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On quadratic families of CM elliptic curves


Author: Ritabrata Munshi
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
Published electronically: March 4, 2011
MathSciNet review: 2792990
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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 $ \vert u\vert, \vert v\vert \leq X$ is at least $ X^{2-\varepsilon}$ for any $ \varepsilon>0$.


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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
Email: rmunshi@math.ias.edu, rmunshi@math.tifr.res.in

DOI: https://doi.org/10.1090/S0002-9947-2011-05433-4
Keywords: CM elliptic curves, $L$-functions, nonvanishing
Received by editor(s): December 1, 2009
Published electronically: March 4, 2011
Additional Notes: The author was supported by NSF grant No. DMS-0635607.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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