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Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales
versão impressa ISSN 0370-3908
Rev. acad. colomb. cienc. exact. fis. nat. vol.36 no.139 Bogotá abr./jun. 2012
MATEMÁTICAS
![](img/revistas/racefn/v36n139/v36n139a08sind.jpg)
* Departamento de Matemáticas, Universidad del Valle, Apartado Aéreo 25360, Cali, Colombia E-mail: alvarogr@univalle.edu.co
Universidad Industrial de Santander. Bucaramanga, Colombia. E-mail: ateheran@gmail.com
AMS Classification 2000: 14G05.
Abstract
In this paper we give a generalization of two results obtained by Garcia and Stichtenoth and use them to exhibit a method to construct curves over finite fields whose number of rational points is large compared to their genus. Such curves are induced by algebraic functions fields obtained from elementary abelian
-extensions of the rational function field
using the trace operator
Key words: Finite Fields, Algebraic Curves, Algebraic Function Fields, Elementary Abelian -Extensions, Rational Points.
Resumen
En este articulo generalizamos dos resultados obtenidos por García & Stictenoth en ([G-S]) y usamos estas generalizaciones para construir curvas sobre cuerpos finitos cuyo número de puntos racionales es grande en comparación con su género. Tales curvas son obtenidas considerando -extensiones abelianas elementales del cuerpo de funciones racionales
usando el operador traza
.
Palabras clave: Cuerpos finitos, curvas algebraicas, cuerpos algebraicos de funciones, âextensions abelianas elementales, puntos racionales
1. 1. Elementary Abelian -Extensions
![](img/revistas/racefn/v36n139/v36n139a08sk.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s1.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sk.jpg)
![](img/revistas/racefn/v36n139/v36n139a08eq1.jpg)
the Artin-Schreier operator.
Definition 1.1. For a subset we denote by
, the splitting field of all polynomials
, with
. For
such that
the extension
is called an Artin-schreier extension of
.
The following theorem provides a complete description of the Artin-schreier extensions. Its proof depends essentially on the following lemma.
Lemma 1.2. (Hilbert's Theorem 90) Let be a finite extension of
whose Galois group G is cyclic generated by
. Then
has the form
for some
, if and only if
.
Prooƒ: see [L-N] Theorem 2.25.
Theorem 1.3. Let be a field of characteristic
. The polynomial
![](img/revistas/racefn/v36n139/v36n139a08e1.jpg)
either splits completely over or else,
is irreducible over . Moreover the following assertions are equivalent:
Prooƒ: Suppose that(1)
is a cyclic extension of degree
.
(2)
, whose minimal polynomial over
is
, where
is defined as (1), for some
.
(3)
is the splitting field of an irreducible polynomial of the form (1), for some
.
![](img/revistas/racefn/v36n139/v36n139a08s17.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s18.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s19.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s14.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sp.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s20.jpg)
Now, its is clear that, if , then
is the splitting field of
.
It remains to consider the case . Let
. To prove that
is irreducible over
it is enough to prove that
, that is to say, that
is the minimal polynomial of
over
(which, from now on, we will denote by min
).
Since is the splitting field of the polynomial
we have that
is a Galois extension, therefore, it is sufficient to show that
.
For this end, observe that since each is completely determined by its action on
and
permutes all the roots of
, then
for some
, hence,
.
Now we will to prove the equivalences:
suppose that
is a cyclic extension of degree
and let
be such that
. Since
, then by Lemma 1.2, there exist
such that
. Moreover, since
and
.
On the other hand, observe that . That is to say,
, then
therefore, there exist
such that
and consequently
satisfies the polynomial
.
Now, since
![](img/revistas/racefn/v36n139/v36n139a08e2.jpg)
and
![](img/revistas/racefn/v36n139/v36n139a08s21.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s42.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s43.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s44.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s45.jpg)
If
, with
we have that
is the splitting field of min
.
Assume that
is the splitting field of an irreducible polynomial of the form
for some
. Again by similar arguments as above we obtain that
, which means that
is cyclic of degree
.
Definition 1.4. An extension is said to be an Elementary Ableian
-Extension of exponent
and degree
if
is Galois with
The following Theorem states a relationship between the additive subgroups of and the elementary abelian
-extensions. To this end, we first need to establish a result.
