Advertisement

Congruences modulo prime powers of Hecke eigenvalues in level 1

  • Nadim RustomEmail author
Research
  • 21 Downloads

Abstract

We continue the study of strong, weak, and dc-weak eigenforms introduced by Chen, Kiming, and Wiese. We completely determine all systems of Hecke eigenvalues of level 1 modulo 128, showing there are finitely many. This extends results of Hatada and can be considered as evidence for the more general conjecture formulated by the author together with Kiming and Wiese on finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed level. We also discuss the finiteness of systems of Hecke eigenvalues of level 1 modulo 9, reducing the question to the finiteness of a single eigenvalue. Furthermore, we answer the question of comparing weak and dc-weak eigenforms and provide the first known examples of non-weak dc-weak eigenforms.

Keywords

Modular forms Congruences 

Mathematics Subject Classification

Primary 11F33 Secondary 11F80 

Notes

Acknowlegements

The author would like to thank Ian Kiming and Gabor Wiese for many interesting discussions on the topic of eigenforms modulo prime powers over the year, as well as Shaunak Deo and Ming-Lun Hsieh for helpful discussions and comments on this work. The author would also like to thank the anonymous referee for thoroughly reading the manuscript and providing numerous valuable comments and suggestions, in particular suggesting how to fill an earlier gap in the proof of Proposition 10.2. All relevant computations were done using Sage. This research was supported by a Postdoctoral Fellowship at the National Center for Theoretical Sciences, Taipei, Taiwan.

