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

Part of: Algebraic number theory: local and $p$-adic fields Other special functions Arithmetic algebraic geometry Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  17 February 2016

RUPAM BARMAN ,
HASANUR RAHMAN  and
NEELAM SAIKIA
RUPAM BARMAN*
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India email [email protected]
HASANUR RAHMAN
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India email [email protected]
NEELAM SAIKIA
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India email [email protected]
*
✉[email protected]
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Abstract

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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 $d$.

Keywords

characters of finite fieldshypergeometric seriesTeichmüller characterp-adic gamma functionDwork hypersurfaces

MSC classification

Primary: 11G25: Varieties over finite and local fields
Secondary: 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 11T24: Other character sums and Gauss sums 33C20: Generalized hypergeometric series, $_pF_q$ 33E50: Special functions in characteristic $p$ (gamma functions, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 208 - 216
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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