On isomorphisms between Siegel modular threefolds

TL;DR

This paper extends the known degree 8 endomorphism of the Satake compactification of moduli spaces of abelian surfaces to other modular threefolds using Siegel modular forms and Satake compactifications.

Contribution

It demonstrates that the degree 8 endomorphism can be constructed on various modular threefolds through isomorphisms of graded rings of modular forms and analyzes the Fricke involution's role.

Findings

Degree 8 endomorphism extends to other modular threefolds.

Construction via isomorphism of graded rings of modular forms.

Analysis of Fricke involution's impact on endomorphisms.

Abstract

The Satake compactification of the moduli space of principally polarized abelian surfaces with a level two structure has a degree 8 endomorphism. The aim of this paper is to show that this result can be extended to other modular threefolds. The main tools are Siegel modular forms and Satake compactifications of arithmetic quotients of the Siegel upper-half space. Indeed, the construction of the degree 8 endomorphism on suitable modular threefolds is done via an isomorphism of graded rings of modular forms. By studying the action of the Fricke involution one gets a further extension of the previous result to other modular threefolds. The possibility of a similar situation in higher dimensions is also discussed.