1 Introduction
In the last few years Betti maps associated to sections of abelian schemes have been extensively studied and applied to problems of diophantine nature. In this article, we present an apparently new approach, already sketched by the third author in [Reference Zannier34], to study Betti maps in a special case, which is strictly related to the polynomial Pell equation. In particular, we explain how some properties of certain ramified covers of the projective line with prescribed ramification and arising from such equations can be used to prove a special case of a result by André, Corvaja and the third named author [Reference André, Corvaja and Zannier2] on the distribution of the rational values of the Betti map given by a particular section of the family of Jacobians of hyperelliptic curves. In this special case, our approach actually gives something more precise than the result in [Reference André, Corvaja and Zannier2] and allows in principle to obtain a description of the preimage of the set of rational points with fixed denominator; see Corollary 1.6 and the subsequent discussion.
Given an abelian scheme
$\mathcal {A}\rightarrow S$
of relative dimension
$g\geq 1$
over a smooth complex algebraic variety S and a section
$\sigma :S\rightarrow \mathcal {A}$
, a Betti map for
$\mathcal {A}$
and
$\sigma $
is a real analytic map
$\tilde {\beta }: \tilde {S}\rightarrow \mathbb {R}^{2g}$
, where
$\tilde {S}$
is the universal covering of
$S(\mathbb {C})$
. This map is of particular relevance in many different contexts; for example, rational points in the image of
$\tilde {\beta } $
correspond to torsion values for the section, thus linking the Betti map to other diophantine problems. For an account about the use and the study of Betti maps, see [Reference André, Corvaja and Zannier2, Section 1].
The rank
$\operatorname {{\mathrm {rk}}} \tilde {\beta }$
of the Betti map, namely the maximal value of the rank of the derivative
$d\tilde {\beta }(\tilde {s})$
when
$\tilde {s}$
runs through
$\tilde {S}$
, is of particular interest. Indeed, if
$\dim S\geq g$
, its maximality is equivalent to the fact that the image of the map contains a dense open subset of
$\mathbb {R}^{2g}$
and implies that the preimage of the set of torsion points of
$\mathcal {A}$
via
$\sigma $
is dense in
$S(\mathbb {C})$
; see [Reference André, Corvaja and Zannier2, 2.1.1 Proposition].
In [Reference Corvaja, Masser and Zannier11], the authors studied the rank of Betti maps associated to abelian surface schemes in the context of a relative Manin–Mumford problem. This work initiated more general investigations that led to [Reference André, Corvaja and Zannier2], in which the authors conjectured a sufficient condition for the maximality of
$\operatorname {{\mathrm {rk}}}\tilde {\beta }$
, proving it under some quite general and natural hypotheses (see also [Reference Gao16] for further results on this topic).
In particular, in [Reference André, Corvaja and Zannier2], the authors handled the case of a specific nontorsion section of the Jacobian of the universal hyperelliptic curve of genus
$g>0$
. This case is relevant by itself, but it is also linked to an issue raised by Serre [Reference Serre30], who noticed a gap in an article by Robinson [Reference Robinson27], in which the family of hyperelliptic curves over the real numbers appears in connection with the Pell equation in polynomials. In [Reference André, Corvaja and Zannier2, Section 9], the authors give an argument fixing this gap using Betti maps; see also [Reference Bogatyrev7, Reference Peherstorfer25, Reference Totik31, Reference Lawrence18] for independent proofs of the same result.
In the first part of this paper, we give an alternative proof of this result in the complex case for a particular but significant section. We obtain this as a consequence of the characterization of the dimension of particular subvarieties arising from solvable Pell equations in the ‘moduli space’ of polynomials of even degree
$\geq 4$
.
Let
$\mathcal {B}_{2d}$
be the Zariski-open subset of
$\mathbb {A}^{2d}_{\mathbb {C}}$
defined by
$$ \begin{align} \mathcal{B}_{2d}:=\{ (s_1,\dots, s_{2d} ) \in \mathbb{A}^{2d}_{\mathbb{C}}: \text{ the discriminant of } t^{2d}+s_{1}t^{2d-1}+\dots +s_{2d} \text{ is nonzero} \}. \end{align} $$
Given a point
$\overline {s}=(s_1,\dots , s_{2d} ) \in \mathcal {B}_{2d}$
, we consider the affine hyperelliptic curve defined by
$y^2=t^{2d}+s_{1}t^{2d-1}+\dots +s_{2d} $
. If we homogenize this equation, we obtain a projective curve which is singular at infinity. There exists, however, a nonsingular model
$H_{\overline {s}}$
with two points at infinity which we denote by
$\infty ^+$
and
$\infty ^-$
. We fix them by stipulating that the function
$t^d\pm y$
has a zero at
$\infty ^{\pm }$
. The curve
$H_{\overline {s}}$
is then a hyperelliptic curve of genus
$d-1$
.
We denote by
$\mathcal {J}_{\overline {s}}$
the Jacobian variety of
$H_{\overline {s}}$
; then, letting
$\overline {s}$
vary in
$\mathcal {B}_{2d}$
, we have an abelian scheme
$\mathcal {J}\rightarrow S:=\mathcal {B}_{2d}$
of relative dimension
$g:=d-1$
. We call
$\sigma :S\rightarrow \mathcal {J}$
the section corresponding to the point
$[\infty ^+-\infty ^-]$
.
Let us give an informal definition of the Betti map associated to
$\sigma $
; for a precise definition we refer to [Reference André, Corvaja and Zannier2, Section 3].
In the constant case (that is, when
$\mathcal {J}$
is a complex abelian variety), an abelian logarithm of a point of
$\mathcal {J}(\mathbb {C})$
can be expressed as an
$\mathbb {R}$
-linear combination of the elements of a basis of the period lattice. These real coordinates are called Betti coordinates. Note that they depend on the choice of the abelian logarithm and of the basis of the period lattice.
In the relative setting, the Betti map describes the variation of the Betti coordinates; locally (in the complex topology) on
$S(\mathbb {C})$
, the Lie algebra
$\text {Lie}(\mathcal {J})$
is the trivial vector bundle of rank g. Moreover, the kernel of the exponential map is a locally constant sheaf on
$S(\mathbb {C})$
. One can then locally define a real analytic function that associates a point
$\overline {s}$
of
$S(\mathbb {C})$
to the Betti coordinates of
$\sigma (\overline {s})$
.
Betti maps are defined for general abelian schemes, and one of the easiest cases which has been deeply studied in literature is that of the Legendre scheme
$\mathcal {L}$
. In this case, the base S is the curve
$\mathbb {P}^1 \setminus \{0,1,\infty \}$
and
$\mathcal {L}\rightarrow \mathbb {P}^1$
is the abelian scheme having fibers in
$\mathbb {P}^2$
defined by
$$\begin{align*}ZY^2=X(X-Z)(X-\lambda Z), \end{align*}$$
for every
$\lambda \in \mathbb {P}^1 \setminus \{0,1,\infty \}$
. Consequently,
$\mathcal {L}$
is embedded in
$\mathbb {P}^1\setminus \{0,1,\infty \}\times \mathbb {P}^2$
. Locally, one can define a basis of periods using the hypergeometric functions. Indeed, for example, for
$\lambda $
in the region
$D=\{|\lambda |<1 \mbox { and } |1-\lambda |<1\}\subseteq S$
, a suitable basis of periods is given by
$f(\lambda ):=\pi F(\lambda )$
and
$g(\lambda ):=\pi iF(1-\lambda )$
, where
$F(\lambda )=\sum _{m=0}^{\infty } \frac {(m!)^2}{2^{4m}(m!)^4}\lambda ^m$
. Given a section
$\sigma : S \rightarrow \mathcal L$
, locally over D one can take its elliptic logarithm z and write it as
$z=u_1f+u_2g$
, where the functions
$u_i$
are real analytic functions. Hence, the Betti map
$\beta : D \rightarrow \mathbb {R}^2$
associated to
$\sigma $
is given by
$\beta (\lambda )=(u_1(\lambda ), u_2(\lambda ))$
.
In general, this map cannot be extended to the whole of
$S(\mathbb {C})$
because of monodromy, but we can pass to the universal covering
$\tilde {S}$
of
$S(\mathbb {C})$
and define a Betti map
$\tilde {\beta }: \tilde {S}\rightarrow \mathbb {R}^{2(d-1)}$
, which is not unique and depends on several choices. The following is our first result.
Theorem 1.1. Let d be an integer
$\geq 2$
, and let
$\mathcal {J}\rightarrow S$
and
$\sigma :S\rightarrow \mathcal {J}$
be the abelian scheme and the section defined above. Then the corresponding Betti map
$\tilde {\beta }: \tilde {S}\rightarrow \mathbb {R}^{2(d-1)}$
satisfies
$\operatorname {{\mathrm {rk}}} \tilde {\beta }\geq 2(d-1)$
; equivalently,
$\tilde {\beta }$
is submersive on a dense open subset of
$\tilde {S}$
. In particular, the set of
$\overline {s}\in S(\mathbb {C})$
such that
$\sigma (\overline {s})$
is torsion on
$\mathcal {J}_{\overline {s}}$
is dense in
$S(\mathbb {C})$
in the complex topology.
