

The path $\mu\xi\mu^{-1}$ is a loop in $\mathcal{B}^{\prime}-A$ based on
$b^{\prime}$: we call such a loop a _small loop on $\mathcal{B}^{\prime}-A$
around $r^{\prime}_{i}$_. Since this loop does not pass through any point of
$q^{-1}(\\{0,1,\infty\\})$, it can be seen as a loop in $B^{\prime}-A$. Since
$q(\xi)$ is homotopically trivial in $S$, then $q(\mu\xi\mu^{-1})$ is also
homotopically trivial in $S$.

Let us consider a small loop around $r^{\prime}_{i}$ for every $i$: they are
loops contained in $B^{\prime}-A$ whose projections via
$q:B^{\prime}\rightarrow S$ are homotopically trivial in $S$; moreover, their
homotopy classes form a set of generators for the fundamental group
$\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})$. This means that the path
$\delta_{i,j}$ can be obtained as a lifting of a composition of small loops
around some points of $A$. Since these loops does not pass through any point
of $q^{-1}(\\{0,1,\infty\\})$, then $\delta_{i,j}$ can be viewed as a path in
$B-\phi^{-1}(A)$. Moreover, by our construction we obtained that
$p(\delta_{i,j})$ is a homotopically trivial loop in $S$.
∎

######  Remark 4.6.

Observe that in Lemma 4.5 we have used in an essential way the construction of
$\mathcal{B}^{\prime}$ and the _‘separation of the branch locus’_.

Fix the paths $\delta_{i-1,i}$ for each $i=2,\ldots,N$ coming from Lemma 4.5
and define the paths $\delta_{i}$ as follows:

$\delta_{i}:=\delta_{1,2}*\cdots*\delta_{i-1,i}.$ (24)

In other words, $\delta_{i}$ is a path joining $b_{1}$ to $b_{i}$ such that
$p(\delta_{i})$ is a homotopically trivial loop in $S$ with base point $s$.

At this points, the proof works exactly as above. For the sake of completeness
we give a brief recap. First of all we can analytically continue
$\log_{\sigma}$ at the $b_{i}$’s as in Equation 16 and we can assume again
Equation 18 to be satisfied. Again, by Mordell-Weil theorem for function
fields we can consider a loop $\alpha$ whose homotopy class is one of the
generators of $\pi_{1}(B,b_{1})$ as in Equation 19 and define the auxiliary
loops $\alpha_{i}$. By Shioda’s theorem, we obtain Lemma 3.5 and define the
loop $\Gamma$ as in Equation 20. With the same calculations as in Equation 21
we get the thesis.

###  4.3 Some comments about the approach and Belyi pairs

Here, we discuss the advantages and limitations of this alternative approach
compared to the first one. In what follows we are going to refer to the form
of the loop $\Gamma$. Notice that the construction of the loops $\alpha_{i}$’s
is the same in both approaches; what is different is the constructions of the
$\delta_{i}$’s. This second approach is advantageous in terms of effectivity:
in fact in Lemma 3.3 we need to construct some loop $\xi$ in $B$ which
annihilates the monodromy of the path $\mu_{i,j}$. In Lemma 4.5, the paths
$\delta_{i,j}$ are provided immediately by the monodromy action of
$\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})$ on the fiber: each lifting which
connects $b_{i}$ to $b_{j}$ is fine, since all these paths have a projection
which is homotopically trivial in $S$.

On the other hand, the main limitation of this approach is clearly the
assumption on the branch locus of sections made in 4.1. Here an example to
underline this limitation showing that the previous proof cannot be extended
without new considerations to the case in which $\sigma$ is ramified at the
whole bad locus $\\{0,1,\infty\\}$.

######  Example 4.7.

Let us consider the algebraic section of $\mathcal{L}\rightarrow S$

$\sigma(\lambda)=\left(x(\lambda),\sqrt{x(\lambda)(x(\lambda)-1)(x(\lambda)-\lambda)}\right),$

where $x(\lambda)$ is defined by

$x(\lambda):=\sqrt{\lambda}+\sqrt{\lambda-1}+\sqrt{\lambda-r},$

for a fixed point $r\in S$.

