An inner product relation on Saito-Kurokawa lifts

Abstract

Let f be a newform of weight 2k−2 and level M with M an odd square-free integer. Via the Saito-Kurokawa correspondence there is associated to f a Siegel newform Ff of weight k and level M. In this paper we provide a formula relating the Petersson products {Ff, Ff} and {f, f}. We use this result to give a new proof of a special case of a well-known result of Shimura on the algebraicity of a special value of a Rankin convolution L-function.