Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections

Kouakou Kouassi Vincent; Edjabrou Ulrich Blanchard Kablam

doi:doi:10.11648/j.ajam.20261406.12

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Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections

Kouakou Kouassi Vincent^*, Edjabrou Ulrich Blanchard Kablam

Published in American Journal of Applied Mathematics (Volume 14, Issue 6)

Received: 16 September 2026 Accepted: 9 October 2026 Published: 10 November 2026

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Abstract

We investigates an arithmetic family of congruent-number elliptic curves arising from the Euclid parametrization of Pythagorean triples, with the aim of determining its generic Mordell-Weil rank and understanding the occurrence of higher ranks under specialization. Our approach combines explicit constructions of rational sections with the geometric theory of elliptic surfaces. A suitable quadratic base change produces a new elliptic surface on which two independent rational sections can be exhibited explicitly. The singular fibres are classified using Kodaira's theory, and the associated global invariants are computed to obtain a geometric upper bound for the Mordell–Weil rank. The Shioda–Tate formula, together with the complex multiplication structure and Galois descent, is then used to establish the exact generic rank over the rational function field. We also examine the geometric origin of a natural non-torsion section arising from an arithmetic pencil of lines and relate its coordinates to a fixed auxiliary cubic curve. Finally, we study the arithmetic behavior of specialized fibres through computational searches and rigorous rank certificates based on rational points, height-pairing matrices, and descent bounds. The results distinguish the generic rank from the higher ranks occurring at particular parameter values and reveal explicit specializations with rank at least three, including examples with rank at least four. We also establish a connection with a previously known rank-three family, interpreting it through a quadratic extension and the decomposition into invariant and anti-invariant sections. These findings illustrate how explicit arithmetic constructions and the geometry of elliptic surfaces can work together to clarify generic rank and rank-jump phenomena in families of congruent-number elliptic curves. We further relate our construction to the rank-three families of Halbeisen, Hungerbühler, and Shamsi Zargar, interpreting their arithmetic through a quadratic extension and the decomposition into invariant and anti-invariant sections. This connection illustrates how dilations of classical congruent numbers (the Euclid areas), explicit arithmetic constructions, and the geometry of elliptic surfaces contribute to the study of generic ranks and rank jumps in families of congruent-number elliptic curves.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 6)
DOI	10.11648/j.ajam.20261406.12
Page(s)	401-412
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Pythagorean Triples, Congruent Numbers, Elliptic Surface, Mordell-Weil Rank, Specialization

References

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[7]	J. Hatley and J. Stack, Two infinite Families of Elliptic Curves With Rank Greater Than One, arXiv:2103.00307v3 [math.NT], 26 Nov 2021. https://doi.org/10.48550/arXiv.2103.00307
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[16]	L. Halbeisen, N. Hungerbühler and A. S. Zargar, A family of congruent number elliptic curves of rank three, Quaestiones Mathematicae, 46(6), 1131–1137 (2023). https://doi.org/10.2989/16073606.2022.2058435
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APA Style

Vincent, K. K., Kablam, E. U. B. (2026). Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections. American Journal of Applied Mathematics, 14(6), 401-412. https://doi.org/10.11648/j.ajam.20261406.12

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ACS Style

Vincent, K. K.; Kablam, E. U. B. Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections. Am. J. Appl. Math. 2026, 14(6), 401-412. doi: 10.11648/j.ajam.20261406.12

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AMA Style

Vincent KK, Kablam EUB. Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections. Am J Appl Math. 2026;14(6):401-412. doi: 10.11648/j.ajam.20261406.12

