Intersection Numbers of Hecke Cycles on Hilbert Modular Varieties

Let O be the ring of integers of a totally real number field E and set G := ResE/Q(GL2). Fix an ideal c ⊂ O. For each ideal m ⊂ O let T (m) denote the mth Hecke operator associated to the standard compact open subgroup U0(c) of G(Af ). Setting X0(c) := G(Q)\G(A)/K∞U0(c), where K∞ is a certain subgroup of G(R), we use T (m) to define a Hecke cycle Z(m) ∈ IH2[E:Q](X0(c)×X0(c)). Here IH• denotes intersection homology. We use Zucker’s conjecture (proven by Looijenga and independently by Saper and Stern) to obtain a formula relating the intersection number Z(m) ·Z(n) to the trace of ∗T (m) ◦ T (n) considered as an endomorphism of the space of Hilbert cusp forms on U0(c).