Theorem 1.5. Let be a field of characteristic
>0. There exist a one to one correspondence between the additive subgroups U of
containing
F which have finite index (U:
F), and the elementary abelian
-extensions. This correspondence is given by
![](img/revistas/racefn/v36n139/v36n139a08eq2.jpg)
In such case
![](img/revistas/racefn/v36n139/v36n139a08eq3.jpg)
The inverse map of is given by
![](img/revistas/racefn/v36n139/v36n139a08ss2.jpg)
Prooƒ: see {16} page 263
Remark 1.6. observe that, regarding abelian P-torsion group as vector spaces over we can as well define
as the map (induced by
) that takes finite-dimensional vector subspaces
(over
) of the quotient space
to finite dimensional subspaces
of the
vector space
(where
F is the inverse image under
in some fixed separable closure
). That is to say
![](img/revistas/racefn/v36n139/v36n139a08eq5.jpg)
Finally if instead of we consider the map
![](img/revistas/racefn/v36n139/v36n139a08eq6.jpg)
one can see that, one such n-dimensional subspace corresponds, in the notation of Theorem 1.5 to a subgroup
with
that is a "section" of
in the sense that
. Therefore, if
and
are subgroups of
such that:
![](img/revistas/racefn/v36n139/v36n139a08e3.jpg)
then, the following sentences are equivalent.
(Observe that
exactly when
( more accurately, when
and
are the same subspace of
.)) Moreover, if
, then
![](img/revistas/racefn/v36n139/v36n139a08e4.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sf.jpg)
![](img/revistas/racefn/v36n139/v36n139a08eq8.jpg)
Then, the extension is an elementary abelian
-extension of
of exponent
.
Prooƒ: first observe that since E is the splitting field of the set of polynomials then the extensions
is a Galois extensions. On the other hand, since U is an additive subgroup of
and char(F) =
, then there exists
nonzero elements such that
![](img/revistas/racefn/v36n139/v36n139a08e5.jpg)
We can find , such that
. Now, by Theorem 1.3 we have that
![](img/revistas/racefn/v36n139/v36n139a08e6.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s80.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s81.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e7.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s82.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s83.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sef.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s84.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e8.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s85.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e10.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e11.jpg)
But
![](img/revistas/racefn/v36n139/v36n139a08e12.jpg)
So, for , since
then
![](img/revistas/racefn/v36n139/v36n139a08e15.jpg)
consequently, . Next we prove that
For
we define
as follows,
![](img/revistas/racefn/v36n139/v36n139a08e16.jpg)
Observe that is the identity on
Therefore by Theorem 1.3, the polynomial is irreducible over
(otherwise
and
which is a contradiction) hence the
are actually well defined. Now it clear that each
and
, also
![](img/revistas/racefn/v36n139/v36n139a08ss4.jpg)
In fact, if then,
![](img/revistas/racefn/v36n139/v36n139a08e17.jpg)
it follows that i = j and (mod
). Therefore
![](img/revistas/racefn/v36n139/v36n139a08ss5.jpg)
The converse of the Theorem 1.7 also holds. To prove it, we need the following lemma.
Lemma 1.8. Suppose that L/M is finite Galois extension with Galois Group of the form
![](img/revistas/racefn/v36n139/v36n139a08e18.jpg)
where
![](img/revistas/racefn/v36n139/v36n139a08s100.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s101.jpg)
![](img/revistas/racefn/v36n139/v36n139a08SS6.jpg)
then
![](img/revistas/racefn/v36n139/v36n139a08SS7.jpg)
Prooƒ: see {Ro} Corollary 5.5.4.
Theorem 1.9. If
![](img/revistas/racefn/v36n139/v36n139a08Sef.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sp.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sf.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s52.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e20.jpg)
for some additive subgroup U of which satisfies (3).
