Proof. For all closed points $x$ of $X$, we have that $P_x(L,t)\in {{\mathbb {Q}}}[t]\subset \overline {\mathbb {Q}}_l[t]$ as $\iota ({{\mathbb {Q}}})={{\mathbb {Q}}}\subset \overline {\mathbb {Q}}_l$. The first statement then follows from the Cebotarev density theorem and the Brauer–Nesbitt theorem.

If ${{\mathcal {E}}}_i$ is an irreducible summand of ${{\mathcal {E}}}$, then $^{\iota }{{\mathcal {E}}}_i$ is an irreducible $\overline {\mathbb {Q}}_l$-sheaf by [Reference KedlayaKed18, Theorem 3.3.1]. As the companions relation commutes with direct sum, it follows that if ${{\mathcal {E}}}\cong \oplus {{\mathcal {E}}}_i^{m_i}$ is the decomposition of ${{\mathcal {E}}}$ into irreducible objects in $\textbf {F-Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$, then $L\cong \oplus (^{\iota }{{\mathcal {E}}}_i)^{m_i}$ is a decomposition of $L$ into irreducible lisse $\overline {\mathbb {Q}}_l$-sheaves on $X$. One may observe that $\det (^{\iota }{{\mathcal {E}}}_i)\cong \overline {\mathbb {Q}}_l(-1)$ because, for every closed point $x$ of $X$, the constant term of $P_x({{\mathcal {E}}}_i,t)$ is $q$ and hence the constant term of $P_x(^{\iota }{{\mathcal {E}}}_i,t)$ is also $q$. Finally, suppose for contradiction that there exists an $i$ with $L_i:=\ ^{\iota }{{\mathcal {E}}}_i$ having finite local monodromy at infinity. Then there exist a smooth projective variety $\bar {X}'/\mathbb {F}_q$, an open dense subscheme $X'\subset \bar {X}'$, and an alteration $f\colon X'\rightarrow X$ such that $f^{*}L_i$ extends to $\bar {X}'$. It follows from [Reference KedlayaKed18, Corollary 3.3.3] that $f^{*}{{\mathcal {E}}}_i$ also extends to $\bar {X}'$, contradicting the hypothesis that ${{\mathcal {E}}}_i$ had infinite local monodromy at infinity.

Proof. We reduce the crystalline case to the étale case. (Note that we could have equivalently proceeded by reduction to curves using [Reference Abe and EsnaultAE19].) As ${{\mathcal {F}}}$ is semi-simple, write an isotypic decomposition:

\[ \displaystyle {{\mathcal{F}}}\cong \bigoplus_{i=1}^{a}{{\mathcal{F}}}_i^{m_i}. \]

Note that each ${{\mathcal {F}}}_i$ is pure by [Reference Abe and EsnaultAE19, Theorem 2.7]. Fix an isomorphism $\sigma \colon \overline {\mathbb {Q}}_p\rightarrow \overline {\mathbb {Q}}_l$. By [Reference Abe and EsnaultAE19, Theorem 4.2] or [Reference KedlayaKed18, Corollary 3.5.3], the $\sigma$-companion to each ${{\mathcal {F}}}_i$ exists as an irreducible lisse $\overline {\mathbb {Q}}_l$-sheaf $L_i$. Setting $L$ to be the semi-simple $\sigma$-companion of ${{\mathcal {F}}}$, we have

\[ \displaystyle L\cong \bigoplus_{i=1}^{a}L_i^{m_i}. \]

Set $\iota \in \text {Aut}_{{{\mathbb {Q}}}}(\overline {\mathbb {Q}}_p)$. Then ${{\mathcal {F}}}_j$ is the $\iota$-companion to ${{\mathcal {F}}}_i$ if and only if $L_j$ is the $\sigma \circ \iota \circ \sigma ^{-1}$-companion to $L_i$. Therefore it suffices to prove the result in the étale setting.

Let $M$ be an irreducible lisse $\overline {\mathbb {Q}}_l$-sheaf on $X$. Then $M$ is pure by [Reference DeligneDel12, Théorème 1.6] and class field theory. Then the multiplicity of $M$ in the semi-simple sheaf $L$ is $\dim (H^{0}(X,M^{*}\otimes L))$. By assumption we have that, for all closed points $x$ of $X$, $P_x(L,t)\in {{\mathbb {Q}}}[t]\subset \overline {\mathbb {Q}}_l[t]$. Let $\iota \in \textrm {Aut}_{{{\mathbb {Q}}}}(\overline {\mathbb {Q}}_l)$, and note that the semi-simple $\iota$-companion to $L$ is again isomorphic to $L$. Then we claim that the $\iota$-companion to $M^{*}\otimes L$ is isomorphic to $(^{\iota } M^{*})\otimes L$. Indeed, this follows from the following two facts. First of all, both $M^{*}\otimes L$ and $(^{\iota } M^{*})\otimes L$, being the tensor product of semi-simple representations of characteristic 0, are semi-simple. Second, it follows from the fundamental theorem of symmetric functions that for fixed $d,e\in \mathbb {N}$ there exist universal polynomials $(u_i)_{i=0}^{de}$ in the ring $\mathbb {Q}[\alpha _1,\dots,\alpha _d,\beta _1,\dots,\beta _e]$ with the following property. Let $V$ and $W$ be finite-dimensional vector spaces over a field $K$ of characteristic 0 and of dimensions $d$ and $e$ and let $A$ and $B$ be linear operators on $V$ and $W$, respectively. Write $P(A,t)=\sum _{i=0}^{d}a_it^{i}$ and $P(B,t)=\sum _{j=0}^{e}b_jt^{j}$ for the reverse characteristic polynomials of $A$ and $B$. Then the reverse characteristic polynomial $P(A\otimes B,t)$ of $A\otimes B$ is equal to

\[ \displaystyle P(A\otimes B,t)=\sum_{k=0}^{de}u_k(a_0,\dots,a_d,b_0,\dots,b_e)t^{k}. \]

