$\underline{f}:=(f^{1},\ldots,f^{n})\in W_{n}(\kappa)$ with only pole $0$,
called the degeneration of the tree. The rational $f^{i}$ is called the $i$-th
degeneration of $\mathcal{T}$. Define

$d:=\max\\{p^{n-l}\deg_{x^{-1}}(f^{l})\mid l=1,\ldots,n\\}.$

These data are required to satisfy all of the following conditions.

  1. (H1)

Let $v\in V$. We have $\delta_{v}\neq 0$ if $v\neq v_{0}$.

  2. (H2)

For each $v\in V\setminus\\{v_{0}\\}$, the differential form $\omega_{v}$ does
not have zeros nor poles on $U_{v}\subsetneq C_{v}$.

  3. (H3)

For every edge $e\in E\setminus\\{e_{0}\\}$, we have the equality

$-\operatorname{ord}_{z_{e}}\omega_{t(e)}-1=\operatorname{ord}_{z_{e}}\omega_{s(e)}+1.$

  4. (H4)

For $v_{0}$, we have $d=\operatorname{ord}_{z_{e_{0}}}\omega_{t(e_{0})}+1$.

  5. (H5)

For every edge $e\in E$, we have

$d_{e}=-\operatorname{ord}_{z_{e}}\omega_{t(e)}-1\underset{s(e)\neq
v_{0}}{\stackrel{{\scriptstyle\ref{c2Hurwitz}}}{{=}}}\operatorname{ord}_{z_{e}}\omega_{s(e)}+1.$

  6. (H6)

For every edge $e\in E$, we have

$\delta_{s(e)}+(p-1)\epsilon_{e}d_{e}=\delta_{t(e)}.$

  7. (H7)

For $b\in B$, let $C_{v}$ be the component containing the point $x_{b}$. Then
the differential $\omega_{v}$ has a pole at $x_{b}$ of order $h_{b}$.

  8. (H8)

For each $v$, we have

$G_{v^{\prime}}\subseteq G_{v},$

for every successor vertex $v^{\prime}$ of $v$. Moreover, we have

$\sum_{v\to v^{\prime}}[G_{v}:G_{v^{\prime}}]>1,$

except if $v=v_{0}$ is the root, in which case there exists exactly one
successor $v^{\prime}$ and we have $G_{v}=G_{v^{\prime}}=G$.

For each $v\in V\setminus\\{v_{0}\\}$, we call $(\delta_{v},\omega_{v})$ the
degeneration type of $v$. For each $e\in E$, we call
$(\delta_{s(e)},d_{s(e)})$ (resp. $(\delta_{t(e)},d_{t(e)})$) the initial
degeneration type (resp. the final degeneration type) of $e$. The positive
integer $\mathfrak{C}:=d+1$ is called the _conductor_ of the Hurwitz tree. The
rational $\delta:=\delta_{v_{0}}$ is the depth of $\mathcal{T}$. We define the
height of the tree to be the maximal positive direction edges from its root to
its leaves. We call $\sum_{b\in B}h_{b}\cdot[B]$ the branching divisor of
$\mathcal{T}$.

######  Remark 4.3.

With the notation as above, fix an edge $e\in E$ and let $C_{e}\subseteq C$ be
the union of all components $C_{v}$ corresponding to vertices $v$ which are
separated from the root $v_{0}$ by the edge $e$. Then

$d_{e}=\sum_{\begin{subarray}{c}b\in B\\\ x_{b}\in
C_{e}\end{subarray}}h_{b}-1>0.$

In particular, we have $\mathfrak{C}=d+1=\sum_{b\in B}h_{b}$.

######  Remark 4.4.

Depending on the study, one may like to add or remove some information from
the Hurwitz tree in Definition 4.2. For instance, the monodromy groups can be
omitted where $G=\mathbb{Z}/p$ as in [Dan20b] and [Hen00], or where
$G=\mathbb{Z}/p\rtimes\mathbb{Z}/m$ as in [BW06]. In our situation, these data
are useful for deducing the Hurwitz tree of a sub-cover as discussed in
§4.4.2. When $G$ is non-abelian (e.g.,
$G=\mathbb{Z}/p\rtimes_{\psi}\mathbb{Z}/m$, where $\psi$ is non-trivial), one
would like to add one extra piece of information coming from $\psi$ to the
Hurwitz tree of the $\mathbb{Z}/p$-sub-cover like in [BW06, Definition 3.2].

###  4.2 Hurwitz trees arise from cyclic covers

Fix a cyclic group $G:=\mathbb{Z}/p^{n}$. Let $R=k[[t]]$. Let $K$ denote the
fraction field of $R$. As usual, we may associate with $D\subset
C^{\operatorname{an}}$ the spectrum of $R[[X]]$. Suppose we are given an
admissible $G$-character $\chi\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$, which
gives rise to an exponent-$p^{n}$-extension of power series

$\Phi:R[[Z]]\xrightarrow{}R[[X]].$

Obviously, $\Phi$ induces a $\mathbb{Z}/p$-cover of the boundary
$\operatorname{Spec}R[[X]]\\{X^{-1}\\}$ of the disc
$\operatorname{Spec}R[[X]]$. Let $h$ be the conductor, and $\delta$ be the
depth of this cover. Following the exact procedure from [Dan20b, §4.2], one
can construct a Hurwitz tree $\mathcal{T}$ from the dual graph of its semi-
stable model (as discussed in §3.3) and the refined Swan conductors. More
precisely, the data of $\mathcal{T}$ are as in Table 4. See also [BW06]
[BW09].

Data on $\mathcal{T}$ |  Degeneration data of the cover
---|---
The decorated tree |  The branching geometry of $\Phi$ (Definition 3.6)
The depth of $v$ |  The depth of the restriction of $\Phi$ to $U_{v}$.
The differential conductor at $v$ |  The differential conductor of the restriction of $\Phi$ to $U_{v}$
The thickness of an edge $e$ |  The thickness of the corresponding annulus divided by $p$
The conductor at a leaf $b$ |  The conductor of $\Phi$ at the branch point associated to $b$
The reduced degeneration at $v_{0}$ |  The reduced degeneration of $\Phi$
The monodromy group at $v$ |  The largest inertia group of the leaves succeeding $v$
Table 4: Assigning a Hurwitz tree to an admissible cover

Recall that $U_{v}$ is the punctured disc corresponding to $v$.

######  Example 4.5.

Let us calculate the Hurwitz tree for the $\mathbb{Z}/4$-cover from Example
2.28. Figure 2(a) (resp. 2(b)) is the one associated with the subcover
$\chi_{1}$ (resp. $\chi_{2}$), which we call $\mathcal{T}_{1}$ (resp.
$\mathcal{T}_{2}$). The monodromy group at each vertex of $\mathcal{T}_{1}$ is
$\mathbb{Z}/2$. The reduction type at $v_{1}$ is
$\big{(}12,\frac{dx}{x^{2}(x-1)^{2}}\big{)}$ and at $v_{0}$ is
$\big{(}0,\frac{1}{x^{3}}\big{)}$.

$v_{0}$$v_{1}$$2[t^{8}]$$2[0]$$e_{0}$$4$ (a) $\mathcal{T}_{1}$

$v_{0}$$v^{\prime}_{1}$$v_{2}$$2[t^{2}(1+t^{2})]$$2[t^{2}]$$e_{2}$$1$$v_{1}$$3[t^{8}]$$3[0]$$e_{0,2}$$3$$e_{0,1}$$1$
(b) $\mathcal{T}_{2}$

Figure 2: The Hurwitz trees for Example 2.28

Regarding $\mathcal{T}_{2}$, it follows from the calculation in Example 3.41
that the degeneration type at $v_{1},v^{\prime}_{1},v_{2}$, and $v_{0}$ are
$\Big{(}24,\frac{dx}{x^{3}(x-1)^{3}}\Big{)}$,
$\Big{(}9,\frac{dx}{x^{6}(x-1)^{4}}\Big{)}$,
$\Big{(}12,\allowbreak\frac{dx}{x^{2}(x-1)^{2}}\Big{)}$, and
$\Big{(}0,\Big{(}\frac{1}{x^{3}},\frac{1}{x^{9}}\Big{)}\Big{)}$, respectively.
The monodromy groups at all vertices but $v_{2}$, and the two leaves $[t^{2}]$
and $[t^{2}(1+t^{2})]$ are $\mathbb{Z}/4$. The monodromy groups at those
vertices are $\mathbb{Z}/2$.

######  Definition 4.6.

Suppose we are given a $\mathbb{Z}/p^{n-1}$-tree $\mathcal{T}_{n-1}$ and a
$\mathbb{Z}/p^{n}$-tree $\mathcal{T}_{n}$. We say $\mathcal{T}_{n}$ extends
$\mathcal{T}_{n-1}$, denoted by $\mathcal{T}_{n-1}\prec\mathcal{T}_{n}$, if
the followings hold.

  1. (1)

The decorated tree of $\mathcal{T}_{n}$ is a refinement of one for
$\mathcal{T}_{n-1}$.

  2. (2)

At each vertex $v$ of $\mathcal{T}_{n-1}$, the depth conductor and the
differential conductor verify the conditions of Theorem 3.42.

  3. (3)

At each vertex $v$ (resp. each leaf $b$) of $\mathcal{T}_{n-1}$, if the
monodromy group is $\mathbb{Z}/p^{i}$ ($i\leq n-1$), then the monodromy group
at the corresponding one on $\mathcal{T}_{n}$ is $\mathbb{Z}/p^{i+1}$.

  4. (4)

At each vertex $v$ (resp. each leaf $b$) of $\mathcal{T}_{n}$ that is not one
of $\mathcal{T}_{n-1}$, the monodromy group is exactly $\mathbb{Z}/p$.

  5. (5)

Suppose $\delta_{\mathcal{T}_{n}}(v_{0})=\delta_{\mathcal{T}_{n-1}}(v_{0})=0$.
Then the reduced degeneration of $\mathcal{T}_{n}$ is a length-$n$ Witt vector
$(f^{1},\ldots,f^{n-1},f^{n})$ such that $(f^{1},\ldots,f^{n-1})$ is the
reduced degeneration of $\mathcal{T}_{n-1}$.

We say $\mathcal{T}_{n}$ extends a $\mathbb{Z}/p^{i}$-tree $\mathcal{T}_{i}$
if there exist a sequence of consecutive $n-i+1$ extending trees
$\mathcal{T}_{i}\prec\mathcal{T}_{i+1}\prec\ldots\prec\mathcal{T}_{n-1}\prec\mathcal{T}_{n}$.

######  Definition 4.7.

Suppose $r$ is a rational place on an edge $e$ of a Hurwitz tree
$\mathcal{T}$. Set

$\mathfrak{C}_{\mathcal{T}}(e):=\sum_{b\in\mathbb{B}_{\mathcal{T}}(e)}h_{b},$

which is equal to $d_{e}+1$ by Remark 4.3. We say a $\mathbb{Z}/p^{n}$-Hurwitz
tree $\mathcal{T}$ is étale if $\delta_{\mathcal{T}}=0$, and is radical
otherwise. We define, for an edge $e$ and $r\in[s(e),t(e)]\cap\mathbb{Q}$, the
depth of $\mathcal{T}$ at the place $r$ of $e$ as follows

$\delta_{\mathcal{T}}(r,e):=\delta_{\mathcal{T}}(s(e))+d_{e}(r-s(e))=\delta_{\mathcal{T}}(t(e))-d_{e}(t(e)-r).$

The existence of a Hurwitz tree $\mathcal{T}$ with particular conditions is
necessary for a cyclic extension to deform non-trivially.

######  Corollary 4.8.

Suppose $\mathcal{T}$ is an étale $\mathbb{Z}/p^{n}$-tree that extends
$\mathcal{T}_{n-1}$, and $\chi\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ is a
character extends $\chi_{n-1}$. Suppose, moreover, that the tree associated
with $\chi$ has the same shape as $\mathcal{T}$ and its branch points’
conductors agree with that of $\mathcal{T}$. Then, on each edge $e$ of
$\mathcal{T}$,

$\delta_{\chi}(r,z_{e})\geq\delta_{\mathcal{T}}(r,e)$

holds for all $r\in[s(e),t(e)]\cap\mathbb{Q}_{\geq 0}$. The equality happens
everywhere if $\chi$ is étale (Definition 3.7).

###### Proof.

The result follows immediately of Remark 3.23. ∎

######  Theorem 4.9.

The $\mathbb{Z}/p^{n}$-cover $Y_{n}\xrightarrow{\phi_{n}}C$ verifying (1) and
(2) in Proposition 2.36 is a deformation of $k[[y_{n}]]/k[[x]]$ over $R$ if it
gives rise to an étale tree $\mathcal{T}_{n}$ and its reduction is in the same
Artin-Schreier-Witt class of one that defines $k[[y_{n}]]/k[[x]]$.

###### Proof.

The assumption about the degeneration of $\mathcal{T}_{n}$ implies that the
cover has étale reduction and the special fiber is birational equivalence to
$k[[y_{n}]]/\allowbreak k[[x]]$. In addition, suppose the $n$-th conductor of
$k[[y_{n}]]/k[[x]]$ is $\iota_{n}$, which is equal to $d+1$ by definition.
Then, by Remark 4.3, the sum of the conductors of $\phi_{n}$’s branch points
is also $\iota_{n}$. The theorem follows immediately from the previous
corollary and Corollary 3.24 (1). ∎

Suppose we are given a $\mathbb{Z}/p^{n}$-extension $\phi_{n}$ of $k[[x]]$
that is defined by $\underline{f}:=(f^{1},\ldots,f^{n})\in W_{n}(k[[x]])$, and
a $\mathbb{Z}/p^{n-1}$-deformation $\chi_{n-1}$ of its sub-cover. The above
theorem implies that, one can extend $\chi_{n-1}$ to a deformation of
$\phi_{n}$ only if the Hurwitz tree it gives rise to can be extended. We call
that obligation the _Hurwitz tree obstruction_ for the deforming in towers
problem.

When $n=1$, it is known that every $\mathbb{Z}/p$-tree arises from some
$\mathbb{Z}/p$-covers of the rigid disc.

######  Theorem 4.10 ([Dan20b, Theorem 1.1]).

Suppose $\mathcal{T}$ is an étale $\mathbb{Z}/p$-tree. Then there exists a
$\mathbb{Z}/p$-covers whose associated tree coincides with $\mathcal{T}$.

######  Remark 4.11.

The proof of Theorem 4.10 utilizes the formal patching technique for
$G\cong\mathbb{Z}/p$-torsor. Recall that a $\mathbb{Z}/p$-cover of the
boundary of a disc is determined by its depth and its boundary Swan conductor
as mentioned in Remark 3.27. Hence, one can “glue” the covers of the discs and
the annuli which partition a unit disc, along their boundaries in a
$G$-equivariant way to get a $\mathbb{Z}/p$-cover of the whole disc. That is
the technique first used by Henrio in [Hen00] to construct a
$\mathbb{Z}/p$-lift from a Hurwitz tree in mixed characteristic.
Unfortunately, we cannot apply the same technique for
$\mathbb{Z}/p^{n}$-covers (where $n>1$), since it is no longer true that the
depth and the boundary conductor determine the action on the boundary as
discussed in the same remark. We can, however, generalize that approach to an
arbitrary $\mathbb{Z}/p\rtimes_{\psi}\mathbb{Z}/m$-cover by adding an
information coming from $\psi$ to the Hurwitz tree of the
sub-$\mathbb{Z}/p$-cover as explained in Remark 4.4. We thus expect an analog
of Theorem 4.10 for $\mathbb{Z}/p\rtimes_{\psi}\mathbb{Z}/m$-covers, which is
also parallel to the main result of [BW06].

######  Definition 4.12.

Suppose $\mathcal{T}_{n}$ is a $\mathbb{Z}/p^{n}$-Hurwitz tree, and $v\neq
v_{0}$ is one of its vertices with $m$ branches branching out
$e_{1},\ldots,e_{m}$, i.e., $v=s(e_{1})=\ldots=s(e_{m})$. Suppose, moreover,
that the differential conductor at $v$ is of the form

$\omega_{\mathcal{T}_{n}}(v)=\frac{c_{v}dx}{\prod_{i=1}^{m}(x-a_{i})^{h_{i}}}=\bigg{(}\sum_{i=1}^{m}\sum_{j=1}^{h_{i}}\frac{a_{i,j}}{(x-a_{i})^{j}}\bigg{)}dx=:\sum_{i=1}^{m}\omega_{i},$

where $a_{i}=[z_{e_{i}}]_{v}$. We call $\omega_{i}$ the $e_{i}$-part of
$\omega_{\mathcal{T}_{n}}(v)$.

###  4.3 Reduction type

Suppose we are given a cyclic character
$\chi_{n}\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ of order $p^{n}$ that is
defined by a length-$n$-Witt-vector
$\underline{F}_{n}:=(F^{1},\ldots,F^{n})\in W_{n}(\mathbb{K})$. Suppose,
moreover, that $\delta_{\chi_{n}}(0)=0$. Then the reduction
$\overline{\chi}_{n}\in\textrm{H}^{1}_{p^{n}}(\kappa)$ of $\chi_{n}$ is well-
defined. One may further assume that $\overline{\chi}_{n}$ is given by a
length-$n$-Witt-vector

$\underline{f}_{n}:=(f^{1},f^{2},\ldots,f^{n})\in W_{n}(\kappa).$

We call $\underline{f}_{n}$ a reduction type of $\chi_{n}$. If
$\underline{f}_{n}$ is reduced, we say it is the reduced reduction type of
$\chi_{n}$, and $f^{i}$ is its $i$-th degeneration. It is clear that the
character $\chi_{i}:=(\chi_{n})^{p^{n-i}}\in\textrm{H}^{1}_{p^{i}}(\kappa)$
has $\underline{f}_{i}=(f^{1},\ldots,f^{i})$ as a reduction type.

