Elliptic Curves with Long Arithmetic Progressions Have Large Rank

Garcia-Fritz, Natalia; Pasten, Hector

doi:10.1093/imrn/rnaa061

Cited by 3 publications

(1 citation statement)

References 52 publications

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“…Recent research in rank studies has tackled three key areas: calculating ranks for specific families of curves [15][16][17][18][19], exploring how rank behaves for curves constructed from special number sequences [20][21][22][23][24][25][26][27][28] and analyzing rank distributions within families and across field extensions [29][30][31]. Dujella [32] provided an enumeration of the strategies for generating high-rank Diaphontine elliptic curves.…”

Section: Introductionmentioning

confidence: 99%

Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks

Dalkılıç,

Altınışık

2024

Journal of New Theory

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The theory of elliptic curves is one of the popular topics of recent times with its unsolved problems and interesting conjectures. In 1922, Mordell proved that the group of $\mathbb{Q}$-rational points on an elliptic curve is finitely generated. However, the rank of this group, signifying the number of independent generators, can be arbitrarily high for certain curves, a fact yet to be definitively proven. This study leverages the computer algebra system Magma to investigate curves with potentially high ranks using a technique developed by Mestre.

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Section: Introductionmentioning

confidence: 99%

Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks

Dalkılıç,

Altınışık

2024

Journal of New Theory

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Intersecting the Torsion of Elliptic Curves

GARCIA-FRITZ,

PASTEN

2023

Bull. Aust. Math. Soc.

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Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree- $2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree- $2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

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Families of cubic elliptic curves containing sequences of consecutive powers

Salami¹,

Zargar²

2021

Rocky Mountain J. Math.

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Elliptic Curves with Long Arithmetic Progressions Have Large Rank

Cited by 3 publications

References 52 publications

Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks

Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks

Intersecting the Torsion of Elliptic Curves

Families of cubic elliptic curves containing sequences of consecutive powers

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