论文标题
代数循环和功能性升降机从$ g_2 $到$ \ mathrm {pgsp} _6 $
Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$
论文作者
论文摘要
我们研究了$ \ mathrm {pgsp} _6 $的贝林森 - 塔特猜想的实例，其自旋$ l $ - 功能的极点为$ s = 1 $。我们在siegel-simura六个尺寸的siegel-simura种类中构建了代数循环，我们将其调节器与残基的$ s = 1 $相关联$ \ $ \ mathrm {pgsp} _6 $的$ s = 1 $。使用类型$ g_2 $和$ \ mathrm {pgsp} _6 $的拆分组之间的非凡theta对应关系，并假设某个Archimedean积分的不变，这使我们能够确认Gross和Savin的猜想，而Savin的等级为$ 7 $ $ g_2 $。
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 $G_2$.