######  Proposition 4.13.

Suppose we are given a character
$\overline{\chi}_{n}\in\textrm{H}^{1}_{p^{n}}(\kappa)$ of order $p^{n}$
defined by a length $n$-Witt vector
$\underline{f}_{n}=(f^{1},f^{2},\ldots,f^{n})\in W_{n}(\kappa)$, a complete
discrete valuation ring $R$ that is finite over $k[[t]]$, and a deformation
$\chi_{n-i}$ of the sub-character
$\overline{\chi}_{n-i}:=\overline{\chi}^{p^{i}}_{n}$ over $R$ given by

$\wp(\underline{Y}_{n-i})=(F^{1},\ldots,F^{n-i}).$

Suppose, moreover, that $\chi_{n}$ is a character given by

$\wp(\underline{Y}_{n})=(F^{1},\ldots,F^{n-1},F^{n}),$

and gives rise to an étale tree $\mathcal{T}_{n}$ that extends
$\mathcal{T}_{n-i}$ (in the sense of Definition 4.6). Then $\chi_{n}$ extends
$\chi_{n-i}$ if it has reduction type $\underline{f}_{n}$.

###### Proof.

It is clear from the definition that $\chi_{n}$ extends $\chi_{n-i}$. Since
the Hurwitz tree $\mathcal{T}_{n}$ to which it gives rise to is étale,
$\chi_{n}$ has good reduction by Theorem 4.9. Moreover, having reduction type
$\underline{f}_{n}$ guarantees that the special fiber of $\chi_{n}$ is
birationally equivalent to $\overline{\chi}_{n}$. It then follows from the
definition that $\chi_{n}$ is a deformation of $\overline{\chi}_{n}$ extending
$\chi_{n-i}$. ∎

######  Remark 4.14.

The above proposition implies that one can extend a
$\mathbb{Z}/p^{n-1}$-deformation by finding a rational function $F^{n}$ that
gives the right degeneration data at the root of the associated tree
$\mathcal{T}_{n}$, and the right branching data to the generic fiber. In
general, it is not easy to calculate the refined Swan conductor of a
character, let alone controlling it to obtain the wanted reduction. To get
around it, we start our construction of $F^{n}$ from the disc corresponds to a
“final vertex” (see Ind 1) of the extending Hurwitz tree $\mathcal{T}_{n}$,
where the degeneration is easy to manage. We then continuously modify $F^{n}$
until we get to the boundary of the unit disc $\operatorname{Spec}R[[X]]$.
More details will be given in §5.4. That strategy again resembles one used by
Obus and Wewers in [OW14] to prove the Oort conjecture.

###  4.4 The compatibility of the differential conductors

######  Definition 4.15.

Suppose $\mathcal{T}$ is a $\mathbb{Z}/p^{n}$-Hurwitz-tree arises from
$\chi\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$, which has conductor $\iota_{n}$
and good reduction. Suppose, moreover, that $v$ is a vertex (which can be the
root) of the étale tree $\mathcal{T}$ initiating $m$ edges
$e_{1},\ldots,e_{m}$, and has a differential conductor (or $n$-th degeneration
type) of the form

$\omega_{\mathcal{T}}(v)=\frac{c_{v}dx}{\prod_{i=1}^{m}(x-a_{i})^{h_{i}}}=\sum_{i=1}^{m}\sum_{j=1}^{h_{i}}\frac{c_{v,j}dx}{(x-a_{i})^{j}}\hskip
5.69054pt\bigg{(}\text{or }\sum_{j=1}^{l}\frac{d_{j}}{x^{j}}\text{, where
}d_{l}\neq 0,p\nmid l\bigg{)},$

where $c_{v,h_{i}}\neq 0$ for $i=1,\ldots,m$ and $a_{i}=[z_{e_{i}}]_{v}$. Then
we call $c_{v}$ (or $-ld_{l}$) the constant coefficient at $v$, and
$c_{v,h_{i}}$ the constant coefficient of its $e_{i}$ part.

Suppose $e$ is an edge in $\mathcal{T}$ and $r\in(s(e),t(e))\cap\mathbb{Q}$.
Then it follows from the discussion in §3.6 that the differential conductor at
$r$ of $\chi$ is a finite sum of the form

$\omega_{\chi}(r)=\sum_{j\geq l}\frac{c_{j}dx}{(x-[z_{e}]_{r})^{j}}$

for some $l\in\mathbb{N}$ where $c_{l}\neq 0$.

######  Definition 4.16.

With the notation above, we say $\omega_{\mathcal{T}}$ has coefficient $c_{l}$
at $r$.

The following result shows that the constant coefficients of the differential
conductors along the Hurwitz tree constructed in §4.2 are “compatible” in some
senses.

######  Theorem 4.17.

Suppose $\mathcal{T}$ is a tree that arises from a cyclic cover with good
reduction. Let $e$ be an edge in $\mathcal{T}$. If
$\delta_{\mathcal{T}}(s(e))>0$, then the followings hold.

  1. (1)

Suppose $s(e)<r<t(e)$ is a rational place on $e$. Then the constant
coefficient at $r$ is equal to that at $t(e)$.

  2. (2)

The constant coefficient at $t(e)$ is equal to that of the $e$-part at $s(e)$.

Suppose $\mathcal{T}$ is an étale tree, and the $n$-th level reduced
degeneration at its root is a polynomial in $x^{-1}$ of degree $l$ with
coefficient $-ld_{l}$. If $l<p\iota_{n-1}-p$, then we only need the
differential conductors starting from $v_{1}$ to be compatible. Otherwise, we
have $\iota_{n}=l+1$, and the differential conductor at $v_{1}$ has the same
coefficient with that at $v_{0}$.

We first prove the result above for the situation of §3.8. The rest of the
proof will be given §5.3, where we have extra tool to generalize Proposition
4.18.

######  Proposition 4.18.

Suppose $\chi\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ satisfies conditions (D1),
(D2), and (D3) of §3.8 with respect to a rational place $r_{0}$. Suppose at
$r_{0}$, the differential conductor $\omega_{\chi}(r_{0})$ has constant
coefficient $c$ and order of infinity $\iota-2>0$. Then there exists
$r\in[0,r_{0})\cap\mathbb{Q}$ such that, for all
$s\in(r,r_{0})\cap\mathbb{Q}$, we have

$\delta_{\chi}(s)=\delta_{\chi}(r_{0})-(\iota-1)(r_{0}-s),\text{ and
}\omega_{\chi}(s)=\frac{cdx}{x^{\iota}},$

In particular, for all $r^{\prime}\in[0,r_{0})\cap\mathbb{Q}$ such that
$\delta_{\chi}(r^{\prime})=\delta_{\chi}(r_{0})-(\iota-1)(r_{0}-r^{\prime})$,
we have

$\omega_{\chi}(r^{\prime})=\frac{cdx}{x^{\iota}}+\sum_{i=1}^{\iota-1}\frac{c_{i}dx}{x^{i}}=:\frac{cdx}{x^{\iota}}+\omega^{\prime}_{\chi}(r^{\prime}),$

where $\omega^{\prime}_{\chi}(r^{\prime})$ is exact, i.e., $c_{i}=0$ for all
$i\equiv 1\pmod{p}$.

###### Proof.

We first assume that $G\cong\mathbb{Z}/p$. Set $Y:=X/t^{pr_{0}}$ and
$\delta:=\delta_{\chi}(r_{0})$. By Proposition 3.26, we may assume the
restriction of $\chi$ to $D[pr_{0}]$ is represented by an Artin-Schreier class
$f_{\chi}=t^{-p\delta}F(Y)$, where the reduction (modulo $t$) of $F$ is
$\overline{F}:=f\not\in\kappa^{p}$. In addition, assumption (D1) allows us to
write

$df=\bigg{(}\sum_{j\geq\iota}\frac{\overline{b}_{j}}{x^{j}}\bigg{)}dx\in\Omega_{\kappa}^{\ast},$

where $\overline{b}_{j}\in k$ and $\overline{b}_{\iota}=c$. Set
$I:=\\{i\in\mathbb{N}\mid\overline{b}_{i}\neq 0\\}$. It then follows from the
discussion in §3.8 that

$f_{\chi}=\sum_{i=0}^{\infty}a_{i}Y^{-i}=\frac{1}{t^{p\delta}}\Big{(}\sum_{j\geq\iota-1}\frac{d_{j}}{Y^{j}}+\sum_{l\not\in
I}\frac{e_{l}}{Y^{l}}\Big{)},$ (34)

where $d_{j}\in R$, $\nu(d_{j})=0$, and $\nu(e_{l})>0$. In addition, using the
assumption about the constant coefficient and Proposition 3.26, we learn that
$c=(1-\iota)\overline{d}_{\iota-1}$. Let $Z:=Yt^{ps}$. We thus have

$f_{\chi}=\sum_{i=0}^{\infty}a_{i}t^{psi}Z^{-i}=\frac{1}{t^{p\delta}}\Big{(}\sum_{j\geq\iota-1}\frac{d_{j}t^{psj}}{Y^{j}}+\sum_{l\not\in
I}\frac{e_{l}t^{psl}}{Y^{l}}\Big{)}.$ (35)

Therefore, there exists $v\in(0,r_{0})\cap\mathbb{Q}$ such that, for all
$w\in[v,r_{0})\cap\mathbb{Q}$, the value
$\nu(d_{\iota-1})+p(r_{0}-w)(\iota-1)$ is strictly smaller than the valuations
of other terms. The first part then immediately follows from Proposition 3.26.
Let $r^{\prime}\geq v$ be the largest kink of $\delta_{\chi}$ on $(0,r_{0}]$.
Apply the process from §3.8 for some $r\in(0,r^{\prime})\cap\mathbb{Q}$, one
may replace $f_{\chi}$ of (34) by another one in the same Artin-Schreier
class, which may assume to have the form of (35), such that Proposition 3.26
applies immediately for any rational place in $(0,v)$. In addition, that
procedure does not change the ${\iota-1}$ coefficient of $f_{\chi}$ and, for
$s:=r_{0}-r$, $\nu(d_{\iota-1}t^{ps(\iota-1)})\leq\nu(e_{l}t^{psl})$ only when
$l<\iota-1$ and $l\not\equiv 0\pmod{p}$. The rest once more follows from an
application of Proposition 3.26 to the place $r^{\prime}$.

We now consider the case $G\cong\mathbb{Z}/p^{n}$ ($n>1$). Suppose the
length-$n$-Witt vector $\underline{f}=(f^{1},\ldots,\allowbreak
f^{n-1},f^{n})$ that represent $\chi$ is best at $r_{0}$, and $n-j+1$ is its
relevance length (see Definition 3.36). Recall from Proposition 3.40 that its
differential Swan conductor at $r_{0}$ is

$\operatorname{dsw}_{\chi}(r_{0})=\sum_{i\geq
j}^{n}[f^{i}]_{r_{0}}^{p^{n-i}-1}d[f^{i}]_{r_{0}}=\bigg{[}\sum_{i=1}^{n}(f^{i})^{p^{n-i}-1}df^{i}\bigg{]}_{r_{0}}.$

The first assertion then follows from the application of the base case’s
argument to $\sum_{i=1}^{n}(f^{i})^{p^{n-i}-1}\allowbreak df^{i}$. Let us
consider the case $\delta_{\chi}(r_{0})>p\delta_{\chi^{p}}(r_{0})$. Then
$\underline{f}$’s relevance length (Definition 3.38) is $1$ and
$\operatorname{dsw}_{\chi}(r_{0})=d[f^{n}]_{r_{0}}$. One then may prove the
second assertion for this case using the same reasoning for the case $n=1$,
where $f^{n}$ plays the role of $f_{\chi}$. Suppose now that
$\delta_{\chi}(r_{0})=p\delta_{\chi^{p}}(r_{0})$, i.e., the relevance length
of $\underline{f}$ is greater than $1$. Then, as before, there exists a
smallest $r^{\prime}\in[0,r_{0})\cap\mathbb{Q}$ such that $\delta_{\chi}(s)$
is linear on $[r^{\prime},r_{0})$ with slope $\iota-1$. In addition, for all
$s\in[r,r_{0})\cap\mathbb{Q}$, it must be true that

$\omega_{\chi}(s)=\frac{c_{s}dx}{x^{\iota}}\text{, and
}\omega_{\chi}(r^{\prime})=\frac{c_{r^{\prime}}dx}{x^{\iota}}+\sum_{i=1}^{\iota-1}\frac{c_{r^{\prime},i}dx}{x^{i}}.$

Note that $c_{s}=c$ for $s\in(r,r_{0})\cap\mathbb{Q}$. If
$\delta_{\chi}(s^{\prime})>p\delta_{\chi}(s^{\prime})$ for
$s^{\prime}\in(r,r_{0})$, then $\delta_{\chi}(s)>p\delta_{\chi}(s)$ for all
$s\in[r^{\prime},r_{0})\cap\mathbb{Q}$ and we are in a situation similar to
one where $\delta_{\chi}(r_{0})>p\delta_{\chi^{p}}(r_{0})$, where $s^{\prime}$
plays the role of $r_{0}$. If
$\delta_{\chi}(s^{\prime})=p\delta_{\chi}(s^{\prime})$ for
$s^{\prime}\in(r,r_{0})$, then, by Theorem 3.42, it must be true that
$\delta_{\chi}(s)=p\delta_{\chi}(s)$ for all $s\in[r,r_{0}]$. Note that, if
$\delta_{\chi^{p}}(\\--)$ concaves down at a place $s$ on $(0,r_{0})$, then
$\operatorname{ord}_{\infty}\omega_{\chi^{p}}(s)+1>-\operatorname{ord}_{0}\omega_{\chi^{p}}(s)-1$
(Proposition 3.15), which implies $\omega_{\chi^{p}}(s)$ has a pole outside
$0$, contradicting Corollary 3.22. Therefore, if
$\omega_{\chi^{p}}(s^{\prime})=mdx/x^{l}$, then
$\omega_{\chi^{p}}(s)=mdx/x^{l}$ for all
$s\in(r^{\prime},r_{0})\cap\mathbb{Q}$ by induction, hence
$\omega_{\chi}(s)=m^{p}dx/x^{pl+1}$. The rest easily follows. ∎

######  Definition 4.19.

With the above notation, we say the conductor at $t(e)$ is _compatible_ with
one at $s(e)$ if the constant coefficient at $t(e)$ is equal to that of the
$e$ part at $s(e)$. Suppose $v\prec v^{\prime}$, then we say $v^{\prime}$ is
compatible with $v$ if, given any pair of adjacent vertices $v\prec v_{1}\prec
v_{2}\prec v^{\prime}$, $v_{1}$ is compatible with $v_{2}$.

####  4.4.1 Partition of a tree by its edges and vertices

Suppose we are given a Hurwitz tree $\mathcal{T}$ and $e$ is one of its edges.
We define $\mathcal{T}(e)$ to be a sub-(non Hurwitz)-tree of $\mathcal{T}$
whose

  * •

edges, vertices and information on them coincide with those succeeding $s(e)$,
and whose

  * •

differential datum at $s(e)$ is equal to the $e$-part of
$\omega_{\mathcal{T}}(s(e))$.

These tree will play a critical role in the extending of Hurwitz tree in §4.5,
and the “partition process” §5.3. More precisely, §5 will construct a cover
that gives rise to a whole tree inductively from its sub-tree. Note that
$\mathcal{T}(e)$ verifies all of the Hurwitz tree’s axioms but ones that
involve the root and the trunk.

####  4.4.2 Deriving a $\mathbb{Z}/p^{n-1}$-tree from a
$\mathbb{Z}/p^{n}$-tree

Suppose $\chi_{n}$ is a $\mathbb{Z}/p^{n}$-character that gives rise to a
Hurwitz tree $\mathcal{T}_{n}$. By following the below process, one can derive
the tree $\mathcal{T}_{n-1}$ for level $n-1$ just from the level-$n$-tree, and
the branching data of the generic fiber.

  1. (1)

Erasing the $\mathbb{Z}/p$-leaves and the edges $e$ of $\mathcal{T}_{n}$ such
that the monodromy group at $t(e)$ is $\mathbb{Z}/p$.

  2. (2)

Set the conductors at the leaves according to the $(n-1)$-branching-data of
the generic fiber.

  3. (3)

If the monodromy group at a vertex $v$ is
$G_{\mathcal{T}_{n}}(v)=\mathbb{Z}/p^{m}$, where $m>1$, then set
$G_{\mathcal{T}_{n-1}}(v)=\mathbb{Z}/p^{m-1}$.

  4. (4)

Starting from the root, assigning the degeneration data at the succeeding
vertices inductively so that the differential conductors are compatible and
verify Theorem 3.42.

One thus can easily derive the degeneration data of the lower (than $(n-1)$)
levels from the highest one by reiterating the above process. The readers can
see that from the examples in §4.5.

###  4.5 The vanishing of the differential Hurwitz tree obstruction

In the below proposition, we discuss the vanishing of the Hurwitz tree
obstruction for the refined deformation problem. In addition, those extending
Hurwitz trees are specially designed to be used as the “skeletons” to build
the lifts in §5. The proof is postponed to §6.2 as they are quite technical.
However, we will provide some generalized enough examples and discuss some of
their features to help the readers follow §5.

######  Proposition 4.20.

Suppose $k[[y_{n}]]/k[[x]]$ is cyclic Galois of order $p^{n}$ and with
conductor $\iota_{n}$. Suppose, moreover, that $R[[Y_{n-1}]]/R[[X]]$ is a
deformation of the $\mathbb{Z}/p^{n-1}$-subextension $k[[y_{n-1}]]/\allowbreak
k[[x]]$ over a finite extension $R$ of $k[[t]]$, and $\mathcal{T}_{n-1}$ is
its associated Hurwitz tree. Then there exists a good
$\mathbb{Z}/p^{n}$-Hurwitz tree $\mathcal{T}_{n}$ of conductor $\iota_{n}$
that extends $\mathcal{T}_{n-1}$.

