Hypergeometric Series and Periods of Elliptic Curves

In [7], Greene introduced the notion of general hypergeometric series over finite fields or Gaussian hypergeometric series, which are analogous to classical hypergeometric series. The motivation for his work was to develop the area of character sums and their evaluations through parallels with the theory of hypergeometric functions. The basis for this parallel was the analogy between Gauss sums and the gamma function as discussed in [5, 11, 14, 20]. Since then, the interplay between ordinary hypergeometric series and Gaussian hypergeometric series has played an important role in character sum evaluations [10], supercongruences [15], finite field versions of the Lagrange inversion formula [8] and the representation theory of SL(2, R) [9]. Recently, the author in [18] has further developed this interplay by providing an expression for the real period of an elliptic curve in Legendre normal form in terms of an ordinary hypergeometric series. This formula is analogous to an expression for the trace of Frobenius of the curve in terms of a Gaussian hypergeometric series. He then displays a striking analogy between binomial coefficients involving rational numbers and those involving multiplicative characters. This paper examines this analogy further using a different family of elliptic curves and is organized as follows. In Section 2 we outline this analogy and state our results. Section 3 recalls some properties of ordinary hypergeometric series, elliptic curves and the arithmetic-geometric mean. In Section 4 we prove our results.