Values of Gaussian hypergeometric series
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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 {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
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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
- MR Author ID: 342109
- Email: ono@math.ias.edu, ono@math.psu.edu
- 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.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 1205-1223
- MSC (1991): Primary 11T24
- DOI: https://doi.org/10.1090/S0002-9947-98-01887-X
- MathSciNet review: 1407498