Document Type

Article

Publication Date

2008

Publication Title

Pure and Applied Mathematics Quarterly

Abstract

In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X0(N) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. Two more recent related works are [KR], which concentrates on computing local intersection multiplicities at ramified primes under quite general hypotheses, and [KRY], which relates intersection numbers on Shimura curves to coefficients of modular forms. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F, working with general Shimura curves. We expect that all of [GKZ] should generalize similarly.

Volume

4

Issue

4

First Page

1165

Last Page

1204

DOI

http://dx.doi.org/10.4310/PAMQ.2008.v4.n4.a8

ISSN

1558-8599

Comments

This is a pre-publication version of an article published in Pure and Applied Mathematics Quarterly. The final authenticated version is available online at: http://dx.doi.org/10.4310/PAMQ.2008.v4.n4.a8

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