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 nonsingular (nonsingular for \(n > 1\)) complex threedimensional 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 The Siegel Modular Variety of Degree Two and Level Four  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
Cohomology of the Siegel Modular Group of Degree Two and Level Four  Introduction
 The building
 Cycles
 The main theorems
 References
