Date: 12 Oct 2006

Torsion points on J 0(N) and Galois representations

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Abstract

Suppose that N is a prime number greater than 19 and that P is a point on the modular curve X 0(N) whose image in J o(N) (under the standard embedding ι: X 0(N)→J 0(N)) has finite order. In [2], Coleman-Kaskel-Ribet conjecture that either P is a hyperelliptic branch point of X 0(N) (so that N∈{23,29,31,41,47,59,71}) or else that ι(P) lies in the cuspidal subgroup C of J 0(N). That article suggests a strategy for the proof: assuming that P is not a hyperelliptic branch point of X 0(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 J 0(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.

To Barry Mazur, for his 60th birthday
The author’s research was partially supported by National Science Foundation contract #DMS 96 22801.