1 Introduction
Let $C/K$ be a proper, smooth geometrically irreducible curve, with $K$ a number field, let $f:E\rightarrow U\subset C$ be a non-isotrivial family of elliptic curves over a non-empty open subset $U$ of $C$ , and let $L=\text{R}^{1}f_{\ast }(\overline{\mathbf{Q}}_{\ell })$ be the associated rank-two lisse $\ell$ -adic sheaf on $U$ . The following properties hold.
(a) There is an isomorphism $\bigwedge ^{2}L\cong \overline{\mathbf{Q}}_{\ell }(1)$ .
(b) There exists a proper smooth model ${\mathcal{C}}$ of $C$ over $\operatorname{Spec}{\mathcal{O}}_{K}[1/N]$ , an open subset ${\mathcal{U}}$ of ${\mathcal{C}}$ extending $U$ , and a lisse sheaf ${\mathcal{L}}$ on ${\mathcal{U}}$ extending $L$ .
(c) For every closed point $x$ of ${\mathcal{U}}$ , the trace of the Frobenius element on ${\mathcal{L}}_{x}$ is a rational number.
(d) There exists a point $x$ of $C_{\overline{K}}$ at which $L_{\overline{K}}$ does not have potentially good reduction, i.e., the restriction to the inertia subgroup at $x$ of the representation of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U_{\overline{K}})$ associated to $L_{\overline{K}}$ does not act through a finite order quotient.
To see (d), take $x$ to be a pole of the $j$ -invariant of $E$ .
The purpose of this paper is to prove the following converse of the above statement.
Theorem 1. Let $C/K$ be as above, and let $L$ be an irreducible rank-two lisse $\overline{\mathbf{Q}}_{\ell }$ -sheaf over an open subset $U\subset C$ . Assume conditions (a)–(d) above hold. Then there exists a family of elliptic curves $f:E\rightarrow U$ and an isomorphism $L\cong \text{R}^{1}f_{\ast }(\overline{\mathbf{Q}}_{\ell })$ .
Remark 2. The Fontaine–Mazur conjecture predicts that representations of $G_{K}$ satisfying certain natural conditions should appear in the étale cohomology of algebraic varieties. It seems reasonable to expect some kind of generalization of this conjecture to higher-dimensional bases. Our theorem can be viewed as confirmation of a very simple case of this.
Remark 3. One can prove a version of Theorem 1 where $C$ is replaced with a higher-dimensional variety. We just treat the case of curves to keep the exposition simpler.
Remark 4. The theorem is not true if one assumes only (a)–(c). Recall that a fake elliptic curve is an abelian surface $A$ such that $\operatorname{End}(A)$ is an order $R$ in a non-split quaternion algebra that is split at infinity. The moduli space of fake elliptic curves corresponding to $R$ is a proper curve. We therefore can construct a family $f:A\rightarrow C$ of fake elliptic curves for some $C$ as above. The sheaf $\text{R}^{1}f_{\ast }\overline{\mathbf{Q}}_{\ell }$ decomposes as $L^{\oplus 2}$ for some rank-two lisse sheaf $L$ . This $L$ satisfies (a)–(c) but not (d), and thus does not come from a non-isotrivial family of elliptic curves. (Note that one can take $f$ so that $L$ is not isotrivial, in which case $L$ does not come from any family of elliptic curves.)
Question 5. Suppose that for each prime number $\ell$ we have an irreducible rank-two $\mathbf{Q}_{\ell }$ sheaf $L_{\ell }$ satisfying (a)–(c) such that $\{L_{\ell }\}$ forms a compatible system (meaning that the ${\mathcal{U}}$ in part (b) can be chosen uniformly and that the traces of Frobenius elements are independent of $\ell$ ). Does the system come from a family of elliptic curves? Note that the fake elliptic curve counterexample does not apply here: if $\ell$ ramifies in $R\otimes \mathbf{Q}$ then $\text{R}^{1}f_{\ast }\mathbf{Q}_{\ell }$ does not decompose as $L^{\oplus 2}$ .
1.1 Summary of the proof
The basic idea is to use Drinfeld’s results on the global Langlands program to construct an elliptic curve over ${\mathcal{C}}_{\mathbf{F}_{v}}$ for most places $v$ of ${\mathcal{O}}_{K}$ , and then piece these together to get one over ${\mathcal{C}}$ . More precisely, we proceed as follows.
