Abstract. The present paper is motivated by Mukai's "Igusa quartic and Steiner surfaces". There he proved that the Satake compactification of the moduli space of principally polarized abelian surfaces with a level two structure has a degree 8 endomorphism. This compactification can be viewed as a Siegel modular threefold. 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 discussed at the end of the paper.

Abstract. We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first look, since they are constructed in two very different ways. If V grad is the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and VΘ is the one generated by vector-valued modular forms arising from second order theta constants with our new construction, we will prove that V grad = VΘ.This result could also be proven as a consequence of the "heat equation" for theta functions.

We use the gradients of theta functions at odd twotorsion points -thought of as vector-valued modular forms -to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of twotorsion points.

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].