Next, we will give three examples of extending a $\mathbb{Z}/p$-tree of
conductor $\iota_{1}$ to certain $\mathbb{Z}/p^{2}$-trees, one for minimal
second upper jump (i.e., $\iota_{2}=p\iota_{1}-p$) and two for non-minimal
second upper jump (i.e., $\iota_{2}=p\iota_{1}-p+a_{2}$, where $p\nmid
a_{2}$).

######  Example 4.21.

Suppose $\overline{\chi}$ is a character in $\textrm{H}^{1}_{7^{2}}(\kappa)$
that is given by the equation

$\wp(y_{1},y_{2})=\bigg{(}\frac{1}{x^{29}},\frac{2}{x^{135}}+\frac{1}{x}\bigg{)}.$

One can think of $\overline{\chi}$ as a $\mathbb{Z}/{49}$-extension of the
complete discrete valuation ring $k[[x]]$. By Theorem 2.4, the extension
$\overline{\chi}$ has upper jumps $u_{1}=29$ and $u_{2}=\max\\{29\cdot
7,135\\}=203$. That means the second jump is minimal. The
$\mathbb{Z}/7$-subextension $\overline{\chi}_{1}$ of $\overline{\chi}$ is
given by

$y_{1}^{7}-y_{1}=\frac{1}{x^{29}}.$

Consider the $\mathbb{Z}/7$-Hurwitz tree $\mathcal{T}_{1}$ in Figure 3(a),
whose degeneration data are on Table 5 .The number $a$ can be either
$2+5\sqrt{6}$ or $2-5\sqrt{6}$. Note that these choices of $a$ make
$dx/(x^{11}(x-1)^{5}(x-a)^{3})$ exact. Note also that the constant coefficient
$1/a^{3}$ of the differential conductor at $t(e_{2})$ is necessary for it to
be compatible with one at $t(e_{1})$. Recall that one can construct from
$\mathcal{T}_{1}$ a $\mathbb{Z}/7$-character $\chi_{1}$ of type
$[5,6,5,3,4,7]^{\top}$ that deforms $\overline{\chi}_{1}$ (Theorem 4.10). The
tree $\mathcal{T}_{2}^{\min}$ in Figure 3(b) with degeneration data in the
third column of Table 5 and with monodromy group $\mathbb{Z}/49$ at each
vertex is a $\mathbb{Z}/49$-tree that extends $\mathcal{T}_{1}$ and has
conductor $204$. One can show
$\delta_{\mathcal{T}_{2}}(v)=7\delta_{\mathcal{T}_{1}}(v)$ and
$\mathcal{C}(\omega_{\mathcal{T}^{\min}_{2}}(v))=\omega_{\mathcal{T}_{1}}(v)$
for each $v\in V_{\mathcal{T}^{\min}_{2}}\setminus\\{v_{0}\\}$. It follows
from the data in Table 5 that $\mathcal{T}_{2}^{\min}$ can represent a
deformation of type

$\begin{bmatrix}30&204\\\
\end{bmatrix}\xrightarrow{}\begin{bmatrix}5&6&5&3&4&7\\\ 35&36&35&21&28&49\\\
\end{bmatrix}^{\intercal}.$

$v_{0}$$v_{1}$$[5]$$[3]$$[6]$$b_{1}$$[5]$$e_{2}$$1$$e_{1}$$1$$[7]$$[4]$$e_{3}$$1$$e_{0}$$1$
(a) $\mathcal{T}_{1}$

$v_{0}$$v_{1}$$[35]$$[21]$$[36]$$b_{1}$$[35]$$e_{2}$$1$$e_{1}$$1$$[49]$$[28]$$e_{3}$$1$$e_{0}$$1$
(b) $\mathcal{T}^{\min}_{2}$

$v_{0}$$v_{1}$$[35]$$[21]$$[45]$$b_{1}$$[35]$$e_{2}$$1$$e_{1}$$1$$[49]$$[28]$$e_{3}$$1$$e_{0}$$1$
(c) $\mathcal{T}^{9}_{2}$

Figure 3: Extending trees

Vertices |  $\mathcal{T}_{1}$ |  $\mathcal{T}^{\min}_{2}$ |  $\mathcal{T}^{9}_{2}$
---|---|---|---
$v_{0}$ |  $\big{(}0,\frac{1}{x^{29}}\big{)}$ |  $\big{(}0,\frac{2}{x^{135}}+\frac{1}{x}\big{)}$ |  $\big{(}0,\frac{3}{x^{212}}+\frac{1}{x^{3}}\big{)}$
$v_{1}$ |  $\big{(}29,\frac{dx}{x^{19}(x-1)^{11}}\big{)}$ |  $\big{(}203,\frac{dx}{x^{127}(x-1)^{77}}\big{)}$ |  $\big{(}212,\frac{3dx}{x^{136}(x-1)^{77}}\big{)}$
$t(e_{3})$ |  $\big{(}39,\frac{dx}{x^{7}(x-1)^{4}}\big{)}$ |  $\big{(}280,\frac{dx}{x^{49}(x-1)^{28}}\big{)}$ |  $\big{(}288,\frac{3dx}{x^{49}(x-1)^{28}}\big{)}$
$t(e_{1})$ |  $\big{(}47,\frac{-dx}{x^{11}(x-1)^{5}(x-a)^{3}}\big{)}$ |  $\big{(}329,\frac{-dx}{x^{71}(x-1)^{35}(x-a)^{21}}\big{)}$ |  $\big{(}347,\frac{-3dx}{x^{80}(x-1)^{35}(x-a)^{21}}\big{)}$
$t(e_{2})$ |  $\big{(}57,\frac{-dx}{a^{3}x^{5}(x-1)^{6}}\big{)}$ |  $\big{(}399,\frac{-dx}{a^{21}x^{35}(x-1)^{36}}\big{)}$ |  $\big{(}426,\frac{-3dx}{a^{21}x^{35}(x-1)^{52}}\big{)}$
Table 5: Degeneration data of the trees in Figure 3

######  Example 4.22.

If we replace the Witt vector that defined $\overline{\chi}$ in the previous
example by

$\wp(y_{1},y_{2})=\bigg{(}\frac{1}{x^{29}},\frac{3}{x^{212}}+\frac{1}{x^{3}}\bigg{)},$

then the upper jumps of the new character is $(30,213)$. Suppose the
deformation on the $\mathbb{Z}/7$-level is the same. Hence, its corresponding
tree $\mathcal{T}_{1}$ is also identical to one in the previous example.
Figure 3(c) is a tree $\mathcal{T}^{9}_{2}$ that extends $\mathcal{T}_{1}$ and
has conductor $213$. Note that
$\delta_{\mathcal{T}_{2}}(v)>7\delta_{\mathcal{T}_{1}}(v)$ and
$\mathcal{C}(\omega_{\mathcal{T}^{9}_{2}}(v))=0$ for each $v\in
V_{\mathcal{T}^{9}_{2}}\setminus\\{v_{0}\\}$.

Note that the extension tree in Example 4.23 is not unique! The following
example illustrates an alternative construction where the extending tree
$\mathcal{T}^{\operatorname{ness}}_{2}$ has no essential jumps from level
$(n-1)$ to level $n$ (_ness_ stands for non-essential). The general
construction will be given in Proposition 6.5.

######  Example 4.23.

Suppose the character $\overline{\chi}$ from the previous example corresponds
to

$\wp(y_{1},y_{2})=\bigg{(}\frac{1}{x^{29}},\frac{2}{x^{211}}+\frac{1}{x}\bigg{)}\bigg{(}\text{resp.
}\bigg{(}\frac{1}{x^{29}},\frac{3}{x^{212}}+\frac{1}{x^{3}}\bigg{)}\bigg{)},$

and the $\mathbb{Z}/7$-deformation is the same. Then the diagram in Figure
4(b) (resp. Figure 4(c)) gives an extending tree of $\mathcal{T}_{1}$ with the
right reduction type. Its degeneration data are in Table 6. The monodromy
groups at all vertices but the ends of the two leaves of conductors $7$ are
isomorphic to $\mathbb{Z}/49$. The monodromy groups at those two vertices are
$\mathbb{Z}/7$. Observe that, in both Figure 4(b) and 4(c), we have
$\delta_{\mathcal{T}_{2}}(v_{1})=7\delta_{\mathcal{T}_{1}}(v_{1})$.

$v_{0}$$v_{1}$$[5]$$[3]$$[6]$$b_{1}$$[5]$$e_{2}$$1$$e_{1}$$1$$[7]$$[4]$$e_{3}$$1$$e_{0}$$1$
(a) $\mathcal{T}_{1}$

$v_{0}$$[7]$$[35]$$[21]$$[36]$$b_{1}$$[35]$$e_{2}$$1$$e_{1}$$1$$[43]$$[28]$$e_{3}$$1$$[7]$$e_{0,1}$$\frac{3}{7}$
(b) $\mathcal{T}^{8\operatorname{ness}}_{2}$

$v_{0}$$[7]$$[35]$$[21]$$[37]$$b_{1}$$[35]$$e_{2}$$1$$e_{1}$$1$$[43]$$[28]$$e_{3}$$1$$[7]$$e_{0,3}$$\frac{5}{14}$
(c) $\mathcal{T}^{9\operatorname{ness}}_{2}$

Figure 4: Extending trees with no essential jumps from level $1$ to $2$ Vertices |  $\mathcal{T}_{1}$ |  $\mathcal{T}^{8\operatorname{ness}}_{2}$ |  $\mathcal{T}^{9\operatorname{ness}}_{2}$
---|---|---|---
$v_{0}$ |  $\big{(}0,\frac{1}{x^{29}}\big{)}$ |  $\big{(}0,\frac{2}{x^{211}}+\frac{1}{x}\big{)}$ |  $\big{(}0,\frac{3}{x^{212}}+\frac{1}{x^{3}}\big{)}$
$t(e_{0,1})$ |  $\big{(}\frac{89}{7},\frac{dx}{x^{30}}\big{)}$ |  $\big{(}\frac{633}{7},\frac{2dx}{x^{198}(x-1)^{7}(x-2)^{7}}\big{)}$ |
$t(e_{0,3})$ |  $\big{(}\frac{145}{14},\frac{dx}{x^{30}}\big{)}$ |  |  $\big{(}\frac{530}{7},\frac{3dx}{x^{199}(x-1)^{7}(x-3)^{7}}\big{)}$
$v_{1}$ |  $\big{(}29,\frac{dx}{x^{19}(x-1)^{11}}\big{)}$ |  $\big{(}203,\frac{dx}{x^{127}(x-1)^{71}}\big{)}$ |  $\big{(}203,\frac{dx}{x^{128}(x-1)^{71}}\big{)}$
$t(e_{3})$ |  $\big{(}39,\frac{dx}{x^{7}(x-1)^{4}}\big{)}$ |  $\big{(}273,\frac{dx}{x^{43}(x-1)^{28}}\big{)}$ |  $\big{(}273,\frac{dx}{x^{43}(x-1)^{28}}\big{)}$
$t(e_{1})$ |  $\big{(}47,\frac{-dx}{x^{11}(x-1)^{5}(x-a)^{3}}\big{)}$ |  $\big{(}329,\frac{-dx}{x^{71}(x-1)^{35}(x-a)^{21}}\big{)}$ |  $\big{(}330,\frac{-dx}{x^{72}(x-1)^{35}(x-a)^{21}}\big{)}$
$t(e_{2})$ |  $\big{(}57,\frac{-dx}{a^{3}x^{5}(x-1)^{6}}\big{)}$ |  $\big{(}399,\frac{-dx}{a^{21}x^{35}(x-1)^{36}}\big{)}$ |  $\big{(}401,\frac{-dx}{a^{21}x^{35}(x-1)^{37}}\big{)}$
Table 6: Degeneration data of the trees in Figure 4

######  Remark 4.24.

Suppose $r$ is an arbitrary rational point on the tree $\mathcal{T}_{2}$. Then
one can observe from the previous examples that there are at most two
succeeding leaves whose conductors are not divisible by $p$. That also holds
for all the trees $\mathcal{T}_{n}$ that we will construct to prove
Proposition 4.20. This fact is essential for the proof of Lemma 5.15, which,
in turn, is a key ingredient of Theorem 2.35’s proof.

######  Remark 4.25.

In the tree $\mathcal{T}_{2}^{\min}$ of Figure 3(b), there is one leaf $b_{1}$
with conductor $l_{1,2}=7l_{1,1}-7+1$. The other leaves $b_{i}$ have conductor
$l_{i,2}=7l_{i,1}$. Furthermore, at each vertex $v$ adjacent to the edges
$e_{2},e_{1}$, and $e_{0}$,
$\delta_{\mathcal{T}_{2}}(v)=7\delta_{\mathcal{T}_{1}}(v)$. We say the tree
$\mathcal{T}^{\min}_{2}$ extends $\mathcal{T}_{1}$ minimally. Note that
$\mathcal{T}^{\min}_{2}(e_{1})$ (resp. $\mathcal{T}^{\min}_{2}(e_{2})$) also
extends $\mathcal{T}_{1}(e_{1})$ (resp. $\mathcal{T}_{1}(e_{2})$) minimally.
In addition, the tree $\mathcal{T}^{9}_{2}(e_{1})$ of Figure 3(c) has one leaf
$b_{1}$ with conductor $l_{1,2}=7l_{1,1}-7+9+1$ and the other leaves $b_{i}$
has conductor $l_{i,2}=7l_{i,1}$. We say $\mathcal{T}^{9}_{2}$ extends
$\mathcal{T}_{1}$ by $9$-additively. Similarly, the tree
$\mathcal{T}^{\min}_{2}(e_{3})$ extends $\mathcal{T}_{1}(e_{3})$
$(7-1)$-additively.

At the vertex $v_{1}$ of the $\mathbb{Z}/49$-trees in Figure 4(b) and 4(c),
all but two sub-trees starting from it extends the corresponding ones of
$\mathcal{T}_{1}$ $(7-1)$-additively. The tree
$\mathcal{T}^{9\operatorname{ness}}_{2}(e_{1})$ extends
$\mathcal{T}_{1}(e_{1})$ $1$-additively,
$\mathcal{T}^{8\operatorname{ness}}_{2}(e_{3})$ and
$\mathcal{T}^{9\operatorname{ness}}_{2}(e_{3})$) extend
$\mathcal{T}_{1}(e_{3})$ minimally. Formal definitions of the above
conventions will be given in §6.2.

##  5 Proof of Theorem 2.35

In this section, we give the final step (Step 4) of the proof of Proposition
2.36, hence Theorem 2.35, hence Theorem 1.2. From now on, we denote
$\mathbb{K}:=K(X)$, and $\kappa:=k(x)$ its residue field.

The inverse process of §4.2, constructing a cover from a Hurwitz tree, was
utilized by Henrio in [Hen00] and by Bouw and Wewers in [BW06] to solve the
lifting problem for $\mathbb{Z}/p$-covers and
$\mathbb{Z}/p\rtimes\mathbb{Z}/m$-covers, respectively. However, the main
technique in these papers is the gluing method, which no longer works in our
situation (discussed in Remark 3.27 and Remark 4.11). We instead generalize
Obus and Wewers’ approach in [OW14].

###  5.1 Geometric set up

In the rest of the paper, we set
$C\cong\mathbb{P}_{K}^{1}=\operatorname{Proj}K[X,\allowbreak V]$. Recall that
$\mathcal{D}$ is the closed unit disc inside
$(\mathbb{A}^{1}_{K})^{\operatorname{an}}\subset C^{\operatorname{an}}$
centered at $X=0$. We are given a character
$\overline{\chi}_{n}\in\textrm{H}^{1}_{p^{n}}(\kappa)$ corresponding to a
cyclic cover of $\mathcal{D}$ over $k$ of order exactly $p^{n}$, branching
only at $0$ with upper ramification breaks $(m_{1},\ldots,m_{n-1},m_{n})$. Set
$(\iota_{1},\ldots,\iota_{n}):=(m_{1}+1,\ldots,m_{n}+1)$. For
$i=1,\ldots,n-1$, we set
$\overline{\chi}_{i}:=(\overline{\chi}_{n})^{p^{n-i}}\in\textrm{H}^{1}_{p^{i}}(\kappa)$.
Suppose there is a character
$\chi_{n-1}\in\textrm{H}^{1}_{p^{n-1}}(\mathbb{K})$ deforming
$\overline{\chi}_{n-1}$ (in the sense of Definition 3.9). In order to prove
the refined local deformation problem, we must show that there exists a
character $\chi_{n}\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ that smoothly
deforms $\overline{\chi}_{n}$ and verifies $\chi_{n}^{p}=\chi_{n-1}$.

