Proof. Let $e_{ij}$ be the $n\times n$ matrix with a 1 in the $(i,j)$ position and 0 elsewhere. By assumption, there exists $m$ such that $\unicode[STIX]{x1D70C}(H)$ contains $1+\ell ^{m}e_{ij}$ for all $i,j$ , and so $e_{ij}\in \mathbf{Q}_{\ell }[\unicode[STIX]{x1D70C}(G)]$ . For any $A\in \unicode[STIX]{x1D70C}(G)$ we have $A_{i,j}=\operatorname{tr}(e_{ii}Ae_{jj})\in \operatorname{tr}(\mathbf{Q}_{\ell }[\unicode[STIX]{x1D70C}(G)])=\mathbf{Q}_{\ell }$ , where the last equality follows from the fact that $\text{tr}\circ \unicode[STIX]{x1D70C}:G\rightarrow \mathbf{Q}_{\ell }$ is continuous and $\operatorname{tr}(\unicode[STIX]{x1D70C}(g_{i}))\in \mathbf{Q}_{\ell }$ for all $i$ . It thus follows that $\unicode[STIX]{x1D70C}(G)\subset \mathbf{GL}_{n}(\mathbf{Q}_{\ell })$ as claimed.◻

Proof. Consider $B=\operatorname{Res}_{K}^{K^{\prime }}E$ , an abelian variety over $K$ of dimension $[K^{\prime }:K]$ [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 7.6]. Write $B=\prod _{i=1}^{r}B_{i}$ (up to isogeny) where each $B_{i}$ is a power of a simple abelian variety. Thus

$$\begin{eqnarray}\operatorname{End}_{K}(B)\otimes \mathbf{Q}=\bigoplus _{i=1}^{r}D_{i},\end{eqnarray}$$

where $D_{i}=\operatorname{End}_{K}(B_{i})\otimes \mathbf{Q}$ is a simple algebra. We have

$$\begin{eqnarray}\operatorname{Hom}_{G_{K}}(\unicode[STIX]{x1D70C},\overline{\mathbf{Q}}_{\ell }\otimes V_{\ell }(B))=\operatorname{Hom}_{G_{K^{\prime }}}(\unicode[STIX]{x1D70C}|_{G_{K^{\prime }}},\overline{\mathbf{Q}}_{\ell }\otimes V_{\ell }(E))=\overline{\mathbf{Q}}_{\ell },\end{eqnarray}$$

where the latter follows from Faltings’ proof of the Tate conjecture [Reference Faltings, Wüstholz, Grunewald, Schappacher and StuhlerFWGSS92, Theorem 1, p. 211] since $E$ is non-CM. We thus see that $\unicode[STIX]{x1D70C}$ occurs uniquely in $\overline{\mathbf{Q}}_{\ell }\otimes V_{\ell }(B_{i})$ for some  $i$ . Let $A$ be this  $B_{i}$ . Since $A$ is a power of a simple abelian variety and $\unicode[STIX]{x1D70C}$ occurs uniquely in its Tate module, $A$ must itself be simple. The endomorphism ring $D_{i}$ must preserve  $\unicode[STIX]{x1D70C}$ , and thus the action of $\unicode[STIX]{x1D70C}$ comes from an algebra homomorphism $\unicode[STIX]{x1D713}:D_{i}\rightarrow \overline{\mathbf{Q}}_{\ell }$ , which is injective since $D_{i}$ is simple. We thus see that $D_{i}=F$ is a number field, and $\unicode[STIX]{x1D713}$ corresponds to a place $w$ of $F$ above  $\ell$ . Since $V_{w}(A)\otimes _{F_{w}}\overline{\mathbf{Q}}_{\ell }$ contains $\unicode[STIX]{x1D70C}$ and is absolutely irreducible by Proposition 7, it is equal to  $\unicode[STIX]{x1D70C}$ . Therefore $V_{w}(A)$ is two dimensional over  $F_{w}$ , and so $[F:\mathbf{Q}]=\dim (A)$ by Proposition 6.◻

Proof. By Lemma 15, we can find an abelian variety $A/K$ with $\operatorname{End}(A)\otimes \mathbf{Q}=F$ a number field of degree $\dim (A)$ , and a place $w_{0}$ of $F$ such that $\unicode[STIX]{x1D70C}\cong \overline{\mathbf{Q}}_{\ell }\otimes _{F_{w_{0}}}V_{w_{0}}(A)$ .

