1. Introduction

Let F denote a totally real number field, and let K/F denote a totally imaginary quadratic extension. We fix an automorphic cuspidal representation π of GL2(F), and a finite order Hecke character χ of K. Thus χ is a representation of GL1(K).

Under certain hypotheses, it is known that the central critical value L(π⊗χ, ½) is algebraic up to a known transcendental factor. Explicit formulae for this value have been given by a number of authors, notably Gross, Waldspurger, and Zhang. Essentially, the work of Gross and Zhang shows that this value is given by the height of a certain CM divisor on a suitable space, while the work of Waldspurger gives a criterion for nonvanishing of this value in terms of a certain linear functional arising from representation theory, and a formula in terms of torus integrals on a quaternion algebra. Our goal in this article is to explain the connections between these works, and to provide a bridge between the general representation-theoretic framework described by Gross (see his article [Gro] in this volume) and the theorems of Zhang [Zha01a] and Waldspurger [Wal85].

We want to point out that the formula we will discuss has numerous applications to arithmetic and Iwasawa theory (see [BD96] and its various sequels). We will therefore attempt to formulate the representation-theoretic results in terms that are familiar to number theorists. We will not however discuss any arithmetic applications directly—the reader will find some of these applications elsewhere in this volume.

Needless to say, the present work is mostly expository. The ideas are largely drawn from [Gro87], [Gro], [Wal85], [Zha01a]. However, the organization here is perhaps novel. Our main contribution is given in Theorem 6.4. While the ingredients in this theorem are all well-known, our formulation seems to be new, and is well-suited for applications to number theory as in [BD96] and [Vat02].

I thank Benedict Gross, Shou-Wu Zhang, and Hui Xue for patiently answering my numerous questions on this subject. The statements in this paper reflect my very incomplete understanding of their work, and the reader interested in the details should consult the original sources. Finally, I would also like to thank Barry Mazur, Christophe Cornut, and the mathematics department at Harvard University for their hospitality in April 2002, during which time most of this article was written.