\section{The Frey motive}\label{sec:motive}

The local system $\VV$ of the introduction comes from a family of surfaces.  Its Frobenius data at the two primes above $7$ are used for the irreducibility of the Frey representation (\cref{sec:irreducibility}) and for the triviality of a limit character at $s=0$ (\cref{sec:limit}).

\subsection{The surfaces and the character}\label{sec:surfaces}
Let $K_9=K(\zeta_9)=\Q(\zeta_9)$, a cyclic cubic extension of $K$, and let $\sigma_0\in\Gal(K_9/K)$ be the generator with $\sigma_0(\zeta_9)=\zeta_9^4$.  Let $\RR_{K_9/K}\A^1$ be the Weil restriction of the affine line, an affine space of dimension $3$ over $K$, let $\Nm\colon\RR_{K_9/K}\A^1\to\A^1$ be the norm, and let $H=\RR_{K_9/K}\mu_9$, a finite étale group scheme of order $729$ over $K$ acting on $\RR_{K_9/K}\A^1$ by multiplication.  Let $\SSS=\PP^1_K\smallsetminus\{0,1,\infty\}$ with coordinate $s$, and let
\[
X=\{(y,s)\in\RR_{K_9/K}\A^1\times\SSS:\ \Nm(1-y^9)=s\},\qquad f\colon X\to\SSS,
\]
with fibers $X_s$.  The action of $H$ preserves $X$.

Fix a $K$-embedding $\tau_0\colon K_9\to\Kbar$ and put $\tau_j=\tau_0\circ\sigma_0^j$ for $j\in\Z/3$.  The isomorphism $K_9\otimes_K\Kbar\to\Kbar^3$, $x\otimes z\mapsto(\tau_j(x)z)_j$, identifies $\RR_{K_9/K}\A^1$ over $\Kbar$ with $\A^3$, $\Nm(1-y^9)$ with $\prod_j(1-y_j^9)$, $H_{\Kbar}$ with $\mu_9^3$, and $X_s\otimes\Kbar$ with the surface $Z_s\colon(1-y_0^9)(1-y_1^9)(1-y_2^9)=s$ of the introduction.  An element $\sigma\in G_K$ with $\sigma\circ\tau_0=\tau_0\circ\sigma_0^k$ acts on $\Kbar$-points by $(y_j)_j\mapsto(\sigma(y_{j-k}))_j$, and on $H(\Kbar)=\mu_9(\Kbar)^3$ by the same formula, where $\sigma(\zeta)=\zeta^{4^k}$ for $\zeta\in\mu_9$.  For $e=(e_0,e_1,e_2)\in(\Z/9)^3$ let $\chi_e(g)=\prod_j\iota_\ell(g_j)^{e_j}$, a character of $H(\Kbar)$ with values in $\Ql^\times$.

\begin{lemma}\label{lem:characters}
The characters $\chi_e$ fixed by $G_K$ are those with $e=m(1,7,4)$, $m\in\Z/9$.  Those of order $9$ on each factor of $\mu_9^3$ are the six characters $\chi^m$, $m\in(\Z/9)^\times$, where $\chi\eqdef\chi_{(1,7,4)}$.  Each of them is trivial on the diagonal $\mu_3$ and nontrivial on the diagonal $\mu_9$.
\end{lemma}

\begin{proof}
By the formulas above, $\chi_e(\sigma g)=\prod_j\iota_\ell(g_j)^{4^ke_{j+k}}$, so $\chi_e$ is fixed by $G_K$ if and only if $e_{j+1}\equiv 7e_j\pmod 9$ for all $j$, that is $e=e_0(1,7,4)$.  Such a character has order $9$ on each factor exactly when $e_0\in(\Z/9)^\times$.  On the diagonal, $\chi^m(\zeta,\zeta,\zeta)=\iota_\ell(\zeta)^{12m}$, which is $1$ for $\zeta\in\mu_3$ and not for a primitive $\zeta\in\mu_9$.
\end{proof}

For a $G_K$-fixed character $\chi'$ of $H(\Kbar)$ the idempotent $\mathbf{1}_{\chi'}=|H|^{-1}\sum_{g}\chi'(g)^{-1}g^*$ acting on the cohomology of $H$-stable varieties commutes with $G_K$; its image on a cohomology group is called the $\chi'$-part and is written with the exponent $\chi'$.  The automorphism $\sigma_0\otimes1$ of $\RR_{K_9/K}\A^1$ is defined over $K$, preserves $X$ and commutes with $f$; on $\Kbar$-points it permutes the coordinates cyclically, so it carries the $\chi^m$-part of the cohomology to the $\chi^{4m}$-part.

