Critical and ramification points of the modular parametrization of an elliptic curve

Christophe Delaunay[1]

  • [1] Institut Camille Jordan Bâtiment Braconnier Université Claude Bernard Lyon 1 43, avenue du 11 novembre 1918 69622 Villeurbanne cedex, France

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 109-124
  • ISSN: 1246-7405

Abstract

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Let E be an elliptic curve defined over ℚ with conductor N and denote by ϕ the modular parametrization: ϕ : X 0 ( N ) → E ( ℂ ) . In this paper, we are concerned with the critical and ramification points of ϕ . In particular, we explain how we can obtain a more or less experimental study of these points.

How to cite

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Delaunay, Christophe. "Critical and ramification points of the modular parametrization of an elliptic curve." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 109-124. <http://eudml.org/doc/249431>.

@article{Delaunay2005,
abstract = {Let $E$ be an elliptic curve defined over $\mathbb\{Q\}$ with conductor $N$ and denote by $\varphi $ the modular parametrization:\[\varphi \; : \; X\_0(N) \rightarrow E(\mathbb\{C\}) \; .\]In this paper, we are concerned with the critical and ramification points of $\varphi $. In particular, we explain how we can obtain a more or less experimental study of these points.},
affiliation = {Institut Camille Jordan Bâtiment Braconnier Université Claude Bernard Lyon 1 43, avenue du 11 novembre 1918 69622 Villeurbanne cedex, France},
author = {Delaunay, Christophe},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {modular elliptic curves; critical points; ramification points; analytic rank; Hecke operator; involutory curve},
language = {eng},
number = {1},
pages = {109-124},
publisher = {Université Bordeaux 1},
title = {Critical and ramification points of the modular parametrization of an elliptic curve},
url = {http://eudml.org/doc/249431},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Delaunay, Christophe
TI - Critical and ramification points of the modular parametrization of an elliptic curve
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 109
EP - 124
AB - Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and denote by $\varphi $ the modular parametrization:\[\varphi \; : \; X_0(N) \rightarrow E(\mathbb{C}) \; .\]In this paper, we are concerned with the critical and ramification points of $\varphi $. In particular, we explain how we can obtain a more or less experimental study of these points.
LA - eng
KW - modular elliptic curves; critical points; ramification points; analytic rank; Hecke operator; involutory curve
UR - http://eudml.org/doc/249431
ER -

References

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  1. A.O.L. Atkin, J. Lehner, Hecke operators on Γ 0 ( N ) . Math. Ann. 185 (1970), 134–160. Zbl0177.34901
  2. C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, pari-gp, available at http://www.math.u-psud.fr/~belabas/pari/ 
  3. C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over ℚ : wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843–939 (electronic). Zbl0982.11033MR1839918
  4. B. Birch, Heegner points of elliptic curves. Symp. Math. Inst. Alta. Math. 15 (1975), 441–445. Zbl0317.14015MR384805
  5. H. Cohen, A course in computational algebraic number theory. Graduate Texts in Math. 138, Springer-Verlag, New-York, 4-th corrected printing (2000). Zbl0786.11071MR1228206
  6. J. Cremona, Algorithms for modular elliptic curves. Cambridge University Press, (1997) second edition. Zbl0758.14042MR1628193
  7. J. Cremona, Elliptic curve data for conductors up to 25000. Available at http://www.maths.nott.ac.uk/personal/jec/ftp/data/INDEX.html Zbl01.0253.01
  8. C. Delaunay, Computing modular degrees using L -functions. Journ. theo. nomb. Bord. 15 (3) (2003), 673–682. Zbl1070.11021MR2142230
  9. B. Gross, Heegner points on X 0 ( N ) . Modular Forms, ed. R. A. Ramkin, (1984), 87–105. Zbl0559.14011MR803364
  10. B. Gross, D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), 225–320. Zbl0608.14019MR833192
  11. B. Mazur, P. Swinnerton-Dyer, Arithmetic of Weil curves. Invent. Math. 25 (1974), 1–61. Zbl0281.14016MR354674
  12. G. Shimura, Introduction to the arithmetic theory of automorphic functions. Math. Soc of Japan 11, Princeton university Press (1971). Zbl0221.10029MR1291394
  13. N. Skoruppa, D. Zagier, Jacobi forms and a certain space of modular forms. Inv. Math. 98 (1988), 113–146. Zbl0651.10020MR958592
  14. R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553–572. Zbl0823.11030MR1333036
  15. M. Watkins, Computing the modular degree. Exp. Math. 11 (4) (2002), 487–502. Zbl1162.11349MR1969641
  16. A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141 (1995), no.3, 443–551. Zbl0823.11029MR1333035
  17. D. Zagier, Modular parametrizations of elliptic curves. Canad. Math. Bull. 28 (3) (1985), 372–384. Zbl0579.14027MR790959

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