$2$-Selmer near-companion curves
HTML articles powered by AMS MathViewer
- by Myungjun Yu PDF
- Trans. Amer. Math. Soc. 372 (2019), 425-440 Request permission
Abstract:
Let $E$ and $A$ be elliptic curves over a number field $K$. Let $\chi$ be a quadratic character of $K$. We prove the conjecture posed by Mazur and Rubin on $n$-Selmer near-companion curves in the case $n=2$. Namely, we show if the difference of the $2$-Selmer ranks of $E^\chi$ and $A^\chi$ is bounded independent of $\chi$, there is a $G_K$-module isomorphism $E[2] \cong A[2]$.References
- Zev Klagsbrun, Barry Mazur, and Karl Rubin, Disparity in Selmer ranks of quadratic twists of elliptic curves, Ann. of Math. (2) 178 (2013), no. 1, 287–320. MR 3043582, DOI 10.4007/annals.2013.178.1.5
- Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 444670, DOI 10.1007/BF01389815
- Barry Mazur and Karl Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799, viii+96. MR 2031496, DOI 10.1090/memo/0799
- B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541–575. MR 2660452, DOI 10.1007/s00222-010-0252-0
- Barry Mazur and Karl Rubin, Selmer companion curves, Trans. Amer. Math. Soc. 367 (2015), no. 1, 401–421. MR 3271266, DOI 10.1090/S0002-9947-2014-06114-X
- Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
- Bjorn Poonen and Eric Rains, Random maximal isotropic subspaces and Selmer groups, J. Amer. Math. Soc. 25 (2012), no. 1, 245–269. MR 2833483, DOI 10.1090/S0894-0347-2011-00710-8
- Myungjun Yu, On 2-Selmer ranks of quadratic twists of elliptic curves, Math. Res. Lett. 24 (2017), no. 5, 1565–1583. MR 3747176, DOI 10.4310/MRL.2017.v24.n5.a11
- Myungjun Yu, Selmer ranks of twists of hyperelliptic curves and superelliptic curves, J. Number Theory 160 (2016), 148–185. MR 3425203, DOI 10.1016/j.jnt.2015.08.009
Additional Information
- Myungjun Yu
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
- MR Author ID: 1136113
- Email: myungjuy@umich.edu
- Received by editor(s): November 8, 2016
- Received by editor(s) in revised form: February 27, 2018
- Published electronically: October 1, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 425-440
- MSC (2010): Primary 11G05
- DOI: https://doi.org/10.1090/tran/7563
- MathSciNet review: 3968774