1. Introduction
Let
be an algebraic curve defined on number field
. We note
be the set of algebraic points on
defined on
and
the set of algebraic points on
to be coordinated in
of degree at most l over
. The degree of an algebraic point R is the degree of its defining field on
;
. A famous theorem of Faltings states that if
then the set
of algebraic points on
defined on
is finite. A generalization to subvarieties of an abelian variety allows a qualitative study of the set
of algebraic points on
of degree at most l over
.
We propose to study in detail the set of algebraic points of any degree given on
on the curve
of affine equation
.
Our affine equation curve
is a special case of quotients of Fermat curves of equations
,
studied in [1].
Let
,
and
denote the point at infinity of
. Consider the Jacobian folding defined by
We will designate J the Jacobian of
and by
the class denoted
of
.
Our approach relies on the knowledge of the Mordell-Weil group of the Jacobian J-variety of
and the condition that it is finite: it consists in using the Abel-Jacobi theorem to plunge the curve into its Jacobian and to study linear systems on the curve
.
The Mordell-Weil group
of rational points of the Jacobian J of
is finite and given by
( [2], p. 219 and [3]).
Our study results from the work of Gross-Rohrlich who determined
the set of algebraic points on
of degree at most 2 on
and given by the following proposition:
Proposition 1.
The set of algebraic points on
of degree at most 2 on
is given by
(1)
We extend these results by giving a geometric parametrization of algebraic points of any given degree on
on the curve
of affine equation
.
Our essential tools are:
1) The Mordell-Weil group
of the Jacobian of
.
2) The Abel-Jacobi theorem (see in [4] page 156).
3) The study of linear systems on the curve
.
4) The theory of intersection.
Our main result is as follows:
Theorem
The set of algebraic points of degree
on
is:
(2)
with
(3)
(4)
2. Auxiliary Results
Let x and y be the rational functions defined on
by:
and
.
For a divisor D on
, let
be the
-vector space of the rational functions f defined by
(5)
The projective equation of the curve
is:
.
We have the following Lemma:
Lemma 1
;
;
.
Proof 1 It is a calculation of type
(6)
From (6), we have
.
For
, the projective equation gives
; and for
, we obtain the point
of multiplicity equal to 11.
For
, the projective equation gives
; and for
, we obtain the point
of multiplicity equal to 11. Thus
.
In the same way we show that
and
.
Consequence 1
;
so
and
generate the same subgroup
.
Lemma 2 A
-base of
is given by :
(7)
Proof 2. It is clear that
is free. It remains to show that
.
By the Riemann-Roch theorem, we have
as soon as
with
Let us consider the following cases:
Case 1: Suppose that l is even, and let
. Then we have
and
.
So we obtain
,
and therefore
.
Case 2: Suppose that l is odd, and let
.
and
So we obtain
,
and therefore
.
3. Demonstration of the Theorem
Let
with
. Let
be the Galois conjugates of R, and let
which is a point of
; so
with
. This gives the relation
(8)
We note that
.
Case
Then there exists a rational function f such that
, so
. According to Lemma 2, we have
with
if l is even (otherwise the
would be equal to
) and
if l is odd (otherwise the
would be equal to
). At the points
we have
hense
and therefore
,
so
.
So the equation
becomes
which is an equation of degree l in y. We thus find a family of points of degree l
In the same way we show that for
with
, the relation (8) gives
. Then there exists a rational function f such that
, so
. According to the Lemma 2, we have
; and as
,
therefore
with
if l is even (otherwise the
would be equal to
) and
if l is odd (otherwise the
would be equal to
). At the points
we have
hense
and therefore
.
So the equation
becomes
which is an equation of degree l in y. We thus find a family of points of degree l