Algebraic Points of Given Degree on the Affine Curve <img src=image/10.5923.j.ajms.20221201.02_001.gif></img>

El Hadji Sow; Moussa Fall; Oumar Sall

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American Journal of Mathematics and Statistics

p-ISSN: 2162-948X e-ISSN: 2162-8475

2022; 12(1): 5-8

doi:10.5923/j.ajms.20221201.02

Received: Feb. 16, 2022; Accepted: Mar. 4, 2022; Published: Mar. 15, 2022

Algebraic Points of Given Degree on the Affine Curve

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El Hadji Sow, Moussa Fall, Oumar Sall

U.F.R. of Science and Technology, Mathematics and Applications Laboratory, Assane SECK University of Ziguinchor, Senegal

Correspondence to: El Hadji Sow, U.F.R. of Science and Technology, Mathematics and Applications Laboratory, Assane SECK University of Ziguinchor, Senegal.

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This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

In this work, we determine the set of algebraic points of given degree over on the curve of affine equation This note extends a result of Booker, Sijsling, Sutherland, Voight and Yasak in [1] who gave a description of the set of -rational points i.e the set of points of degree one over on this curve.

Keywords: Planes curves, Degree of algebraic points, Rationals points, Algebraic extensions, Jacobian

Cite this paper: El Hadji Sow, Moussa Fall, Oumar Sall, Algebraic Points of Given Degree on the Affine Curve , American Journal of Mathematics and Statistics, Vol. 12 No. 1, 2022, pp. 5-8. doi: 10.5923/j.ajms.20221201.02.

Article Outline

1. Introduction

2. Auxiliary Results

3. Proof of Theorem

1. Introduction

Let

be a smooth algebraic curve defined over

Let

be a numbers field. We note by

the set of points of

with coordinates in

and

the set of points of

with coordinates in

of degree at most

over

The goal is to determine the set of algebraic points of given degree over

on the curve

given by the affine equation

The Mordell-Weil group

of rational points of the Jacobian is a finite set (refer to [1,4]).

We denote by:

and

the point at infinity. In [1] Booker, Sijsling, Sutherland, Voight and Yasak gave a description of the rational points over

on this curve. This description is as follows:

Proposition: The

-rational points on

are given by

In this note, we give an explicit description of algebraic points of given degree over

on the curve

Our main result is given by the following theorem:

Theorem: The set of algebraic points of given degree over

on the curve

isgiven by:

With:

and

where

and

2. Auxiliary Results

For a divisor

, we note

the

-vector space of rational functions

defined on

such that

;

designates the

. In [1, 4] the Mordell-Weil group

is isomorph to

and

is a hyperelliptic curve of genus

Let

be two rational functions on

defined as follow:

The projective equation of

We denote by

and let's put

for

Let us designate by

the intersection cycle of algebraic curve

defined on

and

Lemma 1:

Consequence of lemma 1:

Lemma 2:

Proof:

- We have

since if

then the curve

is of genus zero (see [1,4]), which is not the case.

- Since the genus of

is equal to 2, then

is a canonical divisor of

- For the rest we apply the Riemann-Roch theorem which says that

Lemma 3:

-base of

is given by

Proof:

It is clear that

is free and it remains to show that

According to the Riemann-Roch theorem, we have

According to the parity of

we have the following two cases:

Case 1: Suppose that

is even and let

Thus

we have

and we have

. Then we get

we have

Case 2: Suppose that

is odd and let

Thus we have

and

Then we get

We have

Lemma 4:

Proof: (See [1,4]).

3. Proof of Theorem

Given

The work of Booker, Sijsling, Sutherland, Voight and Yasak in [1] allows us to assume that

Note that

are the Galois conjugates of

Let's work with

according to lemma 4 we have

So we have

Our proof is divided in three cases:

Case

We have

then there exist a function

with coefficient in

such that

then

and according to lemma 3 we have

For the points

we have

hence

and the relation

gives the equation

We find a family of points

Cases

and

For

we have

then there exist a function

with coefficient in

such that

then

and according to lemma 3 we have

We have

For the points

we have

hence

and the relation

gives the equation

We find a family of points

For

we have

By a similar argument as in case

we have

Finally, we have the family

Cases

and

For

, we have

then there exist a function

with coefficient in

such that

then

and according to lemma 3 we have

The function

is of order 2 at point P so we must have

For the points

we have

hence

and the relation

gives the equation

We find a family of points

For

we have

By a similar argument as in case

we have

Finally, we have the family

References

[1]	A. R. Booker, J. Sijsling, A. V. Sutherland, J. Voight and D. Yasak, A database of genus-2 curves over the rational numbers. LMS Journal of Computation and Mathematics, 19(A), 235-254, 2016.
[2]	N. Bruin, On powers as sums of two cubes, International Algorithmic Number Theory Symposium. Springer, Berlin, Heidelberg, 2000.
[3]	A. Chenciner, Courbes algébriques planes. Springer, 2008.
[4]	LMFDB Collaboration, The L-functions and Modular Forms Database. Available at: https://www.lmfdb.org. [Online; accessed 8 November 2021].
[5]	E. L. García, Diophantine Geometry,Course notes from the CIMPA school "Functional Equations: Theory, Practice and Interactions" held in Hanoi from 12-23 April 2021.
[6]	P. A. Griffiths, Introduction to algebraic curves, Translations of mathematical monographs volume 76. American Mathematical Society, Providence (1989).
[7]	M. Hindry, J. H. Silverman, Diophantine Geometry, An Introduction, Graduate Texts in Mathematics, January 1, 2000.
[8]	J. TH. Mulholland, Elliptic curves with rational 2-torsion and related ternary Diophantine equations. ProQuest LLC. Ann Arbor, MI (2006).
[9]	S. Siksek, Explicit Chabauty over number fields, Algebra & Number Theory, Volume 7, No. 4, 765-793, 2013.
[10]	E. H. Sow, M. Fall, O. Sall, Points algébriques de degrés au-plus 5 sur la courbe d'équation affine SCIREA Journal of Mathematics, Volume 6, Issue 6, 2021.
[11]	E. H. Sow, P. M. Sarr, O. Sall, Algebraic Points of Degree at Most 5 on the Affine Curve , Asian Research Journal of Mathematics, 51-58, 2021.

Paper Information

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Algebraic Points of Given Degree on the Affine Curve

Article Outline

1. Introduction

2. Auxiliary Results

3. Proof of Theorem

References