Translating back, let $x$ be a closed point of $X$ and write $P_x(M^{*},t)=\sum _{i=0}^{d} a_it^{i}$ and $P_x(L,t)=\sum _{j=0}^{e} b_jt^{j}$. Then we have

\[ \displaystyle P_x(M^{*}\otimes L,t)=\sum_{k=0}^{de}u_k(a_0,\dots,a_d,b_0,\dots,b_e)t^{k}. \]

It follows that

\[ P_x(^{\iota}(M^{*}\otimes L),t)=P_x((^{\iota}M^{*})\otimes\ ^{\iota}L,t)=\sum_{k=0}^{de}u_k(\iota(a_0),\dots,\iota(a_d),\iota(b_0),\dots,\iota(b_e))t^{k}. \]

But $\iota (b_j)=b_j$ because $b_j\in \mathbb {Q}$ for all $j$. Therefore $P_x(^{\iota }(M^{*}\otimes L),t)=P_x(^{\iota }(M^{*})\otimes L,t)$. The semi-simplicity of $^{\iota }(M^{*}\otimes L)$ and $^{\iota }M^{*}\otimes L$ allows us to conclude that $^{\iota }(M^{*}\otimes L)$ is isomorphic to $^{\iota }M^{*}\otimes L$.

On the other hand, the exact argument of [Reference Abe and EsnaultAE19, 3.2] for lisse $l$-adic sheaves implies that $\dim (H^{0}(X,M^{*}\otimes L))=\dim (H^{0}(X,\ ^{\iota }(M^{*} \otimes L))$. Therefore $\dim (H^{0}(X,M^{*}\otimes L))=\dim (H^{0}(X,(^{\iota }M^{*})\otimes L))$, and the result follows.

Proof. There are finitely many ordered $m+1$-tuples of sections $H^{0}(Y,\alpha )\cong H^{0}(C_n,\alpha |_{C_n})$ because $H^{0}(Y,\alpha )$ is a finite-dimensional vector space over $\mathbb {F}_q$. By the pigeonhole principle, in our infinite collection we may find an $m+1$-tuple of sections $\tilde {t}_0,\dots,\tilde {t}_m\in H^{0}(Y,\alpha )$ such that there exists an infinite set $S\subset \mathbb {N}$ with

\[ (\tilde{t}_0,\dots,\tilde{t}_m)|_{C_n}=(t_{n,0},\dots,t_{n,m}) \]

for every $n\in S$. There is therefore an induced rational map $\tilde {f}\colon Y\dashrightarrow \mathbb {P}^{m}$ with $\tilde {f}|_{C_n}=f_n$ for each $n\in S$. On the other hand, the collection $(C_n)_{n\in S}$ is Zariski dense in $Y$ by assumption and $\tilde {f}(C_n)\subset M$; therefore the image of $\tilde {f}$ lands inside of $M$, as desired.

Proof. For any $s>0$, let $E\in |sD|$ be an integral divisor in the linear series. Then there is an exact sequence

\[ 0\rightarrow \alpha(-E)\rightarrow \alpha\rightarrow \alpha|_E\rightarrow 0. \]

If $h^{0}(Y,\alpha (-E))=h^{1}(Y,\alpha (-E))=0$, then by the long exact sequence in cohomology, the restriction map $H^{0}(Y,\alpha )\rightarrow H^{0}(E,\alpha |_E)$ is an isomorphism. Our task is therefore to show that, for all sufficiently large $s$, $h^{0}(Y,\alpha (-sD))=h^{1}(Y,\alpha (-sD))=0$.

Let $\mathfrak {L}$ be the canonical bundle of $Y$. Then by Serre duality, $h^{i}(Y,\alpha (-sD))=h^{d-i}(Y,\alpha ^{\vee }(sD)\otimes \mathfrak {L})$. It follows from Serre vanishing that there exists an $s_0>0$ such that, for any $s\geq s_0$ and for any $i< d$, $h^{d-i}(Y,\alpha ^{\vee }(sD)\otimes \mathfrak {L})=0$. Therefore, for any $s\geq s_0$ and for any $i< d$, $h^{i}(Y,\alpha (-sD))=0$ and the result follows.

Proof. By induction, it suffices to construct a smooth ample divisor $\bar {D}\subset \bar {X}$ such that:

• • $\bar {D}\cap \bar {U}$ has complementary codimension at least $2$ in $\bar {D}$;

• • $\bar {D}$ intersects $Z$ transversely;

• • $\bar {D}$ contains $x_j$, for $j=1,\dots, s$; and

• • the natural map $H^{0}(\bar {X},\alpha )\rightarrow H^{0}(\bar {D},\alpha |_{\bar {D}})$ is an isomorphism.

This is a standard application of Poonen's Bertini theorem over finite fields [Reference PoonenPoo04, Theorem 1.3]. Fix a closed embedding $\bar {X}\hookrightarrow \mathbb {P}^{m}_{\mathbb {F}_q}$ and let $S_{\text {homog}}$ be the set of homogenous polynomials on $\mathbb {P}^{m}_{\mathbb {F}_q}$, as in [Reference PoonenPoo04, p. 1100]. Consider the set $\mathcal {T}$ of those functions $f\in S_{\text {homog}}$ such that $\bar {D}:=V(f)\cap \bar {X}$ is a smooth ample divisor of $\bar {X}$ and the above four properties hold for $\bar {D}$. Our goal is to show that $\mathcal {T}$ is non-empty.

• • Let $\bar E:=\bar {X}{\setminus} \bar {U}$; by hypothesis, $\dim (\bar E)\leq n-2$. If $f\in S_{\text {homog}}$ is such that $V(f)$ does not contain any component of $\bar E$, then $\dim (V(f)\cap \bar E)\leq n-3$. For this to hold, it is sufficient that $V(f)$ avoids at least one given closed point $e_i$ on each connected component of $\bar E$.

