Burns–Macias Castillo equivariant Birch and Swinnerton-Dyer conjecture

Let $A$ be an abelian variety, let $F/k$ be a finite Galois extension with group $G$, let $S$ be the specified finite set of places, and let $\widehat G$ be the set of irreducible complex characters of $G$. For each $\psi\in\widehat G$, let $V_\psi$ afford $\psi$, let $e_\psi$ be the corresponding central primitive idempotent, and let $\check\psi$ be the contragredient character. Let $L_S(A,\psi,z)$ be the truncated Hasse–Weil–Artin $L$-series, $L_S^*(A,\psi,1)$ its leading coefficient, $\Omega_{\omega_\bullet}(A_{F/k})$ the period element, ${\rm SC}_{S,\omega_\bullet}(A_{F/k})$ the Nekovář–Selmer complex, $h_{A,F}$ the Néron–Tate height pairing, and $\mu_S(A_{F/k})$ the Fontaine–Messing correction term. Equivariant Birch and Swinnerton-Dyer conjecture. The following claims are valid: (i) \russ\char88$(A_F)$ is finite; (ii) for every $\psi\in\widehat G$, $L(A,\psi,z)$ has an analytic continuation to $z=1$ and has there a zero of order $\psi(1)^{-1}\operatorname{dim}_{\mathbb C}(e_\psi(\mathbb C\otimes_{\mathbb Z}A^t(F)))$; (iii) for every symplectic $\psi\in\widehat G^{\rm s}$, $L_S^*(A,\psi,1)$ is strictly positive, and there is a unique $L_S^*(A_{F/k},1)\in K_1(\mathbb R[G])$ satisfying

$$\operatorname{Nrd}_{\mathbb R[G]}(L_S^*(A_{F/k},1))_\psi=L_S^*(A,\check\psi,1)$$

for all $\psi$; and (iv)

$$\partial_G\left(\frac{L_S^*(A_{F/k},1)}{\Omega_{\omega_\bullet}(A_{F/k})}\right)=\chi_G({\rm SC}_{S,\omega_\bullet}(A_{F/k}),h_{A,F})+\mu_S(A_{F/k})$$

in $K_0(\mathbb Z[G],\mathbb R[G])$. This is the paper’s central equivariant refinement of Birch and Swinnerton-Dyer, relating Artin $L$-values, Mordell–Weil ranks, Selmer complexes, heights, periods, and local correction terms. The source gives no resolution of the conjecture.