The above theorem is a special case of [Reference André, Corvaja and Zannier2, 2.3.3 Theorem]; however, we are going to give a different proof of it in Section 3. Our proof makes use of the connection between the torsion values of the section
$\sigma : S \rightarrow \mathcal {J}$
corresponding to the point
$[\infty ^+ - \infty ^-]$
and the polynomial Pell equation.
We recall that the classical Pell equation is an equation of the form
$A^2-DB^2=1$
where D is a positive integer to be solved in integers A and B with
$B\neq 0$
. A theorem of Lagrange says that such an equation is nontrivially solvable if and only if D is not a perfect square.
We consider the polynomial analogue of this problem, replacing
$\mathbb {Z}$
by a polynomial ring over a field. This variant is old as well and can be dated back to studies by Abel [Reference Abel1] especially in the context of integration in finite terms of certain algebraic differentials. Lately, these studies have been carried out by several authors (see [Reference Bogatyrev8, Reference Serre30, Reference Masser and Zannier23] for more details). Apart from their link with points of finite order in Jacobians of hyperelliptic curves, the polynomial Pell equation, which from now on we will call a Pell–Abel equation as suggested by Serre in [Reference Serre30], has several connections with other mathematical problems, like polynomial continued fractions [Reference Zannier35] and elliptical billiards [Reference Dragović and Radnovic13, Reference Corvaja and Zannier12] but also problems in mathematical physics [Reference Burskii and Zhedanov9] and dynamical systems [Reference McMullen20].
Let K be a field of characteristic 0. For a nonconstant polynomial
$D(t)\in K[t]$
, we look for solutions of
$$ \begin{align} A(t)^2-D(t)B(t)^2=1, \end{align} $$
where
$A(t),B(t)\in K[t]$
and
$B\neq 0$
; if such a solution exists, we call the polynomial D Pellian. Clearly, necessary conditions for a polynomial D to be Pellian are that D has even degree and it is not a square in
$K[t]$
, but the leading coefficient is a square in K. Unlike the integer case, these conditions are not sufficient to guarantee the existence of a nontrivial solution, and there are examples of non-Pellian polynomials satisfying these conditions (see, for example, [Reference Zannier34]).
If K is algebraically closed (in particular
$K=\mathbb {C}$
) there is a criterion, attributed to Chebyshev in [Reference Berry4], that links the solvability of a Pell–Abel equation to the order of the point
$[\infty ^+-\infty ^-]$
in the Jacobian of the hyperelliptic curve defined by the equation
$y^2=D(t)$
.
Proposition 1.2 (see [Reference Zannier34, Proposition 12.1]).
Let
$D\in K[t]$
be a squarefree polynomial of degree
$2d\geq 4$
. Then the Pell–Abel equation
$A^2-DB^2=1$
has a nontrivial solution
$A,B \in \overline {K}[t]$
with
$B \neq 0$
if and only if the point
$[\infty ^+-\infty ^-]$
has finite order in the Jacobian of the smooth projective model of the affine hyperelliptic curve of equation
$y^2=D(t)$
. Moreover, the order of
$[\infty ^+-\infty ^-]$
is the minimal degree of the polynomial A of a nontrivial solution.
We point out that a similar criterion holds as well in the case of nonsquarefree D, by using generalized Jacobians (for more on this, see [Reference Zannier35]).
As in the integer case, if
$D(t)$
is Pellian, then the associated Pell–Abel equation has infinitely many solutions in
$\overline {K}[t]$
. Indeed, a possible nontrivial solution
$(A,B)$
generates infinitely many ones by taking powers
$A_m+\sqrt {D}B_m:=\pm \left (A\pm \sqrt {D}B\right )^m$
. Moreover, solutions to the Pell–Abel equation correspond to the units of the ring
$\overline {K}[\sqrt {D}]$
which form a group isomorphic to
$\mathbb {Z} \oplus \mathbb {Z}/2\mathbb {Z}$
.
We will call a solution primitive if it has minimal degree among all the nontrivial ones. On the other hand, we say that a solution
$(A_m,B_m)$
is an m-th power when it can be obtained from another one as explained above. Notice that every solution of the Pell–Abel equation will then be a power of a primitive one.
We call degree of a solution
$(A,B)$
the degree of A. Thus, the order of
$[\infty ^+-\infty ^-]$
, if finite, is the degree of a primitive solution of the corresponding Pell–Abel equation.
If we consider the abelian scheme
$\mathcal {J} \rightarrow S=\mathcal {B}_{2d}$
as above and the section
$\sigma : S\rightarrow \mathcal {J}$
corresponding to the point
$[\infty ^+-\infty ^-]$
, we have that the polynomial
$D_{\overline {s}}=t^{2d}+s_{1}t^{2d-1}+\dots +s_{2d}$
is Pellian with primitive solution of degree n if and only if
$\sigma (\overline {s})$
is a torsion point of order n.
We define the Pellian locus
$\mathcal {P}_{2d}$
in
$\mathcal {B}_{2d}$
to be
$$ \begin{align} \mathcal{P}_{2d}:=\{ \overline{s}=(s_1,\dots, s_{2d} ) \in \mathcal{B}_{2d}: \text{ the polynomial }D_{\overline{s}}=t^{2d}+s_{1}t^{2d-1}+\dots +s_{2d} \text{ is Pellian} \}. \end{align} $$
We have the following result.
Theorem 1.3. For
$d\geq 2$
, the set
$\mathcal {P}_{2d}(\mathbb {C})$
of Pellian complex polynomials is dense in
$\mathcal {B}_{2d}(\mathbb {C})$
with respect to the complex topology.
We point out here that the distribution of Pellian polynomials in families, and its connection to problems of unlikely intersections has been investigated (often using Betti maps) in recent years, mainly in the case of curves in
$\mathcal {B}_{2d}$
where the behaviour is completely opposite to what happens in the setting of the above theorem (for a general account on the problems of unlikely intersections, see [Reference Zannier33]). Indeed, for instance, in [Reference Masser and Zannier22] and [Reference Masser and Zannier23], the authors show that a generic curve in
$\mathcal {B}_{2d}$
, where
$d\geq 3$
, contains at most finitely many complex points that correspond to Pellian polynomials. One can see also [Reference Bertrand, Masser, Pillay and Zannier6] and [Reference Schmidt29] for the nonsquarefree case and [Reference Barroero and Capuano3] for similar results for the generalized Pell–Abel equation.
Clearly, Theorem 1.3 is a consequence of Proposition 1.2 and Theorem 1.1 (and thus of [Reference André, Corvaja and Zannier2]); however, in this paper we present a different proof of these two results with the principal aim of showing the link between the Betti maps and some properties of certain ramified covers of the projective line with fixed ramification. Moreover, we are able to study more deeply the Pellian locus
$\mathcal {P}_{2d} \subseteq \mathcal {B}_{2d}$
and to show that it consists of a denumerable union of algebraic subvarieties of
$\mathcal {B}_{2d}$
of dimension at most
$d+1$
(see Proposition 3.1), each one coming from the preimage of a rational point of a Betti map
$\tilde {\beta } $
. If
$\tilde {\beta }$
did not have maximal rank, then these preimages would have dimension strictly larger than
$d+1$
.
To study the Pellian locus
$\mathcal {P}_{2d}$
, we associate to a degree n solution
$(A,B)$
of a Pell–Abel equation the ramified cover of degree
$2n$
given by
$A^2:\mathbb {P}^1(\mathbb {C})\rightarrow \mathbb {P}^1(\mathbb {C})$
. Because of the Pell–Abel equation, the latter has to be ramified above 0, 1 and
$\infty $
and, by the Riemann–Hurwitz formula, at most
$d-1$
additional points, independently of n. For this reason, we call these maps, following the third author in [Reference Zannier34], almost-Belyi maps.
The Riemann existence theorem gives a link between such covers and permutation representations. To be more precise, if our cover of degree
$2n$
has h branch points, an h-tuple
$\Sigma $
of elements of
$S_{2n}$
can be associated to it. These permutations satisfy certain properties, for example, they have very strong constrains on their cycle structure. On the other hand, fixing the branch points and a tuple of permutations satisfying these properties determines the map up to automorphisms of the domain.
It turns out that one can determine whether a solution associated to a certain representation is primitive or a power of another one by looking at the partitions of
$\{1, \dots , 2n\}$
preserved by the subgroup of
$S_{2n}$
generated by the same representation, that is, the monodromy group of the cover.
Theorem 1.4 (Theorem 6.2).
Let
$(A,B)$
be a solution of degree n of a Pell–Abel equation
$A^2-DB^2=1$
with
$\deg D=2d$
, and let
$G_A$
be the monodromy group of
$A^2$
. Then, for every integer
$m\mid n$
with
$\frac {n}{m} \ge d$
,
$(A,B)$
is the m-th power of another solution if and only if
$G_A$
preserves a partition of the set
$\{1,\dots , 2n\}$
in
$2m$
subsets, each of cardinality
$\frac {n}{m}$
.
In [Reference Zannier34], the third author gives an example of a permutation representation associated to a Pell–Abel equation of arbitrary degree; see Example 2.4 below. Using the above criterion it is easy to see that the corresponding solution of the Pell–Abel equation is primitive, and this implies that, after fixing their degree, there are Pellian polynomials with primitive solutions of any possible degree.
Corollary 1.5. Let
$d,n$
be positive integers with
$n\geq d\geq 2$
. Then there exists a squarefree Pellian
$D\in \mathbb {C}[t]$
of degree
$2d$
such that a primitive solution of the corresponding Pell–Abel equation (1.2) has degree n.
