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Pullbacks of Saito-Kurokawa lifts. (English) Zbl 1188.11020

From the introduction: Pullbacks of Siegel Eisenstein series have been studied by Garrett, Böcherer, Heim, and play a key role in the proof of the algebraicity of critical values of certain automorphic \(L\)-functions. More generally, one might consider pullbacks of Siegel cusp forms. For example, Ikeda gave a conjectural formula for pullbacks of Ikeda lifts in terms of critical values of \(L\)-functions for \(\text{Sp}_n \times\text{GL}_2\). Also, the Gross-Prasad conjecture would relate pullbacks of Siegel cusp forms of degree 2 to central critical values of \(L\)-functions for \(\text{GSp}_2 \times\text{GL}_2 \times\text{GL}_2\).
Indeed, S. Böcherer, M. Furusawa and R. Schulze-Pillot [Contributions to automorphic forms, geometry, and number theory. Johns Hopkins University, Baltimore, MD, USA, 2002, 105–130 (2004; Zbl 1088.11036)] gave an explicit formula for pullbacks of Yoshida lifts. In this paper, we give an explicit formula for pullbacks of Saito-Kurokawa lifts and prove the algebraicity of central critical values of certain \(L\)-functions for \(\text{Sp}_1\times\text{GL}_2\).

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F32 Modular correspondences, etc.

Citations:

Zbl 1088.11036
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References:

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