\section{Inertia at the prime above 3}\label{sec:inertia3}

\begin{theorem}\label{thm:inertia3}
Let $\ell\ne3$ be a prime and $s_0\in K_\p^\times$ with $v_3(s_0)>15/2$.  Then the semisimplification of $\VV_{s_0}$ as a representation of $G_{K_\p}$ is $1\oplus\varepsilon_\ell^{-1}\oplus\varepsilon_\ell^{-2}$.  In particular the inertia group $I_\p$ acts unipotently on $\VV_{s_0}$.
\end{theorem}

\emph{Overview.}  For a prime $\q\nmid3\ell$ of $K$ and $s_0$ with $v_\q(s_0)>0$, the analogue of \cref{thm:inertia3} is \cref{lem:tame}~(b): $I_\q$ acts on $\VV_{s_0}$ through powers of the local monodromy $\gamma_0$ at $s=0$, a single unipotent Jordan block. Its proof uses that $\VV$ is lisse on $\SSS$ over $\OO[1/3\ell]$ and tamely ramified along $0,1,\infty$, so that by Abhyankar's lemma the covers trivializing it near $s=0$ are Kummer covers of order prime to the residue characteristic. At $\p$ neither input is available: \cref{prop:compactification} gives a lisse sheaf only over $\OO[1/3\ell]$, and the Kummer covers $y^9=1-x$ from which $\VV$ is built have degree divisible by $3$. For the same reason we cannot use the log-smooth (toroidal) models that describe tame degenerations, and we know of no semistable model of the surfaces over the $3$-adic disc. We use instead one analytic fact, \cref{lem:radius}: over $\C_3$ the torsor $y^9=1-x$ is trivial on the disc $v_3(x)>5/2$, the disc of convergence of $(1-x)^{1/9}$. By \cref{lem:integral} and its proof, $\VV_s$ is $H^2_c$ of the torus $x_0x_1x_2=s$ with coefficients in $\LL_{e_0}(x_0)\otimes\LL_{e_1}(x_1)\otimes\LL_{e_2}(x_2)$. The three valuations $v_3(x_j)$ add up to $v_3(s)$, so if $v_3(s)>15/2$ then at every point of the torus some $v_3(x_j)$ exceeds $5/2$ and the $j$th Kummer factor is constant near it: $15/2=3\cdot5/2$ is exactly the condition for the three regions $\{v_3(x_j)>5/2\}$ to cover the torus. \Cref{lem:rank2,prop:unipotent} carry out this covering one convolution at a time, first for $\LL_{e_1}\star\LL_{e_2}$ on the discs $v_3(z)>d'$ with $d'>5=5/2+5/2$, then for $\VV$ on the discs $v_3(s)>d$ with $d>15/2=5+5/2$, with a Mayer--Vietoris sequence for each cover.  They show that every torsion sheaf $\VV_n$ of \cref{lem:integral} is unipotent of length at most $4$ on the punctured disc $D_{\C_3}$, where $D=\{v_3(s)>d\}\subset\SSS^{\mathrm{an}}$ for a fixed $d>15/2$.

The length bound does not depend on $n$, and this is what passes to $\Ql$-coefficients: for $x\in D(K_\p)$ and $\gamma_1,\dots,\gamma_4$ in the group $\Pi_D$ described below, $(\gamma_1-1)\cdots(\gamma_4-1)$ kills the stalk at $\bar x$ of every $\VV_n$, hence the lattice $\varprojlim_n\VV_{n,\bar x}$ (\cref{cor:unipotent-group}).  The value $4$ comes from the second Mayer--Vietoris sequence, which makes $\VV_n|_{D_{\C_3}}$ an extension of two sheaves, each unipotent of length at most $2$.  This value is not sharp on $\VV_{\bar x}$, since the product of any three factors $\gamma_i-1$ already kills $\VV_{\bar x}$ (\cref{prop:descent}~(a)). Unipotence at each level with lengths growing with $n$ would not suffice: a character with values in $1+\ell\Z_\ell$ and of infinite order is unipotent of length $\le n$ modulo $\ell^n$ for every $n$, but it is not unipotent. We use no description of the étale fundamental group of the punctured disc over $\C_3$ (which has covers of degree divisible by $3$ that are not Kummer covers), only the subgroup $\Pi_D$ of the geometric fundamental group of $\SSS$ that preserves the components over $D_{\C_3}$ of all finite étale covers (\cref{def:PiD}); by the above, it acts unipotently on $\VV_{\bar x}$. \Cref{sec:descent} shows that the decomposition group of a $K_\p$-point of the disc normalizes $\Pi_D$ and agrees, modulo $\Pi_D$, with the decomposition group of the tangential base point at $s=0$ (\cref{lem:paths}), which acts by conjugation on the inertia group $I_0\subset\Pi_D$ at $s=0$ through the cyclotomic character $\chi_{\mathrm{cyc}}\colon G_{K_\p}\to\hat\Z^\times$ (whose $\ell$-adic component is $\varepsilon_\ell$); $I_0$ acts on $\VV$ through the local monodromy $\gamma_0$. As $\gamma_0$ is a single Jordan block, the filtration of $\VV_{s_0}$ defined by $\Pi_D$ is a Galois-stable flag whose graded pieces carry characters $\beta_1,\beta_1\varepsilon_\ell^{-1},\beta_1\varepsilon_\ell^{-2}$ (\cref{prop:descent}). Geometry does not determine $\beta_1$ (\cref{rem:legendre-limit}): \cref{sec:limit} identifies it with the restriction of a global character of $K$ unramified outside $\p$, and the Frobenius data at the primes above $7$ show that this character is trivial.

\subsection{Analytic setting}\label{sec:analytic}
Let $\C_3$ be the completion of an algebraic closure $\overline K_\p$ of $K_\p$.  Analytic spaces are Berkovich spaces, and we use the étale cohomology of \cite{Berkovich93}; theorem numbers in brackets refer to that paper.  For a $K_\p$-analytic space $X$ and a complete field $k\supset K_\p$ we write $X_k=X\widehat\otimes_{K_\p}k$, and for a scheme $\mathcal X$ of finite type over $k$ we write $\mathcal X^{\mathrm{an}}$ for its analytification.  For a point $x$ of an analytic space and an analytic function $f$ we put $v_3(f(x))=-\log_3|f(x)|\in\R\cup\{\infty\}$.  On $\mathbb G_m^{\mathrm{an}}$ with coordinate $x$, and for $-\infty\le d_1<d_3\le\infty$, the open subset $\{d_1<v_3(x)<d_3\}$ is an \emph{open annulus} (a \emph{punctured open disc} if $d_3=\infty$), and the subsets $\{d_2\le v_3(x)<d_3\}$, $\{d_1<v_3(x)\le d_2\}$ with $d_1<d_2<d_3$ are \emph{half-open annuli}.  These sets do not contain the point $0$.  All of them are connected.

\emph{Coefficients.}  Let $E=\Q_\ell(\iota_\ell(\zeta_9))\subset\Ql$, let $\OO_E$ be its ring of integers, $\lambda$ a uniformizer and $\Lambda_n=\OO_E/\lambda^n$.  As $\ell\ne3$, the elements $9$ and $1-\zeta$ ($\zeta\in\mu_9(\OO_E)\smallsetminus\{1\}$) are units of $\OO_E$.  Sheaves are étale sheaves of $\Lambda_n$-modules.  We write $\LL_{j,n}$ for the Kummer sheaf $\LL_j$ defined before \cref{prop:geometry}, with coefficients $\Lambda_n$ and extended by zero at $x=1$.  Statements about $\Ql$-coefficients are deduced at the end from statements at every level $n$ (\cref{cor:unipotent-group}).

\emph{Facts from \cite{Berkovich93}.}  The torsion orders below are powers of $\ell$, prime to the residue \mbox{characteristic}~$3$.
\begin{enumerate}[(B1)]
\item \emph{Base change.} For a Hausdorff morphism $\varphi\colon Y\to X$ of $k$-analytic spaces, a sheaf $\mathcal F$ on $Y$ and a point $x\in X$, the stalk of $\RR^q\varphi_!\mathcal F$ at a geometric point over $x$ is $H^q_c(Y_{\bar x},\mathcal F)$, where $Y_{\bar x}$ is the fiber of $\varphi$ over $x$ base-changed to the completion of an algebraic closure of the completed residue field $\mathcal H(x)$ [Thm.~5.3.1]; and $\RR^q\varphi_!$ commutes with every base change $X'\to X$ [Thm.~7.7.1].  In particular, if $k$ is algebraically closed, $P$ and $B$ are Hausdorff $k$-analytic spaces, $\mathrm{pr}\colon P\times B\to B$ is the projection and $\mathcal G$ is a sheaf on $P$, then $\RR^q\mathrm{pr}_!(\mathrm{pr}_P^*\mathcal G)$ is the constant sheaf on $B$ with value $H^q_c(P,\mathcal G)$ (base change along $B\to\mathcal M(k)$).
\item \emph{Comparison.} For a compactifiable morphism $\varphi$ of $k$-schemes of finite type and a torsion sheaf $\mathcal F$, $(\RR^q\varphi_!\mathcal F)^{\mathrm{an}}\isom\RR^q\varphi^{\mathrm{an}}_!\mathcal F^{\mathrm{an}}$ [Thm.~7.1.1, Cor.~7.1.4].
\item \emph{Excision and Mayer--Vietoris.} If $T\subset X$ is open with closed complement $R$, there is a long exact sequence $\cdots\to H^q_c(T,\mathcal F)\to H^q_c(X,\mathcal F)\to H^q_c(R,\mathcal F)\to H^{q+1}_c(T,\mathcal F)\to\cdots$ [Prop.~5.2.6(ii)].  Relative to a Hausdorff morphism $\varphi$, the exact sequences $0\to j_!j^*\mathcal F\to\mathcal F\to i_*i^*\mathcal F\to0$ and, for an open cover $X=U_1\cup U_2$, $0\to j_{12!}j_{12}^*\mathcal F\to j_{1!}j_1^*\mathcal F\oplus j_{2!}j_2^*\mathcal F\to\mathcal F\to0$ give long exact sequences of the sheaves $\RR^q\varphi_!$ of the restrictions, since $\RR(\varphi j)_!=\RR\varphi_!\circ j_!$ for an open immersion $j$ and $i_*=i_!$ for a closed immersion $i$ [Cor.~5.2.4, Cor.~5.2.5].
\item \emph{Poincaré duality.} If $k$ is algebraically closed, $X$ is a separated smooth $k$-analytic curve and $\mathcal F$ is finite locally constant, then $H^q(X,\mathcal F^\vee(1))\isom H^{2-q}_c(X,\mathcal F)^\vee$ [Thm.~7.4.3].  The trace maps are compatible with composition [Thm.~7.2.1(b),(c)], so the duality is compatible with restriction from $X$ to an open subset on one side and extension by zero on the other.
\item \emph{Affinoid curves.} If $k$ is algebraically closed and $X$ is an affinoid domain in the analytification of a projective curve $\mathcal X$ whose complement is a disjoint union of open discs $D_1,\dots,D_r$, and $x_i\in D_i(k)$, then restriction gives $H^1(\mathcal X\smallsetminus\{x_1,\dots,x_r\},\Z/\ell^n)\isom H^1(X,\Z/\ell^n)$ [Thm.~6.3.9 and the proof of Thm.~6.4.1]; and $H^q(X,\Z/\ell^n)=0$ for $q\ge2$ for every one-dimensional $k$-affinoid space $X$ [Thm.~6.4.1(i)].  If a paracompact space $X$ is an increasing union of closed analytic domains $X_i$ and the images of $H^{q-1}(X_{i+1})$ and $H^{q-1}(X_{i+2})$ in $H^{q-1}(X_i)$ coincide for all $i$, then $H^q(X)=\varprojlim H^q(X_i)$ [Lemma~6.3.12].
\item \emph{Local structure.} Étale morphisms are quasifinite and flat by definition [Def.~3.3.4], hence open maps [Prop.~3.2.7]; finite morphisms are closed maps (they induce compact maps of the underlying spaces [p.~28, after Lemma~1.3.7], and compact maps are closed [p.~14]).  The analytification of an étale, resp.\ finite, resp.\ proper, morphism of schemes of finite type over $k$ is étale [Prop.~3.3.11], resp.\ finite, resp.\ proper [Prop.~2.6.9]; so the analytification of a proper $k$-scheme is compact [Example~1.5.3(iii)].  If $\varphi\colon Y\to X$ is étale, $y\in Y$, and $\mathcal H(\varphi(y))\to\mathcal H(y)$ is an isomorphism, then $\varphi$ maps an open neighborhood of $y$ isomorphically onto an open neighborhood of $\varphi(y)$ [Thm.~3.4.1, and the beginning of its proof: a morphism of $\mathrm{F\acute et}(X,x)$ that becomes an isomorphism in $\mathrm{F\acute et}(\mathcal H(x))$ is an isomorphism of germs].
\end{enumerate}

