2.1 Definitions

Let k be a field and let $D\subset\mathbb{P}^1_k$ be a reduced effective divisor containing $\infty$ . Let $X=\mathbb{P}^1\setminus D$ . There are natural projection maps

\[\pi_i: X\times X\setminus \Delta \to X,\]

for $i=1, 2$ as well as a map

\[m: X\times X\setminus \Delta\to \mathbb{G}_m\]

given by

\[m: (x, y)\mapsto x-y.\]

Let

\[j: X\times X\setminus \Delta\hookrightarrow \mathbb{P}^1_k\times X\]

be the evident inclusion.

Definition 2.1. Let $\chi$ be a rank 1 local system on $\mathbb{G}_m$ , and let $\mathbb{V}$ be a local system on X. The middle convolution $MC_\chi(\mathbb{V})$ is defined to be

\[MC_\chi(\mathbb{V})=R^1(\pi_2)_*j_*(\pi_1^*\mathbb{V}\otimes m^*\chi).\]

This definition is an unwinding of [Reference KatzKat96, 2.8], without the use of the language of perverse sheaves. For a proof that these two definitions agree, see, for example, [Reference KatzKat96, 5.1]. An alternative description well adapted to explicit computation is given by Dettweiler and Reiter; see [Reference Dettweiler and ReiterDR03, Theorem 1.1].

We will also make use of the following variant of middle convolution. Let Y be a smooth proper curve, and let $f: Y\to \mathbb{P}^1$ be a finite, generically étale map, with branch divisor contained in D. Let $I=f^{-1}(\infty)$ . Let $\Gamma$ be the graph of f. Then again there are natural projections

\[\pi_1: Y\times X\setminus (\Gamma\cup I\times X)\to Y, \pi_2: Y\times X\setminus (\Gamma\cup I\times X)\to X\]

and a natural map

\[m: Y\times X\setminus (\Gamma\cup I\times X)\to \mathbb{G}_m\]

\[(y,x)\mapsto f(y)-x.\]

Let

\[i: Y\times X\setminus (\Gamma\cup I\times X)\hookrightarrow Y\times X\]

be the evident inclusion.

Definition 2.3. Let $\chi$ be a rank 1 local system on $\mathbb{G}_m$ , and let $\mathbb{V}$ be a local system on Y. The middle convolution $MC_\chi(f,\mathbb{V})$ is defined to be

\[MC_\chi(f,\mathbb{V})=R^1(\pi_2)_*i_*(\pi_1^*\mathbb{V}\otimes m^*\chi).\]

The next proposition follows from, for example, proper base change.

The definitions as above are convenient for computing in the Betti and $\ell$ -adic settings; in the positive characteristic de Rham and Higgs settings we will use a slightly different formalism.

2.2 Betti and $\ell$ -adic computations

In this section we work over an algebraically closed field k of characteristic different from 2. Suppose D has even degree with $\deg(D)\geq 4$ (with the case $\deg(D)=2$ being uninteresting) and let $f: Y\to \mathbb{P}^1$ be the double cover branched over D. Let $\mathbb{L}$ be a rank 1 local system on Y. Let $\chi$ be the unique non-trivial rank 1 local system on $\mathbb{G}_m$ with $\chi^2=\text{triv}.$ We write $MC_{-1}$ instead of $MC_{\chi}$ to emphasize this particular choice of $\chi$ .

• - At a point $P\neq \infty$ , there is a Jordan block J(1,2), and all other Jordan blocks are J(1,1).

• - At $P=\infty$ , there is a Jordan block $J(-1,2)$ , and all other Jordan blocks are $J(-1,1)$ .

Proof. Given $x\in X$ , the fiber $U_x$ of $\pi_2:Y\times X\setminus (\Gamma\cup I\times X)\to X$ over x is isomorphic to $Y\setminus f^{-1}(\{x\cup \infty\})$ , which has Euler characteristic $1-\deg(D)$ . The restriction $\pi_1^*\mathbb{L}\otimes m^*\chi|_{U_x}$ has non-trivial monodromy at the two points of $f^{-1}(x)$ and trivial monodromy at $f^{-1}(\infty)$ . Hence, the Grothendieck–Ogg–Shafarevich formula yields

\[\chi(j_*(\pi_1^*\mathbb{L}\otimes m^*\chi|_{U_x}))=2-\deg(D).\]

As $\pi_1^*\mathbb{L}\otimes m^*\chi|_{U_x}$ is a non-trivial rank 1 local system, $H^0(\pi_1^*\mathbb{L}\otimes m^*\chi|_{U_x})=H^2(\pi_1^*\mathbb{L}\otimes m^*\chi|_{U_x})=0$ . Hence, $MC_{-1}(f,\mathbb{L})$ has rank equal to $\deg(D)-2$ .

For the computation of local monodromies, we refer the reader to [Reference Dettweiler and ReiterDR07, Lemma 5.1]. Indeed, $MC_{-1}(f, \mathbb{L})$ is equivalent to the standard middle convolution applied to (the restriction to X of) $f_*\mathbb{L}$ , and the latter satisfies the conditions of [Reference Dettweiler and ReiterDR07, Lemma 5.1] by our assumptions that $\mathbb{L}$ is non-trivial.

The upshot of this proposition is that if $D=(0)+(1)+(\lambda)+(\infty)$ , so that $Y=E$ is an elliptic curve, and $\mathbb{L}$ is a rank 1 local system on E, then $MC_{-1}(f,\mathbb{L})$ has rank 2 and satisfies $(\star)$ . In the complex setting, if $\mathbb{L}$ is unitary (hence underlies a complex variation of Hodge structure, henceforth abbreviated as $\mathbb{C}$ -VHS), then $MC_{-1}(f, \mathbb{L})$ underlies a $\mathbb{C}$ -VHS as well.