∙ We first show that we are free to pass to finite covers of $C$ . The main content here is a descent result that shows that if $L$ comes from an elliptic curve over a cover of $C$ then it comes from an elliptic curve over $C$ . Using this, we replace $C$ with a cover so that $L/\ell ^{3}L$ is trivial (after replacing $L$ with an integral form).
∙ We next consider $L$ over ${\mathcal{C}}_{\mathbf{F}_{v}}$ and use Drinfeld’s results on the global Langlands program to produce a $\mathbf{GL}_{2}$ -type abelian variety $A_{v}$ realizing $L$ .
∙ Using hypotheses (c) and (d), we descend the coefficient field of $A_{v}$ to $\mathbf{Q}$ , obtaining an elliptic curve $E_{v}$ . (It is likely this could be obtained directly from Drinfeld’s proof.)
∙ We next consider a certain moduli space ${\mathcal{M}}$ of maps ${\mathcal{U}}\rightarrow Y(\ell ^{3})$ . From the previous step (and the triviality of $L/\ell ^{3}L$ ), we see that ${\mathcal{M}}$ has $\mathbf{F}_{v}$ -points for infinitely many $v$ . Since ${\mathcal{M}}$ is of finite type over ${\mathcal{O}}_{K}$ , it therefore has a $\overline{K}$ -point. This yields an elliptic curve $E_{\overline{K}}$ over $U_{\overline{K}}$ realizing $L_{\overline{K}}$ .
∙ Our hypotheses imply that $L_{\overline{K}}$ is irreducible. A simple representation theory argument thus shows that there is a finite extension $K^{\prime }/K$ such that $E$ descends to $U_{K^{\prime }}$ and its Tate module agrees with $L_{K^{\prime }}$ . We have already shown that it suffices to prove the result over a finite cover of $C$ , so we are now finished.
We note that we use Faltings’ proof of the Tate conjecture in the third and fifth steps.
1.2 Outline
In § 2 we recall the relevant background material. In § 3, we prove a few descent results for abelian varieties. In § 4, we package Drinfeld’s results on the global Langlands program into the form we need; in particular, we use the results of § 3 to produce elliptic curves (as opposed to $\mathbf{GL}_{2}$ -type abelian varieties). In § 5, we construct a mapping space parametrizing maps between two affine curves. Finally, in § 6, we prove Theorem 1.
2 Background
2.1 Abelian varieties
Let $A$ be an abelian variety over a field $K$ such that $\operatorname{End}_{K}(A)\otimes \mathbf{Q}$ contains a number field $F$ . Let $V_{\ell }(A)$ denote the rational Tate module of $A$ at the rational prime $\ell$ . This is a module over $F\otimes \mathbf{Q}_{\ell }=\prod _{w\mid \ell }F_{w}$ , and thus decomposes as $\bigoplus _{w\mid \ell }V_{w}(A)$ where each $V_{w}(A)$ is a continuous representation of $G_{K}$ over the field $F_{w}$ . We recall the following standard results.
Proposition 6. Let $\unicode[STIX]{x1D70E}\in \operatorname{End}_{K}(A)$ commute with $F$ . Then the characteristic polynomial of $\unicode[STIX]{x1D70E}$ on $V_{w}(A)$ (regarded as an $F_{w}$ -vector space) has coefficients in $F$ and is independent of $w$ . In particular, each $V_{w}(A)$ has the same dimension over $F_{w}$ .
Proof. See [Reference ShimuraShi67, § 11.10] and (for $F=\mathbf{Q}$ ) [Reference MilneMil, Proposition 9.23].◻
Proposition 7. Assume $K$ is a number field and $\operatorname{End}_{K}(A)\otimes \mathbf{Q}=F$ . Let $w$ be a place of $F$ above a prime $p$ . Then $\operatorname{End}_{\mathbf{Q}_{p}[G_{K}]}(V_{w}(A))=F_{w}$ . In particular, $V_{w}(A)$ is absolutely irreducible as a representation of $G_{K}$ over $F_{w}$ .