The ramification locus of this section contains the whole set
$\\{0,1,\infty\\}$. Moreover, the section is also ramified at $r$, so it is a
non-torsion section.444Note that we are not applying 1.3 this time: it regards
the reverse implication. However, what we are saying is also true by general
theory. Let $p:B\rightarrow S$ the base change which makes $\sigma$ to become
well-defined and fix a point $s\in S$ which is not in the branch locus of
$\sigma$. The points $b_{i}$ in the fiber $p^{-1}(s)$ correspond to different
choices of the square roots which appear in the coordinates of $\sigma$.

By using Lemma 4.3, we obtain the diagram

${\widetilde{\mathcal{E}}}$${\mathcal{L}^{\prime}}$${\mathcal{L}}$${\widetilde{B}}$${B^{\prime}}$${S,}$$\scriptstyle{\phi}$$\scriptstyle{\widetilde{p}}$$\scriptstyle{q}$

which satisfies the properties described in Equation 22. We use this
construction because we do not want to consider all the points of the fibers
$p^{-1}(s)$ or $\widetilde{p}^{-1}(s)$: we need paths $\delta_{i}$ which
leaves periods unchanged. Thus, we fix a point $b^{\prime}\in q^{-1}(s)$ and
consider only the points of $\phi^{-1}(b^{\prime})$. Choosing only these
points corresponds to fixing a branch of the square root $\sqrt{\lambda}$ and
some other branch corresponding to the ramification at $\infty$; in fact,
recall that the previous diagram is used to ‘separate the branch locus’ of
$\sigma$ and obtain the morphism $\phi$ which is unramified over the preimages
of $0,\infty$. Unfortunately, our previous method does not remove the
ramification over the whole of $\\{0,1,\infty\\}$: this means that into the
fiber $\phi^{-1}(b^{\prime})$ there are points which correspond to different
choices of the square root $\sqrt{\lambda-1}$. If we consider two points
$b_{i},b_{j}\in\phi^{-1}(b^{\prime})$ which correspond to the same branch of
$\sqrt{\lambda}$ and $\sqrt{\lambda-r}$ but to different branches of
$\sqrt{\lambda-1}$, then for each path $\delta$ connecting $b_{i}$ to $b_{j}$
the loop $\widetilde{p}(\delta)$ is not homotopically trivial in $S$.
Therefore, in this situation our previous method fails to provide paths
$\delta_{i}$ which satisfy Lemma 4.5. In what follows we give a method to deal
with some cases like this.

##### Belyi pairs.

Keep in mind the proof of 4.1 and 4.7. The impossibility to remove the
ramification over the whole of $\\{0,1,\infty\\}$ is due to the fact that we
also want the compactification of $B^{\prime}$ to be $\mathbb{P}_{1}$ (we use
this fact in the construction of $\delta_{i}$). We would be able to remove all
three ramifications, but only allowing the genus of $B^{\prime}$ at least $1$.
Anyway, this attempt also fails: in fact, the growth of the genus of
$B^{\prime}$ gives rise to new generators of the fundamental group
$\pi_{1}(\mathcal{B}^{\prime}-A,b^{\prime})$ which act non-trivially by
monodromy on periods. This naturally leads to ask the following question: _in
which case the approach can be extended to deal with sections which are
ramified into the whole of $\\{0,1,\infty\\}$?_

Here, the idea is to deal with the problem by using the theory of Belyi pairs
and dessin d’enfants (see [11] and [14] for details). By the term _Belyi pair_
we will refer to a pair $(\mathcal{B},f)$ in which $\mathcal{B}$ is a compact
Riemann surface and $f$ is a Belyi function, i.e. a morphism
$f:\mathcal{B}\rightarrow\mathbb{P}_{1}$ with exactly three branching values
which we can assume to be $\\{0,1,\infty\\}$; two Belyi pairs will be
considered equivalent when they are equivalent as ramified coverings. There
exists a bijective map between equivalence classes of Belyi pairs and
equivalence classes of dessins d’enfants: moreover the branching properties of
the Belyi function are determined by the ‘graph theory’ properties of the
corresponding dessin d’enfant.

