The cuspidal class number formula for the modular curves X₁(2p)

Abstract

Let p be a prime not equal to 2 or 3. We determine the group of all modular units on the modular curve X₁(2p), and its full cuspidal class number. We mention a fact concerning the non-existence of torsion points of order 5 or 7 of elliptic curves over Q of square-free conductor n as an application of a result by Agashe and the cuspidal class number formula for X₀(n). We also state the formula for the order of the subgroup of the Q-rational torsion subgroup of J₁(2p) generated by the Q-rational cuspidal divisors of degree 0.