# Article

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Summary:
A conjecture due to Honda predicts that given any abelian variety over a number field \$K\$, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda's conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over \$\mathbb Q\$.
References:
[1] Baker, M. H.: Lower bounds for the canonical height on elliptic curves over abelian extensions. Int. Math. Res. Not. 2003 (2003), 1571-1589. DOI 10.1155/S1073792803212083 | MR 1979685 | Zbl 1114.11058
[2] Brumer, A., Kramer, K.: The rank of elliptic curves. Duke Math. J. 44 (1977), 715-743. DOI 10.1215/S0012-7094-77-04431-3 | MR 0457453 | Zbl 0376.14011
[3] Gupta, R., Murty, M. R.: Primitive points on elliptic curves. Compos. Math. 58 (1986), 13-44. MR 0834046 | Zbl 0598.14018
[4] Honda, T.: Isogenies, rational points and section points of group varieties. Jap. J. Math. 30 (1960), 84-101. MR 0155828 | Zbl 0109.39602
[5] Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics 84 Springer, New York (1982). MR 0661047 | Zbl 0482.10001
[6] Lang, S.: Fundamentals of Diophantine Geometry. Springer, New York (1983). MR 0715605 | Zbl 0528.14013
[7] Merel, L.: Bounds for the torsion of elliptic curves over number fields. Invent. Math. 124 French (1996), 437-449. MR 1369424 | Zbl 0936.11037
[8] Murty, M. R.: Problems in Analytic Number Theory (2nd edition). Graduate Texts in Mathematics 206, Readings in Mathematics Springer, New York (2008). MR 1803093
[9] Ooe, T., Top, J.: On the Mordell-Weil rank of an abelian variety over a number field. J. Pure Appl. Algebra 58 (1989), 261-265. MR 1004606 | Zbl 0686.14025
[10] Rubin, K., Silverberg, A.: Ranks of elliptic curves. Bull. Am. Math. Soc., New Ser. 39 (2002), 455-474. DOI 10.1090/S0273-0979-02-00952-7 | MR 1920278 | Zbl 1052.11039
[11] Rubin, K., Silverberg, A.: Ranks of elliptic curves in families of quadratic twists. Exp. Math. 9 (2000), 583-590. DOI 10.1080/10586458.2000.10504661 | MR 1806293 | Zbl 0959.11023
[12] Shimura, G., Taniyama, Y.: Complex Multiplication of Abelian Varieties and Its Applications to Number Theory. Publications of the Mathematical Society of Japan 6 Mathematical Society of Japan, Tokyo (1961). MR 0125113 | Zbl 0112.03502
[13] Silverman, J. H.: The Arithmetic of Elliptic Curves (2nd edition). Graduate Texts in Mathematics 106 Springer, New York (2009). DOI 10.1007/978-0-387-09494-6 | MR 2514094
[14] Silverman, J. H.: A lower bound for the canonical height on elliptic curves over abelian extensions. J. Number Theory 104 (2004), 353-372. DOI 10.1016/j.jnt.2003.07.001 | MR 2029512 | Zbl 1053.11052
[15] Silverman, J. H.: Representations of integers by binary forms and the rank of the Mordell-Weil group. Invent. Math. 74 (1983), 281-292. DOI 10.1007/BF01394317 | MR 0723218 | Zbl 0525.14012

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