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We consider the KZ differential equations over C in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field F_p. We study the polynomial solutions of these differential equations over F_p, constructed in a previous work joint with V. Schechtman and called the F_p-hypergeometric solutions. The dimension of the space of F_p-hypergeometric solutions depends on the prime number p. We say that the KZ equations have ample reduction for a prime p, if the dimension of the space of F_p-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over C. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis F_p-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials (z_i−z_j)^{M_i+M_j} are replaced with (z_i−z_j)^{Mi+Mj−p} and the Euler gamma function Γ(x) is replaced with a suitable F_p-analog defined on F_p
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Distribution of values of Gaussian hypergeometric functions
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 Ap ́ery-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 2F1 functions, the limiting distribution is semicircular (i.e. SU(2)), whereas the distribution for the 3F2 functions is the Batman distribution for the traces of the real orthogonal group O3.
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- Award ID(s):
- 2055118
- PAR ID:
- 10498901
- Publisher / Repository:
- International Press
- Date Published:
- Journal Name:
- Pure and Applied Mathematics Quarterly
- Volume:
- 19
- Issue:
- 1
- ISSN:
- 1558-8599
- Page Range / eLocation ID:
- 371 to 407
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4310/PAMQ.2023.v19.n1.a14