Now, look at the proof of 4.1. In order to allow the idea of the proof to work
we need a Belyi pair $(\mathbb{P}_{1},q)$ which play the role of the morphism
$q:\mathcal{B}^{\prime}\rightarrow\mathbb{P}_{1}$ in Lemma 4.3. Moreover,
since we need the use of Abhyankar’s Lemma, we want the Belyi function $q$ to
be a Galois morphism. This corresponds to considering _regular dessins_. We
say that a regular dessin is of the type $(l,m,n)$ if the ramification indexes
at preimages of $0,1,\infty$ are $l,m,n$ respectively. By an application of
Riemann-Hurwitz formula it’s possible to determine all the possibilities for
the type $(l,m,n)$. First of all we obtain that in each case at least one of
$l,m$ and $n$ must be equal to $2$; we may assume that $m=2$. Simple
arithmetic shows that the only solutions for the type $(l,m,n)=(l,2,n)$ are
the following: $(3,2,3),(3,2,4),(3,2,5)$ and $(2,2,n)$. These are all the
cases where the proof of 4.1 can be extended to a section which ramifies in
the whole of $\\{0,1,\infty\\}$.

##  5 Further results and applications

Here we end with some final considerations and examples. In particular, we
will consider the case of Masser’s section determining generators of the full
monodromy group $M_{\sigma}^{\textnormal{rel}}$. Thus, we obtain stronger
effective results in this specific case and leave some questions which could
be interesting thinking to some possible applications.

###  5.1 Some final questions

We continue denoting by $\mathfrak{a}_{0}$ and $\mathfrak{a}_{1}$ two fixed
and small enough circles centered at $0$ and $1$, respectively. We consider
again a section which ramifies in the whole of $\\{0,1,\infty\\}$. Since the
section is algebraic, then each branch point has finite branching order.
Hence, by Equation 3 there exist loops $\zeta_{0},\zeta_{1}$ in $B$ with base
point $b_{1}$ such that

$p(\zeta_{0})=\mathfrak{a}_{0}^{z},\quad
p(\zeta_{1})=\mathfrak{a}_{1}^{z},\qquad\textrm{for some }z\in\mathbb{Z}.$
(25)

We want to point out that the proof of 4.1 works if we assume we have some
control on the variation of $\log_{\sigma}$. To this end, suppose that
$\log_{\sigma}$ has non-trivial variation along $\zeta_{0}$ or along
$\zeta_{1}$; without loss of generality we put

$c_{\zeta_{1}}(\log_{\sigma})=\log_{\sigma}(b_{1})+\omega_{\zeta_{1}},\qquad\omega_{\zeta_{1}}\neq
0.$

Then we are assuming $\alpha=\zeta_{1}$ in Equation 19; from now on we write
$\alpha$ in place of $\zeta$. The same proof as before continues to hold with
some additional considerations: the branch locus $A$ of the morphism $\phi$ in
Lemma 4.3 contains some preimage of $1$. Hence, the projections of paths
$\delta_{i}$ defined in Equation 24 are not homotopically trivial but they
satisfy

$p(\delta_{i})=\mathfrak{a}_{1}^{k_{i}},\qquad\textrm{for some
}k_{i}\in\mathbb{Z}.$

By Equation 3 we get

$\omega(b_{i}):=c_{\delta_{i}}(\omega)=c_{\mathfrak{a}^{k_{i}}}(\omega),\qquad
c_{\alpha_{i}}(\omega_{j})=c_{\mathfrak{a}^{z}}(\omega_{j}^{\mathcal{L}})$
(26)

for each period function $\omega$ locally defined on $B$ and $j=1,2$.
Analogously to what has been done in Lemma 3.5, since $\omega_{\alpha_{1}}$ is
non-zero there exists an index $i$ satisfying $\omega_{\alpha_{i}}\neq
c_{\delta_{i}}(\omega_{\alpha_{1}})$. Let’s say some words about this point:
if we assume $\omega_{\alpha_{i}}=c_{\delta_{i}}(\omega_{\alpha_{1}})$ for all
$i$’s, thanks to 1.3 we get $\sum_{i}c_{\delta_{i}}(\omega_{\alpha_{1}})=0$.
Since the monodromy action of each $c_{\delta_{i}}$ is represented by a
multiple of the triangular matrix associated to $\mathfrak{a}_{1}$, then the
action of $\sum_{i}c_{\delta_{i}}$ is represented by an invertible matrix and
the thesis follows. Let us fix an index $i$ which satisfies
$\omega_{\alpha_{i}}\neq c_{\delta_{i}}(\omega_{\alpha_{1}})$. Let’s consider
the path $\delta_{i}$ and define the loop

$\Gamma:=\alpha_{1}\delta_{i}\alpha_{i}^{-1}\delta_{i}^{-1}.$

Figure 1 aims at representing the loop $\Gamma$ when section ramifies at $1$.

