  We establish precise relations between Euler systems that are respectively
associated to a $p$-adic representation $T$ and to its Kummer dual $T^*(1)$.
Upon appropriate specialization of this general result, we are able to deduce
the existence of an Euler system of rank $[K:\mathbb{Q}]$ over a totally real
field $K$ that both interpolates the values of the Dedekind zeta function of
$K$ at all positive even integers and also determines all higher Fitting ideals
of the Selmer groups of $\mathbb{G}_m$ over abelian extensions of $K$. This
construction in turn motivates the formulation of a precise conjectural
generalization of the Coleman-Ihara formula and we provide supporting evidence
for this conjecture.
