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Let  p  be a prime and let  J  1 ( p   n  ) denote the Jacobian of the modular curve  X  1 ( p   n  ). The Jacobian  J  1 ( p   n  ) contains a ℚ‐rational torsion subgroup generated by the cuspidal divisor classes [( a / p   n  )−(∞)], where  p ∤  a . In this paper, we determine the structure of the  p ‐primary subgroup of this ℚ‐rational torsion subgroup in the case where  p  is a regular prime.

Structure of the cuspidal rational torsion subgroup of J 1 ( p n )