\begin{definition}\label{def:V}
Fix $m\in(\Z/9)^\times$.  For a prime $\ell$ let
\[
\VV=\VV_{\chi^m}\eqdef\big(\RR^2f_!\,\Ql\big)^{\chi^m},
\]
a sheaf on $\SSS$, with fibers $\VV_s=H^2_c(X_s\otimes\Kbar,\Ql)^{\chi^m}$.  By the preceding paragraph $\VV_{\chi^m}\isom\VV_{\chi^{4m}}\isom\VV_{\chi^{7m}}$, so up to isomorphism there are two such sheaves, $\VV_\chi$ and $\VV_{\chi^{-1}}$.  The statements below hold for both, and where the two differ (only in \cref{prop:frob7}) both cases are listed.  We write $\VV$ for either.
\end{definition}

\subsection{Geometry over \texorpdfstring{$\Kbar$}{the algebraic closure}}
For $j\in\Z/9$, $j\ne0$, let $\LL_j$ be the Kummer sheaf of order $9$ attached to the $\mu_9$-torsor $y^9=1-x$ on $\A^1\smallsetminus\{1\}$ and the character $\zeta\mapsto\iota_\ell(\zeta)^j$ of $\mu_9$, extended by zero to $\A^1$.  For sheaves $\mathcal K,\mathcal K'$ on $\mathbb{G}_m$ let $\mathcal K\star\mathcal K'$ be the multiplicative convolution with compact supports, $(\mathcal K\star\mathcal K')_s=H^*_c(\mathbb{G}_m,\mathcal K(x)\otimes\mathcal K'(s/x))$.

\begin{proposition}\label{prop:geometry}
Over $\Kbar$, and over $\Fbar_q$ for every prime $\q\nmid 3\ell$ of $K$ with residue field $\F_q$:
\begin{enumerate}[(a)]
\item $\VV$ is the convolution $\LL_{m}\star\LL_{7m}\star\LL_{4m}$ restricted to $\SSS$, and $H^i_c(X_s,\Ql)^{\chi^m}=0$ for $i\ne2$;
\item $\VV$ is lisse of rank $3$ on $\SSS$, irreducible, and tamely ramified at $0$, $1$, $\infty$;
\item its local monodromy is unipotent with a single Jordan block at $0$; a pseudo-reflection of order $3$ at $1$; and semisimple of order $9$ at $\infty$, with eigenvalues $\iota_\ell(\zeta)^{m},\iota_\ell(\zeta)^{4m},\iota_\ell(\zeta)^{7m}$, or their inverses (according to the orientation), on a generator $\zeta$ of the tame inertia modulo $9$.
\end{enumerate}
In particular the local monodromy at $\infty$ factors through the quotient $\mu_9$ of the tame inertia group $\hat\Z'(1)=\varprojlim\mu_n$ at $\infty$ (the Kummer character of $1/s$), and $\mu_3\subset\mu_9$ acts on $\VV$ by the scalars $\zeta\mapsto\iota_\ell(\zeta)^{\ninf}$, for an integer $\ninf\in\{1,2\}$ depending only on $\VV$.  The stalks of $j_*\VV$ (for $j\colon\SSS\to\PP^1$) at $0$, $1$, $\infty$ have dimensions $1$, $2$, $0$.
\end{proposition}

\begin{proof}
Put $x_j=1-y_j^9$.  The map $Z_s\to Y_s=\{x_0x_1x_2=s\}$ is a $\mu_9^3$-torsor over the locus where no $x_j$ equals $1$, and on the $\chi^m$-isotypic part the push-forward of $\Ql$ is $\boxtimes_j\LL_{e_j}(x_j)$ with $(e_j)=m(1,7,4)$, extended by zero where some $x_j=1$ (each $e_j$ is nonzero, so the isotypic part has no invariants there).  Hence $H^i_c(Z_s)^{\chi^m}=H^i_c(Y_s,\boxtimes_j\LL_{e_j}(x_j))$, which is the fiber at $s$ of the triple convolution.  Katz's theory of hypergeometric sheaves \cite[Chapter~8]{KatzESDE} shows that, the three characters being nontrivial and the exponents at $\infty$ being distinct from the (trivial) exponents at $0$, the convolution is concentrated in one degree, is lisse of rank $3$ on $\mathbb{G}_m\smallsetminus\{1\}$, irreducible, tame at $0,1,\infty$, with the local monodromies of (c).  At $0$ the monodromy is a single unipotent Jordan block (the three characters at $0$ coincide), at $\infty$ it is the direct sum of the three characters, and at $1$ it is a pseudo-reflection whose nontrivial eigenvalue is the product of the characters, of order dividing $9$ and here of order $3$ since $12m\equiv 3m\pmod 9$.  Over $\C$ this is the hypergeometric local system of Beukers and Heckman \cite{BeukersHeckman89} with parameters $\{m/9,4m/9,7m/9\}$ (or their negatives) and $\{0,0,0\}$.  The degree is $2$ because $Y_s$ is a torus of dimension $2$ and the convolution of two of the rank-one sheaves $\LL_j$ is placed in degree $1$ (so the triple one in degree $2$).  The cube of the monodromy at $\infty$ acts by $\iota_\ell(\zeta^3)^{\pm m}$ on all three eigenlines, because $4\equiv7\equiv1\pmod 3$; so $\ninf\equiv\pm m\pmod 3$.  The dimensions of the stalks of $j_*\VV$ are those of the invariants of the local monodromies: $1$ (one Jordan block), $2$ (a pseudo-reflection) and $0$ (no exponent at $\infty$ is $0$).
\end{proof}

