Rank 2 local systems and abelian varieties II

Abstract

Let $X/\mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. Let $\mathcal {E}$ be a semi-simple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal {E}_i$ of $\mathcal {E}$ have rank 2, determinant $\bar {\mathbb {Q}}_p(-1)$, and infinite monodromy at $\infty$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal {E}$ at $x$ is in $\mathbb {Q}[t]\subset \mathbb {Q}_p[t]$. Then there exists a dense open subset $U\subset X$ such that $\mathcal {E}|_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse $\bar {\mathbb {Q}}_l$ sheaf on $X$ that has rank 2, determinant $\bar {\mathbb {Q}}_l(-1)$, and infinite monodromy at $\infty$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture) if and only if there exist a dense open subset $U\subset X$ and an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\bar {\mathbb {Q}}_l$.

1. Introduction

Throughout this paper, $p$ is a prime number and $q$ is a power of $p$. If $X/k$ is a smooth scheme over a perfect field of characteristic $p$, then $\textbf {F-Isoc}^{{\dagger} }({X})$ denotes the category of overconvergent $F$-isocrystals on $X$ and $\textbf {F-Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$ denotes its $\overline {\mathbb {Q}}_p$-linearization. Overconvergent $F$-isocrystals are a $p$-adic analog of lisse $l$-adic sheaves.

Definition 1.1 Let $X/k$ be a smooth, geometrically connected scheme over a perfect field $k$ of characteristic $p$ and let ${{\mathcal {E}}}\in \textbf {F-Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$. We say that ${{\mathcal {E}}}$ has infinite local monodromy at infinity if, for every triple $(X',\overline {X'},f)$ where $\overline {X'}$ is smooth projective over $k$, $X'\subset \overline {X'}$ is a dense Zariski open subset, and $f\colon X'\rightarrow X$ is an alteration, the overconvergent $F$-isocrystal $f^{*}{{\mathcal {E}}}$ does not extend to an $F$-isocrystal on $\overline {X'}$.

This definition of infinite local monodromy at infinity applies equally well to lisse $\overline {\mathbb {Q}}_l$-sheaves and is compatible with the other notions of infinite local monodromy at infinity.

Theorem 1.2 Let $X/\mathbb {F}_q$ be a smooth, geometrically connected, quasi-projective scheme. Let ${{\mathcal {E}}}\in \mathbf{F\text{-}Isoc}^{{\dagger} }({X})$ be a semi-simple overconvergent $F$-isocrystal. Suppose:

• • for every closed point $x$ of $X$, the polynomial $P_x({{\mathcal {E}}},t)$ has coefficients in ${{\mathbb {Q}}}\subset \mathbb {Q}_p$;

• • every irreducible summand ${{\mathcal {E}}}_i\in \mathbf {F\text{-}Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$ of ${{\mathcal {E}}}$ has rank $2$, determinant $\overline {\mathbb {Q}}_p(-1)$, and infinite local monodromy around infinity.

Then ${{\mathcal {E}}}$ comes from a family of abelian varieties. More precisely, there exist a non-empty open subset $U\subset X$ and an abelian scheme $A_U\rightarrow U$, so that $\mathbb {D} (A_U[p^{\infty }])\otimes \overline {\mathbb {Q}}_p\cong \mathcal {E}|_U$.

Here, if $G\rightarrow X$ is a $p$-divisible group, $\mathbb {D}(G)$ is the (contravariant) Dieudonné crystal attached to $G$. We have the following applications. Deligne formulated what is now called the companions conjecture in [Del80, Conjecture 1.2.10(vi)]. For a guide to the crystalline companions conjecture, see [Ked18, Ked22].

Corollary 1.3 Let $X/\mathbb {F}_q$ be a smooth, geometrically connected, quasi-projective scheme. Let $L_1$ be an irreducible rank $2$ lisse $\overline {\mathbb {Q}}_l$ sheaf on $X$ with infinite monodromy around infinity and determinant $\overline {\mathbb {Q}}_l(-1)$. Then the following are equivalent:

1. (1) there exist a non-empty open subset $U\subset X$ and an abelian scheme $\pi \colon A_U\rightarrow U$ such that $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\overline {\mathbb {Q}}_l$;

2. (2) all crystalline companions to $L_1$ exist (as predicted by Deligne's crystalline companions conjecture).

