HEXAGONbeta

Infinitely many elliptic curves with torsion Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z} and rank at least six

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Manvir Jaswal ↗

Version 1 / Oct 27, 2026 / CC BY 4.0

Abstract

We show that there are infinitely many elliptic curves over Q\mathbb{Q} with torsion subgroup Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z} and rank at least 66, answering the case Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z} of Problem 2.6 in Dujella's list of open problems. The largest rank previously known to occur infinitely often with this torsion was 55, due to Eroshkin. The curves are fibers of a one-parameter family of Dujella and Peral, which we show has rank exactly 33 over the rational function field. The family is a base change of an elliptic K3 surface, and quadratic twists of that surface give it three further independent points over the function field of an elliptic curve of rank one. Congruences then carry the independence of the six points, and the torsion Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}, from one rational point of that curve to infinitely many. The result is unconditional. It contradicts the prediction of Park, Poonen, Voight and Wood that only finitely many curves with this torsion have rank greater than 55, although the curves form a sparse family. A second family, over an elliptic curve of rank two, gives another proof.

Provenance statement

This paper, its results and its Lean formalization were produced by an AI research system that the author designed, built and directs, running on Anthropic's Claude Opus 5.5 in Claude Code and OpenAI's GPT-6 Astra in Codex. The author chooses the problems and prioritizes the work. Separate AI agents checked each result, and a second program confirmed the computations the proofs use, with any exceptions listed in the public deposit. The system also searched the literature and checked the citations against reliable sources. The author read the paper, reran the computations from the deposit and takes full responsibility for all of the paper's contents. Programs and data: https://doi.org/10.5281/zenodo.23437115. Lean formalization: https://doi.org/10.5281/zenodo.23682552.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
CodexVersion GPT-6 Astra
Lean
LeanVersion 4, with Mathlib
SageMath
SageMathVersion 10.9
PARI Group
PARI/GPVersion 2.17.3
Python Software Foundation
Python (python-flint)Version 3.12

