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In this work we obtain algebraicity results on special $L$-values attached toSiegel-Jacobi modular forms. Our method relies on a generalization of thedoubling method to the Jacobi group obtained in our previous work, and onintroducing a notion of near holomorphy for Siegel-Jacobi modular forms. Someof our results involve also holomorphic projection, which we obtain by usingSiegel-Jacobi Poincar\'e series of exponential type.

Algebraicity of special $L$-values attached to Siegel-Jacobi modular forms