Choose an integral scheme $X$ of finite type over $\operatorname{Spec}(\mathbf{Z})$ with fraction field $K$ and a family of abelian varieties ${\mathcal{A}}\rightarrow X$ extending $A$ . The representation of $G_{K}$ on $V_{w}(A)$ factors through $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X)$ , for all  $w$ . Since $V_{w_{0}}(A)$ satisfies ( $\ast$ ), we can replace $X$ with a dense open subscheme such that the Frobenius elements in $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X)$ have rational traces on $V_{w_{0}}(A)$ . By Proposition 6, it follows that the Frobenius elements in $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X)$ have rational traces on $V_{w}(A)$ for all  $w$ .

By assumption, $\unicode[STIX]{x1D70C}|_{G_{K^{\prime }}}\cong \overline{\mathbf{Q}}_{\ell }\,\otimes \,V_{\ell }(E)$ for some non-CM elliptic curve $E/K^{\prime }$ . As above, pick an integral scheme $X^{\prime }$ of finite type over $\operatorname{Spec}(\mathbf{Z})$ with fraction field $K^{\prime }$ such that there is an elliptic curve ${\mathcal{E}}\rightarrow X^{\prime }$ extending  $E$ . Further shrinking $X,X^{\prime }$ we can assume $X^{\prime }$ maps to $X$ inducing the inclusion $K\subset K^{\prime }$ .

Pick a place $w\mid p$ of $F$ . As we saw above, Frobenius elements of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X)$ have equal traces on $\unicode[STIX]{x1D70C}\cong \overline{\mathbf{Q}}_{\ell }\otimes _{F_{w_{0}}}V_{w_{0}}(A)$ and $\overline{\mathbf{Q}}_{p}\otimes _{F_{w}}V_{w}(A)$ . Similarly, the traces of Frobenius elements of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X^{\prime })$ on  $V_{\ell }(E)$ and $V_{p}(E)$ are equal. We thus see by Corollary 9 that $\overline{\mathbf{Q}}_{p}\otimes _{F_{w}}V_{w}(A)$ is isomorphic to $\overline{\mathbf{Q}}_{p}\otimes _{\mathbf{Q}_{p}}V_{p}(E)$ as representations of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X^{\prime })$ . By the Tate conjecture proved by Faltings [Reference Faltings, Wüstholz, Grunewald, Schappacher and StuhlerFWGSS92, Theorem 1, p. 211], the image of $G_{K^{\prime }}$ in $\mathbf{GL}(V_{p}(E))$ contains an open subgroup of $\mathbf{GL}_{2}(\mathbf{Z}_{p})$ .

It follows that the conditions of Lemma 14 are fulfilled for $V_{w}(A)\,\otimes _{F_{w}}\,\overline{\mathbf{Q}}_{p}$ , and so $V_{w}(A)\,\otimes _{F_{w}}\,\overline{\mathbf{Q}}_{p}$ is defined over $\mathbf{Q}_{p}$ ; that is, there exists some representation $V$ of $G_{K}$ over $\mathbf{Q}_{p}$ such that $V_{w}(A)\,\otimes _{F_{w}}\,\overline{\mathbf{Q}}_{p}\cong \overline{\mathbf{Q}}_{p}\otimes _{\mathbf{Q}_{p}}V$ . Since $V_{w}(A)$ and $V\otimes _{\mathbf{Q}_{p}}F_{w}$ have the same character, and are irreducible, it follows that they are isomorphic. We thus see that $\operatorname{End}_{\mathbf{Q}_{p}[G_{K}]}(V_{w}(A))\cong \operatorname{End}_{\mathbf{Q}_{p}}(F_{w})$ , where on the right side we are taking endomorphisms of $F_{w}$ as a vector space. By Proposition 7 we have that $\operatorname{End}_{\mathbf{Q}_{p}[G_{K}]}(V_{w}(A))\cong F_{w}$ , and so $\operatorname{End}_{\mathbf{Q}_{p}}(F_{w})=F_{w}$ , which implies $F_{w}=\mathbf{Q}_{p}$ . We thus see that all places of $F$ are split, and so $F=\mathbf{Q}$ . Thus $A$ is actually an elliptic curve, and the proof is complete.◻

