Hypergeometric Functions over Finite Fields

Fuselier, Jenny G.; Лонг, Линг; Ramakrishna, Ravi; Swisher, Holly; Tu, Fang-Ting

doi:10.1007/978-3-030-04161-8_36

Cited by 23 publications

(47 citation statements)

References 8 publications

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“…al.) in [14], which was an effort to unify and improve on the interplay between classical and finite-field hypergeometric functions in the single-variable setting.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A cubic transformation formula for Appell–Lauricella hypergeometric functions over finite fields

Frechette

Swisher

2018

Res. number theory

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We define a finite field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling recent development by Fuselier, Long, Ramakrishna, and the last two authors in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite field Appell-Lauricella functions to establish a finite field analogue of Koike and Shiga's cubic transformation for the Appell hypergeometric function F 1 , proving a conjecture of Long. We use our multivariable period functions to construct formulas for the number of F p -points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite field Appell-Lauricella functions.

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“…al.) in [14], which was an effort to unify and improve on the interplay between classical and finite-field hypergeometric functions in the single-variable setting.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A cubic transformation formula for Appell–Lauricella hypergeometric functions over finite fields

Frechette

Swisher

2018

Res. number theory

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“…The theory of finite hypergeometric functions was initiated by Greene [24] and Katz [28], with considerable developments over the recent years -see [8,38,50]. An interpretation of the theory in connection with Galois representations [28,20] yields some fruitful results in computing zeta functions of algebraic varieties defined over finite fields.…”

Section: 1mentioning

confidence: 99%

“…Gauss sums are known to be finite-field analogues of the Gamma function values (see [20] for a dictionary between the classical and finite-field settings), therefore, the right-hand side of (26) reminisces (25). Next, we follow [8] to extend the definition to all finite fields F q , without the restriction on q to satisfy (q − 1)α j ∈ Z and (q − 1)β j ∈ Z for j = 1, .…”

Section: 1mentioning

confidence: 99%

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Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

Лонг

Yui

et al. 2019

2017 MATRIX Annals

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We establish the supercongruences for the fourteen rigid hypergeometric Calabi-Yau threefolds over Q conjectured by Rodriguez-Villegas in 2003. Two different approaches are implemented, and they both successfully apply to all the fourteen supercongruences. Our first method is based on Dwork's theory of p-adic unit roots, and it allows us to establish the supercongruences for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over Q. Essential ingredients in executing the both approaches are the modularity of the underlying Calabi-Yau threefolds and a p-adic perturbation method applied to hypergeometric functions.

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“…In [1,2], Ahlgren-Ono expressed the special value 4 F 3 (1) in terms of Fourier coefficients of the Hecke eigenform associated to a certain modular Calabi-Yau manifold. In [3], Ahlgren, Ono, and Penniston studied the zeta functions of a certain family of K3-surfaces, which lead to information about the special values of 3 F 2 (x) at x = 1, 8, 1/8, −4, −1/4, 64, and −1/64 in terms of the trace of Frobenius on suitable elliptic curves with complex multiplication over Q [14]. In other words, these values of the hypergeometric functions can also be expressed in terms of Hecke characters on imaginary quadratic number fields.…”

Section: Introductionmentioning

confidence: 99%