Proof. We have
where the first equality is the Tate conjecture proved by Faltings [Reference Faltings, Wüstholz, Grunewald, Schappacher and StuhlerFWGSS92, Theorem 1, p. 211]. Since the endmost spaces have the same dimension, we conclude that the containments are equalities, and so $\operatorname{End}_{\mathbf{Q}_{p}[G_{K}]}(V_{w}(A))=F_{w}$ .◻
2.2 Arithmetic fundamental groups
Let $X$ be an affine normal integral scheme of finite type over $\mathbf{Z}$ and consider $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X)$ . For each closed point $x$ of $X$ there is a conjugacy class of Frobenius elements $F_{x}$ . We recall the following generalization of the Chebotarev density theorem.
Proposition 8. The elements $\{F_{x}\}_{x\in X}$ are dense in $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X)$ .
Proof. This follows from [Reference SerreSer65, Theorem 7]. ◻
Corollary 9. Suppose that $\unicode[STIX]{x1D70C}_{1}$ and $\unicode[STIX]{x1D70C}_{2}$ are semi-simple continuous representations $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(X)\rightarrow \mathbf{GL}_{n}(\overline{\mathbf{Q}}_{\ell })$ such that $\operatorname{tr}(\unicode[STIX]{x1D70C}_{1}(F_{x}))=\operatorname{tr}(\unicode[STIX]{x1D70C}_{2}(F_{x}))$ for all $x$ . Then $\unicode[STIX]{x1D70C}_{1}$ and $\unicode[STIX]{x1D70C}_{2}$ are equivalent.
2.3 Ramification of characters
Lemma 10. Let $C$ be a curve over a number field $K$ , and let $U$ be an open subset. Suppose that $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D70B}_{1}(U)\rightarrow \overline{\mathbf{Q}}_{\ell }^{\times }$ is a continuous homomorphism. Then for every point $x$ of $C_{\overline{K}}$ , the inertia subgroup of $\unicode[STIX]{x1D70B}_{1}(U_{\overline{K}})$ at $x$ has finite image under $\unicode[STIX]{x1D6FC}$ .
Proof. We are free to replace $K$ with a finite extension, so we may as well assume $x$ is a $K$ -point. The decomposition group at $x$ has the form $\widehat{\mathbf{Z}}\rtimes G_{K}$ , where $\widehat{\mathbf{Z}}$ is the geometric inertia group and $G_{K}$ acts on it through the cyclotomic character $\unicode[STIX]{x1D712}$ . Let $T$ be a topological generator of $\widehat{\mathbf{Z}}$ , written multiplicatively. Then for $\unicode[STIX]{x1D70E}\in G_{K}$ we have $\unicode[STIX]{x1D6FC}(T)=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70E}T\unicode[STIX]{x1D70E}^{-1})=\unicode[STIX]{x1D6FC}(T^{\unicode[STIX]{x1D712}(\unicode[STIX]{x1D70E})})=\unicode[STIX]{x1D6FC}(T)^{\unicode[STIX]{x1D712}(\unicode[STIX]{x1D70E})}$ . It follows that $\unicode[STIX]{x1D6FC}(T)$ has finite order.◻
2.4 Some lemmas from representation theory
Lemma 11. Let $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}^{\prime }$ be finite-dimensional representations of a group $G$ with equal determinant. Suppose there exists a normal subgroup $H$ of $G$ such that $\unicode[STIX]{x1D70C}|_{H}$ and $\unicode[STIX]{x1D70C}^{\prime }|_{H}$ are absolutely irreducible and isomorphic. Then there exists a finite-index subgroup $G^{\prime }$ of $G$ such that $\unicode[STIX]{x1D70C}|_{G^{\prime }}$ and $\unicode[STIX]{x1D70C}^{\prime }|_{G^{\prime }}$ are isomorphic.