Prooƒ: since is an elementary abelian extension of degree
, then
![](img/revistas/racefn/v36n139/v36n139a08e21.jpg)
thus, , where each
has order
. Let us define for
and
, be as in the Lemma 1.8 (a)
![](img/revistas/racefn/v36n139/v36n139a08e22.jpg)
consequently, by Theorem 1.3. there exist such that
for some
with
, which amounts to,
. Observe that
and since
, from Lemma 1.8 (b),
, and therefore
![](img/revistas/racefn/v36n139/v36n139a08e23.jpg)
We now claim that are linearly independent over
. In fact suppose there is a non-trivial linear combination
,with
, then
![](img/revistas/racefn/v36n139/v36n139a08ss8.jpg)
hence . Now if we assume that
then
and since, for
, we have that
, then
. On the other hand by Lemma 1.8,
, consequently
which is a contradiction since
. Therefore
. Similar arguments will lead us to prove that
. Let U be the subgroup generated by
. Then by (23),
therefore only remains to prove that
. In fact, if
, then
with
and each
. Therefore it is enough to prove that
. For this end, observe that
![](img/revistas/racefn/v36n139/v36n139a08e24.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s137.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e25.jpg)
which is a contradiction.
Theorem 1.10. Let U be an additive subgroup of such that
![](img/revistas/racefn/v36n139/v36n139a08ss9.jpg)
If , then there exit
- intermediate fields
such that
, where
with
.
Prooƒ: if , then
with
and by Theorem 1.3,
is either 1 or
. but if
, then
therefore
which is a contradiction with the choice of u, hence
. On the other hand, if
is a subfields such that
and
then by Theorem 1.3,
is the splitting field of one irreducible polynomial of the form
- for some
. Now, since
then by Remark 1.6 we have that
hence
, for some
and
, from which
,for some
for some
. In sum each subfield
such that
and
has the form
for some
. Finally by Remark 1.6 we obtain the number of these subfields.
Theorem 1.11. Let K be a field of characteristic p > 0 an F/K an algebraic function field of transcendence degree one over K, with constant field K and genus g(F). Consider an elementary abelian extension E/F of degree pn such that K is also the constant field of E. denote by the intermediate fields
and by g(E) (resp
) the genus of
(resp
). Then
![](img/revistas/racefn/v36n139/v36n139a08e26.jpg)
Prooƒ: Let
![](img/revistas/racefn/v36n139/v36n139a08s173.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s174.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s175.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s176.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s177.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s178.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss10.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss11.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s179.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss12.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s180.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s181.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s182.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s183.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s184.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s185.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s186.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s187.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s188.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s189.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s190.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s191.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e27.jpg)
Now we shall show that any is contained in precisely t subgroup
. In fact, each
has the form
![](img/revistas/racefn/v36n139/v36n139a08e28.jpg)
where each has order
and the set
is a basis of
over
. Now, if
, then
if and only if
for
. That is to say, there exist
subgroups Hj such that
. In other words σ is contained in precisely
subgroup
of G, therefore
![](img/revistas/racefn/v36n139/v36n139a08e29.jpg)
But from and
![](img/revistas/racefn/v36n139/v36n139a08e32.jpg)
It follows that, . Thus
![](img/revistas/racefn/v36n139/v36n139a08e33.jpg)
Which amounts to, we have the following relation in
![](img/revistas/racefn/v36n139/v36n139a08e36.jpg)
The theorem now follows from Kani's result.
Observe that the intermediate extension mentioned in Theorem 1.10 is an Artin-Schreier extension, whose genus,
can be computed by [[ST], III.7.8]. This takes us to determine explicitly such intermediate field, which we will call Artin-Schreier intermediate subfields, for which we give the following results generalizing Propositions 1.1 and 1.2 in ([G-S]).
