Equivariant Birch and Swinnerton-Dyer relation

Let $G$ be a finite abelian group and let $V\to W$ be a $G$-cover of smooth projective geometrically connected curves of positive genus over ${\bf Q}$. Set $A={\rm Jac}(V)$, let ${\mathcal A}$ be its Néron model over ${\rm Spec}({\bf Z})$, and write ${\rm H}_1(A({\bf C}),{\bf Z})^+$ for the subgroup fixed by complex conjugation. Let $\Phi_l(A)$ and $\Phi_\infty(A)$ denote the finite and real component groups, let $A({\bf Q})$ be the Mordell–Weil group, and let $[M]^\vee$ be the duality involution on ${\rm G}_0({\bf Z}[G])$. Equivariant Birch and Swinnerton-Dyer conjecture. If the Tate–Shafarevich group ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(A)$ is finite, then

$$[{\rm Lie}({\mathcal A})]-[{\rm H}_1(A({\bf C}),{\bf Z})^+]+[\Phi_\infty(A)]+\sum_l[\Phi_l(A)]+[A({\bf Q})]^\vee-[A({\bf Q})]+[{\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(A)]=0$$

in ${\rm G}_0({\bf Z}[G])$. This is a refined equivariant form of the Birch and Swinnerton-Dyer relation, expected to follow from more precise equivariant Tamagawa-number conjectures; the supplied text does not establish it.