Going back to our Betti map
$\tilde {\beta }: \tilde {S}\rightarrow \mathbb {R}^{2(d-1)}$
associated to the abelian scheme
$\mathcal {J} \rightarrow S$
, we can deduce some properties of the rational points in the image
$\tilde {\beta }(\tilde {S})$
. We call the denominator of a rational point of
$\mathbb {R}^{2(d-1)}$
the positive least common denominator of the coordinates of the point.
It is easy to see that there are no nonzero points of denominator
$< d$
, since a nontrivial solution of the Pell–Abel equation has degree at least d. From [Reference André, Corvaja and Zannier2], one can easily deduce that
$\tilde {\beta }(\tilde {S})$
contains rational points of every denominator
$\geq n_0$
for some large enough natural number
$n_0$
. Our Corollary 1.5 implies the following result.
Corollary 1.6. The image
$\tilde {\beta }(\tilde {S})$
of the Betti map contains a point of denominator n for all
$n\geq d$
.
In a recent work [Reference Corvaja, Demeio, Masser and Zannier10], the authors consider the distribution of points of the base of an elliptic scheme where a section takes torsion values. Those are exactly the points of the base where the corresponding Betti map takes rational values. More specifically, they prove that the number of points where the Betti map takes a rational value with denominator (dividing) n is, for large n, essentially
$n^2$
times the area of the base with respect to the measure obtained locally by pulling back the Lebesgue measure on
$\mathbb {R}^2$
by the Betti map. Moreover, they show that the constant of the main term is equal to
$\hat {h}(\sigma )$
, where
$\hat h$
is the canonical height associated to twice the divisor at infinity on the elliptic scheme.
The corresponding result for our family of Jacobians
$\mathcal {J} \rightarrow S$
and our section would probably give an asymptotic formula for the number of equivalence classes of polynomials D of fixed degree
$2d\geq 4$
satisfying a Pell–Abel equation
$A^2-DB^2=1$
with A of degree n, for n tending to infinity.
On the other hand, using the correspondence given by the Riemann existence theorem explained above, one is able to determine the exact number of equivalence classes of Pell–Abel equations of fixed degree n. In Section 8, we compute this number in the case
$d=2$
. We point out that, as d grows, the combinatorics behind the problem becomes more complicated, but this approach would still work in principle.
In turn, this complete classification should allow to compute the number of components of
$\mathcal {P}_{2d}\subseteq \mathcal {B}_{2d}$
that map via the Betti map to rational points of fixed denominator rather than just an asymptotic formula analogous to the one in [Reference Corvaja, Demeio, Masser and Zannier10].
Right before this article was published, Fedor Pakovich pointed out his paper [Reference Pakovitch24] to the authors. It contains several remarks about the Pell–Abel equation, points of finite order on Jacobians and Belyi maps which are deeply related to our work.
2 Almost-Belyi maps
Given a (squarefree) Pellian polynomial
$D(t)$
of degree
$2d$
and a solution
$(A, B)$
of degree n of the corresponding Pell–Abel equation
$A^2-D(t)B^2=1$
, we consider the map
$\phi _A:=A^2 : \mathbb {P}^1(\mathbb {C}) \rightarrow \mathbb {P}^1(\mathbb {C})$
of degree
$2n$
.
It is easy to see that the Pell–Abel equation forces the branching of the map
$\phi _A$
to be ‘concentrated’ in
$ 0,\ 1$
and
$\infty $
. First, since
$\phi _A$
is a polynomial, there is total ramification above
$\infty $
. Moreover, as
$\phi _{A}$
is a square and
$\phi _A-1=DB^2$
, we have that the ramification indices above
$0$
are all even, while the ones above
$1$
are all even with the exception of
$2d$
points (the simple roots of
$D(t)$
). Hence, counting the branching of
$\phi _A$
as the sum of
$e-1$
over the ramification indices e, we have that above
$0$
the branching is at least n, above
$1$
at least
$n-d$
and above
$\infty $
it is exactly
$2n-1$
. On the other hand, by the Riemann–Hurwitz formula (see [Reference Hindry and Silverman17, Theorem A.4.2.5., p. 72]), the total branching is equal to
$4n-2$
. This means that the branching outside
$0,1,\infty $
is at most
$d-1$
, and thus there cannot be more than
$d-1$
further branch points. Following the third author [Reference Zannier34], we call
$\phi _A$
an almost-Belyi map as its branching is ‘concentrated’ above
$0,1, \infty $
(in the sense that the number of branch points outside this set does not depend on the degree n of
$\phi _A$
but only on the degree of D which is fixed)Footnote
1
.
To each such map, we can associate a monodromy permutation representation in the following way (for references see [Reference Fried15], [Reference Miranda21] or [Reference Völklein32]).
Let us call
$\mathcal {B}=\{b_1, \ldots , b_{h}\}$
the set of the branch points of
$\phi _A$
(where
$h\le d+2$
), and let us choose a base point q different from the
$b_i$
. Let us moreover call
$V:=\mathbb {P}^1(\mathbb {C})\setminus \mathcal {B}$
. The fundamental group
$\pi _1(V,q)$
of V is a free group on h generators
$[\gamma _1],\ldots ,[\gamma _{h}]$
modulo the relation
$$\begin{align*}[\gamma_1]\cdots[\gamma_{h}]=1, \end{align*}$$
where each
$\gamma _i:[0,1]\rightarrow V$
is a closed path which winds once around
$b_i$
. Now, consider the fiber
$\phi _A^{-1}(q)$
above q, and denote by
$q_1, \ldots , q_{2n}$
the
$2n$
distinct points in this fiber. Every loop
$\gamma \subseteq V$
based at q and not passing through the
$b_i$
can be lifted to
$2n$
paths
$\tilde {\gamma }_1, \ldots , \tilde {\gamma }_{2n}$
, where
$\tilde {\gamma }_j$
is the unique lift of
$\gamma $
starting at
$q_j$
. Hence,
$\tilde {\gamma }_j(0)=q_j$
for every j. Now, consider the endpoints
$\tilde {\gamma }_j(1)$
; these also lie in the fiber
$\phi _A^{-1}(q)$
and indeed form the entire preimage set
$\{q_1, \ldots , q_{2n}\}$
. We call
$\sigma (j)$
the index in
$\{1, \ldots , 2n\}$
such that
$\tilde {\gamma }_j(1)=q_{\sigma (j)}$
. The function
$\sigma $
is then a permutation of the set
$\{1, \ldots , 2n\}$
which depends only on the homotopy class of
$\gamma $
; therefore, we have a group homomorphism
$\rho :\pi _1(V,q) \longrightarrow S_{2n}$
called the monodromy representation of the covering map
$\phi _A$
. This is clearly determined by the images
$\sigma _i=\rho ([\gamma _i])$
of the generators of
$\pi _1(V,q)$
, and the image of
$\rho $
must be a transitive subgroup of
$S_{2n}$
, as V is connected.
Note that these permutations must have a precise cycle structure depending on the branch points of the map. Following the notation of [Reference Bilu5], we say that a branch point c of a rational map f is of type
$(m_1, \ldots , m_k)$
if, for
$ f^{-1}(c)=\{c_1, \dots , c_k\}$
, the point
$c_j$
has ramification index
$m_j$
for every
$j=1, \ldots , k$
. The type of a branch point corresponds to the cycle structure of the corresponding permutation, meaning that, going back to
$\phi _A$
, if
$b_i$
is of type
$(m_1, \ldots , m_k)$
, then
$\sigma _i$
is the product of k disjoint cycles
$\tau _1,\dots , \tau _k$
with
$\tau _j$
of length
$m_j$
.
We just showed how to associate to a covering a certain tuple of permutations satisfying some specific decomposition properties; the Riemann existence theorem tells us that we can do the opposite, that is, to any tuple of permutations satisfying certain properties we can associate a covering. One can actually say more: There exists a 1-1 correspondence between these two sets if one mods out by the following equivalence relations on both sides.
Definition 2.1. Two rational maps
$F_1,F_2:\mathbb {P}^1(\mathbb {C}) \longrightarrow \mathbb {P}^1(\mathbb {C})$
are called equivalent if there is an automorphism
$\varphi $
of
$\mathbb {P}^1(\mathbb {C})$
such that
$F_1=F_2\circ \varphi $
.
Definition 2.2. If
$\Sigma =(\sigma _1, \sigma _2, \cdots ,\sigma _{h})$
and
$\Sigma '=(\sigma ^{\prime }_1, \sigma ^{\prime }_2, \cdots , \sigma ^{\prime }_{h})$
are two h-tuples of permutations of
$S_{2n}$
, we say that they are conjugated if there exists a
$\tau \in S_{2n}$
such that
$\tau ^{-1}\sigma _i\tau =\sigma _i'$
for every
$i=1, \ldots , h$
. If this holds, we use the notation
$\tau ^{-1}\Sigma \tau =\Sigma '$
.
We can finally state this consequence of the Riemann existence theorem.
Theorem 2.3 ([Reference Miranda21], Corollary 4.10).