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@article{10.11648/j.ajam.20261406.12,
  author = {Kouakou Kouassi Vincent and Edjabrou Ulrich Blanchard Kablam},
  title = {Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {6},
  pages = {401-412},
  doi = {10.11648/j.ajam.20261406.12},
  url = {https://doi.org/10.11648/j.ajam.20261406.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261406.12},
  abstract = {We investigates an arithmetic family of congruent-number elliptic curves arising from the Euclid parametrization of Pythagorean triples, with the aim of determining its generic Mordell-Weil rank and understanding the occurrence of higher ranks under specialization. Our approach combines explicit constructions of rational sections with the geometric theory of elliptic surfaces. A suitable quadratic base change produces a new elliptic surface on which two independent rational sections can be exhibited explicitly. The singular fibres are classified using Kodaira's theory, and the associated global invariants are computed to obtain a geometric upper bound for the Mordell–Weil rank. The Shioda–Tate formula, together with the complex multiplication structure and Galois descent, is then used to establish the exact generic rank over the rational function field. We also examine the geometric origin of a natural non-torsion section arising from an arithmetic pencil of lines and relate its coordinates to a fixed auxiliary cubic curve. Finally, we study the arithmetic behavior of specialized fibres through computational searches and rigorous rank certificates based on rational points, height-pairing matrices, and descent bounds. The results distinguish the generic rank from the higher ranks occurring at particular parameter values and reveal explicit specializations with rank at least three, including examples with rank at least four. We also establish a connection with a previously known rank-three family, interpreting it through a quadratic extension and the decomposition into invariant and anti-invariant sections. These findings illustrate how explicit arithmetic constructions and the geometry of elliptic surfaces can work together to clarify generic rank and rank-jump phenomena in families of congruent-number elliptic curves. We further relate our construction to the rank-three families of Halbeisen, Hungerbühler, and Shamsi Zargar, interpreting their arithmetic through a quadratic extension and the decomposition into invariant and anti-invariant sections. This connection illustrates how dilations of classical congruent numbers (the Euclid areas), explicit arithmetic constructions, and the geometry of elliptic surfaces contribute to the study of generic ranks and rank jumps in families of congruent-number elliptic curves.},
 year = {2026}
}

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TY  - JOUR
T1  - Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections
AU  - Kouakou Kouassi Vincent
AU  - Edjabrou Ulrich Blanchard Kablam
Y1  - 2026/11/10
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261406.12
DO  - 10.11648/j.ajam.20261406.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 401
EP  - 412
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261406.12
AB  - We investigates an arithmetic family of congruent-number elliptic curves arising from the Euclid parametrization of Pythagorean triples, with the aim of determining its generic Mordell-Weil rank and understanding the occurrence of higher ranks under specialization. Our approach combines explicit constructions of rational sections with the geometric theory of elliptic surfaces. A suitable quadratic base change produces a new elliptic surface on which two independent rational sections can be exhibited explicitly. The singular fibres are classified using Kodaira's theory, and the associated global invariants are computed to obtain a geometric upper bound for the Mordell–Weil rank. The Shioda–Tate formula, together with the complex multiplication structure and Galois descent, is then used to establish the exact generic rank over the rational function field. We also examine the geometric origin of a natural non-torsion section arising from an arithmetic pencil of lines and relate its coordinates to a fixed auxiliary cubic curve. Finally, we study the arithmetic behavior of specialized fibres through computational searches and rigorous rank certificates based on rational points, height-pairing matrices, and descent bounds. The results distinguish the generic rank from the higher ranks occurring at particular parameter values and reveal explicit specializations with rank at least three, including examples with rank at least four. We also establish a connection with a previously known rank-three family, interpreting it through a quadratic extension and the decomposition into invariant and anti-invariant sections. These findings illustrate how explicit arithmetic constructions and the geometry of elliptic surfaces can work together to clarify generic rank and rank-jump phenomena in families of congruent-number elliptic curves. We further relate our construction to the rank-three families of Halbeisen, Hungerbühler, and Shamsi Zargar, interpreting their arithmetic through a quadratic extension and the decomposition into invariant and anti-invariant sections. This connection illustrates how dilations of classical congruent numbers (the Euclid areas), explicit arithmetic constructions, and the geometry of elliptic surfaces contribute to the study of generic ranks and rank jumps in families of congruent-number elliptic curves.
VL  - 14
IS  - 6
ER  -

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Author Information

Kouakou Kouassi Vincent

Applied Fundamental Sciences Department, Nangui ABROGOUA University, Abidjan, Côte d’Ivoire

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Edjabrou Ulrich Blanchard Kablam

Applied Fundamental Sciences Department, Nangui ABROGOUA University, Abidjan, Côte d’Ivoire

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APA Style

Vincent, K. K., Kablam, E. U. B. (2026). Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections. American Journal of Applied Mathematics, 14(6), 401-412. https://doi.org/10.11648/j.ajam.20261406.12

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ACS Style

Vincent, K. K.; Kablam, E. U. B. Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections. Am. J. Appl. Math. 2026, 14(6), 401-412. doi: 10.11648/j.ajam.20261406.12

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AMA Style

Vincent KK, Kablam EUB. Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections. Am J Appl Math. 2026;14(6):401-412. doi: 10.11648/j.ajam.20261406.12

