  Let $f:X\to C$ be a family of semistable K3 surfaces with non-empty set $S$
of singular fibres having infinite local monodromy. Then, when the so called
Arakelov-Yau inequality reaches equality, we prove that $C\setminus S$ is a
modular curve and the family comes essentially from a family of elliptic curves
through a so called Nikulin-Kummer construction. In particular, when $C=\BBb
P^1$, the family of elliptic curves must be one of Beauville's 6 examples where
Arakelov inequality reaches equality.