3.1 Preliminaries

Proof. From $(\star)$ , the residue matrices of $\nabla$ at $\{0,1,\lambda\}$ have trace 0, and the residue matrix at $\infty$ has trace 1. This means that $\mathscr{E}$ and hence $F^1\oplus\mathscr{E}/{F}^1$ has degree $-1$ , by [Reference Esnault and ViehwegEV86, B.3], for example. Let

\[\theta: F^1\to \mathscr E/F^1\otimes \Omega^1_{\mathbb{P}^1}(\log D)\]

be the non-zero $\mathscr{O}$ -linear map obtained from $\nabla$ . As in the introduction we consider the Higgs bundle

\[F^1\oplus \mathscr{E}/{F}^1\]

with the Higgs field $\begin{pmatrix} 0\quad & \theta \\ 0\quad & 0\end{pmatrix}.$

By Simpson’s theory [Reference SimpsonSim90, Theorem 8], this Higgs bundle is parabolically stable of parabolic degree 0. Parabolic stability implies the parabolic degree of $F^1$ is positive, and hence the honest degree satisfies $\deg F^1>-1$ . The fact that $\theta$ is non-zero implies that $\deg F^1\leq \deg \mathscr{E}/F^1+\deg\Omega^1_{\mathbb{P}^1}(\log D)$ , and hence that $2\deg F^1\leq \deg \mathscr{E}+2=1$ . Hence, $\deg F^1=0$ , and $\deg\mathscr{E}/F^1=-1$ . Finally, we must have that the extension

\[0\to F^1\to \mathscr{E}\to \mathscr{E}/F^1\to 0\]

splits as $\operatorname{Ext}^1(\mathscr{O}_{\mathbb{P}^1}(-1),\mathscr{O}_{\mathbb{P}^1})=0.$

Let $D=\{0, 1,\infty, \lambda\}$ . Given $\mathbb{V}$ an irreducible local system satisfying $(\star)$ and underlying a (necessarily unique, by irreducibility) $\mathbb{C}$ -VHS, the above proposition shows that we may canonically associate to $\mathbb{V}$ a non-zero map $\theta:\mathscr{O}_{\mathbb{P}^1}\to\mathscr{O}_{\mathbb{P}^1}(-1)\otimes\Omega^1_{\mathbb{P}^1}(\log D)=\mathscr{O}_{\mathbb{P}^1}(1),$ which vanishes at a unique point $w(\mathbb{V})$ of $\mathbb{P}^1$ , as in the introduction.

Proof. The point $w(\mathbb{V})$ determines $\theta$ up to multiplication by a non-zero scalar, which by Proposition 3.1 above determines the stable parabolic Higgs bundle associated to $\mathbb{V}$ up to isomorphism. Now Simpson’s theory [Reference SimpsonSim90, Theorem 8] tells us that this parabolic Higgs bundle determines $\mathbb{V}$ .

3.2 De Rham and Higgs computations

As before, let $f: E\rightarrow \mathbb{P}^1$ denote the double cover ramified $D=0+1+\lambda+\infty$ , viewed as an elliptic curve with origin $e=f^{-1}(\infty)$ . Let $(\mathscr{L}, \nabla)$ be a locally free sheaf on E of rank 1 and degree 0 with integrable unitary connection. If $k=\mathbb{C}$ , the Riemann–Hilbert correspondence gives $\mathbb{L}:=\ker(\nabla)$ a rank 1 unitary local system on E. We may write $\mathscr{L}=\mathcal{O}_E(e-q)$ for some $q\in E$ . In this section we compute the parabolic Higgs bundle on $(\mathbb{P}^1, D)$ obtained from $(\mathscr{L},\nabla)$ via middle convolution. That is, $MC_{-1}(f, \mathbb{L})$ (as defined in § 2.2) underlies a complex variation of Hodge structure $(\mathscr{E}, F, \nabla)$ of rank 2, where $(\mathscr{E}, \nabla)$ is a logarithmic flat vector bundle on $(\mathbb{P}^1, D)$ obtained as the Deligne canonical extension of $MC_{-1}(f, \mathbb{L})$ , and F is the non-trivial piece of the Hodge filtration; we will compute the natural map

\[\theta: F\to \mathscr{E}/F\otimes\Omega^1_{\mathbb{P}^1}(\log D)\]

obtained from $\nabla$ . More precisely, we will compute the fiber $\theta_x$ of $\theta$ , at each point $x\in \mathbb{P}^1\setminus D$ .

Let $x_1, x_2 \in E$ be the preimages of x. Consider the rank 1 locally free sheaf with flat connection on $(E-\{x_1\}-\{x_2\})\times\{x\}$ corresponding to the character $m^*\chi$ under the Riemann–Hilbert correspondence (with notation as in § 2), and let $(\mathscr{M}_0, \nabla)$ denote its Deligne canonical extension to E. The bundle $\mathscr{M}_0$ naturally obtains the structure of a parabolic bundle $\mathscr{M}_\star$ as in [Reference Landesman and LittLL24b, Definition 3.3.1]. (See [Reference Landesman and LittLL24b, 2] or § 4 in this paper for a brief recollection on parabolic bundles.)

We claim that $\mathscr{M}_0\simeq\mathscr{O}_E(-e)$ . Indeed, we have the double cover (by abusing notation) $m:E\rightarrow \mathbb{P}^1$ ramified at $-x, 1-x, \lambda-x, \infty$ , and a parabolic line bundle on $\mathbb{P}^1$ with parabolic weights $1/2$ at $0, \infty$ . [Reference Alfaya and BiswasAB23, Equation (3.4)] shows that the underlying bundle for the parabolic pullback is $\mathscr{O}_E(-e)$ , as claimed.