\subsection{A smooth compactification}\label{sec:compactification}
Let $\bar\SSS=\PP^1\smallsetminus\{0,\infty\}$ with coordinate $s$, so that $\SSS=\bar\SSS\smallsetminus\{1\}$, and
let $\bar X=\{(y,s)\in\RR_{K_9/K}\A^1\times\bar\SSS:\ \Nm(1-y^9)=s\}$, so that $X=\bar X\times_{\bar\SSS}\SSS$.  In this
subsection all schemes are over $\OO[1/3]$.  The model constructed here extends $\VV$ to a lisse sheaf over $\OO[1/3\ell]$, which gives the behavior of $\VV_{s_0}$ at the primes not dividing $3\ell$ (\cref{sec:frey}), and it is the starting point for the crystallinity at $p$ (\cref{sec:crys}).

\begin{proposition}\label{prop:compactification}\leavevmode
\begin{enumerate}[(a)]
\item There are a projective morphism $\tilde f^\flat\colon\tilde X^\flat\to\bar\SSS$ with an action of $H$ over
$\bar\SSS$, and an $H$-equivariant open immersion $\bar X\hookrightarrow\tilde X^\flat$ over $\bar\SSS$ whose
complement $\tilde D^\flat$, with its reduced structure, is a divisor with strict normal crossings relative to
$\bar\SSS$; after base change to $\OO[1/3,\zeta_9]$ each component of $\tilde D^\flat$ is a disjoint union of copies of
$\PP^1_{\bar\SSS}$.  The morphism $\tilde f^\flat$ is smooth outside the section $\{y=0,\ s=1\}$.
\item Let $\tilde f\colon\tilde X\to\SSS$ and $\tilde D$ be the restrictions of $\tilde f^\flat$ and $\tilde D^\flat$ to
$\SSS$.  Then $\tilde f$ is smooth and projective, $X=\tilde X\smallsetminus\tilde D$, and
$H^*(\tilde D^\flat_s,\Ql)^{\chi^m}=0$ for every geometric point $s$ of $\bar\SSS$ of characteristic different
from~$\ell$.
\item Consequently $\VV\isom(\RR^2\tilde f_*\Ql)^{\chi^m}$ extends to a lisse sheaf on $\SSS$ over $\OO[1/3\ell]$, pure
of weight~$2$, and for every point $s$ of $\SSS$ its fiber is the $\chi^m$-part of $H^2$ of the smooth projective surface
$\tilde X_s$.
\end{enumerate}
\end{proposition}

\begin{proof}
\emph{Reduction to the split form.}  The ring $\OO'=\OO[1/3,\zeta_9]$ is finite étale and Galois over $\OO[1/3]$, with
group $\Gal(K_9/K)$, because $K_9/K$ is unramified outside~$3$.  Let $P=\RR_{\OO'/\OO[1/3]}\PP^1$, a smooth projective
$\OO[1/3]$-scheme containing $\RR_{\OO'/\OO[1/3]}\A^1$ and acted on by $H$.  The schemes below are obtained from
$\bar X\subset P\times\bar\SSS$ by taking a scheme-theoretic closure, then reduced loci where a morphism to $\bar\SSS$
is not smooth, and blow-ups along them.  These operations commute with the étale base change $\OO[1/3]\to\OO'$, and the
properties to be proved, other than projectivity (which holds by construction), are local for the étale topology; so we
may check them over $\OO'$.  Over $\OO'$, as in \cref{sec:surfaces}, $P\isom(\PP^1)^3$, $H\isom\mu_9^3$ acting
coordinatewise, and $\bar X\isom\{\prod_j(1-y_j^9)=s\}\subset\A^3\times\bar\SSS$.
In what follows $\{i,j,k\}=\{0,1,2\}$ and $x_j=1-y_j^9$.  For $\zeta,\zeta'\in\mu_9$ let
$L_{i,j,\zeta}=\{y_i=\infty,\ y_j=\zeta\}$, a line with coordinate $y_k$;
$P_{i;\zeta,\zeta'}=\{y_i=\infty,\ y_j=\zeta,\ y_k=\zeta'\}$ (with $j<k$); and
$Q_{j,\zeta}=\{y_j=\zeta,\ y_i=y_k=\infty\}$.