• • Write $Z=\bigcup ^{r}_{j=1} Z_j$ to be the decomposition of $Z$ into connected components. For each $J\subset \{1,2,\dots,r\}$, set $Z_J:=\bigcap _{j\in J}Z_j$ to be the corresponding scheme-theoretic intersection. By assumption, for each $J$, $Z_J$ is a smooth subvariety of $\bar {X}$. The condition that $\bar {D}$ intersects $Z$ in good position means that $\bar {D}$ must intersect each stratum $Z_J$ transversely, that is, that $Z_J\cap \bar {D}$ is a smooth subvariety of $\bar {D}$ of dimension $n-1-|J|$.

Then the positive density (and hence non-emptiness) of $\mathcal {T}$ immediately follows from [Reference PoonenPoo04, Theorem 1.3]: the conditions on $f$ are that $V(f)\cap \bar {X}$ intersect a finite set of smooth subvarieties transversely, avoid a given finite set of points, pass through another given finite set of points, and have sufficiently high degree by Proposition 2.11.

Proof. Set $N=\text {ker}(\psi )$. Then $N$ is a (commutative) finite flat group scheme over $X$ of $r$-primary order. We have a short exact sequence in the category of fppf sheaves:

\[ 0\rightarrow N\rightarrow A[p^{\infty}]\xrightarrow{\psi} G\rightarrow 0. \]

Consider the quotient $A/N$ in the category of fppf sheaves. It follows from, for example, [Reference Chai, Conrad and OortCCO14, 1.4.1.3, 1.4.1.4] that there exists an abelian scheme $B\rightarrow X$ that represents the sheaf $A/N$. We then have the following commutative diagram of fppf sheaves:

where the right vertical arrow exists because $G=\text {coker}(N\rightarrow A[r^{\infty }])$. We claim that the induced map $G\rightarrow B$ yields an isomorphism $G\rightarrow B[r^{\infty }]$. By the snake lemma, $G\rightarrow B$ is injective. However, an injective isogeny of $r$-divisible groups is an isomorphism.

Proof of Theorem 1.2 We proceed in several steps.

Step 1: organizing the summands of $\mathcal {E}$. As ${{\mathcal {E}}}_i$ is irreducible, has determinant $\overline {\mathbb {Q}}_p(-1)$, and has rank $2$, the slopes of $({{\mathcal {E}}}_i)_x$ are in the interval $[0,1]$ for every closed point $x$ of $X$ (see [Reference Drinfeld and KedlayaDK17, § 1.2, pp. 136–137], where it is deduced from Corollary 1.1.7).

Write the isotypic decomposition of ${{\mathcal {E}}}$ in $\textbf {F-Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$:

\[ \displaystyle {{\mathcal{E}}}\cong \bigoplus_{i=1}^{a}({{\mathcal{E}}}_i)^{m_i}. \]

The field generated by the coefficients of $P_x({{\mathcal {E}}},t)$ as $x$ ranges through closed points of $X$ is ${{\mathbb {Q}}}$. Therefore, by [Reference DrinfeldDri18, E.10] and either [Reference Abe and EsnaultAE19, Theorem 4.2] or [Reference KedlayaKed18, Corollary 3.5.3], we can pick an $l$ and a field isomorphism $\sigma \colon \overline {\mathbb {Q}}_p\rightarrow \overline {\mathbb {Q}}_l$ such that the semi-simple $\sigma$ companion $L$ to ${{\mathcal {E}}}$ exists and in fact may be defined over $\mathbb {Q}_l$, that is, corresponds to a representation

\[ \pi_1(X)\rightarrow \mathrm{GL}_N(\mathbb{Q}_l). \]

(We emphasize that $L$ is independent of the choice of $\sigma$ by Proposition 2.4.) By compactness of $\pi _1(X)$, we may conjugate the representation into $\mathrm {GL}_N({{\mathbb {Z}}}_l)$. We refer to the attached lisse ${{\mathbb {Z}}}_l$-sheaf as $\tilde {L}$. Similarly, for each $i$ we denote by $L_i$ the $\sigma$-companion to ${{\mathcal {E}}}_i$ (the $L_i$ indeed do depend on the choice of $\sigma$). The companion relation commutes with direct sum; hence, we have

\[ \displaystyle L\cong \bigoplus_{i=1}^{a} L_i^{m_i}. \]

(See also the proof of Proposition 2.4.) Let $E_i\subset \overline {\mathbb {Q}}_p$ denote the (number) field generated by the coefficients of $P_x({{\mathcal {E}}}_i,t)$ as $x$ ranges through the closed points of $X$. Note that for each ${{\mathcal {E}}}_i$, all $p$-adic companions exist and are summands of ${{\mathcal {E}}}$ by Lemma 2.5. For each ${{\mathcal {E}}}_i$, set ${{\mathcal {F}}}_i$ to be the sum of all distinct $p$-adic companions of ${{\mathcal {E}}}_i$. Note that there are $[E_i\colon {{\mathbb {Q}}}]$ distinct $p$-adic companions of ${{\mathcal {E}}}_i$, parametrized by the embeddings $E_i\hookrightarrow \overline {\mathbb {Q}}_p$. By reordering the indices, we write the decomposition of ${{\mathcal {E}}}$ as

(3.1)\begin{equation} \displaystyle {{\mathcal{E}}}\cong \bigoplus_{i=1}^{b} {{\mathcal{F}}}_i^{m_i} \end{equation}

for some integer $1\leq b\leq a$. (Under this reordering, the collection of $({{\mathcal {E}}}_i)_{i=1}^{b}$ are all mutually not companions and, for each $b+1\leq j \leq a$, there exists a unique $1\leq i\leq b$ such that ${{\mathcal {E}}}_j$ is a companion of ${{\mathcal {E}}}_i$.) Set

(3.2)\begin{equation} g=\sum_{j=1}^{b} m_i[E_i\colon {{\mathbb{Q}}}].\end{equation}

Step 2: the proof in a simplified situation. We first assume that $X$ admits a simple normal crossings compactification $\bar {X}$ such that ${{\mathcal {E}}}$ extends to a logarithmic $F$-isocrystal $\bar {{\mathcal {E}}}$ with nilpotent residues on $\bar {X}$ and, moreover, that $\tilde {L}$ has trivial residual representation. Write $Z:=\bar {X}\backslash X$ for the boundary. (Note that under the above assumption on ${{\mathcal {E}}}$, the $l$-companion $L$ is tamely ramified.)

