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Trilinear forms and the central values of triple product \(L\)-functions. (English) Zbl 1222.11065

The author and Ikeda have a beautiful conjecture explicating the Gross-Prasad conjecture. In this paper the author proves a low rank case of the Ichino-Ikeda conjecture. The main result is an explicit formula relating the global trilinear form on a quaternion algebra and the Petersson inner product. As a corollary, this implies the relation between the global trilinear form and the central value of the triple product \(L-\)function, which is the content of Jacquet’s conjecture, originally established by Harris-Kudla.
The idea of the proof is to apply the theta correspondence and Siegel-Weil formula, and express both the trilinear form and the Petersson product using the Whittaker functions of automorphic forms of \(\text{GL}_2\). The trilinear form part was done in Harris-Kudla’s work, while the Petersson product part was done in Waldspurger’s work on toric periods. The extra work needed is to establish a local identity on Whittaker functions, where the main difficulty lies in convergence issues.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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