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

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.

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References

  1. 1.

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

  2. 2.

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

    Article  Google Scholar 

  3. 3.

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

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 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. 6.

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

    MATH  Google Scholar 

  7. 7.

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

    Book  Google Scholar 

  8. 8.

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

    Google Scholar 

  9. 9.

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

    Google Scholar 

  10. 10.

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

    MATH  Google Scholar 

  11. 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. 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.

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Correspondence to Wang Chung-Hsuan.

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Chung-Hsuan, W. Congruence relations for p-adic hypergeometric functions \(\widehat{{\mathscr {F}}}_{a,...,a}^{(\sigma )}(t)\) and its transformation formula. manuscripta math. (2021). https://doi.org/10.1007/s00229-021-01327-1

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Mathematics Subject Classification

  • 33E50