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Quadratic torsion subgroups of modular Jacobian varieties

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Abstract

Let D be an odd square-free positive integer and C be a divisor of D. For any quadratic character x modulo C, we prove that a large portion of the x-part of the torsion subgroup of J0(DC) coincides with that of its cuspidal subgroup.

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References

  1. N. Billerey and L. V. Dieulefait, Explicit large image theorems for modular forms, Journal of the London Mathematical Society 89 (2014), 499–523.

    Article  MathSciNet  Google Scholar 

  2. H. Carayol, Sur les reprsentations galoisiennes modulo ℓ attachées aux formes modulaires, Duke Mathematical Journal 59 (1989), 785–801.

    Article  MathSciNet  Google Scholar 

  3. H. Darmon, F. Diamond and R. Taylor, Fermat’s last theorem, in Elliptic Curves, Modular Forms and Fermat’s Last Theorem (Hong Kong, 1993), International press, Cambridge, MA, 1997, pp. 2–140.

    Google Scholar 

  4. B. H. Gross, A tameness criterion for Galois representations associated to modular forms (mod p), Duke Mathematical Journal 61 (1990), 445–517.

    Article  MathSciNet  Google Scholar 

  5. S. Ling, On the ℚ-rational cuspidal subgroup and the component group of J0(pr), Israel Journal of Mathematics 99 (1997), 29–54.

    Article  MathSciNet  Google Scholar 

  6. S. Ling and J. Oesterle, The Shimura subgroup of J0(N), Astérisque 196–197 (1991), 171–203.

    MATH  Google Scholar 

  7. R. Livné, On the conductors of mod ℓ Galois representations coming from modular forms, Journal of Number Theory 31 (1989), 133–141.

    Article  MathSciNet  Google Scholar 

  8. D. J. Lorenzini, Torsion points on the modular Jacobian J0(N), Compositio Mathematica 96 (1995), 149–172.

    MathSciNet  MATH  Google Scholar 

  9. B. Mazur, Modular curves and the Eisenstein ideal, Publications Mathématiques de ľInstitut des Hautes Études Scientifiques 47 (1977), 33–186.

    Article  MathSciNet  Google Scholar 

  10. M. Ohta, Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties. II, Tokyo Journal of Mathematics 37 (2014), 273–318.

    Article  MathSciNet  Google Scholar 

  11. K. Ribet, Mod p Hecke operators and congruences between modular forms, Inventiones Mathematicae 71 (1983), 193–205.

    Article  MathSciNet  Google Scholar 

  12. K. Ribet, On modular representations of \({\rm{Gal}}(\bar{\mathbb{Q}}/{\mathbb{Q}})\) arising from modular forms, Inventiones mathematicae 100 (1990), 431–476.

    Article  MathSciNet  Google Scholar 

  13. G. Stevens, Arithmetic of Modular Curves, Progress in Mathematics, Vol. 20, Birkhäuser, Boston, MA, 1982.

    Book  Google Scholar 

  14. G. Stevens, The cuspidal group and special values of L-functions. Transactions of the American Mathematical Society 291 (1985), 519–550.

    MathSciNet  MATH  Google Scholar 

  15. A. Wiles, Modular elliptic curves and Fermat’s last theorem, Annals of Mathematics 141 (1995), 443–551.

    Article  MathSciNet  Google Scholar 

  16. H. Yoo, The index ofan Eisenstein ideal and multiplicity one, Mathematische Zeitschrift 282 (2016), 1097–1116.

    Article  MathSciNet  Google Scholar 

  17. H. Yoo, On rational Eisenstein primes and the rational cuspidal groups ofmodular Jacobian varieties, Transactions of the American Mathematical Societ 372 (2019), 2429–2466.

    Article  Google Scholar 

Download references

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (Grant No. 12001390). The author wishes to express his sincere appreciation to the anonymous referee for invaluable comments and suggestions.

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  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610066, China

    Yuan Ren

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  1. Yuan Ren
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Correspondence to Yuan Ren.

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Ren, Y. Quadratic torsion subgroups of modular Jacobian varieties. Isr. J. Math. 245, 675–710 (2021). https://doi.org/10.1007/s11856-021-2223-0

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  • DOI: https://doi.org/10.1007/s11856-021-2223-0

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