Abstract
We study the Gaussian hypergeometric series of type 3 F 2 over finite fields F p . For each prime p and each λ ∈ F p , we explicitly determine p 2 3 F 2(λ) p (mod p 2). Using this perspective, we are able to give a direct proof of one of Beukers’ conjectured “supercongruences” between certain Apéry numbers and the coefficients of a weight three modular form of CM type. Finally, we record many new supercongruences of this form.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. reine angew. Math. 518 (200), 187–212.
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), 201–210.
S. Chowla, B. Dwork, and R. Evans, On the mod p2 determination of ((p-1)/2 (p-1)/4)J Number Theory 24 (1986), 188–196.
R.J. Evans, Character sums over finite fields, Finite fields, coding theory, and advances in communications and computing,(Las Vegas 1991), Lect. Notes. Pure Appl. Math., Dekker, New York, 141 (1993), 57–73.
J. Greene, Hypergeometric series over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77–101.
B. Gross and N. Koblitz, Gauss sums and the p-adic F-function, Annals of Math. 109 (197), 569–581.
T. Ishikawa, Supercongruence for the Apéry numbers, Nagoya Math. J. 118 (1990), 195–202.
N. Koblitz, p-adic analysis: a short course on recent work, Cambridge Univ. Press, 1980.
M. Karr, Summation in finite terms, J. Assoc. Comput. Mach. 28, no. 2 (1981), 305–350.
M. Koike, Hypergeometric series over finite fields and Apéry numbers, Hiroshima Math. J. 22 (1992), 461–467.
K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), 1205–1223.
J. Silverman, The arithmetic of elliptic curves, Springer Verlag, 1991.
L. Van Hamme, Proof of a conjecture of Beukers on Apéry numbers, Proceedings of the conference on p-adic analysis, Hengelhof, Belgium, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Ahlgren, S. (2001). Gaussian Hypergeometric Series and Combinatorial Congruences. In: Garvan, F.G., Ismail, M.E.H. (eds) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0257-5_1
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0257-5_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4020-0101-7
Online ISBN: 978-1-4613-0257-5
eBook Packages: Springer Book Archive