Besides (B1)--(B6), \cref{lem:locconst}~(b) uses that a morphism of sheaves is an isomorphism if it is one on all stalks [Cor.~4.2.3]; otherwise we use from \cite{Berkovich93} only definitions and elementary properties from its \S\S1--5.

\emph{Locally constant and unipotent sheaves.}  Let $X$ be a connected analytic space over an algebraically closed field.  A sheaf $\mathcal F$ on $X$ is \emph{locally constant} if every point has an étale neighborhood $U$ such that $\mathcal F|_U$ is isomorphic to the constant sheaf $M_U$ of a finitely generated $\Lambda_n$-module $M$, and \emph{constant} if $\mathcal F\isom M_X$.  It is \emph{unipotent of length $\le k$} if it is locally constant and has a filtration $0=\mathcal F_0\subset\mathcal F_1\subset\dots\subset\mathcal F_k=\mathcal F$ by locally constant subsheaves with constant quotients $\mathcal F_i/\mathcal F_{i-1}$.

\begin{lemma}\label{lem:locconst}
Let $X$ be a connected analytic space over an algebraically closed field.
\begin{enumerate}[(a)]
\item Kernels, images and cokernels of morphisms between locally constant sheaves are locally constant, and an extension of locally constant sheaves is locally constant.
\item A morphism between constant sheaves $M_X\to N_X$ is induced by a homomorphism $M\to N$.  A locally constant subsheaf of a constant sheaf is constant, and so is the quotient of a constant sheaf by a locally constant subsheaf.
\item Let $\mathcal F$ be unipotent of length $\le k$.  Every locally constant subsheaf of $\mathcal F$, and the quotient of $\mathcal F$ by every locally constant subsheaf, is unipotent of length $\le k$.  An extension of a sheaf unipotent of length $\le k'$ by $\mathcal F$ is unipotent of length $\le k+k'$.  The kernel and the cokernel of a morphism $\varphi\colon\mathcal F\to\mathcal G$, with $\mathcal G$ unipotent of length $\le k'$, are unipotent of lengths $\le k$ and $\le k'$.
\end{enumerate}
\end{lemma}

\begin{proof}
A section of a constant sheaf $S_U$ of finite sets is, étale-locally on $U$, a constant; since étale morphisms are open maps (B6), a section over $U$ defines a locally constant function on the topological space $|U|$, hence a constant one if $U$ is connected.  As $\mathrm{Hom}(M_U,N_U)$ is the set of sections of the constant sheaf with value $\mathrm{Hom}_{\Lambda_n}(M,N)$, a morphism $M_U\to N_U$ is étale-locally, and on connected $U$ globally, induced by a homomorphism $M\to N$; this proves the first assertion of (b), and the assertions of (a) about kernels, images and cokernels, which may be checked étale-locally.  For an extension $0\to\mathcal A\to\mathcal B\to\mathcal C\to0$ with $\mathcal A=A_U$ and $\mathcal C=C_U$ constant, choose a surjection $\Lambda_n^r\to C$ with kernel $C_0$; étale-locally the $r$ generators lift to sections of $\mathcal B$, giving $\psi\colon\Lambda^r_{n,U}\to\mathcal B$ over $\Lambda_{n,U}^r\to C_U$; then $\mathcal B$ is the quotient of $A_U\oplus\Lambda^r_{n,U}$ by the graph of the morphism $-\psi|_{C_{0,U}}\colon C_{0,U}\to A_U$, which is étale-locally induced by a homomorphism $C_0\to A$; so $\mathcal B$ is étale-locally constant.  For the second assertion of (b), let $\mathcal G\subset M_X$ be locally constant.  By the first assertion, every point has an étale neighborhood $U$ on which $\mathcal G|_U=(G_U)_U$ for a submodule $G_U\subset M$, so the stalk $\mathcal G_{\bar x}\subset M$ is a locally constant function of $x\in|X|$ (images of étale maps are open), hence constant, equal to $G$ say; and $\mathcal G=G_X$ because a subsheaf is determined by its stalks.  The quotient is $(M/G)_X$.  For (c), if $\mathcal F'\subset\mathcal F$ is locally constant, the subsheaves $\mathcal F'\cap\mathcal F_i=\ker(\mathcal F'\to\mathcal F/\mathcal F_i)$ are locally constant by (a), and $(\mathcal F'\cap\mathcal F_i)/(\mathcal F'\cap\mathcal F_{i-1})$ is a locally constant subsheaf of the constant sheaf $\mathcal F_i/\mathcal F_{i-1}$, hence constant by (b).  The quotient case is the same with the filtration $(\mathcal F_i+\mathcal F')/\mathcal F'$, whose graded pieces are quotients of $\mathcal F_i/\mathcal F_{i-1}$ by locally constant subsheaves.  Extensions: concatenate the filtrations.  Kernels and cokernels: $\ker\varphi\subset\mathcal F$ and $\mathrm{im}\,\varphi\subset\mathcal G$ are locally constant by (a), so by the first assertion of (c) $\ker\varphi$ is unipotent of length $\le k$, and $\mathcal G/\mathrm{im}\,\varphi$ of length $\le k'$.
\end{proof}

\begin{lemma}[radius]\label{lem:radius}
Let $k$ be a complete field containing $K_\p$ and $j\in\Z/9\smallsetminus\{0\}$.  Over the open disc $\{v_3(x)>5/2\}$ of $\mathbb A^{1,\mathrm{an}}_k$ the $\mu_9$-torsor $y^9=1-x$ has a section.  Hence, if $k$ is algebraically closed, $\LL_{j,n}$ is constant on $\{v_3(x)>5/2\}$, and for every analytic space $Y$ over $k$ and every analytic function $g$ on $Y$ with $v_3(g)>5/2$ everywhere on $Y$, the pull-back $\LL_{j,n}(g)$ is constant.
\end{lemma}

\begin{proof}
The coefficient of $x^i$ in the binomial series $f(x)=\sum_i\binom{1/9}{i}(-x)^i$ has valuation $v_3\big(\prod_{0\le t<i}(1-9t)/(9^i\,i!)\big)=-2i-v_3(i!)=-(5i-s_3(i))/2$, where $s_3(i)$ is the sum of the ternary digits of $i$ (Legendre).  On a closed disc $\{v_3(x)\ge r\}$ the $i$th term has valuation at least $(r-5/2)i$, so for $r>5/2$ the series converges there; it defines an analytic function on $\{v_3(x)>5/2\}$, with coefficients in $\Q$, and $f^9=1-x$ there (an identity of formal power series).  So $y=f(x)$ is a section of the torsor, and over an algebraically closed $k$ (where $\mu_9$ is constant) the torsor is trivial and its push-out $\LL_{j,n}$ is constant.  The last assertion follows by pull-back.
\end{proof}

\begin{lemma}[annuli]\label{lem:annuli}
Let $k$ be an algebraically closed complete field containing $K_\p$, ${-\infty\le d_1<d_3\le\infty}$, and $A=\{d_1<v_3(x)<d_3\}\subset\mathbb G_m^{\mathrm{an}}$.
\begin{enumerate}[(a)]
\item $H^0(A,\Lambda_n)=\Lambda_n$; $H^1(A,\Lambda_n(1))$ is free of rank one over $\Lambda_n$, generated by the Kummer class of the coordinate $x$; and $H^q(A,\Lambda_n)=0$ for $q\ge2$.  Restriction to an open sub-annulus $A'=\{d'_1<v_3(x)<d'_3\}$, $d_1\le d_1'<d_3'\le d_3$, is an isomorphism on $H^q$ for every $q$.
\item $H^1_c(A,\Lambda_n)\isom\Lambda_n$, $H^2_c(A,\Lambda_n)\isom\Lambda_n(-1)$, and $H^q_c(A,\Lambda_n)=0$ for $q\ne1,2$.  Extension by zero from $A'$ to $A$ is an isomorphism on $H^q_c$ for every $q$.
\item For $d_1<d_2<d_3$ the half-open annuli $Z=\{d_2\le v_3(x)<d_3\}$ and $Z'={\{d_1<v_3(x)\le d_2\}}$ satisfy $H^q_c(Z,\mathcal F)=H^q_c(Z',\mathcal F)=0$ for all $q$ and every sheaf $\mathcal F$ on $A$ that is unipotent of some length.
\end{enumerate}
\end{lemma}

