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.