Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Parity of ranks of elliptic curves with equivalent mod $ p$ Galois representations

Author: Sudhanshu Shekhar
Journal: Proc. Amer. Math. Soc. 144 (2016), 3255-3266
MSC (2010): Primary 14H52, 11F33, 11R23
Published electronically: February 3, 2016
MathSciNet review: 3503694
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given two elliptic curves $ E_1$ and $ E_2$ defined over the field of rational numbers $ \mathbb{Q}$ that have good and ordinary reduction at an odd prime $ p$, and have equivalent, irreducible mod $ p$ Galois representations, we study the variation of the parity of Selmer ranks and analytic ranks of $ E_1$ and $ E_2$ over certain number fields.

References [Enhancements On Off] (What's this?)

  • [BCDT] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $ \mathbf {Q}$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843-939 (electronic). MR 1839918 (2002d:11058),
  • [C] L. Clozel, Base change for $ {\rm GL}(n)$, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 791-797. MR 934282 (89i:22034)
  • [Dr] Michael Jeffrey Drinen, Iwasawa mu-invariants of Selmer groups, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)-University of Washington. MR 2699373
  • [Dr1] Michael J. Drinen, Finite submodules and Iwasawa $ \mu $-invariants, J. Number Theory 93 (2002), no. 1, 1-22. MR 1892927 (2003b:11118),
  • [Gr] Ralph Greenberg, Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997) Lecture Notes in Math., vol. 1716, Springer, Berlin, 1999, pp. 51-144. MR 1754686 (2002a:11056),
  • [Gr1] Ralph Greenberg, Selmer groups and congruences, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 231-248. MR 2827793 (2012j:11119)
  • [Gr2] Ralph Greenberg, Iwasawa theory for $ p$-adic representations, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97-137. MR 1097613 (92c:11116)
  • [GV] Ralph Greenberg and Vinayak Vatsal, On the Iwasawa invariants of elliptic curves, Invent. Math. 142 (2000), no. 1, 17-63. MR 1784796 (2001g:11169),
  • [EPW] Matthew Emerton, Robert Pollack, and Tom Weston, Variation of Iwasawa invariants in Hida families, Invent. Math. 163 (2006), no. 3, 523-580. MR 2207234 (2007a:11059),
  • [CH] J. H. Coates and S. Howson, Euler characteristics and elliptic curves. II, J. Math. Soc. Japan 53 (2001), no. 1, 175-235. MR 1800527 (2001k:11215),
  • [HM] Yoshitaka Hachimori and Kazuo Matsuno, An analogue of Kida's formula for the Selmer groups of elliptic curves, J. Algebraic Geom. 8 (1999), no. 3, 581-601. MR 1689359 (2000c:11086)
  • [HV] Yoshitaka Hachimori and Otmar Venjakob, Completely faithful Selmer groups over Kummer extensions, Doc. Math. Extra Vol. (2003), 443-478 (electronic). Kazuya Kato's fiftieth birthday. MR 2046605 (2005b:11072)
  • [OV] Yoshihiro Ochi and Otmar Venjakob, On the structure of Selmer groups over $ p$-adic Lie extensions, J. Algebraic Geom. 11 (2002), no. 3, 547-580. MR 1894938 (2003m:11082),
  • [DD] Tim Dokchitser and Vladimir Dokchitser, Root numbers and parity of ranks of elliptic curves, J. Reine Angew. Math. 658 (2011), 39-64. MR 2831512 (2012h:11084),
  • [DD1] T. Dokchitser and V. Dokchitser, Computations in non-commutative Iwasawa theory, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 211-272. With an appendix by J. Coates and R. Sujatha. MR 2294995 (2008g:11106),
  • [RS] 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)
  • [W] Andrew Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 1333035 (96d:11071),
  • [SS] Sudhanshu Shekhar and R. Sujatha, Euler characteristic and congruences of elliptic curves, Münster J. Math. 7 (2014), 327-343. MR 3271248

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14H52, 11F33, 11R23

Retrieve articles in all journals with MSC (2010): 14H52, 11F33, 11R23

Additional Information

Sudhanshu Shekhar
Affiliation: Mathematics Center Heidelberg – and – Indian Institute of Science education and Research, Mohali

Keywords: Iwasawa theory, congruences, Selmer groups, elliptic curves
Received by editor(s): June 9, 2015
Received by editor(s) in revised form: August 21, 2015, and September 23, 2015
Published electronically: February 3, 2016
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society