Denniston partial difference sets exist in the odd prime case

Denniston constructed partial difference sets (PDSs) with the parameters $(2^{3m},(2^{m+r}-2^m+2^r)(2^m-1),2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2),(2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m\ge 2,1\le r<m$. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters $(p^{3m},(p^{m+r}-p^m+p^r)(p^m-1),p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2),(p^{m+r}-p^m+p^r)(p^r-1))$ exist in all elementary abelian groups of order $p^{3m}$ for all $m\ge 2,r\in \{1,m-1\}$ where $p$ is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.