Paper detail

Boundary behavior of special cohomology classes arising from the Weil representation

In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p,q)$. This correspondence is realized using theta functions associated to explicitly constructed "special" Schwartz forms. Furthermore, the theta functions give rise to generating series of certain "special cycles" in $X$ with coefficients. In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel-Sere compactification $\bar{X}$ of $X$. However, for the $\Q$-split case for signature $(p,p)$, we have to construct and consider a slightly larger compactification, the "big" Borel-Serre compactification. The restriction to each face of $\bar{X}$ is again a theta series as in [math.NT/0408050], now for a smaller orthogonal group and a larger coefficient system. As application we establish the cohomological nonvanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish.