curves $\varphi : (X', V' \cup D') \to (X, V \cup D)$ (see (4.30)) equipped with a $\Sigma$-isomorphism $\varphi^{-1}(\Sigma) \cong \Sigma'$ of metrized complexes of curves. Since $V' = \varphi^{-1}(V)$ and $D' = \varphi^{-1}(D)$, we will often denote a lifting simply by $X'$. Let $\varphi : X' \to X$ and $\varphi' : X'' \to X$ be two liftings of $\Sigma'$, and let $\psi : X' \xrightarrow{\sim} X''$ be an $X$-isomorphism of curves. Then, $\psi$ restricts to a $\Sigma$-automorphism

$$ \psi|_{\Sigma'} : \Sigma' \cong \varphi^{-1}(\Sigma) \xrightarrow{\sim} \varphi'^{-1}(\Sigma) \cong \Sigma'. $$

We will consider liftings up to $X$-isomorphism preserving $\Sigma'$, i.e. such that $\psi_{\Sigma'}$ is the identity, and we will also consider liftings up to isomorphism as curves over $X$.

For every finite vertex $x \in V$, let $\Sigma(x)$ be the connected component of $x$ in $\{x\} \cup (\Sigma \setminus V)$, let $Y(x) = \tau^{-1}(\Sigma(x))$, and let $D(x) = D \cap Y(x)$, as in Example 6.14. Then, $(Y(x), x, D(x))$ is a punctured star-shaped curve. By Proposition 2.22, for every finite vertex $x' \in V(\Sigma')$ lying above $x$, the morphism $\varphi_{x'} : C_{x'} \to C_x$ is finite, tamely ramified, and branched only over the points of $C_x$ corresponding to tangent directions at $x$ represented by edges of $\Sigma(x)$. Let $\psi(x') : (Y'(x'), x', D(x')) \to (Y(x), x, D(x))$ be the unique lifting of $C_{x'}$ to a punctured star-shaped curve over $(Y(x), x, D(x))$ provided by Theorem 6.18, and let $\Sigma'(x') = \psi(x')^{-1}(\Sigma(x)) = \Sigma(Y', \{x'\} \cup D'(x'))$. Then, $\Sigma'(x')$ is canonically identified with the connected component of $x'$ in $\{x'\} \cup (\Sigma' \setminus V_f(\Sigma'))$ in such a way that for every edge $e'$ of $\Sigma'(x')$, the point $\text{red}_{x'}(e') \in C_{x'}(k)$ is identified with the pointred$(e')$ defined in (6). This induces an identification of $D'(x')$ with $\varphi^{-1}(D) \cap \Sigma'(x')$. Let $\tau_{x'}$ be the canonical retraction $Y'(x') \to \Sigma'(x')$ defined in (6).

Let $e' \in E_f(\Sigma')$ and $e = \varphi(e')$. Choose $a, a' \in K^\times$ with $\text{val}(a) = \ell(e)$ and $\text{val}(a') = \ell(e')$. Let $x', y'$ be the endpoints of $e'$. By Proposition 6.15(3), we can choose isomorphisms $\tau^{-1}(e^\circ) \cong S(a)_+$ and $\tau_{x'}^{-1}(e'^\circ) \cong S(a')_+$, $\tau_{y'}^{-1}(e'^\circ) \cong S(a')_+$ in such a way that the finite morphisms $\tau_{x'}^{-1}(e'^\circ) \to \tau^{-1}(e^\circ)$ and $\tau_{y'}^{-1}(e'^\circ) \to \tau^{-1}(e^\circ)$ are given by $t \mapsto t^{d_{e'}(\varphi)}$, where $d_{e'}(\varphi)$ is the degree of the edge map $e' \to e$. In particular, there exists a $\tau^{-1}(e^\circ)$-isomorphism $\tau_{x'}^{-1}(e'^\circ) \xrightarrow{\sim} \tau_{y'}^{-1}(e'^\circ)$. As explained in (6.9), there are canonical identifications

$$ \mathrm{Aut}_{\tau^{-1}(e^\circ)}(d_{e'}(\varphi)) := \mathrm{Aut}_{\tau^{-1}(e^\circ)}(\tau_{x'}^{-1}(e'^\circ)) = \mathrm{Aut}_{\tau^{-1}(e^\circ)}(\tau_{y'}^{-1}(e'^\circ)) \cong \mathbb{Z}/d_{e'}(\varphi)\mathbb{Z}; $$

hence the set of isomorphisms $\tau_{x'}^{-1}(e'^\circ) \xrightarrow{\sim} \tau_{y'}^{-1}(e'^\circ)$ is a principal homogeneous space under the pre- or post-composition action of $\mathrm{Aut}_{\tau^{-1}(e^\circ)}(d_{e'}(\varphi))$.

Let $E_f^{\pm}(\Sigma')$ denote the set of oriented finite edges of $\Sigma'$, and for $e' \in E_f^{\pm}(\Sigma')$ let $\vec{e}'$ denote the same edge with the opposite orientation. Let $\mathcal{G}(\Sigma', X)$ denote the set of tuples $(\Theta_{e'})_{e' \in E_f^{\pm}(\Sigma')}$ of isomorphisms

$$ \Theta_{e'} : \tau_{x'}^{-1}(e'^\circ) \xrightarrow{\sim} \tau_{y'}^{-1}(e'^\circ) \qquad (7.2.1) $$

such that $\Theta_{\vec{e}'} = \Theta_{e'}^{-1}$, where $e' = \vec{x'}y'$. We call $\mathcal{G}(\Sigma', X)$ the set of gluing data for a lifting of $\Sigma'$ to a tame covering of $(X, V \cup D)$, and we emphasize that $\mathcal{G}(\Sigma', X)$ is nonempty.

Let $\alpha \in \mathrm{Aut}_{\Sigma}(\Sigma')$, so $\alpha$ is a degree-1 finite harmonic morphism $\Sigma' \xrightarrow{\sim} \Sigma'$ preserving $\Sigma' \to \Sigma$. Let $x' \in V_f(\Sigma')$, and let $x'' = \alpha(x')$ and $x = \varphi(x') = \varphi(x'')$. Part of the data of
$\alpha$ is a $C_x$-isomorphism $\alpha_{x'} : C_{x'} \xrightarrow{\sim} C_{x''}$. By Corollary 6.21, there is a unique lift of $\alpha_{x'}$ to a $Y(x)$-isomorphism $Y'(x') \xrightarrow{\sim} Y'(x'')$ of punctured star-shaped curves inducing the isomorphism $C_{x'} \xrightarrow{\sim} C_{x''}$ on residue curves. Let $e' \in E_f(\Sigma')$ be an edge adjacent to $x'$, let $e'' = \alpha(e')$, and let $e = \varphi(e') = \varphi(e'')$. Then, $\alpha_{x'}$ restricts to an isomorphism

$$ \alpha_{x'} : \tau_{x'}^{-1}(e'^{\circ}) \xrightarrow{\sim} \tau_{x''}^{-1}(e''^{\circ}). $$

Define the conjugation action of Aut$_\Sigma(\Sigma')$ on $\mathcal{G}(\Sigma', X)$ by the rule

$$ \alpha \cdot (\Theta_{e'})_{e' \in E_f^{\pm}(\Sigma')} = (\alpha_{y'}^{-1} \circ \Theta_{\alpha(e')} \circ \alpha_{x'})_{e' \in E_f^{\pm}(\Sigma')} \quad (7.3.1) $$

where $e' = \xrightarrow{\rightarrow} x'y'$.

**Theorem 7.4 (Classification of lifts of harmonic morphisms).** Let $(X, V \cup D)$ be a trian-gulated punctured curve with skeleton $\Sigma = \Sigma(X, V \cup D)$. Assume that $\Sigma$ has no loop edges. Let $\varphi : \Sigma' \to \Sigma$ be a tame covering of metrized compexes of curves.

1. There is a canonical bijection between the set of gluing data $\mathcal{G}(\Sigma', X)$ and the set of liftings of $\Sigma'$ to a tame covering of $(X, V \cup D)$, up to $X$-isomorphism preserving $\Sigma'$. In particular, there exists a lifting of $\Sigma'$. Any such lifting has no nontrivial automorphisms which preserve $\Sigma'$.

2. Two tuples of gluing data determine $X$-isomorphic curves if and only if they are in the same orbit under the conjugation action Equation 7.3.1. The stabilizer in Aut$_\Sigma(\Sigma')$ of an element of $\mathcal{G}(\Sigma', X)$ is canonically isomorphic to the $X$-automorphism group of the associated curve.

**Proof.** Given $(\Theta_{e'}) \in \mathcal{G}(\Sigma', X)$, one can glue the local lifts $\{Y(x')\}_{x' \in V_f(\Sigma')}$ via the iso-morphisms Equation 7.2.1 to obtain an analytic space which one easily verifies is smooth and proper, hence arises as the analytification of an algebraic curve $X'$. Moreover, the morphisms $Y'(x') \to Y(x)$ glue to give a morphism $X'^{\text{an}} \to X^{\text{an}}$, which is the analytifica-tion of a morphism $\varphi : X' \to X$. By construction, if $V' = \varphi^{-1}(V)$ and $D' = \varphi^{-1}(D)$, then $(X', V' \cup D') \to (X, V \cup D)$ is a lifting of $\Sigma'$ to a tame covering of $(X, V \cup D)$.

Now, let $\varphi : (X', V' \cup D') \to (X, V \cup D)$ be any lifting of $\Sigma'$ to a tame covering of $(X, V \cup D)$. As explained in Example 6.14, for every $x \in V$, the inverse image $\varphi^{-1}(Y(x))$ is a disjoint union of tame covers of the punctured star-shaped curve $(Y(x), x, D(x))$, one for each $x' \in \varphi^{-1}(x)$. By Theorem 6.18, we have canonical identification $\varphi^{-1}(Y(x)) = \bigsqcup_{x' \mapsto x} Y(x')$. For $x', y' \in V'$, we have $Y(x') \cap Y(y') = \bigsqcup_{e' \in x', y' \in V'} \tau^{-1}(e'^{\circ})$, where $\tau : X'^{\text{an}} \to \Sigma'$ is the retraction. Hence, $X'^{\text{an}}$ is obtained by pasting the local liftings $Y(x')$ via a choice of isomorphisms $(\Theta_{e'}) \in \mathcal{G}(\Sigma', X)$.

Let $\psi : X' \xrightarrow{\sim} X'$ be an $X$-automorphism preserving $\Sigma'$. Since $\psi$ is the identity on the set $\Sigma' \subset X'^{\text{an}}$, for $x' \in V'$, we have $\psi(Y(x')) = Y(x')$. We also have that for $x' \in V'$ the map of residue curves $\psi_{x'} : C_{x'} \to C_{x'}$ is the identity, so by Corollary 6.21, $\psi$ restricts to the identity morphism $Y(x') \to Y(x')$. Since $X'^{\text{an}} = \bigsqcup_{x' \in V'} Y(x')$, we have that $\psi$ is the identity. It follows from this that the tuple $(\Theta_{e'})$ can be recovered from the class of $X'$ modulo $X$-isomorphisms preserving $\Sigma'$, so different gluing data give rise to curves which are not equivalent under such isomorphisms. This proves (1).

Let $\varphi : (X', V' \cup D') \to (X, V \cup D)$ and $\varphi'' : (X'', V'' \cup D'') \to (X, V \cup D)$ be the liftings of $\Sigma'$ associated to the tuples of gluing data $(\Theta_{e'})$, $(\Theta_{e''}) \in \mathcal{G}(\Sigma', X)$, respectively. Suppose
that there exists $\alpha \in \mathrm{Aut}_{\Sigma}(\Sigma')$ such that $\alpha \cdot (\Theta'_{e'}) = (\Theta''_{e'} bridge)$, Then, for all $e' = x'y' \in E_{\Sigma'}^{\pm}(\Sigma')$, we have

$$ \Theta''_{\alpha(e')} \circ \alpha_{x'} = \alpha_{y'} \circ \Theta'_{e'} $$

so the $Y(\varphi(x'))$-isomorphisms $\alpha_{x'} : Y(x') \xrightarrow{\sim} Y(\alpha(x'))$ glue to give an X-isomorphism $\alpha : X' \xrightarrow{\sim} X''$. Conversely, the restriction of an X-isomorphism $\alpha : X' \xrightarrow{\sim} X''$ to $\Sigma'$ is a $\Sigma$-automorphism of $\Sigma'$. It is easy to see that these are inverse constructions. Taking $X' = X''$ we have an injective homomorphism $\mathrm{Aut}_{\chi}(X') \hookrightarrow \mathrm{Aut}_{\Sigma}(\Sigma')$; it follows formally from the above considerations that its image is the stabilizer of $(\Theta'_{e'})$. ■

