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Transactions of the American Mathematical Society

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Values of Gaussian hypergeometric series

Author: Ken Ono
Journal: Trans. Amer. Math. Soc. 350 (1998), 1205-1223
MSC (1991): Primary 11T24
MathSciNet review: 1407498
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Abstract: Let $p$ be prime and let $GF(p)$ be the finite field with $p$ elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions

\begin{equation*}_{2}F_{1}(x)=_{2} F_{1} \left ( \begin{matrix}\phi , & \phi \\ \ & \epsilon \end{matrix} | \ x \right ) \ \ \ {\text{\rm and}} \ \ \ _{3}F_{2}(x)= _{3}F_{2} \left ( \begin{matrix}\phi , & \phi , & \phi \\ \ & \epsilon , & \epsilon \end{matrix} | \ x \right ), \end{equation*}

where $\phi $ and $\epsilon $ respectively are the quadratic and trivial characters of $GF(p).$ For all but finitely many rational numbers $x=\lambda ,$ there exist two elliptic curves $_{2}E_{1}(\lambda )$ and $_{3}E_{2}(\lambda )$ for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes $p$ for which $_{2}F_{1}(\lambda )$ is zero; however if $\lambda \neq -1,0, \frac{1}{2}$ or $2$, then the set of such primes has density zero. In contrast, if $\lambda \neq 0$ or $1$, then there are only finitely many primes $p$ for which $_{3}F_{2}(\lambda ) =0.$ Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that $_{3}E_{2}(8)$ is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating $_{3}F_{2}(4)$ over every $GF(p).$

References [Enhancements On Off] (What's this?)

  • 1. B. Berndt and R. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer, Illinois J. Math 23 (1979), 374-437. MR 81j:10055
  • 2. B.C. Berndt, R.J. Evans, and K.S. Williams, Gauss and Jacobi sums, Wiley Publ., to appear.
  • 3. F. Beukers, Some congruences for Apéry numbers, J. Number Theory 21 (1985), 141-155. MR 87g:11032
  • 4. -, Another congruence for Apéry numbers, J. Number Theory 25 (1987), 201-210. MR 88b:11002
  • 5. F. Beukers and J. Stienstra, On the Picard-Fuchs equation and the formal Brauer group of certain $K3$ surfaces and elliptic curves, Math. Ann. 271 (1985), 269-304. MR 86j:14045
  • 6. J. E. Cremona, Algorithms for modular elliptic curves, Cambridge Univ. Press, 1992. MR 93m:11053
  • 7. N. Elkies, Distribution of supersingular primes, J. Arithmétiques, Astérisque 198-200 (1991), 127-132. MR 93b:11070
  • 8. A. Erdélyi et al., Higher transcendental functions, Vol.1, McGraw-Hill, New York, 1953. MR 15:419i
  • 9. R. Evans, Character sums over finite fields, Finite fields, coding theory, and advances in communications and and computing, Eds. G. Mullen and P. Shiue, Lecture Notes in Pure and Appl. Math., vol. 141, Marcel Dekker, 1993, pp. 57-73. MR 94c:11118
  • 10. -, Identities for products of Gauss sums over finite fields, Enseign. Math. (2) 27 (1981), 197-209. MR 83i:10050
  • 11. R. Evans, J. Pulham, and J. Sheehan, On the number of complete subgraphs contained in certain graphs, J. Combin. Theory Ser. B 30 (1981), 364-371. MR 83c:05075
  • 12. J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77-101. MR 88e:11122
  • 13. J. Greene and D. Stanton, A character sum evaluation and Gaussian hypergeometric series, J. Number Theory 23 (1986), 136-148. MR 88a:11076
  • 14. K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer-Verlag, 1982. MR 88g:12001
  • 15. A. Knapp, Elliptic curves, Princeton Univ. Press, 1992. MR 93j:11032
  • 16. N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag, 1984. MR 86c:11040
  • 17. M. Koike, Hypergeometric series over finite fields and Apéry numbers, Hiroshima Math. J. 22 (1992), 461-467. MR 93i:11146
  • 18. -, Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields, Hiroshima Math. J. 25 (1995), 43-52. MR 96b:11079
  • 19. Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc., to appear.
  • 20. K. Ono, Euler's concordant forms, Acta Arith. 78 (1996), 101-123. CMP 97:05
  • 21. A.R. Rajwade, The Diophantine equation $y^{2}=x(x^{2}+21Dx+112D^{2})$ and the conjectures of Birch and Swinnerton-Dyer, J. Austral. Math. Soc. Ser. A 24 (1977), 286-295. MR 57:12518
  • 22. J. Silverman, The arithmetic of elliptic curves, Springer-Verlag, 1986. MR 87g:11070

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Additional Information

Ken Ono
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540; Department of Mathematics, Penn State University, University Park, Pennsylvania 16802

Keywords: Gaussian hypergeometric series, elliptic curves, Ap\'{e}ry numbers, character sums
Received by editor(s): September 8, 1995
Received by editor(s) in revised form: July 3, 1996
Additional Notes: The author is supported by NSF grants DMS-9304580 and DMS-9508976.
Article copyright: © Copyright 1998 American Mathematical Society

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