# The Siegel Modular Variety of Degree Two and Level Four/Cohomology of the Siegel Modular Group of Degree Two and Level Four

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*Ronnie Lee; Steven H. Weintraub; J. William Hoffman*

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 \(\mathbf M^*_4 \rightarrow \mathbf M^*_2\).

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 \(\mathbf
M_4\).