On central critical values of the degree four L-functions for {text {GSp}}left( 4right) : a simple trace formula

We establish a simple relative trace formula for ${text {GSp}}(4)$ and inner forms with respect to Bessel subgroups to obtain a certain Bessel identity. From such an identity, one can hope to prove a formula relating central values of degree four spinor $L$ -functions to squares of Bessel periods as conjectured by Böcherer. Under some local assumptions, we obtain nonvanishing results, i.e., a global Gross–Prasad conjecture for $(text {SO}(5),text {SO}(2))$ .