Proof. It suffices to treat the case where $(K^{\prime })^{p}=K$ . Let $E/K^{\prime }$ be the elliptic curve giving rise to $\unicode[STIX]{x1D70C}|_{K^{\prime }}$ . Let $E^{(p)}=E\times _{K^{\prime },F_{0}}K^{\prime }$ where $F_{0}:K^{\prime }\rightarrow K^{\prime }$ is the absolute Frobenius map. Then $E^{(p)}$ is defined over  $K$ , and there is a canonical isogeny (relative Frobenius map) $F:E\rightarrow E^{(p)}$ defined over $K^{\prime }$ inducing an isomorphism on rational $\ell$ -adic Tate modules. Thus $V_{\ell }(E^{(p)})|_{G_{K^{\prime }}}\cong \unicode[STIX]{x1D70C}|_{G_{K^{\prime }}}$ and so $V_{\ell }(E^{(p)})\cong \unicode[STIX]{x1D70C}$ since $K^{\prime }/K$ is purely inseparable.◻

Proof. Let $\ell$ be a rational prime that splits completely in $F$ . Then the $\ell$ -adic Tate module ${\mathcal{L}}_{\ell }$ of $A$ decomposes as $\bigoplus _{v\mid \ell }{\mathcal{L}}_{v}$ . The ${\mathcal{L}}_{v}$ form a compatible system with coefficients in $F$ , so for each closed point $x\in U$ there exists $\unicode[STIX]{x1D6FC}\in F$ such that the trace of the Frobenius element at $x$ on ${\mathcal{L}}_{v,x}$ is the image of $\unicode[STIX]{x1D6FC}$ in  $F_{v}$ . By our assumptions, $\unicode[STIX]{x1D6FC}$ is a rational number, and so if $v,w\mid \ell$ then the traces of Frobenius elements on ${\mathcal{L}}_{v,x}$ and ${\mathcal{L}}_{w,x}$ are the same element of $\mathbf{Q}_{\ell }\subset F_{v},F_{w}$ . It follows that the characters of ${\mathcal{L}}_{v}$ and ${\mathcal{L}}_{w}$ are equal at Frobenius elements, and so ${\mathcal{L}}_{v}\cong {\mathcal{L}}_{w}$ . Thus $\dim (\operatorname{End}({\mathcal{L}}_{\ell }))\geqslant g^{2}$ . By Faltings’ isogeny theorem, it follows that $\dim (\operatorname{End}(A)\otimes \mathbf{Q})\geqslant g^{2}$ .

Let $C^{\prime }\rightarrow C$ be a cover such that $A$ has semi-stable reduction, and let $x$ be a point at which $A^{\prime }$ (the pullback of  $A$ ) has bad reduction. Let ${\mathcal{A}}^{\prime }$ be the Néron model of $A^{\prime }$ over  $C^{\prime }$ , and let $T$ be the torus quotient of the identity component of  ${\mathcal{A}}_{x}^{\prime }$ . The dimension $h$ of $T$ is at least 1, and at most  $g$ . Under the map $\operatorname{End}(A)\otimes \mathbf{Q}\rightarrow \operatorname{End}(T)\otimes \mathbf{Q}\subset M_{h}(\mathbf{Q})$ , the field $F$ must inject, and so $h=g$ . Thus $T$ is the entire identity component of ${\mathcal{A}}_{x}^{\prime }$ , and so the map $\operatorname{End}(A)\otimes \mathbf{Q}\rightarrow \operatorname{End}(T)\otimes \mathbf{Q}\subset M_{g}(\mathbf{Q})$ is injective. Combined with the previous paragraph, we find that $\dim (\operatorname{End}(A)\,\otimes \,\mathbf{Q})=g^{2}$ , and so the map $\operatorname{End}(A)\otimes \mathbf{Q}\rightarrow M_{g}(\mathbf{Q})$ is an isomorphism. The statement now follows by projecting under an idempotent.◻