We may assume that $\chi_{n-1}$ corresponds to an extension of $\mathbb{K}$
given by a length-$(n-1)$ Witt vector
$\underline{G}_{n-1}=(G^{1},\ldots,G^{n-1})$. Then any
$\chi_{n}\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ such that
$\chi_{n}^{p}=\chi_{n-1}$ is given by a length-$n$ Witt vector
$\underline{G}_{n}=(G^{1},\ldots,G^{n-1},\allowbreak G^{n})\in
W_{n}(\mathbb{K})$. After a finite extension of $K$, one may assume that
$\underline{G}_{n-1}$ is best (Definition 3.36). Recall that, by Proposition
4.20, one can construct a Hurwitz tree $\mathcal{T}_{n}$ with conductor
$\iota_{n}=m_{n}+1$ that extends $\mathcal{T}_{n-1}$ (Definition 4.6). Suppose
the type of such $\mathcal{T}_{n}$ is $(\iota_{n,1},\ldots,\iota_{n,r})$,
where $\sum_{i}\iota_{n,i}=\iota_{n}$. We would then like $G^{n}$ to be of the
form

$G^{n}=\sum_{j=1}^{r}\sum_{i=1}^{\iota_{n,j}-1}\frac{a_{j,i}}{(X-b_{j})^{i}}\in\mathbb{K},$

where $a_{i,j}\in k((t))$, $b_{i}$’s are pairwise distinct $K$-points of the
open unit disc $D$, and the geometry of the poles (Definition 3.6) of
$\underline{G}_{n}$ agrees with that of $\mathcal{T}_{n}$. We say that the
polynomial $G^{n}$ gives rise to the character $\chi_{n}$ (from $\chi_{n-1}$).

###  5.2 The base case

The case $n=1$ is trivial as one can always deform an Artin-Schreier cover
trivially over $k[[t]]$ (using the same Artin-Schreier equation that defines
the special fiber to represent the generic fiber). Recall that Theorem 4.10
asserts that the existence of a deformation of Artin-Schreier covers with a
given type is fully determined by the presence of $\mathbb{Z}/p$-differential-
Hurwitz trees of the same type. One may utilize this fact to study the
deformation ring $R_{\mathbb{Z}/p}$ (§2.3.2).

###  5.3 Partition of the extending Hurwitz tree

In this section, we discuss a strategy to translate our situation to one that
can be solved using the tools from [OW14]. Suppose that $\chi_{n-1}$ branches
at $s$ points $b_{1},\ldots,b_{s}$ where $b_{i}\in\mathfrak{m}R$. We first
show that its corresponding Witt vector
$\underline{G}_{n-1}=(G^{1},\ldots,G^{n-1})$ can be partitioned with respect
to a rational place on the associated Hurwitz tree.

######  Proposition 5.1.

With the notation as in the geometric setup, suppose $A$ and $B$ partition the
branch locus $\mathbb{B}(\chi_{n-1})=\\{b_{1},\ldots,b_{s}\\}$. Then there
exist two length-$(n-1)$ Witt vectors

$(G^{1}_{A},\ldots,G^{n-1}_{A}),\hskip 5.69054pt\text{ and }\hskip
5.69054pt(G^{1}_{B},\ldots,G^{n-1}_{B})$

such that the former vector only has poles in $A$, the latter one only has
poles in $B$, and

$(G^{1},\ldots,\allowbreak
G^{n-1})\allowbreak=(G^{1}_{A},\ldots,G^{n-1}_{A})+(G^{1}_{B},\ldots,G^{n-1}_{B}).$

###### Proof.

We prove by induction on $n$. Let us first assume that $n=2$. We then can
write $G^{1}$ as the sum of $G^{1}_{A}$ and $G^{1}_{B}$ simply by splitting
its partial fraction decomposition. When $n>2$, we have

$\begin{split}(G^{1},G^{2},\ldots,G^{n-1})-(G^{1}_{A},0,\ldots,0)&=(G^{1}_{B},G^{2}_{1},\ldots,G^{n-1}_{1})\\\
&=(G^{1}_{B},0,\ldots,0)+(0,G^{2}_{1},\ldots,G^{n-1}_{1}).\\\ \end{split}$

The second equation comes from the “decomposition” property of Witt vector
[Lor08, §26, (32)]. The second term is
$\mathbb{V}(G^{2}_{1},\ldots,G^{n-1}_{1})$, where $\mathbb{V}$ is the
Verschiebung operation for Witt-vectors. By induction, we may write
$(G^{2}_{1},\ldots,G^{n-1}_{1})=(G^{2}_{1,A},\ldots,G^{n-1}_{1,A})+(G^{2}_{1,B},\ldots,G^{n-1}_{1,B})$,
using the obvious notation. Therefore,

$(G^{1},\ldots,G^{n-1})=(G^{1}_{A},G^{2}_{1,A},\ldots,G^{n-1}_{1,A})+\allowbreak(G^{1}_{B},G^{2}_{1,B},\ldots,G^{n-1}_{1,B})$

as we want, completing the proof. ∎

######  Definition 5.2.

Let $s$ be a rational place on an edge $e$ of a Hurwitz tree
$\mathcal{T}_{n-1}$, and $\overline{z}\in k$ or $\overline{z}=\infty$. We
define $\underline{G}_{n-1,s,\overline{z}}$ to be a partition of
$\underline{G}_{n-1}$ whose poles specify to $\overline{z}$ on the canonical
reduction of $\mathcal{D}[s,z_{e}]$. For any $s(e)<r<t(e)$, set

$\underline{G}_{n-1,r}:=\underline{G}_{n-1,r,0},\hskip 8.53581pt\text{ and
}\hskip
8.53581pt\underline{G}_{n-1,s(e)}=\underline{G}_{n-1,t(e)}:=G_{n-1,r}.$

If $v=t(e)$ is a vertex, we set $\underline{G}_{v}:=\underline{G}_{t(e)}$.
Define

$\chi_{n-1,s}:=\mathfrak{K}_{n-1}(\underline{G}_{n-1,s}),\hskip
8.53581pt\text{ and }\hskip
8.53581pt\chi_{n-1,s,\infty}:=\mathfrak{K}_{n-1}(\underline{G}_{n-1,s,\infty}).$

######  Remark 5.3.

Observe that $\underline{G}_{n-1,s_{1}}=\underline{G}_{n-1,s_{2}}$ for all
$s(e)<s_{1}\leq s_{2}<t(e)$. In addition, if $s(e)<s<t(e)$ then

$\underline{G}_{n-1,t(e)}=\underline{G}_{n-1,s}+\underline{G}_{n-1,s,\infty}.$

Finally, if $r=t(e)=s(e_{1})=\ldots=s(e_{l})$, then
$G_{n-1,t(e)}=\sum_{j=1}^{l}G_{n-1,s(e_{j})}$.

The following result is simple but will turn out to be critical for our
approach.

######  Proposition 5.4.

Suppose we are given a character
$\chi_{n-1}\in\textrm{H}^{1}_{p^{n-1}}(\mathbb{K})$, which is defined by
$\underline{G}_{n-1}:=(G^{1},\ldots,G^{n-1})\in W_{n-1}(\mathbb{K})$, has good
reduction, and gives rise to a $\mathbb{Z}/p^{n-1}$-Hurwitz tree
$\mathcal{T}_{n-1}$. Then the followings hold.

  1. 1.

Suppose $s(e)<s<t(e)\in\mathbb{Q}_{\geq 0}$, where $e$ is an edge of
$\mathcal{T}_{n-1}$, and
$\underline{G}_{n-1}=\underline{G}_{{n-1},s}+\underline{G}_{{n-1},s,\infty}$
as in Proposition 5.1. Then:

$\delta_{\chi_{n-1}}(s)=\delta_{\chi_{n-1,s}}(s),\hskip 8.53581pt\text{ and
}\hskip 8.53581pt\omega_{\chi_{n-1}}(s)=\omega_{\chi_{n-1,s}}(s).$

  2. 2.

Suppose $s=t(e)=s(e_{1})=\ldots=s(e_{l})$. Then

$\delta_{\chi_{n-1}}(s)=\delta_{\chi_{n-1,s(e_{i})}}(s),\hskip 8.53581pt\text{
and }\hskip
8.53581pt\omega_{\chi_{n-1}}(s)=\sum_{i=1}^{l}\omega_{\chi_{n-1,s(e_{i})}}(s).$

In particular, the constant coefficient (§4.4) of the $e_{i}$-part of
$\omega_{\chi_{n-1}}(s)$ is equal to the constant coefficient of
$\omega_{\chi_{n-1,s(e_{i})}}(s)$.

###### Proof.

Suppose $\delta_{\chi_{n-1}}(r)\neq\delta_{\chi_{n-1,s}}(r)$ at a rational
place $s(e)<s\leq r<t(e)$ in $e$. Since $\chi_{n-1}$ has good reduction, it
follows from Proposition 3.20 and Remark 3.23 that $\delta_{\chi_{n-1}}(r)$ is
the minimal depth possible at $r$ with the given positions and conductors of
the branch points whose valuations (with respect to $z_{e}$) are greater than
$r$. Note that the restrictions of $\chi_{n-1,s}$ and $\chi_{n}$ to
$D[pr,z_{e}]$ have the same branching data. Hence
$\delta_{\chi_{n-1}}(r)<\delta_{\chi_{n-1,s}}(r)$. Therefore, it must be true
that $\delta_{\chi_{n-1,s}}(r)=\delta_{\chi_{n-1,s,\infty}}(r)$, else
$\delta_{\chi_{n-1}}(r)=\max(\delta_{\chi_{n-1,s}}(r),\delta_{\chi_{n-1,s,\infty}}(r))$
by Proposition 3.12, which contradicts the previous sentence. On the other
hand, by applying Proposition 3.20, we obtain

$\operatorname{ord}_{0}(\omega_{\chi_{n-1,s,\infty}}(w))\geq 0\text{, and
}\operatorname{ord}_{\infty}(\omega_{\chi_{n-1,s}}(w))\geq 0,$

for any $w\in\mathbb{Q}\cap[s(e),t(e)]$. That means, on the direction from
$s(e)$ to $t(e)$ in $e$, $\delta_{\chi_{n-1,s}}$ is stricly increasing, and
$\delta_{\chi_{n-1,s,\infty}}$ is strictly decreasing (recall Corollary 3.16
which says the differential determines the rate of change of the depth). Thus,
it must be true that
$\delta_{\chi_{n-1,s,\infty}}(w)\leq\delta_{\chi_{n-1,s}}(w)$, and the
equality can only happen when $w=s(e)$. That is a contradiction. The claim
about the differential conductor then follows immediately from Lemma 3.12.

The rest is straightforward, following a similar line of reasoning. ∎

######  Remark 5.5.

Suppose $\chi_{n-1}:=\mathfrak{K}_{n-1}(\underline{G}_{n-1})$ is a
$\mathbb{Z}/p^{n-1}$-cover that has $\mathcal{T}_{n-1}$ as the corresponding
tree, and $\mathcal{T}_{n}$ is a $\mathbb{Z}/p$-extension of
$\mathcal{T}_{n-1}$. Consider a vertex $v=t(e)$ on $\mathcal{T}$ with $m$
branches $e_{1},e_{2},\ldots,e_{m}$ on the directions away from the root. As
before we can partition $\underline{G}_{n-1}$ into a sum as below

$\underline{G}_{n-1}=\underline{G}_{n-1,v,\infty}+\sum_{i=1}^{m}\underline{G}_{n-1,v,i}.$

Each $\underline{G}_{n-1,v,i}$ here is a length-$(n-1)$-Witt-vector whose
roots are the leaves of $\mathcal{T}_{n-1}$ succeeding $e_{i}$. The poles of
$\underline{G}_{n-1,v,\infty}$ are the leaves of $\mathcal{T}_{n-1}$ that are
not ones of the $\mathcal{T}_{n-1}(e_{i})$’s. Set
$\chi_{n-1,v,i}:=\mathfrak{K}_{n-1}(\underline{G}_{n-1,v,i})\in\textrm{H}^{1}_{p^{n-1}}(\mathbb{K})$.
For any rational place $r$ on $\mathcal{T}_{n}(e_{i})$, we have
$\delta_{\chi_{n-1,v,i}}(r)=\delta_{\chi_{n-1}}(r)$ by Proposition 5.4. Hence,
we obtain full information about the depth and the differential conductor of
$\chi_{n-1,v,i}$. Moreover, all of the branch points of $\chi_{n-1,v,i}$ lie
in the closed sub-disc $D[pv,z_{e}]$. That is similar to the situation
considered in [OW14], thus allows us to apply many technique from that paper.
More precisely, we will extend each of the $\chi_{n-1,v,i}$’s at a time. Their
sum then gives an extension of $\chi_{n-1,v}$ with tree $\mathcal{T}_{n}(v)$.

######  Definition 5.6.

Suppose we are given a tree $\mathcal{T}_{n}$ that extends
$\mathcal{T}_{n-1}$, and $r$ is a rational place on an edge $e$ of
$\mathcal{T}_{n}$. When $s(e)<r\leq t(e)$, we define $\mathcal{H}_{r}$ to be
the collection of all $H\in\mathbb{K}$ that gives rise to
$\mathfrak{K}_{1}(H)\in\textrm{H}^{1}_{p}(\mathbb{K})$ whose branching divisor
is everywhere smaller than or equal to that of
$\mathcal{T}_{n}(s(e))=\mathcal{T}_{n}(r)$. We define
$\mathcal{H}_{s(e)}=\mathcal{H}_{r}$ for a rational $s(e)<r<t(e)$. Hence
$\mathcal{H}_{r}=\mathcal{H}_{s}$ for all $s(e)\leq s\leq r\leq t(e)$. As
before, if $v=t(e)$ is a vertex, then we set
$\mathcal{H}_{v}:=\mathcal{H}_{t(e)}$.

######  Corollary 5.7.

With the notation as in Definition 5.6, a polynomial
$H\in\mathcal{H}_{s(e_{0})}$ extends $\chi_{n-1}$ to a character $\chi_{n}$
that deforms $\overline{\chi}_{n}$ if and only if its reduction type coincides
with $\overline{\chi}_{n}$’s reduced representation.

###### Proof.

“$\Leftarrow$” Since $H\in\mathcal{H}_{s(e_{0})}$ and $\underline{G}_{n-1}$ is
best (as we assume in the geometric setup), it follows from Theorem 2.4 that
$\mathfrak{C}(\chi_{n},e_{0},0)\leq\iota_{n}$. Moreover, as the different of
the generic fiber is at least that of the special fiber by Proposition 2.26,
the assumption on $\chi_{n}$’s reduction type implies that the conductor of
the special fiber is $\iota_{n}$, hence enforcing the equality. That
information, combining with Proposition 4.13, proves “$\Leftarrow$”.

The “$\Rightarrow$” direction is immediate from the definition. ∎

######  Definition 5.8.

Given a $\mathbb{Z}/p^{n}$-tree $\mathcal{T}_{n}$ that extends
$\mathcal{T}_{n-1}$, and a vertex $v$ of $\mathcal{T}_{n}$. We define
$\mathcal{G}_{v}\subseteq\mathcal{H}_{v}$ to be the collection of
$G\in\mathbb{K}$ such that the branching datum of the character
$\chi_{n,G}\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ to which it gives rise fits
into $\mathcal{T}_{n}(v)$, and such that

$\delta_{\chi_{n,G}}(v)=\delta_{\mathcal{T}_{n}}(v).$

We denote by $\mathcal{G}^{\ast}_{v}\subseteq\mathcal{G}_{v}$ the collection
of $G$ such that $\omega_{\chi_{n,G}}(v)$ is a nonzero multiple of
$\omega_{\mathcal{T}_{n}}(v)$.

######  Definition 5.9.

Suppose we are given a rational $G\in\mathcal{G}_{v}$, and $v^{\prime}$ is a
vertex succeeding $v$ in $\mathcal{T}_{n-1}$. Then, as before, one may right
$G=G_{v^{\prime}}+G_{v^{\prime},\infty}$. We set
$G\lvert_{v^{\prime}}:=G_{v^{\prime}}$.

######  Proposition 5.10.

Suppose $\mathcal{T}_{n}$ is a good $\mathbb{Z}/p^{n}$-Hurwitz tree that
extends $\mathcal{T}_{n-1}$, and $G\in\mathcal{G}_{v}$ for some vertex $v$ of
$\mathcal{T}_{n}$. Let $\chi$ be the extension of $\chi_{n-1}$ by $G$. Then

$\delta_{\chi}(r)=\delta_{\mathcal{T}_{n}}(r)$

for every rational place $r>v$. In particular, at every vertex $v^{\prime}$
succeeding $v$, we have $G\lvert_{v^{\prime}}\in\mathcal{G}_{v^{\prime}}$.

###### Proof.

Suppose $\delta_{\chi}(r)\neq\delta_{\mathcal{T}_{n}}(r)$ for some $r>v$.
Then, it follows from Corollary 4.8 that
$\delta_{\chi}(r)>\delta_{\mathcal{T}_{n}}(r)$. On the other hand, the right
derivative of $\delta_{\chi}$ at any rational place $r$ is
$-\operatorname{ord}_{0}(\omega_{\chi}(r))-1$ by Corollary 3.16, which is at
most $\mathfrak{C}(\chi,r,\overline{0})-1$ by Proposition 3.20. Hence, as
$\delta_{\chi}(v)=\delta_{\mathcal{T}_{n}}(v)$, it must be true that
$\delta_{\chi}(r)\leq\delta_{\mathcal{T}_{n}}(r)$, which negates the
assumption. The last assertion follows easily from Proposition 5.4 (2). ∎

######  Definition 5.11.

At a vertex $v=t(e)$ with $r$ direct successor edges
$e_{1},e_{2},\ldots,e_{l}$, we define

$\sum_{i=1}^{r}\mathcal{G}_{s(e_{i})}:=\bigg{\\{}\sum_{i=1}^{l}G_{s(e_{i})}\mid
G_{s(e_{i})}\in\mathcal{G}_{s(e_{i})}\bigg{\\}}.$

######  Corollary 5.12.

With the notation as in the above definition, we have
$\mathcal{G}_{v}=\mathcal{G}_{t(e)}=\sum_{i=1}^{l}\mathcal{G}_{s(e_{i})}$.