Fix a finite set
$\mathcal B=\{b_1, \ldots , b_h\}\subseteq \mathbb {P}^1(\mathbb {C})$
. Then there is a 1-1 correspondence between
$$\begin{align*}\begin{Bmatrix} \text{equivalence classes}\\ \text{ of rational maps} \\ F:\mathbb{P}^1(\mathbb{C}) \longrightarrow \mathbb{P}^1(\mathbb{C}) \\ \text{of degree }2n \\ \text{whose branch points lie in }\mathcal{B} \end{Bmatrix} \text{ and } \begin{Bmatrix} \text{conjugacy classes } (\sigma_1, \ldots, \sigma_{h})\in S_{2n}^h \\ \text{such that }\sigma_1\cdots \sigma_{h}=\mathrm{id},\\ \text{the subgroup generated by the } \sigma_i \\ \text{is transitive and } \sum_{i,j}(m_{ij}-1)=4n-2, \\ \text{where } (m_{i1},\dots, m_{ik_i}) \text{ is the cycle structure of } \sigma_i \end{Bmatrix}. \end{align*}$$
Moreover, given the tuple of permutations
$(\sigma _1, \ldots , \sigma _{h})$
, for each
$i=1,\dots , h$
there are
$k_i$
preimages
$b_{i1}, \ldots , b_{ik_i}$
of
$b_i$
for the corresponding cover
$F: \mathbb {P}^1(\mathbb {C}) \longrightarrow \mathbb {P}^1(\mathbb {C}) $
, with
$e_{F}(b_{ij})=m_{ij}$
.
As seen before, given a Pell–Abel equation (1.2) with A of degree n and D of degree
$2d$
, we get the map
$\phi _A$
that is branched in
$\{0,1,\infty \}$
and in further
$k $
points, where
$0\leq k \le d-1$
. Call
$\mathcal {B}:=\{0,1,\infty ,b_1, \ldots , b_{k}\}$
and
$V:=\mathbb {P}^1(\mathbb {C})\setminus \mathcal {B}$
. We fix a
$q\in V$
, and we let
$\rho :\pi _1(V,q) \longrightarrow S_{2n}$
be the associated monodromy representation (as explained above). We now let
$\gamma _0,\gamma _{\infty },\gamma _{1}$
be closed paths winding once around
$0,\infty ,1$
, respectively, and
$\delta _i$
be a closed path winding once around
$b_i$
for all
$i=1, \ldots , k$
. Then
$[\gamma _0] [\gamma _{\infty }] [\gamma _{1}] [\delta _1] \cdots [\delta _{k}]=1$
and all these classes generate
$\pi _1(V,q)$
.
We let
$\sigma _0=\rho (\gamma _0)$
,
$\sigma _{\infty }=\rho (\gamma _{\infty })$
,
$\sigma _1=\rho (\gamma _1)$
and
$\tau _i=\rho (\delta _i)$
for
$i=1, \ldots , k$
. Then we have that
-
1 -
$\sigma _0\sigma _{\infty }\sigma _1\tau _1\cdots \tau _{k}=\mathrm {id}$
; -
2 - the subgroup of
$S_{2n}$
generated by these permutations is transitive; -
3 -
$\sigma _0$
and
$\sigma _{\infty }$
do not fix any index; -
4 -
$\sigma _1$
fixes exactly
$2d$
indexes.
Moreover, concerning the decomposition of these permutations in disjoint cycles we have that
-
5 - all nontrivial cycles in
$\sigma _0$
and
$\sigma _1$
have even length; -
6 -
$\sigma _{\infty }$
must be a
$2n$
-cycle; -
7 - the sum over all cycles of
$\sigma _0, \dots , \tau _k$
of their lengths minus 1 must give
$4n-2$
.
The permutations
$\sigma _0, \dots , \tau _k$
depend on the chosen labelling of the preimages of q, so a Pell–Abel equation gives exactly one conjugacy class of
$(k+3)$
-tuples and a
$\mathcal {B}$
for some
$k\in \{0,\dots , d-1\}$
.
On the other hand, fixing
$k\in \{0,\dots , d-1\}$
and
$\mathcal {B}$
, a conjugacy class of
$(k+3)$
-tuples satisfying the above conditions gives a cover
$\mathbb {P}^1(\mathbb {C})\rightarrow \mathbb {P}^1(\mathbb {C})$
that is a polynomial branched exactly in
$\mathcal {B}$
. Moreover, the cycle structures of
$\sigma _0$
and
$\sigma _1$
make this polynomial satisfy a Pell–Abel equation.
We see an example.
Example 2.4. Let us consider the case in which the branching outside
$0,1, \infty $
is maximal, that is,
$k=d-1$
. In this case,
$\sigma _1$
is the product of
$n-d$
transpositions,
$\sigma _0$
is the product of n transpositions and each
$\tau _i$
consists of a single transposition.
A possible choice given by the third author in [Reference Zannier34] and leading to a Pell–Abel equation (1.2), is
$\sigma _{\infty }=(2n, \ldots , 1)$
,
$\sigma _0=(1,2n)(2,2n-1)\cdots (n,n+1)$
,
$\sigma _1=(1,2n-1)(2,2n-2)\cdots (n-d,n+d)$
and
$\tau _i=(n-i,n+i)$
,
$i=1, \ldots , d-1$
. We will prove in the next sections that this example corresponds to a primitive solution of a Pell–Abel equation.
3 Proof of Theorem 1.1
This section is devoted to proving Theorem 1.1. For the reader’s convenience, we recall the notations and the statement of the theorem. Given a point
$\overline {s}=(s_1, \ldots , s_{2d})\in \mathcal {B}_{2d}$
, we denote by
$H_{\overline s}$
a nonsingular model of the hyperelliptic curve defined by
$y^2=t^{2d}+ \cdots +s_{2d}$
with two points at infinity that we call
$\infty ^+$
and
$\infty ^-$
, and by
$\mathcal {J}_{\overline s}$
its Jacobian variety. We denote by
$\mathcal {J} \rightarrow S=\mathcal {B}_{2d}$
the abelian scheme of relative dimension
$d-1$
whose fibers are the
$\mathcal {J}_{s}$
and by
$\sigma :S \rightarrow \mathcal {J}$
the section corresponding to the point
$[\infty ^+-\infty ^-]$
. We recall moreover that
$\tilde {S}$
is the universal covering of
$S(\mathbb {C})$
.
Theorem. Let
$\tilde {\beta }: \tilde {S} \rightarrow \mathbb {R}^{2(d-1)}$
be the Betti map associated to the section
$\sigma $
; then
$\operatorname {{\mathrm {rk}}} \tilde {\beta }\ge 2(d-1)$
and, equivalently,
$\tilde {\beta }$
is submersive on a dense open subset of
$\tilde S$
. In particular, the set of
$\overline {s}\in S(\mathbb {C})$
such that
$\sigma (\overline s)$
is torsion on
$\mathcal {J}_{\overline s}$
is dense in
$S(\mathbb {C})$
with respect to the complex topology.
Proof. The rank of
$\tilde {\beta }$
is trivially bounded from above by
$2(d-1)$
, so it is enough to show that there exists a point
$\tilde {s}\in \tilde {S}$
where the rank is
$\geq 2(d-1)$
.
Recall that S has dimension
$2d$
. Suppose by contradiction that the maximal rank is
$ r< 2(d-1)$
; then the nonempty fibers of
$\tilde {\beta }$
are complex analytic varieties of dimension
$\ge 2d-r/2>d+1$
; see [Reference Corvaja, Masser and Zannier11, Propositions 2.1 and 2.2].
Now, by applying Example 2.4 and Theorem 2.3, we have that the set of Pellian polynomials of degree
$2d$
is not empty, and this gives the existence of a rational point, with denominator say n, in the image of
$\tilde {\beta }$
. We call
$\tilde {V}$
an irreducible component of the preimage of that point; we may assume it has dimension
$>d+1$
. Moreover, we let V be its image in
$S(\mathbb {C})$
; this must again have dimension
$>d+1$
and we have that each point of
$\sigma (V)$
is a torsion point of order n in the respective fiber.
Then we have
$V\subseteq \mathcal {P}_{2d}(\mathbb {C})$
, where
$\mathcal {P}_{2d}$
is the Pellian locus defined in equation (1.3) as, by Proposition 1.2, points of V correspond to some Pellian polynomials with a solution of degree n.
The following Proposition 3.1 concludes the proof of Theorem 1.1 since it contradicts the lower bound
$\dim V>d+1$
.
Proposition 3.1. The Pellian locus
$\mathcal {P}_{2d}$
consists of denumerably many algebraic subvarieties of
$\mathcal {B}_{2d}$
of dimension at most
$d+1$
.
Proof. As the irreducible components of
$\mathcal {P}_{2d}$
correspond to rational values of the Betti map, we clearly have that the number of components of
$\mathcal {P}_{2d}$
is countable. Moreover, they are components of projections on
$\mathcal {B}_{2d}$
of intersections of torsion subgroup schemes of
$\mathcal {J}$
with the image of the section
$\sigma $
. Therefore, they are algebraic subvarieties of
$\mathcal {B}_{2d}$
.