Proof. Let $V$ and $V^{\prime }$ be the spaces for $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}^{\prime }$ and let $f:V\rightarrow V^{\prime }$ be an isomorphism of $H$ representations. One easily verifies that for $g\in G$ the endomorphism $g^{-1}f^{-1}gf$ of $V$ commutes with $H$ , and is therefore given by multiplication by some scalar $\unicode[STIX]{x1D712}(g)$ . Thus we have $f^{-1}gf=\unicode[STIX]{x1D712}(g)g$ for all $g\in G$ . It follows easily from this that $\unicode[STIX]{x1D712}$ is a homomorphism, i.e., $\unicode[STIX]{x1D712}(gg^{\prime })=\unicode[STIX]{x1D712}(g)\unicode[STIX]{x1D712}(g^{\prime })$ . We thus see that $\unicode[STIX]{x1D712}\otimes \unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D70C}^{\prime }$ are isomorphic as representations of $G$ . Taking determinants, we see that $\unicode[STIX]{x1D712}^{n}$ is trivial, where $n$ is the dimension of $\unicode[STIX]{x1D70C}$ . The result follows by taking $G^{\prime }$ to be the kernel of $\unicode[STIX]{x1D712}$ ; this has finite index since the image of $\unicode[STIX]{x1D712}$ is contained in the group of $n$ th roots of unity of $k$ . (In fact, the index of $G^{\prime }$ divides $n$ .)◻
Lemma 12. Let $G$ be a group and $H$ a normal subgroup of $G$ . Consider a two-dimensional irreducible representation $\unicode[STIX]{x1D70C}:G\rightarrow \mathbf{GL}_{2}(\overline{\mathbf{Q}}_{\ell })$ . Assume that for some element $h\in H$ , the matrix $\unicode[STIX]{x1D70C}(h)$ is a non-trivial unipotent element. Then $\unicode[STIX]{x1D70C}|_{H}$ is irreducible.
Proof. Suppose not. Then by the existence of $h$ , there exists a unique one-dimensional subspace $V$ invariant under $H$ . But since $H$ is normal in $G$ , it follows that $V$ is invariant under $G$ as well. This contradicts the supposition.◻
3 Descent results for abelian varieties
For this section, fix a finitely generated field $K$ . We consider the following condition on a continuous representation $\unicode[STIX]{x1D70C}:G_{K}\rightarrow \mathbf{GL}_{n}(\overline{\mathbf{Q}}_{\ell })$ .
(∗) There exists an integral scheme $X$ of finite type over $\operatorname{Spec}(\mathbf{Z})$ with function field $K$ and a lisse $\overline{\mathbf{Q}}_{\ell }$ -sheaf $L$ on $X$ with generic fiber $\unicode[STIX]{x1D70C}$ such that at every closed point $x$ of $X$ the trace of the Frobenius element on $L_{x}$ is rational.
In our proof of Theorem 1, we will show that $\unicode[STIX]{x1D70C}$ comes from an elliptic curve over some finite extension of $K^{\prime }$ . We use the following result to conclude that we actually get an elliptic curve over $K$ .
Proposition 13. Let $\unicode[STIX]{x1D70C}:G_{K}\rightarrow \mathbf{GL}_{2}(\overline{\mathbf{Q}}_{\ell })$ be a Galois representation satisfying condition ( $\ast$ ). Suppose that there exists a finite extension $K^{\prime }/K$ such that $\unicode[STIX]{x1D70C}|_{K^{\prime }}$ comes from a non-CM elliptic curve $E$ . Then $\unicode[STIX]{x1D70C}$ comes from an elliptic curve.Footnote 1
We proceed with a number of lemmas.
Lemma 14. Let $\unicode[STIX]{x1D70C}:G\rightarrow \mathbf{GL}_{n}(\overline{\mathbf{Q}}_{\ell })$ a continuous representation of the profinite group $G$ . Suppose there exists an open subgroup $H$ of $G$ such that $\unicode[STIX]{x1D70C}(H)$ is contained in $\mathbf{GL}_{n}(\mathbf{Z}_{\ell })$ and contains an open subgroup of $\mathbf{GL}_{n}(\mathbf{Z}_{\ell })$ . Suppose also that there is a dense set of elements $\{g_{i}\}_{i\in I}$ of $G$ such that $\operatorname{tr}\unicode[STIX]{x1D70C}(g_{i})\in \mathbf{Q}_{\ell }$ for all $i\in I$ . Then $\unicode[STIX]{x1D70C}(G)$ is contained in $\mathbf{GL}_{n}(\mathbf{Q}_{\ell })$ .
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.◻
The following lemma is basically [Reference TaylorTay02, Corollary 2.4].
