Compositio Mathematica

Towards the triviality of $X_0^+ (p^r) (\mathbb{Q})$ for r > 1

Pierre J. R. Parent a1
a1 A2X, UFR mathématiques-informatique, Université de Bordeaux 1, 351 cours de la libération, 33405 Talence cedex, France

Article author query
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We give a criterion to check if, given a prime power pr with r > 1, the only rational points of the modular curve X0+ (pr) are trivial (i.e. cusps or points furnished by complex multiplication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit (p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves.

(Received November 6 2003)
(Accepted March 1 2004)
(Published Online April 21 2005)

Key Words: elliptic curves; modular curves; rational points; Gross–Heegner points.

Maths Classification

11G18; 11G05; 14G10.