We recall the following proposition, obtained by specializing [Reference Landesman and LittLL24a, Theorem 5.1.6] to our setting:Footnote ³

Proposition 3.3. The fiber $\theta_x$ of the Higgs field associated to the complex variation of Hodge structure $MC_{-1}(f, \mathbb{L})$ at x is computed via the map

\[H^0(E, \mathscr{L}\otimes \mathscr M_0\otimes \omega_E(x_1+x_2))\to H^1(E, \mathscr{L}\otimes \mathscr M_0)\otimes T^*_xX\]

given as adjoint (via Serre duality) to the multiplication map

\[H^0(E, \mathscr{L}\otimes \mathscr M_0\otimes \omega_E(x_1+x_2))\otimes H^0(E, \mathscr{L}^\vee\otimes \mathscr{M}_0^\vee\otimes \omega_E)\to H^0(\omega_E^{\otimes 2}(x_1+x_2))\to T_x^*X,\]

where the map $H^0(\omega_E^{\otimes2}(x_1+x_2))\to T_x^*X$ is the one given by pullback on cotangent spaces along the map $E\rightarrow \mathcal{M}_{1,2}$ given by

\[ x_1 \mapsto (E, x_1, -x_1).\]

The following is the key computation required for our proofs.

Proof. As above, we let $\{x_1, x_2\}=f^{-1}(x)$ , and assume $x_1=q$ . By Proposition 3.3, the Higgs field factors through

\[H^0(\omega_E(-q+x_1+x_2)) \otimes H^0(\omega_E(q)) \rightarrow H^0(\omega_E^{\otimes 2}(x_1+x_2)).\]

Since $x_1=q$ , there is a natural inclusion $\omega_E\hookrightarrow\omega_E(x_2)=\omega_E(-q+x_1+x_2)$ , so we have the following commutative diagram.

(3.1)

Indeed, the natural map $H^0(\omega_E)\rightarrow H^0(\omega(-q+x_1+x_2))$ is an isomorphism, and the same is true of the map $H^0(\omega_E)\rightarrow H^0(\omega_E(q))$ . Finally, the composition $H^0(\omega_E^{\otimes2}) \rightarrow H^0(\omega^{\otimes 2}(x_1+x_2))\rightarrow T^*_xX$ is zero, since this is the map induced by pullback of differentials along the composite map

which is constant; here, the map $\mathcal{M}_{1,2}\rightarrow \mathcal{M}_{1,1}$ is given by forgetting the second marked point. This implies $\theta_{f(q)}=0$ , as required.

3.3 Proofs

We now begin with the proofs of our main theorems in characteristic 0.

Proof. Torsion evidently implies geometric origin, as all torsion rank 1 local systems arise as summands of $\pi_*\mathbb{C}$ for $\pi: Y\to X$ some cyclic étale cover, so we prove the converse.

Suppose $\mathbb{L}$ is rank 1 and of geometric origin. Then the monodromy representation of $\mathbb{L}$ is defined over the ring of integers $\mathscr{O}_K$ of some number field K. Moreover, for each embedding $\mathscr{O}_K\hookrightarrow\mathbb{C}$ , $\mathbb{L}\otimes_{\mathscr{O}_K}\mathbb{C}$ is unitary, as it underlies a polarizable $\mathbb{C}$ -VHS of rank 1. Hence, if $\gamma\in\pi_1(X)$ is a loop, the scalar in $\mathscr{O}_K^\times$ given by the monodromy of $\gamma$ is an algebraic integer all of whose Galois conjugates have absolute value 1, hence a root of unity.

Proof. Let q be a point of E. Then as the Narasimhan–Seshadri correspondence [Reference Narasimhan and SeshadriNS65] is a tensor functor, the unique unitary connection $\nabla$ on $\mathscr{L}=\mathscr{O}(q-e)$ has torsion monodromy if and only if q itself is torsion; in particular, the sheaf $\mathbb{L}$ of flat sections of $(\mathscr{L}, \nabla)$ is of geometric origin if and only if q is torsion, by Lemma 3.5.

Suppose q, and hence $\mathbb{L}$ , is torsion. Now $\mathbb{V}=MC_{-1}(f,\mathbb{L})=MC_{-1}(f_*\mathbb{L}|_X)$ is evidently of geometric origin (as middle convolution preserves the property of being of geometric origin), is irreducible [Reference KatzKat96, Theorem 2.9.8(2)], and $w(\mathbb{V})=q$ by Lemma 3.4. As $w(\mathbb{V})$ uniquely determines $\mathbb{V}$ by Proposition 3.2, we have shown that if $w(\mathbb{V})$ is torsion, then $\mathbb{V}$ is of geometric origin.

Conversely, suppose $\mathbb{V}$ is of geometric origin and $w(\mathbb{V})=q$ is not torsion; we have that $\mathbb{V}=MC_{-1}(f,\mathbb{L})=MC_{-1}(f_*\mathbb{L}|_X)$ for $\mathbb{L}$ the sheaf of flat sections to the unitary flat line bundle $(\mathscr{O}(q-e),\nabla)$ as above, by Lemma 3.4 and Proposition 3.2. Hence, as [Reference KatzKat96, Theorem 2.9.8(1)] shows that $MC_{-1}$ is involutive, we have $f_*\mathbb{L}|_X=MC_{-1}(\mathbb{V})$ . So if $\mathbb{V}$ was of geometric origin, the same would be true for $f_*\mathbb{L}|_X$ , and hence for $f^*f_*\mathbb{L}|_X=(\mathbb{L}\oplus\mathbb{L}^\vee)|_{f^{-1}(X)}$ . Hence, $\mathbb{L}$ itself would be of geometric origin, and so have finite monodromy, by Lemma 3.5. But this contradicts our assumption that q is not torsion.

