The Siegel Modular Variety of Degree Two and Level Four, by Ronnie Lee and Steven H. Weintraub

Let \(\mathbf M_n\) denote the quotient of the degree two Siegel space by the principal congruence subgroup of level \(n\) of \(Sp_4(\mathbb Z)\). \(\mathbf M_n\) is the moduli space of principally polarized abelian surfaces with a level \(n\) structure and has a compactification \(\mathbf M^*_n\) first constructed by Igusa. \(\mathbf M^*_n\) is an almost non-singular (non-singular for \(n > 1\)) complex three-dimensional projective variety (of general type, for \(n > 3\)).

The authors analyze the Hodge structure of \(\mathbf M^*_4\), completely determining the Hodge numbers \(h^{p,q} = \dim H^{p,q}(\mathbf M^*_4)\). Doing so relies on the understanding of \(\mathbf M^*_2\) and exploitation of the regular branched covering \(\mathbf M^*_4 \rightarrow \mathbf M^*_2\).

Cohomology of the Siegel Modular Group of Degree Two and Level Four, by J. William Hoffman and Steven H. Weintraub

The authors compute the cohomology of the principal congruence subgroup \(\Gamma_2(4) \subset S{_p4}(\mathbb Z)\) consisting of matrices \(\gamma \equiv \mathbf 1\) mod 4. This is done by computing the cohomology of the moduli space \(\mathbf M_4\). The mixed Hodge structure on this cohomolgy is determined, as well as the intersection cohomology of the Satake compactification of \(\mathbf M_4\).

Readership

Graduate students and research mathematicians working in algebraic geometry.

Table of Contents

• Introduction
• Algebraic background
• Geometric background
• Taking stock
• Type III A
• Type II A
• Type II B
• Type IV C
• Summing up
• Appendix. An exact sequence in homology
• References

• Introduction
• The building
• Cycles
• The main theorems
• References