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@article{10.11648/j.ajam.20261406.12,
  author = {Kouakou Kouassi Vincent and Edjabrou Ulrich Blanchard Kablam},
  title = {Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {6},
  pages = {401-412},
  doi = {10.11648/j.ajam.20261406.12},
  url = {https://doi.org/10.11648/j.ajam.20261406.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261406.12},
  abstract = {We investigates an arithmetic family of congruent-number elliptic curves arising from the Euclid parametrization of Pythagorean triples, with the aim of determining its generic Mordell-Weil rank and understanding the occurrence of higher ranks under specialization. Our approach combines explicit constructions of rational sections with the geometric theory of elliptic surfaces. A suitable quadratic base change produces a new elliptic surface on which two independent rational sections can be exhibited explicitly. The singular fibres are classified using Kodaira's theory, and the associated global invariants are computed to obtain a geometric upper bound for the Mordell–Weil rank. The Shioda–Tate formula, together with the complex multiplication structure and Galois descent, is then used to establish the exact generic rank over the rational function field. We also examine the geometric origin of a natural non-torsion section arising from an arithmetic pencil of lines and relate its coordinates to a fixed auxiliary cubic curve. Finally, we study the arithmetic behavior of specialized fibres through computational searches and rigorous rank certificates based on rational points, height-pairing matrices, and descent bounds. The results distinguish the generic rank from the higher ranks occurring at particular parameter values and reveal explicit specializations with rank at least three, including examples with rank at least four. We also establish a connection with a previously known rank-three family, interpreting it through a quadratic extension and the decomposition into invariant and anti-invariant sections. These findings illustrate how explicit arithmetic constructions and the geometry of elliptic surfaces can work together to clarify generic rank and rank-jump phenomena in families of congruent-number elliptic curves. We further relate our construction to the rank-three families of Halbeisen, Hungerbühler, and Shamsi Zargar, interpreting their arithmetic through a quadratic extension and the decomposition into invariant and anti-invariant sections. This connection illustrates how dilations of classical congruent numbers (the Euclid areas), explicit arithmetic constructions, and the geometry of elliptic surfaces contribute to the study of generic ranks and rank jumps in families of congruent-number elliptic curves.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections
AU  - Kouakou Kouassi Vincent
AU  - Edjabrou Ulrich Blanchard Kablam
Y1  - 2026/11/10
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261406.12
DO  - 10.11648/j.ajam.20261406.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 401
EP  - 412
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261406.12
AB  - We investigates an arithmetic family of congruent-number elliptic curves arising from the Euclid parametrization of Pythagorean triples, with the aim of determining its generic Mordell-Weil rank and understanding the occurrence of higher ranks under specialization. Our approach combines explicit constructions of rational sections with the geometric theory of elliptic surfaces. A suitable quadratic base change produces a new elliptic surface on which two independent rational sections can be exhibited explicitly. The singular fibres are classified using Kodaira's theory, and the associated global invariants are computed to obtain a geometric upper bound for the Mordell–Weil rank. The Shioda–Tate formula, together with the complex multiplication structure and Galois descent, is then used to establish the exact generic rank over the rational function field. We also examine the geometric origin of a natural non-torsion section arising from an arithmetic pencil of lines and relate its coordinates to a fixed auxiliary cubic curve. Finally, we study the arithmetic behavior of specialized fibres through computational searches and rigorous rank certificates based on rational points, height-pairing matrices, and descent bounds. The results distinguish the generic rank from the higher ranks occurring at particular parameter values and reveal explicit specializations with rank at least three, including examples with rank at least four. We also establish a connection with a previously known rank-three family, interpreting it through a quadratic extension and the decomposition into invariant and anti-invariant sections. These findings illustrate how explicit arithmetic constructions and the geometry of elliptic surfaces can work together to clarify generic rank and rank-jump phenomena in families of congruent-number elliptic curves. We further relate our construction to the rank-three families of Halbeisen, Hungerbühler, and Shamsi Zargar, interpreting their arithmetic through a quadratic extension and the decomposition into invariant and anti-invariant sections. This connection illustrates how dilations of classical congruent numbers (the Euclid areas), explicit arithmetic constructions, and the geometry of elliptic surfaces contribute to the study of generic ranks and rank jumps in families of congruent-number elliptic curves.
VL  - 14
IS  - 6
ER  -

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