Proof of Theorems 1.5 and 1.7. Theorem 1.5 is immediate from what we have proven above; we may simply consider $MC_{-1}(f, \mathbb{L})$ as $\mathbb{L}$ varies over all rank 1 local systems on E. These are of geometric origin if and only if $\mathbb{L}$ has finite monodromy, and there are infinitely many such, in which case MCG-finiteness follows from the fact that the middle convolution is equivariant for the action of the mapping class group [Reference Dettweiler and ReiterDR00, Theorem 5.1], and local systems with finite monodromy are evidently MCG-finite. On the other hand, again by the equivariance of middle convolution for the action of the mapping class group, $MC_{-1}(f,\mathbb{L})$ is MCG-finite if and only if the same is true for $\mathbb{L}$ , which happens if and only if $\mathbb{L}$ has finite image, by [Reference Biswas, Gupta, Mj and WhangBGMW22, Lemma 3.2], for example.

To prove Theorem 1.7, note that the proof above tells us that each $\mathbb{V}$ of geometric origin satisfying $(\star)$ is of the form $R^1(\pi_2)_*i_*(\pi_1^*\mathbb{L}\otimes m^*\chi)$ for $\chi$ of order 2 and $\mathbb{L}$ a finite order rank 1 local system on E, say of order s. But the local system $\pi_1^*\mathbb{L}\otimes m^*\chi$ on $U=E\times\mathbb{P}^1\setminus (\Gamma_f \cup E\times\{\infty\})$ is trivialized on the preimage of U in the variety $Z'_s$ defined in the statement of Theorem 1.7, by construction. Hence, its cohomology appears in the cohomology of $Z'_s \to \mathbb{P}^1$ , as desired.

4.1 Parabolic de Rham and Higgs bundles

Let $\mathcal{C}/k$ be a smooth proper curve over a field k, and $\mathcal{D}=\sum \mathcal{D}_i \subset\mathcal{C}$ a reduced effective divisor of degree l; we denote by $(\mathcal{C}, \mathcal{D})$ the logarithmic curve with log structure specified by $\mathcal{D}$ . We refer the reader to [Reference Krishnamoorthy and ShengKS20, Definition 2.3] for the definition of parabolic (respectively de Rham, respectively Higgs) bundles on $(\mathcal{C}, \mathcal{D})$ ; roughly this consists of bundles $(V_{\alpha})_{\alpha}$ indexed by $(\alpha_1, \ldots , \alpha_l)$ with $\alpha_i\in\mathbb{R}$ , such that:

• - each $V_{\alpha}$ is itself a (respectively de Rham, respectively Higgs) bundle; and

• - they are equipped with maps of (respectively de Rham, respectively Higgs) bundles $V_{\alpha}\hookrightarrow{} V_{\beta}$ for all $\alpha\geq \beta$ (where we write $\alpha\geq \beta$ whenever $\alpha_i\geq \beta_i$ for all i).

The bundles $(V_{\alpha})_{\alpha}$ are required to satisfy several more conditions, and the parabolic weights of $(V_{\alpha})_{\alpha}$ at $\mathcal{D}_i$ are, roughly, the values $\alpha_i$ such that the bundle changes at $\alpha_i$ : see [Reference Krishnamoorthy and ShengKS20, Definition 2.3] for details.

Following [Reference Krishnamoorthy and ShengKS20, Example 2.5], we have the following construction of parabolic bundles.

Definition 4.1. Let V be an arbitrary vector bundle on $\mathcal{C}$ . For $(\alpha_1, \ldots , \alpha_l) \in\mathbb{Q}^{l}$ , we define the parabolic bundle $V(-\sum_{i=1}^l \alpha_i\mathcal{D}_i)$ by

\[V\bigg(-\sum_{i=1}^l \alpha_i\mathcal{D}_i\bigg)_{\beta}=V\bigg(\sum_{i=1}^l -\lceil \alpha_i+\beta_i \rceil \mathcal{D}_i\bigg),\]

for any $\beta=(\beta_1, \ldots, \beta_{l})\in \mathbb{R}^{l}$ .

The following relates parabolic objects on a curve with those on a cover; see [Reference BiswasBis97] for the analogous result in characteristic 0.

4.2 Conjecture of Sun, Yang and Zuo

We now recall the statement of [Reference Sun, Yang and ZuoSYZ21, Conjecture 4.8]. We consider logarithmic graded semistable Higgs bundles on $(\mathbb{P}^1,D=0+1+\lambda + \infty)$ such that the underlying graded vector bundle is isomorphic to $\mathscr{O}\oplus \mathscr{O}(-1)$ . By [Reference Sun, Yang and ZuoSYZ21], the moduli space $\mathcal{M}_{\mathrm{HIG}}$ of such is isomorphic to $\mathbb{P}^1$ , by taking the unique zero of the Higgs field; they then consider the twisted Higgs–de Rham flow on $\mathcal{M}_{\mathrm{HIG}}$ , which in general depends on a lift of $(\mathbb{P}^1, D)$ to $W_2$ , inducing a self-map $\psi_p: \mathbb{P}^1\rightarrow \mathbb{P}^1$ .

(4.1)

where $f: E\to \mathbb{P}^1$ is the elliptic curve double cover branched over D, with $f^{-1}(\infty)$ being the identity.

