Skip to main content
SpringerLink
Log in

Differential modular forms over totally real fields of integral weights

  • Research in Number Theory
  • Published:
Research in Number Theory Aims and scope Submit manuscript

Abstract

In this article, we construct a differential modular form of non-zero order and integral weight for compact Shimura curves over totally real fields bigger than \(\mathbb {Q}\). The construction uses the theory of lifting ordinary mod p Hilbert modular forms to characteristic 0 as well as the theory of Igusa curve. This is the analogue of the construction of Buium in the case of modular curves parametrizing elliptic curves with level structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Banerjee, D.: Differential modular forms on Shimura curves over totally real fields. J. Number Theory 135, 353–373 (2014)

    Article  MathSciNet  Google Scholar 

  2. Barcau, M.A.: Isogeny covariant differential modular forms and the space of elliptic curves up to isogeny. Compos. Math. 137(3), 237–273 (2003)

    Article  MathSciNet  Google Scholar 

  3. Barcau, M., Buium, A.: Siegel differential modular forms. Int. Math. Res. Not. 28, 1457–1503 (2002)

    Article  MathSciNet  Google Scholar 

  4. Bhatt, B., Scholze, P.: Prisms and prismatic cohomology. arXiv preprint (2019). arXiv:1905.08229

  5. Borger, J.: The basic geometry of Witt vectors, I: the affine case. Algebra Number Theory 5(2), 231–285 (2011)

    Article  MathSciNet  Google Scholar 

  6. Borger, J.: The basic geometry of Witt vectors. II: spaces. Math. Ann. 351(4), 877–933 (2011)

    Article  MathSciNet  Google Scholar 

  7. Borger, J., Saha, A.: Differential characters of Drinfeld modules and de Rham cohomology. Algebra Number Theory 13(4), 797–837 (2019)

    Article  MathSciNet  Google Scholar 

  8. Borger, J., Saha, A.: Isocrystals associated to arithmetic jet spaces of abelian schemes. Adv. Math. 351, 388–428 (2019)

    Article  MathSciNet  Google Scholar 

  9. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21. Springer, Berlin (1990)

  10. Buium, A.: Differential characters of abelian varieties over \(p\)-adic fields. Invent. Math. 122(2), 309–340 (1995)

    Article  MathSciNet  Google Scholar 

  11. Buium, A.: Differential modular forms. J. Reine Angew. Math. 520, 95–167 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Buium, A.: Differential modular forms on Shimura curves. I. Compos. Math. 139(2), 197–237 (2003)

    Article  MathSciNet  Google Scholar 

  13. Buium, A.: Differential modular forms on Shimura curves. II. Serre operators. Compos. Math. 140(5), 1113–1134 (2004)

    Article  MathSciNet  Google Scholar 

  14. Buium, A.: Differential eigenforms. J. Number Theory 128(4), 979–1010 (2008)

    Article  MathSciNet  Google Scholar 

  15. Buium, A.: Differential modular forms attached to newforms mod \(p\). J. Number Theory 155, 111–128 (2015)

    Article  MathSciNet  Google Scholar 

  16. Buium, A., Dupuy, T.: Arithmetic differential equations on \(GL_n\), I: differential cocycles. J. Algebra 454, 273–291 (2016)

    Article  MathSciNet  Google Scholar 

  17. Buium, A., Dupuy, T.: Arithmetic differential equations on \(GL_n\), II: arithmetic Lie–Cartan theory. Sel. Math. (N.S.) 22(2), 447–528 (2016)

    Article  Google Scholar 

  18. Buium, A., Dupuy, T.: Arithmetic differential equations on \(GL_n\), III. Galois groups. Sel. Math. 22(2), 529–552 (2016)

    Article  Google Scholar 

  19. Buium, A., Miller, L.E.: Perfectoid spaces arising from arithmetic jet spaces. arXiv preprint (2019). arXiv:1911.00113

  20. Buium, A., Poonen, B.: Independence of points on elliptic curves arising from special points on modular and Shimura curves. II. Local results. Compos. Math. 145(3), 566–602 (2009)

    Article  MathSciNet  Google Scholar 

  21. Buium, A., Saha, A.: Differential overconvergence. In: Algebraic Methods in Dynamical Systems. Banach Center Publications, vol. 94, pp. 99–129. Institute of Mathematics of the Polish Academy of Sciences, Warsaw (2011)

  22. Buium, A., Saha, A.: Hecke operators on differential modular forms mod \(p\). J. Number Theory 132(5), 966–997 (2012)

    Article  MathSciNet  Google Scholar 

  23. Buium, A., Saha, A.: The ring of differential Fourier expansions. J. Number Theory 132(5), 896–937 (2012)

    Article  MathSciNet  Google Scholar 

  24. Carayol, H.: Sur la mauvaise réduction des courbes de Shimura. Compos. Math. 59(2), 151–230 (1986)

    MATH  Google Scholar 

  25. Faltings, G., Chai, C.-L.: Degeneration of Abelian Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22. Springer, Berlin. With an appendix by D. Mumford (1990)

  26. Gee, T.: Companion forms over totally real fields. Manuscr. Math. 125(1), 1–41 (2008)

    Article  MathSciNet  Google Scholar 

  27. Gross, B.H.: A tameness criterion for Galois representations associated to modular forms (mod \(p\)). Duke Math. J. 61(2), 445–517 (1990)

    Article  MathSciNet  Google Scholar 

  28. Hesselholt, L.: The big de Rham–Witt complex. Acta Math. 214(1), 135–207 (2015)

    Article  MathSciNet  Google Scholar 

  29. Hida, H.: On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves. Am. J. Math. 103(4), 727–776 (1981)

    Article  MathSciNet  Google Scholar 

  30. Kassaei, P.L.: \(p\)-adic modular forms over Shimura curves over totally real fields. Compos. Math. 140(2), 359–395 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We wish to thank the anonymous referees for carefully reading our article and for the suggestions which led to deeper clarifications and enrichment of this paper. The first author was partially supported by the SERB grant CRG/2020/000223 and MTR/2017/000357. The second author is also grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. He was partially supported by the SERB Grant SRG/2020/002248. The authors wish to thank Alexandru Buium and James Borger for several inspiring discussions and clarifications. We would also like to thank Alexei Pantchichkine and Jack Shotton for the discussions and insights that took place during the preparation of this paper.

Author information

Authors and Affiliations

  1. Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune, 411 008, India

    Debargha Banerjee

  2. Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, 382355, India

    Arnab Saha

Authors
  1. Debargha Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arnab Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debargha Banerjee.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Banerjee, D., Saha, A. Differential modular forms over totally real fields of integral weights. Res. number theory 7, 42 (2021). https://doi.org/10.1007/s40993-021-00269-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40993-021-00269-7

Keywords

Mathematics Subject Classification

Search

Navigation