Proof. By Drinfeld’s theorem ([Reference DrinfedDri83, Main theorem, Remark 5], see also [Reference DrinfeldDri78]) there is a cuspidal automorphic representation $\unicode[STIX]{x1D70B}$ of $\mathbf{GL}_{2}(\mathbf{A}_{k(C)})$ which is compatible with  $L$ . Since inertia at $x$ does not have finite order, $\unicode[STIX]{x1D70B}$ must be special at $x$ . It follows by another theorem of Drinfeld [Reference DrinfeldDri77, Theorem 1] that there exists a number field $E$ and a $\mathbf{GL}_{2}(E)$ -type abelian variety $A$ over $U$ which is compatible with $\unicode[STIX]{x1D70B}$ and thus also with  $L$ , in the sense that the $\ell$ -adic Tate module $L^{\prime }$ of $A$ is isomorphic with $L$ when tensored up to $\overline{\mathbf{Q}}_{\ell }$ ; that is, $L^{\prime }\otimes _{E}\overline{\mathbf{Q}}_{\ell }\cong L$ . By Proposition 18, we may take $A$ to be an elliptic curve.◻

Proof. Let $P$ be the image of a section $S\rightarrow C_{1}$ . Define $\widetilde{P}$ to be the $d$ th nilpotent thickening of  $P$ . Precisely, if $P$ is defined by the ideal sheaf ${\mathcal{I}}_{P}$ then $\widetilde{P}$ is defined by ${\mathcal{I}}_{P}^{d}$ . Let $Q$ be the image of a section $S\rightarrow C_{2}$ , and define $\widetilde{Q}$ similarly. Let $0\leqslant e\leqslant d$ be an integer. For an $S$ -scheme $S^{\prime }$ , let ${\mathcal{C}}_{P,Q,e}(S^{\prime })$ be the set of maps $f:\widetilde{P}_{S^{\prime }}\rightarrow \widetilde{Q}_{S^{\prime }}$ such that $f^{\ast }({\mathcal{I}}_{Q})\subset {\mathcal{I}}_{P}^{e}$ . As these are finite schemes over  $S$ , this functor is represented by a scheme ${\mathcal{C}}_{P,Q,E}$ of finite type over  $S$ .

Write $Z_{1}=\coprod _{i=1}^{n}P_{i}$ and $Z_{2}=\coprod _{j=1}^{m}Q_{j}$ . (Here $\coprod$ denotes disjoint union.) By a ramification datum we mean a tuple $\unicode[STIX]{x1D70C}=(\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{n})$ where each $\unicode[STIX]{x1D70C}_{i}$ is either null (denoted  $\emptyset$ ), or a pair $(k_{i},e_{i})$ , where $1\leqslant k_{i}\leqslant m$ and $0\leqslant e_{i}\leqslant d$ , such that the following condition holds: for any $1\leqslant j\leqslant m$ , we have $\sum _{k_{i}=j}e_{i}=d$ (the sum taken over $i$ for which $\unicode[STIX]{x1D70C}_{i}\neq \emptyset$ ). Obviously, there are only finitely many ramification data. For a ramification datum $\unicode[STIX]{x1D70C}$ , define ${\mathcal{C}}_{\unicode[STIX]{x1D70C}}=\prod _{\unicode[STIX]{x1D70C}_{i}\neq \emptyset }{\mathcal{C}}_{P_{i},Q_{k_{i}},e_{i}}$ . Finally, define ${\mathcal{C}}=\coprod _{\unicode[STIX]{x1D70C}}{\mathcal{C}}_{\unicode[STIX]{x1D70C}}$ , where the union is taken over all ramification data  $\unicode[STIX]{x1D70C}$ .

For an $S$ -scheme $S^{\prime }$ , let ${\mathcal{A}}(S^{\prime })$ be the set of morphisms $(C_{1})_{S^{\prime }}\rightarrow (C_{2})_{S^{\prime }}$ having degree $d$ in each geometric fiber. The theory of the Hilbert scheme shows that ${\mathcal{A}}$ is represented by a scheme of finite type over  $S$ . For $1\leqslant i\leqslant n$ , let ${\mathcal{B}}_{i}(S^{\prime })$ be the set of all maps $(\widetilde{P}_{i})_{S^{\prime }}\rightarrow (C_{2})_{S^{\prime }}$ . This is easily seen to be a scheme of finite type over  $S$ . Let ${\mathcal{B}}=\coprod _{i=1}^{n}\widetilde{{\mathcal{B}}}_{i}$ .

