(In)dependence of the conductor of a Galois representation and the choice of l

Question

In the work of A. P. Ogg, Elliptic Curves and Wild Ramification, he proves that the conductor of an elliptic curve is independent of the choice of $\ell$. That is, for example, if $E$ is an elliptic curve defined over $Q$ and if $G_{Q_p}$ is the Galois group of the local field $Q_p$. When $p\neq \ell$, then the conductor of the $\ell$-adic $G_{Q_p}$-representation induced by the $\ell$-adic Tate module of $E$ is independent of the choice of $\ell$.

My question is: is the statement "the conductor of the $\ell$-adic $G_{Q_p}$-representation $\rho_{\ell}$ is independent of the choice of $\ell$" always true as long as we know that $\rho_{\ell}$ is coming from an etale cohomology of a smooth algerabic variety? If not, how much do we know about it? Any comments and suggestions are welcome.

Answer 1

This is expected, but I believe it is not known in general.

It's more generally believed that if $X$ is a smooth variety over a $p$-adic local field $F$, and we denote by $\rho_\ell$ the Galois representation $H^i(X, \mathbf{Q}_{\ell})$ for $\ell \ne p$, then the representation $WD(\rho_\ell)$ of the Weil--Deligne group of $F$ produced from $\rho_\ell$ by Grothendieck's abstract monodromy theorem should be independent of $\ell$. Since you can read off the conductor from $WD(\rho_\ell)$ this gives your conjecture as a corollary. (This is also expected to hold for $\ell = p$ too, if you define $WD(\rho_p)$ using Fontaine's $D_{\mathrm{pst}}$ functor.)

However, this independence-of-$\ell$ result for Weil-Deligne reps is open in general, although there are some substantial partial results. This MO question has some results and references: https://mathoverflow.net/questions/191715/when-is-independence-of-l-known.