Intersection numbers of Hecke cycles on Hilbert modular varieties

Let Script O sign be the ring of integers of a totally real number field E and set G := ResE/ℚ( GL2). Fix an ideal c ⊂ Script O sign. For each ideal m ⊂ Script O sign let T(m) denote the mth Hecke operator associated to the standard compact open subgroup Uo(c) of G(double-struck Af). Setting X0(c) := G(ℚ)\G(double-struck 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) x X 0(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). © 2007 by The Johns Hopkins University Press.

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Author Jayce Robert Getz Mathematics