\begin{proof}
It suffices to take coefficients $\Z/\ell^n$: as $\ell\ne3$, $E$ is unramified over $\Q_\ell$, so $\lambda^n\OO_E=\ell^n\OO_E$ and $\Lambda_n$ is free of finite rank over $\Z/\ell^n$.
(a) Write $A$ as the union of the increasing sequence of closed annuli $B_i=\{a_i\le v_3(x)\le b_i\}$ with $a_i,b_i\in\Q$, $a_i\downarrow d_1$, $b_i\uparrow d_3$.  The complement of $B_i$ in $\PP^{1,\mathrm{an}}$ is the disjoint union of the open discs $\{v_3(x)>b_i\}\cup\{0\}$ and $\{v_3(x)<a_i\}\cup\{\infty\}$, so by (B5), with $x_1=0$ and $x_2=\infty$, restriction gives $H^1(\mathbb G_m,\mu_{\ell^n})\isom H^1(B_i,\mu_{\ell^n})$, and $H^q(B_i,\mu_{\ell^n})=0$ for $q\ge2$.  By Kummer theory on $\mathbb G_m$ over $k$, $H^1(\mathbb G_m,\mu_{\ell^n})=k[x^{\pm1}]^\times/(k[x^{\pm1}]^\times)^{\ell^n}=\Z/\ell^n$, generated by the class of $x$.  The $B_i$ are connected, so $H^0(B_i)=\Z/\ell^n$, and the restriction maps $H^q(B_{i+1})\to H^q(B_i)$ are isomorphisms for $q=0,1$ (the Kummer class of $x$ restricts to the Kummer class of $x$).  The space $A$ is paracompact (locally compact, Hausdorff and a countable union of compact sets), and the hypothesis of the last statement of (B5) holds in every degree $q\ge1$, so $H^q(A)=\varprojlim_iH^q(B_i)$; with $H^0(A)=\Z/\ell^n$ ($A$ is connected) this gives the first three assertions.  The last one follows, as $A'$ has the same description and restriction carries constants to constants and the Kummer class of $x$ to that of $x$.
(b) By Poincaré duality (B4), $H^q_c(A,\Z/\ell^n)$ is dual to $H^{2-q}(A,\mu_{\ell^n})$, and extension by zero is dual to restriction; so (b) follows from (a).
(c) $Z$ is closed in $A$ with complement $\{d_1<v_3(x)<d_2\}$, and extension by zero from the complement is an isomorphism by (b); by excision (B3), $H^*_c(Z,\Z/\ell^n)=0$.  The same holds for $Z'$, with complement $\{d_2<v_3(x)<d_3\}$.  Every finitely generated $\Lambda_n$-module has a composition series with quotients $\Lambda_1$, a finite-dimensional $\F_\ell$-vector space, so $H^*_c(Z,M_Z)=0$ for every constant sheaf, and by induction on the length $H^*_c(Z,\mathcal F|_Z)=0$ for every unipotent $\mathcal F$.  Likewise for $Z'$.
\end{proof}

\begin{lemma}\label{lem:exterior}
Let $k$ be as in \cref{lem:annuli}, $j\in\Z/9\smallsetminus\{0\}$, $r>5/2$ and $U_r=\{v_3(x)<r\}\subset\mathbb G_m^{\mathrm{an}}$.  Then $H^1_c(U_r,\LL_{j,n})$ is free of rank one over $\Lambda_n$, and $H^q_c(U_r,\LL_{j,n})=0$ for $q\ne1$.
\end{lemma}

\begin{proof}
The complement of $U_r$ in $\mathbb G_m^{\mathrm{an}}$ is the closed half-open annulus $Z=\{r\le v_3(x)<\infty\}$, contained in $\{v_3(x)>5/2\}$, where $\LL_{j,n}$ is constant (\cref{lem:radius}).  So $H^*_c(Z,\LL_{j,n})=0$ by \cref{lem:annuli}~(c), and excision (B3) and comparison (B2) give $H^q_c(U_r,\LL_{j,n})\isom H^q_c(\mathbb G_m^{\mathrm{an}},\LL_{j,n})\isom H^q_c(\mathbb G_m,\LL_{j,n})=H^q_c(\mathbb G_m\smallsetminus\{1\},\LL_{j,n})$.  On the curve $\mathbb G_m\smallsetminus\{1\}=\PP^1\smallsetminus\{0,1,\infty\}$ the sheaf $\LL_{j,n}$ is lisse of rank one, and its local monodromy at $x=1$ is multiplication by $\zeta^j$ for a primitive $9$th root of unity $\zeta$; since $\zeta^j-1\in\Lambda_n^\times$, the coinvariants $H^2_c$ vanish, and $H^0_c=0$ as the curve is affine.  The same holds with coefficients $\Lambda_1$.  The complex $P=\RR\Gamma_c(\mathbb G_m\smallsetminus\{1\},\LL_{j,n})$ is perfect and $P\otimes^{\mathbb L}_{\Lambda_n}\Lambda_1$ computes the cohomology with coefficients $\Lambda_1$, which is therefore concentrated in degree $1$; so $P$ is a free $\Lambda_n$-module placed in degree $1$, of rank $-\chi_c=1$ (the curve has Euler characteristic $-1$ and the sheaf is tame of rank one).
\end{proof}

\begin{lemma}[integral convolution]\label{lem:integral}
Let $(e_0,e_1,e_2)=m(1,7,4)$.  Over an algebraically closed field of characteristic $0$:
\begin{enumerate}[(a)]
\item for every $z\in\mathbb G_m$, $H^q_c(\mathbb G_m,\LL_{e_1,n}(x)\otimes\LL_{e_2,n}(z/x))$ vanishes for $q\ne1$ and is free of rank $2$ for $z\ne1$, of rank $1$ for $z=1$; so, with $\mathrm{pr}(x,z)=z$, the sheaf $\MM_n=\RR^1\mathrm{pr}_!\big(\LL_{e_1,n}(x)\otimes\LL_{e_2,n}(z/x)\big)$ on $\mathbb G_m$ is the only nonzero direct image;
\item the sheaf $\VV_n=(\RR^2f_!\Lambda_n)^{\chi^m}$ on $\SSS$ is defined and lisse over $K$, $\big(\varprojlim_n\VV_n\big)\otimes_{\OO_E}\Ql=\VV$, and $\VV_n\isom\RR^1\mathrm{pr}_!\big(\LL_{e_0,n}(x)\otimes\MM_n(s/x)\big)|_{\SSS}$, with $\mathrm{pr}(x,s)=s$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) For $z\ne1$ the sheaf $\LL_{e_1,n}(x)\otimes\LL_{e_2,n}(z/x)$ is the extension by zero of a lisse sheaf of rank one on $\mathbb G_m\smallsetminus\{1,z\}$; at $x=1$ only the first factor is ramified, with monodromy $\zeta^{e_1}$; for $z=1$ it is the extension by zero of a lisse sheaf on $\mathbb G_m\smallsetminus\{1\}$ with monodromy $\zeta^{e_1+e_2}=\zeta^{2m}$ at $x=1$.  In both cases the monodromy at $1$ is a nontrivial root of unity $\zeta'$ with $\zeta'-1\in\Lambda_n^\times$, and the argument of \cref{lem:exterior} applies (Euler characteristics $-2$ and $-1$).  The second assertion follows by proper base change.
(b) The boundary argument in the proof of \cref{prop:compactification} works with coefficients $\Lambda_n$ ($9$ is invertible in $\Lambda_n$, so the $\chi^m$-projector is defined and exact), and gives $(\RR^2f_!\Lambda_n)^{\chi^m}\isom(\RR^2\tilde f_*\Lambda_n)^{\chi^m}$, which is lisse by smooth and proper base change.  The second assertion is the definition of $\VV$ (\cref{def:V}).  For the third, the computation in the proof of \cref{prop:geometry}~(a) also works with coefficients $\Lambda_n$: on the $\chi^m$-isotypic part, the direct image of $\Lambda_n$ along the finite map $y\mapsto x=(1-y_j^9)_j$ is $\boxtimes_j\LL_{e_j,n}(x_j)$, extended by zero where some $x_j=1$ (there the stabilizer contains a factor $\mu_9$ on which $\chi^m$ is nontrivial).  Hence $(\RR f_!\Lambda_n)^{\chi^m}$ is the direct image of $\boxtimes_j\LL_{e_j,n}(x_j)$ along $(x_0,x_1,x_2)\mapsto x_0x_1x_2$, which by proper base change and (a) is $\RR\mathrm{pr}_!\big(\LL_{e_0,n}(x)\otimes\MM_n(s/x)\big)[-1]$.  Taking $\RR^2$ gives the claim.
\end{proof}

\subsection{Unipotence on the disc}\label{sec:unipotent}
Let $k$ be an algebraically closed complete field containing $K_\p$ (below, $k=\C_3$, or the completion of an algebraic closure of the residue field of a point of a disc over $\C_3$).  Let $(e_0,e_1,e_2)=m(1,7,4)$ and let $\MM_n$, $\VV_n$ be as in \cref{lem:integral}; we use the same letters for their analytifications, which by (B2) are given by the same formulas with the analytic $\mathbb G_m$.

\begin{lemma}\label{lem:rank2}
Let $d'>5$ and $D'=\{v_3(z)>d'\}\subset\mathbb G_m^{\mathrm{an}}$ over $k$.  For every $n$, $\MM_n|_{D'}$ has a subsheaf $\mathcal C_1$ such that $\mathcal C_1$ and $\MM_n|_{D'}/\mathcal C_1$ are constant, free of rank one over $\Lambda_n$.  In particular $\MM_n|_{D'}$ is unipotent of length $\le2$.
\end{lemma}

