On the torsion of Jacobians of principal modular curves of level 3^n

Details On the torsion of Jacobians of principal modular curves of level 3^n On the torsion of Jacobians of principal modular curves of level 3^n

2005-10-03

arxiv.org/abs/math/0510023v1

Arch. Math. (Basel) 88 (2007), 19-28

Mathematics

Number Theory

8 pages

Scientific paper

We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3^n) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of the criterion are provided; the first relies on a theorem of Shimura, while the second uses the moduli interpretation.

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