**Remark 7.5.** Let $X'$ be a lifting of $\Sigma'$ to a tame covering of $(X, V \cup D)$. It follows from Theorem 7.4(1) that the natural homomorphism

$$ \mathrm{Aut}_{\chi}(X') \to \mathrm{Aut}_{\Sigma}(\Sigma') $$

is injective. It is not in general surjective: see Example 7.8 (where all the stabilizer groups of elements of $\mathcal{G}(\Sigma', X)$ are proper subgroups of $\mathrm{Aut}_{\Sigma}(\Sigma')$).

**Remark 7.6.** It is worth mentioning that the question of lifting (and classification of all possible liftings) in the wildly ramified case is more subtle and cannot be guaranteed in general. Some lifting results in the wildly ramified case are known: e.g., Liu proves in [33], Proposition 5.4, that any finite surjective generically étale admissible cover of proper semistable curves over an algebraically closed field $k$ lifts to a finite morphism of smooth proper curves over $K$. However, the same statement for metrized complexes cannot be true in general: consider the metrized complex $\mathcal{C}$ consisting of a single finite vertex $v$ and an infinite vertex $u$ attached to $v$ by an infinite edge $e$ with $C_v \cong \mathbf{P}_k^1$ and $\mathrm{red}_v(e) = \infty$. Consider the degree $p$ generically étale morphism of metrized complexes $\mathcal{C} \to \mathcal{C}$ which restricts to a degree $p$ Artin-Schreier cover $\varphi : \mathbf{P}_k^1 \to \mathbf{P}_k^1$, étale over $\mathbf{A}_k^1 \to \mathbf{A}_k^1$ and ramified at $\infty$ (with ramification index $p$) and which is linear of slope $p$ on the infinite edge $e$. This map cannot be lifted to a degree $p$ cover $\mathbf{P}_k^1 \to \mathbf{P}_k^1$ in characteristic zero; otherwise, we would obtain a (connected) étale degree $p$ cover of $\mathbf{A}_k^1$ which is impossible since $\mathbf{A}_k^1$ is simply connected.

We now state a variant of Theorem 7.4 in which we allow loop edges.

**Theorem 7.7.** Let $(X, V \cup D)$ be a triangulated punctured K-curve, let $\Sigma = \Sigma(X, V \cup D)$, and let $\bar{\varphi} : \Sigma' \to \Sigma$ be a tame covering of $\Lambda$-metrized complexes of k-curves. Then, there exists a tame covering $\varphi : (X', V' \cup D') \to (X, V \cup D)$ of triangulated punctured K-curves lifting $\varphi$.

**Proof.** The only issue is that in Theorem 7.7, we merely require that $V$ is a semistable vertex set whereas; whereas in Theorem 7.4, we require it to be strongly semistable. However, we can modify $\Sigma$ by inserting a (valence 2) vertex $v$ along each loop edge, with $C_v \cong \mathbf{P}_k^1$, and with the marked points on $C_v$ being 0 and $\infty$. Call the resulting metrized complex $\tilde{\Sigma}$, and let $\bar{V}_0$ denote the set of vertices which have been added to the vertex set $\bar{V}$ for $\Sigma$. We construct a new metrized complex $\tilde{\Sigma}'$ from $\Sigma'$ by adding $\bar{V}'_0 := \bar{\varphi}^{-1}(\bar{V}_0)$
to the vertex set $\bar{V}'$ for $\Sigma'$ and letting $C'_{v'} \cong \mathbf{P}_k^1$ for $v' \in \bar{V}'_0$, with the marked points on $C'_{v'}$ being 0 and $\infty$. We can extend the tame covering $\bar{\varphi}: \Sigma' \to \Sigma$ to a tame covering $\tilde{\Sigma}' \to \tilde{\Sigma}$ by letting the map from $C'_{v'}$ to $C_{\bar{\varphi}(v')}$, for $v' \in \bar{V}'_0$, be $z \mapsto z^d$, where $d$ is the local degree of $\bar{\varphi}$ along the loop edge corresponding to $v'$. By Theorem 7.4, there is a tame covering $(X', V' \cup V'_0 \cup D') \to (X, V \cup V_0 \cup D)$ lifting $\tilde{\Sigma}' \to \tilde{\Sigma}$, where $V'_0$ (resp. $V_0$) corresponds to $\bar{V}'_0$ (resp. $\bar{V}_0$). Removing the vertices in $V'_0$ and $V_0$ gives a tame covering $(X', V' \cup D') \to (X, V \cup D)$ lifting $\bar{\varphi}$.

<figure><img src="image_7.png" /></figure>

**Example 7.8.** In this example, we suppose that $\mathrm{char}(k) \neq 2$. Let $E$ be a Tate curve over $K$, let $\Sigma$ be its (set-theoretic) skeleton, and let $\tau: E^{\mathrm{an}} \to \Sigma$ be the canonical retraction. Let $U: \mathbf{G}_m^{\mathrm{an}}/q^Z \xrightarrow{\sim} E^{\mathrm{an}}$ be the Tate uniformization of $E^{\mathrm{an}}$. The two-torsion subgroup of $E$ is $U(\{\pm 1, \pm \sqrt{q}\})$; choose a square root of $q$, let $y = \tau(U(1))$ and $z = \tau(U(\sqrt{q}))$, and let $V = \{y, z\} \subset \Sigma$. This is a semistable vertex set of $E$ and $\Sigma = \Sigma(E, V)$ is the circle with circumference $\mathrm{val}(q)$; the points $y$ and $z$ are antipodal on $\Sigma$. Let $e_1, e_2$ be the edges of $\Sigma$; orient $e_1$ so that $y$ is the source vertex and $e_2$ so that $z$ is the source vertex. The residue curves $C_y$ and $C_z$ are both isomorphic to $\mathbf{P}_k^1$; fix isomorphisms $C_y \cong \mathbf{P}_k^1$ and $C_z \cong \mathbf{P}_k^1$ such that the tangent direction at a vertex in the direction of the outgoing (resp. incoming) edge corresponds to $\infty$ (resp. 0).

Let $\Sigma'$ be a circle of circumference $\frac{1}{2}\mathrm{val}(q)$, let $V' = \{y', z'\}$ be a pair of antipodal points on $\Sigma'$, and let $e'_1 = \vec{y'z'}$ and $e'_2 = \vec{z'y'}$ be the two edges of $\Sigma'$, with the indicated orientations. Enrich $\Sigma'$ with the structure of a metrized complex of curves by setting $C_{y'} = C_{z'} = \mathbf{P}_k^1$ and letting the tangent direction at a vertex in the direction of the outgoing (resp. incoming) edge correspond to $\infty$ (resp. 0). Define a morphism $\varphi: \Sigma' \to \Sigma$ of metric graphs by

$$ \varphi(y') = y, \quad \varphi(z') = z, \quad \varphi(e'_1) = e_1, \quad \varphi(e'_2) = e_2, $$

with both degrees equal to 2. See Figure 3.

Topologically, $\Sigma' \to \Sigma$ is a homeomorphism. Let $\varphi_{y'}: C_{y'} \to C_y$ and $\varphi_{z'}: C_{z'} \to C_z$ both be the morphism $t \mapsto t^2: \mathbf{P}_k^1 \to \mathbf{P}_k^1$. This makes $\varphi$ into a degree-2 tame covering of metrized complexes of curves.

For $x \in V$, the analytic space $Y(x)$ is an open annulus of logarithmic modulus $\mathrm{val}(q)$; fix an isomorphism $Y(x) \cong \mathbf{S}(q)_+$ for each $x$. For $x' \in V'$, the analytic space $Y'(x')$ is an open annulus of logarithmic modulus $\frac{1}{2}\mathrm{val}(q)$; fix isomorphisms $Y'(x') \cong \mathbf{S}(\sqrt{q})_+$ such that the morphisms $Y'(x') \to Y(\varphi(x'))$ are given by $t \mapsto t^2: \mathbf{S}(\sqrt{q})_+ \to \mathbf{S}(q)_+$. For $e' \in E(\Sigma')$, we have $d_{\varphi}(e') = 2$ by definition. Hence, $\mathcal{G}(\Sigma', E)$ has four elements, which

<figure><img src="image_8.png" /><figcaption>**Figure 3** This figure illustrates Example 7.8. The morphism $\varphi$ is a homeomorphism of underlying sets but has a degree of 2 on $e'_1$ and $e'_2$. The arrows on $e_1, e_2, e'_1, e'_2$ represent the chosen orientations and the indicated tangent vectors at $y, z, y', z'$ represent the tangent direction corresponding to $\infty$ in the corresponding $k$-curve.</figcaption></figure>
we label $(\pm 1, \pm 1)$. By Theorem 7.4(1), there are four corresponding classes of liftings of $\Sigma'$ to a tame covering of $(E, V)$ up to isomorphism preserving $\Sigma'$.

Any $\Sigma$-automorphism of $\Sigma'$ is the identity on the underlying topological space. Hence, an element of Aut$_{\Sigma}(\Sigma')$ is a pair of automorphisms $(\psi_{y'}, \psi_{z'}) \in \text{Aut}_{C_y}(C_{y'}) \times \text{Aut}_{C_z}(C_{z'})$ such that $\psi_{y'}$, $\psi_{z'}$ fix the points $0, \infty \in \mathbb{P}_k^1$. Thus, $\psi_{y'}, \psi_{z'} = \pm 1$, so $\text{Aut}_{\Sigma}(\Sigma') = \{\pm 1\} \times \{\pm 1\}$. For $x' = y', z'$, the automorphism $-1 : C_{x'} \xrightarrow{\sim} C_{x'}$ lifts to the automorphism $-1 : \mathbf{S}(\sqrt{q})_+ \xrightarrow{\sim} \mathbf{S}(\sqrt{q})_+$. Therefore, the conjugation action of Aut$_{\Sigma}(\Sigma')$ on $\mathcal{G}(\Sigma', E)$ is given as follows:

$$ (1, 1) \cdot (\pm 1, \pm 1) = (-1, -1) \cdot (\pm 1, \pm 1) = (\pm 1, \pm 1) \\ (-1, 1) \cdot (\pm 1, \pm 1) = (1, -1) \cdot (\pm 1, \pm 1) = -(\pm 1, \pm 1). $$

By Theorem 7.4(2), there are two isomorphism classes of lifts of $\Sigma'$ to a tame covering of $(E, V)$, and each such lift has two automorphisms.

These liftings can be described concretely as follows. Fix a square root of $q$, let $E_{\pm}$ be the algebraization of the analytic elliptic curve $\mathbf{G}_{m}^{\text{an}} / (\pm \sqrt{q})^{2}$, and let $\psi_{\pm} : E_{\pm} \to E$ be the morphism $t \mapsto t^2$ on uniformizations. Let $\Sigma_{\pm} = \psi_{\pm}^{-1}(\Sigma)$. Then, $\Sigma_{\pm}$ is isomorphic to $\Sigma'$ as a tame covering of $\Sigma$ and $E_{\pm}$ is a lifting of $\Sigma'$ to a tame covering of $(E, D)$. The elliptic curves $E_{\pm}$ are not isomorphic (as $K$ schemes) because they have different $q$ invariants, so they represent the two isomorphism classes of liftings of $\Sigma'$. The nontrivial automorphism of $E_{\pm}$ is given by translating by the image of $-1 \in \mathbf{G}_{m}^{\text{an}}(K)$ (this is not a homomorphism). In fact, since $\psi_{\pm} : E_{\pm} \to E$ is an étale Galois cover of degree 2, this is the only nontrivial automorphism of $E_{\pm}$ as an $E$-curve; so the homomorphism

$$ \text{Aut}_{E}(E_{\pm}) \to \text{Aut}_{\Sigma}(\Sigma') \cong \{\pm 1\} \times \{\pm 1\} $$

is injective but not surjective (its image is $\{\pm(1, 1)\}$).

