Modified Lichtenbaum–Gross conjecture

Let $S$ be a finite set of places of $k$ containing $S_\infty$ and all places ramified in $F/k$. Let $\lambda:Y_r\to X_r$ be a homomorphism with finite kernel and cokernel, let $R_r^\lambda(\psi)$ be the associated regulator, and let $t^j_{\lambda,\psi}:Y^j_{r,\psi}\to X^{j,\psi}_r$ be the composite defined above for $1\leq j\leq\chi(1)$. Write $q(f)={\rm Fit}_{\mathcal O}(\operatorname{coker}(f)){\rm Fit}_{\mathcal O}(\ker(f))^{-1}$. Modified Lichtenbaum–Gross conjecture. For every such $\lambda$ and $j$,

$$\frac{L_S^*(r,\check\chi^\alpha)}{R_r^\lambda(\chi^\alpha)}=\frac{L_S^*(r,\check\chi)^\alpha}{R_r^\lambda(\chi)^\alpha}$$

for all $\alpha\in\operatorname{Aut}(\mathbb C)$, and

$$\frac{L_S^*(r,\check\chi^\gamma)}{R_r^\lambda(\chi^\gamma)}\mathcal O=q(t^j_{\lambda,\chi^\gamma})^{-1}$$

for all $\gamma\in\Gamma$. This conjecture connects leading terms of Artin $L$-functions with regulators and Fitting ideals; the first equality is identified with Gross's conjecture, while the second was formulated by Chinburg, Kolster, Pappas, and Snaith.