Selmer companion curves

Mazur, Barry; Rubin, Karl

doi:10.1090/S0002-9947-2014-06114-X

Selmer companion curves
HTML articles powered by AMS MathViewer

by Barry Mazur and Karl Rubin PDF

Trans. Amer. Math. Soc. 367 (2015), 401-421 Request permission

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.

References

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
J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
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 179169, DOI 10.1515/crll.1965.217.180
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, DOI 10.4153/CMB-2005-002-x
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, DOI 10.4007/annals.2010.172.567
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, DOI 10.1090/S0894-0347-1991-1080648-7
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, DOI 10.1007/BF01231536
E. Kani and W. Schanz, Modular diagonal quotient surfaces, Math. Z. 227 (1998), no. 2, 337–366. MR 1609061, DOI 10.1007/PL00004379
H. Kisilevsky, Rank determines semi-stable conductor, J. Number Theory 104 (2004), no. 2, 279–286. MR 2029506, DOI 10.1016/S0022-314X(03)00157-4
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
Z. Klagsbrun, B. Mazur and K. Rubin, Selmer ranks of quadratic twists of elliptic curves, to appear in Annals of Math.
Kenneth Kramer, Arithmetic of elliptic curves upon quadratic extension, Trans. Amer. Math. Soc. 264 (1981), no. 1, 121–135. MR 597871, DOI 10.1090/S0002-9947-1981-0597871-8
A. Kraus and J. Oesterlé, Sur une question de B. Mazur, Math. Ann. 293 (1992), no. 2, 259–275 (French). MR 1166121, DOI 10.1007/BF01444715
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
B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
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
PARI/GP, version 2.4.3. The PARI Group (2011) http://pari.math.u-bordeaux.fr.
A. N. Parshin and Yu. G. Zarhin, Finiteness problems in Diophantine geometry, Amer. Math. Soc. Transl. 143 (1989) 35–102.
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
Michel Raynaud, Schémas en groupes de type $(p,\dots , p)$, Bull. Soc. Math. France 102 (1974), 241–280 (French). MR 419467, DOI 10.24033/bsmf.1779
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
Sage Mathematics Software, Version 4.7.2. The Sage Development Team (2011) http://www.sagemath.org.
Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
I. R. Šafarevič, Algebraic number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 163–176 (Russian). MR 0202709
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
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
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, DOI 10.1017/S0305004108001588
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) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039
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