###### Proof.

“$\supseteq$” is immediate from the definition and Lemma 3.12 ((2)). Suppose
$G\in\mathcal{G}_{t(e)}$. Then, as discussed in Remark 5.5, we may write
$G=\sum_{i=1}^{r}G_{s(e_{i})}$, where $G_{s(e_{i})}\in\mathcal{H}_{s(e_{i})}$.
Finally, as
$\delta_{\chi_{s(e_{i})}}(r)=\delta_{\chi}(r)=\delta_{\mathcal{T}_{n}}(r)$ for
all $s(e)<r$ by Proposition 5.10, Proposition 3.15 asserts that
$\delta_{\chi_{s(e_{i})}}(s(e_{i}))=\delta_{\chi}(s(e_{i}))=\delta_{\mathcal{T}_{n}}(s(e_{i}))$.
Therefore, $G_{s(e_{i})}\in\mathcal{G}_{s(e_{i})}$. ∎

Note that, we also have
$\mathcal{H}_{t(e)}=\sum_{i=1}^{l}\mathcal{H}_{s(e_{i})}$.

####  5.3.1 Proof of Theorem 4.17

We now have the necessary tools to complete the proof of the compatibility
theorem. Recall that the situation of §3.8 has been solved by Proposition
4.18. Let us now consider an arbitrary rational place $r\in(s(e),t(e)]$. We
may again write $\underline{G}=\underline{G}_{r}+\underline{G}_{r,\infty}$.
Applying Proposition 5.4, with $\chi_{r}:=\mathfrak{K}(\underline{G}_{r})$,
one obtains

$\delta_{\chi}(r)=\delta_{\chi_{r}}(r)\text{, and
}\omega_{\chi}(r)=\omega_{\chi_{r}}(r).$

Hence, as the differential conductor of $\chi_{r}$ follow the compatibility
rules, so is $\chi$. ∎

######  Remark 5.13.

Most of the results in §4 and §5 until this point easily translate to the case
where $R$ is of mixed characteristic. The most important among them is
Proposition 5.4 (1). The only problematic part is Proposition 5.4 (2), hence
so is Theorem 4.17, since we not yet have an exact analog of Theorem 3.39 in
mixed characteristic.

###  5.4 The induction process

Following is the outline of the induction process to construct a character
$\chi_{n}$ that deforms $\overline{\chi}_{n}$, extends $\chi_{n-1}$, and gives
rise to $\mathcal{T}_{n}$. One may assume that the length-$(n-1)$ Witt-vector
that defines $\chi_{n-1}$ is best. We would like to find a rational function
$G^{n}\in\mathbb{K}$ that extends $\chi_{n-1}$ to a deformation $\chi_{n}$ of
$\overline{\chi}_{n}$. Below is the induction (ind) process.

  1. Ind 1

We start at a vertex $v=t(e)$ of $\mathcal{T}_{n}$ that only precedes leaves.
We call $v$ a final vertex of $\mathcal{T}_{n}$. In §5.6.1, we consider the
affinoid $\mathcal{G}_{t(e)}$. It is a collection of
$G^{n}_{t(e)}\in\mathbb{K}$ that gives rise to a cover $\chi_{n,t(e)}$ with
desired depth and (generic) branching datum at $t(e)$. We then show that the
subset $\mathcal{G}^{\ast}_{t(e)}\subsetneq\mathcal{G}_{t(e)}$, whose elements
give rise to the characters with the right differential conductor at $t(e)$,
is non-empty. Recall that $\omega_{\mathcal{T}_{n}}(t(e))$ is either exact or
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(t(e)))=\omega_{\chi_{n-1}}(t(e))$
(Theorem 3.42). The case $\omega_{\mathcal{T}_{n}}(v)$ is exact is considered
in §5.6.2, the non-exact case is examined in §5.6.3.

  2. Ind 2

This step is carried out in §5.7 and §5.8. Let $e$ be an edge of
$\mathcal{T}_{n}$ that is the direct predecessor of a final vertex. Then
$\underline{G}_{n-1,t(e)}=\underline{G}_{n-1,s(e)}$ and there are no leaves in
$\mathcal{T}_{n}$ with valuation (with respect to $z_{e}$) greater than $s(e)$
besides one at $t(e)$. In order for the depth of $\chi_{n,s(e)}$ at $s(e)$ to
be equal to $\delta_{\mathcal{T}_{n}}(s(e))$, we continuously control its
differential conductor along $e$ by modifying $G^{n}_{t(e)}$. The key
ingredient is Proposition 5.18, which says that one can always replace
$G^{n}_{t(e)}$ by another element of $\mathcal{G}_{t(e)}$ that “fits” better
into $\mathcal{T}_{n}$. Furthermore, we prove that the collection of
$G^{n}_{t(e)}$ whose depth at $s(e)$ is $\delta_{\mathcal{T}_{n}}(s(e))$,
i.e., the quasi-compac quasi-separated (qcqs) set $\mathcal{G}_{s(e)}$, is a
non-empty subset of $\mathcal{G}^{\ast}_{t(e)}$ (Proposition 5.20). That shows
one can secure the wanted ramification data over the edge $e$, completing the
base case.

  3. Ind 3

Suppose $e$ is an edge of $\mathcal{T}_{n}$ where $t(e)$ is _not_ final, and
$e_{1},\ldots,e_{l}$ are the direct successors of $t(e)$. As in Remark 5.5, we
may write $\underline{G}_{n-1}$ as

$\underline{G}_{n-1}=\sum_{i=1}^{l}\underline{G}_{n-1,s(e_{i})}+\underline{G}_{n-1,t(e),\infty}.$

By induction, for each $i$, there exists a qcqs $\mathcal{G}_{s(e_{i})}$ such
that for each $G^{n}_{s(e_{i})}\in\mathcal{G}_{s(e_{i})}$ and for each
$i=1,\ldots,l$, the following holds.

    1. (a)

The branching datum of
$\chi_{n,s(e_{i})}=\mathfrak{K}_{n}\big{(}\underline{G}_{n-1,s(e_{i})},G^{n}_{s(e_{i})}\big{)}$
coincides with $\mathcal{T}_{n}(e_{i})$.

    2. (b)

The depth of $\chi_{n,s(e_{i})}$ at $t(e)$ is equal to
$\delta_{\mathcal{T}_{n}}(t(e)\big{)}$.

Moreover, it follows from Corollary 5.12 that
$\mathcal{G}_{t(e)}=\sum_{i=1}^{l}\mathcal{G}_{s(e_{i})}$. Hence,
$\mathcal{G}_{t(e)}$ is also qcqs. We call $t(e)$ an exact vertex if
$\delta_{\mathcal{T}_{n}}(t(e))>p\delta_{\mathcal{T}_{n-1}}(t(e))$. Otherwise,
$\delta_{\mathcal{T}_{n}}(t(e))=p\delta_{\mathcal{T}_{n-1}}(t(e))$, and we
name $t(e)$ a Cartier vertex. The controlling of exact vertices (resp. Cartier
vertices) to achieve
$\omega_{\mathcal{T}_{n}}(t(e))=\omega_{\chi_{n,t(e)}}(t(e))$ is similar to
Ind 2. We again obtain a non-empty open subset
$\mathcal{G}^{\ast}_{t(e)}\subset\mathcal{G}_{t(e)}$ whose elements gives the
desired differential conductor at $t(e)$.

  4. Ind 4

Let $e$ be the edge from the previous step. We can simply repeat the process
from Ind 2 to achieve the right degeneration data over $e$. That completes the
inductive step.

  5. Ind 5

Iterating the previous two steps, we acquire a nonempty qcqs
$\mathcal{G}_{s(e_{0})}$ (recall that $e_{0}$ is the trunk of
$\mathcal{T}_{n}$). Finally, §5.10 shows that $\mathcal{G}_{s(e_{0})}$
contains a non-empty subset $\mathcal{G}^{\ast}_{s(e_{0})}$ whose elements
give rise to characters with desired reduction type. That result, together
with Corollary 5.7, complete the proof of the main theorem.

###  5.5 Some natural degree $p$ characters

######  Definition 5.14.

Suppose $\mathcal{T}_{n}$ is a tree that extends $\mathcal{T}_{n-1}$ in
Proposition 4.20, and $r$ is a rational place on an edge $e$ of
$\mathcal{T}_{n}$. We may assume that $z_{e}=0$. Suppose, moreover, that
$b_{1},\ldots,b_{m}$ are the successor leaves of $e$ on $\mathcal{T}_{n}$,
$l_{j}$ is the conductor at $b_{j}$, and set $l:=\sum_{j=1}^{m}l_{j}$. Define
$\mathcal{W}_{r}$ to be the collection of exact differentials forms

$\omega_{r}=\sum_{i=1}^{l}\frac{c_{i}dx}{x^{i}},$ (36)

where $c_{i}\in k$. Note that, as $\omega_{r}$ is exact, and $c_{qp-1}=0$ for
all $q\in\mathbb{Z}$.

######  Lemma 5.15.

Suppose $\mathcal{T}_{n}$ is a tree extending $\mathcal{T}_{n-1}$ in
Proposition 4.20, and $r$ is a rational place on an edge $e$ of
$\mathcal{T}_{n}$. Suppose, moreover, that we are given
$\delta\in\mathbb{Q}_{>0}$ and $\omega$ is an exact differential form in
$\mathcal{W}_{r}$. Then there exists $H_{r}\in\mathcal{H}_{r}$ (defined in
Definition 5.6) such that, for $\psi:=\mathfrak{K}_{1}(H_{r})$, we have
$\delta_{\psi}(r)=\delta$ and $\omega_{\psi}(r)=\omega$.

###### Proof.

Suppose $\omega$ has the form as in (36). Suppose, for each $i$, there exists
a rational function $H_{r,i}\in\mathcal{H}_{r}$ so that, for
$\psi_{i}:=\mathfrak{K}_{1}(H_{r,i})$, we have $\delta_{\psi_{i}}(r)=\delta$
and $\omega_{\psi_{i}}(r)=\frac{c_{i}dx}{x^{i}}$. Then, by Lemma 3.12 (2), the
character $\psi:=\sum_{i=1}^{r}\psi_{i}$ works. Therefore, we may further
assume that

$\omega=\frac{c_{i}dx}{x^{i}},$

where $c_{i}\in k$, $p\nmid(i-1)$, and $i\leq l$. By the construction of
$\mathcal{T}_{n}$, all but at most two of the $l_{i}$’s are divisible by $p$
(mentioned in Remark 4.24). Without loss of generality, we may assume that
they are $l_{1}$ and $l_{2}$. We will treat each case separately.

Suppose first that $p\mid l_{2}$. That is the situation on most of the places
of $\mathcal{T}^{\min}_{2}$ in Example 4.21. Consider the rational function

$H:=\frac{x^{l-i}}{t^{p(\delta-r(i-1))}(x-b_{1})^{l_{1}-1}\prod_{j=2}^{m}(x-b_{j})^{l_{j}}}\in\mathbb{K}.$

Then it is straightforward to check that $H\in\mathcal{H}_{r}$,
$[H]_{r}=\frac{1}{x_{r}^{i-1}}\notin(k(x_{r}))^{p}$, and
$\nu_{r}(H)=-p\delta$. Hence, by Proposition 3.26, it is exactly what we are
seeking.

Suppose $p\nmid l_{1}$ and $p\nmid l_{2}$, for instance, where $r$ is a place
on $e_{0,1}$ of $\mathcal{T}^{8\operatorname{ness}}_{2}$ in Example 4.23 and
$\mathcal{T}_{n}=\mathcal{T}^{8\operatorname{ness}}_{2}(r)$. We first regard
the case $l_{1}=m_{1}p+1$ and $l_{2}=m_{2}p+1$. Then, as
$\omega\in\mathcal{W}_{r}$ and $l=pq+2$, we may assume $i\leq l-2$. One can
show that

$H:=\frac{x^{l-1-i}}{t^{p(\delta-r(i-1))}(x-b_{1})^{mp_{1}}(x-b_{2})^{mp_{2}}\prod_{j=3}^{m}(x-b_{j})^{l_{j}}}$

works as before.

Finally, let us consider the case $l_{2}=m_{2}p+1$ and $1<\overline{l}_{1}<p$
as on the edge $e_{0,1}$ of $\mathcal{T}^{9\operatorname{ness}}_{2}$ in
Example 4.23. Then, similar to the other cases, the rational function that
does the job is

$H:=\frac{x^{l-1-i}}{t^{p(\delta-r(i-1))}(x-b_{1})^{l_{1}-1}(x-b_{2})^{m_{2}p}\prod_{j=3}^{m}(x-b_{j})^{l_{j}}}.$

We hence have solutions for all the cases. ∎

######  Corollary 5.16.

Suppose $\iota_{n}$ is the conductor of $\mathcal{T}_{n}$,
$g=\sum_{j=1}^{m_{n}}c_{j}/x^{j}$ is a polynomial in $k[x^{-1}]$ that consists
of only terms of prime-to-$p$ degree. Then, after a possible finite extension
of $\mathbb{K}$, there exists $G^{\prime}_{s(e_{0})}\in\mathcal{H}_{s(e_{0})}$
such that the character $\psi:=\mathfrak{K}_{1}(G^{\prime}_{s(e_{0})})$
satisfies the followings

  * •

$\delta_{\psi}(0)=0$, and

  * •

its reduction $\overline{\psi}$, considered as a character in
$\textrm{H}^{1}_{p}(\kappa)$, corresponds to the extension $y^{p}-y=g$.

###### Proof.

The proof is almost identical to one for Lemma 5.15. In particular,
$c_{j}/x^{j}$ in this case plays the role of $-jc_{j}dx/x^{j+1}$ in the proof
of that lemma. ∎

###  5.6 Controlling a final vertex

Suppose $v=t(e)$ is a final vertex of $\mathcal{T}_{n}$ with differential
conductor

$\omega_{\mathcal{T}_{n}}(v)=\frac{cdx}{\prod_{j=1}^{r}(x-[b_{j}]_{v})^{l_{j}}},$

where $l_{j}$ is the $n$-th conductor of the branch point associated to the
leaf $b_{j}$ on $\mathcal{T}_{n}$. It follows from the rules asserted in
Theorem 3.42 that either
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(v))=\omega_{\mathcal{T}_{n-1}}(v)=\omega_{\chi_{n-1}}(v)$,
or $\mathcal{C}(\omega_{\mathcal{T}_{n}}(v))=0$. We discuss the former case in
§5.6.2 and the latter one in §5.6.3.

Recall from §5.3 that $\chi_{n-1}$ can be represented by a best length-$(n-1)$
Witt vector $(G^{1},\ldots,\allowbreak
G^{n-1})\allowbreak=(G^{1}_{v},\ldots,G^{n-1}_{v})+(G^{1}_{\infty},\ldots,G^{n-1}_{\infty})$.
The discussion in Remark 5.5 allow us to reduce our study to the case where
$\underline{G}_{n-1}$ is equal to $(G^{1}_{v},\ldots,G^{n-1}_{v})$ and is also
best. We first set $G^{n}=0$. Then, it follows from Theorem 3.39 and Theorem
3.42 that the character $\chi_{n}$ it gives rise to verifies

$\delta_{\chi_{n}}(v)=p\delta_{\mathcal{T}_{n-1}}(v),\text{ and
}\mathcal{C}(\omega_{\chi_{n}}(v))=\omega_{\mathcal{T}_{n-1}}(v).$

Furthermore, Theorem 2.4 shows that the $n$-th conductor of $\chi_{n}$ at
$b_{j}$ is $\iota_{j,n}=p\iota_{j,n-1}$ for all $j$. Therefore,
$0\in\mathcal{H}_{v}$ and
$\delta_{\chi_{n}}(v)\leq\delta_{\mathcal{T}_{n}}(v)$. In the rest of this
subsection, we show that one can replace $0$ by some $G^{n}\in\mathcal{H}_{v}$
such that $\delta_{\chi_{n}}(v)=\delta_{\mathcal{T}_{n}}(v)$ and
$\omega_{\chi_{n}}(v)=\omega_{\mathcal{T}_{n}}(v)$.

####  5.6.1 The final qcqs

Suppose there are $m$ leaves $b_{1},\ldots,b_{m}$ with $n$-th conductors
$l_{1},\ldots,l_{m}$, respectively, that are attached to $v$. As discussed in
Remark 4.24, one may assume that $p$ divides $l_{i}$ for all $i$’s but maybe
$l_{1}$ and $l_{2}$. Consider the subset $\mathcal{H}^{\prime}_{v}$ of
$\mathcal{H}_{v}$ that consists of the elements of the form

$G=\frac{1}{t^{p\delta_{\mathcal{T}_{n}}(v)}}\bigg{(}\sum_{j=1}^{m}\sum_{i=1}^{l_{j}-1}\frac{a_{j,i}}{(x_{v}-(b_{j})_{v})^{i}}\bigg{)},$
(37)

where $a_{j,i}\in k((t))$, $\min_{i\not\equiv 0\pmod{p}}(\nu(a_{j,i}))=0$, and
${a}_{j,i}=0$ where $i\equiv 0\pmod{p}$ for each $j$. Note that
$\mathcal{H}^{\prime}_{v}$ can be identified with an affinoid subset of
$\mathbb{A}_{\mathbb{K}}^{\sum_{j=1}^{m}(l_{j}-1)}$ via the $a_{j,i}$’s. We
first show that this collection is exactly the $\mathcal{G}_{v}$ defined in
Definition 5.8.

######  Proposition 5.17.

The subset $\mathcal{H}^{\prime}_{v}$ coincides with $\mathcal{G}_{v}$. In
particular, $\mathcal{G}_{v}$ is an affinoid (hence qcqs).