Corollary 1.4 Let $X/\mathbb {F}_q$ be a smooth, geometrically connected, quasi-projective scheme. Let ${{\mathcal {E}}}_1$ be an irreducible rank $2$ object of $\mathbf {F\text{-}Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$ with infinite monodromy around infinity and determinant $\overline {\mathbb {Q}}_p(-1)$. Suppose the (number) field $E_1\subset \overline {\mathbb {Q}}_p$ generated by the coefficients of $P_x({{\mathcal {E}}}_1,t)$ as $x$ ranges through the closed points of $X$ has a single prime over $p$. Then ${{\mathcal {E}}}_1$ comes from a family of abelian varieties: there exist a non-empty open subset $U\subset X$ and an abelian scheme $A_U\rightarrow U$ such that ${{\mathcal {E}}}_1|_U$ is a summand of $\mathbb {D}(A_U[p^{\infty }])\otimes \overline {\mathbb {Q}}_p$.

In particular, Corollaries 1.3 and 1.4 provide some evidence for a question of Drinfeld [Dri12, Question 1.4] and a conjecture of the authors [KP21, Conjecture 1.2]. Our motivation for formulating this conjecture was a celebrated theorem of Corlette and Simpson over $\mathbb {C}$ [CS08, Theorem 11.2], the proof of which uses non-abelian Hodge theory. In contrast to our earlier work [KP21], this paper does not use Serre–Tate deformation theory nor does it use the algebraization/globalization techniques of [Har70].

We briefly sketch the proof. Drinfeld's first work on the Langlands correspondence for $\text {GL}_2$, together with Abe's work on the $p$-adic Langlands correspondence and Lemma 2.5, implies Theorem 1.2 when $\dim (X)=1$. (The precise argument is given in Step $2$ of the proof and also uses Remark 2.8 to organize the summands, as explained in Step $1$.) Note that the resulting abelian scheme is not unique, but it is unique up to isogeny.

To deal with the higher-dimensional case, we first assume that $X$ admits a simple normal crossings compactification $\bar {X}$ and ${{\mathcal {E}}}$ is a logarithmic $F$-isocrystal with nilpotent residues. (We recall the notion of logarithmic $F$-isocrystals in Appendix A.) A technique of Katz, combined with slope bounds originally due to Lafforgue, allows one to construct a (non-canonical) logarithmic Dieudonné crystal on an open set $U$ of the compactification $\bar {X}$ whose associated logarithmic $F$-isocrystal is isomorphic to the restriction ${{\mathcal {E}}}|_U$. After the work of Kato and Trihan, this logarithmic Dieudonné crystal yields a natural line bundle, which we call the Hodge bundle $\omega$ of the logarithmic Dieudonné crystal, on $\bar {X}$.

For any odd prime $l\neq p$, let $\mathscr {A}_{h,1,l}$ denote the moduli space of principally polarized abelian schemes of dimension $h$ equipped with full level $l$ structure over $\operatorname {Spec}({{\mathbb {Z}}}[1/l])$. It is well known that the Hodge line bundle is ample on $\mathscr {A}_{h,1,l}$ over $\operatorname {Spec}({{\mathbb {Z}}}[1/l])$. We use Poonen's Bertini theorem over finite fields together with Drinfeld's result and Zarhin's trick to find a well-adapted family of extremely ample space-filling curves $\bar {C}_n$ of $\bar {X}$ that each map to the minimal compactification $\mathscr {A}^{*}_{h,1,l}\subset \mathbb {P}^{m}$ via some fixed power of the Hodge bundle $\omega |_{\bar {C}_n}^{r}$. (This step uses foundational work of Étesse, Kato, Kedlaya, and Trihan that we explain in Appendix A.) Note that $H^{0}(\bar {X},\omega ^{r})$ is a finite-dimensional vector space over a finite field and is hence a finite set. We use this finiteness together with the pigeonhole principle to prove that infinitely many of these maps can be pieced together into a rational map $\bar {X}\dashrightarrow \mathscr {A}_{h,1,l}\subset \mathbb {P}^{m}$. Therefore we obtain an abelian scheme $\psi _U\colon B_U\rightarrow U$ over some open $U\subset X$. The space-filling properties of the $\bar {C}_n$ and Zarhin's work on the Tate isogeny theorem for fields finitely generated over $\mathbb {F}_q$ then allow us to conclude.

