On Siegel modular forms of weight 12 (Q2703584)

The authors work on the complex vector space \(V\) of all formal linear combinations of the isometry classes of even unimodular lattices in \(m\)-dimensional Euclidean space (\(m\) a fixed multiple of 8). They consider the filtration \(V\supseteq V_0\supseteq V_1\supseteq\dots\) given by the subspaces \(V_n\) of elements whose \(n\)th degree Siegel theta series vanishes, so that \(V_{n-1}/V_n\) is the space of those Siegel cusp forms of level 1, weight \(m/2\) and degree \(n\) which are linear combinations of theta series. For \(m=24\), some Hecke operators defined on \(V\) and preserving the filtration are explicitly calculated. These computations lead to much information on the Siegel cusp forms of weight 12. In particular, the dimension of \(V_{n-1}/V_n\) is found for most (conjecturally all) degrees \(n\).