2008/11/06 by Jens Funke, John J. Millson, Funke, Jens +1 · 1 citation
Mathematics · #11F27 #11F46 #22E40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.0811.0856
openalex publication_date 2008/11/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 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 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.