References

  1. 1.
    Ashworth, M.H.: Congruence and identical properties of modular forms. PhD thesis, Oxford (1968)Google Scholar
  2. 2.
    Bambah, R.P., Chowla, S.: The residue of Ramanujan’s function \(\tau (n)\) to the modulus \(2^8\). J. Lond. Math. Soc. 22, 140–147 (1947)CrossRefGoogle Scholar
  3. 3.
    Bellaïche, J.: Pseudodeformations. Math. Z. 270(3–4), 1163–1180 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bellaïche, J., Khare, C.: Level 1 Hecke algebras of modular forms modulo \(p\). Compos. Math. 151(3), 397–415 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Buzzard, K.: Questions about slopes of modular forms. Automorphic forms I. Astérisque 298, 1–15 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Calegari, F., Emerton, M.: The Hecke algebra \(T_k\) has large index. Math. Res. Lett. 11(1), 125–137 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chenevier, G.: The \(p\)-adic analytic space of pseudo characters of a profinite group and pseudo representations over arbitrary rings. In: Automorphic Forms and Galois Representations, vol. 1. London Mathematical Society Lecture Note Series, vol. 414, pp. 221–285. Cambridge University Press, Cambridge (2014)Google Scholar
  8. 8.
    Chen, I., Kiming, I., Wiese, G.: On modular Galois representations modulo prime powers. Int. J. Number Theory 9(1), 91–113 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coleman, R.F., Stein, W.A.: Approximation of eigenforms of infinite slope by eigenforms of finite slope. In: Geometric Aspects of Dwork Theory, vol. I, II, pp. 437–449. Walter de Gruyter, Berlin (2004)Google Scholar
  10. 10.
    Deligne, P.: Formes modulaires et représentations \(l\)-adiques. In: Séminaire Bourbaki, vol. 1968/69: exposés 347–363. Lecture Notes in Mathematical, vol. 175, Exp. No. 355, pp. 139–172. Springer, Berlin (1971)Google Scholar
  11. 11.
    Deligne, P.: Courbes elliptiques: formulaire d’après J. Tate, Modular functions of one variable, IV. In: Proceeding of International Summer School, University of Antwerp, Antwerp, 1972. Lecture Notes in Mathematical, vol. 476, pp. 53–73 (1975)Google Scholar
  12. 12.
    Deligne, P., Serre, J.-P.: Formes modulaires de poids \(1\). Ann. Sci. Écol. Norm. Supér. 7(4), 507–530 (1975)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Deo, S.V.: Structure of Hecke algebras of modular forms modulo \(p\). Algebra Number Theory 11(1), 1–38 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Emerton, M.: Local-global compatibility in the \(p\)-adic Langlands programme for \(\text{GL}_{2/\mathbb{Q}}\). http://www.math.uchicago.edu/emerton/pdffiles/lg.pdf (2011)
  15. 15.
    Emerton, M.: \(p\)-adic families of modular forms (after Hida, Coleman, and Mazur). Astérisque (2011), no. 339, Exp. No. 1013, vii, 31–61. Séminaire Bourbaki, vol. 2009/2010. Exposés 1012–1026Google Scholar
  16. 16.
    Fontaine, J.-M., Mazur, B.: Geometric Galois Representations Elliptic Curves Modular Forms & Fermat’s Last Theorem (Hong Kong, 1993). Number Theory I, pp. 41–78. International Press, Cambridge (1995)Google Scholar
  17. 17.
    Gouvêa, F.Q., Mazur, B.: On the density of modular representations. In: Computational Perspectives on Number Theory (Chicago, IL, 1995). AMS/IP Studies in Advanced Mathematics, vol. 7, pp. 127–142. American Mathematical Society, Providence (1998)Google Scholar
  18. 18.
    Gouvêa, F.Q.: Arithmetic of \(p\)-Adic Modular Forms. Lecture Notes in Mathematics, vol. 1304. Springer, Berlin (1988)CrossRefGoogle Scholar
  19. 19.
    Hatada, K.: Eigenvalues of Hecke operators on \({\rm SL}(2,\,{ \mathbf{Z}})\). Math. Ann. 239(1), 75–96 (1979)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hatada, K.: Congruences for eigenvalues of Hecke operators on \({\rm SL}_{2}({ \mathbf{Z}})\). Manuscr. Math. 34(2–3), 305–326 (1981)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jochnowitz, N.: Congruences between systems of eigenvalues of modular forms. Trans. Am. Math. Soc. 270(1), 269–285 (1982)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Katz, N.M.: \(p\)-adic properties of modular schemes and modular forms. In: Modular Functions of One Variable, III (Proceedings of International Summer School, University of Antwerp, Antwerp, 1972), pp. 69–190. Lecture Notes in Mathematics, vol. 350 (1973)Google Scholar
  23. 23.
    Nicholas, M.K.: Higher congruences between modular forms. Ann. Math. (2) 101, 332–367 (1975)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Katz, N.M.: \(p\)-adic \(L\)-functions via moduli of elliptic curves. Algebraic geometry. In: Proceedings of Symposia in Pure Mathematics, vol. 29, Humboldt State University, Arcata, 1974, pp. 479–506 (1975)Google Scholar
  25. 25.
    Kilford, L.J.P.: Slopes of 2-adic overconvergent modular forms with small level. Math. Res. Lett. 11(5–6), 723–739 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kilford, L.J.P.: Modular Forms. A Classical and Computational Introduction. Imperial College Press, London (2008)CrossRefGoogle Scholar
  27. 27.
    Kisin, M.: The Fontaine-Mazur conjecture for \({\rm GL}_2\). J. Am. Math. Soc. 22(3), 641–690 (2009)CrossRefGoogle Scholar
  28. 28.
    Kolberg, O.: Congruences for Ramanujan’s function \(\tau (n)\). Arbok Univ. Bergen Mat. Nat. Ser. 11, 8 (1962)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kiming, I., Rustom, N., Wiese, G.: On certain finiteness questions in the arithmetic of modular forms. J. Lond. Math. Soc. (2) 94(2), 479–502 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture. In: Proceedings of the International Congress of Mathematicians, vol. 2, pp. 280–293. Hindustan Book Agency, New Delhi (2010)Google Scholar
  31. 31.
    Laures, G.: \(K(1)\)-local topological modular forms. Invent. Math. 157(2), 371–403 (2004)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Merel, L.: Universal Fourier Expansions of Modular Forms on Artin’s Conjecture for Odd 2-Dimensional Representations. Lecture Notes in Mathematics, vol. 1585. Springer, Berlin (1994)Google Scholar
  33. 33.
    Nicolas, J.-L., Serre, J.-P.: Formes modulaires modulo 2: l’ordre de nilpotence des opérateurs de Hecke. C. R. Math. Acad. Sci. Paris 350(7–8), 343–348 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ramanujan, S.: On Certain Arithmetical Functions [Trans. Cambridge Philos. Soc. 22 (1916), no. 9, 159–184]. Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publication, Providence (2000)Google Scholar
  35. 35.
    Rustom, N.: Code and data for the paper Congruences modulo prime powers of hecke eigenvalues in level 1. https://rustomn.wordpress.com/hecke-congruences-level-1/ (2017). Accessed 31 Oct 2017
  36. 36.
    Rustom, N.: Filtrations of dc-weak eigenforms. Acta Arith. 180(4), 297–318 (2017)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Serre, J.-P.: Formes Modulaires et Fonctions zêta \(p\)-Adiques. Lecture Notes in Mathematics, vol. 350, pp. 191–268. Springer, New York (1973)Google Scholar
  38. 38.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106, 2nd edn. Springer, Dordrecht (2009)CrossRefGoogle Scholar
  39. 39.
    Stein, W.: Modular Forms, a Computational Approach. Graduate Studies in Mathematics, vol. 79. American Mathematical Society, Providence (2007), with an appendix by Paul E. Gunnells (2007)Google Scholar
  40. 40.
    Swinnerton-Dyer, H.P.F.: On \(l\)-Adic Representations and Congruences for Coefficients of Modular Forms. In: Modular Functions of One Variable, III (Proc. Internat. Summer School, University of Antwerp, 1972) (1973), pp. 1–55. Lecture Notes in Mathematics, vol. 350Google Scholar
  41. 41.
    Tsaknias, P., Wiese, G.: Topics on modular Galois representations modulo prime powers. J. Lond. Math. Soc. 94(2), 479–502 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentKoç University, Rumelifeneri YoluSarıyerTurkey

Personalised recommendations