Fix now a component U of
$\mathcal {P}_{2d}$
of dimension
$>d+1$
mapping to a rational point of denominator n. Then every point of U corresponds to a Pellian polynomial with a solution of degree n. We let W be the closure of
$$ \begin{align*} \{(a_0,\dots ,a_{2n})\in \mathbb{A}_{\mathbb{C}}^{2n+1}: a_{0}t^{2n}+a_{1}t^{2n-1}+\dots+ a_{2n} &=A^2 \mbox{ is a square} \\ &\qquad\text{and } A^2-D_{\overline{s}}B^2=1 \mbox{ for some }\overline{s}\in U \} \end{align*} $$
in
$$ \begin{align*} \mathcal{A}_{2n}:=\{(a_0,\dots ,a_{2n})\in \mathbb{A}_{\mathbb{C}}^{2n+1}: a_{0}\neq 0 \}. \end{align*} $$
Note that W is the closure of the projection on
$\mathcal {A}_{2n}$
of
$$ \begin{align*} \{ (a_0,\dots ,a_{2n},&b_0,\dots, b_{2n-2d},\overline{s})\in \mathcal{A}_{2n}\times \mathbb{A}_{\mathbb{C}}^{2n-2d+1}\times U: \\ &a_{0}t^{2n}+\dots+ a_{2n} =A^2 \text{ and } b_{0}t^{2n-2d}+\dots+ b_{2n-2d} =B^2 \mbox{ are squares} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{and } A^2-D_{\overline{s}}B^2=1 \}. \end{align*} $$
Such projection has finite fibers, therefore W has dimension
$>d+1$
.
Note that it is possible to compose any degree
$2n$
complex polynomial f with a linear polynomial and obtain a monic polynomial with no term of degree
$2n-1$
. Moreover, there are at most
$2n$
possible polynomials of this special form that can be obtained in this way from a given f. We let
$$ \begin{align*} \mathcal{C}_{2n}:=\{(c_2,\dots ,c_{2n})\in \mathbb{A}_{\mathbb{C}}^{2n-1} \} \text{ and } \mathcal{L}:=\{(a,b)\in \mathbb{A}_{\mathbb{C}}^2: a\neq 0\} \end{align*} $$
and define the morphism
$\varphi :\mathcal {C}_{2n}\times \mathcal {L}\rightarrow \mathcal {A}_{2n}$
,
$$ \begin{align*} \varphi((c_2,\dots ,c_{2n}),(a,b)) &=(a_0,\dots ,a_{2n}) \iff \\ & a_0t^{2n}+\dots +a_{2n}=(at+b)^{2n}+c_{2}(at+b)^{2n-2}+\cdots+ c_{2n-1}(at+b)+ c_{2n}. \end{align*} $$
By the above considerations,
$\varphi $
is surjective and the fibers are finite of cardinality at most
$2n$
.
If we let
$\pi $
be the projection
$ \mathcal {C}_{2n}\times \mathcal {L}\to \mathcal {C}_{2n}$
and
$Z=\pi (\varphi ^{-1}(W))$
, as the fibers of
$\pi $
have dimension 2, Z must have dimension
$>d-1$
.
Now, for
$\overline {c}=(c_2,\dots ,c_{2n})\in Z$
, the corresponding polynomial
$f_{\overline {c}}(t)=t^{2n}+c_{2}t^{2n-2}+\cdots + c_{2n}$
is a square and we have
$f_{\overline {c}}(t)-D(t)B(t)^2=1$
for some polynomials
$D(t),B(t)$
with D of degree
$2d$
. We let
$$ \begin{align*} \tilde{Z}&= \{ (\overline{c},c^{\prime}_{2},\dots, c^{\prime}_{2n-1},r_1,\dots, r_{2n-1})\in Z \times \mathbb{A}_{\mathbb{C}}^{2n-2}\times \mathbb{A}_{\mathbb{C}}^{2n-1} : f^{\prime}_{\overline{c}}= 2n t^{2n-1}+ c^{\prime}_{2} t^{2n-3}+\dots+ c^{\prime}_{2n-1} \\ &\qquad\qquad\qquad\qquad\text{and } f^{\prime}_{\overline{c}} = 2n(t^{2n-1}+ \sigma_2(r_1,\dots, r_{2n-1})t^{2n-3}+\dots+ \sigma_{2n-1}(r_1,\dots, r_{2n-1})) \}, \end{align*} $$
where
$\sigma _i(X_1,\dots , X_{2n-1})$
is the i-th elementary symmetric polynomial in
$2n-1$
variables and
$f^{\prime }_{\overline {c}}$
is the derivative of
$f_{\overline {c}}$
. In other words, given
$\overline c \in Z$
,
$c_2',\ldots , c_{2n-1}'$
are the coefficients of the derivative of the polynomial
$f_{\overline c}(t)$
and
$r_1,\ldots , r_{2n-1}$
are the ramification points of the same polynomial (not necessarily distinct). Clearly,
$\tilde {Z}$
must have dimension
$>d-1$
.
Finally, if we consider the morphism
$\psi :\tilde {Z} \to \mathbb {A}_{\mathbb {C}}^{2n-1} $
defined by
$$ \begin{align*}(\overline{c},c^{\prime}_{2},\dots, c^{\prime}_{2n-1},r_1,\dots, r_{2n-1})\mapsto (f_{\overline{c}}(r_1), \dots , f_{\overline{c}}( r_{2n-1})),\end{align*} $$
then
$f_{\overline {c}}(r_1), \dots , f_{\overline {c}}( r_{2n-1})$
are the (not necessarily distinct) branch points of
$f_{\overline {c}}$
. By the Riemann existence theorem and the above considerations, the fibers of
$\psi $
are finite and have cardinality at most
$2n$
, and therefore
$\psi (\tilde {Z})$
has dimension
$>d-1$
.
This gives a contradiction because the considerations of the previous section imply that the polynomial
$f_{\overline {c}}$
, since it fits in a Pell–Abel equation, may have not more than
$d+1$
finite branch points, two of which are 0 and 1. Therefore, not more than
$d-1$
branch points are allowed to vary and
$\psi (\tilde {Z})$
must have dimension
$\leq d-1$
, as wanted.
4 Chebychev polynomials and powers of solutions of the Pell–Abel equation
This section is devoted to describing the group of solutions of a Pell–Abel equation. As seen in the introduction, if a Pell–Abel equation
$A^2-DB^2=1$
has a nontrivial solution, then it has infinitely many ones, obtained by taking powers
$A_m+\sqrt D B_m=(A+\sqrt D B)^m$
. Our goal for this section is to prove that
$A_m^2=f_m(A^2)$
, for some polynomial
$f_m$
related to the m-th Chebychev polynomial.
We recall that Chebychev polynomials of the first kind are defined recursively as
$$ \begin{align*} \begin{cases} T_0(t)=1, \\ T_1(t)=t, \\ T_{n+1}(t)=2tT_n(t)-T_{n-1}(t), \end{cases} \end{align*} $$
so
$T_n$
is a polynomial of degree n, and, for every
$n>1$
, the two terms of highest degree of
$T_n$
are
$2^{n-1}t^n$
and
$-n2^{n-3}t^{n-2}$
. This polynomial satisfies many important properties; for example, for every
$n\ge 0$
, we have that
$T_n(t+t^{-1})=t^n+t^{-n}$
. Explicitly, we have that
$$ \begin{align} T_{n}(t):=\sum _{h=0}^{[n/2]}\binom{n}{2h}t^{n-2h}(t^{2}-1)^{h}, \end{align} $$
where
$[\cdot ]$
denotes the floor function. For a survey on these topics, see [Reference Lidl, Mullen and Turnwald19].
Suppose now that
$(A,B)$
is a solution of the Pell equation
$A^2-DB^2=1$
. Then
$(A_m,B_m)$
defined by
$A_m+\sqrt D B_m=(A+\sqrt D B)^m$
is another solution of the same Pell–Abel equation, and
$A_m=\sum _{j=0}^{[m/2]} \binom {m}{2j} (DB^2)^j A^{m-2j}$
. Using that
$DB^2=A^2-1$
, we have
$$ \begin{align} A_m=\sum_{j=0}^{[m/2]} \binom{m}{2j} (A^2-1)^j A^{m-2j}=T_m(A). \end{align} $$
For the rest of the paper, we will denote by
$\phi $
the square function
$\phi (t)=t^2$
and by
$\phi _g$
the composition
$\phi \circ g=g^2$
.
For our purposes, it will be useful to express
$\phi _{A_m}$
as a function of
$\phi _A$
. By equation (4.2), one can see that
$\phi _{A_m}=f_m(\phi _A)$
, where
$$ \begin{align} f_{2k}(w):=\left ( \sum_{j=0}^{k} \binom{2k}{2j} (w-1)^j w^{k-j} \right )^2, \end{align} $$
and
$$ \begin{align} f_{2k+1}(w):=w\left ( \sum_{j=0}^{k} \binom{2k+1}{2j} (w-1)^j w^{k-j} \right )^2. \end{align} $$
For every
$m\ge 1$
we call
$f_m$
the m-th power polynomial. Notice that
$f_m \circ \phi =\phi \circ T_m$
.
For our purposes we will also need a description of the branch locus of the
$f_m$
; this can be easily done by looking at the branch locus of the Chebychev polynomials which is well-known in the literature. We recall that, given a polynomial f, we say that a branch point b of f is of type
$(\mu _1, \ldots , \mu _k)$
if the
$\mu _i$
are the ramification indexes of the points in the preimage of b. This is also equal to the array of the multiplicities of the roots of
$f(t)-b$
.