4.3 Translation to parabolic Higgs bundles

Following [Reference Krishnamoorthy and ShengKS20, Reference Lin, Sheng and WangLSW22], we translate Conjecture 4.4 into the language of parabolic Higgs–de Rham flows, which is more natural for our purposes. Let k denote an algebraic closure of $\mathbb{F}_p$ , with p different from 2, and let $F_k: {\rm Spec}(k)\rightarrow {\rm Spec}(k)$ denote absolute Frobenius.

From [Reference Krishnamoorthy and ShengKS20, Theorem 2.10], we have the parabolic version of the inverse Cartier transform:

\[C^{-1}_{\mathrm{par}}: HIG_{p-1,N}^{\mathrm{par}}(\mathcal{C}, \mathcal{D}) \rightarrow MIC_{p-1,N}^{\mathrm{par}}(\mathcal{C}, \mathcal{D}).\]

This in general depends on a chosen lift of $(\mathcal{C}, \mathcal{D})$ to $W_2$ . Note that if $(\mathscr{E}_\star, \nabla)$ is an object of $MIC_{p-1,N}^{\mathrm{par}}(\mathcal{C}, \mathcal{D})$ of rank 2, the bundle $\operatorname{Gr}(\mathscr{E})$ obtained by taking the associated graded with respect to the Harder–Narasimhan filtration on $\mathscr{E}_0$ has a nilpotent Higgs field $\theta$ induced by $\nabla$ and hence gives rise to an object $\operatorname{Gr}(\mathscr{E},\nabla)\in HIG_{p-1,N}^{\mathrm{par}}(\mathcal{C}, \mathcal{D})$ .

5.1 Middle convolution, again

We now reinterpret the constructions of § 2 to define de Rham and Higgs analogues of the local systems $MC_{-1}(f_*\mathbb{L})$ constructed earlier, in positive characteristic.

We fix for the rest of this section a cyclic étale cover $\eta: \tilde{E}\rightarrow E$ of degree m, for some m odd. We denote by S the surface $E\times \mathbb{P}^1$ , and by $p_1, p_2$ the projection from S to E and $\mathbb{P}^1$ , respectively. Let $D_S$ be the divisor $\Gamma +E\times {\infty}$ on S, where $\Gamma$ denotes the graph of $f: E\rightarrow \mathbb{P}^1$ .

We now give a more concrete description of the families of curves defined above. For $x\in X$ , let $Z_x, W_x$ denote the fibers of Z’, W’ at x; these fit into the Cartesian square.

(5.1)

Then the map $W_x\rightarrow E$ is branched over $f^{-1}(x)\subset E$ .

Notation 5.2. Denote by $Z_X, W_X$ the fiber products $Z'\otimes_{\mathbb{P}^1} X, W' \otimes_{\mathbb{P}^1} X$ , respectively; these are families of curves over X, and we denote by $J_{Z, X}, J_{W, X}$ the relative Jacobians of $Z_X, W_X$ over X. Also we write $z_X: Z_X\rightarrow X, w_X:W_X\rightarrow X$ for the natural maps.

• - the natural $(\mathbb{Z}/m\times\mathbb{Z}/2)\times \mathbb{Z}/2$ -action Footnote ⁴ on $Z_{E^{\circ}}$ extends to $Z_E$ ;

• - the natural $\mathbb{Z}/2\times\mathbb{Z}/2$ -action on $W_{E^{\circ}}$ extends to $W_E$ ; and

• - there is a commutative diagram

(5.2)

extending the natural one with $Z_{E^{\circ}}, W_{E^{\circ}}, E^{\circ}$ , and such that $\theta$ is equivariant for the first $\mathbb{Z}/2$ -action, and the entire diagram is equivariant for the second $\mathbb{Z}/2$ -action.

Proof. The desired compactifications are obtained by normalizing W’, Z’ in the function fields of $W_{E^\circ}, Z_{E^\circ}$ . The three bullets are clear. One may prove semistability in a number of ways, for example by computing with local models; we give a proof by analyzing the monodromies of the local systems $R^1w_{E^\circ, *}\overline{\mathbb{Q}}_{\ell},R^1z_{E^\circ, *}\overline{\mathbb{Q}}_{\ell}$ about the points of E[2]. As a curve has semistable reduction if and only if its Jacobian does, it suffices to show, by the Néron–Ogg–Shafarevich criterion, that these local monodromies are unipotent.

Recall that we have the map $z_X: Z_X\rightarrow X$ . It suffices to show (as $E\to \mathbb{P}^1$ is ramified to order 2 at $0,1,\infty,\lambda$ ) that the local monodromies of $R^1z_{X,*}\overline{\mathbb{Q}}_{\ell}$ around all punctures have generalized eigenvalues 1 or $-1$ . This is true for $R^1z_{X, *}\overline{\mathbb{Q}}_{\ell}/R^1w_{X, *}\overline{\mathbb{Q}}_{\ell}$ , since this is a sum of middle convolutions as in Theorem 1.7, and the claim about local monodromies follows from Proposition 2.5. It remains to show the same for $R^1w_{X, *}\overline{\mathbb{Q}}_{\ell}$ .

Denote the map $E^{\circ}\rightarrow X$ by $\pi^{\circ}$ , and let $\tau$ denote a rank 1 local system on X, given by the non-trivial summand of $\pi^{\circ}_*\overline{\mathbb{Q}}_{\ell}$ . Then $R^1w_{X*}\overline{\mathbb{Q}}_{\ell}$ is a direct sum of the local system associated to the constant family $E\times X$ , and $MC_{-1}(\tau)$ . For $MC_{-1}(\tau)$ , the local monodromies at the punctures satisfy $(\star)$ , again by [Reference Dettweiler and ReiterDR07, Lemma 5.1].