To deduce the general case, we use Kedlaya's semi-stable reduction theorem for overconvergent $F$-isocrystals.

Remark 1.5 We comment on the relation of this paper to [KP21]. In [KP21] we prove a Lefschetz-style theorem for families of $\mathrm {GL}_2$-type abelian schemes over finite fields. This has the following implication for [KP21, Conjecture 1.2]: if $X/\mathbb {F}_q$ is a smooth projective variety, then there exists an ample curve $C\subset X$ such that if ${{\mathcal {E}}}\in \textbf {F-Isoc}({X})_{\overline {\mathbb {Q}}_p}$ and ${{\mathcal {E}}}|_C$ comes from an abelian scheme $A_C\rightarrow C$ of $\mathrm {GL}_2$-type, then there is an open subset $U\subset X$ such that ${{\mathcal {E}}}|_U$ comes from an abelian scheme $B_U\rightarrow U$ of $\mathrm {GL}_2$-type. (It follows from Zarhin's work on the Tate isogeny conjecture that $B_C\rightarrow C$ is indeed isogenous to $A_C\rightarrow C$.) To prove this, we use Serre–Tate deformation theory and globalization results of [Har70], the latter of which critically uses the positivity of $C$ in $X$. In this paper, we only deal with non-proper varieties $X/\mathbb {F}_q$ and we use infinitely many (space-filling, affine) curves together with a result of Drinfeld, which is only known for affine curves. In particular, the main results of [KP21] do not imply the main result of this paper.

2. Preliminaries

Before proving Theorem 1.2, we need several preliminary results. A key ingredient in the proof is the following theorem, which is a byproduct of Drinfeld's first work on the Langlands correspondence for $\mathrm {GL}_2$.

Theorem 2.1 (Drinfeld)

Let $C/\mathbb {F}_q$ be a smooth affine curve and let $L_1$ be a rank $2$ irreducible $\overline {\mathbb {Q}}_l$ sheaf with determinant $\overline {\mathbb {Q}}_l(-1)$. Suppose $L_1$ has infinite local monodromy around some point at $\infty \in \overline {C}\backslash C$. Then $L_1$ comes from a family of abelian varieties in the following sense. Let $E$ be the field generated by the Frobenius traces of $L_1$ and suppose $[E:{{\mathbb {Q}}}]=g$. Then there exist an abelian scheme

\[ \pi_C\colon A_{C}\rightarrow C \]

of dimension $g$ and an isomorphism $E\cong \textrm {End}_{C}(A)\otimes {{\mathbb {Q}}}$, realizing $A_C$ as a $\mathrm {GL}_{2}$-type abelian scheme, such that $L_1$ occurs as a summand of $R^{1}(\pi _C)_*\overline {\mathbb {Q}}_l$. Moreover, $A_{C}\rightarrow C$ is totally degenerate around $\infty$.

See [ST18, Proof of Proposition 19, Remark 20] for how to recover this result from Drinfeld's work. This amounts to combining [Dri83, Main Theorem, Remark 5] with [Dri77, Theorem 1].

For completeness, we briefly recall the theory of companions and what is known about them. For a thorough summary about the definitions and also what is known, we refer the reader to [Ked18]. Alternatively, the reader may see [KP21, § 4].

Definition 2.2 Let $X/\mathbb {F}_q$ be a smooth, geometrically connected variety. Let $\lambda$ be a prime number and let $\mathcal {E}$ denote either a smooth $\overline {\mathbb {Q}}_{\lambda }$ sheaf on $X$ if $\lambda \neq p$ or an overconvergent $F$-isocrystal with coefficients in $\overline {\mathbb {Q}}_p$ if $\lambda =p$. Following Kedlaya [Ked18, § 1], we call such ${{\mathcal {E}}}$ coefficient objects.

