Values of Gaussian hypergeometric series

Abstract

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 functionswhere φ and respectively are the quadratic and trivial characters of GF (p). For all but finitely many rational numbers x = λ, there exist two elliptic curves 2 E 1 (λ) and 3 E 2 (λ) 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 (λ) is zero; however if λ = −1, 0, 1 2 or 2, then the set of such primes has density zero. In contrast, if λ = 0 or 1, then there are only finitely many primes p for which 3 F 2 (λ) = 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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“…then −φ(−1)p 2 F 1 (λ) p is the p-th Fourier coefficient of the weight 2 newform associated to E(λ) by the Shimura-Taniyama correspondence [12], [17]. Similarly, if λ ∈ −1, 4, 1 4 , −8, − 1 8 , 64, 1 64 , then, for all but finitely many primes p, it turns out that 3 F 2 (λ) p is essentially the p-th Fourier coefficient of an explicit weight 3 newform with complex multiplication that is associated to a certain singular K3 surface X λ (see Corollary 11.20 of [18]).…”

“…Other works by Ahlgren [1], Koike [12], and the second author [17] provide further examples of p-adic results for combinatorial expressions whose proofs require these functions.…”

“…The classical functions satisfy many interesting properties, such as transformation and summation formulas, and Greene showed that their finite field analogues enjoyed many similar properties. Koike [12] and the second author [17] further explored the arithmetic properties of Gaussian hypergeometric functions, including the number-theoretic significance of certain special values of these functions. We continue this study below in Section 5, proving Theorem 1.1 and Koike's conjecture.…”

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“…Greene [1987] defined Gaussian hypergeometric functions over arbitrary finite fields and showed that they have properties analogous to those of classical hypergeometric functions. We recall some definitions and notation from [Ono 1998] in the case of fields of prime order.…”

Evaluation of Certain Hypergeometric Functions over Finite Fields

“…For convenience, we denote by n+1 F n (x) the hypergeometric functions with A i = φ and B j = ε for all i and j. The evaluations of 2 F 1 (x), 3 F 2 (x) at certain rational numbers have been studied by numerous mathematicians such as Barman-Kalita [4,5], Evans-Lam [9], Greene-Stanton [17], Koike [20,21], and Ono [25]. They gave explicit relationship between values of these hypergeometric functions and arithmetic of elliptic curves.…”