By Lemma A.7, there exist a Zariski open $\bar {U}\subset \bar {X}$ with complementary codimension at least 2, and a logarithmic Dieudonné crystal $(M_{\bar {U}},F,V)$ on $\bar {U}$ (with the logarithmic structure coming from $Z\cap \bar {U}$) such that the associated logarithmic $F$-isocrystal is isomorphic to $\overline {{{\mathcal {E}}}}|_{\bar {U}}$. (In other words, $M_{\bar {U}}$ is an $F$ and $p\circ F^{-1}$ stable lattice in $\overline {{{\mathcal {E}}}}|_{\bar {U}}$.) Let

\[ (N_{\bar{U}},F,V):=(M_{\bar{U}},F,V)^{4}\oplus ((M_{\bar{U}},F,V)^{t})^{4}, \]

where the $t$ denotes the dual logarithmic Dieudonné crystal. We also consider this logarithmic Dieudonné crystal as we will need to use Zarhin's trick. We set $U:=\bar {U}\backslash (\bar {U}\cap Z)$.

After Remark A.8, it follows that we may define Hodge line bundles $\omega _M$ and $\omega _N$ on $\bar {U}$ attached to the two logarithmic Dieudonné crystals. As $\bar {U}\subset \bar {X}$ has complementary codimension at least $2$ and $\bar {X}$ is smooth, it follows that $\omega _M$ and $\omega _N$ extend canonically to line bundles on all of $\bar {X}$.

The Hodge line bundle $\alpha$ on the fine moduli scheme $\mathscr {A}_{8g,1,l}\otimes \mathbb {F}_q$ is ample by [Reference Moret-BaillyMB85, Ch. IX, Théorème 3.1, p. 210] or [Reference Faltings and ChaiFC90, Ch. V, Theorem 2.5(i)]. Let $g$ be as in (3.2) and choose an $r$ so that the $\alpha ^{r}$ is very ample on $\mathscr {A}_{8g,1,l}$. As $8g>1$, it follows from the Koecher principle that $H^{0}(\mathscr {A}_{8g,1,l}\otimes \mathbb {F}_q,\alpha ^{r})$ is a finite-dimensional $\mathbb {F}_q$-vector space for all $r\in \mathbb {Z}$ [Reference Faltings and ChaiFC90, Ch. V, Theorem 1.5(ii)]. Fix a basis $s_0,\dots,s_m$ of the vector space

(3.3)\begin{equation} s_0,\dots,s_m\in H^{0}(\mathscr{A}_{8g,1,l}\otimes \mathbb{F}_q,\alpha^{r})\end{equation}

once and for all. There is an induced embedding $\mathscr {A}_{8g,1,l}\subset \mathbb {P}^{m}$. As is customary, denote by $\mathscr {A}^{*}_{8g,1,l}$ the Zariski closure of $\mathscr {A}_{8g,1,l}$ in $\mathbb {P}^{m}$; we call this the minimal compactification. In an abuse of notation, we also denote by $\alpha$ the Hodge line bundle on $\mathscr {A}^{*}_{8g,1,l}$. The Koecher principle implies that $H^{0}(\mathscr {A}_{8g,1,l}\otimes \mathbb {F}_q,\alpha ^{r})=H^{0}(\mathscr {A}^{*}_{8g,1,l}\otimes \mathbb {F}_q,\alpha ^{r})$; this follows from [Reference Faltings and ChaiFC90, Ch. V, Theorem 1.5(ii), Theorem 2.5(iii)].

It follows from [Reference DeligneDel12, Proposition 3.4] that there exist a finite number of closed points $(x_j)^{s}_{j=1}$ of $U$ such that, for each ${{\mathcal {E}}}_i$, the field generated by the coefficients of $P_{x_j}({{\mathcal {E}}}_i,t)\in \overline {\mathbb {Q}}_p[t]$ as $j=1,\dots, s$ is $E_i\subset \overline {\mathbb {Q}}_p$. We call this fact $\blacklozenge$.

If $\bar {C}\subset \bar {U}$ is a good curve for the quintuple $(\bar {X}, X, \bar {U}, \omega _N^{r}, (x_j)^{s}_{j=1})$ as in Definition 2.10, set $C:=\bar {C}\cap X$. Then the following three properties hold.

• • Each ${{\mathcal {E}}}_i|_C$ is irreducible by [Reference Abe and EsnaultAE19, Theorem 2.6].

• • The field generated by Frobenius traces of ${{\mathcal {E}}}_i|_C$ is $E_i$ by $\blacklozenge$.

• • Each ${{\mathcal {E}}}_i|_C$ has infinite monodromy around $\infty$. Indeed, from the positivity of $\bar {C}$, and the good position assumption, it follows that $\bar {C}$ intersects each irreducible component $Z_m$ of $Z$ in a non-empty and transverse way; moreover, $\bar {C}$ does not intersect the codimension $2$ strata $Z_{m}\cap Z_n$. By assumption, for each ${{\mathcal {E}}}_i$, there exists a component $Z_m$ around which the monodromy around $Z_m$ of ${{\mathcal {E}}}_i$ (equivalently, of $L_i$) is infinite. On the other hand, there is a surjective morphism of tame fundamental groups

  \[ \pi_1^{\text{tame}}(C)\twoheadrightarrow \pi_1^{\text{tame}}(X) \]
  by [Reference Esnault and KindlerEK16, Theorem 1.1(a)]. Moreover, for each $m$, we may restrict the above surjection to a surjective map of tame inertia groups
  \[ I_{Z_m\cap \bar{C}}^{\text{tame}}(C)\twoheadrightarrow I_{Z_m}^{\text{tame}}(X) \]
  around $Z_m\cap \bar {C}$ and $Z_m$, respectively. By the assumption that $L_i$ had infinite monodromy around $Z_m$ and the fact that wild inertia is a pro-$p$ group, it follows that the image of $I_{Z_m}^{\text {tame}}(X)$ in the $l$-adic representation corresponding to $L_i$ is infinite. Therefore, the image of $I_{Z_m\cap \bar {C}}^{\text {tame}}(C)$ in the $l$-adic representation corresponding to $L_i|_C$ is also infinite, or equivalently, ${{\mathcal {E}}}_i|_C$ has infinite monodromy around $Z_m\cap \bar {C}$, as desired.