Consider now the automorphism group $\text{Aut}_{\Sigma}^{0}(\Sigma') \subset \text{Aut}_{\Sigma}(\Sigma')$ consisting of all degree-1 finite harmonic morphisms $\alpha : \Sigma' \to \Sigma'$ respecting $\varphi : \Sigma' \to \Sigma$ and inducing the identity on the metric graph $\Gamma'$ underlying $\Sigma'$. The restriction of the conjugacy action of $\text{Aut}_{\Sigma}(\Sigma')$ on $\mathcal{G}(\Sigma', X)$ to the subgroup $\text{Aut}_{\Sigma}^{0}(\Sigma')$ admits a simplified description that we describe now. Combining this with arguments similar to those in the proof of Theorem 7.4 yields a classification of the set of liftings of $\Sigma'$ up to isomorphism as liftings of the metric graph underlying $\Sigma'$; see Theorem 7.10.

First, for each $x' \in V(\Sigma')$ with image $x = \varphi(x')$, let $\text{Aut}_{C_{x'}}^{0}(C_{x'})$ be the subgroup of $\text{Aut}_{C_{x}}(C_{x'})$ that fixes every point of $C_{x'}$ of the form $\text{red}(e')$ for some edge $e'$ of $\Sigma'$ adjacent to $x'$. Then,

$$ \text{Aut}_{\Sigma}^{0}(\Sigma') = \prod_{x' \in V_{f}(\Sigma')} \text{Aut}_{C_{\varphi(x)}}^{0}(C_{x'}) =: \mathcal{E}^{0}. $$

Denote by $\mathcal{E}^1$ the finite abelian group

$$ \mathcal{E}^1 = \prod_{e' \in E_f(\Sigma')} \text{Aut}_{\tau^{-1}(\varphi(e')^{\circ})} (d_{e'}(\varphi)). $$

The discussion preceding Theorem 7.4 shows that the set of gluing data $\mathcal{G}(\Sigma', X)$ is canonically a principal homogeneous space under $\mathcal{E}^1$.
The subgroup Aut<sub>C<sub>x</sub></sub><sup>0</sup>(C<sub>x'</sub>) ⊂ Aut<sub>C<sub>x</sub></sub>(C<sub>x'</sub>) corresponds to the subgroup Aut<sub>Y<sub>(x)</sub></sub><sup>0</sup>(Y'(x')) of automorphisms in Aut<sub>Y<sub>(x)</sub></sub>(Y'(x')) which act trivially on the skeleton Σ'(x'). Restriction of a Y(x)-automorphism of Y'(x') to τ<sub>x'</sub><sup>-1</sup>(e<sup>°</sup>) defines a homomorphism

$$ \rho_{x',e'} : \mathrm{Aut}_{C_x}^0(C_{x'}) = \mathrm{Aut}_{Y(x)}^0(Y'(x')) \to \mathrm{Aut}_{\tau^{-1}(e^0)}(d_{e'}(\varphi)). $$

Fix an orientation of each finite edge of Σ'. For x' ∈ V<sub>f</sub>(Σ') with image x = φ(x'), let ρ<sub>x'</sub>: Aut<sub>C<sub>x</sub></sub><sup>0</sup>(C<sub>x'</sub>) → E<sup>1</sup> be the homomorphism whose e' coordinate is

$$ (\rho_{x'}(\alpha))_{e'} = \begin{cases} \rho_{x',e'}(\alpha) & \text{if } x' \text{ is the source vertex of } e' \\ \rho_{x',e'}(\alpha)^{-1} & \text{if } x' \text{ is the target vertex of } e' \\ 1 & \text{if } x' \text{ is not an endpoint of } e'. \end{cases} $$

Taking the product over all x' ∈ V<sub>f</sub>(Σ') yields a homomorphism ρ: E<sup>0</sup> → E<sup>1</sup>. The kernel and cokernel of ρ are independent of the choice of orientations. Viewing E<sup>0</sup> → E<sup>1</sup> as a two-term complex E<sup>•</sup> of groups, its cohomology groups are

$$ H^0(E^\bullet) = \ker(\rho) \quad \text{and} \quad H^1(E^\bullet) = \mathrm{coker}(\rho). $$

**Theorem 7.10.** Let (X, V ∪ D) be a triangulated punctured curve with skeleton Σ = Σ(X, V ∪ D). Assume that Σ has no loop edges. Let φ : Σ' → Σ be a tame covering of metrized complexes of curves.

1. G(Σ', X) is canonically a principal homogeneous space under E<sup>1</sup>, and the conjugacy action of Aut<sub>Σ</sub><sup>0</sup>(Σ') on G(Σ', X) is given by the action of the subgroup ρ(Aut<sub>Σ</sub><sup>0</sup>(Σ')) = ρ(E<sup>0</sup>) ⊆ E<sup>1</sup> on G(Σ', X).

2. The set of liftings of Σ' up to isomorphism as liftings of the metric graph underlying Σ' is a principal homogeneous space under H<sup>1</sup>(E<sup>•</sup>), and the group of automorphisms of a given lifting as a lifting of the metric graph underlying Σ' is isomorphic to H<sup>0</sup>(E<sup>•</sup>).

### 7.11 Descent to a general ground field

Let K<sub>0</sub> be a subfield of K, let X<sub>0</sub> be a smooth, projective, geometrically connected K<sub>0</sub>-curve, and let D ⊂ X<sub>0</sub>(K<sub>0</sub>) be a finite set. Let X = X<sub>0</sub>⊗<sub>K<sub>0</sub></sub>K, let V be a strongly semistable vertex set of (X, D), let Σ = Σ(X, V ∪ D), and let φ : Σ' → Σ be a tame covering of metrized complexes of curves, as in the statement of Theorem 7.4. Let φ : X' → X be a lifting of φ to a tame covering of (X, V ∪ D). Whereas we take the data of the morphism Σ' → Σ to be geometric, i.e. only defined over K, the covering X' → X is in fact defined over a finite, separable extension of K<sub>0</sub>. This follows from the fact that if U = X \ D and U' = X' \ φ<sup>-1</sup>(D), then φ : U' → U is a tamely ramified cover of U over X relative to D (see Remarks 2.20(1) and 4.32(1)), along with the following lemma.

