Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T22:18:46.724Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Quadratic Twists of a Family of Elliptic Curves

Part of: Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  21 December 2009

Gang Yu
Gang Yu
Affiliation:
Department of Mathematics, University of Michigan, 525 E. University Ave., Ann Arbor, MI 48109–1109, USA. E-mail: [email protected].
Get access

Abstract

In this paper is considered the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over ℚ with ℚ-torsion group ℤ2 × ℤ2. The existence is shown of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.

MSC classification

Secondary: 14H52: Elliptic curves 11G05: Elliptic curves over global fields
Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 139 - 154
Copyright
Copyright © University College London 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Burgess, D. A.. On character sums and L-series, II. Proc. Lond. Math. Soc., (3), (1963), 524–536.CrossRefGoogle Scholar
2Goldfeld, D.. Conjectures on elliptic curves over quadratic fields. Number Theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture Notes in Math., 751, Springer (Berlin, 1979), 108–118.Google Scholar
3Gouvea, F. and Mazur, B.. The square-free sieve and the rank of elliptic curves. J. Amer. Math. Soc., 4 (1991), 1–23.CrossRefGoogle Scholar
4Heath-Brown, D. R.. The size of Selmer groups for the congruent number problem, I. Invent. Math., 111 (1993), 171–195.CrossRefGoogle Scholar
5Heath-Brown, D. R.. The size of Selmer groups for the congruent number problem, II. Invent. Math., 118 (1994), 331–370.CrossRefGoogle Scholar
6James, K.. An example of an elliptic curve with a positive density of prime quadratic twists which have rank zero. Topics in Number Theory (University Park, PA 1997) (1999), 223–227.Google Scholar
7James, K.. L-series with non-zero central critical value. J. Amer. Math. Soc., 11 (1998), 635–641.CrossRefGoogle Scholar
8Kolyvagin, V. A.. Finiteness of E(ǁ) and the Tate-Shafarevich group of E(ℚ) for a subclass of Weil curves. Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 522–540, 670–671.Google Scholar
9Mazur, B.. Modular curves and the Eisenstein ideal. IHES Publ. Math., 47 (1977), 33–186.CrossRefGoogle Scholar
10Mazur, B.. Rational isogenies of prime degree. Invent. Math., 44 (1978), 129–162.CrossRefGoogle Scholar
11Monsky, P.. Generalizing the Birch-Stephens theorem. I: modular curves. Math. Z., 221 (1996), 415–420.Google Scholar
12Ono, K.. Rank zero quadratic twists of modular elliptic curves. Compositio Math., 104 (1996), 293–304.Google Scholar
13Ono, K.. Twists of elliptic curves. Compositio Math., 106 (1997), 349–360.CrossRefGoogle Scholar
14Ono, K. and Skinner, C.. Fourier coefficients of half-integral weight modular forms modulol. Ann. Math., (2), 147 (1998), 453–476.CrossRefGoogle Scholar
15Ono, K. and Skinner, C.. Non-vanishing of quadratic twists of modular L-functions. Invent. Math., 134 (1998), 651–660.CrossRefGoogle Scholar
16Shimura, G.. On modular forms of half integral wieght. Ann. Math., (2), 97 (1973), 440–481.CrossRefGoogle Scholar
17Silverman, J.. The arithmetic of elliptic curves. GTM 106, Springer (1986).Google Scholar
18Waldspurger, J. L.. Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl., 60 (1981), 375–484.Google Scholar
19Wong, S.. Elliptic curves and class number divisibility. Internat. Math. Res. Notices (1999), 661–672.CrossRefGoogle Scholar
20Yu, G.. Rank 0 quadratic twists of a family of elliptic curves. Compositio Math., 135 (2003), 331–356.CrossRefGoogle Scholar