We have restriction maps ${\mathcal{A}}\rightarrow {\mathcal{B}}$ and ${\mathcal{C}}\rightarrow {\mathcal{B}}$ . Define ${\mathcal{M}}$ to be the fiber product ${\mathcal{A}}\times _{{\mathcal{B}}}{\mathcal{C}}$ , which is a scheme of finite type over $S$ . We can write ${\mathcal{M}}=\coprod _{\unicode[STIX]{x1D70C}}{\mathcal{M}}_{\unicode[STIX]{x1D70C}}$ , where ${\mathcal{M}}_{\unicode[STIX]{x1D70C}}={\mathcal{A}}\times _{{\mathcal{B}}}{\mathcal{C}}_{\unicode[STIX]{x1D70C}}$ . If $s\in S(k)$ then ${\mathcal{M}}_{\unicode[STIX]{x1D70C},s}(k)$ is the set of degree $d$ maps $f:C_{1,s}\rightarrow C_{2,s}$ satisfying the following conditions at the  $P_{i}$ : if $\unicode[STIX]{x1D70C}_{i}=\emptyset$ then there is no condition at  $P_{i}$ ; otherwise, $f(P_{i})=Q_{k_{i}}$ and the ramification index $e(P_{i}\mid Q_{k_{i}})$ is at least $e_{i}$ . (In fact, the ramification index is exactly $e_{i}$ , since the total ramification is $d$ and the $e_{i}$ with $k_{i}=j$ add up to  $d$ .) It is clear that such a map carries $U_{1,s}$ into $U_{2,s}$ , and that any map $U_{1,s}\rightarrow U_{2,s}$ of degree $d$ comes from a point of some ${\mathcal{M}}_{\unicode[STIX]{x1D70C},s}$ . Thus every map $f:U_{1,s}\rightarrow U_{2,s}$ comes from some $k$ -point of  ${\mathcal{M}}_{s}$ . Finally, note that the universal map $(C_{1})_{{\mathcal{M}}}\rightarrow (C_{2})_{{\mathcal{M}}}$ carries $(U_{1})_{{\mathcal{M}}}$ to $(U_{2})_{{\mathcal{M}}}$ , as this can be checked at field points of  ${\mathcal{M}}$ . This proves the proposition.◻

Proof. Suppose not. Then there exists an open normal subgroup $H$ of $G=\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U_{K})$ such that $L|_{H}$ is reducible. It is semi-simple by Lemma 12, and therefore a sum of two characters. Since characters have finite ramification by Lemma 10, we have contradicted assumption (d).◻

Proof. Note that for every field-valued point $f$ of ${\mathcal{M}}$ there are only finitely many choices for $\unicode[STIX]{x1D713}$ and so the map $\widetilde{{\mathcal{M}}}_{n}\rightarrow {\mathcal{M}}$ is quasi-finite. Thus, to prove the lemma it is sufficient to show that $\unicode[STIX]{x1D70B}$ is proper.

We use the valuative criterion. Let $R$ be a discrete valuation ring with fraction field $F$ . Let $f\in {\mathcal{M}}(R)$ and $(\unicode[STIX]{x1D713},f)\in \widetilde{{\mathcal{M}}}_{n}(F)$ . Thus, $f$ corresponds to a map $f:U_{R}\rightarrow Y_{R}$ and $\unicode[STIX]{x1D713}:f^{\ast }({\mathcal{T}}_{n})_{F}\rightarrow ({\mathcal{L}}_{n})_{F}$ is an isomorphism. Since $f^{\ast }({\mathcal{T}}_{n})$ and ${\mathcal{L}}_{n}$ are finite étale sheaves on  $U_{R}$ , which is a normal scheme, $\unicode[STIX]{x1D713}$ extends uniquely over  $U_{R}$ .◻

Proof. Let $s$ be a closed point of $S$ of characteristic $p$ . By Lemma 12, ${\mathcal{L}}_{\overline{K}}$ is an irreducible sheaf. Let $\overline{S}$ be the strict Hensilization of $S$ at  $s$ , $\overline{s}$ the geometric point corresponding to  $s$ , and $K_{s}$ the fraction field of  $\overline{S}$ . By [Reference Grothendieck and RaynaudSGA1, XIII, 2.10, p. 289],