$b_{1}$$b_{i}$$\alpha_{1}$$\alpha_{i}$$\delta_{i}$$\bm{\Gamma}$

Figure 1: A visualization of the loop $\Gamma$.

By Equation 26, we get

$c_{\Gamma}(\omega)=\omega(b_{1})$

for any period function $\omega$, i.e. periods don’t change value in $b_{1}$
along $\Gamma$. With the same calculations as in Equation 21 we obtain

$c_{\Gamma}(\log_{\sigma})=\log_{\sigma}(b_{1})+c_{\delta_{i}^{-1}}c_{\alpha_{i}^{-1}}(c_{\delta_{i}}(\omega_{\alpha_{1}})-\omega_{\alpha_{i}}).$

Since the variation is non-trivial, we have obtained the thesis.

Anyway, we have no guarantees that the hypothesis for the variation of
$\log_{\sigma}$ along $\zeta_{0}$ or $\zeta_{1}$ occur. Thus, the previous
argument only apply to some particular cases. However it gives rise to some
questions:

  1. 1.

Are we able to obtain some control on the variation of logarithms along
$\zeta_{0}$ or/and $\zeta_{1}$?

  2. 2.

Are we able to find two loops of the type Equation 20 which give rise to
independent variations of $\log_{\sigma}$?

The first question aims at extending effectivity towards other ends, for
example thinking to some applications in transcendence theory or finding
explicit expressions of the monodromy action of $\pi_{1}(B,b)$ on
determinations of the Betti map. The second question is a further push towards
effectivity: thanks to our previous proofs we are now able to find an explicit
loop which realizes the non-triviality of $M_{\sigma}^{\textnormal{rel}}$ but
we also know that $M_{\sigma}^{\textnormal{rel}}\cong\mathbb{Z}^{2}$. Answer
the second question correspond to finding two loops which generate the full
relative monodromy group (or a full-rank subgroup of it). In the final section
we apply the techniques developed above, giving a concrete example and
answering the previous questions in the particular case of Masser’s section.

###  5.2 The case of Masser’s section

We precise that the following apply to all non-torsion algebraic sections with
constant abscissa. For the sake of simplicity we choose one of them to give a
specific example. Masser’s section is an algebraic section of the Legendre
scheme $\mathcal{L}\rightarrow S$ defined by

$\sigma(\lambda)=\left(2,\sqrt{2(2-\lambda)}\right).$

In order to make it well defined we need the ramified cover $p:B\rightarrow S$
defined by taking the square root of $2-\lambda$. This cover has degree $2$
and is ramified above $\lambda=2$. We consider the elliptic scheme
$\mathcal{E}$ obtained as pull-back of Legendre scheme via $p$: it can be seen
as the elliptic curve over $\mathbb{C}(\psi)$ defined by

$y^{2}=x(x-1)(x+\psi^{2}-2),$

where $\psi^{2}=2-\lambda$. The section $\sigma$ is pulled back to a well
defined (rational) section $\widetilde{\sigma}(\psi)=(2,\sqrt{2}\psi)$, which
is not identically torsion, since it is ramified over $S$. When we refer to
$\log_{\sigma}$ (resp. $\sigma$) on $B$, we mean $\log_{\widetilde{\sigma}}$
(resp. $\widetilde{\sigma}$). Therefore, we have a diagram as in Equation 13.
Moreover, observe that a different choice of the square root would merely
change sign to the section.

Our aim is finding a loop in $B$ where logarithm has a non-trivial variation
but which leaves periods unchanged.

##### Choice of the right loop.

Fix a base point $s$ in $S-\\{2\\}$ and consider the loops $\mathfrak{a}_{0}$
and $\mathfrak{a}_{1}$ with base point $s$. Moreover, according to the
notation used in Equation 12, call $\mathfrak{d}$ a small loop in $S$ around
$2$. The fibre $p^{-1}(s)$ contains two points, say $b_{+},b_{-}$, which
correspond to the two opposite square roots. Let $\delta$ be the lifting (via
$p$) of $\mathfrak{d}$ with starting point $b_{+}$.