1. (1) Let $l\neq p$ be a prime number and let $L$ be a lisse $\overline {\mathbb {Q}}_l$-sheaf on $X$. Fix a (possibly non-continuous) field isomorphism $\iota \colon \overline {\mathbb {Q}}_{\lambda }\rightarrow \overline {\mathbb {Q}}_l$. We say that $L$ is an $\iota$-companion of $\mathcal {E}$ if, for all closed points $x\in X$, we have

   \[ \iota(P_x(\mathcal{E},t))=P_x(L,t)\in \overline{\mathbb{Q}}_l[t], \]
   where $P_x(-,t)$ denotes the reverse characteristic polynomial at the closed point $x$.

2. (2) Let $\mathcal {F}$ be an overconvergent $F$-isocrystal on $X$ with coefficients in $\overline {\mathbb {Q}}_p$ and fix an isomorphism $\iota \colon \overline {\mathbb {Q}}_{\lambda }\rightarrow \overline {\mathbb {Q}}_p$. We say $\mathcal {F}$ is an $\iota$-companion of $\mathcal {E}$ if, for all closed points $x\in X$, we have

   \[ \iota(P_x(\mathcal{E},t))=P_x(\mathcal{F},t)\in \overline{\mathbb{Q}}_p[t]. \]

In either of these cases, we say that the $\iota$-companion to $\mathcal {E}$ exists.

Suppose $\mathcal {E}$ is semi-simple and each irreducible summand has algebraic determinant. Then Deligne's conjecture, together with Crew's $p$-adic enhancement, predict that all $\iota$-companions to ${{\mathcal {E}}}$ exist. It follows from work of Abe, Abe and Esnault, Deligne, Drinfeld, Kedlaya, and Lafforgue [AE19, Abe18, Del12, Dri12, Laf02] that this conjecture is known to hold in the following cases.

Theorem 2.3 Let $X/\mathbb {F}_q$ be a smooth, geometrically connected variety. Let $\mathcal {E}$ be a semi-simple coefficient object on $X$ such that the irreducible summands have algebraic determinant.

• • If $\dim (X)=1$, then all $\iota$-companions exist ([Laf02, Théorème VII.6] and [Abe18, Theorem 4.4.1]).

• • For any $l\neq p$ and any isomorphism $\iota \colon \overline {\mathbb {Q}}_{\lambda }\rightarrow \overline {\mathbb {Q}}_l$, the $\iota$-companion to $\mathcal {E}$ exists ([Dri12, Theorem 1.1] and [AE19, Theorem 4.2] or [Ked18, Theorem 0.4.1]).

In particular, $p$-adic companions are not known to exist when $\dim (X)>1$, although Kedlaya has recently proposed a promising strategy [Ked21].

Proposition 2.4 Maintain the hypotheses of Theorem 1.2. Let $\iota \colon \overline {\mathbb {Q}}_p\rightarrow \overline {\mathbb {Q}}_l$ be a field isomorphism and let $L:=\ ^{\iota }{{\mathcal {E}}}$ be the (semi-simple) $\iota$-companion to ${{\mathcal {E}}}$.

• • The isomorphism class of $L$ is independent of the choice of $\iota$.

• • Let $L_i$ be an irreducible summand of $L$. Then $L_i$ has rank 2, determinant $\overline {\mathbb {Q}}_l(-1)$, and infinite monodromy at infinity.

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 [Ked18, 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 [Ked18, 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.

We will need the following lemma to ensure that, given the hypotheses of Theorem 1.2, every $p$-adic companion of ${{\mathcal {E}}}_i$ is again a summand of ${{\mathcal {E}}}$; moreover, the companion relation preserves multiplicity in the isotypic decomposition of ${{\mathcal {E}}}$.

Lemma 2.5 Let $X/\mathbb {F}_q$ be a smooth, geometrically connected scheme.

1. (1) Let $l\neq p$ be a prime and let $L$ be a lisse, semi-simple $\overline {\mathbb {Q}}_l$-sheaf on $X$, all of whose irreducible summands $L_i$ have algebraic determinant. Suppose that, for all closed points $x$ of $X$, we have

   \[ P_x(L,t)\in {{\mathbb{Q}}}[t]\subset \overline{\mathbb{Q}}_l[t]. \]
   Let $L_i$ be an irreducible summand of $L$ that occurs with multiplicity $m_i$ and $\iota \in \text {Aut}_{{{\mathbb {Q}}}}(\overline {\mathbb {Q}}_l)$ be a field automorphism. Then the $\iota$-companion to $L_i$, denoted $^{\iota }L_i$, is isomorphic to an irreducible summand of $L$ that occurs with multiplicity $m_i$.

