Abstract

This article presents a new proof of a theorem of Karl Rubin relating values of the Katz p-adic L-function of an imaginary quadratic field at certain points outside its range of classical interpolation to the formal group logarithms of rational points on CM elliptic curves. The approach presented here is based on the p-adic Gross-Zagier type formula proved by the three authors in previous work. As opposed to the original proof which relied on a comparison between Heegner points and elliptic units, it only makes use of Heegner points, and leads to a mild strengthening of Rubin's original result. A generalization to the case of modular abelian varieties with complex multiplication is also included.

We prove integrality of the ratio f, f / g, g (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and , denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain level-lowering congruences satisfied by f and to the central values of a family of Rankin-Selberg L-functions. Finally we give two applications, the first to proving the integrality of a certain triple product L-value and the second to the computation of the Faltings height of Jacobians of Shimura curves.

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