Strongly regular Cayley graphs over the group Zpn⊕Zpn

Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [J.A. Davis, Partial difference sets in p-groups, Arch. Math. 63 (1994) 103-110; K.H. Leung, S.L. Ma, Partial difference sets with Paley parameters, Bull. London Math. Soc. 27 (1995) 553-564]. It was shown that such graphs exist iff the corresponding group is isomorphic to Zpn⊕Zpn, where p is an odd prime. In this paper we classify all strongly regular Cayley graphs over this group using Schur ring method. As a consequence we obtain a complete classification of strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2.