\begin{proof}
Let $\mathcal U=\mathbb G_m^{\mathrm{an}}\times D'$ with coordinates $(x,z)$, $\mathrm{pr}(x,z)=z$, and $\mathcal F=\LL_{e_1,n}(x)\otimes\LL_{e_2,n}(z/x)$.  By \cref{lem:integral}~(a), (B1) and (B2), $\MM_n|_{D'}=\RR^1\mathrm{pr}_!\mathcal F$ and $\RR^q\mathrm{pr}_!\mathcal F=0$ for $q\ne1$.  As $5/2<d'-5/2$, the open sets $\mathcal U_1=\{v_3(x)>5/2\}$ and $\mathcal U_2=\{v_3(x)<d'-5/2\}$ cover $\mathcal U$, and $\mathcal U_{12}=\mathcal U_1\cap\mathcal U_2=A\times D'$ with $A=\{5/2<v_3(x)<d'-5/2\}$.  Write $\mathcal F_\ast$ for the restriction of $\mathcal F$ to $\mathcal U_\ast$.
\begin{itemize}
\item On $\mathcal U_2=U\times D'$, $U=\{v_3(x)<d'-5/2\}$, we have $v_3(z/x)>d'-(d'-5/2)=5/2$, so $\LL_{e_2,n}(z/x)$ is constant there (\cref{lem:radius}) and $\mathcal F_2\isom\mathrm{pr}_x^*(\LL_{e_1,n}|_U)$.  By (B1), $\RR^q\mathrm{pr}_!\mathcal F_2$ is constant with value $H^q_c(U,\LL_{e_1,n})$, free of rank one for $q=1$ and zero otherwise (\cref{lem:exterior}, $r=d'-5/2$).
\item On $\mathcal U_{12}$ both factors are constant, and $\RR^q\mathrm{pr}_!\mathcal F_{12}$ is constant with value $H^q_c(A,\Lambda_n)$: $\Lambda_n$ for $q=1$, $\Lambda_n(-1)$ for $q=2$, zero otherwise (\cref{lem:annuli}~(b)).
\item On $\mathcal U_1$, $\LL_{e_1,n}(x)$ is constant (\cref{lem:radius}), so $\mathcal F_1\isom\LL_{e_2,n}(z/x)$.  The automorphism $(x,z)\mapsto(x',z)=(z/x,z)$ of $\mathcal U$ over $D'$ carries $\mathcal U_1$ onto $\mathcal W=\{v_3(x')<v_3(z)-5/2\}$ and $\mathcal F_1$ onto $\mathrm{pr}_{x'}^*\LL_{e_2,n}$.  The open subset $\mathcal W'=\{v_3(x')<d'-5/2\}\times D'$ lies in $\mathcal W$, because $v_3(z)>d'$.  The fiber of the closed complement $\mathcal Z=\mathcal W\smallsetminus\mathcal W'$ over a point $z$ of $D'$ is, over the completion $k_z$ of an algebraic closure of $\mathcal H(z)$, the half-open annulus $\{d'-5/2\le v_3(x')<v_3(z)-5/2\}$, which lies in $\{v_3(x')>5/2\}$ since $d'>5$; there $\LL_{e_2,n}$ is constant.  By \cref{lem:annuli}~(c) over $k_z$ and (B1), $\RR\mathrm{pr}_!$ of the restriction to $\mathcal Z$ vanishes, so by excision (B3) and (B1) $\RR^q\mathrm{pr}_!\mathcal F_1\isom\RR^q\mathrm{pr}_!(\mathrm{pr}_{x'}^*\LL_{e_2,n}|_{\mathcal W'})$ is constant with value $H^q_c(\{v_3(x')<d'-5/2\},\LL_{e_2,n})$, free of rank one for $q=1$ and zero otherwise (\cref{lem:exterior}).
\end{itemize}
The Mayer--Vietoris sequence (B3) for $\mathcal U=\mathcal U_1\cup\mathcal U_2$ reads
\[
0=\RR^0\mathrm{pr}_!\mathcal F\to\RR^1\mathrm{pr}_!\mathcal F_{12}\xrightarrow{\ \alpha\ }\RR^1\mathrm{pr}_!\mathcal F_1\oplus\RR^1\mathrm{pr}_!\mathcal F_2\to\MM_n|_{D'}\to\RR^2\mathrm{pr}_!\mathcal F_{12}\to0 .
\]
All terms other than $\MM_n|_{D'}$ are constant, so $\MM_n|_{D'}$ is an extension of constant sheaves and is locally constant (\cref{lem:locconst}~(a)).  So $\mathcal C_1=\mathrm{coker}\,\alpha$ is constant (\cref{lem:locconst}~(b)), and $\MM_n|_{D'}/\mathcal C_1\isom\RR^2\mathrm{pr}_!\mathcal F_{12}$ is constant, free of rank one.  On stalks $\alpha$ is an injective map $\Lambda_n\to\Lambda_n^2$; the image of $1$ has a unit coordinate (otherwise it would be killed by $\lambda^{n-1}$), so $\mathrm{coker}\,\alpha$ is free of rank one.
\end{proof}

\begin{proposition}\label{prop:unipotent}
Let $d>15/2$ and $D=\{v_3(s)>d\}\subset\SSS^{\mathrm{an}}$ over $K_\p$, so that $D_k={\{v_3(s)>d\}}\subset\mathbb G_m^{\mathrm{an}}$ over $k$.  For every $n$, $\VV_n|_{D_k}$ is unipotent of length $\le4$.
\end{proposition}

\begin{proof}
Choose $d'$ with $5<d'<d-5/2$.  Let $\mathcal U=\mathbb G_m^{\mathrm{an}}\times D_k$ with coordinates $(x,s)$, $\mathrm{pr}(x,s)=s$, and $\mathcal F=\LL_{e_0,n}(x)\otimes\MM_n(s/x)$, so that $\VV_n|_{D_k}=\RR^1\mathrm{pr}_!\mathcal F$ (\cref{lem:integral}~(b), (B1), (B2)).  As $d-d'>5/2$, the open sets $\mathcal U_1=\{v_3(x)>5/2\}$ and $\mathcal U_2=\{v_3(x)<d-d'\}$ cover $\mathcal U$, and $\mathcal U_{12}=A'\times D_k$ with $A'=\{5/2<v_3(x)<d-d'\}$.
\begin{itemize}
\item On $\mathcal U_2=U\times D_k$, $U=\{v_3(x)<d-d'\}$, we have $v_3(s/x)>d'$, so $\phi(x,s)=s/x$ maps $\mathcal U_2$ into $D'=\{v_3(z)>d'\}$, and by \cref{lem:rank2} $\MM_n(s/x)|_{\mathcal U_2}=\phi^*(\MM_n|_{D'})$ is an extension of a constant sheaf with value $C_2$ by a constant sheaf with value $C_1$, where $C_1$ and $C_2$, free of rank one over $\Lambda_n$, are the values of the constant sheaves $\mathcal C_1$ and $\MM_n|_{D'}/\mathcal C_1$.  So $\mathcal F_2$ is an extension of $\mathrm{pr}_x^*(\LL_{e_0,n}|_U)\otimes C_2$ by $\mathrm{pr}_x^*(\LL_{e_0,n}|_U)\otimes C_1$.  By (B1) the $\RR^q\mathrm{pr}_!$ of these two sheaves are constant, so by the long exact sequence $\RR^q\mathrm{pr}_!\mathcal F_2$ is an extension of the kernel of a morphism between constant sheaves by the cokernel of another, hence unipotent of length $\le2$ (\cref{lem:locconst}).
\item On $\mathcal U_{12}$, $\LL_{e_0,n}(x)$ is constant and $\MM_n(s/x)$ is an extension of constant sheaves as above, so $\RR^q\mathrm{pr}_!\mathcal F_{12}$ is unipotent of length $\le2$ in the same way.
\item On $\mathcal U_1$, $\LL_{e_0,n}(x)$ is constant and $\mathcal F_1\isom\MM_n(s/x)$.  The automorphism $(x,s)\mapsto(z,s)=(s/x,s)$ carries $\mathcal U_1$ onto $\mathcal W=\{v_3(z)<v_3(s)-5/2\}$ and $\mathcal F_1$ onto $\mathrm{pr}_z^*\MM_n$.  The open subset $\mathcal W'=\{v_3(z)<d-5/2\}\times D_k$ lies in $\mathcal W$.  The fiber of $\mathcal Z=\mathcal W\smallsetminus\mathcal W'$ over a point $s$ of $D_k$ is the half-open annulus $\{d-5/2\le v_3(z)<v_3(s)-5/2\}$ over $k_s$, contained in the open annulus $\{d'<v_3(z)<v_3(s)-5/2\}\subset D'_{k_s}$ (as $d-5/2>d'$), on which $\MM_n$ is unipotent by \cref{lem:rank2} over $k_s$.  By \cref{lem:annuli}~(c) and (B1), $\RR\mathrm{pr}_!$ of the restriction to $\mathcal Z$ vanishes, so by excision $\RR^q\mathrm{pr}_!\mathcal F_1\isom\RR^q\mathrm{pr}_!(\mathrm{pr}_z^*\MM_n|_{\mathcal W'})$, which is constant by (B1).
\end{itemize}
In the Mayer--Vietoris sequence (B3), $\RR^1\mathrm{pr}_!\mathcal F$ is an extension of $\ker\big(\RR^2\mathrm{pr}_!\mathcal F_{12}\to\RR^2\mathrm{pr}_!\mathcal F_1\oplus\RR^2\mathrm{pr}_!\mathcal F_2\big)$ by $\mathrm{coker}\big(\RR^1\mathrm{pr}_!\mathcal F_{12}\to\RR^1\mathrm{pr}_!\mathcal F_1\oplus\RR^1\mathrm{pr}_!\mathcal F_2\big)$.  The sheaves $\RR^q\mathrm{pr}_!\mathcal F_1\oplus\RR^q\mathrm{pr}_!\mathcal F_2$ and $\RR^q\mathrm{pr}_!\mathcal F_{12}$ are unipotent of length $\le2$, so by \cref{lem:locconst}~(c) the kernel and the cokernel are unipotent of length $\le2$, and $\VV_n|_{D_k}$ is unipotent of length $\le4$.
\end{proof}

\subsection{A substitute for the fundamental group of the disc}\label{sec:descent}
\Cref{prop:unipotent} is a statement about sheaves on the punctured disc $D_{\C_3}$, on which $G_{K_\p}$ acts only through the homeomorphisms $\tilde\sigma=\mathrm{id}\otimes\sigma$ defined below.  This subsection turns it into a statement about the representations $\VV_{s_0}$ of $G_{K_\p}$.  For this we need a group that acts on the fibers of $\VV$, is normalized by the decomposition groups of the $K_\p$-points of the disc, and contains, up to conjugation, the inertia group at $s=0$.  Rather than a fundamental group of the punctured disc, we use the étale fundamental group of $\SSS_{K_\p}$, which contains these decomposition groups and admits the tangential base point at $s=0$, and in it a closed subgroup $\Pi_D$ of the geometric fundamental group: the elements that, on every finite étale cover, preserve the connected component over the disc attached to each point of the fiber (\cref{def:PiD}); it is the image of the profinite fundamental group of $D_{\C_3}$, but we do not need that group (see the paragraph after \cref{cor:unipotent-group}).  It acts on the fibers of the $\VV_n$ unipotently, with length at most $4$ for every $n$, because the $\VV_n$ are unipotent of length at most $4$ on the disc (\cref{prop:unipotent,lem:dictionary,cor:unipotent-group}).  As $D$ is defined over $K_\p$, Galois permutes these components, so the image of the section of $G_{K_\p}$ at a $K_\p$-point of the disc normalizes $\Pi_D$.  As every component over the disc reaches $s=0$, a suitable path from that point (\cref{lem:paths}~(a)) to the tangential base point at $s=0$ conjugates $\Pi_D$ to the corresponding group there, which contains the local inertia group $I_0$, and carries one section of $G_{K_\p}$ to the other modulo $\Pi_D$ (\cref{lem:components,lem:paths}).  This carries the action of $G_{K_\p}$ on $I_0$, through the cyclotomic character, to $\VV_{s_0}$ (\cref{prop:descent}).