References

  1. Bekker, Boris M.; Zarhin, Yuri G. Division by 2 of rational points on elliptic curves. St. Petersburg Math. J., vol. 29, no. 4, pp. 683–713. 2018. \DOI10.1090/spmj/1512; \arXiv1702.02255v2
  2. Cremona, John E. Algorithms for Modular Elliptic Curves. Cambridge University Press. 1997
  3. Cremona, John E. Classical invariants and 2-descent on elliptic curves. J. Symbolic Comput., vol. 31, no. 1–2, pp. 71–87. 2001. \DOI10.1006/jsco.1998.1004
  4. Cremona, John E.; Fisher, Tom A.; Stoll, Michael. Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves. Algebra Number Theory, vol. 4, no. 6, pp. 763–820. 2010. \DOI10.2140/ant.2010.4.763; \arXiv0908.1741v1
  5. Dujella, Andrej. High rank elliptic curves with prescribed torsion. https://web.math.pmf.unizg.hr/~duje/tors/tors.html (accessed October 15, 2026)link
  6. Dujella, Andrej. Infinite families of elliptic curves with high rank and prescribed torsion. https://web.math.pmf.unizg.hr/~duje/tors/generic.html (accessed October 15, 2026)link
  7. Dujella, Andrej. Open problems on Diophantine mm-tuples and elliptic curves. https://web.math.pmf.unizg.hr/~duje/pdf/open2.pdf, version of October 15, 2026link
  8. Dujella, Andrej. Torsion group Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}, rank =9=9. https://web.math.pmf.unizg.hr/~duje/tors/z2z4.html (accessed October 15, 2026)link
  9. Dujella, Andrej. On Mordell–Weil groups of elliptic curves induced by Diophantine triples. Glas. Mat. Ser. III, vol. 42, no. 1, pp. 3–18. 2007. \DOI10.3336/gm.42.1.01; \arXiv0705.1875v1
  10. Dujella, Andrej. High-rank elliptic curves with given torsion group and some applications. Banach Center Publ., vol. 126, pp. 9–26. 2023. \DOI10.4064/bc126-1
Show all 39 referencesShow fewer references
  1. Dujella, Andrej; Kazalicki, Matija; Peral, Juan Carlos. Elliptic curves with torsion groups Z/8Z\mathbb{Z}/8\mathbb{Z} and Z/2Z×Z/6Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, vol. 115, no. 4, pp. Paper No. 169. 2021. \DOI10.1007/s13398-021-01112-5; \arXiv2105.06215v2
  2. Dujella, Andrej; Peral, Juan Carlos. High-rank elliptic curves with torsion Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z} induced by Diophantine triples. LMS J. Comput. Math., vol. 17, no. 1, pp. 282–288. 2014. \DOI10.1112/S1461157014000023
  3. Dujella, Andrej; Peral, Juan Carlos. Elliptic curves with torsion group Z/8Z\mathbb{Z}/8\mathbb{Z} or Z/2Z×Z/6Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}. Trends in Number Theory, vol. 649. 2015. pp. 47–62, \DOI10.1090/conm/649/13019; \arXiv1306.0027v1, where §§ 5.4 and 6.6 are §§ 4.4 and 5.5
  4. Dujella, Andrej; Peral, Juan Carlos. Elliptic curves induced by Diophantine triples. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, vol. 113, no. 2, pp. 791–806. 2019. \DOI10.1007/s13398-018-0513-0; \arXiv1712.02082v2
  5. Dujella, Andrej; Peral, Juan Carlos. Construction of high rank elliptic curves. J. Geom. Anal., vol. 31, no. 7, pp. 6698–6724. 2021. \DOI10.1007/s12220-020-00373-7
  6. Dujella, Andrej; Peral, Juan Carlos. An elliptic curve over Q(u)\mathbb{Q}(u) with torsion Z/4Z\mathbb{Z}/4\mathbb{Z} and rank 6. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., vol. 28, pp. 185–192. 2024. \DOI10.21857/y54jof4o2m; \arXiv2207.08206v2
  7. Elkies, Noam D. Three lectures on elliptic surfaces and curves of high rank. 2007. Preprint, \arXiv0709.2908v1
  8. Elkies, Noam D. An elliptic K3 surface X/Q(t)X/\mathbb{Q}(t) with Mordell–Weil rank 17, I: Formulas for XX and base changes of rank 18 and 19. 2026. Preprint, \arXiv2608.25406v1
  9. Faltings, Gerd. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math., vol. 73, no. 3, pp. 349–366. 1983. \DOI10.1007/BF01388432
  10. The FLINT team. FLINT: Fast Library for Number Theory. 2025. https://flintlib.org, version 3.3.1, with the Python interface python-flint 0.8.0link
  11. Gusić, Ivica; Tadić, Petra. Injectivity of the specialization homomorphism of elliptic curves. J. Number Theory, vol. 148, pp. 137–152. 2015. \DOI10.1016/j.jnt.2014.09.023; \arXiv1409.7189v2
  12. Harron, Robert; Snowden, Andrew. Counting elliptic curves with prescribed torsion. J. Reine Angew. Math., vol. 729, pp. 151–170. 2017. \DOI10.1515/crelle-2014-0107; \arXiv1311.4920v1
  13. Jaswal, Manvir. Infinitely many elliptic curves with torsion Z/2Z×Z/6Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z} and rank at least four. 2026. Zenodo, \DOI10.5281/zenodo.23364157
  14. Jaswal, Manvir. Lean formalization: ``Infinitely many elliptic curves with torsion Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z} and rank at least six''. 2026. Zenodo, \DOI10.5281/zenodo.23682552
  15. Jaswal, Manvir. Programs and data for ``Infinitely many elliptic curves with torsion Z/2Z×Z/4Z\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z} and rank at least six''. 2026. Zenodo, \DOI10.5281/zenodo.23437115
  16. Lang, Serge; Néron, André. Rational points of abelian varieties over function fields. Amer. J. Math., vol. 81, no. 1, pp. 95–118. 1959. \DOI10.2307/2372851
  17. Lecacheux, Odile. Rang de courbes elliptiques avec groupe de torsion non trivial. J. Théor. Nombres Bordeaux, vol. 15, no. 1, pp. 231–247. 2003. \DOI10.5802/jtnb.400
  18. The LMFDB Collaboration. The LL-functions and modular forms database. https://www.lmfdb.org, elliptic curve 9240.bf4 (accessed October 16, 2026)link
  19. Mazur, Barry. Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math., vol. 47, pp. 33–186. 1977. \DOI10.1007/BF02684339
  20. Miranda, Rick. The Basic Theory of Elliptic Surfaces. ETS Editrice. 1989
  21. Nagao, Koh-ichi. Q(T)\mathbb{Q}(T)-rank of elliptic curves and certain limit coming from the local points. Manuscripta Math., vol. 92, no. 1, pp. 13–32. 1997. \DOI10.1007/BF02678178
  22. Néron, André. Problèmes arithmétiques et géométriques rattachés à la notion de rang d'une courbe algébrique dans un corps. Bull. Soc. Math. France, vol. 80, pp. 101–166. 1952. \DOI10.24033/bsmf.1427
  23. Ono, Ken. Euler's concordant forms. Acta Arith., vol. 78, no. 2, pp. 101–123. 1996. \DOI10.4064/aa-78-2-101-123
  24. The PARI Group. PARI/GP version 2.17.3. 2025. Univ. Bordeaux, https://pari.math.u-bordeaux.fr/link
  25. Park, Jennifer; Poonen, Bjorn; Voight, John; Wood, Melanie Matchett. A heuristic for boundedness of ranks of elliptic curves. J. Eur. Math. Soc. (JEMS), vol. 21, no. 9, pp. 2859–2903. 2019. \DOI10.4171/JEMS/893; \arXiv1602.01431v3
  26. Rosen, Michael; Silverman, Joseph H. On the rank of an elliptic surface. Invent. Math., vol. 133, no. 1, pp. 43–67. 1998. \DOI10.1007/s002220050238
  27. The Sage Developers. SageMath, the Sage Mathematics Software System (Version 10.9). 2026. https://www.sagemath.orglink
  28. Silverman, Joseph H. Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math., vol. 342, pp. 197–211. 1983. \DOI10.1515/crll.1983.342.197
  29. Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer, vol. 106. 2009. \DOI10.1007/978-0-387-09494-6

Version history

  1. v1Submitted by Manvir JaswalInitial depositCurrentOct 27, 2026