## History of elliptic curves rank records

Let E be an elliptic curve over Q. By Mordell's theorem, E(Q) is a finitely generated abelian group. This means that E(Q) = E(Q)tors × Zr. By Mazur's theorem, we know that E(Q)tors is one of the following 15 groups:

Z/nZ   with 1 ≤ n ≤ 10   or   n = 12,
Z/2Z × Z/2mZ   with 1 ≤ m ≤ 4.

On the other hand, it is not known what values of rank r are possible for elliptic curves over Q. The "folklore" conjecture is that a rank can be arbitrary large. However there are also recent heuristic arguments that suggest the boundedness of the rank of elliptic curves. The current record is an example of elliptic curve with rank ≥ 28, found by Elkies in 2006 (the previous record was rank ≥ 24, found by Martin and McMillen in 2000).

The highest rank of an elliptic curve which is (unconditionally) known exactly (not only a lower bound for rank) is equal to 20, and it is found by Elkies-Klagsbrun in 2020. It improves previous records due to Kretschmer (rank = 10), Schneiders-Zimmer (rank = 11), Fermigier (rank = 14), Dujella (rank = 15) and Elkies (rank = 17, rank = 18, rank = 19).

The following table contains some historical data on elliptic curve rank records.

```________________________________________________________________________________

rank >=         year                 Author(s)
________________________________________________________________________________

3             1938            Billing

4             1945            Wiman

6             1974            Penney - Pomerance

7             1975            Penney - Pomerance

8             1977            Grunewald - Zimmert

9             1977            Brumer - Kramer

12             1982            Mestre

14             1986            Mestre

15             1992            Mestre

17             1992            Nagao

19             1992            Fermigier

20             1993            Nagao

21             1994            Nagao - Kouya

22             1997            Fermigier

23             1998            Martin - McMillen

24             2000            Martin - McMillen

28             2006            Elkies

________________________________________________________________________________

```

Click on rank r to see the corresponding curve(s) and independent points P1, P2, ... , Pr of infinite order.

### References:

1. G. Billing, Beitrage zur arithmetischen Theorie der ebenen kubishen Kurven vom Geschlecht Eins, Nova Acta Soc. Sci. Upsal. (4) 11 (1938), Nr.1, Diss.165.

2. A. Wiman, Uber den Rang von Kurven y2 = x(x+a)(x+b), Acta Math. 76 (1945), 225-251.

3. A. Wiman, Uber rationale Punkte auf Kurven y2 = x(x2-c2), Acta Math. 77 (1945), 281-320.

4. D.E. Penney and C. Pomerance, A search for elliptic curves with large rank, Math. Comp. 28 (1974), 851-853.

5. D.E. Penney and C. Pomerance, Three elliptic curves with rank at least seven, Math. Comp. 29 (1975), 965-967.

6. F.J. Grunewald and R. Zimmert, Uber einige rationale elliptische Kurven mit treiem Rang ≥ 8, J. Reine Angew. Math. 296 (1977), 100-107.

7. A. Brumer and K. Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), 715-743.

8. J.-F. Mestre, Construction d'une courbe elliptique de rang ≥ 12, C. R. Acad. Sci. Paris Ser I Math. 295 (1982), 643-644.

9. J.-F. Mestre, Courbes elliptiques et formules explicites, Theorie des nombres, Semin. Delange-Pisot-Poitou, Paris 1981/82, Prog. Math. 38, Birkhauser, Boston, 1983, pp. 179-187

10. J.-F. Mestre, Formules explicites et minorations de conducteurs de varietes algebriques, Compositio Math. 58 (1986), 209-232.

11. J.-F. Mestre, Un exemple de courbe elliptique sur Q de rang ≥ 15, C. R. Acad. Sci. Paris Ser I Math. 314 (1992), 453-455.

12. K. Nagao, Examples of elliptic curves over Q with rank ≥ 17, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), 287-289.

13. S. Fermigier, Un exemple de courbe elliptique definie sur Q(t) de rang ≥ 19, C. R. Acad. Sci. Paris Ser. I 315 (1992), 719-722.

14. K. Nagao, An example of elliptic curve over Q with rank ≥ 20, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 291-293.

15. K. Nagao and T. Kouya, An example of elliptic curve over Q with rank ≥ 21, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 104-105.

16. K. Nagao, Construction of high-rank elliptic curves, Kobe J. Math. 11 (1994), 211-219.

17. S. Fermigier, Une courbe elliptique definie sur Q(t) de rang ≥ 22, Acta Arith. (1997), 359-363.

18. R. Martin and W. McMillen, An elliptic curve over Q with rank at least 23, Number Theory Listserver, March 1998.

19. R. Martin and W. McMillen, An elliptic curve over Q with rank at least 24, Number Theory Listserver, May 2000.

20. N. D. Elkies, Z28 in E(Q), etc., Number Theory Listserver, May 2006.

21. N. D. Elkies, Three lectures on elliptic surfaces and curves of high rank, Lecture notes, Oberwolfach, 2007, arXiv:0709.2908.

22. T.J. Kretschmer, Construction of elliptic curves with large rank, Math. Comp. 46 (1986), 627-635.

23. U. Schneiders and H.G. Zimmer, The rank of elliptic curves upon quadratic extensions, in: Computational Number Theory (A. Petho, H.C. Williams, H.G. Zimmer, eds.), de Gruyter, Berlin, 1991, pp. 239-260.

24. S. Fermigier, Exemples de courbes elliptiques de grand rang sur Q(t) et sur Q possedant des points d'ordre 2, C. R. Acad. Sci. Paris Ser. I 322 (1996), 949-952.

25. A. Dujella, An example of elliptic curve over Q with rank equal to 15, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), 109-111.

26. N. D. Elkies, E(Q) = (Z/2Z) * Z17, Number Theory Listserver, Jun 2005.

27. N. D. Elkies, Some more rank records: E(Q) = (Z/2Z) * Z18, (Z/4Z) * Z12, (Z/8Z) * Z6, (Z/2Z) * (Z/6Z) * Z6, Number Theory Listserver, Jun 2006.

28. N. D. Elkies, Personal communication, 2009.

29. N. D. Elkies and Z. Klagsbrun, New rank records for elliptic curves faving rational torsion, preprint, 2020, arXiv: 2003.00077.