The N\'eron component series of an abelian variety

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.