Fix $d>15/2$ and $D=\{v_3(s)>d\}\subset\SSS^{\mathrm{an}}$ over $K_\p$.  For a geometric point $\bar x$ of $\SSS_{K_\p}$ let $F_{\bar x}$ be the fiber functor on finite étale covers of $\SSS_{K_\p}$, $\pi_1(\bar x)=\pi_1(F_{\bar x})=\mathrm{Aut}(F_{\bar x})$ the étale fundamental group, and $\pi_1^{\mathrm{geo}}(\bar x)=\pi_1^{\mathrm{geo}}(F_{\bar x})$ the kernel of $\pi_1(\bar x)\to G_{K_\p}$, that is $\pi_1(\SSS_{\overline K_\p},\bar x)$ \cite[Exposé~V, \S\S4--7, and Exposé~IX, Théorème~6.1]{SGA1}.  The two kinds of base points give the following fiber functors.
\begin{itemize}
\item For $x\in D(K_\p)$, $F_{\bar x}(Y)$ is the set of $\overline K_\p$-points of $Y$ above $x$; $G_{K_\p}$ acts on it by conjugating coordinates, and this defines a section $s_x\colon G_{K_\p}\to\pi_1(\bar x)$.
\item The tangential base point $\vec t=\partial/\partial s$ at $s=0$ \cite[\S15]{Deligne89}.  Let $\Omega=\bigcup_{e\ge1}\overline K_\p((s^{1/e}))$, with roots chosen so that $(s^{1/ee'})^{e'}=s^{1/e}$; it is algebraically closed (Puiseux).  Then $F_{\vec t}(Y)=\mathrm{Hom}_{\SSS}(\mathrm{Spec}\,\Omega,Y)$.  The group $G_{K_\p}$ acts on $\Omega$ through the coefficients, fixing every $s^{1/e}$, which defines a section $s_{\vec t}$; the inertia group $I_0=\mathrm{Gal}(\Omega/\overline K_\p((s)))\isom\hat\Z(1)$ lies in $\pi_1^{\mathrm{geo}}(\vec t)$, and $s_{\vec t}(\sigma)\,\theta\,s_{\vec t}(\sigma)^{-1}=\theta^{\chi_{\mathrm{cyc}}(\sigma)}$ for $\theta\in I_0$, where $\chi_{\mathrm{cyc}}$ is the cyclotomic character, because $\sigma$ fixes $s^{1/e}$ and acts on the $e$th roots of unity.  (Points act on rings of functions on the left: $(\gamma y)^\#=\gamma\circ y^\#$.)  If $Y$ is defined over $K$, every point of $F_{\vec t}(Y)$ is defined over $\Omega_K=\bigcup_e\Kbar((s^{1/e}))\subset\Omega$: the points of $Y$ over $\Omega_K$ form a subset of $F_{\vec t}(Y)$, and both sets have $\deg Y$ elements, since $\Omega_K$ is algebraically closed as well (Puiseux).  So the fiber $\VV_{\vec t}$ of \cref{sec:limit}, restricted to $G_{K_\p}$, is the fiber of $\VV$ at $F_{\vec t}$ with $G_{K_\p}$ acting through $s_{\vec t}$, and $I_0$ acts on it through the local monodromy $\gamma_0$ at $s=0$.  This is the fiber functor at the tangent vector $\partial/\partial s$ of \cite[\S15]{Deligne89}: there a cover $R$ gives the cover $R_{(s)}=\coprod_yT_y^*$ of the punctured tangent line at $0$, mapping to it by the principal parts $z\mapsto a_yz^{e_y}$ of $R\to\PP^1$ at the points $y$ above $0$ \cite[15.14--15.15, 15.18, 15.20--15.21]{Deligne89}; here $z$ is a local coordinate at $y$, defined over the field of definition of $y$, with $s=a_yz^{e_y}+\cdots$.  A point of $R$ over $\Omega$ specializing to $y$ has a Puiseux expansion $z=c\,s^{1/e_y}+\cdots$ with $a_yc^{e_y}=1$, and sending it to the point $c$ of $T_y^*$ is a bijection from $F_{\vec t}(R)$ onto the fiber of $R_{(s)}$ over $\partial/\partial s$, compatible with $G_{K_\p}$ (which acts on the coefficients) and with $I_0$ (which multiplies $s^{1/e_y}$, hence $c$, by roots of unity).
\end{itemize}
Fix $b\in D(K_\p)$.  For an open normal subgroup $N$ of $\pi_1(\bar b)$ let $Y_N\to\SSS_{K_\p}$ be the corresponding connected Galois cover, $G_N$ its automorphism group, $\bar Y_N$ the normalization of $\PP^1_{K_\p}$ in $Y_N$, and $Y_{N,D}\subset(Y_N\otimes_{K_\p}\C_3)^{\mathrm{an}}$ the preimage of $D_{\C_3}$, a finite étale cover of $D_{\C_3}$.  The group $G_N$ acts on $(\bar Y_N\otimes\C_3)^{\mathrm{an}}$ through the first factor, and $\sigma\in G_{K_\p}$ through $\tilde\sigma=\mathrm{id}\otimes\sigma$, a homeomorphism which commutes with $G_N$, preserves $Y_{N,D}$, and on $\C_3$-points is the conjugation of coordinates.

\begin{lemma}\label{lem:components}\leavevmode
\begin{enumerate}[(a)]
\item $Y_{N,D}$ has finitely many connected components; each is open and closed and maps onto $D_{\C_3}$.  The groups $G_N$ and $G_{K_\p}$ (through $\tilde\sigma$) permute them.
\item Every point $z\in\bar Y_N(\overline K_\p)$ above $s=0$ has a basis of open neighborhoods $U$ in $(\bar Y_N\otimes\C_3)^{\mathrm{an}}$ with $U\smallsetminus\{z\}$ connected and contained in $Y_{N,D}$.  Let $c_N(z)$ be the component of $Y_{N,D}$ containing these sets.  Then $c_N(gz)=g\,c_N(z)$ for $g\in G_N$, $c_N(\sigma z)=\tilde\sigma\,c_N(z)$ for $\sigma\in G_{K_\p}$, and every component of $Y_{N,D}$ is $c_N(z)$ for some $z$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) The cover $Y_{N,D}\to D_{\C_3}$ is finite étale of some degree $r$, as $D_{\C_3}$ is connected.  If $Y_{N,D}$ is a disjoint union of nonempty open subsets, each of them is finite étale over $D_{\C_3}$, so its image is open and closed (B6), hence equal to $D_{\C_3}$, and the degrees add up to $r$.  So there is a decomposition into at most $r$ nonempty open and closed subsets that cannot be refined, and these are the connected components.  The last assertion holds because $g$ and $\tilde\sigma$ are homeomorphisms preserving $Y_{N,D}$.
(b) The curve $\bar Y_N\otimes\C_3$ is smooth.  Let $t$ be a uniformizer at $z$; on an affine open neighborhood $\mathcal Z$ of $z$, $t$ is regular, étale as a map $\mathcal Z\to\mathbb A^1$, and vanishes only at $z$.  By (B6), $t$ maps an open neighborhood of $z$ in $\mathcal Z^{\mathrm{an}}$ isomorphically onto an open neighborhood of $0$, so $z$ has a basis of open neighborhoods $U$ isomorphic to discs $\{|t|<\rho\}$, with $U\smallsetminus\{z\}\isom\{0<|t|<\rho\}$ connected.  For small $\rho$, $U$ meets the fiber above $0$ only in $z$ and maps into $D_{\C_3}\cup\{0\}$, so $U\smallsetminus\{z\}\subset Y_{N,D}$.  Two such neighborhoods contain a third, so $c_N(z)$ is well defined, and the equivariance holds because $g$ and $\tilde\sigma$ are homeomorphisms carrying $z$ to $gz$, $\sigma z$ and preserving $Y_{N,D}$.  Let $T$ be a component.  If no point $z$ above $0$ lay in the closure of $T$ in the compact space $(\bar Y_N\otimes\C_3)^{\mathrm{an}}$, each such $z$ would have an open neighborhood disjoint from $T$, the complement of their union would be a compact set containing $T$ whose image in $\PP^{1,\mathrm{an}}$ misses $0$, and $T$ would not meet the preimage of some punctured disc $\{0<|s|<\rho\}$, contradicting (a).  So some $z$ lies in the closure of $T$; then $U\smallsetminus\{z\}$ meets $T$ for every $U$ as above, and being connected it lies in $T$.
\end{proof}

For $x\in D(K_\p)$ and $y\in F_{\bar x}(Y_N)$, let $c(y)$ be the component of $Y_{N,D}$ containing $y$.  For $y\in F_{\vec t}(Y_N)$, the morphism $\mathrm{Spec}\,\Omega\to Y_N\to\bar Y_N$ extends, by the valuative criterion, to the valuation ring $\bigcup_e\overline K_\p[[s^{1/e}]]$ of $\Omega$; let $\mathrm{sp}(y)\in\bar Y_N(\overline K_\p)$ be the image of its closed point, a point above $0$, and put $c(y)=c_N(\mathrm{sp}(y))$.  In both cases $c(gy)=g\,c(y)$ for $g\in G_N$ and
\begin{equation}\label{eq:c-sigma}
c\big(s_F(\sigma)y\big)=\tilde\sigma\,c(y)\qquad(\sigma\in G_{K_\p}),
\end{equation}
where $F$ is the fiber functor and $s_F$ its section: for $F_{\bar x}$, $s_x(\sigma)y$ is the conjugate point $\tilde\sigma(y)$; for $F_{\vec t}$, $\mathrm{sp}(s_{\vec t}(\sigma)y)=\sigma(\mathrm{sp}(y))$ and \cref{lem:components}~(b) applies.  For $\theta\in I_0$, $\mathrm{sp}(\theta y)=\mathrm{sp}(y)$, since $\theta$ acts trivially on the residue field $\overline K_\p$.

\needspace{5\baselineskip}\begin{definition}\label{def:PiD}
For $F=F_{\bar x}$ ($x\in D(K_\p)$) or $F=F_{\vec t}$, let $\Pi_D(F)$ be the closed subgroup of $\pi_1^{\mathrm{geo}}(F)$ of the elements $\gamma$ with $c(\gamma y)=c(y)$ for every $N$ and every $y\in F(Y_N)$.
\end{definition}

So $I_0\subset\Pi_D(F_{\vec t})$.  For $x\in D(K_\p)$, if $\gamma\in\pi_1^{\mathrm{geo}}(\bar x)$ satisfies $c(\gamma y_N)=c(y_N)$ for one point $y_N\in F_{\bar x}(Y_N)$, then it does for all, because $\gamma$ commutes with $G_N$ and $G_N$ acts transitively on $F_{\bar x}(Y_N)$.

\begin{lemma}\label{lem:dictionary}
Let $x\in D(K_\p)$, let $\mathcal F$ be a lisse sheaf of $\Lambda_n$-modules on $\SSS_{K_\p}$, $T=\mathcal F_{\bar x}$, and $\mathcal G'\subset\mathcal G$ locally constant subsheaves of the analytification of $\mathcal F$ restricted to $D_{\C_3}$.  Then $\mathcal G_{\bar x}$ and $\mathcal G'_{\bar x}$ are $\Pi_D(F_{\bar x})$-stable submodules of $T$, and if $\mathcal G/\mathcal G'$ is constant then $\Pi_D(F_{\bar x})$ acts trivially on $\mathcal G_{\bar x}/\mathcal G'_{\bar x}$.  Hence, if $\mathcal F|_{D_{\C_3}}$ is unipotent of length $\le k$, then $(\gamma_1-1)\cdots(\gamma_k-1)T=0$ for all $\gamma_1,\dots,\gamma_k\in\Pi_D(F_{\bar x})$.
\end{lemma}

