# Intersection numbers of Hecke cycles on Hilbert modular varieties

Published

Journal Article

Let Script O sign be the ring of integers of a totally real number field E and set G := Res E/ℚ ( GL 2 ). 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 U o (c) of G(double-struck A f ). Setting X 0 (c) := G(ℚ)\G(double-struck A)/K ∞ U 0 (c), where K ∞ is a certain subgroup of G(ℝ), we use T(m) to define a Hecke cycle Z(m) ∈ IH 2[E:ℚ] (X 0 (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 U 0 (c). © 2007 by The Johns Hopkins University Press.

### Full Text

### Duke Authors

### Cited Authors

- Getz, J

### Published Date

- December 1, 2007

### Published In

### Volume / Issue

- 129 / 6

### Start / End Page

- 1623 - 1658

### International Standard Serial Number (ISSN)

- 0002-9327

### Digital Object Identifier (DOI)

- 10.1353/ajm.2007.0041

### Citation Source

- Scopus