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1 results

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

  • Pierre J. R. Parent
  • Journal:
    Compositio Mathematica / Volume 141 / Issue 3 / May 2005
    Published online by Cambridge University Press:
    21 April 2005, pp. 561-572
    Print publication:
    May 2005
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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.