\emph{The closure.}  Write $y_j=Y_j/X_j$.  The hypersurface $\bar Z\subset(\PP^1)^3\times\bar\SSS$ defined by
$\prod_j(X_j^9-Y_j^9)=s\prod_jX_j^9$ is a relative effective Cartier divisor, flat over $\bar\SSS$, since its equation
is nonzero on every fiber.  It contains $\bar X$ as the open subscheme $X_0X_1X_2\ne0$, and it is the scheme-theoretic
closure of $\bar X$: it has no embedded points (a Cartier divisor in a regular scheme is Cohen--Macaulay), all its irreducible
components have the same dimension, and by (ii)--(iv) below its boundary part has dimension at most~$1$ in every fiber,
so it contains no component.  Let $s$ be a geometric point of
$\bar\SSS$; its characteristic is not~$3$, and $s\ne0$.
\begin{enumerate}[(i)]
\item \emph{No coordinate infinite.}  The partial derivatives of $\prod_j(1-y_j^9)$ are $-9y_j^8\prod_{i\ne j}x_i$.  At a
point with $\prod_jx_j=s\ne0$ they all vanish only if $y=0$, where the value is~$1$.  So $\bar Z_s\cap\A^3$ is smooth if
$s\ne1$, and for $s=1$ its only singular point is $y=0$.
\item \emph{Exactly one coordinate, $y_i$, infinite.}  With $w=1/y_i$ the equation is
$g:={(w^9-1)}x_jx_k-sw^9=0$.  On $w=0$ it reads $x_jx_k=0$, so $\bar Z_s\cap\{y_i=\infty\}$ is the union of the $18$ lines
$L_{i,j,\zeta}$ and $L_{i,k,\zeta}$, $\zeta\in\mu_9$.  At $w=0$ we have $\partial g/\partial w=0$,
$\partial g/\partial y_j=9y_j^8x_k$ and $\partial g/\partial y_k=9y_k^8x_j$.  If $x_j=0\ne x_k$, then $y_j\ne0$ and
$\partial g/\partial y_j\ne0$; similarly if $x_k=0\ne x_j$.  So the singular points with exactly one infinite
coordinate are the $3\cdot81=243$ points $P_{i;\zeta,\zeta'}$.
\item \emph{Exactly two coordinates, $y_i,y_k$, infinite.}  With $w_i=1/y_i$ and $w_k=1/y_k$ the equation is
$(w_i^9-1)(w_k^9-1)x_j=sw_i^9w_k^9$.  At $w_i=w_k=0$ it forces $y_j\in\mu_9$, and there
$\partial/\partial y_j=-9y_j^8(w_i^9-1)(w_k^9-1)\ne0$.  So $\bar Z_s$ is smooth at the $27$ points $Q_{j,\zeta}$, with local
coordinates $(w_i,w_k)$, in which $L_{i,j,\zeta}=\{w_i=0\}$ and $L_{k,j,\zeta}=\{w_k=0\}$ meet transversally.
\item \emph{All coordinates infinite.}  The equation $\prod_j(w_j^9-1)=s\prod_jw_j^9$ fails at $w=0$.
\end{enumerate}

\emph{The points $P_{i;\zeta,\zeta'}$.}  Near the section $P_{i;\zeta,\zeta'}\times\bar\SSS$ put
$x_j'=x_j(1-w^9)/(-s)$.  Then
\[
(w^9-1)\,x_jx_k-sw^9=s\,(x_j'x_k-w^9),
\]
and $(x_j',x_k,w)$, together with the projection to $\bar\SSS$, is an étale coordinate system on
$(\PP^1)^3\times\bar\SSS$ near this section: at the point, $dx_j'\equiv(9\zeta^8/s)\,dy_j$ modulo $dw$ and $ds$, and
$dx_k=-9\zeta'^8\,dy_k$.  So near the section $\bar Z$ is étale-locally isomorphic to $A_8\times\bar\SSS$, where
$A_8=\{uv=w^9\}\subset\A^3$, the section corresponding to the origin and the two lines $L_{i,j,\zeta}$,
$L_{i,k,\zeta'}$ through $P_{i;\zeta,\zeta'}$ to the axes $\{u=w=0\}$ and $\{v=w=0\}$ respectively.

\emph{The resolution.}  For $e\ge3$ the blow-up of the origin of $\{uv=w^e\}$ is covered by three charts:
$v=uv_1$, $w=uw_1$, with equation $v_1=u^{e-2}w_1^e$ (smooth); the symmetric chart $u=vu_1$, $w=vw_1$ (smooth); and
$u=wu_2$, $v=wv_2$, with equation $u_2v_2=w^{e-2}$.  The exceptional divisor consists of two copies of $\PP^1$
($\{u=v_1=0\}$ and $\{v=u_1=0\}$, completed in the third chart by $\{w=v_2=0\}$ and $\{w=u_2=0\}$ respectively), which meet only at the origin of the
third chart; this point is smooth if $e=3$ (and the two curves meet transversally there) and is a singular point of
the form $\{u_2v_2=w^{e-2}\}$ if $e\ge4$.  The strict transforms of the two axes meet the exceptional divisor
transversally, one in each copy, at the origin of the first chart and at the origin of the second chart, away from the third chart.  Let
$\tilde X^\flat$ be obtained from $\bar Z$ by blowing up four times the reduced locus of boundary points at which the
morphism to $\bar\SSS$ is not smooth (at the first step, the $243$ sections $P_{i;\zeta,\zeta'}\times\bar\SSS$; the
other non-smooth point, $y=0$ over $s=1$, is not on the boundary).  Since blowing up commutes with flat base change,
the computation with $e=9,7,5,3$ shows: $\tilde X^\flat$ is smooth over $\bar\SSS$ along the boundary; over each
section $P_{i;\zeta,\zeta'}\times\bar\SSS$ lies a chain $E_1\cup\dots\cup E_8$ of copies of $\PP^1_{\bar\SSS}$, with
$E_r\cap E_{r+1}$ a section and no other intersections between them; and the strict transforms of $L_{i,j,\zeta}$ and
$L_{i,k,\zeta'}$ are disjoint near the chain and meet $E_1$ and $E_8$ (in some order) transversally, in one section
each.  The morphism $\tilde f^\flat\colon\tilde X^\flat\to\bar\SSS$ is projective, as a composite of blow-ups and a
projective morphism, and by (i)--(iv) it is smooth outside the section $\{y=0,\ s=1\}$.  The group $H$ preserves $\bar Z$
and each blow-up center, so it acts on $\tilde X^\flat$ over $\bar\SSS$; the same holds for $\Gal(K_9/K)$, which is how
the construction descends to $\OO[1/3]$.