\begin{proof}
Choose $N$ such that $\mathcal F$ becomes constant on the connected scheme $Y_N$ (the $Y_N$ are cofinal among finite étale covers), a point $y\in F_{\bar x}(Y_N)$, and let $C=c(y)$.  For $y'\in F_{\bar x}(Y_N)$ the stalk of the pull-back of $\mathcal F$ at $y'$ is $T$; for a section $f$ we write $f(y')\in T$ for its value there.  A section $f$ of $\mathcal F$ over the connected space $C$ is the restriction of a section over $Y_N$ (the pull-back of $\mathcal F$ to $Y_N$ is the constant sheaf $T$, and $Y_N$ and $C$ are connected, so restriction from $Y_N$ to $C$ is the identity of $T$), that is, of a morphism $\varphi\colon Y_N\to\mathcal E$ to the finite étale cover $\mathcal E$ of $\SSS_{K_\p}$ that represents $\mathcal F$; so $f(y')=F_{\bar x}(\varphi)(y')$, and naturality of $\gamma\in\pi_1(\bar x)$ gives $f(\gamma y)=\gamma f(y)$ whenever $\gamma y\in C$, in particular for $\gamma\in\Pi_D(F_{\bar x})$.  Now let $\mathcal G$ be a locally constant subsheaf.  On the connected space $C$ it is a locally constant subsheaf of the constant sheaf $\mathcal F|_C$, hence constant (\cref{lem:locconst}~(b)); so every element of $\mathcal G_{\bar x}=\mathcal G_y$ is $f(y)$ for a section $f$ of $\mathcal G$ over $C$, and then $\gamma f(y)=f(\gamma y)\in\mathcal G_{\gamma y}=\mathcal G_{\bar x}$.  So $\mathcal G_{\bar x}$ is $\Pi_D(F_{\bar x})$-stable, and likewise $\mathcal G'_{\bar x}$.  If $\mathcal G/\mathcal G'$ is constant, every element $q$ of its stalk at $\bar x$ is the value of a section over $D_{\C_3}$.  On $C$ this section lifts to a section $f$ of $\mathcal G$ (both $\mathcal G|_C$ and $(\mathcal G/\mathcal G')|_C$ are constant on the connected space $C$), and the images of $f(y)$ and of $f(\gamma y)=\gamma f(y)$ in $(\mathcal G/\mathcal G')_{\bar x}$ are both $q$.  So $\gamma$ acts trivially on $\mathcal G_{\bar x}/\mathcal G'_{\bar x}$.  The last assertion follows by applying this to the filtration.
\end{proof}

\begin{corollary}\label{cor:unipotent-group}
For $x\in D(K_\p)$ and $\gamma_1,\dots,\gamma_4\in\Pi_D(F_{\bar x})$, $(\gamma_1-1)\cdots(\gamma_4-1)$ is zero~on~$\VV_{\bar x}$.
\end{corollary}

\begin{proof}
By \cref{prop:unipotent} with $k=\C_3$ and \cref{lem:dictionary}, the product kills $\VV_{n,\bar x}$ for every $n$, hence $\varprojlim_n\VV_{n,\bar x}$ and $\VV_{\bar x}=(\varprojlim_n\VV_{n,\bar x})\otimes_{\OO_E}\Ql$ (\cref{lem:integral}~(b)).
\end{proof}

\emph{$\Pi_D$ and fundamental groups of the disc.}  Let $x\in D(K_\p)$.  The finite étale covers of the connected analytic space $D_{\C_3}$ have a profinite fundamental group $\pi_1^{\mathrm{alg}}(D_{\C_3},\bar x)$.  De Jong's étale fundamental group $\pi_1(D_{\C_3},\bar x)$ classifies a larger class of covers and maps to it with dense image \cite[\S2, Theorem~2.10]{deJong95}.  Pulling finite étale covers of $\SSS_{K_\p}$ back to $D_{\C_3}$ gives a homomorphism from $\pi_1^{\mathrm{alg}}(D_{\C_3},\bar x)$ to $\pi_1(\bar x)$, and its image is $\Pi_D(F_{\bar x})$.  Indeed, an automorphism of a fiber functor preserves the fiber of each connected component of a cover.  So the image preserves $c$, and hence lies in $\Pi_D(F_{\bar x})$ by ($\ast$) and ($\ast\ast$) in the proof of \cref{lem:paths}.  Conversely, $\pi_1^{\mathrm{alg}}(D_{\C_3},\bar x)$ acts transitively on the fiber of each connected cover \cite[Theorem~2.10~(i)]{deJong95}, so for every $N$ and $y\in F_{\bar x}(Y_N)$ the orbit of $y$ under the image is $\{y'\in F_{\bar x}(Y_N):c(y')=c(y)\}$, which contains $\gamma y$ for every $\gamma\in\Pi_D(F_{\bar x})$.  An element of $\pi_1(\bar x)$ commutes with $G_N$, so it is determined on $F_{\bar x}(Y_N)$ by the image of $y$; hence, for each $N$, $\gamma$ agrees on $F_{\bar x}(Y_N)$ with some element of the image; as the image is compact, hence closed, $\gamma$ lies in it.  So $\Pi_D(F_{\bar x})$ is also the closure of the image of $\pi_1(D_{\C_3},\bar x)$.  \Cref{lem:dictionary} is then the case we need of the correspondence between locally constant sheaves of finite sets on $D_{\C_3}$ and finite sets with a continuous action of $\pi_1^{\mathrm{alg}}(D_{\C_3},\bar x)$, a correspondence that follows from \cite[Lemma~2.3, Theorem~2.10]{deJong95}.  By contrast, we know of no published form of \cref{lem:paths}.  Finally, $\Pi_D$ is not generated by conjugates of the inertia group at $s=0$, which act trivially on the fibers of the covers that are étale over $s=0$.  For an integer $j>d-1/2$, the cover $w^{3^j}=1-s$ of $\SSS_{K_\p}$ is étale over $s=0$, but it has no section over $D_{\C_3}$: a section would extend over $s=0$ and be a root of unity times the binomial series of $(1-s)^{1/3^j}$, which converges only on $v_3(s)>j+1/2$ (as in the proof of \cref{lem:radius}).  So $\Pi_D(F_{\bar x})$ fixes no point of the fiber of this cover at $\bar x$.

\begin{lemma}[paths inside $D$]\label{lem:paths}
Let $F,F'$ be fiber functors of the two kinds above.
\begin{enumerate}[(a)]
\item There is an isomorphism of fiber functors $\delta\colon F\Rightarrow F'$ with $c\circ\delta=c$ on every $F(Y_N)$.  For every such $\delta$, $\delta^{-1}\Pi_D(F')\delta=\Pi_D(F)$.
\item For every $\sigma\in G_{K_\p}$, $s_F(\sigma)$ normalizes $\Pi_D(F)$, and $\delta^{-1}s_{F'}(\sigma)\delta\in\Pi_D(F)\,s_F(\sigma)$.
\item If $F'=F_{\vec t}$, then $\delta^{-1}I_0\,\delta\subset\Pi_D(F)$.
\end{enumerate}
\end{lemma}

\begin{proof}
We first describe $\pi_1^{\mathrm{geo}}$ through the covers $\SSS_L=\SSS\otimes_{K_\p}L$, $L/K_\p$ finite Galois.  Such a cover is connected and Galois over $\SSS_{K_\p}$ with group $\mathrm{Gal}(L/K_\p)$, so it is one of the $Y_N$.  For both kinds of $F$, $F(\SSS_L)=\mathrm{Hom}_{K_\p}(L,\overline K_\p)$ (for $F_{\vec t}$, $\mathrm{Hom}_{K_\p}(L,\Omega)=\mathrm{Hom}_{K_\p}(L,\overline K_\p)$, because $\overline K_\p$ is algebraically closed in each field $\overline K_\p((s^{1/e}))$), and $s_F(\sigma)$ acts on it by $\tau\mapsto\sigma\circ\tau$.  The projection $\pi_1(F)\to G_{K_\p}$ of \cite[Exposé~IX, Théorème~6.1]{SGA1} is induced by $\SSS_{K_\p}\to\mathrm{Spec}\,K_\p$, that is, by the action of $\pi_1(F)$ on the sets $F(\SSS_L)$; so
\begin{equation*}
\text{($\ast$)}\qquad \gamma\in\pi_1(F)\ \text{lies in}\ \pi_1^{\mathrm{geo}}(F)\iff\gamma\ \text{acts trivially on every}\ F(\SSS_L).
\end{equation*}
Moreover $L\otimes_{K_\p}\C_3=\prod_\tau\C_3$, the product over $\tau\in\mathrm{Hom}_{K_\p}(L,\overline K_\p)$, so the preimage of $D_{\C_3}$ in $(\SSS_L\otimes_{K_\p}\C_3)^{\mathrm{an}}$ is the disjoint union of copies $D_\tau$ of $D_{\C_3}$, and $c(\tau)=D_\tau$ for $\tau\in F(\SSS_L)$: for $F=F_{\bar x}$ the point $(x,\tau)$ lies in $D_\tau$; for $F=F_{\vec t}$, the normalization of $\PP^1_{K_\p}$ in $\SSS_L$ is $\PP^1_L$, and $\mathrm{sp}(\tau)$ is the point $0$ of the copy of $\PP^1_{\C_3}$ indexed by $\tau$, whose punctured neighborhoods lie in $D_\tau$.  In particular
\begin{equation*}
\text{($\ast\ast$)}\qquad c\ \text{is injective on}\ F(\SSS_L).
\end{equation*}
Together, ($\ast$) and ($\ast\ast$) show that an element $\gamma\in\pi_1(F)$ with $c(\gamma y)=c(y)$ for every $N$ and every $y\in F(Y_N)$ fixes every $F(\SSS_L)$ pointwise, so it is geometric and lies in $\Pi_D(F)$; this is how the two sections are compared in (b).
(a) Fix compatible points $y_N\in F(Y_N)$.  An isomorphism $\delta$ is the same as a compatible family of points $y'_N\in F'(Y_N)$, with $\delta(gy_N)=gy'_N$ \cite[Exposé~V, \S\S4--7]{SGA1}; since $c$ is $G_N$-equivariant, $c\circ\delta=c$ means $c(y'_N)=c(y_N)$ for all $N$.  The set $P_N$ of $y'\in F'(Y_N)$ with $c(y')=c(y_N)$ is finite and nonempty: for $F'=F_{\bar x'}$ because the component $c(y_N)$ maps onto $D_{\C_3}\ni x'$ (\cref{lem:components}~(a)); for $F'=F_{\vec t}$ because $c(y_N)=c_N(z)$ for some $z$ above $0$ (\cref{lem:components}~(b)), and some $y'\in F_{\vec t}(Y_N)$ has $\mathrm{sp}(y')=z$ (embed the completion of the function field of $\bar Y_N\otimes\overline K_\p$ at $z$, a field $\overline K_\p((u))$ with $u^e\in s\cdot\overline K_\p[[u]]^\times$, into $\Omega$ over $\overline K_\p((s))$).  For $N'\subset N$ the map $Y_{N'}\to Y_N$ carries $P_{N'}$ into $P_N$, as it carries components into components and commutes with $\mathrm{sp}$.  So $\varprojlim P_N\ne\emptyset$, which gives $\delta$.  For the second assertion: by $c\circ\delta=c$ and the bijectivity of $\delta$ on each $F(Y_N)$, $\gamma'\in\pi_1(F')$ preserves $c$ if and only if $\delta^{-1}\gamma'\delta$ does, and by ($\ast$) $\gamma'\in\pi_1^{\mathrm{geo}}(F')$ if and only if $\delta^{-1}\gamma'\delta\in\pi_1^{\mathrm{geo}}(F)$.
(b) For $\gamma\in\Pi_D(F)$ and $y\in F(Y_N)$, by \eqref{eq:c-sigma}, $c(s_F(\sigma)\gamma s_F(\sigma)^{-1}y)=\tilde\sigma\,c(\gamma s_F(\sigma)^{-1}y)=\tilde\sigma\,c(s_F(\sigma)^{-1}y)=c(y)$; and $s_F(\sigma)\gamma s_F(\sigma)^{-1}$ lies in the normal subgroup $\pi_1^{\mathrm{geo}}(F)$.  So $s_F(\sigma)$ normalizes $\Pi_D(F)$.  Next let $h=\delta^{-1}s_{F'}(\sigma)\delta\,s_F(\sigma)^{-1}\in\pi_1(F)$.  For every $N$ and $y\in F(Y_N)$, using $c\circ\delta^{-1}=c$ on $F'(Y_N)$, then \eqref{eq:c-sigma} for $F'$, then $c\circ\delta=c$, then \eqref{eq:c-sigma} for $F$,
\[
c(hy)=c\big(s_{F'}(\sigma)\,\delta s_F(\sigma)^{-1}y\big)=\tilde\sigma\,c\big(\delta s_F(\sigma)^{-1}y\big)=\tilde\sigma\,c\big(s_F(\sigma)^{-1}y\big)=c(y).
\]
For $y\in F(\SSS_L)$ this gives $hy=y$ by ($\ast\ast$).  So $h$ acts trivially on every $F(\SSS_L)$, hence $h\in\pi_1^{\mathrm{geo}}(F)$ by ($\ast$), and $h\in\Pi_D(F)$.
(c) This follows from $I_0\subset\Pi_D(F_{\vec t})$ and (a).
\end{proof}