**Lemma 7.12.** Let K<sub>0</sub> be any field, let K be a separably closed field containing K<sub>0</sub>, let X<sub>0</sub> be a smooth, projective, geometrically connected K<sub>0</sub>-curve, and let D ⊂ X<sub>0</sub>(K<sub>0</sub>) be a finite set. Let X = X<sub>0</sub>⊗<sub>K<sub>0</sub></sub>K, let φ : X' → X be a finite morphism with X' smooth and (geometrically) connected, and suppose that φ is branched only over D, with all ramification degrees prime
to the characteristic of $K$. Then, there exists a finite, separable extension $K_1$ of $K$ and a
morphism $\varphi_1 : X'_1 \to X_0 \otimes_{K_0} K_1$ descending $\varphi$.

**Proof.** Let $U_0 = X_0 \setminus D$, and let $U = X \setminus D$ and $U' = X' \setminus \varphi^{-1}(D)$. First, suppose that $K_0$ is separably closed. By [40], Exposé XIII, Corollaire 2.12, the tamely ramified étale fundamental groups $\pi_1^t(U_0)$ and $\pi_1^t(U)$ are isomorphic (with respect to some choice of base point). Since $\varphi : U' \to U$ is a tamely ramified cover of $U$ over $X$ relative to $D$, it is classified by a finite-index subgroup of $\pi_1^t(U) = \pi_1^t(U_0)$, so there exists a tamely ramified cover $\varphi_0 : U'_0 \to U_0$ of $U_0$ over $X_0$ relative to $D$ descending $\varphi$.

Now, we drop the hypothesis that $K_0$ is separably closed. By the previous paragraph, we may assume that $K$ is a separable closure of $K_0$. By general principles, the projective morphism $X' \to X$ descends to a subfield of $K$ which is finitely generated (i.e. finite) over $K_0$. ■

**Remark 7.13.** With the notation in (7.11), suppose that $K_0$ is a complete valued field with value group $\Lambda_0 = \text{val}(K_0^\times)$ and *algebraically closed* residue field $k$; that $\Sigma$ is 'rational over $K_0'$ in that it comes from a (split) semistable formal $R_0$-model of $X_0$ in the sense of Section 5, and that $\Sigma'$ has edge lengths contained in $\Lambda_0$. With some extra work, it is possible to carry out the gluing arguments of Theorem 7.7 directly over the field $K_0$ (in this context Lemma 6.10(1) still holds), which shows that the cover $X' \to X$ is in fact defined over $K_0$. In the case of a discrete valuation, this also follows from [47] or from [39] if $D = \emptyset$.

**7.14 Liftings of tame harmonic morphisms**

Theorem 7.7 implies the existing of liftings for (finite) tame harmonic morphisms of
metrized complexes which are not necessarily tame coverings, the difference being
generic étaleness. See Definition 2.21 for both definitions.

**Proposition 7.15.** Let $(X, V \cup D)$ be a triangulated punctured $K$-curve, let $\Sigma = \Sigma(X, V \cup D)$, and let $\varphi : \Sigma' \to \Sigma$ be a tame harmonic morphism of metrized complexes of curves. Then, there exists a triangulated punctured $K$-curve $(X', V' \cup D')$ and a finite morphism $\psi : (X', V' \cup D') \to (X, V \cup D)$ lifting $\varphi$.

**Proof.** For any finite vertex $x'$ of $\Sigma'$ at which $\varphi$, seen as a morphism of augmented metric graphs, is ramified, let $q_1, \dots, q_r \in C_v$ be all the branch points of $\varphi_{x'}$ which do not correspond to any edge of $\Sigma$, and let $p_{ij}$ denote the preimages of $q_i$ under $\varphi_{x'}$. We modify $\Sigma'$ and $\Sigma$ by attaching infinite edges $e_i$ to $\Sigma$ at $x = \varphi(x')$ for each $q_i$, infinite edges $e'_{ij}$ to $\Sigma'$ at $x'$ for each $p_{ij}$, and defining $\text{red}_x(e_i) = q_i$ and $\text{red}_{x'}(e'_{ij}) = p_{ij}$. Since each $\varphi_{x'}$ is tamely ramified, the harmonic morphism $\varphi$ naturally extends to a tame covering $\tilde{\varphi} : \tilde{\Sigma}' \to \tilde{\Sigma}$ between the resulting modifications. Enlarging $D$ to $\tilde{D}$ by choosing points in $X(K) \setminus D$ with reduction $q_i$, we can assume that $\tilde{\Sigma} = \Sigma(X, V \cup \tilde{D})$. The result now follows from Theorem 7.7 by first lifting $\tilde{\varphi}$ to a tame covering $\psi : (X', V' \cup \tilde{D}) \to (X, V \cup \tilde{D})$ and then taking the restriction of $\psi$ to $(X', V' \cup D') \to (X, V \cup D)$, where $D' = \psi^{-1}(D)$. ■

**Remark 7.16.** As mentioned above, in Proposition 7.15 we do not require that $\varphi : \Sigma' \to \Sigma$ be generically étale. This corresponds to not requiring that $D \subset X(K)$ contain the
branch locus of the lift $\psi: X' \to X$: indeed, by Lemma 4.33, if $D$ contains the branch locus then $\varphi$ is a tame covering. In the situation of Proposition 7.15, the set of liftings of $\Sigma'$ to a tame covering of $(X, V \cup D)$ can be infinite. For example, if $\Gamma' = \{p'\}$ and $\Gamma = \{p\}$ are both points and the morphism $\varphi_{p'}: C_{p'} \cong \mathbf{P}^1 \to C_p \cong \mathbf{P}^1$ is $z \mapsto z^2$, with $(X, V)$ a minimal triangulation of $\mathbf{P}^1$ and $D = \emptyset$, then there are infinitely many such lifts, corresponding to the different ways of lifting the critical points and critical values of $\varphi_{p'}$ from $k$ to $K$.

