Advertisement

Modular eigenforms at the boundary of weight space

  • Jan VonkEmail author
Research
  • 63 Downloads

Abstract

Andreatta, Iovita, and Pilloni recently introduced \({{\mathrm{\mathbf {F}}}}_p(({{\mathrm{\mathrm {t}}}}))\)-Banach spaces of overconvergent \({{\mathrm{\mathrm {t}}}}\)-adic modular forms, whose weight may be considered a “boundary” point of weight space. In an effort to make them concrete and accessible to explicit experimentation, we construct orthonormal bases, deduce \({{\mathrm{\mathrm {t}}}}\)-adic analogues of certain p-adic results in the literature, and exhibit explicit examples.

References

  1. 1.
    Andreatta, F., Iovita, A., Pilloni, V.: Le halo spectral. Preprint (2015)Google Scholar
  2. 2.
    Andreatta, F., Iovita, A., Stevens, G.: Overconvergent Eichler–Shimura isomorphisms. ArXiv preprint (2013)Google Scholar
  3. 3.
    Buzzard, K., Calegari, F.: Slopes of overconvergent \(2\)-adic modular forms. Compos. Math. 141(3), 591–604 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bergdall, J., Pollack, R.: Arithmetic properties of Fredholm series for \(p\)-adic modular forms. Proc. Lond. Math. Soc. 113(4), 419–444 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Calegari, F.: Congruences between modular forms. Arizona Winter School (2013)Google Scholar
  7. 7.
    Coleman, R.: On the coefficients of the characteristic series of the U-operator. Proc. Natl. Acad. Sci. USA 94, 11129–11132 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coleman, R.: \(p\)-Adic Banach spaces and families of modular forms. Invent. Math. 127, 417–479 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Coleman, R.: The Eisenstein family. Proc. Am. Math. Soc. 141(9), 2945–2950 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Coleman, R., Stein, W.: Approximation of eigenforms of infinite slope by eigenforms of finite slope. Geometric Aspects of Dwork Theory, vol. 1, pp. 437–449. De Gruyter, Berlin (2004)Google Scholar
  11. 11.
    Johansson, C., Newton, J.: Extended eigenvarieties for overconvergent cohomology. ArXiv preprint (2017) arXiv:1604.07739
  12. 12.
    Katz, N.: \(p\)-Adic properties of modular schemes and modular forms. In: Deligne, P., Kuyk, W. (eds.) Modular Forms in One Variable III, Volume 350 of LNM, pp. 69–190. Springer, Berlin (1973)Google Scholar
  13. 13.
    Lauder, A.: Computations with classical and \(p\)-adic modular forms. LMS J. Comput. Math. 14, 214–231 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liu, R., Wan, D., Xiao, L.: The eigencurve over the boundary of weight space. Duke Math. J. 166(9), 1739–1787 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pilloni, V.: Formes modulaires surconvergentes. Ann. Inst. Four. 63(1), 219–239 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Roe, D.: The 3-adic eigencurve at the boundary of weight space. Int. J. Number Theory 10(7), 1791–1806 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Vonk, J.: Computing overconvergent forms for small primes. LMS J. Comput. Math. 18(1), 250–257 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wan, D.: Dimension variation of classical and \(p\)-adic modular forms. Invent. Math. 133, 449–463 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBurnside HallMontrealCanada

Personalised recommendations