\needspace{5\baselineskip}\begin{proposition}\label{prop:descent}
Let $x\in D(K_\p)$, write $V=\VV_{\bar x}$ and $\Pi_D=\Pi_D(F_{\bar x})$, and for a subspace $W\subset V$ let $IW$ be the span of the $(\gamma-1)w$ with $\gamma\in\Pi_D$ and $w\in W$; put $I^0V=V$ and $I^{j+1}V=I(I^jV)$.
\begin{enumerate}[(a)]
\item $V\supsetneq IV\supsetneq I^2V\supsetneq I^3V=0$, the quotients are lines, and $I^2V=V^{\Pi_D}$.
\item The $I^jV$ are stable under $s_x(G_{K_\p})$, which acts on $I^2V$, $IV/I^2V$ and $V/IV$ by characters $\beta_1$, $\beta_1\varepsilon_\ell^{-1}$ and $\beta_1\varepsilon_\ell^{-2}$.  So the semisimplification of $\VV_x$ as a $G_{K_\p}$-representation is $\beta_1\oplus\beta_1\varepsilon_\ell^{-1}\oplus\beta_1\varepsilon_\ell^{-2}$.
\item The character $\beta_1$ of $G_{K_\p}$ does not depend on $x\in D(K_\p)$.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) By \cref{cor:unipotent-group}, $I^4V=0$; hence $IW\ne W$ for every nonzero $\Pi_D$-stable subspace $W$, and the chain $I^jV$ decreases strictly until it reaches $0$.  Let $\delta\colon F_{\bar x}\Rightarrow F_{\vec t}$ be as in \cref{lem:paths} and $\gamma'=\delta^{-1}\theta_0\delta$ for a topological generator $\theta_0$ of $I_0$.  Then $\gamma'\in\Pi_D$, and through $\delta$ it acts on $V$ as $\gamma_0$ acts on $\VV_{\vec t}$: a single unipotent Jordan block of length $3$ (\cref{prop:geometry}~(c)).  So $0\ne(\gamma'-1)^2V\subset I^2V$, and as $\dim V=3$ the dimensions are $3,2,1,0$.  Finally $I^2V\subset V^{\Pi_D}\subset\ker(\gamma'-1)$, and the last space is a line.
(b) By \cref{lem:paths}~(b), $s_x(\sigma)$ normalizes $\Pi_D$, so $s_x(\sigma)I^jV=I^jV$ by induction on $j$, as $g(\gamma-1)w=(g\gamma g^{-1}-1)gw$ for $g=s_x(\sigma)$.  Let $b_j(\sigma)$ be the scalar by which $s_x(\sigma)$ acts on the line $\mathrm{gr}^j=I^jV/I^{j+1}V$; on $\mathrm{gr}^j$ the group $\Pi_D$ acts trivially.  By \cref{lem:paths}~(b),~(c), $\delta^{-1}s_{\vec t}(\sigma)\delta=h\,s_x(\sigma)$ with $h\in\Pi_D$, and applying the isomorphism $g\mapsto\delta^{-1}g\delta$ from $\pi_1(F_{\vec t})$ onto $\pi_1(F_{\bar x})$ to the relation $s_{\vec t}(\sigma)\,\theta_0\,s_{\vec t}(\sigma)^{-1}=\theta_0^{\chi_{\mathrm{cyc}}(\sigma)}$ (from the description of $F_{\vec t}$), with $\delta^{-1}\theta_0\delta=\gamma'$, gives $(hs_x(\sigma))\gamma'(hs_x(\sigma))^{-1}=\gamma'^{\chi_{\mathrm{cyc}}(\sigma)}$.  The image of $I_0$ in $\mathrm{GL}(V)$ is topologically generated by the unipotent element $\gamma'$, so it is the group $\{\exp(aN):a\in\Z_\ell\}$, $N=\log\gamma'$, through which $I_0\isom\hat\Z(1)$ acts via its quotient $\Z_\ell(1)$; hence $\gamma'^{\chi_{\mathrm{cyc}}(\sigma)}=\exp(\varepsilon_\ell(\sigma)N)=1+\varepsilon_\ell(\sigma){(\gamma'-1)}+{(\gamma'-1)^2}(\cdots)$.  The map $\mathrm{gr}^j\to\mathrm{gr}^{j+1}$ induced by $\gamma'-1$ is nonzero for $j=0,1$, since the composite of the two is induced by $(\gamma'-1)^2$, which is nonzero on $V$ and kills $IV$.  Comparing the maps $\mathrm{gr}^j\to\mathrm{gr}^{j+1}$ induced by both sides of $(hs_x(\sigma))(\gamma'-1)(hs_x(\sigma))^{-1}=\gamma'^{\chi_{\mathrm{cyc}}(\sigma)}-1$, on which $h$ acts trivially, gives $b_{j+1}(\sigma)b_j(\sigma)^{-1}=\varepsilon_\ell(\sigma)$.  With $\beta_1=b_2$ this is (b).
(c) Let $x'\in D(K_\p)$ and $\delta\colon F_{\bar x}\Rightarrow F_{\bar x'}$ as in \cref{lem:paths}.  Then $\delta^{-1}$ carries $\VV_{\bar x'}^{\Pi_D(F_{\bar x'})}$ onto $V^{\Pi_D}$ and $s_{x'}(\sigma)$ to $h\,s_x(\sigma)$ with $h\in\Pi_D$, which acts on $V^{\Pi_D}$ as $s_x(\sigma)$ does.
\end{proof}

Since $\varepsilon_\ell$ is unramified at $\p$, $I_\p$ acts on the semisimplification of $\VV_{s_0}$ through the single character $\beta_1|_{I_\p}$, for every $s_0\in K_\p^\times$ with $v_3(s_0)>15/2$.  \Cref{thm:inertia3} is therefore equivalent to $\beta_1=1$.

\subsection{The limit at \texorpdfstring{$s=0$}{s=0}}\label{sec:limit}
Let $\VV_{\vec t}$ be the fiber of $\VV$ at the tangential base point $\vec t=\partial/\partial s$ at $s=0$, defined over $K$ \cite[\S15]{Deligne89}, with its actions of $G_K$ and of the local inertia group $I_0\isom\hat\Z(1)$ at $0$, through the local monodromy $\gamma_0$.  Let $\beta^{\lim}\colon G_K\to\Ql^\times$ be the character by which $G_K$ acts on the line $\VV_{\vec t}^{I_0}=\ker(\gamma_0-1)$.

\needspace{4\baselineskip}\begin{lemma}\label{lem:beta-lim}
The character $\beta^{\lim}$ has the following properties.
\begin{enumerate}[(a)]
\item $\beta_1=\beta^{\lim}|_{G_{K_\p}}$.
\item For every prime $\q\nmid3\ell$ of $K$, $\beta^{\lim}$ is unramified at $\q$ and $\beta^{\lim}(\Frob_\q)=\alpha_0(\q)$, the eigenvalue of $\Frob_\q$ on the stalk at $0$ of $j_*\VV$ over $\Fbar_q$.
\item $\alpha_0(\q)$ is the unique root of weight $0$ of the polynomial $\Omega_\q(T)$ of \cref{prop:frob7}, and $\alpha_0(\q)\in\mu_6$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) Let $x\in D(K_\p)$, $\Pi_D=\Pi_D(F_{\bar x})$, and $\delta\colon F_{\bar x}\Rightarrow F_{\vec t}$ as in \cref{lem:paths}.  By \cref{lem:paths}~(c), $\delta^{-1}I_0\delta\subset\Pi_D$, so $\delta^{-1}(\VV_{\vec t}^{I_0})=\VV_{\bar x}^{\delta^{-1}I_0\delta}$ contains $\VV_{\bar x}^{\Pi_D}$.  Both are lines (a single Jordan block, and \cref{prop:descent}~(a)), so they are equal.  For $\sigma\in G_{K_\p}$, $\beta^{\lim}(\sigma)$ is the scalar by which $s_{\vec t}(\sigma)$ acts on $\VV_{\vec t}^{I_0}$, that is, by which $\delta^{-1}s_{\vec t}(\sigma)\delta=h\,s_x(\sigma)$, with $h\in\Pi_D$ (\cref{lem:paths}~(b)), acts on $\VV_{\bar x}^{\Pi_D}$.  As $h$ acts trivially on $\VV_{\bar x}^{\Pi_D}$, this scalar is $\beta_1(\sigma)$.