Before that, we should give a definition. We call a polynomial of the specific form
![](img/revistas/racefn/v36n139/v36n139a08sS13.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sP.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s209.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s210.jpg)
Theorem 1.12. Let
![](img/revistas/racefn/v36n139/v36n139a08ss19.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s260.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sP.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s260.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s211.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s212.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sf.jpg)
![](img/revistas/racefn/v36n139/v36n139a08E37.jpg)
Prooƒ: Let us consider the set
![](img/revistas/racefn/v36n139/v36n139a08s213.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s56.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sP.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s214.jpg)
![](img/revistas/racefn/v36n139/v36n139a08E38.jpg)
and, the set
![](img/revistas/racefn/v36n139/v36n139a08s215.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s56.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s216.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s217.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss20.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s89.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s218.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s89.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sp.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e39.jpg)
On the other hand, since
![](img/revistas/racefn/v36n139/v36n139a08s219.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e40.jpg)
With
![](img/revistas/racefn/v36n139/v36n139a08s220.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s221.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e41.jpg)
where . It follows from (40) that
if and only if
for k = 1,2,â¦,n, that is,
if and only if
, and therefore
. On the other hand, if
and
then, since
, for some
, we have
![](img/revistas/racefn/v36n139/v36n139a08ss14.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s233.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s234.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s14.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s235.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s236.jpg)
Reciprocally we have:
Theorem 1.3. Let be a separable, monic, additive polynomial of degree
n, with its roots in
and
such that the polynomial
is irreducible over
, then the extension
where
is an elementary abelian
-extension of degree
n. The intermediate subfield
with
, have the form
where,
with
and each
satisfies the equation
therefore
, with
Prooƒ: It is clear that
![](img/revistas/racefn/v36n139/v36n139a08sphi.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sf.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s251.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sphi.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s252.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s253.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sf.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sphi.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s254.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sphi.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s255.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s256.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s257.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s258.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s259.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sp.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s260.jpg)
Now, if , then
and
it follow that
. On the other hand, since
then
and therefore
![](img/revistas/racefn/v36n139/v36n139a08ss15.jpg)
consequently It is say,
Now, by Remark 1.6 there exist
such subextensions
and therefore
where
. Finally by Theorem 1.10 there exist exactly
intermediate fields
with
therefore such
must be one of the
.
2. An application to the construction of curves over finite fields
It is well known that algebraic function fields over finite fields have many applications in coding theory, and the latter is closely related to cryptography, see for example [N-Ch]. In this section we exhibit a method to construct algebraic function fields over finite fields (algebraic curves) with many rational places (rational points).
Let be a prime number,
the finite field with
elements and
the rational function field over the finite field
. By E/K we mean a function field of transcendence degree one over K , with constant field K. We denote by
the maximum number of rational places of the function field E/K of genus g(E/K) = g. The Hasse-Weil bound implies
![](img/revistas/racefn/v36n139/v36n139a08e42.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s282.jpg)
In 1981 Ihara showed in [1] that
![](img/revistas/racefn/v36n139/v36n139a08e43.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s285.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e44.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e45.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s286.jpg)
To begin our construction, let us benote by the additive polynomial
![](img/revistas/racefn/v36n139/v36n139a08e46.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s287.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s288.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s289.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e47.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss21.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss22.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss23.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss24.jpg)
Theorem 2.1. The polynomial defined as (47)has the following property:
![](img/revistas/racefn/v36n139/v36n139a08ss26.jpg)
Proof. It is enough to prove that
![](img/revistas/racefn/v36n139/v36n139a08ss27.jpg)
Since
![](img/revistas/racefn/v36n139/v36n139a08ss28.jpg)
for some polynomial , then
![](img/revistas/racefn/v36n139/v36n139a08s300.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss30.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss40.jpg)
Remark 2.2. Observe that in accordance with Theorem 2.1, we have that, for , the equation
![](img/revistas/racefn/v36n139/v36n139a08e48.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s304.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s305.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s306.jpg)
The following result provides us a relationship among the genus of the function field E/K and the genus of the Artin-Schreier intermediate subfields .