###### Proof.

Suppse $G\in\mathcal{H}^{\prime}_{v}$. It is immediate from (37) that
$[G]_{v}\in\kappa_{v}\setminus\kappa_{v}^{p}$. Hence, we have
$\delta_{G}(v)=\delta_{\mathcal{T}_{n}}(v)$ by Proposition 3.26. Therefore,
$G\in\mathcal{G}_{v}$, hence
$\mathcal{H}^{\prime}_{v}\subseteq\mathcal{G}_{v}$. Conversely, each
$H\in\mathcal{G}_{v}$ is in the same Artin-Schreier class of one of the form

$\tilde{H}=\frac{1}{t^{p\delta_{\mathcal{T}_{n}(v)}}}(H^{\prime})_{v}=\frac{1}{t^{p\delta_{\mathcal{T}_{n}(v)}}}\bigg{(}\sum_{j=1}^{m}\sum_{i=1}^{l_{j}-1}\bigg{(}\frac{b_{j,i}}{(x-b_{j})^{i}}\bigg{)}\bigg{)}_{v},$

where $[H^{\prime}]_{v}\not\in\kappa^{p}$, again by Proposition 3.26. It is
straight forward to realize that $\tilde{H}$ must be of the form (37). ∎

####  5.6.2 The exact case

Suppose first that $\omega_{\mathcal{T}_{n}}(v)$ is exact. Then, by the
conditions on an extending tree (Definition 4.6), we have

$\delta_{\mathcal{T}_{n}}(v)>p\delta_{\mathcal{T}_{n-1}}(v)\text{, and
}\omega_{\mathcal{T}_{n}}(v)=\frac{cdx}{\prod_{j=1}^{m}(x-[b_{j}]_{v})^{l_{j}}}=:d\overline{h}_{v}.$

for some $\overline{h}_{v}\in k(x)\setminus k(x)^{p}$. We may further assume
that

$\overline{h}_{v}=\sum_{j=1}^{m}\sum_{i=1}^{l_{j}-1}\frac{b_{j,i}}{(x-[b_{j}]_{v})^{i}}$

where $b_{j,i}\in k$, and $b_{j,l_{j}-1}\neq 0$ for each $j$. One can consider
$\overline{h}_{v}$, after replacing $x$ by $x_{v}$, as an element of
$k[[t]]](x_{v})$. Hence, the rational function

$H_{v}:=\frac{1}{t^{p\delta_{\mathcal{T}_{n}}(v)}}\bigg{(}\sum_{j=1}^{m}\sum_{i=1}^{l_{j}-1}\frac{b_{j,i}}{(x_{v}-(b_{j})_{v})^{i}}\bigg{)}\in\mathbb{K}$

is an element of $\mathcal{H}_{v}$ that verifies, for
$\psi:=\mathfrak{K}_{1}(H_{v})$,

$\delta_{\psi}(v)=\delta_{\mathcal{T}_{n}}(v),\text{ and
}\omega_{\psi}(v)=\omega_{\mathcal{T}_{n}}(v).$

It follows from Lemma 3.12 (1) that, adding $H_{v}$ to $G^{n}=0$ makes

$\delta_{\chi_{n}}=\delta_{\mathcal{T}_{n}}(v),\text{ and
}\omega_{\chi_{n}}=\omega_{\mathcal{T}_{n}}(v).$

We thus achieve the desired branching data at $v$. Define

$\mathcal{G}^{\ast}_{t(e)}:=\big{\\{}G\in\mathcal{G}_{t(e)}\mid\omega_{\chi_{n}}(v)=\omega_{\mathcal{T}_{n}}(v)\big{\\}},$

which is non-empty as the above construction suggests.

####  5.6.3 The non-exact case

Let us now consider the case
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(v))=\omega_{\mathcal{T}_{n-1}}(v)=\omega_{\chi_{n-1}}(v)$.
Recall that, we start with $G^{n}=0$. Then, as $\underline{G}_{n-1}$ is best,
it follows from Theorem 3.39 that

$\delta_{\chi_{n}}(v)=p\delta_{\chi_{n-1}}(v)=\delta_{\mathcal{T}_{n}}(v).$

Thus,
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(v)-\omega_{\chi_{n}}(v))=\omega_{\mathcal{T}_{n-1}}(v)-\omega_{\mathcal{T}_{n-1}}(v)=0$,
i.e., the difference between the two differential forms is exact. Therefore,
we may assume

$\omega_{\mathcal{T}_{n}}(v)-\omega_{\chi_{n}}(v)=d\overline{h}_{v}$

for some $\overline{h}_{v}\in k(x)\setminus k(x)^{p}$. In addition, Theorem
2.4 again tells us that the $n$-th conductor at the branch point $b_{j}$ is
$\iota_{j,n}=p\iota_{j,n-1}\leq l_{j}$ for $1\leq j\leq m$. It then follows
from Proposition 3.20 that

$\operatorname{ord}_{[b_{j}]_{v}}(d\overline{h}_{v})\geq-l_{j}$

for all $j$. One thus may assume that

$\overline{h}_{v}=\sum_{j=1}^{m}\sum_{i=1}^{l_{j}-1}\frac{b_{j,i}}{(x-[b_{j}]^{i})}.$

Hence, $d\overline{h}_{v}\in\mathcal{W}_{v}$. By the same argument as before,
there exists an $H_{v}\in\mathcal{H}_{v}$ such that, for
$\psi:=\mathfrak{K}_{1}(H_{v})$ ,

$\delta_{\psi}(v)=\delta_{\mathcal{T}_{n}}(v)\text{ and
}\omega_{\psi}(v)=d\overline{h}_{v}.$

Again, replacing $G^{n}$ by $G^{n}+H_{v}$ results to

$\omega_{\chi_{n}}(v)=\omega_{\mathcal{T}_{n}}(v).$

Thus, the set $\mathcal{G}^{\ast}_{t(e)}$ defined in the previous subsection
is not empty, completing Ind 1.

###  5.7 Controlling an edge

Suppose $e$ is an edge where $t(e)$ is final. We again may assume that
$z_{e}=0$. Recall that $\mathcal{G}_{t(e)}^{\ast}$, which is the collection of
$G\in\mathbb{K}$ that gives rise to a character $\chi_{n}$ verifying

$\delta_{\chi_{n}}(t(e))=\delta_{\mathcal{T}_{n}}(t(e))\text{, and
}\omega_{\chi_{n}}(t(e))=\omega_{\mathcal{T}_{n}}(t(e)),$

is non-empty by Ind 1. Suppose, moreover, that the order of $\infty$ of
$\omega_{\mathcal{T}_{n}}(t(e))$ is $l-2$. Then, by Corollary 3.16, the left
derivative of $\delta_{\chi_{n}}$ at $t(e)$ is $l-1$. Therefore, as
$\delta_{\chi_{n}}$ is piece-wise linear by Proposition 3.15, there exists a
rational $s(e)\leq\lambda<t(e)$ such that

$\delta_{\chi_{n}}(s)=\delta_{\mathcal{T}_{n}}(t(e))-(l-1)(t(e)-s)=\delta_{\mathcal{T}_{n}}(s)$

for all $s\in[\lambda,t(e)]$. Let $\lambda_{e}(G)$ be the minimal value of
$\lambda$ with this property. That means, $\lambda_{e}(G)$ is the largest
“kink” of the function $\delta_{\chi_{n}}$ on $(s(e),t(e))$ (or is $s(e)$ if
$\delta_{\chi_{n}}$ is linear on $[s(e),t(e))$). Our goal is proving that
there exists a $G\in\mathcal{G}^{\ast}_{t(e)}$ satisfying
$\lambda_{e}(G)=s(e)$. First, we will show that one can always reduce the kink
on $e$, i.e., making $\lambda_{e}(G)$ strictly smaller (if it is greater than
$s(e)$) by replacing it with another element of $\mathcal{G}^{\ast}_{t(e)}$.

######  Proposition 5.18.

Suppose $G\in\mathcal{G}^{\ast}_{t(e)}$ with $\lambda_{e}(G)>s(e)$. Then there
exists $G^{\prime}\in\mathcal{G}^{\ast}_{t(e)}$ with
$\lambda_{e}(G^{\prime})<\lambda_{e}(G)$.

###### Proof.

Suppose $\lambda:=\lambda_{e}(G)$ is the largest kink of $\chi_{n}$, which is
the extension of $\chi_{n-1}$ by $G$, on $e$. We know from Proposition 3.20
that $\operatorname{ord}_{\infty}(\omega_{\chi_{n}}(\lambda))$ is equal to the
left-derivative-of-$\delta_{\chi_{n}}$-plus-one, which is at most $l-2$.
Moreover, it follows from the same corollary that

$\operatorname{ord}_{0}(\omega_{\chi_{n}}(\lambda))=-l,\text{ and
}\operatorname{ord}_{\overline{z}}(\omega_{\chi_{n}}(\lambda))\geq 0,\hskip
5.69054pt\forall\overline{z}\neq 0,\infty.$

Let us first consider the case
$\delta_{\chi_{n}}(\lambda)>p\delta_{\chi_{n-1}}(\lambda)$. Then
$\omega_{\chi_{n}}(s)$ is exact and $(l-1)$ is prime to $p$. Therefore, we may
assume that $\omega_{\chi}(\lambda)$ is of the form

$\omega_{\chi}(\lambda)=\frac{cdx}{x^{l}}+d\overline{F}$

where $0\neq d\overline{F}\in\mathcal{W}_{\lambda}$. Applying Lemma 5.15, we
obtain $F\in\mathcal{H}_{\lambda}$, such that, for
$\psi:=\mathfrak{K}_{1}(F)$,

$\delta_{\psi}(\lambda)=\delta_{\chi_{n}}(\lambda),\text{ and
}\omega_{\psi}(\lambda)=-d\overline{F}.$

One may check that $G^{\prime}:=G+F$ lies in $\mathcal{G}_{t(e)}$. Finally, it
follows from Lemma 3.12 (2) that replacing $G$ by $G^{\prime}$ changes
$\chi_{n}$ as follows

$\delta_{\chi_{n}}(\lambda)=\delta_{\mathcal{T}_{n}}(\lambda),\text{ and
}\omega_{\chi_{n}}(\lambda)=\frac{cdx}{x^{l}}.$

Thus, $\chi_{n}$ has no kink at $\lambda$, whence
$\lambda_{e}(G^{\prime})<\lambda_{e}(G)$.

Let us now consider the case
$\delta_{\chi_{n}}(\lambda)=p\delta_{\chi_{n-1}}(\lambda)$. The same line of
reason as in §5.6.3 gives

$\omega_{\chi_{n}}(\lambda)-\omega_{\mathcal{T}_{n}(\lambda)}=d\overline{F},$

where $\overline{F}$ is as in the previous case. Repeating the above process,
we obtain a $G^{\prime}\in\mathcal{G}^{\ast}_{t(e)}$ that verifies
$\lambda_{e}(G^{\prime})<\lambda$. ∎

######  Remark 5.19.

If $\lambda_{e}(G^{\prime})$ is once more strictly greater than $s(e)$, we can
repeat the procedure from the above proof to obtain an element of
$\mathcal{G}^{\ast}_{t(e)}$ with strictly smaller kink (on $e$). Thus, if we
can show that $\lambda_{e}$ achieves a minimum value then it must be equal to
$s(e)$. That is the goal of the next section.

###  5.8 The minimal depth Swan conductor

This section adapts [OW14, §6.4]. We would like show that the function
$\lambda_{e}:\mathcal{G}_{t(e)}\rightarrow\mathbb{Q}_{\geq 0}$ defined in §5.7
takes the value $s(e)$ for some $G_{t(e)}\in\mathcal{G}^{\ast}_{t(e)}$. As
discussed in Remark 5.19, it suffices to prove the following.

######  Proposition 5.20.

The function $G_{t(e)}\mapsto\lambda_{e}(G_{t(e)})$ takes a minimal value on
$\mathcal{G}_{t(e)}$ (i.e. $\mathcal{G}_{s(e)}$ is nonempty). Moreover,
$\mathcal{G}_{s(e)}$ can be identified with a qcqs set.

####  5.8.1 A lemma from rigid analysis

The following lemma will be a crucial ingredient in the proof of Proposition
5.20.

######  Lemma 5.21 (c.f. [OW14, Lemma 6.16]).

Let $X$ be qcqs over $K$ and $f^{1},\ldots,f^{n}\allowbreak\in A$ be analytic
functions on $X$. Then the function

$\phi:X\rightarrow\mathbb{R},\hskip 28.45274ptx\mapsto\max_{1\leq i\leq
n}\sqrt[i]{\lvert f^{i}(x)\rvert}$

takes a minimal value on $X$. Equivalently, the function

$x\mapsto\tau(x):=\min_{1\leq i\leq n}\frac{\nu(f^{i}(x))}{i}$

takes a maximal value on $X$. Furthermore, the subset of $X$ on which the
minimum (maximum) is attained is qcqs.

###### Proof.

The proof adapts [OW14, Lemma 6.16] and [OW16, Lemma 4.19]. It suffices to
show that, if the following inequality holds

$\gamma:=\sup_{a\in X}\bigg{(}\min_{1\leq i\leq
n}\frac{\nu(f^{i}(a))}{i}\bigg{)}\geq 0,$

then $\gamma$ is achieved on a qcqs subset of $X$. We may assume that $X$ is
an affinoid. We set $g_{i}(x):=\nu(f^{i}(x))/i$, for $1\leq i\leq n$. For each
$i$, let $X_{i}\subseteq X$ be the rational sub-domain where $g_{i}$ is
minimal among all the $g_{j}$’s. Then the restriction of $\tau$ to $X_{i}$ is
simply equal to $g_{i}$, which attains its maximum on $X_{i}$ by the maximum
modulus principle. Furthermore, the subspace of $X_{i}$ where this maximum is
attained is a Weierstrass domain in $X_{i}$, hence qcqs. Thus, the subspace of
$X$ where $\gamma$ is attained is a union of finitely many qcqs spaces, whence
also qcqs, completing the proof. ∎

####  5.8.2 A quasi-compact quasi-separated contained in $\mathcal{G}_{t(e)}$

Consider an edge $e$ of the tree $\mathcal{T}_{n}$. Assume for a moment that
we got the desired refined Swan conductor at every $r\geq t(e)$. Suppose,
moreover, that there are $m$ edges $e_{1},e_{2},\ldots,e_{m}$ where
$s(e_{i})=t(e)$ for each $i$. Set $I_{e_{j}}$ to be an indexing set of leaves
succeeding $e_{j}$. It follows from the previous constructions that an element
$G_{t(e)}\in\mathcal{G}_{t(e)}$ can be represented by

$G_{t(e)}=\sum_{j=1}^{m}G_{s(e_{j})}=\sum_{j=1}^{m}\Bigg{(}\sum_{h\in\cup
I_{e_{j}}}\bigg{(}\sum_{i=1}^{l_{h}-1}\frac{a_{h,i}}{(x-b_{h})^{i}}\bigg{)}\Bigg{)},$
(38)

where $G_{s(e_{j})}\in\mathcal{G}_{s(e_{j})}$, and $a_{h,i}=0$ for $i\equiv
0\pmod{p}$. Set $m_{t(e)}:=\sum_{j=1}^{m}\sum_{h\in\cup
I_{e_{j}}}(l_{h}-1-\lfloor l_{h}/p\rfloor)$. As before, we can think of
$\mathcal{G}_{t(e)}$ as a subset of an affine $m_{t(e)}$-space over $K$ via
the coordinates $a_{h,i}$ and $G_{t(e)}$ as one of its $K$-rational point.
Moreover, by the induction hypothesis in §5.4, the subset
$\mathcal{G}_{s(e_{j})}$ is qcqs for $j=1,\ldots,l$. Hence,
$\mathcal{G}_{t(e)}=\sum_{j=1}^{m}\mathcal{G}_{s(e_{j})}$ is also a qcqs
subset of $(\mathbb{A}_{K}^{m_{t(e)}})^{\operatorname{an}}$.

It is a consequence of §5.9 that
$\varnothing\neq\mathcal{G}^{\ast}_{t(e)}\subsetneq\mathcal{G}_{t(e)}$ where

$\mathcal{G}^{\ast}_{t(e)}=\\{G\in\mathcal{G}_{t(e)}\mid\omega_{\chi_{n}(G)}(t(e))=\omega_{\mathcal{T}_{n}}(t(e))\\}.$

As a rigid analytic space, $\mathcal{G}^{\ast}_{t(e)}$ is an open subset of
$\mathcal{G}_{t(e)}$. The goal of this section is to show that
$\lambda_{e}(\mathcal{G}_{t(e)})$ takes the minimal value on
$\mathcal{G}_{t(e)}$ and that the points where that minimum is achieved form a
qcqs subset of $\mathcal{G}^{\ast}_{t(e)}$.

####  5.8.3 Analysis

Let $\phi_{n-1}:Y_{n-1}\rightarrow X$ be the Galois cover corresponding to the
character $\chi_{n-1}$. By our induction hypothesis, it has good reduction and
is totally ramified above $x=z_{e}$ which we may assume to be $0$. As usual,
$D\subset X^{\operatorname{an}}$ is an open unit disc center at $0$. It
follows that the rigid analytic subspace $C:=\phi_{n-1}^{-1}(D)\subseteq
Y_{n-1}$ is another unit disc and contains the unique point $y_{n-1}\in
Y_{n-1}$ above $x=0$. We choose a parameter $\widetilde{x}$ for the disc $C$
such that $\widetilde{x}(y_{n-1})=0$. Then

$x=\widetilde{x}^{p^{n-1}}u(\widetilde{x}),\hskip 5.69054pt\text{ with
}u(\widetilde{x})\in R[[\widetilde{x}]]^{\times}.$

One sees that for $r>0$, the inverse image of the closed disc $D[pr]\subset D$
defined by the condition $\nu(x)\geq r$ is the closed disc $C[\widetilde{r}]$
defined by $\nu(\widetilde{x})\geq\widetilde{r}:=p^{-n+1}r$. Set
$\widetilde{r}_{n-1}:=p^{-n+1}r_{n-1}$. Let $\mathbb{K}_{n-1}$ denote the
function field of $Y_{n-1}$.

