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 | References | Similar Articles | Additional Information
Abstract: Given a CM elliptic curve with Weierstrass equation , and a positive definite binary quadratic form
, we show that there are infinitely many reduced integer pairs
such that the twisted elliptic curve
has analytic rank (and consequently Mordell-Weil rank) one. In fact it follows that the number of such pairs with
is at least
for any
.
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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.