Proposition 4.1. The polynomials
$f_m$
are branched only at
$0$
,
$1$
and
$\infty $
. Moreover,
-
• if m is even, then
$0$
is of type
$(2,2, \ldots , 2)$
and
$1$
is of type
$(1,1,2, \ldots ,2)$
; furthermore, the two unramified points above
$1$
are
$0$
and
$1$
; -
• if m is odd, then
$0$
is of type
$(1,2, \ldots , 2)$
, and
$0$
is the only unramified point above
$0$
; furthermore,
$1$
is of type
$(1,2, \ldots , 2)$
, and
$1$
is the only unramified point above
$1$
.
Proof. For
$m=2$
we have that
$f_m(t)=(2t-1)^2$
, and the proposition holds trivially. We will then assume that
$m\ge 3$
.
The Chebychev polynomials are strictly related to Dickson polynomials
$D_m(x,a)$
(for a definition of these polynomials and their properties, see [Reference Bilu5, Section 3]). Indeed, we have that
$T_m(t)=\frac {1}{2}D_m(2t,1)$
. Using this relation and [Reference Bilu5, Proposition 3.3, (b)] we have that, if
$m\ge 3$
, the finite branching of
$T_m$
happens only in
$\pm 1$
. If m is odd, both
$\pm 1$
are of type
$(1,2, \ldots , 2)$
, while if m is even we have that
$1$
is of type
$(1,1,2, \ldots , 2)$
and
$-1$
is of type
$(2,\ldots ,2)$
. Using that
$\phi \circ T_m= f_m \circ \phi $
, the finite branch points of
$f_m$
are exactly the ones of
$\phi \circ T_m$
,that is,
$0$
and
$1$
. Moreover, since
$0$
is not a branch point for
$T_m$
, then the ramification points of
$f_m\circ \phi $
have ramification index
$2$
.
Assume first that m is even. By equation (4.3), since
$f_m$
is a square, then
$0$
is a branch point for
$f_m$
. As
$0$
is not contained
$f_m^{-1}(0)$
, then it is of type
$(2,2, \ldots , 2)$
. Let us now consider the branch point
$1$
; notice that
$\{0,1\}\subseteq f_m^{-1}(1)$
. Since
$\phi $
is ramified in
$0$
, we have that
$f_m$
is unramified in
$0$
. Using that
$f_m \circ (1-t)=f_m$
, this implies that
$f_m$
is unramified also in
$1$
. By comparing the other ramification indexes, it follows that
$1$
is of type
$(1,1,2, \ldots , 2)$
as wanted.
Assume now m odd. By equation (4.4), we have
$f_m$
is unramified in
$0$
, and all the other points in the preimage of
$0$
have ramification index
$2$
, so
$0$
is of type
$(1,2, \ldots , 2)$
. Using the relation
$$ \begin{align} (1-t) \circ f_m \circ (1-t)= f_m, \end{align} $$
we have that the type of
$0$
and the type of
$1$
are the same, and
$1$
is the unramified point over
$1$
, concluding the proof.
5 Monodromy groups and polynomial decompositions
The problem of finding (functional) decompositions of polynomials is very classical and was studied first by Ritt in [Reference Ritt26] and then by several authors (see, for instance, [Reference Fried14]). In this section, we recall some basic facts that will be used in the paper.
Let f be a nonconstant polynomial in
$\mathbb {C}[t]$
, let u be transcendental over
$\mathbb {C}$
and L be the splitting field of
$f(t)-u$
over
$\mathbb {C}(u)$
. The monodromy group
$\text {Mon}(f)$
of f is the Galois group of L over
$\mathbb {C}(u)$
, viewed as a group of permutations of the roots of
$f(t)-u$
. By Gauss’ lemma, it follows that
$f(t) - u$
is irreducible over
$\mathbb {C}(u)$
, so
$\text {Mon}(f)$
is a transitive permutation group. If x is a root of
$f(t)-u$
in L, then
$u=f(x)$
and
$\text {Mon}(f)=\text {Gal}(L/\mathbb {C}(f(x)))$
. We denote moreover by H the stabilizer of x, that is,
$H=\text {Gal}(L/\mathbb {C}(x))$
. Note that, by Theorem 2.3, the polynomial f of degree n seen as a map
$\mathbb {P}^1(\mathbb {C})\rightarrow \mathbb {P}^1(\mathbb {C})$
corresponds to a conjugacy class of h-tuples of permutations in
$S_n$
, where h is the cardinality of the branch locus of f. Then, once we fix a labelling of the roots of
$f(t)-u$
(which corresponds to fixing a representative
$(\sigma _1, \ldots , \sigma _h)$
in the conjugacy class), the monodromy group of f is isomorphic to the subgroup of
$S_n$
generated by
$\sigma _1, \ldots , \sigma _h$
. Applying Lüroth’s theorem [Reference Schinzel28, Theorem 2] and [Reference Schinzel28, Theorem 4], we have the following proposition.
Proposition 5.1. Given a polynomial
$f\in \mathbb {C}[x]$
, there is a correspondence between polynomial decompositions of f and subfields of
$\mathbb {C}(x)$
containing
$\mathbb {C}(f(x))$
. Moreover, if
$\mathbb {C}(f(x)) \subseteq K \subseteq \mathbb {C}(x)$
, then
$f=g \circ h$
with
$g,h$
polynomials and
$\deg h=[\mathbb {C}(x): K]$
.
Using the Galois correspondence, this shows that the study of the polynomial decompositions of f reduces to the study of subgroups of the monodromy group of f containing H.
We give the following definition.
Definition 5.2. We say that two polynomials
$f,g\in \mathbb {C}[t]$
are linearly equivalent if there exists two linear polynomials
$\ell _1, \ell _2\in \mathbb {C}[t]$
such that
$f= \ell _1 \circ g \circ \ell _2$
.
Notice that, if
$f=g\circ h$
, we can always change the decomposition up to composing with linear polynomials, so it makes sense to study the polynomial decompositions up to linear equivalence. Moreover, notice that two linearly equivalent polynomials have the same monodromy group.
We are now interested in computing the monodromy group of the polynomials
$f_m$
defined in the previous section.
Proposition 5.3. For every
$m\ge 2$
, the monodromy group of
$f_m$
is exactly
$D_{2m}$
, where
$D_{2m}$
denotes the dihedral group of order
$2m$
.
Proof. By Proposition 4.1, the polynomials
$f_m$
are branched only in
$0$
and
$1$
, both of type
$(1,2 \ldots , 2)$
if m is odd or of type
$(2,\ldots , 2)$
and
$(1,1,2\ldots , 2)$
, respectively, if m is even. Applying [Reference Bilu5, Theorem 3.4], this implies that
$f_m$
is linearly equivalent to the Dickson polynomial
$D_m(x,a)$
with
$a \neq 0$
. By [Reference Bilu5, Theorem 3.6], the monodromy group of
$D_m(x,a)$
with
$a \neq 0$
is the dihedral group
$D_{2m}$
, which gives that
$\text {Mon} (f_m)=D_{2m}$
as wanted.
6 Characterization of primitivity
In this section, we want to give a criterion to detect the primitivity of a solution of the Pell–Abel equation in terms of the associated conjugacy class of permutations. We start with a definition.
Definition 6.1. For positive integers
$n,\ell $
with
$\ell \mid 2 n$
, we call a partition
$\mathcal {F}=\{F_1,\ldots , F_{\ell }\}$
of
$\{1,\dots , 2n \}$
an
$\ell $
-partition if
$\mathcal {F}$
consists of
$\ell $
subsets each of cardinality
$\frac {2n}{\ell }$
. A subgroup
$G < S_{2n}$
is said to be
$\ell $
-imprimitive if there exists an
$\ell $
-partition
$\mathcal {F}=\{F_1,\ldots , F_{\ell }\}$
which is preserved by the action of G, that is, for every
$\sigma \in G$
and every
$a\in \{1, \ldots , \ell \}$
, there exists
$b\in \{1, \ldots , \ell \}$
such that
$\sigma (F_a)=F_b$
.
It is well-known (see [Reference Ritt26]) that, if
$f= g \circ h$
is a polynomial of degree r which is the composition of two polynomials g and h, where
$\deg h=s$
and
$\deg g=r$
, then
$\text {Mon}(f)$
is r-imprimitive. Indeed, if q is not a branch point for f, then the set of its preimages via f is partitioned in subsets whose elements map via h to the same preimage of q via g.
We now give a criterion for determining whether a solution
$(A,B)$
of a Pell–Abel equation is primitive or an m-th power for some
$m\geq 2$
, and this behavior is completely determined by the monodromy group
$G_A$
of
$\phi _A$
. Note that
$G_A$
is always
$2$
-imprimitive since
$\phi _A$
is the square of a polynomial.
Theorem 6.2. Let
$(A,B)$
be a solution of degree n of a Pell–Abel equation
$A^2-DB^2=1$
with
$\deg D=2d$
, and let
$G_A$
be the monodromy group of
$\phi _A$
. Then, for every integer
$m\mid n$
with
$\frac {n}{m} \ge d$
,
$(A,B)$
is the m-th power of another solution if and only if
$G_A$
is
$2m$
-imprimitive.
We will prove the above theorem in the next section. Before this, we show how Corollary 1.5 can be deduced from Theorem 6.2.
Proof of Corollary 1.5.