2. (2) Let ${{\mathcal {F}}}$ be a semi-simple object of $\,\mathbf {F\text{-}Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$, all of whose irreducible summands ${{\mathcal {F}}}_i$ have algebraic determinant. Suppose that, for all closed points $x$ of $X$, we have

   \[ P_x({{\mathcal{F}}},t)\in {{\mathbb{Q}}}[t]\subset \overline{\mathbb{Q}}_p[t]. \]
   Let ${{\mathcal {F}}}_i$ be an irreducible summand of ${{\mathcal {F}}}$ that occurs with multiplicity $m_i$. Let $\iota \in \text {Aut}_{{{\mathbb {Q}}}}(\overline {\mathbb {Q}}_p)$. Then the $\iota$-companion of ${{\mathcal {F}}}_i$, denoted by $^{\iota }{{\mathcal {F}}}_i$, exists and is isomorphic to a direct summand of ${{\mathcal {F}}}$ that occurs with multiplicity $m_i$.

Proof. We reduce the crystalline case to the étale case. (Note that we could have equivalently proceeded by reduction to curves using [AE19].) 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 [AE19, Theorem 2.7]. Fix an isomorphism $\sigma \colon \overline {\mathbb {Q}}_p\rightarrow \overline {\mathbb {Q}}_l$. By [AE19, Theorem 4.2] or [Ked18, 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 [Del12, 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 [AE19, 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.

Remark 2.6 The argument of [AE19, 3.2] cited in the proof of Lemma 2.5 is based on [Laf02, Corollary VI.3] and uses $L$-functions. A similar idea is used in the proof that the companions relations preserves irreducibility, which was crucial to Proposition 2.4. See also [Ked18, Lemma 3.1.5, Theorem 3.3.1].

Remark 2.7 It follows from the argument of Lemma 2.5 that if $X/\mathbb {F}_q$ is smooth and geometrically connected and if ${{\mathcal {E}}}, {{\mathcal {F}}}\in \textbf {F-Isoc}^{{\dagger} }({X})_{\overline {\mathbb {Q}}_p}$ are semi-simple objects, all of whose summands are algebraic, then, for any field isomorphism $\iota \colon \overline {\mathbb {Q}}_p\rightarrow \overline {\mathbb {Q}}_l$, we have $^{\iota }({{\mathcal {E}}}\otimes {{\mathcal {F}}})\cong \ ^{\iota }{{\mathcal {E}}}\otimes \ ^{\iota }{{\mathcal {F}}}$, that is, the relation of being $\iota$-companions commutes with tensor product.

Remark 2.8 It follows from Lemma 2.5 that, in the context of Theorem 2.1, there is a decomposition

\[ \displaystyle R^{1}(\pi_C)_*\overline{\mathbb{Q}}_l\cong \bigoplus_{i=1}^{g}(L_i) \]

where the $L_i$ form a complete set of $\overline {\mathbb {Q}}_l$ companions. There are exactly $g$ non-isomorphic companions because the field generated by Frobenius traces of $L_1$ is isomorphic to $E$ and the $l$-adic companions are in bijective correspondence with the embeddings $E\hookrightarrow \overline {\mathbb {Q}}_l$. In particular, each companion occurs with multiplicity 1. In fact, as $E\cong \text {End}_C(A_C)\otimes {{\mathbb {Q}}}$, it follows that $E\otimes \overline {\mathbb {Q}}_l$ acts on $R^{1}(\pi _C)_*\overline {\mathbb {Q}}_l$. On the other hand, $E\otimes \overline {\mathbb {Q}}_l\cong \prod _{i} \overline {\mathbb {Q}}_l$, where $i$ runs over the embeddings $E\hookrightarrow \overline {\mathbb {Q}}_l$. For each $i$, pick a non-trivial idempotent $e_i\in E\otimes \overline {\mathbb {Q}}_l$ whose image is the $i$th component of the direct product decomposition. The above direct sum decomposition is induced by these $e_i$.

