Cyclotomic matrices and hypergeometric functions over finite fields

TL;DR

This paper explores the arithmetic properties of cyclotomic matrices over finite fields using hypergeometric functions, confirming several conjectures and advancing understanding of their algebraic structure.

Contribution

It introduces new methods connecting hypergeometric functions over finite fields with cyclotomic matrices and verifies conjectures related to their properties.

Findings

Confirmed several conjectures by Zhi-Wei Sun.

Established new relationships between hypergeometric functions and cyclotomic matrices.

Provided insights into the structure of matrices involving characters and quadratic forms.

Abstract

With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by Zhi-Wei Sun.