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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 $GF(p).$

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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