\emph{The boundary.}  $\tilde D^\flat$ is the union of the strict transforms of the $54$ lines $L_{i,j,\zeta}$ (each
isomorphic to $L_{i,j,\zeta}$, a smooth curve) and of the $243\cdot8$ curves $E_r$.  By (ii)--(iv), two distinct lines
$L_{i,j,\zeta}$ and $L_{i',j',\zeta'}$ meet only if $i'=i$ and $j'\ne j$ (at a point $P_{i;\cdot,\cdot}$, where the
resolution separates them) or $(i',j',\zeta')=(k,j,\zeta)$ (at $Q_{j,\zeta}$, transversally).  Hence the double points of each fiber
$\tilde D^\flat_s$ are the $27$ points $Q_{j,\zeta}$ and, on each of the $243$ chains, its $7$ inner double points and
its $2$ ends; there are no triple points.  The generator of $\Gal(K_9/K)$ maps $L_{i,j,\zeta}$ to a line
$L_{i+1,j+1,\zeta'}$ and the chain over $P_{i;\zeta,\zeta'}$ to the chain over a point whose infinite coordinate is
$y_{i+1}$; since two lines with the same difference $j-i$ never meet and chains over distinct points are disjoint, the
conjugates of a component are pairwise disjoint, so over $\OO[1/3]$ every component of $\tilde D^\flat$ is still smooth
over $\bar\SSS$.  This proves (a), and $X=\tilde X\smallsetminus\tilde D$.

\emph{The $\chi^m$-part of the cohomology of the boundary.}  Let $s$ be a geometric point of $\bar\SSS$ of
characteristic $\ne\ell$, and let $\mu_9^{(i)}$ be the factor of $\mu_9^3$ that multiplies $y_i$.  It fixes every point
of $L_{i,j,\zeta}$ and every point $P_{i;\zeta,\zeta'}$.  Near $P_{i;\zeta,\zeta'}$ it fixes $x_j'$ and $x_k$ and
multiplies $w$ by a ninth root of unity: an automorphism of $A_8$ that preserves both axes, hence each blow-up center,
each curve $E_r$ over $P_{i;\zeta,\zeta'}$ and each double point of that chain.  At $Q_{j,\zeta}$ it fixes the point and
maps both lines through it to themselves ($L_{i,j,\zeta}$ pointwise).  Hence every component $D_a$ of
$\tilde D^\flat_s$ is mapped to itself by a factor $\mu_9^{(i)}$ acting trivially on $H^*(D_a,\Ql)$ (pointwise on the
lines; on $E_r\isom\PP^1$ every automorphism acts trivially on cohomology), and every double point is fixed by a factor
$\mu_9^{(i)}$ that maps each of the two components through it to itself.  Order the components, and let
$j_a\colon D_a\to\tilde D^\flat_s$ and $j_{ab}\colon D_a\cap D_b\to\tilde D^\flat_s$ be the inclusions.  As there are no
triple points, the sequence of sheaves
\[
0\to\Ql\to\textstyle\bigoplus_aj_{a*}\Ql\to\bigoplus_{a<b}j_{ab*}\Ql\to0,\qquad (f_a)_a\mapsto(f_a-f_b)_{a<b},
\]
is exact; it is $H$-equivariant when $g\in H$ acts on the last term by pull-back followed by the sign $-1$ on the
summands where $g$ reverses the order of the two components.  Let $v=(v_a)_a$ lie in the $\chi^m$-part of
$\bigoplus_aH^q(D_a)$.  Given $a$, choose $g\in\mu_9^{(i)}$, with $\mu_9^{(i)}$ as above for $D_a$, such that
$\chi^m(g)\ne1$; this is possible because $\chi^m$ has order $9$ on each factor (\cref{lem:characters}).  The
$a$-component of $g^*v=\chi^m(g)\,v$ reads $v_a=\chi^m(g)\,v_a$, so $v_a=0$.  The same argument, with an element of a
factor that fixes the double point and preserves both components through it (so that no sign occurs), shows that the
$\chi^m$-part of $\bigoplus_{a<b}H^0(D_a\cap D_b)$ vanishes.  Taking $\chi^m$-parts is exact because $|H|$ is invertible
in $\Ql$, so the long exact cohomology sequence gives $H^*(\tilde D^\flat_s,\Ql)^{\chi^m}=0$.  This proves (b).