As before, for $x\in X$ , let $W_x$ be the fiber of W’ above x; $W_x$ is a smooth curve and is moreover equipped with a $\mathbb{Z}/2$ -covering map $h_{W, x}: W_x\rightarrow E_x=E\times \{x\}$ .

By Proposition 4.3, the $\mathbb{Z}/2$ -equivariant bundle with connection $\mathscr{O}_{W_x}$ descends to a parabolic logarithmic connection $J_x$ on $E_x$ . Let $L_{\chi, x}$ denote the $\chi$ -isotypic component of $\eta_*\mathscr{O}_{\tilde{E}}$ . Note that the isomorphism classes of $E_x$ and $L_{\chi,x}$ are independent of the choice of x, and we simply denote them by $E, L_{\chi}$ , respectively. The line bundle $L_{\chi}$ has degree 0, and so is isomorphic to $\mathscr{O}_E(e-\tilde{y})$ for a unique $\tilde{y}\in E$ ; we say that $y{\unicode{x2254}}f(\tilde{y})$ is the point corresponding to $L_{\chi}$ .

Proposition 5.7. The fiber of $MC^{\mathrm{HIG}}_{-1}(\chi)$ at x is given by

(5.3) \begin{equation}H^1(L_{\chi}\otimes J_x) \oplus H^0(L_{\chi}\otimes J_x \otimes \Omega^1_{E}(f^{-1}(x))),\end{equation}

with the first (respectively, second) factor being the degree 1 (0) piece.

The Higgs field vanishes only at the point y corresponding to $L_{\chi}$ . Moreover, the graded vector bundle underlying $MC^{\mathrm{HIG}}_{-1}(\chi)$ is isomorphic to $\mathscr{O}\oplus \mathscr{O}(-1)$ .

Proof. Since the maps $Z_X, W_X \rightarrow X$ are smooth, the restriction $MC^{\mathrm{dR}}_{-1}(\chi)|_X$ is given by the quotient of the relative de Rham cohomologies of $Z_X$ and $W_X$ over X. Therefore, the fiber $MC_{-1}^{\mathrm{dR}}(\chi)|_x$ is canonically identified with the $\chi \otimes(-1)$ -isotypic component of $H^1_{\mathrm{dR}}(Z_x)$ .

Let $h_{Z, x}: Z_x\rightarrow E$ be the natural map. To take the $\chi\otimes (-1)$ isotypic component, we can first take the $\chi\otimes(-1)$ -component of $h_{Z,x,*}^{\mathrm{dR}}\mathscr{O}_{Z_x}$ ; the latter is a logarithmic connection on the line bundle $L_{\chi}\otimes J_x$ , with poles along $f^{-1}(x)$ . The fiber of $MC_{-1}^{\mathrm{dR}}(\chi)$ at x is then given by taking $\mathbb{H}^1$ of the associated de Rham complex

\[L_{\chi}\otimes J_x \rightarrow L_{\chi}\otimes J_x\otimes \Omega^1_{E}(f^{-1}(x)),\]

with the Hodge filtration given by the naive filtration. Therefore, the fiber of the Higgs cohomology is given by (5.3).

Denote the graded vector bundle underlying $MC^{\mathrm{HIG}}_{-1}(\chi)$ by $\mathcal{F}$ ; we now prove that $\mathcal{F}\simeq \mathscr{O}\oplus \mathscr{O}(-1)$ as a graded bundle.

Suppose $\lambda_W\in W(\mathbb{F}_q)$ is a lift of $\lambda$ , so that the divisor D lifts canonically to a divisor $D_W\subset \mathbb{P}^1_W$ , and we also have the pair $(E_W, E_W[2])$ lifting (E, E[2]), as well as the cyclic étale covering $\tilde{E}_W\rightarrow E_W$ . Fixing a lift $\chi_W: \mathbb{Z}/m \rightarrow W({\overline{\mathbb{F}}_{p}})^{\times}$ of $\chi$ , we can then define a graded logarithmic Higgs bundle $MC^{\mathrm{HIG}}_{-1,\mathbb{P}^1_W}(\chi_W)$ on $(\mathbb{P}^1_W, D_W)$ exactly as in Definition 5.4, which lifts $MC^{\mathrm{HIG}}_{-1}(\chi)$ .

Let us write the generic fiber $MC^{\mathrm{HIG}}_{-1,\mathbb{P}^1_W}(\chi_W)[1/p]$ as $M\oplus N $ , with M (respectively, N) being the degree 1 (respectively, 0) piece for the Hodge grading; this is a stable Higgs bundle of degree 0 on $\mathbb{P}^1_{W[1/p]}$ , with non-vanishing Higgs field. By our geometric construction, under the usual Simpson correspondence, this Higgs bundle corresponds to the local system $MC_{-1}(f,\mathbb{L})$ in the notation of Definition 2.3, with $\mathbb{L}$ being a rank 1 local system on $E_W$ (which is in turn specified by the cover $\tilde{E}_W\rightarrow E_W$ and $\chi: \mathbb{Z}/m\rightarrow W(\overline{\mathbb{F}}_p)^{\times}$ ). Now $MC_{-1}(f, \mathbb{L})$ has monodromy $(\star)$ around by Proposition 2.5, and therefore the parabolic weights of $MC^{\mathrm{HIG}}_{-1,\mathbb{P}^1_W}(\chi_W)[1/p]$ are zero at $0, 1,\lambda_W$ and $1/2$ at $\infty$ . This forces $\deg(M)=0,$ $\deg(N)=-1$ by Proposition 3.1; since Chern classes are invariant under specialization, the same is true of $\mathcal{F}$ , as required.

