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• W2248048385 abstract "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, 1/2) 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 explicate 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 the papers [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 theo-" @default.
• W2248048385 created "2016-06-24" @default.
• W2248048385 creator A5063751719 @default.
• W2248048385 date "2004-06-21" @default.
• W2248048385 modified "2023-10-17" @default.
• W2248048385 title "Special Value Formulae for Rankin L-Functions" @default.
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• W2248048385 doi "https://doi.org/10.1017/cbo9780511756375.007" @default.
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