We prove that a solution
$(A,B)$
associated to the conjugacy class of the tuple
$(\sigma _0,\sigma _{\infty },\sigma _1,\tau _1, \ldots ,\tau _{d-1})$
given in Example 2.4 is primitive for every
$n\ge d \ge 2$
.
We recall that
$\sigma _{\infty }=(2n, \ldots , 1)$
,
$\sigma _0=(1,2n) (2,2n-1)\cdots (n,n+1)$
,
$\sigma _1=(1,2n-1) (2,2n-2)\cdots (n-d,n+d)$
and
$\tau _i=(n-i,n+i)$
, for
$i=1, \ldots , d-1$
.
Suppose by contradiction that
$(A,B)$
is the m-th power of another solution
$(A',B')$
for some
$m \mid n$
,
$1<m<n$
. By Theorem 6.2,
$G_A$
has to preserve a
$2m$
-partition of
$\{1, \ldots , 2n\}$
. We see that the only
$2m$
-partition preserved by
$\sigma _{\infty }$
is
$\mathcal F_{2m}:=\{F_1,\ldots , F_{2m}\}$
, where for every
$1\le h\le 2m$
, we set
$$ \begin{align*}F_h= \{ a \in \{1,\dots , 2n \}: a \equiv h\ \ \mod 2m \}.\end{align*} $$
On the other hand,
$\tau _{1}=(n-1, n+1)$
and
$n-1,\ n+1$
are not in the same congruence class modulo
$2m$
if
$m\ge 2$
, so
$\mathcal F_{2m}$
is not preserved by
$G_A$
. This implies that
$G_A$
does not preserve any
$2m$
-partition, hence proving the theorem.
7 Proof of Theorem 6.2
In view of what we explained in the previous section, one direction is quite easy. Indeed, let
$(A,B)$
be a solution of a Pell–Abel equation that is the m-th power of another solution
$(A', B')$
. In Section 4, we showed that
$\phi _{A}=f_m(\phi _{A'})=f_m \circ \phi \circ A'$
, where
$\phi (t)=t^2$
. As
$f_m$
is a polynomial of degree m, it follows immediately that
$G_{A}$
is
$2m$
-imprimitive.
The proof of the converse is much more involved.
Let us consider a solution
$(A,B)$
with A of degree n and the associated almost-Belyi map
$\phi _A$
. By Theorem 2.3, the map
$\phi _A$
corresponds to the conjugacy class of a
$(k+3)$
-tuple of permutations in
$S_{2n}$
given by
$\Sigma _A:=(\sigma _0,\sigma _{\infty },\sigma _1,\tau _1, \ldots ,\tau _{k})$
with
$k \le d-1$
, where we recall that
$\sigma _{\infty }$
is a
$2n$
-cycle,
$\sigma _0$
is the product of disjoint cycles of even length,
$\sigma _1$
fixes exactly
$2d$
indexes and is the product of disjoint cycles of even length; see properties 1 to 7 after Theorem 2.3. We call
$b_1, \ldots , b_k$
the further k branch points different from
$0,1, \infty $
.
Conjugating by a suitable permutation, we can and will always assume that
$\sigma _{\infty }=(2n,2n-1,\ldots , 1)$
.
We recall that the sum over all cycles of
$\sigma _0, \dots , \tau _k$
of their lengths minus 1 must give
$4n-2$
. Looking at the cycle structure of
$\sigma _0$
,
$\sigma _\infty $
and
$\sigma _1$
we have that
$\sigma _1$
fixes
$2d$
indexes and
$\sum _{i,j}(m_{ij}-1)\leq d-1$
, where
$(m_{i1},\dots , m_{ih_i})$
is the cycle structure of
$\tau _i$
. Therefore, there are at least two indexes which are fixed by
$\sigma _1$
and, at the same time, by every
$\tau _i$
. Without loss of generality, we are going to assume that one of them is
$2n$
. Moreover, each
$\tau _i$
must fix at least
$2n-2(d-1)$
indexes.
The monodromy group
$G_A$
of
$\phi _A$
is the subgroup of
$S_{2n}$
generated by
$\sigma _0,\sigma _{\infty },\sigma _1,\tau _1, \ldots ,\tau _{k}$
; by assumption,
$G_A$
preserves a
$2m$
-partition of
$\{1, \ldots ,2n\}$
. As mentioned before, the only
$2m$
-partition preserved by
$\sigma _{\infty }$
is
$\mathcal F_{2m}=\{F_1,\ldots , F_{2m}\}$
, where, for every
$h=1,\ldots , 2m$
, we defined
$F_h=\{j\in \{1, \ldots , 2n\} : j\equiv h\ \ \mod 2m\}$
; we can then assume that
$G_A$
preserves this specific partition.
For
$h=1, \ldots , m$
we let
$E_h=\{ j\in \{1, \ldots , 2n\} : j\equiv h\ \ \mod m \}$
. Then
$\mathcal E_m=\{E_1, \ldots , E_m\}$
gives another partition of
$\{1,\ldots , 2n\}$
and we have that
$E_h=F_h \cup F_{m+h}$
. We start by proving the following lemma.
Lemma 7.1. Suppose
$G_A$
preserves the
$2m$
-partition
$\mathcal F_{2m}$
, then
$G_A$
preserves the m-partition
$\mathcal E_m$
. More precisely, the action of
$\sigma _0,\sigma _{\infty },\sigma _1,\tau _1, \ldots ,\tau _{k}$
on
$\mathcal F_{2m}$
and
$\mathcal E_m$
is the following:
$$ \begin{align} \begin{cases} \sigma_{\infty}(F_1)=F_{2m} \mbox{ and } \sigma_{\infty}(F_i)= F_{i-1} \ \forall{i=2,\ldots, 2m};\\ \sigma_1(F_{2m})=F_{2m} \mbox{ and } \sigma_1(F_i)=F_{2m-i}\ \forall i=1, \ldots, 2m-1; \\ \sigma_0(F_i)=F_{2m-i+1} \quad \forall i=1,\ldots, 2m; \\ \tau_j(F_i)=F_i \ \forall i=1, \ldots, 2m \mbox{ and } j=1,\ldots, k. \end{cases} \end{align} $$
and:
$$ \begin{align*} \begin{cases} \sigma_{\infty}(E_1)=E_{m} \mbox{ and } \sigma_{\infty}(E_i)= E_{i-1} \ \forall{i=2,\ldots, m};\\ \sigma_1(E_m)=E_{m} \mbox{ and } \sigma_1(E_i)=E_{m-i}\ \forall i=1, \ldots, m-1; \\ \sigma_0(E_i)=E_{m-i+1}\quad \forall i=1,\ldots, m; \\ \tau_j(E_i)=E_i \ \forall i=1, \ldots, m \mbox{ and } j=1,\ldots, k. \end{cases} \end{align*} $$
In particular, by equation (7.1),
$\mathrm {Fix}(\sigma _1):=\{a\in \{1,\ldots , 2n\}\ |\ \sigma _1(a)=a\}\subseteq E_m$
.
Proof. By construction
$\sigma _1(2n)=2n$
, thus we have
$\sigma _1(F_{2m})=F_{2m}$
. Moreover, we have that
$|F_h|=\frac {n}{m} \ge d$
for all h, and for all i at most
$2d-2$
indexes are not fixed by
$\tau _i$
. This implies that
$\tau _i(F_h)=F_h$
for every
$i=1, \ldots , k$
and
$h=1, \ldots , 2m$
.
In what follows the indexes of the
$F_h$
are considered modulo
$2m$
. Using
$\sigma _0\sigma _{\infty }\sigma _1\prod _{i=1}^{k}\tau _i=\mathrm {id}$
, we deduce that
$\sigma _0\sigma _{\infty }\sigma _1(F_h)=F_h$
for every
$h=1, \ldots , 2m$
. As we know that
$\sigma _{\infty }( F_{h})=F_{h-1}$
, this gives that
$$ \begin{align} \begin{cases} \sigma_0(F_{i_1})=F_{i_2} \ \Rightarrow\ \sigma_1(F_{i_2-1})=F_{i_1}; \\ \sigma_1(F_{j_1})=F_{j_2} \ \Rightarrow\ \sigma_0(F_{j_2})=F_{j_1+1}. \end{cases} \end{align} $$
Moreover, we claim that, if
$\sigma _1(F_{k_1})=F_{k_2}$
for some indexes
$k_1$
and
$k_2$
, we must have that
$\sigma _1(F_{k_2})=F_{k_1}$
. Indeed, suppose by contradiction that this is not the case; since the nontrivial cycles appearing in
$\sigma _1$
have even length, there exist pairwise distinct
$k_1, k_2,k_3, k_4\in \{1,\dots , 2m\}$
such that
$\sigma _1(F_{k_1})=F_{k_2}$
,
$\sigma _1(F_{k_2})=F_{k_3}$
and
$\sigma _1(F_{k_3})=F_{k_4}$
. Now, each of the
$n/m$
elements of
$F_{k_1}$
gives a contribution of at least
$3$
to the branching above 1, where we recall that we count the branching of
$\phi _A$
above a point p as the sum of
$e-1$
over the ramification indices e of the preimages of p.
Using moreover that
$\sigma _1$
fixes
$2d$
points, we have that the branching above
$1$
must be at least
$3\frac {n}{m}+(n-2\frac {n}{m}-d)\ge n$
since
$\frac {n}{m}\ge d$
, and this is impossible as the branching above
$1$
can be at most
$n-1$
(recall properties 1–7 after Theorem 2.3).
