Integer points on elliptic curves

Chi, Wen-Chen; Lai, King Fai; Tan, Ki-Seng

doi:10.2140/pjm.2005.222.237

Pacific J. Math.

2005

DOI: 10.2140/pjm.2005.222.237

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Integer points on elliptic curves

Wen-Chen Chi¹,

King Fai Lai

Ki-Seng Tan

Abstract: We study Lang's conjecture on the number of S-integer points on an elliptic curve over a number field. We improve the exponent of the bound of Gross and Silverman from quadratic to linear by using the S-unit equation method of Evertse and a formula on 2-division points.

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Integral points on the congruent number curve

Chan

2022

Trans. Amer. Math. Soc.

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We study integral points on the quadratic twists E D : y 2 = x 3 − D 2 x \mathcal {E}_D:y^2=x^3-D^2x of the congruent number curve. We give upper bounds on the number of integral points in each coset of 2 E D ( Q ) 2\mathcal {E}_D(\mathbb {Q}) in E D ( Q ) \mathcal {E}_D(\mathbb {Q}) and show that their total is ≪ ( 3.8 ) rank ⁡ E D ( Q ) \ll (3.8)^{\operatorname {rank} \mathcal {E}_D(\mathbb {Q})} . We further show that the average number of non-torsion integral points in this family is bounded above by 2 2 . As an application we also deduce from our upper bounds that the system of simultaneous Pell equations a X 2 − b Y 2 = d , b Y 2 − c Z 2 = d aX^2-bY^2=d,\ bY^2-cZ^2=d for pairwise coprime positive integers a , b , c , d a,b,c,d , has at most ≪ ( 3.6 ) ω ( a b c d ) \ll (3.6)^{\omega (abcd)} integer solutions.

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Integral points on the congruent number curve

Chan

2022

Trans. Amer. Math. Soc.

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ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y² = x³ + Ax

Draziotis

2011

Int. J. Number Theory

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Integer points on elliptic curves

Abstract: We study Lang's conjecture on the number of S-integer points on an elliptic curve over a number field. We improve the exponent of the bound of Gross and Silverman from quadratic to linear by using the S-unit equation method of Evertse and a formula on 2-division points.

Cited by 2 publications

References 8 publications

Integral points on the congruent number curve

Integral points on the congruent number curve

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y² = x³ + Ax

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Integer points on elliptic curves

Abstract: We study Lang's conjecture on the number of S-integer points on an elliptic curve over a number field. We improve the exponent of the bound of Gross and Silverman from quadratic to linear by using the S-unit equation method of Evertse and a formula on 2-division points.

Cited by 2 publications

References 8 publications

Integral points on the congruent number curve

Integral points on the congruent number curve

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y2 = x3 + Ax

Contact Info

Product

Resources

About

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y² = x³ + Ax