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Torsion points on J0(N) and Galois representations

  • Kenneth A. Ribet
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1716)

Abstract

Suppose that N is a prime number greater than 19 and that P is a point on the modular curve X0(N) whose image in Jo(N) (under the standard embedding ι: X0(N)→J0(N)) has finite order. In [2], Coleman-Kaskel-Ribet conjecture that either P is a hyperelliptic branch point of X0(N) (so that N∈{23,29,31,41,47,59,71}) or else that ι(P) lies in the cuspidal subgroup C of J0(N). That article suggests a strategy for the proof: assuming that P is not a hyperelliptic branch point of X0(N), one should show for each prime number ℓ that the ℓ-primary part of ι(P) lies in C. In [2], the strategy is implemented under a variety of hypotheses but little is proved for the primes ℓ=2 and ℓ=3. Here I prove the desired statement for ℓ=2 whenever N is prime to the discriminant of the ring End J0(N). This supplementary hypothesis, while annoying, seems to be a mild one; according to W.A. Stein of Berkeley, California, in the range N<5021, it is false only in case N=389.

Keywords

Exact Sequence Prime Number Galois Group Finite Order Galois Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Kenneth A. Ribet
    • 1
  1. 1.University of CaliforniaBerkeley

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