Trace formula for component groups of N\'eron models

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 $\mathrm{\ell }$-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.