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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20181010075702120-0797:S0010437X18007315:S0010437X18007315_eqnU1.gif?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20181010075702120-0797:S0010437X18007315:S0010437X18007315_eqnU2.gif?pub-status=live)
where
$D_{i}=\operatorname{End}_{K}(B_{i})\otimes \mathbf{Q}$
is a simple algebra. We have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20181010075702120-0797:S0010437X18007315:S0010437X18007315_eqnU3.gif?pub-status=live)
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],
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20181010075702120-0797:S0010437X18007315:S0010437X18007315_eqn1.gif?pub-status=live)
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.