## 8 Application: component groups of Néron models

In contrast to the rest of the paper, we assume in this section that $R$ is a complete discrete valuation ring with fraction field $K$ and algebraically closed residue field $k$. In this case, we take the value group $\Lambda$ to be $Z$.

There is a natural notion of harmonic 1 forms on a metric graph $\Gamma$ of genus $g$ (see [37]). The space $\Omega^1(\Gamma)$ is a $g$-dimensional real vector space which can be canonically identified with $H^1(\Gamma, \mathbf{R})$, but we write elements of $\Omega^1(\Gamma)$ as $\omega = \sum_e \omega_e de$ as in [7], where the sum is over all edges with respect to a fixed vertex set for $\Gamma$. There is a canonical lattice $\Omega_{\mathbb{Z}}^1(\Gamma)$ of integer harmonic 1-forms inside $\Omega^1(\Gamma)$; these are the harmonic 1-forms for which every $\omega_e$ is an integer. A harmonic morphism $\varphi: \Gamma' \to \Gamma$ induces a natural pullback map on harmonic 1-forms via the formula

$$ \varphi^* \left( \sum_e \omega_e de \right) = \sum_{e'} d_{e'}(\varphi) \cdot \omega_{\varphi(e')} de'. $$

Recall that a $Z$-metric graph (i.e. a $\Lambda$-metric graph for $\Lambda = Z$) with no infinite vertices is a (compact and finite) metric graph whose edge lengths are all positive integers (or equivalently, having a vertex set with respect to which all edge lengths are 1). If $X/K$ is a smooth, proper, geometrically connected analytic curve and $\mathcal{X}$ is a semistable $R$-model for $X$, the skeleton $\Gamma_{\mathcal{X}}$ of $\mathcal{X}$ is naturally a $Z$-metric graph. Moreover, as we have seen earlier in this paper, a finite morphism of semistable models induces in a natural way a harmonic morphism of $Z$-metric graphs.

Let $\Gamma$ be a $Z$-metric graph with no infinite vertices. We define the regularized Jacobian $Jac_{\text{reg}}(\Gamma)$ of $\Gamma$ to be the group $Jac(G)$, where $G$ is the ‘regular model’ for $\Gamma$ (the unique graph induced by a vertex set with all edge lengths equal to 1). The group $Jac(G)$ can be described explicitly as

$$ Jac(G) = \Omega^1(G)^\# / H_1(G, Z), $$

where $\Omega^1(G)^\#$ denotes the linear functionals $\Omega^1(G, \mathbf{R}) \to \mathbf{R}$ of the form $\int_\alpha$ with $\alpha \in C_1(G, Z)$ (see [7]). There is a canonical isomorphism between $Jac(G)$ and $Pic^0(G)$, the group of divisors of degree 0 on $G$ modulo the principal divisors (see [5]) as well as a canonical isomorphism

$$ Jac(G) \cong H^1(G, Z)^\# / \Omega_{\mathbb{Z}}^1(G), $$

where

$$ H^1(G, Z)^\# = \left\{ \omega \in \Omega^1(G, \mathbf{R}) : \int_{\gamma} \omega \in Z \quad \forall \gamma \in H_1(G, Z) \right\}. $$

We recall (in our own terminology) the following result of Raynaud (cf. [38] and ([6], Appendix A)):
**Theorem 8.2** (Raynaud). If $X/K$ is a semistable curve, then the component group of the Néron model of $Jac(X)$ over $R$ is canonically isomorphic to $Jac_{reg}(\Gamma_X)$ (for any semistable model $\mathcal{X}$ of $X$).

A harmonic morphism $\varphi : \Gamma' \to \Gamma$ of $Z$-metric graphs induces in a functorial way homomorphisms $\varphi_* : Jac_{reg}(\Gamma') \to Jac_{reg}(\Gamma)$, defined by

$$ \varphi_* \left( \left[ \int_{\alpha'} \right] \right) = \left[ \int_{\varphi_* (\alpha')} \right], $$

where

$$ \varphi_* \left( \sum \alpha_{e'} e' \right) = \sum \alpha_{e'} \varphi(e'), $$

and $\varphi^* : Jac_{reg}(\Gamma) \to Jac_{reg}(\Gamma')$, defined by

$$ \varphi^*([\omega]) = [\varphi^* \omega], $$

where

$$ \varphi^* \left( \sum \omega_e de \right) = \sum d_{e'}(\varphi) \omega_{\varphi(e')} de'. $$

We have the following elementary result, whose proof we omit:

**Lemma 8.3.** Under the canonical isomorphism between $Jac(G)$ and $Pic^0(G)$, the homomorphism $\varphi_* : Jac(G') \to Jac(G)$ corresponds to the map $[D'] \mapsto [\varphi_*(D')]$ from $Pic^0(G')$ to $Pic^0(G)$, where $\varphi_* (\sum a_{v'}(v')) = \sum a_{v'} \varphi(v')$.

The following is a 'relative' version of Raynaud's theorem; it implies that the covariant functor which takes a semistable $R$-model $\mathcal{X}$ for a $K$-curve $X$ to the component group of the Néron model of the Jacobian of $X$ factors as the 'reduction graph functor' $\mathcal{X} \mapsto \Gamma_X$ followed by the 'regularized Jacobian functor' $\Gamma \mapsto Jac_{reg}(\Gamma)$. It is a straightforward consequence of the analytic description of Raynaud's theorem given in [6], Appendix A (see also [13]):

**Theorem 8.4.** If $f_K : X' \to X$ is a finite morphism of curves over $K$, the induced maps $f_* : \Phi_{f(X')} \to \Phi_{f(X)}$ and $f^* : \Phi_{f(X)} \to \Phi_{f(X')}$ on component groups coincide with the induced maps $\varphi_* : Jac_{reg}(\Gamma_X') \to Jac_{reg}(\Gamma_X)$ and $\varphi^* : Jac_{reg}(\Gamma_X) \to Jac_{reg}(\Gamma_X')$ for any morphism $f : \mathcal{X}' \to \mathcal{X}$ of semistable models extending $f_K$. (Here, $\varphi$ denotes the harmonic morphism of skeleta induced by $f$; see Remark 5.14.)