Finally, let $L_{\chi_W}$ denote the $\chi_W$ -isotypic component of $\eta_*\mathscr{O}_{\tilde{E}_W}$ , which is a line bundle lifting $L_{\chi}$ . By Lemma 3.4, $\theta$ vanishes at the point $\tilde{x} \in \mathbb{P}^1_{W[1/p]}$ corresponding to $L_{\chi_W}$ , and hence the Higgs field of $MC^{\mathrm{HIG}}_{-1}(\chi)$ vanishes at the point x corresponding to $L_{\chi}$ , as claimed.

5.2 Proof of Conjecture 4.4

Since $z_E: Z_E\rightarrow E$ and $w_E:W_E\rightarrow E$ are semistable families of curves, we may define their relative log-Higgs cohomologies (i.e., the associated graded of relative log-de Rham cohomology with respect to the Hodge filtration), and by Proposition 4.3 we may descend to obtain parabolic logarithmic Higgs bundles on $\mathbb{P}^1$ , which we denote by $R^1z_*^{\mathrm{HIG}}\mathscr{O}_Z$ and $R^1w_*^{\mathrm{HIG}}\mathscr{O}_W$ , respectively. Let $(\mathcal{E}, \theta)$ denote the quotient Higgs bundle $R^1z_*^{\mathrm{HIG}}\mathscr{O}_Z/R^1w_*^{\mathrm{HIG}}\mathscr{O}_W$ . Similarly, we have the logarithmic flat bundles $R^1z_*^{\mathrm{dR}}\mathscr{O}_Z, R^1w_*^{\mathrm{dR}}\mathscr{O}_W,(\mathcal{E}_{\mathrm{dR}}, \nabla){\unicode{x2254}}R^1z_*^{\mathrm{dR}}\mathscr{O}_Z/R^1w_*^{\mathrm{dR}}\mathscr{O}_W$ .

Proof. We have $(\mathcal{E}, \theta)|_X\simeq R^1z_{X*}^{\mathrm{HIG}} \mathscr{O}_{Z_X}/R^1w_{X*}^{\mathrm{HIG}}\mathscr{O}_{W_X}$ ; here $R^1z_{X*}^{\mathrm{HIG}}$ denotes the pushforward in the category of Higgs bundles, along $z_X: Z_X\rightarrow X$ , defined for example in [Reference Ogus and VologodskyOV07, Definition 3.2, Remark 3.3], and similarly for $R^1w_{X*}^{\mathrm{HIG}}$ . The first statement now follows from [Reference Ogus and VologodskyOV07, Theorem 3.8] since for any lift $\tilde{X}$ of X to $W_2$ , we have corresponding lifts of $Z_X, W_X$ to $W_2$ , by repeating Definition 5.1. The second statement is immediate.

Proof of Conjecture 4.4. Let $g: \mathbb{P}^1\rightarrow \mathbb{P}^1$ be the map making (4.1) commute. Then g has degree $p^2$ , and the same is true of $\mathrm{Gr}\circ C^{-1}_{\mathrm{par}}$ by [Reference Sun, Yang and ZuoSYZ21, 4.3]; therefore it suffices to show that $\mathrm{Gr}\circ C^{-1}_{\mathrm{par}}$ and g agree on infinitely many closed points. For any $m\geq 1$ odd, let $\tilde{E}\rightarrow E$ be a non-trivial étale $\mathbb{Z}/m$ -covering, and for any character $\chi: \mathbb{Z}/m \rightarrow{\overline{\mathbb{F}}_p}$ we have the Higgs bundles $MC^{\mathrm{HIG}}_{-1}(\chi)$ on $(\mathbb{P}^1, D)$ , defined in Definition 5.4. By Lemma 5.8, these Higgs bundles correspond to points in $\mathcal{M}_{\mathrm{HIG}}\simeq\mathbb{P}^1$ and we will show that g and $\mathrm{Gr}\circ C^{-1}_{\mathrm{par}}$ agree on all of these points.

Recall that $F_k: k\rightarrow k$ denotes absolute Frobenius. The Higgs–de Rham functor $\mathrm{Gr}\circ C^{-1}$ is $F_k$ -linear, and by Proposition 5.9Footnote ⁷ $\mathrm{Gr}\circ C^{-1}(\mathcal{E}, \theta)|_X \simeq(\mathcal{E}, \theta)|_X$ . Now $MC^{\mathrm{HIG}}_{-1}(\chi)|_X$ is the $\chi$ -isotypic component for the $\mathbb{Z}/m$ -action on $(\mathcal{E},\theta)|_X$ , and therefore $\mathrm{Gr}\circ C^{-1}(MC^{\mathrm{HIG}}_{-1}(\chi))|_X$ is the $\chi^p$ -isotypic component. On the other hand, by definition of the $MC^{\mathrm{HIG}}_{-1}(\chi)$ the $\chi^p$ -isotypic component is given by $MC^{\mathrm{HIG}}_{-1}(\chi^p)|_X$ . Therefore, $\mathrm{Gr} \circ C^{-1}(MC^{\mathrm{HIG}}_{-1}(\chi))|_X\simeq MC^{\mathrm{HIG}}_{-1}(\chi^p)|_X$ , and hence $\mathrm{Gr} \circ C_{\mathrm{par}}^{-1}(MC^{\mathrm{HIG}}_{-1}(\chi))\simeq MC^{\mathrm{HIG}}_{-1}(\chi^p)$ since they are both isomorphic to $\mathscr{O}\oplus \mathscr{O}(-1)$ as graded bundles and hence are determined by the vanishing locus of the Higgs field, which is a reduced point of X. By the explicit description of $MC^{\mathrm{HIG}}_{-1}(\chi)$ given in Proposition 5.7, in particular the claim about the vanishing locus of the Higgs field, we may conclude.

