Algebraic cycles and functorial lifts from G2 to PGSp6

Abstract

We study instances of Beilinson–Tate conjectures for automorphic representations of ${\mathrm{PGSp}<!--FUNCTION\; APPLICATION-->}_6$ whose spin $L$-function has a pole at $s=1$. We construct algebraic cycles of codimension 3 in the Siegel–Shimura variety of dimension 6 and we relate its regulator to the residue at $s=1$ of the $L$-function of certain cuspidal forms of ${\mathrm{PGSp}<!--FUNCTION\; APPLICATION-->}_6$. Using the exceptional theta correspondence between the split group of type $G_2$ and ${\mathrm{PGSp}<!--FUNCTION\; APPLICATION-->}_6$ and assuming the nonvanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank-$7$ motives of type $G_2$.