On the subring of special cycles

Abstract

For a totally real field F of degree d>1 and a quadratic space V of signature (m,2)^{d_+} x (m+2,0)^{d-d_+} with associated Shimura variety Sh(V), we consider the subring of cohomology generated by the classes of weighted special cycles. We assume that d_+<d. We take the quotient SC(V) of this ring by the radical of the restriction of the intersection pairing to it. We show that the inner products of classes in SC(V) are determined by Fourier coefficients of pullbacks of Hilbert-Siegel Eisenstein series of genus m to products of smaller Siegel spaces and that the products of classes in SC(V) are determined by Fourier coefficients of pullbacks to triple products of smaller Siegel spaces. As a consequence, we show that, for quadratic spaces V and V' over F that are isomorphic at all finite places, but with no restriction on d_+(V) and d_+(V') other than the necessary condition that they have the same parity, the special cycles rings SC(V) and SC(V') are isometrically isomorphic. This is a consequence of the Siegel-Weil formula and the matching principle. Finally, we give a combinatorial construction of a ring SC(V_+) associated to a totally positive definite quadratic space V_+ of dimension m+2 over F and show that the comparison isomorphism extends to this case.