Pure and Applied Mathematics Quarterly
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.
Kevin Keating and David P. Roberts. Intersection Numbers of Heegner Divisors on Shimura Curves. Pure and Applied Mathematics Quarterly 4 (2008), no. 4, 1165-1204.