\section{Conventions}\label{sec:conventions}

\subsection{The field \texorpdfstring{$K$}{K}}
Throughout, $K=\Q(\omega)$ with $\omega^2+\omega+1=0$, $\OO=\Z[\omega]$, and $\cc$ is complex conjugation, with a lift $\tilde\cc\in G_\Q$, which we take to be a complex conjugation of $\Qbar$; so $\tilde\cc^2=1$.  The prime of $K$ above $3$ is $\p=(1-\omega)=(\sqrt{-3})$; $K_\p$ is the completion, and $v_3$ is the valuation on an algebraic closure of $\Q_3$ (or on its completion $\C_3$) with $v_3(3)=1$.  The ring $\OO$ is a principal ideal domain with unit group $\mu_6=\{\pm1,\pm\omega,\pm\omega^2\}$, and $\mu_6\to(\OO/\p^2)^\times$ is an isomorphism.  The prime $7$ splits in $K$ as $7=\lp\lpb$ with
\[
\lp=(2-\omega),\qquad \lpb=(3+\omega)=\cc(\lp);
\]
the residue fields are $\F_7$, with $\omega\equiv 2\pmod{\lp}$ and $\omega\equiv 4\pmod{\lpb}$.  For a prime $p\equiv1\pmod 3$ we write $v,\bar v$ for the two primes of $K$ above $p$; then $K_v=K_{\bar v}=\Q_p$.

\subsection{Galois groups}
For a field $F$ we write $G_F$ for its absolute Galois group.  For a prime $\q$ of $K$, $G_{K_\q}\subset G_K$ is a decomposition group, $I_\q$ its inertia subgroup, and $\Frob_\q$ a \emph{geometric} Frobenius element.  Local and global Artin maps send uniformizers to geometric Frobenius elements.  (With the other normalization some characters below are replaced by their inverses; the statements are unchanged.)  For a prime $\ell$, $\varepsilon_\ell$ is the $\ell$-adic cyclotomic character and $\bar\varepsilon_\ell$ its reduction.  Hodge--Tate weights are normalized so that $\varepsilon_p$ has weight $-1$.  Then the $p$-adic cohomology $H^2$ of a smooth projective surface over a $p$-adic field has Hodge--Tate weights in $\{0,1,2\}$, the weight $i$ occurring with multiplicity $h^{i,2-i}$.

For $u\in K^\times$ the \emph{cubic Kummer character} $\kappa_u\colon G_K\to\mu_3\subset\OO^\times$ is $\kappa_u(\sigma)=\sigma(u^{1/3})/u^{1/3}$.  If $u\in\Q^\times$, then $\kappa_u^\cc=\kappa_u^{-1}$, where $\kappa^\cc(\sigma)=\kappa(\tilde\cc\sigma\tilde\cc^{-1})$: for $u^{1/3}$ real, $\tilde\cc$ fixes $u^{1/3}$ and inverts $\mu_3$.

\subsection{Coefficients}
For each prime $\ell$ we fix an embedding $\iota_\ell\colon\Qbar\hookrightarrow\Ql$.  Through $\iota_\ell$ we regard elements of $K$ as elements of $\Ql$.  A representation of $G_K$ over $\Ql$ \emph{has coefficients in $K$} if the characteristic polynomials of the Frobenius elements at almost all primes lie in $K[T]$.  For a continuous representation $\rho$ of a profinite group on a finite-dimensional $\overline\Q_p$-vector space, $\rhob$ denotes the semisimplification of the reduction of a stable lattice, a representation over $\Fbar_p$.  By the Brauer--Nesbitt theorem it is independent of the lattice.  A continuous representation of $G_K$ with finitely many ramified primes is determined up to semisimplification by the characteristic polynomials of the $\Frob_\q$ at the unramified primes (Chebotarev).

\subsection{Local systems}
For a connected scheme $Y$, a lisse $\Ql$-sheaf on $Y$ is a continuous representation of its étale fundamental group (for the analytic spaces of \cref{sec:inertia3} see \cref{sec:analytic}).  A lisse sheaf is \emph{constant} if the geometric fundamental group acts trivially (over a base field $F$ the Galois group $G_F$ may still act on the fiber), and \emph{unipotent} if it has a finite filtration by lisse subsheaves whose graded pieces are constant.  Sub-objects, quotients and extensions of unipotent lisse sheaves are unipotent.  For a representation $\rho$ of $G_{K_\q}$ with $\q\nmid\ell$, Grothendieck's monodromy theorem attaches a Weil--Deligne representation $\WD(\rho)=(\phi,N)$, with $\phi$ of finite image on $I_\q$ and $N$ nilpotent, such that $\rho(\sigma)=\phi(\sigma)\exp(t_\ell(\sigma)N)$ for $\sigma\in I_\q$, where $t_\ell\colon I_\q\to\Z_\ell(1)$ is the $\ell$-adic tame character.