Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T22:20:37.587Z Has data issue: false hasContentIssue false

COUNTING POINTS ON DWORK HYPERSURFACES AND $p$-ADIC HYPERGEOMETRIC FUNCTIONS

Published online by Cambridge University Press:  17 February 2016

RUPAM BARMAN*
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India email rupam@maths.iitd.ac.in
HASANUR RAHMAN
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India email hasrah93@gmail.com
NEELAM SAIKIA
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India email nlmsaikia1@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We express the number of points on the Dwork hypersurface $X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$ over a finite field of order $q\not \equiv 1\,(\text{mod}\,d)$ in terms of McCarthy’s $p$-adic hypergeometric function for any odd prime $d$.

Type
Research Article
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Barman, R. and Saikia, N., ‘ p-Adic gamma function and the trace of Frobenius of elliptic curves’, J. Number Theory 140(7) (2014), 181195.Google Scholar
Barman, R. and Saikia, N., ‘Certain transformations for hypergeometric series in the p-adic setting’, Int. J. Number Theory 11(2) (2015), 645660.Google Scholar
Barman, R., Saikia, N. and McCarthy, D., ‘Summation identities and special values of hypergeometric series in the p-adic setting’, J. Number Theory 153 (2015), 6384.Google Scholar
Berndt, B., Evans, R. and Williams, K., Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts (A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1998).Google Scholar
Fuselier, J., ‘Hypergeometric functions over F p and relations to elliptic curve and modular forms’, Proc. Amer. Math. Soc. 138 (2010), 109123.Google Scholar
Goodson, H., ‘Hypergeometric functions and relations to Dwork hypersurfaces’. arXiv:1510.07661v1.Google Scholar
Greene, J., ‘Hypergeometric functions over finite fields’, Trans. Amer. Math. Soc. 301(1) (1987), 77101.CrossRefGoogle Scholar
Gross, B. H. and Koblitz, N., ‘Gauss sum and the p-adic 𝛤-function’, Ann. of Math. (2) 109 (1979), 569581.CrossRefGoogle Scholar
Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory, Springer International Edition (Springer, New York, 2005).Google Scholar
Koblitz, N., p-Adic Analysis: A Short Course on Recent Work, London Mathematical Society Lecture Note Series, 46 (Cambridge University Press, Cambridge–New York, 1980).CrossRefGoogle Scholar
Koblitz, N., ‘The number of points on certain families of hypersurfaces over finite fields’, Compositio Math. 48(1) (1983), 323.Google Scholar
McCarthy, D., ‘On a supercongruence conjecture of Rodriguez-Villegas’, Proc. Amer. Math. Soc. 140 (2012), 22412254.Google Scholar
McCarthy, D., ‘The trace of Frobenius of elliptic curves and the p-adic gamma function’, Pacific J. Math. 261(1) (2013), 219236.Google Scholar