Let us fix a character $\chi\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ that is
given rise from $\chi_{n-1}$ by a rational $G\in\mathcal{G}_{t(e)}$. Let
$\widetilde{\chi}:=\chi\lvert_{\mathbb{K}_{n-1}}\in\textrm{H}^{1}_{p}(\mathbb{K}_{n-1})$
denote the restriction of $\chi$ to $\mathbb{K}_{n-1}$. If $\chi$ corresponds
to a cover $Y_{n}\rightarrow X$, then $\widetilde{\chi}$ correspond to the
cover $Y_{n}\rightarrow Y_{n-1}$. In analogy to $\lambda_{e}(\chi)$, we define
the function
$\lambda_{e}(\widetilde{\chi}):\textrm{H}^{1}_{p}(\mathbb{K}_{n-1]})\xrightarrow[]{}\mathbb{Q}$
as follows

$\lambda_{e}(\widetilde{\chi})=\begin{cases}\min(\widetilde{s(e)}\leq\widetilde{r}<\widetilde{t(e)}\mid\text{$\delta_{\widetilde{\chi}}$
linear on
$[\widetilde{s(e)},\widetilde{t(e)}]$}),&\chi\in\mathcal{G}^{\ast}_{t(e)}\\\
\widetilde{t(e)},&\chi\in\mathcal{G}_{t(e)}\setminus\mathcal{G}^{\ast}_{t(e)}.\end{cases}$

Let $m_{i}$ (resp. $m$) be the left derivative (on $e$) of $\delta_{\chi_{i}}$
(resp. $\delta_{\chi}$) at $t(e)$. The following result suggests that one may
apply the tools from §3.8 to determine the kinks of $\chi$ on $e$.

######  Lemma 5.22.

  1. 1.

For $\chi\in\mathcal{G}^{\ast}_{t(e)}$, we have
$\lambda_{e}(\widetilde{\chi})=p^{-n+1}\lambda_{e}(\chi)$.

  2. 2.

Let

$\widetilde{m}=p^{n-1}m-\sum_{i=1}^{n-1}m_{i}(p-1)p^{i-1}$

Then $p\nmid\widetilde{m}$ and the character
$\widetilde{\chi}\in\textrm{H}^{1}_{p}(\mathbb{K}_{n-1})$ satisfies the
conditions of (a),(b), and (c) of §3.8 with respect to $\widetilde{m}$, the
open disc $C\in Y^{\operatorname{an}}_{n-1}$ and the family of subdisc
$C[\widetilde{r}]$ for $\widetilde{r}\in[\widetilde{s(e)},\widetilde{t(e)}]$.

  3. 3.

If $\lambda_{\widetilde{m},\widetilde{s(e)}}(\widetilde{\chi})$ is defined as
in Corollary 3.33, and if we set $r_{0}$ in Proposition 3.31 to be equal to
$\widetilde{t(e)}$, then
$\lambda_{e}(\widetilde{\chi})=\lambda_{\widetilde{m},\widetilde{s(e)}}(\widetilde{\chi})$
for all $\chi\in\mathcal{G}_{t(e)}$.

###### Proof.

For $r>s(e)$, we systematically use the notation $\widetilde{r}:=p^{-n+1}r$.
Let the valuation $\nu_{\widetilde{r}}$ of $\mathbb{K}_{n-1}$, which
corresponds to the Gauss valuation on $C[\widetilde{r}]$ (§3.2), be the unique
extension of $\nu_{r}$. By [Wew14, §7.1] we have the following equation

$\begin{split}\delta_{\widetilde{\chi}}(\widetilde{r})&=\psi_{\mathbb{K}_{n-1}/\mathbb{K}}(\delta_{\chi}(r))\\\
&=\delta_{\chi}(r)-\bigg{(}\delta_{1}(r)\frac{p-1}{p^{n-1}}+\ldots+\delta_{n-1}(r)\frac{p-1}{p}\bigg{)},\end{split}$
(39)

where $\psi$ is the inverse Herbrand function [Ser79, IV, §3]. Since all the
characters $\chi_{i}:=\chi^{p^{n-i}}(1\leq i<n)$ have good reduction by the
hypothesis and their branch points are all contained in $\mathcal{D}[t(e)]$,
each $\delta_{i}:=\delta_{\chi_{i}}(1\leq i<n)$ is linear of slope $m_{i}$ on
the interval $[s(e),t(e)]$. Therefore, the left and the right derivative of
$\delta_{i}(r)$ is equal to $m_{i}$ for all $r\in(s(e),t(e))$. Let $c$ be the
left slope of $\delta_{\chi}$ at $r$. Then the left slope of
$\delta_{\widetilde{\chi}}$ at $\widetilde{r}$ is equal to

$p^{n-1}c-\sum_{i=1}^{n-1}m_{i}(p-1)p^{i-1}=p^{n-1}c+\widetilde{m}-p^{n-1}m.$

Part (1) then follows immediately. Part (2) follows from the fact that $c\leq
m$. Part (3) also follows from this property, along with Proposition 3.15 and
the fact that $\operatorname{sw}_{\chi}(t(e),\infty)=m$ if and only if
$\chi\in\mathcal{G}^{\ast}_{t(e)}$. ∎

One may assume that the character $\widetilde{\chi}$ is the Artin-Schreier
class of the rational function

$\widetilde{G}:=F^{n}(y_{1},\ldots,y_{n-1})+G^{n}\in\mathbb{K}^{\times}_{n-1},$

where $F^{n}$ is a polynomial over $\mathbb{F}_{p}$ in $(n-1)$ variables as in
[Obu12, Proposition 6.5]. We thus may write $\widetilde{G}$ as a power series
in the parameters $\widetilde{x}$ as follows

$\widetilde{G}=\sum_{l=1}^{\infty}a_{l}\widetilde{x}^{-l}.$

Since $G^{1},\ldots,G^{n-1}$ are fixed, $\widetilde{G}$ is determined by the
choice of $G^{n}$. So, we may think of the $a_{l}$’s as analytic functions on
the space $\mathcal{G}_{t(e)}$. In fact, $a_{l}$ is a polynomial in the
coordinate $a_{k,i}$ (as in (38)) with coefficients in $R$.

#### 5.8.4

We continue with the process of proving Proposition 5.20. Suppose $G_{0}$ is
an arbitrary element of $\mathcal{G}^{\ast}_{t(e)}$ and $\chi_{0}$ is the
character it gives rise to. Then it is immediate that
$\lambda_{e}(G_{0})<t(e)$. We therefore may choose a rational number
$s\in(\lambda(G_{0}),t(e))$. With the notation from Section 5.8.3, recall that
$\widetilde{\chi}$ is the restriction of $\chi$ to the function field
$\mathbb{K}_{n-1}$ of $Y_{n-1}$. It is an Artin-Schreier cover defined by a
rational function $\widetilde{G}\in\mathbb{K}_{n-1}^{\times}$. By Lemma 5.22,
we have

$\lambda(\widetilde{G})<\widetilde{s}:=p^{1-n}s<\widetilde{t(e)}.$

Set $\widetilde{\delta}:=\delta_{\widetilde{\chi}}(\widetilde{t(e)})$. Let $N$
be an integer such that

$Np\geq\frac{\widetilde{\delta}}{\widetilde{t(e)}-\widetilde{s}}.$

We hence arrive at the situation of (27). The following is parallel to [OW14,
Lemma 6.8]

######  Lemma 5.23.

There exists a finite cover
$\mathcal{G}^{\prime}_{t(e)}\rightarrow\mathcal{G}_{t(e)}$ and analytic
functions $b_{1},\ldots,b_{N}$ on $\mathcal{G}^{\prime}_{t(e)}$ with the
following property. Set

$d:=\sum_{j=0}^{N}b_{j}\widetilde{x}_{1}^{-j}$

and write

$F+d^{p}-d=\sum_{l=1}^{\infty}c_{l}\widetilde{x}_{1}^{-l},$

where the $c_{l}$ are now analytic functions on $\mathcal{G}^{\prime}_{t(e)}$.
Then

  1. 1.

for all $l\geq 1$ and all points $y\in\mathcal{G}^{\prime}_{t(e)}$, we have
$\nu(c_{l}(y))\geq p(\widetilde{t(e)}l-\widetilde{\delta})$;

  2. 2.

we have $c_{pl}=0$ for $l\leq N$.

###### Proof.

The lemma follows immediately from Proposition 3.31 and Remark 3.32. ∎

####  5.8.5 Proof of Proposition 5.20

We can now complete the proof of Proposition 5.20. Let $\widetilde{m}$ be as
in part (2) of Lemma 5.22. Define the function
$\mu_{\widetilde{m},\widetilde{s(e)}}:\mathcal{G}^{\prime}_{t(e)}\rightarrow\mathbb{R}$
as follows

$\mu_{\widetilde{m},\widetilde{s(e)}}(x):=\max\bigg{(}\bigg{\\{}\frac{\nu(c_{\widetilde{m}}(x))-\nu(c_{l}(x))}{\widetilde{m}-l}\mid
1\leq l<\widetilde{m}\bigg{\\}}\cup\\{\widetilde{s(e)}\\}\bigg{)}.$

Let $\chi\in\mathcal{G}_{t(e)}$, write
$\widetilde{\chi}:=\chi\lvert_{\mathbb{K}_{n-1}}$ for its restriction to the
function field of $Y_{n-1}$, and let $x\in\mathcal{G}^{\prime}_{t(e)}$ be an
arbitrary point over $\chi$. Thanks to Lemma 5.23, we can apply Proposition
3.31 to compare $\mu_{\widetilde{m},\widetilde{s(e)}}(x)$ to
$\lambda_{\widetilde{m},\widetilde{s(e)}}(\widetilde{\chi})$, which, in turn,
is equal to $\lambda_{e}(\widetilde{\chi})$ by Lemma 5.22 (3). We thus
conclude that $\mu_{\widetilde{m},\widetilde{s(e)}}(x)<\widetilde{s}$ if and
only if $\lambda_{e}(\widetilde{\chi})<\widetilde{s}$. Moreover, if this is
the case, then we have
$\mu_{\widetilde{m},\widetilde{s(e)}}(x)=\lambda_{e}(\widetilde{\chi})$. In
any case, by Lemma 5.22 (1), we have
$\lambda_{e}(\chi)=p^{n-1}\lambda_{e}(\widetilde{\chi})$ when
$\chi\in\mathcal{G}^{\ast}_{t(e)}$.

The rest of the proof is identical to one in [OW14, §6.4.5]. Let
$G_{0}\in\mathcal{G}^{\ast}_{t(e)}$ be the rational function at the beginning
of §5.8.4. Let $G^{\prime}_{0}\in\mathcal{G}^{\prime}_{t(e)}$ be a point above
$G_{0}$. Because $\lambda_{e}(G_{0})<s$, we have
$\mu_{\widetilde{m},\widetilde{s(e)}}(G^{\prime}_{0})<\widetilde{s}$ as
discussed above.

It follows from Lemma 5.21 that the function
$\mu_{\widetilde{m},\widetilde{s(e)}}$ takes a minimum on
$\mathcal{G}^{\prime}_{t(e)}$. Let $w\in\mathcal{G}^{\prime}_{t(e)}$ be a
point where this minimum is achieved, and let $W\in\mathcal{G}_{t(e)}$ be its
image under $\mathcal{G}^{\prime}_{t(e)}\rightarrow\mathcal{G}_{t(e)}$. We
have
$\mu_{\widetilde{m},\widetilde{s(e)}}(w)\leq\mu_{\widetilde{m},\widetilde{s(e)}}(G^{\prime}_{0})<\widetilde{s}$.
Since

$\lambda_{\widetilde{m},\widetilde{s(e)}}(\widetilde{W})=\mu_{\widetilde{m},\widetilde{s(e)}}(w)<\widetilde{s}<\widetilde{t(e)},$

we see that $W\in\mathcal{G}^{\ast}_{t(e)}$. Applying the above arguments a
second time, we conclude that
$\lambda_{e}(W)=p^{n-1}\mu_{\widetilde{m},\widetilde{s(e)}}(w)$, and this is
actually the minimal value of the function
$\lambda_{e}:\mathcal{G}^{\ast}_{t(e)}\rightarrow\mathbb{R}$. Moreover, the
subset on which the minimum is attained is qcqs by the Lemma 5.21. This
completes the proof of Proposition 5.20.∎

###  5.9 Controlling a non-final vertex

Suppose $t(e)=s(e_{1})=\ldots=s(e_{m})$ is a non-final-vertex of
$\mathcal{T}_{n}$. By the induction process, we obtain a bunch of qcqs sets
$\mathcal{G}_{s(e_{1})},\ldots,\mathcal{G}_{s(e_{m})}$. Recall that each
$G_{i}\in\mathcal{G}_{s(e_{1})}$ has (generic) branching datum fits into
$\mathcal{T}_{n}(e_{i})$ and verifies, for $\chi_{G_{i}}$ be the extension of
$\chi_{n-1,e_{i}}$ by $G_{i}$, the equation
$\delta_{\chi_{G_{i}}}(s(e_{i}))=\delta_{\mathcal{T}_{n}}(s(e_{i}))$. Suppose,
moreover, that all the branch points of $\chi_{G_{i}}$ specialize to
$\overline{a}_{i}\in k$ at $t(e)$. Then the $\overline{a}_{i}$’s are distinct,
and the differential conductor $\omega_{\chi_{G_{i}}}(t(e))$ is of the form

$\omega_{\chi_{G_{i}}}(t(e))=\frac{f^{i}(x)dx}{(x-\overline{a}_{i})^{l_{i}}},$

where $f^{i}(x)\in k[x]$ and
$l_{i}:=\sum_{b\in\mathbb{B}(\mathcal{T}_{n}(e_{i}))}h_{b}$. Set
$G:=\sum_{i=1}^{m}G_{i}$ and $\chi_{G}$ is the character that is given rise
from $\chi_{n-1,e}$ by $G$. By Lemma 3.12 (2), we have

$\omega_{\chi_{G}}(t(e))=\sum_{i=1}^{m}\omega_{\chi_{G_{i}}}(t(e))\text{, and
}\delta_{\chi_{G}}(t(e))=\delta_{\chi_{G_{i}}}(t(e))\hskip 5.69054pt\forall
i=1,\ldots,m.$

Hence, the collection of such $G$, which we call $\mathcal{G}_{t(e)}$, is
equal to $\sum_{i=1}^{m}\mathcal{G}_{s(e_{i})}$ and thus is qcqs. Its elements
give rise to characters with the desired depth and the geometry of the branch
locus at $t(e)$. Suppose the wanted differential form at $t(e)$ is

$\omega_{\mathcal{T}_{n}}(t(e))=\frac{cdx}{\prod_{i=1}^{m}(x-\overline{a}_{i})^{l_{i}}}\hskip
5.69054pt(c\neq 0).$

Recall that
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(t(e)))=\omega_{\mathcal{T}_{n-1}}(t(e))$
when $\delta_{\mathcal{T}_{n}}(t(e))=p\delta_{\mathcal{T}_{n-1}}(t(e))$, and
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(t(e)))=0$ otherwise. Either cases, as we
achieve the right depth Swan conductor at $t(e)$, we may assume that
$\omega_{\chi_{G}}$ has the form

$\omega_{\chi_{G}}(t(e))=\frac{cdx}{\prod_{i=1}^{m}(x-\overline{a}_{i})^{l_{i}}}+\sum_{i=1}^{m}\omega^{\prime}_{\chi_{G}}(s(e_{i})),$

where $\omega^{\prime}_{\chi_{G}}(s(e_{i}))\in\mathcal{W}_{s(e_{i})}$ is
exact. Now, by repeating the process in §5.6 for each $i$, we replace $G$ by
another one in $\mathcal{G}_{t(e)}$ where
$\omega^{\prime}_{\chi_{G}}(s(e_{i}))=0$ for all $i$. That shows
$\mathcal{G}^{\ast}_{t(e)}$ is non-empty.

###  5.10 Controlling the root

By the previous steps of the induction process, we get to the point where
$\mathcal{G}_{s(e_{0})}\neq\varnothing$. That means there exists a
$G_{\min}\in\mathcal{G}_{s(e_{0})}$ that gives rise to a character
$\chi_{\min}$ whose braching datum fits to $\mathcal{T}_{n}$ and whose depth
is zero at $s(e_{0})$, i.e., $\chi_{\min}$ has good reduction.

Recall that $\overline{\chi}_{n}$ is our original character. Assume that
ramification breaks of the one point cover corresponding to
$\overline{\chi}_{n}$ is $(m_{1},\ldots,m_{n})$, and it is represented (upon
completion at $x=0$) by a Witt vector
$\underline{g}_{n}:=(g^{1},\ldots,g^{n})\in W_{n}(k(x))$. As discussed in
§2.4.1, one may regard $\overline{\chi}_{n}$ as a one-point-cover of
$\mathbb{P}^{1}_{k}$. Therefore, we further assume that each $g^{i}$ is a
polynomial in $k[x^{-1}]$, all of whose terms have prime-to-$p$ degree. On the
other hand, it follows from the previous constructions that
$\overline{\chi}_{\min}$ corresponds to a Witt vector
$\overline{g}_{\min}=(g^{1},\ldots,g^{n-1},g_{\min})$ where $g_{\min}\in
k[x^{-1}]$ has degree less than (only when $m_{n}=pm_{n-1}$) or equal to
$m_{n}$ and consists of only terms of prime-to-$p$ degree. Subtracting Witt
vectors yield

$\underline{g}_{n}-\underline{g}_{\min}=(0,\ldots,0,g^{n}-g_{\min}).$

We define $g:=g^{n}-g_{\min}$. Recall that Corollary 5.7 asserting
$\chi_{\min}$ deforms $\overline{\chi}_{n}$ if and only if $g=0$.

