On the modular curves $Y_E(7)$
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- by Emmanuel Halberstadt and Alain Kraus PDF
- Math. Comp. 69 (2000), 1193-1206 Request permission
Abstract:
Let $E$ denote an elliptic curve over $\mathbf {Q}$ and $Y_E(7)$ the modular curve classifying the elliptic curves $E’$ over $\mathbf {Q}$ such that the representations of $\operatorname {Gal}(\overline {\mathbf Q}/\mathbf {Q})$ in the 7-torsion points of $E$ and of $E’$ are symplectically isomorphic. In case $E$ is given by a Weierstraßequation such that the $c_4$ invariant is a square, we exhibit here nontrivial points of $Y_E(7)(\mathbf {Q})$. From this we deduce an infinite family of curves $E$ for which $Y_E(7)(\mathbf {Q})$ has at least four nontrivial points.References
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Additional Information
- Emmanuel Halberstadt
- Affiliation: Université Paris VI, Laboratoire de Mathématiques Fondamentales, UFR 921, 4, place Jussieu, 75252 Paris Cedex 05, France
- Email: halberst@math.jussieu.fr
- Alain Kraus
- Affiliation: Université Paris VI, Institut de Mathématiques, Case 247, 4, place Jussieu, 75252 Paris Cedex 05, France
- MR Author ID: 263331
- Email: kraus@math.jussieu.fr
- Received by editor(s): August 8, 1997
- Received by editor(s) in revised form: July 24, 1998
- Published electronically: May 21, 1999
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 1193-1206
- MSC (1991): Primary 11Gxx
- DOI: https://doi.org/10.1090/S0025-5718-99-01123-0
- MathSciNet review: 1651758