On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized Böcherer conjecture

Abstract

A. We investigate the Gross-Prasad conjecture and its refinement for the Bessel periods in the case of (SO (5), SO(2)). In particular, by combining several theta correspondences, we prove the Ichino-Ikeda type formula for any tempered irreducible cuspidal automorphic representations. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two which are Hecke eigenforms to central special values of -functions. The formula is regarded as a natural generalization of Böcherer's conjecture to the non-trivial toroidal character case.1.2. Measures. Throughout the paper, for an algebraic group G defined over , we write G for G ( ) and we always take the measure on G (A) to be the Tamagawa measure unless specified otherwise. For each , we take the self-dual measure with respect to on . Then recall that the product measure on A is the self-dual measure with respect to and is also the Tamagawa measure since Vol (A/ ) = 1. For a unipotent algebraic group U defined over , we also specify the local measure on U( ) to be the measure corresponding to the gauge form defined over , together with our choice of the measure on , at each place of . Thus in particular we have = and Vol (U ( ) \U (A) , ) = 1.1.3. Similitudes. Various similitude groups appear in this article. Unless there exists a fear of confusion, we denote by ( ) the similitude of an element of a similitude group for simplicity.