ELEMENTARY ABELIAN P-EXTENSIONS AND CURVES WITH MANY POINTS

Garzón R., Álvaro; Teherá Herrera, Arnoldo

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Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales

versión impresa ISSN 0370-3908

Rev. acad. colomb. cienc. exact. fis. nat. vol.36 no.139 Bogotá abr./jun. 2012

MATEMÁTICAS

ELEMENTARY ABELIAN P-EXTENSIONS AND CURVES WITH MANY POINTS

Álvaro Garzón R.^*, Arnoldo Teherán Herrera

^* 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

Throughout this note we denote by

a perfect field of characteristic

, by

an algebraic function field with constant field

and by

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

either splits completely over or else, is irreducible over . Moreover the following assertions are equivalent:

(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 .

Prooƒ: Suppose that

, then for

we have

and since

is a separable polynomial of degree

, it follows that

are all its roots.

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

and

then

. Thus

and (2) ensures that

, which is the same as,

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

In such case

The inverse map of is given by

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

Finally if instead of we consider the map

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:

then, the following sentences are equivalent.

(Observe that exactly when ( more accurately, when and are the same subspace of .)) Moreover, if , then

Theorem 1.7. Let U be an additive subgroup of such that

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

We can find , such that . Now, by Theorem 1.3 we have that

and since

for each

(Remark 1.6) we obtain

Now, since

is a root of

, then the extension

is finite. In order to prove that

, first observe that,

Since

, by Theorem 1.5 we obtain

as well as

But

So, for , since then

consequently, . Next we prove that For we define as follows,

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

In fact, if then,

it follows that i = j and (mod ). Therefore

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

where is the coordinate and

then

Prooƒ: see {Ro} Corollary 5.5.4.

Theorem 1.9. If is an elementary abelian -extension of of degree , then

for some additive subgroup U of which satisfies (3).

Prooƒ: since is an elementary abelian extension of degree , then

thus, , where each has order . Let us define for and , be as in the Lemma 1.8 (a)

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

We now claim that are linearly independent over . In fact suppose there is a non-trivial linear combination ,with , then

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

therefore

, we have that

which is a contradiction.

Theorem 1.10. Let U be an additive subgroup of such that

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

Prooƒ: Let

. For a subgroup

consider the fixed field

whose genus

is denoted by

and the trace idempotent

The idea of he proof is to construct a relationship of the kind

With

, because in this case genera would satisfy the same relation (see[Ka])

First, observe that G has exactly t subgroups with order

. In fact, by one side, for

the Galois group

of the extension

is one of such subgroups. On the other hand, if

is subgroup such that

then since

is a Galois extension with

, we have

from which we get

and

for some

consequently,

Now we shall show that any is contained in precisely t subgroup . In fact, each has the form

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 H_j such that . In other words σ is contained in precisely subgroup of G, therefore

But from and

It follows that, . Thus

Which amounts to, we have the following relation in

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

(where

= char (K)) an additive polynomial over K. observe that

is separable if and only if

.
Theorem 1.12. Let be a separable, monic, additive polynomial of degree , with its roots in F. If E/F is an elementary abelian -extension of degree , then there exists an element such that whose minimal polynomial over has the form

Prooƒ: Let us consider the set

, it is clear that W is a vectorial space over

, moreover W is an additive finite subgroup of F. Now, since each cyclic subgroup of W is order 1 or

, then there exist nonzero elements

such that

and, the set

is a basis of W over

. Now from Lemma 1.8 we can choose

such that

, with

. If we define the

for

as in (16), it is clear that each

is an element of order

of Gal (E/F) and therefore

On the other hand, since

, then σ has a unique representation

With

, then the action of σ over the element

is given by

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

That is to say

and therefore

, consequently y is a root of the monic polynomial

whose degree is

. This implies

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

is the minimal polynomial for y over

. Now, for each

then all roots of the polynomial

have the form

with

, and therefore

(y) is the splitting field of the polynomial

. On the other hand since

permutes the roots of

then

for some

, therefore, the application

, into group

is an isomorphism, that is to say,

is an elementary abelian

-extension of degree

Now, if , then and it follow that . On the other hand, since then and therefore

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

After Weil proved his bound around 1940 the question how many rational places may lie on a function field over a finite field

remained untouched for many years. In 1980 Goppa came up with the beautiful idea to associate an error-correcting code to a linear system on a curve over a finite field, see [Go]. In order to construct good codes one needs function fields with many places and thus Goppa's work led to a revival of interest in rational points on function fields (algebraic curves) over finite fields. Applications in cryptography and recent constructions of quasi-random points sets also require curves with many points and added further impetus to work in the field.

