Summation identities and special values of hypergeometric series in the p-adic setting

“…These properties include transformation laws, explicit evaluations, and contiguous relations. These functions have played central roles in the study of combinatorial supercongruences [1,3,36,43,46,47,51,54,55,56,57,58], Dwork hypersurfaces [9,45], Galois representations [40,41], L-functions of elliptic curves [6,10,11,25,39,44,52,60,63], hyperelliptic curves [7,8], K3 surfaces [4,19,52], Calabi-Yau threefolds [2,3,64], the Eichler-Selberg trace formula [24,25,26,27,38,48,58,59], among other topics.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distribution of values of Gaussian hypergeometric functions

Ono¹,

Saad²,

Saikia³

2021

Preprint

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In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2 F 1 functions, the limiting distribution is semicircular, whereas the distribution for the 3 F 2 functions is the more exotic Batman distribution.

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“…This technique is also often used to establish results involving finite field hypergeometric functions [2,3,11,13,19,21]. We define the Teichmüller character to be the primitive character ω :…”

Section: Theorem 41 (Cf Koblitzmentioning

confidence: 99%