  We study a trace formula for tamely ramified abelian varieties $A$ over a
complete discretely valued field, which expresses the Euler characteristic of
the special fiber of the N\'eron model of $A$ in terms of the Galois action on
the $\ell$-adic cohomology of $A$. If $A$ has purely additive reduction, the
trace formula yields a cohomological interpretation for the number of connected
components of the special fiber of the N\'eron model.