In 1981 Ihara showed in [1] that

For

this bound is better that Weil`s bound and gives the asymptotic bound

Refining Ihara's idea to derive (44) Drinfeld and Vladut proved that

Since the asymptotic bound (45) of Drifeld-Vladut is approximately

times the asymptotic Ihara bond (44) we think that it is reasonable to put this qualification as requirement to filter out curves which should be considered 'poor'.

To begin our construction, let us benote by the additive polynomial

We will consider extensions of the rational function field

of the kind

where

is defined by the equation:

where

denote the remainder of the Euclidean division of the polynomial

. That is to say extensions of the kind,

The reason to consider this type of extensions is that the number of rational places of E/K is related with the image of the function

. More precisely we have.

Theorem 2.1. The polynomial defined as (47)has the following property:

Proof. It is enough to prove that

Since

for some polynomial , then

Now, since

and

is additive, the result follows.

Remark 2.2. Observe that in accordance with Theorem 2.1, we have that, for , the equation

has

solutions in

, therefore the induced curve by the function field E/K has at least

places of degree one. This leads us to expect to get good curves.

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

where is defined as follows:

and

with

Remark 2.4. Lemma III.7.7 in [ST], guarantees that we can exclude the case

for an integer

in the above definition for

Indeed, first observe that from ([ST], i.4.18) together with Theorem 1.11 we have that:

On the other hand, by Theorem 1.13, each the Artin-Schreier subfield

has the form

where and

and

. Now, since

, then each of

different from

is unramified in

, in this way, from ([ST], III.7.8) the genus of

is given by

Now, since there exist exactly

different subfields of the Artin-Schreier, then there are

in W such that, each one those the Artin-Schreier subfields have the form

. Finally, from (53) and (51) we have the result.

Next, we exhibit a technique that allow us to count the rational places of E/K. for this, we will denote by

, the induced curve by the function field E/K.

Lemma 2.5. Let us consider the polynomial

where and are defined as in (47). Then, for , there exist such that - belongs to , if and only if , for some

.

Prooƒ: Let us suppose that exists

such that (x, y) belongs to

, it is to say

. Since

, then by taking

we obtain the desired result. Reciprocally, if

for some

and since the function a is surjective ([L-N], 2.23), then

for some

, it is to say,

.

Lemma 2.6. With the above notations,

for all

Prooƒ: By the division algorithm, there exists such that

whit degree

degree

. Then

and since

for all

, we have

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

Where, γ denotes the number of rational places of E coming from the ramification, which is one or zero.

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

therefore

where

, then

and therefore (x, y) is a rational point of

Now, for each

such that

, we have

places of degree one of E. Additionally, since for all place

different of

, P is unramified in E, then

or 0. Therefore,

where

. On the other hand, if

, then

if and only if

and

and since

is separable, then

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,

in this case, there are exactly t = 3 different subfields E which are the Artin-Schreier extensions over F. For all

we have

and since

then, from ([ST], III.7.8), we obtain

. Observe that this value is independ of the root λ, therefore there exist exactly 3 different the Artin-Schreier extensions generated by such roots, and in accordance with (49) we have

Now, for the number of rational places we have that

Since, the place

of F is the only place that ramifies in the extension E/F then by (57) we have

This is the best value known. See [VV]

Example 3.2. Taking p = n = 3, then q = 27 and k = 1. Also,

Then t = 4 and for all

we have,

. Therefore

Now,

and since

is totally ramified and rational in

then we have

We do not known any function field over

of genus 36 having more that 244 rational places (see [VV]).

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

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

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Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales

versión impresa ISSN 0370-3908

Rev. acad. colomb. cienc. exact. fis. nat. vol.36 no.139 Bogotá abr./jun. 2012