6.1 Motivicity

A well-known consequence of the Langlands correspondence is that all semisimple local systems (with finite order determinant, say) on curves over finite fields are of geometric origin. However, one expects there to be far fewer geometric local systems in characteristic 0, and that the local systems in characteristic p will not in general lift ‘motivically’ to characteristic 0. We show that the case studied in this paper is an exception to this expectation.

For each $\lambda \in \mathbb{F}_q$ , suppose we have a rank 2 tame $\overline{\mathbb{Q}}_{\ell}$ -local system $\mathbb{V}$ on $X=\mathbb{P}^1-\{0, 1, \lambda, \infty\}$ satisfying $(\star)$ . Let $\lambda_W\in W(\mathbb{F}_q)$ be a lift of $\lambda$ , and let $X_W=\mathbb{P}^1-\{0, 1, \lambda_W, \infty\}$ be the corresponding lift of X, and $\tilde{f}$ the lift of f. By definition $\mathbb{V}$ corresponds to a representation of the tame fundamental group $\pi_1^t(X)$ ; the specialization map $sp: \pi_1^t(X_W)\rightarrow \pi_1^t(X)$ gives a local system on $X_W$ , which we denote by $\mathbb{V}_W$ .

Proof. Note that, by using the Weil restriction of abelian schemes [Reference Bosch, Lütkebohmert and RaynaudBLR12, 7.6], it suffices to show that the pullback of $\mathbb{V}_W$ to $X_W\otimes {\rm Spec}(W(\mathbb{F}_{q^2}))$ appears as a direct summand of an abelian scheme. This now follows by applying $MC_{-1}$ to $\mathbb{V}$ . Indeed, by Proposition 2.5 and invertibility of middle convolution, $MC_{-1}(\mathbb{V})$ has rank 2, and the local monodromies at $0,1, \lambda, \infty$ are each semisimple with eigenvalues 1 and $-1$ . As in the rest of the paper, let $f: E\rightarrow \mathbb{P}^1$ be the elliptic curve branched at $0, 1, \lambda, \infty$ , and let $f^{\circ}: E^{\circ}=E-E[2]\rightarrow X$ denote the restriction to X. Therefore, $f^{\circ *}MC_{-1}(\mathbb{V})$ extends to a local system on E, and let us denote this by $\mathbb{W}$ . Then either $\mathbb{W}$ is reducible, and $\mathbb{W}\simeq \mathbb{L}\oplus [-1]^*\mathbb{L}$ for a rank 1 local system $\mathbb{L}$ on E, or it is irreducible and of the form $h_*\mathbb{L}$ for a rank 1 local system $\mathbb{L}$ on $E_{\mathbb{F}_{q^2}}$ , and h the natural map $E_{\mathbb{F}_{q^2}}\rightarrow E$ . Now let $\tilde{f}: E_W\rightarrow \mathbb{P}^1$ denote the elliptic curve branched at $0,1, \lambda_W, \infty$ . In the first case above, $\mathbb{L}$ lifts canonically to a rank 1 local system $\widetilde{\mathbb{L}}$ on $E_W$ , and we have $MC_{-1}(\tilde{f}, \widetilde{\mathbb{L}}) \simeq \mathbb{V}_W$ , and therefore the latter appears in an abelian scheme. In the second case, the same argument applies after we pull $\mathbb{V}_W$ back to $X_W\otimes W(\mathbb{F}_{q^2})$ .

6.2 Lifting to motivic Higgs bundles over W

Fix $\lambda\in \mathbb{F}_q$ , $\lambda_W\in W(\mathbb{F}_q)$ lifting $\lambda$ , and set $X=\mathbb{P}^1\setminus \{0,1,\lambda, \infty\}, X_W=\mathbb{P}^1\setminus \{0,1,\lambda_W, \infty\}$ .

We remark that each parabolic Higgs bundle $MC_{-1}^{\mathrm{HIG}}(\chi)$ , as constructed in Definition 5.4, lifts to W motivically. In fact, this was used in the proof of Proposition 5.7. More precisely, there is a parabolic Higgs bundle $MC_{-1,\mathbb{P}^1_W}^{\mathrm{HIG}}(\chi_W)$ on $\mathbb{P}^1$ , with:

1. - logarithmic poles along the divisor $(0)+(1)+(\lambda_W)+(\infty)$ whose reduction mod p is $MC_{-1}^{\mathrm{HIG}}(\chi)$ ;

2. - and which comes from an abelian scheme, in the sense that, after pulling back along the double cover $E_W\rightarrow \mathbb{P}^1_W$ , this Higgs bundle occurs in the Higgs cohomology of a semistable family of abelian varieties (i.e., with semistable reduction at $0, 1, \lambda_W, \infty$ ).

Assuming that the parabolic Higgs bundles for a semistable family of abelian varieties are periodic for the parabolic Higgs–de Rham flow over W, and that the twisted Higgs–de Rham flow of [Reference Sun, Yang and ZuoSYZ21] can be identified with the parabolic Higgs–de Rham flow, this proves [Reference Sun, Yang and ZuoSYZ21, Conjecture 4.10]. Note that the mod p version of these statements has been worked out; see, for example, [Reference Krishnamoorthy and ShengKS20, Proposition 5.6] and [Reference Lin, Sheng and WangLSW22, Proposition 2.7]. The statements over W seem not to have been written down, but see [Reference Liu, Yang and ZuoLYZ23, Lemma 2.35], which indicates that they follow by the methods developed in [Reference Lan, Sheng and ZuoLSZ19].