Lemma 15. Let $\unicode[STIX]{x1D70C}:G_{K}\rightarrow \mathbf{GL}_{2}(\overline{\mathbf{Q}}_{\ell })$ be a continuous irreducible Galois representation. Suppose that there is a finite separable extension $K^{\prime }/K$ such that $\unicode[STIX]{x1D70C}|_{G_{K^{\prime }}}$ comes from a non-CM elliptic curve $E$ . Then there is an abelian variety $A/K$ with $F=\operatorname{End}(A)\otimes \mathbf{Q}$ a number field of degree $\dim (A)$ such that $\unicode[STIX]{x1D70C}\cong \overline{\mathbf{Q}}_{\ell }\otimes _{F_{w}}V_{w}(A)$ for some place $w$ of $A$ .
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
where $D_{i}=\operatorname{End}_{K}(B_{i})\otimes \mathbf{Q}$ is a simple algebra. We have
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.◻
Lemma 16. Proposition 13 holds if $K^{\prime }/K$ is separable.
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.◻
Lemma 17. Proposition 13 holds if $K^{\prime }/K$ is purely inseparable.
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.◻
Proposition 13 in general follows from the previous two lemmas. We now prove a slightly different descent result. Recall that for an elliptic curve $E$ and a number field $F$ , one has an abelian variety $E\otimes {\mathcal{O}}_{F}$ with multiplication by ${\mathcal{O}}_{F}$ : as an abelian variety, $E\otimes {\mathcal{O}}_{F}$ is simply $E^{n}$ where $n=[F:\mathbf{Q}]$ .
Proposition 18. Let $k$ be a finite field, let $C/k$ be a proper smooth geometrically irreducible curve, and let $f:A\rightarrow U$ be a family of $g$ -dimensional abelian varieties over a non-empty open subset $U$ of $C$ such that $\operatorname{End}(A)\otimes \mathbf{Q}$ contains a number field $F$ of degree $g$ . For a finite place $v$ of $F$ , let ${\mathcal{L}}_{v}$ be the $v$ -adic Tate module of $A$ . Assume that for all closed points $x$ of $U$ , the trace of the Frobenius element at $x$ on ${\mathcal{L}}_{v,x}$ belongs to $\mathbf{Q}$ . Also assume that there is some place of $C$ where $A$ does not have potentially good reduction. Then there exists an elliptic curve $E\rightarrow U$ such that $A$ is isogenous to $E\otimes {\mathcal{O}}_{F}$ .
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.◻
4 Drinfeld’s work on the global Langlands program
Proposition 19. Let $k$ be a finite field, and $C/k$ be a smooth proper geometrically irreducible curve. Let $L$ be an irreducible rank-two lisse $\overline{\mathbf{Q}}_{\ell }$ -sheaf over a non-empty open subset $U\subset C$ , such that the following hold.
(a) There is an isomorphism $\bigwedge ^{2}L\cong \overline{\mathbf{Q}}_{\ell }(1)$ .
(b) For every closed point $x$ of $C$ , the trace of the Frobenius element on $L_{x}$ is a rational number.
(c) There exists a point $x$ of $C_{\overline{k}}$ at which $L_{\overline{k}}$ does not have potentially good reduction.
Then there exists an elliptic curve $f:E\rightarrow U$ such that $\text{R}^{1}f_{\ast }(\overline{\mathbf{Q}}_{\ell })\cong L$ .
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.◻
Remark 20. By following Drinfeld’s proof carefully, one may directly see that we can take $E$ to be the field generated by the Frobenius traces of $L$ , and is thus $\mathbf{Q}$ , which can replace the use of Proposition 18.
5 Mapping spaces
Let $S$ be a noetherian scheme. For $i=1,2$ , let $C_{i}$ be a proper smooth scheme over $S$ with geometric fibers irreducible curves, let $Z_{i}$ be a closed subscheme of $C_{i}$ that is a finite union of sections $S\rightarrow C_{i}$ , and let $U_{i}$ be the complement of $Z_{i}$ in $C_{i}$ . Fix $d\geqslant 1$ .