Theorem 2.3. With the previous notations, the genus of E/K is given by
![](img/revistas/racefn/v36n139/v36n139a08e49.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s309.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e50.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s310.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s311.jpg)
Remark 2.4. Lemma III.7.7 in [ST], guarantees that we can exclude the case
![](img/revistas/racefn/v36n139/v36n139a08s312.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s313.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s314.jpg)
Indeed, first observe that from ([ST], i.4.18) together with Theorem 1.11 we have that:
![](img/revistas/racefn/v36n139/v36n139a08e51.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s315.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e52.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s316.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s317.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s318.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sf.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s377.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s319.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s320.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e53.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s321.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s322.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s323.jpg)
Next, we exhibit a technique that allow us to count the rational places of E/K. for this, we will denote by
![](img/revistas/racefn/v36n139/v36n139a08s324.jpg)
Lemma 2.5. Let us consider the polynomial
![](img/revistas/racefn/v36n139/v36n139a08e54.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s325.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s326.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s327.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s328.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s329.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s324.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s330.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s325.jpg)
Prooƒ: Let us suppose that exists
![](img/revistas/racefn/v36n139/v36n139a08s328.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s324.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s332.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s333.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s334.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s335.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s336.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s338.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s328.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s339.jpg)
Lemma 2.6. With the above notations,
![](img/revistas/racefn/v36n139/v36n139a08s340.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s341.jpg)
Prooƒ: By the division algorithm, there exists such that
![](img/revistas/racefn/v36n139/v36n139a08s343.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s344.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss31.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e55.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s345.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s346.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss32.jpg)
As consequence of all the above mentioned we have,
Theorem 2.7. The number of rational places of the elementary abelian p-extension defined by (47) is given by
![](img/revistas/racefn/v36n139/v36n139a08e57.jpg)
Prooƒ: For fixed , we have that (x, y) is a rational point of the curve
if and only if
where
. In fact, if (x, y) is a rational point of
then
![](img/revistas/racefn/v36n139/v36n139a08e58.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss37.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss36.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e59.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss38.jpg)
Now, for each
![](img/revistas/racefn/v36n139/v36n139a08ss33.jpg)
![](img/revistas/racefn/v36n139/v36n139a08ss39.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s358.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sp.jpg)
![](img/revistas/racefn/v36n139/v36n139a08sf.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s359.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s360.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e60.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s361.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s362.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s363.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s364.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s365.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s366.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s367.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s368.jpg)
3. Examples
In this section we give examples of elementary the Abelian p-extensions of the kind given by (47). We will to consider the particular case when n is odd and and we will determine the genus and the number of rational places of these extensions using the formulas (49) and (57).
Example 3.1 It p = 2 and n =3, then k =1, q =8. Also,
![](img/revistas/racefn/v36n139/v36n139a08e62.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s370.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s371.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s372.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s373.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e65.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e66.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s374.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e68.jpg)
Example 3.2. Taking p = n = 3, then q = 27 and k = 1. Also,
![](img/revistas/racefn/v36n139/v36n139a08e69.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s375.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s376.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e72.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e73.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s377.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s378.jpg)
![](img/revistas/racefn/v36n139/v36n139a08e76.jpg)
![](img/revistas/racefn/v36n139/v36n139a08s379.jpg)
The following table contains the values obtained for the genus and the number of rational points by taking different values for p and n, also we compare this values obtained with the Ihara's bound
![](img/revistas/racefn/v36n139/v36n139a08e77.jpg)
Acknowledgements. The authors deeply appreciate the helpful comments suggestions made by the referees.
References [ Links ] 1991.
[L-N] Lidl Rudolf and Niederreiter Harald. Introduction to finite fields and their applications. Cambridge university press, [ Links ] 1994.
[Go] V.D. Goppa, Codes on algebraic curves. Sov Math.Dokl 24 (1981), 170-172. [ Links ]
[La] Lang Serge, Algebra, Adisson Wesley Publishing Company, [ Links ] 1970.
[Ka] Kani Ernest, Relations between the genera and between the Hasse-Witt invariants of Galois covering of curves, Canad. Math. Bull, Vol 28, pag 321-327, [ Links ]1985.
[N-Ch] Harald Niederreiter, huaxiong Wang and Chaoping Xing, Function Fields over Finite Fields and their applications to Cryptography, Springer- verlag. [ Links ] 2007
[I] Ihara Y. Some remarks on number of rational points of algebraic curves over finite fields. J Fac Sci Tokyo 28 (1981), p.721-724. [ Links ] [Ro] Roman Steve. Field Theory. Springer-Verlag, [ Links ] 1991
[ST] Stichtenoth Hennig. Algebraic functions fields and codes. Springer- verlag, [ Links ] 1993.
[VV] Van Der Geer Gerard and Van Der Vlugt Marcel. Tables of curves with many points. [Online], http://www.science.uva.nl/ geer. [ Links ]
Recibido el 4 de noviembre de 2009
Aceptado para su publicación el 21 de junio de 2010
Versión revisada recibida el 5 de marzo de 2012