Congruences for $\sb 3F\sb 2$ hypergeometric functions over finite fields

Ono, Ken; Penniston, David

doi:10.1215/ijm/1258130978

Cited by 2 publications

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“…There are a number of papers that discuss relations between hypergeometric functions over finite fields and modular forms; see for example [2], [9], [14], [24], [25], [26], [27] and the references therein. To our knowledge, the five eigenforms in Sections 2-3 have not arisen in previous work on this topic.…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric $\mathbf {{}_3{\it F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms

Evans

2009

Proc. Amer. Math. Soc.

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Let H denote the hypergeometric 3 F 2 function over F p whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating H at the argument 1/4. This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Ono's result by evaluating an infinite family of 3 F 2 (1/4) over F q in terms of Jacobi sums. Here we present five new 3 F 2 (1/4) over F q (involving characters of orders 3, 4, 6, and 8) which are conjecturally evaluable in terms of eigenvalues for Hecke eigenforms of weights 2 and 3. There is ample numerical evidence for these evaluations. We motivate our conjectures by proving a connection between 3 F 2 (1/4) and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.

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Section: Introductionmentioning

confidence: 99%

Hypergeometric $\mathbf {{}_3{\it F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms

Evans

2009

Proc. Amer. Math. Soc.

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“…In addition to our main purpose, there are several points of interest in the present study. First, we note that there are a number of interesting studies of the Apéry numbers concerning congruence properties and their analogues (see, e.g., [6,7,9,3,4,1,16,12,13] and references therein). These studies suggest that the congruence originates from automorphic forms associated with some algebraic surface (see, e.g., [5,20]).…”

Section: Introductionmentioning

confidence: 99%

Apéry-Like Numbers Arising From Special Values of Spectral Zeta Functions for Non-Commutative Harmonic Oscillators

Kimoto¹,

Wakayama

2006

Kyushu J. Math.

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Abstract.We derive an expression for the value ζ Q (3) of the spectral zeta function ζ Q (s) for the non-commutative harmonic oscillator using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted J 2 (n) and J 3 (n), which can be regarded as analogues of the Apéry numbers, naturally arise and play a key role in obtaining the expressions for the values ζ Q (2) and ζ Q (3). We also show that the numbers J 2 (n) and J 3 (n) have congruence relations such as those of the Apéry numbers.

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Congruences for $\sb 3F\sb 2$ hypergeometric functions over finite fields

Cited by 2 publications

References 0 publications

Hypergeometric $\mathbf {{}_3{\it F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms

Hypergeometric $\mathbf {{}_3{\it F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms

Apéry-Like Numbers Arising From Special Values of Spectral Zeta Functions for Non-Commutative Harmonic Oscillators

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