\emph{Consequences.}  For $s\in\SSS$ the $H$-equivariant exact sequence
$H^{q-1}(\tilde D_s)\to H^q_c(X_s)\to H^q(\tilde X_s)\to H^q(\tilde D_s)$ gives
$H^q_c(X_s)^{\chi^m}\isom H^q(\tilde X_s)^{\chi^m}$.  In families, over $\OO[1/3\ell]$, apply $\RR\tilde f_*$ to the
$H$-equivariant triangle $j_!\Ql\to\Ql\to i_*\Ql\to$ on $\tilde X$, where $j\colon X\to\tilde X$ and
$i\colon\tilde D\to\tilde X$ are the inclusions, and take $\chi^m$-parts.  By proper base change
\cite[Exposé~XII, Corollaire~5.2~(iii)]{SGA4} the stalks of $(\RR^q(\tilde fi)_*\Ql)^{\chi^m}$ are the groups
$H^q(\tilde D_s)^{\chi^m}=0$, so $(\RR^qf_!\Ql)^{\chi^m}\isom(\RR^q\tilde f_*\Ql)^{\chi^m}$.  Since $\tilde f$ is smooth
and proper, $\RR^q\tilde f_*\Ql$ is lisse with fibers $H^q(\tilde X_s)$ \cite[Exposé~XVI, Corollaire~2.2]{SGA4}, and so
is its direct summand, the $\chi^m$-part; at closed points its Frobenius eigenvalues have absolute value $|k(s)|^{q/2}$ in
every complex embedding \cite[Théorème~(1.6)]{DeligneWeilI}, $\tilde X_s$ being smooth and projective.
\end{proof}

\subsection{Hodge numbers and duality}

\begin{lemma}\label{lem:hodge}
For $s\in K\smallsetminus\{0,1\}$ and any embedding $K\hookrightarrow\C$, the Hodge numbers of $H^2(\tilde X_s(\C),\C)^{\chi^m}$ are $h^{2,0}=h^{1,1}=h^{0,2}=1$.
\end{lemma}

\begin{proof}
By \cref{prop:geometry} the local system $H^2(\tilde X_s)^{\chi^m}$, $s\in\SSS(\C)$, is the irreducible hypergeometric local system with parameters $\{m/9,4m/9,7m/9\}$ (or their negatives) and $\{0,0,0\}$, and the $\chi^m$-part of the variation of Hodge structure $\RR^2\tilde f_*\C$ is a polarizable variation of Hodge structure on it.  An irreducible local system underlies at most one such variation up to a shift of the Hodge filtration \cite[Proposition~1.13]{Deligne87}, and the Hodge numbers of the hypergeometric variations were computed by Fedorov \cite{Fedorov18} (see also Dettweiler--Sabbah \cite{DettweilerSabbah13}): when all the parameters at $0$ are equal, the Hodge numbers are all equal to $1$.  The weight is $2$, so they are $(1,1,1)$ in degrees $(2,0),(1,1),(0,2)$.
\end{proof}

\needspace{5\baselineskip}\begin{lemma}\label{lem:duality}
For $s\in\Q\smallsetminus\{0,1\}$ there is an isomorphism of $G_K$-representations $\VV_s^\cc\isom\VV_s^\vee\otimes\varepsilon_\ell^{-2}$.
\end{lemma}

\begin{proof}
Write $\VV_{\chi',s}$ for the $\chi'$-part.  Poincaré duality on the geometrically irreducible surface $\tilde X_s$ is a perfect $G_K$-equivariant pairing $H^2\times H^2\to H^4=\Ql(-2)$, invariant under $H$, which acts trivially on $H^4$.  Hence it pairs the $\chi'$-part with the $\chi''$-part trivially unless $\chi'\chi''=1$, and $\VV_{\chi^{-m},s}\isom\VV_{\chi^m,s}^\vee(-2)$.  On the other hand, because $s\in\Q$ and $K_9/\Q$ is Galois, $\tilde\cc$ induces an isomorphism of $K$-varieties between the conjugate $X_s\times_{K,\cc}K$ and $X_s$, compatible with the actions of $H$ through an automorphism $g\mapsto g^{\tilde\cc}$ of $H(\Kbar)=\mu_9(\Kbar)^3$ which permutes the three factors and acts on each by $\zeta\mapsto\tilde\cc(\zeta)$.  Transport of structure therefore identifies $\VV_{\chi^m,s}^\cc$ with the $\chi'$-part of $H^2_c(X_s)$, where $\chi'(g)=\chi^m(g^{\tilde\cc^{-1}})$.  The character $\chi'$ is fixed by $G_K$ (as $\tilde\cc$ normalizes $G_K$) and of order $9$ on each factor, as $\chi^m$ is, so $\chi'=\chi^n$ for some $n\in(\Z/9)^\times$ by \cref{lem:characters}.  On the diagonal, $\tilde\cc^{-1}(\zeta)=\zeta^{j}$ with $j\equiv-1\pmod 3$, so $\iota_\ell(\zeta)^{12n}=\chi'(\zeta,\zeta,\zeta)=\iota_\ell(\zeta)^{12mj}$ and $n\equiv -m\pmod 3$.  Hence $\chi'\in\{\chi^{-m},\chi^{-4m},\chi^{-7m}\}$ and $\VV_{\chi^m,s}^\cc\isom\VV_{\chi^{-m},s}$.  Combining the two isomorphisms gives the claim.
\end{proof}

