×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A. Agashe, Rational torsion in elliptic curves and the cuspidal subgroup, preprint, · Zbl 1288.11058
[2] Y. Chen, Cuspidal \(\bm{Q}\)-rational torsion subgroup of \(J(\Gamma)\) of level \(p\), Taiwanese J. Math., 15 (2011), 1305-1323. · Zbl 1333.11055
[3] B. Conrad, B. Edixhoven and W. Stein, \(J_{1}(p)\) has connected fibers, Doc. Math., 8 (2003), 331-408. · Zbl 1101.14311
[4] V. G. Drinfeld, Two theorems on modular curves, Funct. Anal. Appl., 7 (1973), 155-156. · Zbl 0285.14006
[5] S. Klimek, Thesis, Berkeley, 1975.
[6] D. Kubert, The square root of the Siegel group, Proc. London Math. Soc. (3), 43 (1981), 193-226. · Zbl 0473.12002
[7] D. Kubert and S. Lang, Modular Units, Grundlehren der Mathematischen Wissenschaften, 244 , Springer-Verlag, Berlin, 1981. · Zbl 0492.12002
[8] J. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 19-64. · Zbl 0243.14008
[9] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes √Čtudes Sci. Publ. Math., 47 (1977), 33-186. · Zbl 0394.14008
[10] A. Ogg, Rational points on certain elliptic modular curves, AMS Conference, St. Louis, 1972, pp.,211-231. · Zbl 0273.14008
[11] A. Ogg, Diophantine equations and modular forms, Bull. Amer. Math. Soc., 81 (1975), 14-27. · Zbl 0316.14012
[12] G. Stevens, The cuspidal group and special values of \(L\)-functions, Trans. Amer. Math. Soc., 291 (1985), 519-550. · Zbl 0579.10011
[13] T. Takagi, Cuspidal class number formula for the modular curves \(X_{1}(p)\), J. Algebra, 151 (1992), 348-374. · Zbl 0773.11040
[14] T. Takagi, The cuspidal class number formula for the modular curves \(X_{1}(p^{m})\), J. Algebra, 158 (1993), 515-549. · Zbl 0811.11045
[15] T. Takagi, The cuspidal class number formula for the modular curves \(X_{1}(3^{m})\), J. Math. Soc. Japan, 47 (1995), 671-686. · Zbl 0888.11022
[16] T. Takagi, The cuspidal class number formula for the modular curves \(X_{0}(M)\) with \(M\) square-free, J. Algebra, 193 (1997), 180-213. · Zbl 0888.11021
[17] T. Takagi, The cuspidal class number formula for the modular curves \(X_{1}(2^{2n+1})\), J. Algebra, 319 (2008), 3535-3566. · Zbl 1155.11031
[18] T. Takagi, Modified Siegel functions relative to the principal congruence subgroups of \(G(\sqrt{M})\), J. Faculty of Arts and Sciences at Fujiyoshida, Showa University, 4 (2009), 1-16.
[19] T. Takagi, The cuspidal class number formula for certain quotient curves of the modular curve \(X_{0}(M)\) by Atkin-Lehner involutions, J. Math. Soc. Japan, 62 (2010), 13-47. · Zbl 1273.11095
[20] Y. Yang, Modular units and cuspidal divisor class groups of \(X_{1}(N)\), J. Algebra, 322 (2009), 514-553. · Zbl 1208.11076
[21] J. Yu, A cuspidal class number formula for the modular curves \(X_{1}(N)\), Math. Ann., 252 (1980), 197-216. · Zbl 0426.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.