When GL(2) cusp forms meet CM points in GSp(4)

Abstract

Duke's equidistribution theorem shows that the nonconstant contribution of the $\mathrm{GL}(2)$ automorphic spectrum to CM packets becomes asymptotically negligible. CM packets on the Levi base index the fibers of Bessel quotients in $\mathrm{GSp}(4)$, and the symmetric cube correspondence carries $\mathrm{GL}(2)$ cusp forms to $\mathrm{GL}(4)$. In the holomorphic setting considered here, these two constructions meet inside $\mathrm{GSp}(4)$. We prove averaged decay for the resulting class group periods over imaginary quadratic fields, expressed as averages of conditional fiber integrals. More precisely, let $f$ be a non-CM holomorphic Hecke newform with trivial central character. The mean absolute value of the associated class-group averages tends to zero along the transverse unipotent direction.