\subsection{Integrality and traces}

\begin{lemma}\label{lem:traces-in-K}
Let $\q\nmid3\ell$ be a prime of $K$ with residue field $\F_q$, and let $s\in\F_{q^k}\smallsetminus\{0,1\}$.  Then $\tr(\Frob_\q^k\mid\VV_s)\in\OO$, and
\[
\tr(\Frob_\q^k\mid\VV_s)=\frac1{729}\sum_{g\in H(\Fbar_q)}\chi^m(g)^{-1}\,\#\{y\in X_s(\Fbar_q):\ (F^k\circ g)(y)=y\},
\]
where $F$ is the $q$-power Frobenius of $X\otimes\F_q$, which maps $X_s$ to $X_{s^q}$.
\end{lemma}

\begin{proof}
The formula is the Lefschetz trace formula for the twisted Frobenius $F^k\circ g$ (the Frobenius of a twisted form of $X_s$) combined with $H^i_c(X_s)^{\chi^m}=0$ for $i\ne2$.  The right side lies in $\Q(\zeta_9)$ (through $\iota_\ell$); the automorphism $\zeta_9\mapsto\zeta_9^n$ with $n\equiv1\pmod3$ carries it to the trace on the $\chi^{nm}$-part, which is the same trace, since $\sigma_0\otimes1$ commutes with Frobenius and carries the $\chi^m$-part to the $\chi^{4m}$-part.  So the trace lies in the fixed field $K$.  It is a sum of eigenvalues of Frobenius on $H^2_c(X_s)$, which are algebraic integers \cite[Exposé~XXI, 5.5.3]{SGA7II}.
\end{proof}

\subsection{The data at the primes above 7}\label{sec:data7}
The primes $\lp,\lpb$ above $7$ are inert in $K_9/K$ (since $7$ has order $3$ modulo $9$).  For $\q\in\{\lp,\lpb\}$ let $\mathfrak Q$ be the prime of $K_9$ above $\q$, which is unique; we identify its residue field $\OO_{K_9}/\mathfrak Q$, of order $7^3$, with $\F_{343}$.  Every prime of $\Kbar$ above $\q$ lies above $\mathfrak Q$; so $\F_{343}$ is the subfield of order $343$ of the residue field $\Fbar_7$ of the prime used to define $\Frob_\q$, and on $\mu_9\subset K_9$ the reduction at that prime is the reduction modulo $\mathfrak Q$.  As $7\nmid9$, reduction modulo $\mathfrak Q$ maps $\mu_9(K_9)$ isomorphically onto $\mu_9(\F_{343})$.  Let $\vartheta\colon\F_{343}^\times\to\mu_9(K_9)$ be the character with
\[
\vartheta(z)\equiv z^{(343-1)/9}=z^{38}\pmod{\mathfrak Q}
\]
(the ninth-power residue symbol modulo $\mathfrak Q$), extended by $\vartheta(0)=0$, and $\vartheta_6=\vartheta\circ\Nm_{\F_{7^6}/\F_{343}}$.

\begin{lemma}\label{lem:trace7}
Let $\q\in\{\lp,\lpb\}$.  There is $m'\in(\Z/9)^\times$, with $m'$ replaced by $-m'$ when $m$ is replaced by $-m$, such that for $s\in\F_{7^k}\smallsetminus\{0,1\}$:
\begin{align*}
\tr(\Frob_\q\mid\VV_s)&=\textstyle\sum_{x\in\F_{343},\ \Nm_{\F_{343}/\F_7}(x)=s}\vartheta(1-x)^{m'} &(k=1),\\
\tr(\Frob_\q^2\mid\VV_s)&=\textstyle\sum_{x\in\F_{7^6},\ \Nm_{\F_{7^6}/\F_{49}}(x)=s}\vartheta_6(1-x)^{m'} &(k=2),\\
\tr(\Frob_\q^3\mid\VV_s)&=\textstyle\sum_{x_0x_1x_2=s,\ x_j\in\F_{343}}\vartheta(1-x_0)^{m'}\vartheta(1-x_1)^{4m'}\vartheta(1-x_2)^{7m'} &(k=3).
\end{align*}
\end{lemma}

