Beilinson–Tate conjecture for the Spin motive of a cohomological representation

Let $\pi = \pi_\infty \otimes \pi_f$ be a cohomological cuspidal automorphic representation of $\operatorname{PGSp}_6(\mathbf{A})$. Let $M(\pi_f)$ be the Spin Chow motive with coefficients in a number field $L$, let $L(s,M(\pi_f)(3))$ be its Hasse–Weil $L$-function, and let

$$r_\mathcal{H}:H^1_\mathcal{M}(M(\pi_f)(4))\oplus N(M(\pi_f)(3))\longrightarrow H^1_\mathcal{H}(M(\pi_f)(4))$$

be the Beilinson–Deligne regulator, where $N(M(\pi_f)(3))$ is the group of algebraic cycles modulo homological equivalence. Beilinson–Tate conjecture. The map $r_\mathcal{H}$ induces an isomorphism

$$\bigl(H^1_\mathcal{M}(M(\pi_f)(4))\oplus N(M(\pi_f)(3))\bigr)\otimes_{\mathbf{Q}}\mathbf{R}\longrightarrow H^1_\mathcal{H}(M(\pi_f)(4)),$$

moreover

$$\operatorname{ord}_{s=0}L(s,M(\pi_f)(3))=\dim_L H^1_\mathcal{M}(M(\pi_f)(4)),$$ $$-\operatorname{ord}_{s=1}L(s,M(\pi_f)(3))=\dim_L N(M(\pi_f)(3)),$$

and

$$\mathrm{det}(\operatorname{Im}r_\mathcal{H})=L^*(1,M(\pi_f)(3))\mathcal{D}(M(\pi_f)(4)),$$

where $\mathcal{D}(M(\pi_f)(4))$ is the Deligne $L$-structure of $\mathrm{det}(H^1_\mathcal{H}(M(\pi_f)(4)))$. This is the Beilinson–Tate prediction relating motivic cohomology, algebraic cycles, regulators, and special values of the $L$-function; the paper studies the algebraic-cycle contribution when the $L$-function has a simple pole at $s=1$.