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2-Selmer groups of quadratic twists of elliptic curves


Authors: George Boxer and Peter Diao
Journal: Proc. Amer. Math. Soc. 138 (2010), 1969-1978
MSC (2010): Primary 11G05, 11G40
DOI: https://doi.org/10.1090/S0002-9939-10-10273-1
Published electronically: February 11, 2010
MathSciNet review: 2596031
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate families of quadratic twists of elliptic curves. Addressing a speculation of Ono, we identify a large class of elliptic curves for which the parities of the ``algebraic parts'' of the central values $ L(E^{(d)}/\mathbb{D}{Q},1)$, as $ d$ varies, have essentially the same multiplicative structure as the coefficients $ a_d$ of $ L(E/\mathbb{D}{Q},s)$. We achieve this by controlling the 2-Selmer rank (à la Mazur and Rubin) when the Tamagawa numbers do not already dictate the parity.


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

  • [Kr] K. Kramer, Arithmetic of elliptic curves upon quadratic extensions, Trans. Amer. Math. Soc. 264 (1981) 121-135. MR 597871 (82g:14028)
  • [MR] B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert's tenth problem, preprint, available at http://arxiv.org/0904.3709.
  • [O] K. Ono, Nonvanishing of quadratic twists of modular $ L$-functions with applications for elliptic curves, J. Reine Angew. Math. 533 (2001) 81-97. MR 1823865 (2002a:11051)
  • [OS] K. Ono and C. Skinner, Non-vanishing of quadratic twists of modular $ L$-functions, Invent. Math. 134 (1998) 651-660. MR 1660945 (2000a:11077)
  • [Se] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972) 259-331. MR 0387283 (52:8126)
  • [Sh] T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975) 83-126. MR 0389772 (52:10603)
  • [S1] J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, New York: Springer-Verlag (1986). MR 817210 (87g:11070)
  • [S2] J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151, New York: Springer-Verlag (1994). MR 1312368 (96b:11074)

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Additional Information

George Boxer
Affiliation: Department of Mathematics, Mailbox 2704, Frist Center, Princeton University, Princeton, New Jersey 08544
Email: gboxer@princeton.edu

Peter Diao
Affiliation: Department of Mathematics, Mailbox 2704, Frist Center, Princeton University, Princeton, New Jersey 08544
Email: pdiao@princeton.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10273-1
Received by editor(s): October 27, 2009
Published electronically: February 11, 2010
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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