\begin{proof}
Over $\F_7$ the geometric Frobenius $F$ permutes the three coordinates cyclically, $F(y)_j=y_{j-1}^7$ (for a suitable numbering, since $\q$ is inert in $K_9$).  For $g\in\mu_9^3$ a fixed point of $F\circ g$ is determined by $y_0$, which satisfies $y_0=\delta(g)\,y_0^{343}$ with $\delta(g)=g_0^{343}g_1^{49}g_2^{7}$ (exponents modulo $9$: $g_0g_1^4g_2^7$), and the equation of $X_s$ becomes $\Nm_{\F_{343}/\F_7}(1-y_0^9)=s$ with $y_0^9\in\F_{343}$.  For $y_0^9=z\ne0$ the condition $y_0^{342}=\delta(g)^{-1}$ reads $z^{38}=\delta(g)^{-1}$ (as $342=9\cdot38$), and there are then $9$ choices of $y_0$; $y_0=0$ only occurs for $s=1$.  The $G_K$-fixed characters are the powers of $\delta$ (\cref{lem:characters}, in the numbering in which Frobenius shifts the coordinates), so averaging $\chi^m(g)^{-1}$ over the $81$ elements $g$ with given $\delta(g)$ in \cref{lem:traces-in-K} leaves $\sum_{z}\vartheta(z)^{m'}$ over $z=1-x$ with $\Nm(x)=s$.  The same computation with $F^2$ (which still permutes the coordinates cyclically) gives the second formula, and $F^3$ fixes the coordinates, which gives the triple sum.  Replacing $\chi^m$ by $\chi^{-m}$ replaces every exponent by its negative.
\end{proof}

We write $\omega$ also for the image of $\omega\in K$ in the coefficients, and let
\begin{align*}
P^{+}(T)&=T^3-(6-3\omega)T^2+(63+21\omega)T-343,\\
P^{-}(T)&=T^3-(9+3\omega)T^2+(42-21\omega)T-343=\overline{P^{+}}(T).
\end{align*}

\begin{proposition}\label{prop:frob7}
Let $P_\q(T)=\det(T-\Frob_\q\mid\VV_2)$ for $\q\in\{\lp,\lpb\}$, and for $\q\nmid 3\ell$ let $\Omega_\q(T)=\det(T-\Frob_\q\mid H^1_c(\SSS\otimes\Fbar_q,\VV))$.  Then:
\begin{enumerate}[(a)]
\item $(P_\lp,P_\lpb)=(P^+,P^-)$ for one of the two sheaves $\VV_\chi,\VV_{\chi^{-1}}$ and $(P^-,P^+)$ for the other;
\item $\Omega_\lp(T)$ and $\Omega_\lpb(T)$ are $(T-1)(T^2-2\omega T+49\omega^2)$ and $(T-1)(T^2-2\omega^2T+49\omega)$ in some order; in particular $T=1$ is a root of both, and the other roots have absolute value $7$.
\end{enumerate}
\end{proposition}

\begin{proof}
By the Grothendieck--Lefschetz formula and $H^0_c=H^2_c=0$ (\cref{prop:geometry}: $\VV$ is irreducible and nontrivial), $\tr(\Frob_\q^k\mid H^1_c)=-\sum_{s\in\F_{7^k}\smallsetminus\{0,1\}}\tr(\Frob^k_\q\mid\VV_s)$.  We evaluated the sums of \cref{lem:trace7} at both primes and for all six values of $m'$ (\cref{app:computations}, item~C1); every value lies in $\Z[\omega]$.  At $\lp$, the traces at $s=2$ over $\F_7,\F_{49},\F_{343}$ are $6-3\omega$, $-99-87\omega$, $-267-486\omega$ for the three values of $m'$ in one class modulo $3$ (the class $m'\equiv2\pmod3$, called class~I in item~C1), and their complex conjugates $9+3\omega$, $-12+87\omega$, $219+486\omega$ for the other class; Newton's identities turn them into $P^+$ and $P^-$ respectively.  Since $m'$ changes class with $m$, one sheaf has $P_\lp=P^+$ and the other $P_\lp=P^-$.  For $s\in\Q$, \cref{lem:duality} and purity give $\tr(\Frob_\lpb\mid\VV_s)=\tr(\Frob_\lp\mid\VV_s^\cc)=49\,\tr(\Frob_\lp^{-1}\mid\VV_s)=\overline{\tr(\Frob_\lp\mid\VV_s)}$, so $P_\lpb=\overline{P_\lp}$.  This proves (a).  The power sums of $\Frob_\q$ on $H^1_c$ are $(1+2\omega,\,95+94\omega,\,-285)$ or $(-1-2\omega,\,1-94\omega,\,-285)$, at both primes and for both classes, and Newton's identities give the two cubics in (b).  The roots of $T^2-2\omega T+49\omega^2$ are $\omega(1\pm4\sqrt{-3})$, of absolute value $7$, and those of the other quadratic are their conjugates.
\end{proof}