Author:
Pujahari Sudhir,Saikia Neelam
Abstract
AbstractRecently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the p-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the p-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight $$k\ge 4$$
k
≥
4
and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1, 2, 6].
Funder
Austrian Science Fund FWF grant P32305
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference22 articles.
1. Ahlgren, S.: The points of a certain fivefold over finite fields and the twelfth power of the eta function. Finite Fields Appl. 8(1), 18–33 (2002)
2. Ahlgren, S., Ono, K.: Modularity of a certain Calabi-Yau threefold. Montash. Math. 129(3), 177–190 (2000)
3. Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. Wiley, New York (1998)
4. Birch, B.J.: How the number of points of an elliptic curve over a fixed prime field varies. J. Lond. Math. Soc. 43, 57–60 (1968)
5. Bringmann, K., Kane, B., Pujahari, S.: Odd moments for the trace of Frobenius and the Sato–Tate conjecture in arithmetic progressions. arXiv:2112.08205