Skip to main content
SpringerLink
Log in

Congruence relations for p-adic hypergeometric functions \(\widehat{{\mathscr {F}}}_{a,...,a}^{(\sigma )}(t)\) and its transformation formula

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We introduce new kinds of p-adic hypergeometric functions. We show these functions satisfy congruence relations similar to Dwork’s p-adic hypergeometric functions, so they are convergent functions. We also show that there is a transformation formula between our new p-adic hypergeometric functions and p-adic hypergeometric functions of logarithmic type defined in Asakura (New p-adic hypergeometric functions and syntomic regulators. arXiv:1811.03770) in a particular case.

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. Asakura, M.: New p-adic hypergeometric functions and syntomic regulators. arXiv:1811.03770

  2. Asakura, M.: Regulators of \(K_2\) of hypergeometric fibrations. Res. Number Theory 4(2), 22 (2018)

    Article  Google Scholar 

  3. Asakura, M., Miyatani, K.: Milnor K-theory, F-isocrystals and syntomic regulators. arXiv:2007.14255

  4. Dwork, B.: p-adic cycles. Publ. Math. IHES 37, 27–115 (1969)

    Article  MathSciNet  Google Scholar 

  5. Fresnel, J., van der Put, M.: Rigid Analytic Geometry and Its Applications. Progress in Mathematics, vol. 218. Birkhäuser Boston, Inc., Boston (2004)

    Book  Google Scholar 

  6. Lazda, C., Pál, A.: Rigid Cohomology over Laurent Series Fields. Algebra and Applications, vol. 21. Springer, Cham (2016)

    MATH  Google Scholar 

  7. Le Stum, B.: Rigid Cohomology. Cambridge Tracts in Mathematics, vol. 172. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  8. Robert, A.M.: A Course in p-Adic Analysis. Springer, Cham (2013)

    Google Scholar 

  9. Serre, J.-P.: Local Fields. Graduate Texts in Mathematics, vol. 67. Springer, Cham (1979)

    Google Scholar 

  10. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)

    MATH  Google Scholar 

  11. Van der Put, M.: The cohomology of Monsky and Washnitzer. Introductions aux cohomologies \(p\)-adiques (Luminy, 1984). Memoires de la Société Mathématique de France (N.S.), No. 23, pp. 33–59 (1986)

  12. Washington, L.C.: Introduction to Cyclotomic Fields, vol. 83. Springer, Cham (1997)

    Book  Google Scholar 

Download references

Acknowledgements

I truly appreciate the help of Professor Masanori Asakura. He gave the definition of our new p-adic hypergeometric functions and the conjectures of transformation formulas with the aid of computer. Also, he gave a lot of advice of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Cheng Kung University, Tainan City, Taiwan

    Wang Chung-Hsuan

Authors
  1. Wang Chung-Hsuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wang Chung-Hsuan.

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

Chung-Hsuan, W. Congruence relations for p-adic hypergeometric functions \(\widehat{{\mathscr {F}}}_{a,...,a}^{(\sigma )}(t)\) and its transformation formula. manuscripta math. 169, 565–602 (2022). https://doi.org/10.1007/s00229-021-01327-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-021-01327-1

Mathematics Subject Classification

Search

Navigation