The generalized Herbrand–Ribet conjecture

Let $K$ be a CM Galois extension of $\mathbb Q$, let $K^+$ be its totally real subfield, and let $\tau\in\operatorname{Gal}(K/K^+)$ be the nontrivial element. Let $\rho$ be an irreducible odd representation of $\operatorname{Gal}(K/\mathbb Q)$, with $\bar\rho$ its reduction modulo $\mathfrak p$, and write $\bar\rho=\sum n_i\bar\rho_i$. Let $\omega$ be the action on the $p$-th roots of unity in $K$. Generalized Herbrand–Ribet conjecture. If $\bar\rho_j\ne\omega^{-1}$ and $\bar L(0,\bar\rho_j)=0$, then

$$(H_K/H_{K^+})\otimes\bar{\mathbb F}_p$$

contains $n_j\dim(\rho)\bar\rho_j^\vee$ in its semisimplification; contributions from different $\rho$ are independent and fill the semisimplification except for the $\omega$-component. If $\omega\ne1$, no assertion is made about that component, while if $\omega=1$ it is absent. This is a proposed extension of Ribet's theorem relating mod-$p$ special values to class-group representations; it remains open.