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 with conductor N and denote by ϕ the modular parametrization: ϕ : X 0 ( N ) E ( ) . 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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  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
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  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
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