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

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 $G_2$.

57 pages. To appear in Algebra & Number Theory

Cited by in corpus (1)

• Tempered currents and Deligne cohomology of Shimura varieties, with an application to ${\mathrm{GSp}}_6$