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

Şeyda Dalkılıç; Ercan Altınışık

doi:10.53570/jnt.1467401

EN

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

Abstract

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.

Keywords

Mestre method, elliptic curve, rank, height matrix, magma

References

D. Penney, C. Pomerance, A search for elliptic curves with large rank, Mathematics of Computation 28 (127) (1974) 851–853.
D. Penney, C. Pomerance, Three elliptic curves with rank at least seven, Mathematics of Computation 29 (131) (1975) 965–967.
F. Grunewald, R. Zimmert, Uber einige rationale elliptische Kurven mit treiem Rang ≥ 8, Journal für die Reine und Angewandte Mathematik 1977 (296) (1977) 100–107.
A. Brumer, K. Kramer, The rank of elliptic curves, Duke Mathematical Journal 44 (1977) 715– 743.
J.-F. Mestre, Construction d’une courbe elliptique de rang ≥ 12, Comptes Rendus de l’Academie des Sciences Paris 295 (1982) 643–644.
J.-F. Mestre, Courbes elliptiques et formules explicites, Seminaire de Theorie des Nombres de Grenoble 10 (1982) 1–10.
J.-F. Mestre, Courbe elliptiques de rang ≥ 11 sur Q(t), Comptes Rendus de l Academie des Sciences 313 (1991) 139–142.
J.-F. Mestre, Courbe elliptiques de rang ≥ 12 sur Q(t), Comptes Rendus de l Academie des Sciences 313 (1991) 171–174.
J.-F. Mestre, Un exemple de courbe elliptique sur Q de rang ≥ 15, Comptes Rendus de l Academie des Sciences Paris Serie I Mathematics 314 (1992) 453–455.
K. Nagao, Examples of elliptic curves over Q with rank ≥ 17, Proceedings of the Japan Academy Serie A Mathematical Sciences 68 (1992) 287–289.

K. Nagao, An example of elliptic curve over Q with rank ≥ 20, Proceedings of the Japan Academy Serie A Mathematical Sciences 69 (1993) 291–293.
K. Nagao, T. Kouya, An example of elliptic curve over Q with rank ≥ 21, Proceedings of the Japan Academy Serie A Mathematical Sciences 70 (1994) 104–105.
S. Fermigier, Une courbe elliptique definie sur Q de rang ≥ 22, Acta Arithmetica 82 (4) (1997) 359–363.
A. Dujella, History of elliptic curves rank records (nd), https://web.math.pmf.unizg.hr/ duje/tors/rankhist.html, Accessed 8 April 2024.
S.-W. Kim, Searching the ranks of elliptic curves y2 = x3 − px, International Journal of Algebra 12 (8) (2018) 311–318.
S.-W. Kim, Ranks in elliptic curves of the forms y2 = x3 + Ax2 + Bx, International Journal of Algebra 12 (8) (2018) 311–318.
S.-W. Kim, Ranks in elliptic curves of the forms y2 = x3 ∓Ax, International Journal of Contemporary Mathematical Sciences 18 (1) (2023) 19–31.
S.-W. Kim, Ranks in elliptic curves y2 = x3 − 37px and y2 = x3 − 61px and y2 = x3 − 67px and y2 = x3 − 947px, International Journal of Algebra 17 (3) (2023) 121–142.
R. Mina, J. Bacani, Elliptic curves of type y2 = x3 − 3pqx having ranks zero and one, Malaysian Journal of Mathematical Sciences 17 (1) (2023) 67–76.
F. Khoshnam, D. Moody, High rank elliptic curves with torsion Z/4Z induced by Kihara’s elliptic curves, Integers: The Electronic Journal of Combinatorial Number Theory 16 (8) (2016) A70 12 pages.
G. Celik, G. Soydan, Elliptic curves containing sequences of consecutive cubes, Rocky Mountain Journal of Mathematics 47 (7) (2018) 2163–2174.
G. Celik, M. Sadek, G. Soydan, Rational sequences on different models of elliptic curves, Glasnik Matematicki 54 (74) (2019) 53–64.
A. Dujella, M. Kazalicki, J. C. Peral, Elliptic curves with torsion groups Z/8Zand Z/2Z×Z/6Z, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales Serie A Matematicas 115 (4) (2021) 169 24 pages.
A. Dujella, M. Mıkıc, Rank zero elliptic curves induces by rational diophantine triples, Rad Hrvatske Akademije Znanosti i Umjetnosti Matematicke Znanosti 24 (2020) 29–37.
A. Dujella, G. Soydan, On elliptic curves induced by rational Diophantine quadruples, Proceedings of the Japan Academy Mathematical Sciences, Series A; Ueno Park 98 (1) (2022) 1–6.
L. Halbeisen, N. Hungerbühler, Heron triangles and their elliptic curves, Journal of Number Theory 213 (2020) 232–253.
L. Halbeisen, N. Hungerbühler, A. Zargar, A family of congruent number elliptic curves of rank three, Quaestiones Mathematicae 46 (6) (2023) 1131–1137.
N. Garcia-Fritz, H. Pasten, Elliptic curves with long arithmetic progressions have large rank, International Mathematics Research Notices 2021 (10) (2021) 7394–7432.
A. Dujella, J. Peral, An elliptic curve over Q(u) with torsion Z/4Z and rank 6, Rad Hrvatske Akademije Znanosti i Umjetnosti Matematicke Znanosti 28 (2024) 185–192.
L. Beneish, K. Debanjana, R. Anwesh, Rank jumps and growth of Shafarevich-Tate Groups for elliptic curves in Z/pZ extensions, Journal of the Australian Mathematical Society 116 (2024) 1–38.
P. J. Cho, K. Jeong, On the distribution of analytic ranks of elliptic curves, Mathematische Zeitschrift 305 (3) (2023) 42 20 pages.
A. Dujella, J. Peral, Construction of high rank elliptic curves, The Journal of Geometric Analysis 31 (7) (2021) 6698–6724.
N. Elkies, Z. Klagsbrun, New rank records for elliptic curves having rational torsion, in: S. Galbraith (Ed.), Ants XIV: Proceedings of the Fourteenth Algorithmic Number Theory Symposium, Berkeley, 2020, pp. 233–250.
M. Kazalicki, D. Vlah, Ranks of elliptic curves and deep neural networks, Research in Number Theory 9 (3) (2023) 53 20 pages.
M. Schütt, T. Shioda, Mordell–Weil Lattices, Springer, Singapore, 2019.
J. H. Silverman, J. T. Tate, Rational Points on Elliptic Curves, 2nd Edition, Springer International Publishing, Switzerland, 2009.
B. Mazur, Modular curves and the Eisenstein ideal, Publications Mathematiques de l IHES 47 (1977) 133–186.
B. Mazur, D. Goldfeld, Rational isogenies of prime degree, Inventiones Mathematicae 44 (2) (1978) 129–162.
G. Campell, Finding elliptic curves and families of elliptic curves over Q of large rank, Doctoral Dissertation The State University of New Jersey (1999) New Brunswick.
A. Lozano-Robledo, Elliptic curves, modular forms, and their L-functions, 1st Edition, American Mathematical Society, United States of America, 2011.
B. Bırch, H. Swinnerton-Dyer, Notes on elliptic curves. I., Journal für die reine und angewandte Mathematik 212 (1963) 7–25.
K. Rubin, A. Silverberg, Ranks of elliptic curves, Bulletin of the American Mathematical Society 39 (2002) 455–474.

Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Şeyda Dalkılıç
0000-0002-3858-8351
Türkiye

Ercan Altınışık ^*
0000-0002-0476-9429
Türkiye

Publication Date

June 30, 2024

Submission Date

April 10, 2024

Acceptance Date

May 23, 2024

Published in Issue

Year 2024 Number: 47

DOI

https://doi.org/10.53570/jnt.1467401

IZ

https://izlik.org/JA84UJ55YL

APA

Dalkılıç, Ş., & Altınışık, E. (2024). Mestre’s Finite Field Method for Searching Elliptic Curves with High Ranks. Journal of New Theory, 47, 20-27. https://doi.org/10.53570/jnt.1467401

AMA

1.Dalkılıç Ş, Altınışık E. Mestre’s Finite Field Method for Searching Elliptic Curves with High Ranks. JNT. 2024;(47):20-27. doi:10.53570/jnt.1467401

Chicago

Dalkılıç, Şeyda, and Ercan Altınışık. 2024. “Mestre’s Finite Field Method for Searching Elliptic Curves With High Ranks”. Journal of New Theory, nos. 47: 20-27. https://doi.org/10.53570/jnt.1467401.

EndNote

Dalkılıç Ş, Altınışık E (June 1, 2024) Mestre’s Finite Field Method for Searching Elliptic Curves with High Ranks. Journal of New Theory 47 20–27.

IEEE

[1]Ş. Dalkılıç and E. Altınışık, “Mestre’s Finite Field Method for Searching Elliptic Curves with High Ranks”, JNT, no. 47, pp. 20–27, June 2024, doi: 10.53570/jnt.1467401.

ISNAD

Dalkılıç, Şeyda - Altınışık, Ercan. “Mestre’s Finite Field Method for Searching Elliptic Curves With High Ranks”. Journal of New Theory. 47 (June 1, 2024): 20-27. https://doi.org/10.53570/jnt.1467401.

JAMA

1.Dalkılıç Ş, Altınışık E. Mestre’s Finite Field Method for Searching Elliptic Curves with High Ranks. JNT. 2024;:20–27.

MLA

Dalkılıç, Şeyda, and Ercan Altınışık. “Mestre’s Finite Field Method for Searching Elliptic Curves With High Ranks”. Journal of New Theory, no. 47, June 2024, pp. 20-27, doi:10.53570/jnt.1467401.

Vancouver

1.Şeyda Dalkılıç, Ercan Altınışık. Mestre’s Finite Field Method for Searching Elliptic Curves with High Ranks. JNT. 2024 Jun. 1;(47):20-7. doi:10.53570/jnt.1467401