Number Theory
Integral j-invariants and Cartan structures for elliptic curves
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 599-602.

We bound the j-invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that, under the GRH, the modular curve Xsplit(p5) has no non-trivial rational point if p is a sufficiently large prime number.

On borne l'invariant j des points entiers des courbes modulaires, en fonction du groupe de congruence définissant la courbe. Sous l'hypothèse de Riemann généralisée, on en déduit que, si p est un nombre premier suffisamment grand, la courbe modulaire Xsplit(p5) n'a pas de point rationnel non trivial.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.002
Bilu, Yu. 1; Parent, Pierre 1

1 Institut de mathématiques de Bordeaux, 351, cours de la Libération, 33405 Talence cedex, France
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Bilu, Yu.; Parent, Pierre. Integral j-invariants and Cartan structures for elliptic curves. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 599-602. doi : 10.1016/j.crma.2008.04.002. http://www.numdam.org/articles/10.1016/j.crma.2008.04.002/

[1] Yu. Bilu, P. Parent, Explicit bounds for integral j-invariants and level of Cartan structures for elliptic curves, in preparation

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