The equivariant Bloch–Kato conjecture for motives

Let $M$ be a motive as in the paper's notation, let $p eq 2$ be a \prime, and let $k$ be big enough. The fundamental line is

$$\Delta_f(G,M^\lor(1-k))=\operatorname{det}^{-1}_{{\Bbb{Q}}[G]} H_{\mathcal{M}}^1({\Bbb{Z}},{\Bbb{Q}}[G]\otimes M(k))\otimes \operatorname{det}_{{\Bbb{Q}}[G]} ({\Bbb{Q}}[G]\otimes M_B(k-1))^+.$$

Equivariant Bloch–Kato conjecture. The Beilinson regulator induces an isomorphism $\Delta_f(G,M^\lor(1-k))\otimes {\Bbb{R}}\cong \1_{{\Bbb{R}}[G]}$, under which $(\lambda L_S(G,M^\lor(1-k))^*)^{-1}$ is induced by a unique generator $(\lambda^{-1}\delta(G,M,k))\in\Delta_f(G,M^\lor(1-k))$. The reduced norm is an isomorphism $K_1({\Bbb{Q}}_p[G])\cong Z({\Bbb{Q}}_p[G])^*$, and hence

$$\delta_p(G,M,k):=(\lambda^{-1}\delta(G,M,k))\lambda\in \Delta_f(G,M^\lor(1-k))\otimes{\Bbb{Q}}_p.$$

The $p$-adic regulator induces the stated determinant isomorphism, and for a Galois-stable lattice $T_B\subset M_B$ with $T_p=T_B\otimes{\Bbb{Z}}_p$, $\delta_p(G,M,k)$ is induced by a generator $\widetilde{\delta}_p(G,M,k)$ in the corresponding integral determinant line.

This is the equivariant formulation of the Bloch–Kato conjecture, generalizing the original conjecture to non-abelian coefficients. The paper states that the resulting generators are compatible in towers and uses this conjecture as the finite-level input for the non-abelian Iwasawa Main Conjecture.