Let $\bar {C}\subset \bar {U}$ be a good curve for the quintuple $(\bar {X}, X, \bar {U}, \omega _N^{r}, (x_j)^{s}_{j=1})$. Recall the decomposition from (3.1): $\displaystyle {{\mathcal {E}}}\cong \oplus _{i=1}^{b} {{\mathcal {F}}}_i^{m_i}$, where each ${{\mathcal {F}}}_i$ is the sum of the distinct companions of ${{\mathcal {E}}}_i$ under the reordering specified in Step 1. (Note that ${{\mathcal {F}}}_i$ has Frobenius traces in ${{\mathbb {Q}}}$.) By Theorem 2.1 and Remark 2.8, for each $i\in \{1,\dots,b\}$, there exists an abelian scheme $A_i\rightarrow C$ of dimension $g_i=[E_i:{{\mathbb {Q}}}]$ such that ${{\mathcal {F}}}_i|_C$ is compatible with $A_i$. By taking the iterated fiber product over $C$, it therefore follows from (3.2) that there exists an abelian scheme $\pi _C\colon A_C\rightarrow C$ of relative dimension $g$ such that

\[ R^{1}(\pi_C)_*\overline{\mathbb{Q}}_l\cong L|_C. \]

As $l$ is prime to $p$, it follows from the Galois correspondence for $\pi _1(X)$ that the category of (necessarily étale) $l$-divisible groups on $X$ is equivalent to the category of lisse $\mathbb {Z}_l$ sheaves on $X$ (see, for example, [Reference Chai, Conrad and OortCCO14, pp. 147–148], where they explain that the functor is explicitly given as the Tate $l$-group). Write $\Phi$ for an inverse functor. We have assumed that the ${{\mathbb {Z}}}_l$-lattice $\tilde {L}$ has trivial residual representation, that is, the following map is trivial:

\[ \pi_1(X)\rightarrow \mathrm{GL}_{2g}({{\mathbb{Z}}}/l{{\mathbb{Z}}}). \]

Then it follows from the Galois correspondence that $\Phi (\tilde {L})[l]$ is isomorphic to the split étale group scheme $({{\mathbb {Z}}}/l{{\mathbb {Z}}})^{2g}$ (see, for example, the explicit formula on [Reference Chai, Conrad and OortCCO14, p. 148]). On the other hand, $\Phi (\tilde {L})$ is isogenous to $A_C[l^{\infty }]$ because $\tilde {L}$ is isogenous to $T_l(A_C)$. It follows from Lemma 2.13 that there exists an $l$-primary isogeny $A_C\rightarrow A'_C$ over $C$ such that $A'_C[l]\rightarrow C$ is isomorphic to the split étale group scheme $({{\mathbb {Z}}}/l{{\mathbb {Z}}})^{2g}$. The abelian scheme $A'_C\rightarrow C$ therefore has a full collection of $l$-torsion sections, that is, it has trivial $l$-torsion. Replacing $A_C$ by $A_C'$, we may assume that $A_C$ has trivial $l$-torsion.

Similarly, we claim that $\mathbb {D}(A_C[p^{\infty }])\otimes \overline {\mathbb {Q}}_p\cong {{\mathcal {E}}}|_C$. Indeed, $\mathbb {D}(A_C[p^{\infty }])\otimes \overline {\mathbb {Q}}_p$ is a semi-simple object of $\textbf {F-Isoc}^{{\dagger} }({C})_{\overline {\mathbb {Q}}_p}$ by [Reference PálPál15] and is compatible with $L|_C$ by [Reference Katz and MessingKM74]. Therefore, $\mathbb {D}(A_C[p^{\infty }])$ is isogenous to $(M,F,V)_C$ as Dieudonné crystals on $C$. We claim that we may replace $A_C$ by an ($p$-primarily) isogenous abelian scheme in order to ensure that

\[ \mathbb{D}(A_C[p^{\infty}])\cong (M,F,V)_C \]

as Dieudonné crystals on $C$. To see this, use [Reference de JongdJ95] to construct a $p$-divisible group $G_C$ on $C$ where $\mathbb {D}(G_C)\cong (M_C,F,V)$. It follows that $A_C[p^{\infty }]$ and $G_C$ are isogenous. Applying Lemma 2.13, we see that there is a $p$-primary isogeny $A_C\rightarrow A'_C$ such that $A'_C[p^{\infty }]\cong G_C$. As the group of $l$-torsion points of an abelian scheme is a finite flat $l$-primary group scheme, it follows that $A'_C$ also has trivial $l$-torsion. Replace $A_C$ by $A'_C$. We emphasize that this choice of $A_C$ is not canonical!

By construction, the $l$-torsion of $A_C\rightarrow C$ is trivial; it follows that $A_C\rightarrow C$ has semi-stable reduction along $\bar {C}\cap Z$. (Use that the monodromy representation $\pi _1(C)\rightarrow \mathrm {GL}_{2g}(\mathbb {Z}_l)$ has image in $\Gamma (l):=\{1+M\,|\ M\in lM_{n\times n}(\mathbb {Z}_l)\}\subset \mathrm {GL}_{2g}(\mathbb {Z}_l)$, and the fact that if $l>2$, the group $(1+l\overline {\mathbb {Z}}_l)^{\times }$ is torsion-free. Therefore, if $\gamma \in \pi _1(C)$ has quasi-unipotent image in the representation, it then in fact has unipotent image. The claim then follows from Grothendieck's semi-stable reduction theorem for abelian varieties.)

