  We define Schatten classes of adjointable operators on Hilbert modules over
abelian $C^*$-algebras. Many key features carry over from the Hilbert space
case. In particular, the Schatten classes form two-sided ideals of compact
operators and are equipped with a Banach norm and a $C^*$-valued trace with the
expected properties. For trivial Hilbert bundles, we show that our
Schatten-class operators can be identified bijectively with
Schatten-norm-continuous maps from the base space into the Schatten classes on
the Hilbert space fiber, with the fiberwise trace. As applications, we
introduce the $C^*$-valued Fredholm determinant and operator zeta functions,
and propose a notion of $p$-summable unbounded Kasparov cycles in the
commutative setting.