It follows easily from Lemma 8.3 and Theorem 8.4 that if $f_K : X' \to X$ is regularizable, i.e., if $f_K$ extends to a morphism of regular semistable models, then the induced map $f_* : \Phi_{f(X')} \to \Phi_X$ on component groups is surjective. Thus whenever $f_*$ is not surjective, it follows that $f_K$ does not extend to a morphism of regular semistable models. One can obtain a number of concrete examples of this situation from modular curves (e.g. the map $X_0(33) \to E$ over $\mathbb{Q}_3^{\text{unr}}$, where $E$ is the optimal elliptic curve of level 33.)

**Remark 8.5.** One can show that if $\varphi : \Gamma' \to \Gamma$ is a harmonic morphism of $Z$-metric graphs, then $\varphi_* : \Phi_{f(X')} \to \Phi_X$ is surjective iff $\varphi^* : \Phi_X \to \Phi_{X'}$ is injective. Indeed,
it is not hard to check that the maps $\varphi_*$ and $\varphi^*$ are adjoint with respect to the com-
binatorial monodromy pairing, the non-degenerate symmetric bilinear form $\langle \cdot, \cdot \rangle :$
Jac$(G) \times Jac$(G) $\to$ $\mathbb{Q}/\mathbb{Z}$ defined by $\langle [\omega], [\int_\alpha \omega] \rangle = [\int_\alpha \omega]$, where $\omega \in H_1(G, \mathbb{Z})^\#$ and $\int_\alpha \in \Omega^1(G)^\#$. Thus, the groups ker$(\varphi^*)$ and coker$(\varphi_*)$ are canonically dual. This is a combi-
natorial analogue of results proved by Grothendieck in SGA7 on the (usual) monodromy
pairing.

As an application of Theorem 8.4 and our results on lifting harmonic morphisms, one
can construct many examples of harmonic morphisms of Z-metric graphs for which $\varphi_*$ is
not surjective. For example, consider the following question posed by Ken Ribet in a 2007
email correspondence with the second author (Baker):

Suppose $f : X' \to X$ is a finite morphism of semistable curves over a complete discretely valued field $K$ with $g(X) \ge 2$. Assume that the special fiber of the minimal regular model of $X'$ consists of two projective lines intersecting transversely. Is the induced map $f_* : \Phi_{X'} \to \Phi_X$ on component groups of Néron models necessarily surjective?

We now show that the answer to Ribet’s question is **no**.

**Example 8.6.** Consider the ‘banana graph’ $B(\ell_1, \ldots, \ell_{g+1})$ consisting of two vertices and $g+1$ edges of length $\ell_i$. This is the reduction graph of a semistable curve whose reduction has two $\mathbf{P}^1$’s crossing transversely at singular points of thickness $\ell_1, \ldots, \ell_{g+1}$. If we set $G' = B(1, 1, 1, 1)$ and $G = B(1, 2, 2)$ and let $\Gamma', \Gamma$ be the geometric realizations of $G'$ and $G$, respectively, then there is a degree 2 harmonic morphism of $\mathbb{Z}$-metric graphs $\varphi : \Gamma' \to \Gamma$ taking $e'_1$ and $e'_2$ to $e_1$, $e'_3$ to $e_2$, and $e'_4$ to $e_3$. The homomorphism $\varphi_*$ is non-surjective since $|\mathrm{Jac}_{\mathrm{reg}}(\Gamma')| = 4$ and $|\mathrm{Jac}_{\mathrm{reg}}(\Gamma')| = 8$. The map $\varphi$ can be enriched to a harmonic morphism $\tilde{\varphi}$ of metrized complexes of curves by attaching a $\mathbf{P}^1$ to each vertex and letting each morphism $\mathbf{P}^1 \to \mathbf{P}^1$ be $z \mapsto z^2$, with 0, $\pm 1$, and $\infty$ being the marked points upstairs. By Remark 7.13, the morphism $\tilde{\varphi}$ lifts to a morphism $\psi : X' \to X$ of curves over $K$. Since all edges of $G'$ have length 1, the minimal proper regular model of $X'$ has a special fiber consisting of two $\mathbf{P}^1$'s crossing transversely. By Theorem 8.4, the induced covariant map on component groups of Néron models is not surjective.

**Endnotes**

**Acknowledgments**

The authors thank Amaury Thuillier and Antoine Ducros for their help with some of the descent arguments in (5.2). We are grateful to Andrew Obus for a number of useful comments based on a careful reading of the first arXiv version of this manuscript. We are also very grateful to the referee for quickly and thoroughly checking all of the technical arguments in the paper, pointing out several errors and suggesting many improvements. MB was partially supported by NSF grant DMS-1201473. EB was partially supported by the ANR-09-BLAN-0039-01.

<footnote><sup>a</sup>This isomorphism is not canonical. However, there is an isomorphism Aut<sub>A</sub>(A') ≅ μ<sub>δ</sub> which is canonical up to the action of Z/2Z on Aut<sub>A</sub>(A') sending σ to σ<sup>-1</sup>. Equivalently, the parameters t and t<sup>-1</sup> give rise to different homomorphisms σ ↦ σ(t<sup>±1</sup>)/t<sup>±1</sup> : Aut<sub>A</sub>(A') → μ<sub>δ</sub>, but the choice of an orientation for (the skeleton of) A induces a canonical isomorphism Aut<sub>A</sub>(A') ≅ μ<sub>δ</sub>.</footnote>

<footnote><sup>b</sup>This $\delta$ need not be prime to $\mathrm{char}(k)$.</footnote>

<footnote><sup>c</sup>We thank the referee for greatly simplifying our original proof of Lemma 6.19.</footnote>
Author details

¹CNRS - Département de mathématiques et applications, École Normale Supérieure, 45 Rue d'Ulm, Paris. ²School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA. ³Université Pierre et Marie Curie, Paris 6, 4 Place Jussieu, 75 005 Paris, France.

Received: 9 December 2014 Accepted: 30 December 2014

Published online: 10 June 2015

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