Let $A_{\bar {C}}\rightarrow \bar {C}$ be the Néron model and let $A^{o}_{\bar {C}}\rightarrow \bar {C}$ denote the associated semi-abelian scheme, that is, the open subset of $A_{\bar {C}}\rightarrow C$ obtained by removing the non-identity components along $\bar {C}{\setminus} C$. It follows from the third part of Proposition A.11 that the logarithmic Dieudonné crystal of $A_{\bar {C}}\rightarrow \bar {C}$ constructed in Remark A.9 is isomorphic to $(M,F,V)_{\bar {C}}$. Then, by the second part of Proposition A.11, the Hodge bundle of the $A^{o}_{\bar {C}}\rightarrow \bar {C}$ is isomorphic to $\omega _M|_{\bar {C}}$.

Set $B_C:=(A_C\times _C A^{t}_C)^{4}$. By Zarhin's trick [Reference Moret-BaillyMB85, Chapitre IX, Lemme 1.1, p. 205], $B_C$ admits a principal polarization. By construction, we have that

• • $B_C$ has trivial $l$-torsion, and

• • $\mathbb {D}(B_C[p^{\infty }])\cong (N_C,F,V)$.

By the uniqueness part of Proposition A.11 it follows that there is an isomorphism of logarithmic Dieudonné crystals:

\[ \mathbb{D}^{\log}(B_{\bar C})\cong (N,F,V)_{\bar{C}}. \]

Hence, the Hodge line bundle of $B^{o}_{\bar {C}}\rightarrow \bar {C}$ is isomorphic to $\omega _N|_{\bar {C}}$ again by Proposition A.11. However, we emphasize again that the choice $B_C\rightarrow C$ is not canonical!

We have an induced morphism to a fine moduli scheme $C\rightarrow \mathscr {A}_{8g,1,l}$. This extends to a morphism from $\bar{C}$ to the minimal compactification $\mathscr {A}_{8g,1,l}^{*}/\mathbb {F}_q$ because the latter is proper and the former is a smooth curve:

(3.4)

We now claim the pullback of $\alpha$, the Hodge line bundle on $\mathscr {A}^{*}_{8g,1,l}$, is isomorphic to $\omega _N|_{\bar {C}}$. Here is the reason. Choose a toroidal compactification $\bar {\mathscr {A}}_{8g,1,l}$. We then have a commutative diagram

(3.5)

again, because $\bar {\mathscr {A}}_{8g,1,l}/\mathbb {F}_q$ is proper and $\bar {C}/\mathbb {F}_q$ is a smooth curve. By [Reference Faltings and ChaiFC90, Ch. V, Theorem 2.5], there is a semi-abelian scheme $G\rightarrow \bar {\mathscr {A}}_{8g,1,l}$ such that $\varphi ^{*}\alpha$ is isomorphic to the Hodge line bundle of $G\rightarrow \bar {\mathscr {A}}_{8g,1,l}$. Now, [Reference Faltings and ChaiFC90, Ch. I, Proposition 2.7] implies that $h^{*}G$ is isomorphic to $A^{o}_{\bar {C}}\rightarrow \bar {C}$, that is, the semi-abelian scheme given by the open subset of $A_{\bar {C}}\rightarrow \bar {C}$ obtained by removing the non-identity components along $\bar {C}\backslash C$. In particular, it follows from part (2) of Proposition A.11 that the Hodge line bundle of $h^{*}G$ is compatible with the Hodge line bundle constructed in Remark A.8.

In (3.3), we have already fixed a basis of sections

\[ s_0,\dots,s_m \in H^{0}(\mathscr{A}_{8g,1,l}\otimes \mathbb{F}_q,\alpha^{r})=H^{0}(\mathscr{A}^{*}_{8g,1,l}\otimes \mathbb{F}_q,\alpha^{r}); \]

after pulling back the sections to $\bar {C}$ via (3.4), we obtain an $m+1$-tuple of sections $t_0,\dots,t_m$ in $H^{0}(\bar {C},\omega ^{r}_N|_{\bar {C}})$ that define the morphism $\bar {C}\rightarrow \mathscr {A}^{*}_{8g,1,l}\subset \mathbb {P}^{m}$.

In conclusion, for every good curve $\bar {C}\subset \bar {U}$ for the quintuple $(\bar {X}, X, \bar {U}, \omega _N^{r},(x_j)^{s}_{j=1})$, we have constructed an $m+1$-tuple of globally generating sections $t_0,\dots,t_m\in H^{0}(\bar {C},\omega ^{r}_N|_{\bar {C}})$ such that

• • the induced map lands in $\mathscr {A}^{*}_{8g,1,l}\subset \mathbb {P}^{m}$;

• • the image of $C$ under the induced map lands in $\mathscr {A}_{8g,1,l}\subset \mathscr {A}_{8g,1,l}^{*}$;

• • and such that the induced abelian variety on $B_C\rightarrow C$ is isomorphic to $(A_C\times _C A_C^{t})^{4}$ where $A_C\rightarrow C$ is an abelian scheme with $\mathbb {D}(A_C[p^{\infty }])\cong (M,F,V)|_C$ as Dieudonné crystals on $C$. (Therefore we also have that $\mathbb {D}(B_C[p^{\infty }])\cong (N,F,V)|_C$.)

In particular, setting $M=\mathscr {A}_{8g,1,l}^{*}\subset \mathbb {P}^{m}$, condition (3) of Lemma 2.9 holds for $\bar {C}\subset \bar {X}$ (corresponding to the symbols $C\subset Y$ in Lemma 2.9). Note that for two such good curves $C$ and $C'$, there is no reason that the induced maps to $\mathscr {A}_{8g,1,l}$ match up on the intersection $C\cap C'$ because our choices of abelian schemes were not canonical.

For each $n>0$, let $P_n$ denote the union of the set of closed points of $U$ whose residue field is contained in $\mathbb {F}_{q^{n!}}$. Note that, for any infinite subset $S\subset \mathbb {N}$, the set $\bigcup _{n\in S}P_n$ is Zariski dense in $X$; indeed, any given closed point $x$ of $U$ is an element of $P_n$ for all $n\gg 0$. By Lemma 2.12, it follows that, for each $n>0$, there exists a good curve $\bar {C}_n\subset \bar {U}$ for the quintuple $(\bar {X}, X, \bar {U}, \omega _N^{r}, P_n)$.

