On quadratic families of CM elliptic curves
Author:
Ritabrata Munshi
Journal:
Trans. Amer. Math. Soc. 363 (2011), 43374358
MSC (2000):
Primary 11F67; Secondary 11M41, 11G40
Published electronically:
March 4, 2011
MathSciNet review:
2792990
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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 MordellWeil rank) one. In fact it follows that the number of such pairs with is at least for any .
 1.
B.
J. Birch and N.
M. Stephens, The parity of the rank of the MordellWeil group,
Topology 5 (1966), 295–299. MR 0201379
(34 #1263)
 2.
Daniel
Bump, Solomon
Friedberg, and Jeffrey
Hoffstein, Eisenstein series on the metaplectic group and
nonvanishing theorems for automorphic 𝐿functions and their
derivatives, Ann. of Math. (2) 131 (1990),
no. 1, 53–127. MR 1038358
(92e:11053), 10.2307/1971508
 3.
Max
Deuring, Die Typen der Multiplikatorenringe elliptischer
Funktionenkörper, Abh. Math. Sem. Hansischen Univ.
14 (1941), 197–272 (German). MR 0005125
(3,104f)
 4.
D.
Goldfeld, J.
Hoffstein, and S.
J. Patterson, On automorphic functions of halfintegral weight with
applications to elliptic curves, Number theory related to
Fermat’s last theorem (Cambridge, Mass., 1981), Progr. Math.,
vol. 26, Birkhäuser, Boston, Mass., 1982, pp. 153–193.
MR 685295
(84i:10031)
 5.
D.
R. HeathBrown, A mean value estimate for real character sums,
Acta Arith. 72 (1995), no. 3, 235–275. MR 1347489
(96h:11081)
 6.
Henryk
Iwaniec and Emmanuel
Kowalski, Analytic number theory, American Mathematical
Society Colloquium Publications, vol. 53, American Mathematical
Society, Providence, RI, 2004. MR 2061214
(2005h:11005)
 7.
Henryk
Iwaniec and Ritabrata
Munshi, Cubic polynomials and quadratic forms, J. Lond. Math.
Soc. (2) 81 (2010), no. 1, 45–64. MR 2580453
(2011d:11228), 10.1112/jlms/jdp056
 8.
Ritabrata
Munshi, The level of distribution of the special values of
𝐿functions, Acta Arith. 138 (2009),
no. 3, 239–257. MR 2520081
(2011b:11087), 10.4064/aa13832
 9.
Munshi, R. On mean values and nonvanishing of derivatives of functions in a nonlinear family. Compositio Math. 147 (2011), no. 1, 1934.
 10.
Munshi, R. Inequalities for divisor functions. (To appear in Ramanujan Journal.)
 11.
M.
Ram Murty and V.
Kumar Murty, Mean values of derivatives of modular
𝐿series, Ann. of Math. (2) 133 (1991),
no. 3, 447–475. MR 1109350
(92e:11050), 10.2307/2944316
 12.
K.
Soundararajan, Nonvanishing of quadratic Dirichlet
𝐿functions at 𝑠=\frac12, Ann. of Math. (2)
152 (2000), no. 2, 447–488. MR 1804529
(2001k:11164), 10.2307/2661390
 1.
 Birch, B. J.; Stephens, N. M. The parity of the rank of the MordellWeil group. Topology 5 (1966), 295299. MR 0201379 (34:1263)
 2.
 Bump, D.; Friedberg, S.; Hoffstein, J. Eisenstein series on the metaplectic group and non vanishing theorems for automorphic functions and their derivatives. Ann. of Math. (2) 131 (1990), no. 1, 53127. MR 1038358 (92e:11053)
 3.
 Deuring, M. Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Zem. Hansischen Univ. 14 (1941), 197272. MR 0005125 (3:104f)
 4.
 Goldfeld, D.; Hoffstein, J.; Patterson, S. J. On automorphic functions of halfintegral weight with applications to elliptic curves. Number theory related to Fermat's last theorem (Cambridge, Mass., 1981), pp. 153193, Progr. Math., 26, Birkhäuser, Boston, Mass., 1982. MR 685295 (84i:10031)
 5.
 HeathBrown, D. R. A mean value estimate for real character sums. Acta Arith. 72 (1995), no. 3, 235275. MR 1347489 (96h:11081)
 6.
 Iwaniec, H.; Kowalski, E. Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp. MR 2061214 (2005h:11005)
 7.
 Iwaniec, H.; Munshi, R. Cubic polynomials and quadratic forms. J. Lon. Math. Soc. (2) 81 (2010), 4564 MR 2580453
 8.
 Munshi, R. The level of distribution of the special values of functions. Acta Arith. 138 (2009), no. 3, 239257. MR 2520081
 9.
 Munshi, R. On mean values and nonvanishing of derivatives of functions in a nonlinear family. Compositio Math. 147 (2011), no. 1, 1934.
 10.
 Munshi, R. Inequalities for divisor functions. (To appear in Ramanujan Journal.)
 11.
 Murty, M. R.; Murty, V. K. Mean values of derivatives of modular series. Ann. of Math. (2) 133 (1991), no. 3, 447475. MR 1109350 (92e:11050)
 12.
 Soundararajan, K. Nonvanishing of quadratic Dirichlet functions at . Ann. of Math. (2) 152 (2000), no. 2, 447488. MR 1804529 (2001k:11164)
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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:
http://dx.doi.org/10.1090/S000299472011054334
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. DMS0635607.
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
