Mathematics of Computation

Author(s): Emmanuel Halberstadt; Alain Kraus.
Journal: Math. Comp. 69 (2000), 1193-1206.
MSC (1991): Primary 11Gxx
Posted: May 21, 1999
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Abstract: Let

denote an elliptic curve over $\mathbf{Q}$ and

the modular curve classifying the elliptic curves

over $\mathbf{Q}$ such that the representations of $\operatorname{Gal}(\overline{\mathbf Q}/\mathbf{Q})$ in the 7-torsion points of

and of

are symplectically isomorphic. In case

is given by a Weierstraß equation such that the

invariant is a square, we exhibit here nontrivial points of $Y_E(7)(\mathbf{Q})$ . From this we deduce an infinite family of curves

for which $Y_E(7)(\mathbf{Q})$ has at least four nontrivial points.

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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

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