  We find explicit projective models of a compact Shimura curve and of a
(non-compact) surface which are the moduli spaces of principally polarised
abelian fourfolds with an automorphism of order five. The surface has a
24-nodal canonical model in IP^4 which is the complete intersection of two
S_5-invariant cubics. It is dominated by a Hilbert modular surface and we give
a modular interpretation for this. We also determine the L-series of these
varieties as well as those of several modular covers of the Shimura curve.