Therefore, we can rewrite equation (7.2) as
$$ \begin{align} \begin{cases} \sigma_0(F_{i_1})=F_{i_2} \ \Rightarrow\ \sigma_1(F_{i_1})=F_{{i_2}-1}; \\ \sigma_1(F_{j_1})=F_{j_2} \ \Rightarrow\ \sigma_0(F_{j_2})=F_{j_1+1}. \end{cases} \end{align} $$
Now, since
$\sigma _1(F_{2m})=F_{2m}$
, using equation (7.3) we have that
$$ \begin{align*}\sigma_1(F_{2m})=F_{2m} \quad \mbox{ and } \quad \sigma_1(F_i)=F_{2m-i}\ \forall i=1, \ldots, 2m-1,\end{align*} $$
and
$$ \begin{align*}\sigma_0(F_i)=F_{2m-i+1} \quad \forall i=1,\ldots, 2m.\end{align*} $$
Now, for every
$h=1,\ldots , m$
, the set
$E_h$
is equal to
$F_h \cup F_{m+h}$
. Using the previous relations, we have that
$\sigma _1(E_h)=F_{2m-h}\cup F_{m-h}=E_{m-h}$
;
$\sigma _0(E_h)=F_{2m-h+1}\cup F_{m-h+1}=E_{m-h+1}$
,
$\tau _i$
stabilizes
$\{E_1, \ldots , E_m\}$
for every
$i=1, \ldots , d-1$
and
$\sigma _{\infty }(E_h)=F_{h-1}\cup F_{m+h-1}=E_{h-1}$
, which concludes the proof.
For
$G < S_{2n}$
and
$C_1,\ldots , C_{\ell }\subseteq \{1, \ldots , 2n\}$
, we define
$$ \begin{align*}\mathrm{Stab}_G(C_1, \ldots, C_{\ell}):=\bigcap_{i=1}^{\ell} \mathrm{Stab}_G(C_i),\end{align*} $$
where, for every
$i=1, \ldots , \ell $
,
$$ \begin{align*}\mathrm{Stab}_G(C_i)=\{\sigma\in G : \sigma(C_i)=C_i \}.\end{align*} $$
First, we prove this preliminary lemma.
Lemma 7.2. Let us assume that
$G_A$
preserves
$\mathcal F_{2m}$
, and set
$S:=\mathrm {Stab}_{G_A}(\{2n\})$
. Then there exist
$H_1, H_2$
subgroups of
$G_A$
satisfying
$S \subseteq H_1< H_2 < G_A$
with
$[G_A:H_2]=m$
and
$[H_2:H_1]=2$
. Moreover, if
$m>2$
, then there exists a normal subgroup K of
$G_A$
containing S and contained in
$H_2$
such that
$G_A/K \cong D_{2m}$
.
Proof. As by assumption
$G_A$
preserves
$\mathcal {F}_{2m}$
, by Lemma 7.1 it preserves also
$\mathcal {E}_m$
, hence it induces an action on
$\{F_1, \ldots , F_{2m}\}$
and on
$\{E_1, \ldots , E_m\}$
.
Let us define
$H_1:=\mathrm {Stab}_{G_A}(F_{2m})$
and
$H_2:=\mathrm {Stab}_{G_A}(E_m)$
. As
$2n \in F_{2m} \subseteq E_m$
, by Lemma 7.1 it follows that
$S\subseteq H_1 \subseteq H_2$
.
The group
$G_A$
acts transitively on
$\{1,\ldots , 2n\}$
; therefore, the action on
$\{F_1, \ldots , F_{2m}\}$
and on
$\{E_1, \ldots , E_m\}$
is transitive as well, so the orbits of
$F_i$
and
$E_j$
have cardinality
$2m$
and m, respectively. This implies that
$|G_A|/|H_2|=|G_A\cdot E_m|=m$
and
$|G_A|/|H_1|=|G_A\cdot F_{2m}|=2m$
, where
$G_A \cdot C$
denotes the orbit of C with respect to the action of
$G_A$
. This proves that
$[G_A:H_2]=m$
and
$[H_2:H_1]=2$
as wanted.
We now prove the second part of the statement.
For
$m>2$
let us define
$K:= \mathrm {Stab}_{G_A}(E_1, \ldots , E_m)$
. Note that
$K \subseteq H_2$
and, if
$m>2$
, then
$K\neq H_2$
(as
$\sigma _1 \in H_2$
and
$\sigma _1 \not \in K$
). To conclude the proof, we have to show that K is a normal subgroup of
$G_A$
and
$G_A/K\cong D_{2m}$
.
First, let us prove that
$K\lhd G_A$
. Take
$h\in K$
and
$g\in G_A$
, then, if
$g(E_i)=E_{j_i}$
for some
$j_i$
, we have that
$$\begin{align*}g^{-1}hg(E_i)=g^{-1}h(E_{j_i})=g^{-1}(E_{j_i})=E_i \quad \mbox{for all } i=1,\ldots, m, \end{align*}$$
so
$g^{-1}hg \in K$
as wanted.
Finally, we show that
$G_A/K\cong D_{2m}$
. As
$G_A$
induces an action on
$\{E_1,\ldots , E_m\}$
, we can define a homomorphism
$\varphi : G_A \rightarrow S_m$
given by
$\varphi (\alpha )=\beta $
where
$\beta \in S_m$
is defined by
$\beta (i)=j$
if
$\alpha (E_i)=E_j$
. Since
$K=\mathrm {Stab}_{G_A}(E_1, \ldots , E_m)$
, then
$\varphi $
induces an injective homomorphism
$\tilde \varphi : G_A/K \rightarrow S_m$
. We want now to prove that
$\tilde \varphi (G_A/K)\cong D_{2m}$
. By Lemma 7.1,
$\tilde \varphi (\tau _i)=\mathrm {id}$
for every
$i=1, \ldots , k$
, so
$\tilde \varphi (G_A/K)$
is generated by the images of
$\sigma _{\infty }$
and
$\sigma _1$
, which we denote by r and s, respectively. We have to prove that
$r^m=\mathrm {id}$
,
$s^2=\mathrm {id}$
and
$srsr=\mathrm {id}$
. Notice that, if
$\sigma _{\infty }=(2n, 2n-1, \ldots , 1)$
, then
$r=(m,m-1, \ldots , 1)$
, so
$r^m=\mathrm {id}$
. Moreover, by Lemma 7.1, s is a product of transpositions, so it has order
$2$
as wanted. Let us finally prove that
$srsr=\mathrm {id}$
. By Lemma 7.1 we have that
$\sigma _1(E_m)=E_m$
,
$\sigma _1(E_i)=E_{m-i}$
for all
$i=1,\dots , m-1$
and
$\sigma _{\infty }(E_1)=E_{m}$
and
$\sigma _{\infty }(E_j)=E_{j-1}$
for all
$j=2, \ldots , m$
. So, for every
$i=1,\ldots , m$
,
$$\begin{align*}srsr(E_i)=rsr(E_{m-i})=sr(E_{m-i-1})=r(E_{i+1})=E_i. \end{align*}$$
This shows that
$G_A/K\cong D_{2m}$
, concluding the proof.
Using the previous lemma, we can finally prove the following result about the polynomial decomposition of
$\phi _A$
, which concludes the proof of Theorem 6.2.
Proposition 7.3. Suppose
$G_A$
preserves
$\mathcal {F}_{2m}$
with
$m\ge 2$
and
$n/m \ge d$
. Then
$\phi _A=f_m(\phi _{A'})$
, where
$A'$
is another solution of the same Pell–Abel equation.
Proof. By Lemma 7.2, there exists a chain of groups
$$ \begin{align*}\mathrm{Stab}_{G_A}(\{2n\}) < H_1 < H_2 < G_A,\end{align*} $$
where
$[G_A:H_2]=m$
and
$[H_2:H_1]=2$
, where we recall that
$H_1=\mathrm {Stab}_{G_A}(F_{2m})$
and
$H_2=\mathrm {Stab}_{G_A}(E_m)$
. By Galois theory, this implies that there exists a tower of subfields of
$\mathbb {C}(t)$
$$ \begin{align*}T:=\mathbb{C}(\phi_A) \subset L_2 \subset L_1 \subset \mathbb{C}(t),\end{align*} $$
with
$[L_2:\mathbb {C}(\phi _A)]=m$
and
$[L_1:L_2]=2$
. Together with Proposition 5.1, this implies that
$$ \begin{align*}\phi_A=h_2 \circ h_1 \circ z,\end{align*} $$
where
$h_1,h_2$
and z are polynomials with
$\deg h_1=2$
and
$\deg h_2=m$
.
Let us first prove that
$h_2$
is linearly equivalent to
$f_m$
. If
$m=2$
, this is trivial since polynomials of degree
$2$
are linearly equivalent to each other, so let us assume that
$m>2$
. By Lemma 7.2, the subgroup
$K=\mathrm {Stab}_{G_A}(E_1, \ldots , E_m)$
is a normal subgroup of
$G_A$
contained in
$H_2$
and such that
$G_A/K\cong D_{2m}$
. Then K corresponds to a field
$F\subseteq \mathbb {C}(t)$
containing
$L_2$
and such that