Sato-Tate distribution of p-adic hypergeometric functions

Area of Science:

• Number Theory
• Algebraic Geometry
• Harmonic Analysis

Background:

• Ono, Saad, and others previously studied value distributions of Gaussian hypergeometric functions over finite fields.
• Their work identified semicircular and Batman distributions for specific function families.
• This research extends these investigations into the p-adic domain.

Purpose of the Study:

• To investigate the value distributions of hypergeometric functions in the p-adic setting over large finite fields.
• To analyze two- and six-parameter families of hypergeometric functions.
• To establish connections between p-adic hypergeometric functions and traces of Hecke operators.

Main Methods:

• Analysis of two- and six-parameter hypergeometric functions in the p-adic setting.
• Application of techniques from p-adic analysis and number theory.
• Expression of traces of p-th Hecke operators acting on spaces of cusp forms.

Main Results:

• The limiting distributions of the studied hypergeometric function families are semicircular over large finite fields.
• Formulas are derived for the traces of p-th Hecke operators in terms of p-adic hypergeometric functions.
• These findings represent p-adic analogues of existing trace formulas.

Conclusions:

• The study successfully extends the understanding of hypergeometric function distributions to the p-adic realm.
• The established connections between p-adic hypergeometric functions and Hecke operators offer new insights.
• The results provide valuable p-adic analogues for trace formulas in number theory.