Shimura varieties for abelian fourfolds with an automorphism of order five

I will report on joint work with Bert van Geemen concerning moduli of principally polarised abelian fourfolds with an automorphism of order five. This moduli problem gives rise to a compact Shimura curve and a Shimura surface. I will explain a new technique to exhibit explicit projective models for them. The Shimura varieties possess a rich arithmetic and geometric structure that will be described in detail.