To apply Drinfeld's Theorem 2.1, we will use the following lemma.

Lemma 2.9 Let $Y/\mathbb {F}_q$ be a smooth, geometrically connected, projective scheme and let $\alpha$ be a line bundle on $Y$. Let $M\subset \mathbb {P}^{m}_{\mathbb {F}_q}$ be a closed subset. Suppose there exists an infinite collection $(C_n)_{n\in \mathbb {N}}$ of smooth, projective, geometrically connected, closed subcurves $C_n\subset Y$ such that:

1. (1) for each $n\in \mathbb {N}$, the natural map $H^{0}(Y,\alpha )\rightarrow H^{0}(C_n,\alpha |_{C_n})$ is an isomorphism;

2. (2) for any infinite subset $S\subset \mathbb {N}$, the union

   \[ \displaystyle \bigcup_{n\in S}C_n \]
   is Zariski dense in $Y$;

3. (3) for each curve $C_n$, there exist $m+1$ globally generating sections

   \[ t_{n,0},\dots,t_{n,m}\in H^{0}(C_n,\alpha|_{C_n}) \]
   such that the induced morphism to $\mathbb {P}^{m}$ factors through $M$:
   

Then there exist global sections $\tilde {t}_0,\dots,\tilde {t}_m\in H^{0}(Y,\alpha )$ such that the induced rational map $\tilde {f}\colon Y\dashrightarrow \mathbb {P}^{m}$ has image in $M$. Moreover, $\tilde {f}$ can be chosen to be compatible with infinitely many of the maps $f_n$.

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.

Lemma 2.9 has two key ingredients. The first ingredient is that if $X/\mathbb {F}_q$ is a projective variety and $\alpha$ is a coherent sheaf on $X$, then $H^{0}(X,\alpha )$ is a finite set. The second ingredient is the pigeonhole principle. To use Lemma 2.9, the following definition will be useful.

Definition 2.10 Let $\bar {X}/k$ be a smooth, geometrically connected, projective scheme of dimension at least 2, let $Z\subset \bar {X}$ be a reduced simple normal crossings divisor, and set $X:=X\backslash Z$. Let $\bar {U}\subset \bar {X}$ be an open subset whose complement has codimension at least 2. Let $(x_j)_{j=1}^{s}$ be a finite collection of closed points of $U:=\bar {U}\cap X$. Let $\alpha$ be a line bundle on $\bar {X}$. We say that $\bar {C}\subset \bar {U}$ is a good curve for the quintuple $(\bar {X}, X, \bar {U}, \alpha, (x_j)^{s}_{j=1})$ if:

• • $\bar {C}$ is the smooth complete intersection of smooth ample divisors of $\bar {X}$ that intersect $Z$ in good position;

• • $\bar {C}$ contains each of the closed points $x_j$, for $j=1,\dots, s$;

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

In the proof of Theorem 1.2, we will need to know that good curves exist. This is guaranteed by the following two results.

Proposition 2.11 Let $Y/k$ be a smooth, geometrically connected, projective scheme of dimension $d\geq 2$ and let $\alpha$ be a line bundle on $Y$. Let $D\subset Y$ be an ample divisor. Then there exists an $s_0>0$ such that, for any $s\geq s_0$, and for any integral divisor $E\in |sD|$ in the linear series, the natural map

\[ H^{0}(Y,\alpha)\rightarrow H^{0}(E,\alpha|_E) \]

is an isomorphism.

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.

Lemma 2.12 Let $\bar {X}/\mathbb {F}_q$ be a smooth, geometrically connected, projective scheme of dimension at least $2$, let $Z\subset \bar {X}$ be a reduced simple normal crossings divisor, and set $X:=\bar {X}\backslash Z$. Let $\bar {U}\subset \bar {X}$ be an open subset whose complement has codimension at least $2$. Let $(x_j)_{j=1}^{s}$ be a finite collection of closed points of $U:=\bar {U}\cap X$. Let $\alpha$ be a line bundle on $\bar {X}$. Then there is a good curve $\bar {C}\subset \bar {U}$ for the quintuple $(\bar {X}, X, \bar {U}, \alpha, (x_j)^{s}_{j=1})$

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 [Poo04, 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 [Poo04, 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 [Poo04, 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.

