Let $C/\mathbb Q_p$ (or a p-adic local field more generally) be a smooth projective curve with split semistable reduction over $\mathbb Z_p$. What can we say about the action of the Galois group $\Gamma = \mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)$ on the 'etale cohomology $H^1(C,\mathbb Q_\ell)$?
For example, if $C$ is an elliptic curve with split nodal reduction, I believe that the Galois group acts in an upper triangular way with off diagonal entries $\chi_{cycl}$ and $1$.
In general, I would like to relate the action of the Galois group to the irreducible components that occur in the reduction of the Neron model. More specifically, let $g \in \Gamma$ be any lift of the Frobenius. I would like to compute the trace of $g$ on the first cohomology group above in terms of the trace of the Frobenius on the irreducible components that occur over the special fiber.
For a very specific example, suppose $C$ is a genus $2$ curve with the special fiber being two elliptic curves $E_1,E_2/\mathbb F_p$ joined at a point. Is it true that $$tr(g|H^1(C,\mathbb Q_\ell)) = tr(\mathrm{Frob}_p|H^1(E_1,\mathbb Q_\ell)) + tr(\mathrm{Frob}_p|H^1(E_2,\mathbb Q_\ell))?$$ I would also be interested in the analogous question with $\mathbb Q_p$ coefficients instead of $\mathbb Q_\ell$.
