Two moduli spaces of abelian fourfolds with an automorphism of order five

TL;DR

This paper constructs explicit models of moduli spaces of abelian fourfolds with an automorphism of order five, revealing their geometric structure and modular properties.

Contribution

It provides explicit projective models of these moduli spaces, including a Shimura curve and a surface with a detailed geometric description and modular interpretation.

Findings

The surface is a 24-nodal canonical model in IP^4.

The surface is a complete intersection of two S_5-invariant cubics.

The varieties' L-series are explicitly determined.

Abstract

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.