Fix a branch $\log_{\sigma}$ of logarithm of $\sigma$ at $b_{+}$ and define
$\log_{\sigma}(b_{-}):=c_{\delta}(\log_{\sigma})$. Since a different choice of
the square root changes sign to $\sigma$, we have

$c_{\delta}(\sigma)=\sigma(b_{-})=-\sigma(b_{+}).$

Hence $\log_{\sigma}(b_{-})$ is a determination of $-\log_{\sigma}(b_{+})$, so
they differ by a period. In other words, we get

$c_{\delta}(\log_{\sigma})=\log_{\sigma}(b_{-})=-\log_{\sigma}(b_{+})+\omega_{\delta},$
(27)

where $\omega_{\delta}$ is a period. In order to simplify the calculations, we
can replace the section $\sigma$ by a section $\widetilde{\sigma}$ so that we
have

$\log_{\widetilde{\sigma}}=\log_{\sigma}-\frac{\omega_{\delta}}{2}.$

Observe that this transformation is made by adding a two-torsion section to
$\sigma$ and since the two-torsion is well-defined over the base $B$ we don’t
need to extend the base. Moreover, note that adding a torsion section does not
change our conclusions: in fact we are interesting in analytic continuations
along loops which leaves unchanged periods. Let’s continue denoting the new
section $\widetilde{\sigma}$ by $\sigma$; thus we can assume
$\omega_{\delta}=0$ in Equation 27, i.e.

$c_{\delta}(\log_{\sigma})=-\log_{\sigma}(b_{+}).$ (28)

Now, let’s consider $\mathfrak{a}_{0}$ and $\mathfrak{a}_{1}$. Put
$\alpha_{+},\alpha_{-}$ to be the liftings of $\mathfrak{a}_{1}$ with base
points $b_{+},b_{-}$, respectively. By 1.4 the variation of logarithm has to
be non-trivial along one of the liftings of $\mathfrak{a}_{0}$ or
$\mathfrak{a}_{1}$. Thus, without loss of generality we can assume

$c_{\alpha_{+}}(\log_{\sigma})=\log_{\sigma}(b_{+})+\omega_{+},$ (29)

where $\omega_{+}$ is a non-zero period. Let’s consider the loop

$\gamma:=\mathfrak{a}_{1}\mathfrak{d}\mathfrak{a}_{1}^{-1}\mathfrak{d}^{-1}$

whose homotopy class in $\pi_{1}(S,s)$ is the identity. We define the loop
$\Gamma$ as the lifting of $\gamma$ via $p$ with base point $b_{+}$, which is
equivalent to putting

$\Gamma=\alpha_{+}\delta\alpha_{-}^{-1}\delta^{-1}.$ (30)

Appealing to the notations of 2.3, note that $\Gamma\in p_{*}^{-1}(D^{(1)})$.
The situation is clarified in Figure 2.

$2$$1$$0$$s$$b+$$b-$$O$$\mathfrak{a}_{1}$$\mathfrak{d}$$\alpha_{+}$$\alpha_{-}$$\delta$

S

B

Figure 2: A visualization of the covering map $B\rightarrow S$ and the
relevant loops.

##### Monodromy of periods and logarithm along the loop.

Let us study the monodromy of periods along $\Gamma$. As usual, we define a
basis of periods by the equations

$\omega_{1}=\omega_{1}^{\mathcal{L}}\circ
p,\qquad\omega_{2}=\omega_{2}^{\mathcal{L}}\circ p.$

Hence, we get

$c_{\delta}(\omega_{i})=c_{\mathfrak{d}}(\omega_{i}^{\mathcal{L}})=\omega_{i}(b_{+})\qquad\textrm{for
}i=1,2.$

In other words, this means that $\omega_{i}(b_{+})=\omega_{i}(b_{-})$.
Analogously, we obtain

$c_{\alpha_{+}}(\omega_{i})=c_{\alpha_{-}}(\omega_{i})=c_{\mathfrak{a}_{1}}(\omega_{i}^{\mathcal{L}}).$

Therefore, for $i=1,2$ we have

$c_{\Gamma}(\omega_{i})=c_{\delta^{-1}}c_{\alpha_{-}^{-1}}c_{\delta}c_{\alpha_{+}}(\omega_{i})=c_{\alpha_{+}^{-1}}c_{\alpha_{+}}(\omega_{i})=\omega_{i}(b_{+}).$

In other words, periods have no monodromy along $\Gamma$. Now, we want to
study the analytic continuation of logarithm along $\Gamma$. Let us consider
the loops $\alpha_{+},\alpha_{-}:I\rightarrow B$. Since they are liftings of
the same loop $\mathfrak{a}_{1}$ and the section changes sign when changing
branch of the square root, we get