(*)

where $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t},(p)}$ denotes the prime to $p$ quotient of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}$ . We can regard ${\mathcal{L}}$ as a representation of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}({\mathcal{U}})$ and then restrict it to a representation of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}({\mathcal{U}}_{\overline{S}})$ ; since the image of the representation has order $M\ell ^{\infty }$ , which is prime to  $p$ , it factors through $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t},(p)}({\mathcal{U}}_{\overline{S}})$ . We thus obtain a representation of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t},(p)}({\mathcal{U}}_{\overline{s}})$ . The pullback of this representation to $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U_{\overline{K}})$ is irreducible. It follows now that ${\mathcal{L}}_{s}$ is an irreducible sheaf on  ${\mathcal{U}}_{s}$ .

By Proposition 19, we can find a family of elliptic curves ${\mathcal{E}}\rightarrow {\mathcal{U}}_{s}$ and an isomorphism ${\mathcal{L}}_{s}|_{{\mathcal{U}}_{s}}\cong T({\mathcal{E}})\otimes {\mathcal{O}}_{E}$ , where $T({\mathcal{E}})$ is the relative Tate module of  ${\mathcal{E}}$ . It follows that $T({\mathcal{E}})/\ell ^{r}$ is the trivial sheaf, and so we can find a basis for ${\mathcal{E}}[\ell ^{r}]$ over ${\mathcal{U}}_{s}$ . We thus have a map $f:{\mathcal{U}}_{s}\rightarrow Y$ such that $T({\mathcal{E}})\cong f^{\ast }({\mathcal{T}})$ . Factor $f$ as $g\circ F^{n}$ , where $g:{\mathcal{U}}_{s}\rightarrow Y$ is separable and $F:Y_{\unicode[STIX]{x1D705}(s)}\rightarrow Y_{\unicode[STIX]{x1D705}(s)}$ is the absolute Frobenius element. Note that $F^{\ast }({\mathcal{T}})\cong {\mathcal{T}}$ , and so $T({\mathcal{E}})\cong g^{\ast }({\mathcal{T}})$ . Let $\overline{g}$ be the extension of $g$ to a map $C_{s}\rightarrow X$ . Since $\overline{g}$ is separable, it has degree ${\leqslant}D$ . Thus  $\overline{g}$ , and the isomorphism ${\mathcal{L}}\cong g^{\ast }({\mathcal{T}})\otimes {\mathcal{O}}_{E}$ , define a $\unicode[STIX]{x1D705}(s)$ points of ${\mathcal{M}}_{n}$ for all  $n$ , which proves the lemma.◻

Proof of Theorem 1.

Since ${\mathcal{M}}_{\infty }$ is finite type over $S$ and all of its fibers over closed points are non-empty, it follows that the generic fiber of ${\mathcal{M}}_{\infty }$ is non-empty. Choose a point $x$ in ${\mathcal{M}}_{\infty }(L^{\prime })$ , for some finite extension $L^{\prime }/L$ , corresponding to a family of elliptic curves ${\mathcal{E}}\rightarrow U_{L^{\prime }}$ . Now, $x$ lifts to $\widetilde{{\mathcal{M}}}_{n}(\overline{L})$ for all  $n$ . Thus ${\mathcal{L}}_{n}$ and $T({\mathcal{E}})\otimes {\mathcal{O}}_{E}/\ell ^{n}$ are isomorphic for all $n$ , as sheaves on $U_{\overline{L}}$ . It follows that ${\mathcal{L}}$ and $T({\mathcal{E}})\otimes {\mathcal{O}}_{E}$ are isomorphic over $U_{\overline{L}}$ (by compactness). By Lemma 11, ${\mathcal{L}}[1/\ell ]$ and $T({\mathcal{E}})\otimes E$ are isomorphic over $U_{L^{\prime \prime }}$ , for some finite (even quadratic) extension $L^{\prime \prime }$ of  $L^{\prime }$ . Passing to the generic fibers, we see that $\unicode[STIX]{x1D70C}|_{KL^{\prime \prime }}$ comes from an elliptic curve, which completes the proof by Proposition 13.◻