Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$

Abstract

We establish instances of Beilinson-Tate conjectures for automorphic representations of PGSp 6 whose Spin L-function has a pole at s = 1. Using the exceptional theta correspondence between the split group of type G2 and PGSp 6 and assuming the non-vanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank 7 motives of type G2.