For each $n\in \mathbb {N}$, by the above remarks we obtain an $m+1$-tuple of globally generating sections

\[ t_{n,0},\dots, t_{n,m}\in H^{0}(\bar{C}_n,\omega^{r}_N|_{\bar{C}_n}) \]

such that the induced map factors $f_n\colon \bar {C}_n\rightarrow \mathscr {A}^{*}_{8g,1,l}\subset \mathbb {P}^{m}$. Moreover, any infinite subcollection of the $\bar {C}_n$ is Zariski dense because they are space-filling: if $x$ is a closed point of $U$ with residue field $\mathbb {F}_{q^{e}}$, then $x$ is contained in $C_n$ for all $n\geq e$. By Lemma 2.9, it follows that there exist an infinite set $S\subset \mathbb {N}$ and sections $\tilde {t}_0,\dots,\tilde {t}_m\in H^{0}(\bar {X},\omega ^{r}_N)$ such that the induced rational map $\tilde {f}\colon \bar {X}\dashrightarrow \mathbb {P}^{m}$ lands in $\mathscr {A}^{*}_{8g,1,l}$ and, moreover, for each $n\in S$, we have an equality of morphisms $\tilde {f}|_{\bar {C}_n}=f_n$.

By shrinking $U$, we therefore obtain a map $\tilde {f}\colon U\rightarrow \mathscr {A}_{8g,1,l}$ and hence an abelian scheme ${B_U\rightarrow U}$ such that $B_U[l]$ is a trivial étale cover of $U$. The maps $f_n\colon \bar {C}_n\rightarrow \mathscr {A}_{8g,1,l}^{*}$ were all constructed such that the induced abelian scheme $B_{C_n}\rightarrow C_n$ is compatible with

\[ (N_{C_n},F,V)\otimes \mathbb{Q}_p\cong ({{\mathcal{E}}}\oplus {{\mathcal{E}}}^{*}(-1))^{4}|_{C_n}. \]

On the other hand, if $u$ is a closed point of $U$, then $u$ lies on $C_n$ for all $n\gg 0$. We claim that it follows that $B_U\rightarrow U$ is compatible with $(L\oplus L^{*}(-1))^{4}|_U$. Indeed, it suffices to show that for every closed point $u$ of $U$, $B_u\rightarrow u$ and $(L\oplus L^{*}(-1))^{4}|_u$ are compatible, that is, that the characteristic polynomials of Frobenius match up. Pick $n\in S$ with $C_n$ containing $u$. As the map $\tilde {f}\colon U\rightarrow \mathscr {A}_{8g,1,l}$ extends the map $f_n\colon C_n\rightarrow \mathscr {A}_{8g,1,l}$ by the definition of $S$ in Lemma 2.9, the induced abelian scheme $B_U\rightarrow U$ extends the abelian scheme $B_{C_n}\rightarrow C_n$ constructed above, which is compatible with $(L\oplus L^{*}(-1))^{4}|_{C_n}$ by construction. Therefore $B_u\rightarrow u$ is compatible with $(L\oplus L^{*}(-1))^{4}|_{u}$, as desired.

For each $n\in S$ we have that $\tilde {f}|_{\bar {C}_n}=f_n$. By construction there exists an abelian scheme $A_{C_n}\rightarrow C_n$ of dimension $g$ with

\[ B_U|_{C_n}\cong (A_{C_n}\times_{C_n}A^{t}_{C_n})^{4}. \]

Consider the map of representations induced by the first ${{\mathbb {Z}}}_l$-cohomology of the abelian schemes $B_U\rightarrow U$ and $B_U|_{C_n}\rightarrow C_n$:

(3.6)

Then [Reference KatzKatz01, Lemma 6(b)] implies that for $n\gg 0$, the two representations have the same image (which lands in $\mathrm {GL}_{2g}({{\mathbb {Z}}}_l)^{8}$). By the fundamental work of Tate and Zarhin on Tate's isogeny theorem for abelian varieties over finitely generated fields of positive characteristic [Reference Moret-BaillyMB85, Ch. XII, Théorème 2.5(i), p. 244], it follows that, for all $n\gg 0$, the natural injective map $\text {End}_U(B_U)\hookrightarrow \text {End}_{C_n}({B_U|_{C_n}})$ is an isomorphism when tensored with ${{\mathbb {Z}}}_l$ and hence also when tensored with ${{\mathbb {Q}}}_l$. Therefore, for all $n\gg 0$, the map

\[ \text{End}_U(B_U)\otimes {{\mathbb{Q}}}\rightarrow \text{End}_{C_n}(B_U|_{C_n})\otimes {{\mathbb{Q}}} \]

is an isomorphism as both sides are finite-dimensional semi-simple ${{\mathbb {Q}}}$-algebras of the same rank.

We know that $\text {End}_{C_n}(B_{U}|_{C_n})$ has a non-trivial idempotent $e_{C_n}$ that projects onto a copy of $A_{C_n}$. After replacing $e_{C_n}$ by a high integer multiple, we may lift $e_{C_n}$ to $e_U\in \text {End}_U(B_U)$. Set the image of $e_U$ to be the abelian scheme $\pi _U\colon A_U\rightarrow U$. Finally, we claim that $A_U$ is compatible with $L$ (equivalently, ${{\mathcal {E}}}$). Indeed, the image $A_U\rightarrow U$ is an abelian scheme of dimension $g$ that extends $A_{C_n}\rightarrow C_n$. On the other hand, in (3.6) the two images in $\mathrm {GL}_{16g}(\mathbb {Z}_l)$ are the same (as we have assumed $n\gg 0$) and hence have corresponding decompositions in irreducible $\mathbb {Q}_l$ representations.

