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Transactions of the American Mathematical Society

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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
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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  • 1. Birch, B. J.; Stephens, N. M. The parity of the rank of the Mordell-Weil group. Topology 5 (1966), 295-299. MR 0201379 (34:1263)
  • 2. Bump, D.; Friedberg, S.; Hoffstein, J. Eisenstein series on the metaplectic group and non- vanishing theorems for automorphic $ L$-functions and their derivatives. Ann. of Math. (2) 131 (1990), no. 1, 53-127. MR 1038358 (92e:11053)
  • 3. Deuring, M. Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Zem. Hansischen Univ. 14 (1941), 197-272. MR 0005125 (3:104f)
  • 4. Goldfeld, D.; Hoffstein, J.; Patterson, S. J. On automorphic functions of half-integral weight with applications to elliptic curves. Number theory related to Fermat's last theorem (Cambridge, Mass., 1981), pp. 153-193, Progr. Math., 26, Birkhäuser, Boston, Mass., 1982. MR 685295 (84i:10031)
  • 5. Heath-Brown, D. R. A mean value estimate for real character sums. Acta Arith. 72 (1995), no. 3, 235-275. 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), 45-64 MR 2580453
  • 8. Munshi, R. The level of distribution of the special values of $ L$-functions. Acta Arith. 138 (2009), no. 3, 239-257. MR 2520081
  • 9. Munshi, R. On mean values and nonvanishing of derivatives of $ L$-functions in a nonlinear family. Compositio Math. 147 (2011), no. 1, 19-34.
  • 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 $ L$-series. Ann. of Math. (2) 133 (1991), no. 3, 447-475. MR 1109350 (92e:11050)
  • 12. Soundararajan, K. Nonvanishing of quadratic Dirichlet $ L$-functions at $ s=\frac12$. Ann. of Math. (2) 152 (2000), no. 2, 447-488. 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

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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