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Takagi, Toshikazu

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

J. Math. Soc. Japan 64, No. 1, 23-85, erratum 87-89 (2012).
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.

Cited in 5 Documents

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

Keywords:

modular curve; modular unit; cuspidal class number; elliptic curve; Jacobian variety; torsion subgroup
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\textit{T. Takagi}, J. Math. Soc. Japan 64, No. 1, 23--85 (2012; Zbl 1277.11068)
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References:

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