Note that Lemma 2.12 also holds with $\mathbb {F}_q$ replaced by any infinite field $k$ by the usual Bertini theorems. Finally, the following lemma is surely well known but we could not find a reference for exactly the statement we need. (The essential content is contained in [CCO14, § 3.3].) We will use this lemma to make a particular choice of $A_C\rightarrow C$ in the isogeny class from Drinfeld's Theorem 2.1 (though this choice will not be unique).

Lemma 2.13 Let $X$ be a scheme and let $A\rightarrow X$ be an abelian scheme. Let $r$ be a prime and let $G$ be an $r$-divisible group on $X$. Suppose there exists an isogeny $\psi \colon A[r^{\infty }]\rightarrow G$ of $r$-divisible groups on $X$ (as in [CCO14, 3.3.5]). Then there exist an $r$-primary isogeny $\varphi \colon A\rightarrow B$ of abelian schemes over $X$ and an isomorphism $\varepsilon \colon B[r^{\infty }]\rightarrow G$ such that the following diagram commutes:

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, [CCO14, 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.

3. Proofs of Theorem 1.2 and Corollaries 1.3, 1.4

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 [DK17, § 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 [Dri18, E.10] and either [AE19, Theorem 4.2] or [Ked18, 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 [MB85, Ch. IX, Théorème 3.1, p. 210] or [FC90, 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 $r\in \mathbb {Z}$ [FC90, 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 [FC90, Ch. V, Theorem 1.5(ii), Theorem 2.5(iii)].

It follows from [Del12, 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 [AE19, 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 [EK16, 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, [CCO14, 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 [CCO14, 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 [Pál15] and is compatible with $L|_C$ by [KM74]. 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 [dJ95] 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 [MB85, 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 [FC90, 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, [FC90, 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 [Katz01, 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 [MB85, 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 [Ked22, 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 [Ked18, 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 [Dri12, 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 [Poo04, Theorem 1.3].)

Each $L_i|_{C_n}$ has infinite monodromy around $\infty$. By [Katz01, Lemma 6(b)], it follows that for all $n\gg 0$, $C_n$ satisfies (4). For $n\gg 0$, [Katz01, Lemma 6(b)] and [Del12] 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 [MB85, 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.

Proof of Corollary 1.3 Suppose there exists $\pi _U\colon A_U\rightarrow U$ such that $R^{1}(\pi _U)_*\overline {\mathbb {Q}}_l$ has $L_1$ as a summand. By the assumption that $X$ is smooth and geometrically connected, it is irreducible; hence $U\subset X$ is dense. A theorem of Zarhin implies that $R^{1}(\pi _U)_*\mathbb {Q}_l$ is semi-simple [MB85, Chapitre XII, Theorem 2.5, pp. 244–245]. The field generated by the characteristic polynomials of $R^{1}(\pi _U)_*\mathbb {Q}_l$ is clearly ${{\mathbb {Q}}}$; indeed, this follows Weil's theorem that the characteristic polynomial of Frobenius acting on the Tate module of an abelian variety over a finite field has coefficients in $\mathbb {Z}$ [Wei48, IX,X].

We claim that $\mathbb {D}(A_U[p^{\infty }])\otimes \mathbb {Q}_p$ is a semi-simple object of $\textbf {F-Isoc}^{{\dagger} }({U})$. This is essentially contained in [Pál15, Remark 4.11], but some comments are in order.

While the statement of [Pál15, Remark 4.11] assumes that $U$ is a smooth curve, this assumption is unnecessary. Indeed, the only point where this assumption is used is in the citation of [KT03, 4.3–4.8], to argue that the associated $F$-isocrystal is overconvergent. However, [Éte02, Théorème 7] essentially states and proves exactly this: if $S/k$ is a smooth separated scheme over a field $k$ of characteristic $p$ and $A\rightarrow S$ is an abelian scheme, then $R^{1}f_{\text {rig}}(\mathcal {O}_{X/K})$ is an overconvergent $F$-isocrystal on $S$. When $k$ is perfect, this $F$-isocrystal is isomorphic to $R^{1}f_{\text {crys}}(\mathcal {O}_{X/W})\otimes \mathbb {Q}$ because $A\rightarrow S$ is smooth and proper (see, for example, [Ber97, Proposition 1.9]). On the other hand, $R^{1}f_{\text {crys}}(\mathcal {O}_{X/W})\cong \mathbb {D}(A[p^{\infty }])$. In particular, to obtain the semi-simplicity, one simply combines Corollary 4.9 and Proposition 3.5 of [Pál15] with the fact that $\mathbb {D}(A_U[p^{\infty }])\in \textbf {F-Isoc}^{{\dagger} }({U})$, exactly as explained in [Pál15, Remark 4.11].