Proposition 21. There exists a scheme ${\mathcal{M}}$ of finite type over $S$ and a map $\unicode[STIX]{x1D719}:(U_{1})_{{\mathcal{M}}}\rightarrow (U_{2})_{{\mathcal{M}}}$ with the following property: if $k$ is a field, $s\in S(k)$ , and $f:U_{1,s}\rightarrow U_{2,s}$ is a map of curves over $k$ of degree $d$ (meaning the corresponding function field extension has degree $d$ ), then there exists $t\in {\mathcal{M}}(k)$ over $s$ such that $f=\unicode[STIX]{x1D719}_{t}$ .
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.◻
6 Proof of Theorem 1
Keep notation as in Theorem 1, and put $S=\operatorname{Spec}({\mathcal{O}}_{K}[1/N])$ .
Lemma 22. The restriction of $L$ to any open subgroup of $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U_{K})$ is irreducible.
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).◻
Choose $r\gg 0$ so that $X=X(\ell ^{r})$ has genus at least 2 and $Y=Y(\ell ^{r})$ is a fine moduli space; in fact, $r=3$ suffices for any $\ell$ . We replace ${\mathcal{L}}$ with a rank-two ${\mathcal{O}}_{E}$ -sheaf, where $E/\mathbf{Q}_{\ell }$ is a finite extension. Proposition 13 and Lemma 22 show that it suffices to prove the proposition after passing to a finite cover of $C$ . By passing to an appropriate cover, we can therefore assume that ${\mathcal{L}}/\ell ^{r}$ is trivial. The image of the Galois representation $\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}({\mathcal{U}})\rightarrow \mathbf{GL}_{2}({\mathcal{O}}_{E})$ has order $M\cdot \ell ^{\infty }$ (in the sense of profinite groups) for some positive integer $M$ . By enlarging $N$ , we can assume that $M\cdot \ell \mid N$ and that the complement of $U$ in $C$ spreads out to a divisor on ${\mathcal{C}}$ that is smooth over $S$ .
We make the following definitions.
∙ Let $D$ be an integer greater than $(g(C)-1)/(g(X)-1)$ , where $g(-)$ denotes genus. Let ${\mathcal{M}}_{d}$ be the space of maps ${\mathcal{U}}\rightarrow Y$ of degree $d$ , in the sense of Proposition 21, and let ${\mathcal{M}}=\coprod _{d=1}^{D}{\mathcal{M}}_{d}$ .
∙ Let ${\mathcal{T}}$ be the (integral) $\ell$ -adic Tate module of the universal elliptic curve over $Y$ , and let ${\mathcal{T}}^{\prime }={\mathcal{T}}\otimes {\mathcal{O}}_{E}$ . Also let ${\mathcal{L}}_{n}={\mathcal{L}}/\ell ^{n}{\mathcal{L}}$ and let ${\mathcal{T}}_{n}^{\prime }={\mathcal{T}}^{\prime }/\ell ^{n}{\mathcal{T}}^{\prime }$ .
∙ Let $\widetilde{{\mathcal{M}}}_{n}$ be the moduli space of pairs $(f,\unicode[STIX]{x1D713})$ where $f\in {\mathcal{M}}$ and $\unicode[STIX]{x1D713}$ is an isomorphism of ${\mathcal{O}}_{E}$ -sheaves $f^{\ast }({\mathcal{T}}_{n})\rightarrow {\mathcal{L}}_{n}$ .
Lemma 23. The map $\unicode[STIX]{x1D70B}:\widetilde{{\mathcal{M}}}_{n}\rightarrow {\mathcal{M}}$ is finite.
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 $U_{R}$ .◻
Let ${\mathcal{M}}_{n}$ be the image of $\widetilde{{\mathcal{M}}}_{n}$ in ${\mathcal{M}}$ , which is closed by Lemma 23. We endow it with the reduced subscheme structure. As the ${\mathcal{M}}_{n}$ form a descending chain of closed subschemes of ${\mathcal{M}}$ , they stabilize. Let ${\mathcal{M}}_{\infty }$ be ${\mathcal{M}}_{n}$ for $n\gg 0$ .
Lemma 24. The fiber of ${\mathcal{M}}_{\infty }$ over all closed points of $S$ is non-empty.
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.◻
Acknowledgement
We thank the referee for helpful comments, which improved the exposition of the paper.