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Selmer companion curves


Authors: Barry Mazur and Karl Rubin
Journal: Trans. Amer. Math. Soc. 367 (2015), 401-421
DOI: https://doi.org/10.1090/S0002-9947-2014-06114-X
Published electronically: September 4, 2014
MathSciNet review: 3271266
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Abstract | References | Additional Information

Abstract: We say that two elliptic curves $ E_1, E_2$ over a number field $ K$ are $ n$-Selmer companions for a positive integer $ n$ if for every quadratic character $ \chi $ of $ K$, there is an isomorphism $ \operatorname {Sel}_n(E_1^\chi /K) \cong \operatorname {Sel}_n(E_2^\chi /K)$ between the $ n$-Selmer groups of the quadratic twists $ E_1^\chi $, $ E_2^\chi $. We give sufficient conditions for two elliptic curves to be $ n$-Selmer companions, and give a number of examples of non-isogenous pairs of companions.


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  • [1] Spencer Bloch and Kazuya Kato, $ L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400. MR 1086888 (92g:11063)
  • [2] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151 (93m:11053)
  • [3] J. W. S. Cassels, Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180-199. MR 0179169 (31 #3420)
  • [4] Alina Carmen Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic curves, Canad. Math. Bull. 48 (2005), no. 1, 16-31. With an appendix by Ernst Kani. MR 2118760 (2005k:11109), https://doi.org/10.4153/CMB-2005-002-x
  • [5] Tim Dokchitser and Vladimir Dokchitser, On the Birch-Swinnerton-Dyer quotients modulo squares, Ann. of Math. (2) 172 (2010), no. 1, 567-596. MR 2680426 (2011h:11069), https://doi.org/10.4007/annals.2010.172.567
  • [6] 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. MR 1080648 (92b:11039), https://doi.org/10.2307/2939253
  • [7] D. R. Heath-Brown, The size of Selmer groups for the congruent number problem. II, Invent. Math. 118 (1994), no. 2, 331-370. With an appendix by P. Monsky. MR 1292115 (95h:11064), https://doi.org/10.1007/BF01231536
  • [8] E. Kani and W. Schanz, Modular diagonal quotient surfaces, Math. Z. 227 (1998), no. 2, 337-366. MR 1609061 (99a:14031), https://doi.org/10.1007/PL00004379
  • [9] H. Kisilevsky, Rank determines semi-stable conductor, J. Number Theory 104 (2004), no. 2, 279-286. MR 2029506 (2005h:11137), https://doi.org/10.1016/S0022-314X(03)00157-4
  • [10] Zev Klagsbrun, Selmer Ranks of Quadratic Twists of Elliptic Curves, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)-University of California, Irvine. MR 2890124
  • [11] Z. Klagsbrun, B. Mazur and K. Rubin, Selmer ranks of quadratic twists of elliptic curves, to appear in Annals of Math.
  • [12] Kenneth Kramer, Arithmetic of elliptic curves upon quadratic extension, Trans. Amer. Math. Soc. 264 (1981), no. 1, 121-135. MR 597871 (82g:14028), https://doi.org/10.2307/1998414
  • [13] A. Kraus and J. Oesterlé, Sur une question de B. Mazur, Math. Ann. 293 (1992), no. 2, 259-275 (French). MR 1166121 (93e:11074), https://doi.org/10.1007/BF01444715
  • [14] Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183-266. MR 0444670 (56 #3020)
  • [15] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129-162. MR 482230 (80h:14022), https://doi.org/10.1007/BF01390348
  • [16] 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 (2012a:11069), https://doi.org/10.1007/s00222-010-0252-0
  • [17] PARI/GP, version 2.4.3. The PARI Group (2011) http://pari.math.u-bordeaux.fr.
  • [18] A. N. Parshin and Yu. G. Zarhin, Finiteness problems in Diophantine geometry, Amer. Math. Soc. Transl. 143 (1989) 35-102.
  • [19] Bjorn Poonen and Eric Rains, Random maximal isotropic subspaces and Selmer groups, J. Amer. Math. Soc. 25 (2012), no. 1, 245-269. MR 2833483, https://doi.org/10.1090/S0894-0347-2011-00710-8
  • [20] Michel Raynaud, Schémas en groupes de type $ (p,\dots , p)$, Bull. Soc. Math. France 102 (1974), 241-280 (French). MR 0419467 (54 #7488)
  • [21] K. Rubin and A. Silverberg, Families of elliptic curves with constant mod $ p$ representations, Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 148-161. MR 1363500 (96j:11078)
  • [22] Sage Mathematics Software, Version 4.7.2. The Sage Development Team (2011) http://www.sagemath.org.
  • [23] Jean-Pierre Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259-331 (French). MR 0387283 (52 #8126)
  • [24] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492-517. MR 0236190 (38 #4488)
  • [25] I. R. Šafarevič, Algebraic number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 163-176 (Russian). MR 0202709 (34 #2569)
  • [26] Alice Silverberg, Explicit families of elliptic curves with prescribed mod $ N$ representations, Modular forms and Fermat's last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 447-461. MR 1638488
  • [27] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)
  • [28] Peter Swinnerton-Dyer, The effect of twisting on the 2-Selmer group, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 513-526. MR 2464773 (2010d:11059), https://doi.org/10.1017/S0305004108001588
  • [29] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1975, pp. 33-52. Lecture Notes in Math., Vol. 476. MR 0393039 (52 #13850)
  • [30] John Tate, A review of non-Archimedean elliptic functions, Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 162-184. MR 1363501 (97d:11096)


Additional Information

Barry Mazur
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: mazur@math.harvard.edu

Karl Rubin
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697
Email: krubin@math.uci.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06114-X
Received by editor(s): August 18, 2012
Received by editor(s) in revised form: February 28, 2013
Published electronically: September 4, 2014
Additional Notes: This material is based upon work supported by the National Science Foundation under grants DMS-1065904 and DMS-0968831
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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