## The cuspidal class number formula for the modular curves $$X_1(2p)$$.(English)Zbl 1277.11068

Summary: Let $$p$$ be a prime not equal to 2 or 3. We determine the group of all modular units on the modular curve $$X_1(2p)$$, 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 $$\mathbb Q$$ of square-free conductor $$n$$ as an application of a result by A. Agashe [Rational torsion in elliptic curves and the cuspidal subgroup, preprint, http://arxiv-web3.library.cornell.edu/abs/0810.5181] and the cuspidal class number formula for $$X_0(n)$$. We also state the formula for the order of the subgroup of the $$\mathbb Q$$-rational torsion subgroup of $$J_1(2p)$$ generated by the $$\mathbb Q$$-rational cuspidal divisors of degree 0.
In the erratum we correct a theorem on the conductor of elliptic curves over $$\mathbb Q$$ given in the introduction.

### MSC:

 11G18 Arithmetic aspects of modular and Shimura varieties 11G05 Elliptic curves over global fields 11G16 Elliptic and modular units 14G05 Rational points 14G35 Modular and Shimura varieties
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### References:

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