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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On quadratic families of CM elliptic curves
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by Ritabrata Munshi PDF
Trans. Amer. Math. Soc. 363 (2011), 4337-4358 Request permission

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 $\varepsilon >0$.
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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