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

Abstract

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.

8 pages

Cited by in corpus (3)

• Picard curves over Q with good reduction away from 3
• Arithmetic of abelian varieties with constrained torsion
• Abelian surfaces good away from 2