Global epsilon-conjecture

Let $S$ be a finite set of places containing $p$, let $T$ be an $R$-representation of $G_{\mathbb{Q},S}$, and let $T^*$ denote its relevant dual. Let $z_R^S(T_0):\bold{1}_R\overset{\sim}{\rightarrow}\Delta_R^S(T_0)$ be the conjectural global zeta isomorphism for $T_0=T,T^*$, and let $\varepsilon_{R,\zeta^{(l)}}^{(l)}(T):\bold{1}_R\overset{\sim}{\rightarrow}\Delta_R^{(l)}(T)$ be the local epsilon-isomorphisms, with the one at $p$ conjectural and those for $l\ne p$ supplied by Yasuda. Global epsilon-conjecture. Under the canonical Poitou–Tate isomorphism

$$\Delta_R^S(T^*)\overset{\sim}{\longrightarrow}\boxtimes_{l\in S}\Delta_R^{(l)}(T)\boxtimes\Delta_R^S(T),$$

one has

$$z_R^S(T^*)=\boxtimes_{l\in S}\varepsilon_{R,\zeta^{(l)}}^{(l)}(T)\boxtimes z_R^S(T).$$

This is the global functional equation relating zeta and local epsilon-isomorphisms through Poitou–Tate duality; its validity depends on the conjectural global and $p$-adic local isomorphisms.