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Quadratic twists of elliptic curves with 3-Selmer rank 1

Zane Kun Li

Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

https://doi.org/10.1142/S1793042114500213Cited by:2 (Source: Crossref)

Abstract

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve E/ℚ, a positive proportion of its quadratic twists E^(d) have rank 1. Using tools from Galois cohomology, we give criteria on E and d which force a positive proportion of the quadratic twists of E to have 3-Selmer rank 1 and global root number -1. We then give four nonisomorphic infinite families of elliptic curves E_m,n which satisfy these criteria. Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this verifies the aforementioned conjecture for infinitely many elliptic curves. Our elliptic curves are easy to give explicitly and we state precisely which quadratic twists d to use. Furthermore, our methods have the potential of being generalized to elliptic curves over other number fields.

Keywords:

AMSC: 14H52, 11G05

References

M. Bhargava, D. M. Kane, H. W. Lenstra Jr., B. Poonen and E. Rains, Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves, preprint (2013) , arXiv:1304.3971 . Google Scholar
D. Byeon, Acta Arith. 114(4), 391 (2004). Crossref, Web of Science, Google Scholar
D. Byeon , D. Jeon and C. H. Kim , J. Reine Angew. Math. 633 , 67 ( 2009 ) . Web of Science, Google Scholar
D. Byeon and D. Yhee, J. Number Theory 131(3), 552 (2011). Crossref, Web of Science, Google Scholar
H. Cohen and F. Pazuki, Acta Arith. 140(4), 369 (2009). Crossref, Web of Science, Google Scholar
I. Connell, Elliptic curve handbook, Preprint, McGill University, Montreal (1996); http://www.math.mcgill.ca/connell/public/ECH1/ . Google Scholar
H. Darmon, F. Diamond and R. Taylor, Current Developments in Mathematics, 1995 (International Press, Cambridge, MA, 1994) pp. 1–154. Google Scholar
T. Dokchitser and V. Dokchitser, Ann. of Math. (2) 172(1), 567 (2010). Crossref, Web of Science, Google Scholar
P. Erdös , J. London Math. Soc. 28 , 416 ( 1953 ) . Crossref, Google Scholar
D. Goldfeld, Number Theory, Lecture Notes in Mathematics 751 (Springer, Berlin, 1979) pp. 108–118. Crossref, Google Scholar
J. Nakagawa and K. Horie, Proc. Amer. Math. Soc. 104(1), 20 (1988). Crossref, Web of Science, Google Scholar
K. Ono , The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series , CBMS Regional Conference Series in Mathematics 102 ( American Mathematical Society , Providence, RI , 2004 ) . Google Scholar
K. Ono and C. Skinner, Invent. Math. 134(3), 651 (1998). Crossref, Web of Science, Google Scholar
A. Perelli and J. Pomykała, Acta Arith. 80(2), 149 (1997). Crossref, Web of Science, Google Scholar
J. H. Silverman , Advanced Topics in the Arithmetic of Elliptic Curves , Graduate Texts in Mathematics 151 ( Springer , New York , 1994 ) . Crossref, Google Scholar
J. H. Silverman , The Arithmetic of Elliptic Curves , 2nd edn. , Graduate Texts in Mathematics 106 ( Springer , Dordrecht , 2009 ) . Crossref, Google Scholar
S. Schmitt and H. G. Zimmer , Elliptic Curves. A Computational Approach , de Gruyter Studies in Mathematics 31 ( Walter de Gruyter & Co. , Berlin , 2003 ) . Google Scholar
H. Taya, Proc. Amer. Math. Soc. 128(5), 1285 (2000). Crossref, Web of Science, Google Scholar
V. Vatsal, Math. Ann. 311(4), 791 (1998). Crossref, Web of Science, Google Scholar
V. Vatsal, Duke Math. J. 98(2), 397 (1999). Crossref, Web of Science, Google Scholar
T. Wang, Selmer groups and ranks of elliptic curves, Senior thesis, Princeton University (2012) . Google Scholar
L. C. Washington, Modular Forms and Fermat's Last Theorem (Springer, New York, 1997) pp. 101–120. Crossref, Google Scholar
E. Weiss , Cohomology of Groups , Pure and Applied Mathematics 34 ( Academic Press , New York , 1969 ) . Google Scholar