Beilinson–Deligne conjecture for arithmetic surfaces at $s=1$

Let $S=\operatorname{Spec}\mathcal O_F$ for a number field $F$, and let $X$ be an arithmetic surface over $S$, with special zeta value $\zeta^*(X,1)$ and generalized Euler characteristic $\chi(X,1)=\chi_{A,C}(X,1)/\chi_B(X,1)$. Beilinson–Deligne conjecture. Up to powers of two, one has

$$\zeta^*(X,1)=\pm\chi(X,1)=\pm\frac{\chi_{A,C}(X,1)}{\chi_B(X,1)}.$$

This is a global refinement of the Fontaine–Perrin-Riou and Bloch–Kato conjectures, expressing the special value intrinsically through arithmetic and cohomological invariants of $X$.