$\sigma(\alpha_{+}(t))+\sigma(\alpha_{-}(t))=0\quad\textrm{for each }t\in I.$

In terms of logarithms, this means that

$\log_{\sigma}(\alpha_{+}(t))+\log_{\sigma}(\alpha_{-}(t))=\omega(t),$

where $\omega$ is a period function. By Equation 28, we get $\omega(0)=0$.
Since $\omega(t)$ is an integral combination of a basis of periods, it follows
that $\omega(t)=0$ for each $t\in I$. This implies

$\log_{\sigma}(\alpha_{-}(t))=-\log_{\sigma}(\alpha_{+}(t))\quad\textrm{for
each }t\in I.$

In particular this means that the analytic continuation of $\log_{\sigma}$
along $\alpha_{-}$ is the same as the analytic continuation of
$-\log_{\sigma}$ along $\alpha_{+}$, i.e.

$c_{\alpha_{-}^{-1}}(\log_{\sigma})=-c_{\alpha_{+}^{-1}}(\log_{\sigma}).$

Moreover, by Equation 29 and Lemma 3.6 we get

$c_{\alpha_{+}^{-1}}(\log_{\sigma})=\log_{\sigma}(b_{+})-c_{\alpha_{+}^{-1}}(\omega_{+}).$

Thus, appealing to all previous considerations we obtain

$\displaystyle c_{\delta}c_{\alpha_{+}}(\log_{\sigma})$
$\displaystyle=c_{\delta}(\log_{\sigma}+\omega_{+})=c_{\delta}(\log_{\sigma})+c_{\delta}(\omega_{+})=$
$\displaystyle=\log_{\sigma}(b_{-})+\omega_{+}(b_{+}).$

Using again the previous relations we get

$\displaystyle c_{\alpha_{-}^{-1}}c_{\delta}c_{\alpha_{+}}(\log_{\sigma})$
$\displaystyle=c_{\alpha_{-}^{-1}}(\log_{\sigma})+c_{\alpha_{-}^{-1}}(\omega_{+})=$
$\displaystyle=-c_{\alpha_{+}^{-1}}(\log_{\sigma})+c_{\alpha_{+}^{-1}}(\omega_{+})=$
$\displaystyle=-\log_{\sigma}(b_{+})+2c_{\alpha_{+}^{-1}}(\omega_{+})=$
$\displaystyle=\log_{\sigma}(b_{-})+2c_{\alpha_{+}^{-1}}(\omega_{+}).$

Finally we have

$\displaystyle c_{\Gamma}(\log_{\sigma})$
$\displaystyle=c_{\delta^{-1}}c_{\alpha_{-}^{-1}}c_{\delta}c_{\alpha_{+}}(\log_{\sigma})=c_{\delta^{-1}}(\log_{\sigma})+2c_{\alpha_{+}^{-1}}(\omega_{+})=$
(31) $\displaystyle=\log_{\sigma}(b_{+})+2c_{\alpha_{+}^{-1}}(\omega_{+}).$

Therefore, the value of the logarithm in $b_{+}$ has changed by a period
$2c_{\beta_{+}^{-1}}(\bar{\omega}_{+})$ which is non-zero by Equation 29. The
loop $\Gamma$ provides the non-triviality of the group
$M_{\sigma}^{\textnormal{rel}}$.

To be more precise, we have just proved the existence of a loop
$\Gamma_{1}:=\Gamma$ which generates a non-trivial variation of logarithm of
Masser’s section leaving periods of the corresponding scheme unchanged.
Analogously to Equation 30, we can define a loop $\Gamma_{0}$ by replacing
$\alpha_{+}$ and $\alpha_{-}$ with liftings of $\mathfrak{a}_{0}$. Since
$M_{\sigma}^{\textnormal{rel}}\cong\mathbb{Z}^{2}$ and by 2.3 $\log_{\sigma}$
must have non-trivial variation along $\Gamma_{0}$ too. Hence $\Gamma_{0}$ and
$\Gamma_{1}$ are the loops which generate $M_{\sigma}^{\textnormal{rel}}$.
Thus, in this case we are able to exhibit explicit generators of the full
relative monodromy group.

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F. Tropeano, Università della Calabria, Italy

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