Step 3: the proof in the general case via reduction to Step 2. There exists a projective divisorial compactification $\bar {X}$ of $X$. (This means that $\bar {X}$ is normal and the boundary is an effective Cartier divisor.) By Kedlaya's semi-stable reduction theorem (see [Reference KedlayaKed22, Theorem 7.6] for a meta-reference), there is a generically étale alteration $\varphi \colon X'\rightarrow X$ together with a simple normal crossings compactification $\overline {X'}$ such that the overconvergent pullback ${{\mathcal {E}}}'$ extends to a logarithmic $F$-isocrystal with nilpotent residues. After replacing $X'$ with a further finite étale cover, we may guarantee that the residual representation of $L'$ is trivial.

We have proven the theorem for ${{\mathcal {E}}}'$ on $X'$: there exist an open subset $W'\subset X'$ and an abelian scheme $A_{W'}\rightarrow W'$ of relative dimension $g$ with $\mathbb {D}(A_{W'}[p^{\infty }])\cong {{\mathcal {E}}}'|_{W'}$. After shrinking $W'$ and $W$, we may assume that $\varphi |_{W'}\colon W'\rightarrow W$ is finite étale, of degree $d$.

Set $B_W:=\mathfrak {Res}^{W'}_{W}(A_{W'})$ to be the Weil restriction of scalars. This is an abelian scheme over $W$ of dimension $dg$. We claim that $B_W$ is compatible with $L^{d}$. One way to see this is as follows. Consider the short exact sequence of abelian sheaves in the étale topology:

\[ 0\rightarrow A_{W'}[l^{n}]\rightarrow A_{W'}\xrightarrow{l^{n}}A_{W'}\rightarrow 0. \]

As $W'\rightarrow W$ is finite étale, Weil restriction is exact on the level of abelian étale sheaves. Therefore $\mathfrak {Res}^{W'}_W(A_{W'}[l^{n}])\cong B_W[l^{n}]$. As $A_{W'}[l^{n}]$ is a finite étale group over $W$, one deduces that the representation associated to $B_W$ is isomorphic to

\[ \text{Ind}_{\pi_1(W)}^{\pi_1(W')}L'. \]

However, $L'$, as a representation, is the restriction of $L$ along the inclusion $\pi _1(W')\hookrightarrow \pi _1(W)$. Then the desired compatibility follows from the following fact: if $H\subset G$ is the inclusion of a subgroup of finite index $d$, and if $V$ is a finite-dimensional representation of $G$, then

\[ \text{Ind}_H^{G}\text{Res}^{G}_H(V)\cong V^{\oplus d}. \]

Recall that we wrote an isotypic decomposition:

\[ \displaystyle L\cong \bigoplus_{i=1}^{a} (L_i)^{m_i} \]

where each $L_i$ is irreducible on $X$ (and hence on $W$ by [Reference KedlayaKed18, Lemma 1.1.2]). Let $E_i\subset \overline {\mathbb {Q}}_l$ denote the field generated by the traces of Frobenius on $L_i$ as $x$ ranges through the closed points of $W$. We claim that we may find a smooth curve $C\subset W$ with the following properties:

1. (1) each $L_i|_C$ is irreducible;

2. (2) the field generated by Frobenius traces of $L_i|_C$ is $E_i\subset \overline {\mathbb {Q}}_l$;

3. (3) each $L_i|_{C}$ has infinite monodromy around $\infty$; and

4. (4) the induced monodromy representations coming from $B_W\rightarrow W$ and $B_W|_C\rightarrow C$

   
   have the same image.

We have a projective normal compactification $\bar {X}$ of $X$, which is smooth away from a closed subset of codimension at least $2$. Let $F=\bar {X}\backslash X$ and let $F'\subset F$ be the singular locus of $\bar {X}$. For each $L_i$, there is an irreducible component $F_j$ of $F$ that witnesses the fact that $L_i$ has infinite monodromy at $\infty$: having infinite monodromy at $\infty$ means that a certain inertia group has infinite image in the representation.

Pick a closed point $y_j\in F_j\backslash (F_j\cap F')$ for each $j$. Then, by using [Reference DrinfeldDri12, C.2], we may construct an infinite set of curves $(C_n)_{n\in {{\mathbb {N}}}}$ where each $C_n\subset W$ is a smooth, geometrically connected curve that contains all closed points of $W$ whose residue fields are contained in $\mathbb {F}_{q^{n!}}$ and that pass through the $y_j$ transversally (i.e., with a tangent direction that is not contained in $F_j$). (We remark that this is a consequence of Poonen's Bertini theorem [Reference PoonenPoo04, Theorem 1.3].)

Each $L_i|_{C_n}$ has infinite monodromy around $\infty$. By [Reference KatzKatz01, Lemma 6(b)], it follows that for all $n\gg 0$, $C_n$ satisfies (4). For $n\gg 0$, [Reference KatzKatz01, Lemma 6(b)] and [Reference DeligneDel12] guarantee that setting $C:=C'_n$ satisfies the above four conditions.

Again, by using Drinfeld's Theorem 2.1, Remark 2.8, and (3.1) as in Step 2, there exists an abelian scheme $A_{C}\rightarrow C$ that is compatible with $L|_{C}$. On the one hand, using the Tate isogeny theorem [Reference Moret-BaillyMB85, Ch. XII, Théorème 2.5], it follows that $A^{d}_C$ is isogenous to $B_W|_C$. On the other hand, another application the Tate isogeny theorem together with property (4) of $C$ implies that the natural map

\[ \text{End}_{W}(B_W)\rightarrow \text{End}_{C}(B_W|_C) \]

is an isomorphism after tensoring with ${{\mathbb {Q}}}$. As $B_W|_C$ is isogenous to $A_C^{d}$, it follows that $\text {End}_{C}(B_W|_C)\otimes {{\mathbb {Q}}}$ has an element $e_{C}$ projecting onto a factor of $A_C$. After replacing $e_C$ with a high integer multiple, we may lift to $e_W\in \text {End}_W(B_W)$. Set the image of $e_W$ to be the abelian scheme $A_W\rightarrow W$; this is compatible with $L|_W$, as desired.