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

TL;DR

This paper links algebraic cycles, automorphic forms, and L-functions for groups G_2 and PGSp_6, confirming a conjecture relating motives and special values of L-functions.

Contribution

It constructs algebraic cycles in Siegel-Shimura varieties and relates their regulators to L-function residues, using the exceptional theta correspondence to confirm a conjecture of Gross and Savin.

Findings

Confirmed a conjecture of Gross and Savin on G_2 motives.

Constructed algebraic cycles in Siegel-Shimura varieties.

Related regulators of cycles to residues of L-functions.

Abstract

We study instances of Beilinson-Tate conjectures for automorphic representations of $\mathrm{PGSp}_6$ whose Spin $L$-function has a pole at $s=1$. We construct algebraic cycles of codimension three in the Siegel-Shimura variety of dimension six and we relate its regulator to the residue at $s=1$ of the $L$-function of certain cuspidal forms of $\mathrm{PGSp}_6$. Using the exceptional theta correspondence between the split group of type $G_2$ and $\mathrm{PGSp}_6$ 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 $G_2$.