  We introduce the N\'eron component series of an abelian variety $A$ over a
complete discretely valued field. This is a power series in $\Z[[T]]$, which
measures the behaviour of the number of components of the N\'eron model of $A$
under tame ramification of the base field. If $A$ is tamely ramified, then we
prove that the N\'eron component series is rational. It has a pole at T=1,
whose order equals one plus the potential toric rank of $A$. This result is a
crucial ingredient of our proof of the motivic monodromy conjecture for abelian
varieties. We expect that it extends to the wildly ramified case; we prove this
if $A$ is an elliptic curve, and if $A$ has potential purely multiplicative
reduction.
