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Intersection theory on Shimura surfaces

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 March 2009

Benjamin Howard
Benjamin Howard*
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA, 02467, USA (email: [email protected])
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Abstract

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Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycle classes constructed by Kudla, Rapoport and Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

Keywords

Shimura varietiesArakelov theoryKudla’s program

MSC classification

Secondary: 14G35: Modular and Shimura varieties 14G40: Arithmetic varieties and schemes; Arakelov theory; heights 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 423 - 475
Copyright
Copyright © Foundation Compositio Mathematica 2009

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