On quadratic families of CM elliptic curves
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Abstract:
Given a CM elliptic curve with Weierstrass equation $y^2=f(x)$, and a positive definite binary quadratic form $Q(u,v)$, we show that there are infinitely many reduced integer pairs $(u,v)$ such that the twisted elliptic curve $Q(u,v)y^2=f(x)$ has analytic rank (and consequently Mordell-Weil rank) one. In fact it follows that the number of such pairs with $|u|, |v| \leq X$ is at least $X^{2-\varepsilon }$ for any $\varepsilon >0$.References
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Additional Information
- Ritabrata Munshi
- Affiliation: Institute for Advanced Study, Einstein Drive, Princeton New Jersey 08540
- Address at time of publication: School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai 400005, India
- MR Author ID: 817043
- Email: rmunshi@math.ias.edu, rmunshi@math.tifr.res.in
- Received by editor(s): December 1, 2009
- Published electronically: March 4, 2011
- Additional Notes: The author was supported by NSF grant No. DMS-0635607.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 4337-4358
- MSC (2000): Primary 11F67; Secondary 11M41, 11G40
- DOI: https://doi.org/10.1090/S0002-9947-2011-05433-4
- MathSciNet review: 2792990