(b) The geometric point $\mathrm{Spec}\,\Omega_K$ at which $\VV_{\vec t}$ is the fiber (\cref{sec:descent}) lies over $\mathrm{Spec}\,\Kbar\{s\}[1/s]$, where $\Kbar\{s\}\subset\Kbar[[s]]$ is the strict henselization of $\PP^1_K$ at $0$.  The fundamental group of $\mathrm{Spec}\,\Kbar\{s\}[1/s]$ is $I_0$, and $G_K$ acts on $\Kbar\{s\}$ and on $\Omega_K$ through the coefficients; so $\VV_{\vec t}^{I_0}=\Gamma(\mathrm{Spec}\,\Kbar\{s\}[1/s],\VV)$ is the stalk at $0$ of $j_*\VV$ over $\Kbar$, with its natural action of $G_K$.  Let $R\subset\Kbar$ be the valuation ring, for the place of $\Kbar$ that defines $G_{K_\q}$, of the field fixed by $I_\q$, a strict henselization of $\OO$ at $\q$; let $A$ be the henselization of $R[s]$ at $(\mathfrak m_R,s)$, a strict henselization of $\PP^1_{\OO[1/3\ell]}$ at the point $s=0$ of the fiber at $\q$; and let $U_A=\mathrm{Spec}\,A[1/s]$.  The group $G_{K_\q}$ acts on $A$ through its action on $R$, and $I_\q$ acts trivially.  By Abhyankar's lemma \cite[Exposé~XIII, Proposition~5.2 and Corollary~5.3]{SGA1}, applied to the regular strictly local scheme $\mathrm{Spec}\,A$ and the divisor $s=0$ (the tameness it assumes at the generic point of this divisor holds because the residue field there has characteristic $0$), every finite étale cover of $U_A$ becomes trivial over $A[s^{1/n}]$ for some $n$ prime to $q$.  So $\pi_1(U_A)\isom\varprojlim_{(n,q)=1}\mu_n$, and both $I_0$, through the strict localization of $\mathrm{Spec}\,A$ at the generic point of $s=0$, and the inertia group at $0$ of $\PP^1_{\Fbar_q}$, through the special fiber, map onto it.  As $\VV$ is lisse on $\SSS$ over $\OO[1/3\ell]$ (\cref{prop:compactification}), restriction gives $G_{K_\q}$-equivariant isomorphisms from $\Gamma(U_A,\VV)$ onto the stalks at $0$ of $j_*\VV$ over $\Kbar$ and over $\Fbar_q$, where $G_{K_\q}$ acts on the latter through $\Gal(\Fbar_q/\F_q)$.  Hence $I_\q$ acts trivially on $\VV_{\vec t}^{I_0}$, and $\Frob_\q$ acts on it as on the stalk at $0$ of $j_*\VV$ over $\Fbar_q$.

(c) Over $\Fbar_q$, $j_*\VV$ is irreducible on $\PP^1$ with stalks of dimensions $1,2,0$ at $0,1,\infty$ (\cref{prop:geometry}).  Its Euler characteristic is $3\cdot(-1)+1+2+0=0$, and $H^0$ and $H^2$ vanish, so $H^*(\PP^1,j_*\VV)=0$ and the sequence $0\to j_!\VV\to j_*\VV\to\bigoplus_x(j_*\VV)_x\to0$ gives $H^1_c(\SSS,\VV)\isom(j_*\VV)_0\oplus(j_*\VV)_1$.  By Deligne's theory of weights \cite[1.8.4]{DeligneWeilII}, since $\VV$ is pure of weight $2$ and $\gamma_0$ has a single Jordan block of length three, $(j_*\VV)_0$ is pure of weight $0$; and $(j_*\VV)_1$, the invariants of a monodromy of finite order, is pure of weight $2$.  So $\alpha_0(\q)$ is the unique root of $\Omega_\q$ of weight $0$, that is, of absolute value $1$ in every complex embedding.  The coefficients of $\Omega_\q$ lie in $K$ (\cref{lem:traces-in-K} and Newton's identities), and the roots of weight $0$ are permuted by $G_K$, so $\alpha_0(\q)\in K$.  It is an algebraic integer, being an eigenvalue of Frobenius on $H^1_c(\SSS,\VV)$, which is the $\chi^m$-part of $H^3_c$ of the variety $X$ over $\F_q$ \cite[Exposé~XXI, 5.5.3]{SGA7II}.  An algebraic integer of $K$ all of whose conjugates have absolute value $1$ is a root of unity, so $\alpha_0(\q)\in\mu_6$.
\end{proof}

\begin{proposition}\label{prop:limit-trivial}
The character $\beta^{\lim}$ is trivial.  Hence $\beta_1=1$, and \cref{thm:inertia3} holds.
\end{proposition}

\begin{proof}
By \cref{lem:beta-lim}~(b)--(c), $\beta^{\lim}(\Frob_\q)\in\mu_6$ for a set of primes $\q$ of density one, so $(\beta^{\lim})^6=1$ by the Chebotarev density theorem.  The values $\alpha_0(\q)$ do not depend on $\ell$ (they are determined by the traces of \cref{lem:traces-in-K}), so for two primes $\ell,\ell'$ the characters $\beta^{\lim}_\ell$ and $\beta^{\lim}_{\ell'}$, both of finite order, agree on almost all Frobenius elements and are equal.  Hence $\beta^{\lim}$ is unramified outside $\p$.  The Galois group of the maximal abelian extension of $K$ unramified outside $\p$ is the inverse limit of the ray class groups $\Cl_{\p^k}(K)$, which have order $3^{k-2}$ for $k\ge2$ (the class number is one, $|(\OO/\p^k)^\times|=2\cdot3^{k-1}$, and $\mu_6$ injects); so it is a pro-$3$ group, and $\beta^{\lim}$ has order dividing $3$.  A cubic character of $K_\p^\times$ is trivial on $1+\p^4$, because the binomial series of $(1+x)^{1/3}$ converges for $v_3(x)>3/2$, that is for $x\in\p^4$.  So $\beta^{\lim}$ factors through $\Cl_{\p^4}(K)\isom(\Z/3)^2$.  This group is generated by the classes of $\lp$ and $\lpb$ (\cref{app:computations}, item~C3).  If $\ell\ne7$, then $\lp,\lpb\nmid3\ell$, and \cref{lem:beta-lim}~(b),~(c) and \cref{prop:frob7}~(b) give $\beta^{\lim}(\Frob_\lp)=\alpha_0(\lp)=1$ and $\beta^{\lim}(\Frob_\lpb)=\alpha_0(\lpb)=1$, so $\beta^{\lim}=1$.  If $\ell=7$, then $\beta^{\lim}_7=\beta^{\lim}_5$, as shown above, and $\beta^{\lim}_5=1$ by the case $\ell=5$.  Then $\beta_1=1$ by \cref{lem:beta-lim}~(a).  Finally, given $s_0\in K_\p^\times$ with $v_3(s_0)>15/2$, choose $d$ with $15/2<d<v_3(s_0)$; then $s_0\in D(K_\p)$, and \cref{prop:descent}~(b) with $\beta_1=1$ is \cref{thm:inertia3}.
\end{proof}

\begin{remark}[the radius]\label{rem:radius}
We do not know whether $15/2$ is optimal, but it is the natural constant for this argument. The holomorphic solution at $s=0$ of the hypergeometric differential equation underlying $\VV$ is ${}_3F_2(a_0,a_1,a_2;1,1;s)$ with $\{a_0,a_1,a_2\}=\{1/9,4/9,7/9\}$ or $\{2/9,5/9,8/9\}$ (see the proof of \cref{prop:geometry}). Its $n$th coefficient $\prod_j\big((a_j)_n/n!\big)$ is, up to sign, the product of the three binomial coefficients $\binom{-a_j}{n}$, each of the kind in \cref{lem:radius}; so its $3$-adic valuation is $-(15n-3s_3(n))/2$, and the series converges exactly on $v_3(s)>15/2$. The same argument applied to a convolution of two Kummer sheaves gives, for the Legendre family $y^2=x(x-1)(x-\lambda)$ (two quadratic characters, at the prime $2$, each trivial on $v_2(x)>2$), the disc $v_2(\lambda)>4$, and for the Hesse family $x^3+y^3+z^3=3\mu xyz$ with $s=\mu^{-3}$ (two cubic characters, at the prime $3$, each trivial on $v_3(x)>3/2$), the disc $v_3(s)>3$. Since $v_2(j)=8-2v_2(\lambda)$ and $v_3(j)=3-v_3(s)$ when these valuations are positive, these discs are exactly the parts in $v_2(\lambda)>0$, resp.\ $v_3(s)>0$, of the loci $v(j)<0$ of potentially multiplicative reduction.
\end{remark}

\begin{remark}[a numerical check]\label{rem:inertia3-check}
For $s_0\in\Z\smallsetminus\{0,1\}$ let $L(s)$ be the $L$-function of $\mathrm{Ind}_K^{\Q}\VV_{s_0}$, of degree $6$, with gamma factor $\Gamma_\C(s)^2\Gamma_\C(s-1)$ (\cref{lem:hodge}) and conductor $27\cdot3^{f}N'$, where $f$ is the conductor exponent of $\VV_{s_0}$ at $\p$ and $N'$, prime to $3$, is the product of $\Nm\q^2$ over the primes $\q\nmid3$ of $K$ dividing $s_0$ and of $\Nm\q$ over those dividing $s_0-1$ to a power prime to $3$ (\cref{prop:geometry}~(c) and the proof of \cref{lem:tame}). Assume that $\Lambda(s)=\pm\Lambda(3-s)$ for the completed $L$-function.  We computed the Euler factors at the primes $q\le X$ other than $3$, taking at the primes dividing $s_0(s_0-1)$ the factors given by specialization, and tested the functional equation numerically, with both signs: for $f\le2$, with a list of candidate Euler factors at $3$ that contains every factor allowed by \cref{thm:inertia3}, and, at the points outside the disc, also for larger $f$.  The Supplement, item~S25, gives the details.  For the choice of $f$, Euler factor at $3$ and sign in the table, the functional equation holds to within about $2^{-66}\approx10^{-20}$; every other choice tested misses it by at least $2^{-17}\approx10^{-5}$.
\begin{center}\small
\begin{tabular}{@{}lccl@{}}
\toprule
$s_0$ & $v_3(s_0)$ & $X$ & local data at $\p$ singled out \\
\midrule
$3^8$, $3^{10}$ & $8$, $10$ & $5\cdot10^5$, $5.2\cdot10^5$ & $f=2$, Euler factor $1-3^{-s}$, sign $+1$\\
$9$ & $2$ & $5\cdot10^5$ & $f=14$, Euler factor $1$, sign $+1$\\
$-27$ & $3$ & $10^5$ & $f=6$, Euler factor $1$, sign $+1$\\
$-2\cdot3^7$ & $7$ & $3\cdot10^5$ & $f=4$, Euler factor $1$, sign $+1$\\
$-9,\ 27,\ 81,\ 243,\ 729$ & $2$--$6$ & $10^5$ & none with $f\le2$ (nor with the larger $f$ tested)\\
\bottomrule
\end{tabular}
\end{center}
\medskip
Unipotent inertia forces $f\le2$ (the Swan conductor vanishes and there are nonzero invariants). So at $s_0=3^8$ and $3^{10}$ the data agree with \cref{thm:inertia3}, and the factor $1-3^{-s}$ suggests moreover that the monodromy operator is a single Jordan block.  At the point $s_0=9$ of the Catalan solution, at $s_0=-27$, and at $s_0=-2\cdot3^7$ (where $v_3(s_0)=7$, just outside the disc), the data indicate that inertia does not act unipotently. These computations are numerical and conditional on the functional equation and on the Euler factors at the primes dividing $s_0(s_0-1)$.
\end{remark}

\needspace{5\baselineskip}\begin{remark}\label{rem:legendre-limit}
The arithmetic input at the primes above $7$ in the proof of \cref{prop:limit-trivial} cannot be replaced by geometry.  For the Legendre family $y^2=x(x-1)(x-\lambda)$ the character on the invariant line of the limit at $\lambda=0$ is the quadratic character of $-1$, which is nontrivial.  The analogous character for a twist of a family is changed by the twist, and only an arithmetic computation, here the one at the primes above $7$, decides it.
\end{remark}