As $\mathbb {D}(A_U[p^{\infty }])\otimes \mathbb {Q}_p$ is isomorphic to the rational crystalline cohomology of $A_U\rightarrow U$, it follows from [KM74] that $\mathbb {D}(A_U[p^{\infty }])\otimes \mathbb {Q}_p$ and $R^{1}(\pi _U)_*\mathbb {Q}_l$ are companions. It follows from Lemma 2.5 that all crystalline companions of $L_1|_U$ exist and, moreover, are summands of $\mathbb {D}(A_U[p^{\infty }])\otimes \overline {\mathbb {Q}}_p$. Then, by [Ked18, Corollary 3.3.3], all crystalline companions to $L_1$ exist.

Conversely, suppose all crystalline companions $({{\mathcal {E}}}_i)^{b}_{i=1}$ to $L_1$ exist. We first of all claim that each ${{\mathcal {E}}}_i$ has infinite monodromy at $\infty$. Indeed, suppose for contradiction that there existed an alteration $f\colon X'\rightarrow X$ and a compactification $\overline {X'}$ such that $f^{*}{{\mathcal {E}}}_i$ extends to an object ${{\mathcal {F}}}'$ of $\textbf {F-Isoc}^{{\dagger} }({\overline {X'}})_{\overline {\mathbb {Q}}_p}$. Then $f^{*}L_1$ would also extend to $\overline {X'}$ by [Ked18, Corollary 3.3.3], contradicting the assumption that $L_1$ had infinite monodromy at $\infty$. Moreover, each ${{\mathcal {E}}}_i$ is irreducible by [Ked18, Lemma 3.3.1]. Similarly, tensorial operations respect the companion relation, hence $\det ({{\mathcal {E}}}_i)\cong \overline {\mathbb {Q}}_p(-1)$. There exists a $p$-adic local field $K$ with each ${{\mathcal {E}}}_i$ an object of $\textbf {F-Isoc}^{{\dagger} }({X})_{K}$. Set ${{\mathcal {E}}}:=\bigoplus _{i=1}^{b} {{\mathcal {E}}}_i$, considered as an object of $\textbf {F-Isoc}^{{\dagger} }({X})$ (by restricting scalars from $K$ to $\mathbb {Q}_p$, so the rank of ${{\mathcal {E}}}$ is $2b[K:{{\mathbb {Q}}}]$). Note that ${{\mathcal {E}}}$, being the sum of irreducible objects, is semi-simple. Then ${{\mathcal {E}}}$ satisfies the hypotheses of Theorem 1.2, and $L_1$ is a companion of a summand of ${{\mathcal {E}}}$. It follows that there is an open set $U\subset X$ together with an abelian scheme $\pi _U\colon A_U\rightarrow U$ such that ${{\mathcal {E}}}\cong \mathbb {D}(A_U[p^{\infty })\otimes \mathbb {Q}_p$. Again using Zarhin's semi-simplicity, $L_1|_U$ is a summand of $R^{1}(\pi _U)_*\overline {\mathbb {Q}}_l$, as desired.

Proof of Corollary 1.4 Under the assumption on $E_1$, all $p$-adic companions to ${{\mathcal {E}}}_1$ exist by [KP21, Corollary 4.16]. (This result is straightforward; they are all Galois twists of each other.) Fix $\sigma \colon \overline {\mathbb {Q}}_p \rightarrow \overline {\mathbb {Q}}_l$. Then the $\sigma$-companion to ${{\mathcal {E}}}_1$ exists by [AE19, Theorem 4.2] or [Ked18, Corollary 3.5.3]. Apply Corollary 1.3.