Proceedings of the Japan Academy, Series A, Mathematical Sciences

Infinitely many elliptic curves of rank exactly two

Dongho Byeon and Keunyoung Jeong

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In this note, we construct an infinite family of elliptic curves $E$ defined over $\mathbf{Q}$ whose Mordell-Weil group $E(\mathbf{Q})$ has rank exactly two under the parity conjecture.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 5 (2016), 64-66.

First available in Project Euclid: 28 April 2016

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Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

Elliptic curve rank


Byeon, Dongho; Jeong, Keunyoung. Infinitely many elliptic curves of rank exactly two. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 5, 64--66. doi:10.3792/pjaa.92.64.

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  • D. Byeon, D. Jeon and C. H. Kim, Rank-one quadratic twists of an infinite family of elliptic curves, J. Reine Angew. Math. 633 (2009), 67–76.
  • D. Byeon and K. Jeong, Sums of two rational number with many prime factors. (Preprint).
  • J. Brüdern, K. Kawada and T. D. Wooley, Additive representation in thin sequences. II. The binary Goldbach problem, Mathematika 47 (2000), no. 1–2, 117–125.
  • B. J. Birch and N. M. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5 (1966), 295–299.
  • F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), no. 1, 1–23.
  • L. Mai, The analytic rank of a family of elliptic curves, Canad. J. Math. 45 (1993), no. 4, 847–862.
  • K. Rubin and A. Silverberg, Ranks of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 4, 455–474 (electronic).
  • P. Satgé, Groupes de Selmer et corps cubiques, J. Number Theory 23 (1986), no. 3, 294–317.
  • Y. Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121–1174.