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

Abstract

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 Bocherer. Under some local assumptions, we obtain nonvanishing results, i.e., a global Gross–Prasad conjecture for \((\text {SO}(5),\text {SO}(2))\).