######  Proposition 5.24.

There is a character $\chi_{n}$ that is a deformation of
$\overline{\chi}_{n}$.

###### Proof.

Applying Corollary 5.16 for $g$, we obtain $H\in\mathcal{H}_{s(e_{0})}$ such
that $\psi:=\mathfrak{K}_{1}(H)\in\textrm{H}^{1}_{p}(\mathbb{K})$ satisfying
the following

$\delta_{\psi}(0)=0\text{, and }\overline{\psi}\text{ is given by }y^{p}-y=g.$

Hence, if
$\psi^{\prime}:=\mathfrak{K}_{n}((0,\ldots,0,H))\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$,
then we also have $\delta_{\psi^{\prime}}(0)=0$ and the reduction
$\overline{\psi}^{\prime}$ corresponds to the same extension, which is
represented by the Witt vector $(0,\ldots,g)\in W_{n}(\kappa)$.

On the other hand, recall that $\chi_{\min}$ corresponds to the
length-$n$-Witt-vector

$(G^{1},G^{2},\ldots,G^{n-1},G_{\min})\in
W_{n}(\mathbb{K})/\wp(W_{n}(\mathbb{K})).$

Therefore, replacing $G_{\min}$ by $G^{n}:=G_{\min}+H$ equates to multiplying
$\psi^{\prime}$ to $\chi_{\min}$. By Lemma 3.12, the result is an étale
character $\chi_{n}\in\textrm{H}^{1}_{p^{n}}(\mathbb{K})$ with reduction
$\overline{\chi}_{\min}\cdot\overline{\psi}^{\prime}$, which, in turn, is
defined by the Witt vector $(g^{1},\ldots,g^{n-1},g_{\min}+g=g^{n})$. Finally,
since $G^{n}\in\mathcal{G}_{s(e_{0})}$ as $H\in\mathcal{H}_{s(e_{0})}$, it
follows from Corollary 5.7 that $\chi_{n}$ is a lift of $\overline{\chi}_{n}$.
∎

That completes the proof of the Main Theorem. ∎

######  Remark 5.25.

It follows from the construction and the compatibility of the differential
conductors that the $\mathbb{Z}/p^{n}$-tree arising from $\chi_{n}$ coincides
with the tree $\mathcal{T}_{n}$.

##  6 Proofs of Technical Results

###  6.1 A solution to the Cartier operator equation

######  Proposition 6.1.

Suppose we are given a differential form as below

$\omega=\frac{cdx}{\prod_{i=1}^{r}(x-d_{i})^{\iota_{i}}},$

where $c$ and the $d_{i}$’s are in $k$, $d_{i}\neq d_{j}$ for $i\neq j$,
$r\geq 2$, and a fixed integer $1\leq a\leq p-1$. Set
$\iota^{\prime}_{1}:=p\iota_{1}-p+a+1$ ,$\iota^{\prime}_{2}:=p\iota_{2}-p+1$,
$\iota_{i}^{\prime}:=p\iota_{i}$ for each $i\neq 1,2$. Then the differential
form

$\omega^{\prime}=\frac{c^{p}dx}{(-d_{1}-d_{2})^{p-a-1}\prod_{i=1}^{r}(x-d_{i})^{\iota^{\prime}_{i}}}$

verifies $\mathcal{C}(\omega^{\prime})=\omega$.

###### Proof.

The differential form $\omega^{\prime}$ above can be rewritten as

$\omega^{\prime}=\frac{c^{p}(x-d_{1})^{p-a-1}dx}{(x-d_{2})(-d_{1}-d_{2})^{p-a-1}(x-d_{2})^{p(\iota_{2}-1)}(x-d_{1})^{p\iota_{1}}\prod_{i=3}^{r}(x-d_{i})^{p\iota_{i}}}.$

The proposition then follows from the fact that
$\mathcal{C}(u^{p}\omega)=u\mathcal{C}(\omega)$ for all $u\in K$. ∎

###  6.2 Construction of the extending Hurwitz tree

We dedicate this section for the proof of Proposition 4.20. In each proof, we
will only demonstrate the construction of the extending Hurwitz trees. The
readers can easily check that these trees satisfy the axioms in Definition 4.2
together with the conditions for an extension asserted by Theorem 3.42 and
Theorem 4.17.

Recall the set up of Proposition 4.20. We are give a $\mathbb{Z}/p^{n-1}$-tree
$\mathcal{T}_{n-1}$ and an $n$-th dengeneration data
$(\delta_{n},\omega_{n})\in\mathbb{Q}_{>0}\times\Omega^{1}_{\kappa}$ (or
$(0,f_{n})$, where $f_{n}\in\kappa$) at its root. The goal is to construct a
$\mathbb{Z}/p^{n}$-tree $\mathcal{T}_{n}$ that extends $\mathcal{T}_{n-1}$ and
whose degeneration data at its root is the same as the given one. Note that
the original result only requires $\delta_{n}=0$ but we will prove the general
case in order to utilize an induction technique that we will soon see.

Suppose $\mathcal{T}_{n-1}$ is a $\mathbb{Z}/p^{n-1}$-tree that has $m$ leaves
$b_{1},\ldots,b_{m}$ ($m\geq 2$), and the conductor at $b_{i}$ is
$\iota_{i,n-1}$. As usual, $v_{0}$ and $e_{0}$ denote the root and the trunk
of $\mathcal{T}_{n-1}$. Set $v_{1}:=t(e_{0})$. Suppose, moreover, that the
differential conductor (the $(n-1)$th degeneration) at $v_{1}$ and $v_{0}$ of
$\mathcal{T}_{n-1}$ are

$\begin{split}\omega_{\mathcal{T}_{n-1}}(v_{1})&=\frac{c_{v_{1}}dx}{\prod_{i=1}^{r}(x-[b_{i}]_{v_{1}})^{\iota_{i,n-1}}},\\\
\omega_{\mathcal{T}_{n-1}}(v_{0})&=\frac{c_{v_{0}}dx}{x^{\iota_{n-1}}}+\sum_{j<\iota_{n-1}}\frac{c_{j}dx}{x^{j}}\hskip
5.69054pt\bigg{(}g^{n-1}=\sum_{j=1}^{l}\frac{d_{j}}{x^{j}}\bigg{)},\end{split}$
(40)

respectively, where $l<\iota_{n-1}$. We may write
$\omega_{\mathcal{T}_{n-1}}(v_{0})$ in (40) as $f(x)dx/x^{\iota_{n-1}}$. Then,
when $\delta_{\mathcal{T}_{n-1}}(v_{0})>0$, as the two forms are compatible
(§4.4), we have the relation $c_{v_{0}}=c_{v_{1}}=:c$. When
$\delta_{\mathcal{T}_{n-1}}(v_{0})=0$ and $\iota_{n-1}=p\iota_{n-2}-p+1$,
where $\iota_{n-2}$ is the conductor of the sub-$\mathbb{Z}/p^{n-2}$-tree, as
discussed in Theorem 4.17, we don’t have to worry about the compatibility at
the root. Otherwise, the same theorem asserts that $l=\iota_{n-1}-1$ and
$c_{v_{1}}=-ld_{l}$.

Suppose, moreover, that we are given a level $n$ depth
$\delta_{\mathcal{T}_{n}}(v_{0})\geq p\delta_{\mathcal{T}_{n-1}}(v_{0})$ and a
level $n$ differential conductor (or degeneration) at the root as follows

$\omega_{\mathcal{T}_{n}}(v_{0})=\frac{C_{v_{0}}dx}{x^{\iota_{n}}}+\sum_{j<\iota_{n}}\frac{C_{j}dx}{x^{j}}\hskip
5.69054pt\bigg{(}\text{or }g^{n}=\sum_{j=1}^{L}\frac{D_{j}}{x^{j}}\bigg{)},$
(41)

where $L<\iota_{n}$. When $\delta_{\mathcal{T}_{n}}(v_{0})>0$, it is required
that
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(v_{0}))=\omega_{\mathcal{T}_{n-1}}(v_{0})$
if $\delta_{\mathcal{T}_{n}}(v_{0})=p\delta_{\mathcal{T}_{n-1}}(v_{0})$, or
$\omega_{\mathcal{T}_{n}}(v_{0})$ is exact if
$\delta_{\mathcal{T}_{n}}(v_{0})>p\delta_{\mathcal{T}_{n-1}}(v_{0})$. When
$\delta_{\mathcal{T}_{n}}(v_{0})=p\delta_{\mathcal{T}_{n-1}}(v_{0})=0$, then
$L=\iota_{n}-1$ if $\iota_{n}>\iota_{n-1}-p+1$.

Just as the main theme of the whole paper, we first construct the extensions
of certain sub-trees of the given tree $\mathcal{T}_{n-1}$.

######  Definition 6.2.

With the notation as above, let $\mathcal{T}_{n}$ be an extension of
$\mathcal{T}_{n-1}$ with the same underlying tree and such that the depth at
the root of $\mathcal{T}_{n}$ is _$p$ times_ that of $\mathcal{T}_{n-1}$. Let
$a_{n}\in\mathbb{Z}_{>0}$ and $a_{n}=:a^{\prime}_{n}+pu_{n}$, $1\leq
a^{\prime}_{n}<p$. We say

  * •

$\mathcal{T}_{n}$ is an additive $a_{n}$ extension of $\mathcal{T}_{n-1}$ if,
at all but one leaf (which may assume to be $b_{1}$), we have
$\iota_{i,n}=p\iota_{i,n-1}$, hence $\iota_{1,n}=p\iota_{1,n-1}-p+1+a_{n}$,
and

  * •

$\mathcal{T}_{n}$ is a minimum extension of $\mathcal{T}_{n-1}$ if
$\iota_{1,n}=p\iota_{1,n-1}-p+1$ and $\iota_{i,n}=p\iota_{i,n-1}$ for $i\neq
2$.

We first show that one can always construct an additive extension for the tree
$\mathcal{T}_{n-1}$. That is the extension of $\mathcal{T}_{1}(e_{3})$ in
Example 4.21.

######  Proposition 6.3.

With the above settings, there exists a Hurwitz tree $\mathcal{T}_{n}$ that
extends $\mathcal{T}_{n-1}$ $a_{n}$-additively. In particular, Proposition
4.20 holds when $\iota_{n}=p\iota_{n-1}-p+1+a_{n}$.

###### Proof.

First, we set the decorated tree of $\mathcal{T}_{n}$ and the thickness of the
edges to be that of $\mathcal{T}_{n-1}$. We then assign to $b_{1}$ the
conductor $\iota_{1,n}=p\iota_{i,n-1}-p+a_{n}+1$, to each leaf $b_{i}$ ($i\neq
1$) a conductor $\iota_{i,n}=p\iota_{i,n-1}$, to each edge $e$ a slope
$d_{e}:=\sum_{h\succ e}\iota_{h,n}-1$, and to the root $v_{0}$ the depth
$p\delta_{\mathcal{T}_{n-1}}(v_{0})$. At each vertex or leaf of
$\mathcal{T}_{n}$ where the corresponding one in $\mathcal{T}_{n-1}$ has
monodromy group $\mathbb{Z}/p^{i}$, we equip it with the monodromy group
$\mathbb{Z}/p^{i+1}$. We then assign the depth at each vertex $v\neq v_{0}$
inductively on the positive direction starting from $v_{0}$ so that (H5) is
verified. Then it is straight forward to check that
$\delta_{\mathcal{T}_{n}}(v)>p\delta_{\mathcal{T}_{n-1}}(v)$ at all vertices
$v\neq v_{0}$. Finally, we equip each vertex $v$ away from the root with an
_exact_ differential form as follows

$\omega_{\mathcal{T}_{n}}(v)=\frac{c_{v}dx}{\prod_{b\succ
v}(x-[b]_{v})^{\iota_{b,n}}},$ (42)

where $c_{v}\in k^{\times}$ is determined inductively from the root so that
the differential conductors at all vertices are compatible. More precisely, if
$e$ is an edge of $\mathcal{T}_{n-1}$, then we set the coefficient of
$\omega_{\mathcal{T}_{n}}(t(e))$ to be the $e$-part of
$\omega_{\mathcal{T}_{n}}(s(e))$ (resp. the coefficient of the $n$-th
degeneration) when $\delta_{\mathcal{T}_{n}}(s(e))>0$ (resp. when
$\delta_{\mathcal{T}_{n}}(s(e))=0$). ∎

We then construct a minimum extension. That is the extending of
$\mathcal{T}_{1}$ or $\mathcal{T}_{1}(e_{1})$ in Example 4.21.

######  Proposition 6.4.

With the above settings, there exists a Hurwitz tree $\mathcal{T}_{n}$ that
minimally extends $\mathcal{T}_{n-1}$. In particular, Proposition 4.20 holds
for the case $\iota_{n}=p\iota_{n-1}-p+1$.

###### Proof.

Suppose first that the height of the tree $\mathcal{T}_{n-1}$ is $1$. We
assign conductor $\iota_{1,n}:=p\iota_{1,n-1}-p+1$ to $b_{1}$ and
$\iota_{i,n}:=p\iota_{i,n-1}$ to $b_{i}$ where $i\neq 1$, making the sum of
all conductors to be $\iota_{n}$. We then set the depth at $v_{1}$ to be
$p\delta_{\mathcal{T}_{n-1}}(v_{1})$, the slope of $e_{0}$ to be
$\iota_{n}-1$, and the differential conductor at $v_{1}$ to be

$\omega_{\mathcal{T}_{n}}(v_{1})=\frac{c^{p}dx}{\prod_{i=1}^{r}(x-[b_{i}]_{v_{1}})^{\iota_{i,n}}},$

which verifies
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(v_{1}))=\omega_{\mathcal{T}_{n-1}}(v_{1})$
as shown in Proposition 6.1. As before, a vertex’s (or leaf’s) monodromy group
is cyclic of order $p$ times that of the corresponding one in
$\mathcal{T}_{n-1}$. The hypothesis $\iota_{n}=p\iota_{n-1}-p+1$ also implies
that the differential conductor (resp. the $n$-th degeneration) is of the form

$\omega_{\mathcal{T}_{n}(x)}(v_{0})=\frac{c^{p}dx}{x^{\iota_{n}}}+\sum_{j\leq\iota_{n-1}}\frac{c_{j}^{p}dx}{x^{pj-p+1}}+dg(x)\hskip
5.69054pt(\text{resp. }g^{n}(x)),$ (43)

where $g(x)$ (resp. $g^{n}(x)$) is a polynomial in $x^{-1}$ of degree at most
$\iota_{n}-2$. That makes $v_{0}$ compatible with $v_{1}$ in $\mathcal{T}_{n}$
when $\delta_{\mathcal{T}_{n}}(v_{0})>0$, completing the base case. Recall
that, in our minimal jump assumption and when
$\delta_{\mathcal{T}_{n}}(v_{0})=0$, we don’t need the degenerations at
$v_{0}$ and $v_{1}$ to be compatible.

Let us now consider the case where the height of $\mathcal{T}_{n-1}$ is
greater than one. Suppose there are $m$ edges $e_{1},\ldots,e_{m}$ ($m\geq 2$)
starting from $v_{1}$. We then first assign the degeneration data at $v_{0}$
and $v_{1}$ identical to the base case. Note that the Hurwitz tree
$\mathcal{T}_{n-1}(e_{i})$ has height at most $h(\mathcal{T}_{n-1})-1$. We
thus can extend $\mathcal{T}_{n-1}(e_{1})$ to a minimum tree by induction, and
$\mathcal{T}_{n-1}(e_{i})$, $i\neq 1$, to an additive tree as in Proposition
6.3, all with depth $\delta_{\mathcal{T}_{n}}(v_{1})$. The conductor of
$\mathcal{T}_{n}(e_{1})$ (resp. $\mathcal{T}_{n}(e_{i})$) is
$\iota_{n,e_{1}}:=p\mathfrak{C}_{\mathcal{T}_{n-1}}(e_{1})-p+1$ (resp.
$\iota_{n,e_{i}}:=p\mathfrak{C}_{\mathcal{T}_{n-1}}(e_{i})$). Hence, the sum
of conductors are $\iota_{n}$. That completes the construction of the tree
$\mathcal{T}_{n}$. ∎

We have just proved Proposition 4.20. One can also do the following
alternative construction, which is utilized in Example 4.23.

######  Proposition 6.5.

There exists a $\mathbb{Z}/p^{n}$-tree $\mathcal{T}_{n}$ that extends
$\mathcal{T}_{n-1}$ with no essential jumps.

###### Proof.

Suppose there are $m$ edges $e_{1},\ldots,e_{m}$ succeeding $v_{1}$ with
conductors $\mathfrak{C}_{1},\ldots,\mathfrak{C}_{m}$, respectively, on
$\mathcal{T}_{n-1}$. Hence $\sum_{i=1}^{m}\mathfrak{C}_{i}=\iota_{n-1}$. Set
$\mathfrak{C}^{\prime}_{1}:=p\mathfrak{C}_{1}-p+a^{\prime}_{n}$,
$\mathfrak{C}^{\prime}_{2}:=p\mathfrak{C}_{2}-p+1$, and
$\mathfrak{C}^{\prime}_{i}:=p\mathfrak{C}_{i}$. We first assign
$p\delta_{\mathcal{T}_{n-1}}(v_{1})$ as the depth at $v_{1}$ on
$\mathcal{T}_{n}$, and the non-exact differential form

$\omega_{\mathcal{T}_{n}}(v_{1})=\frac{C_{v_{1}}dx}{\prod_{i=1}^{s}(x-[b_{i}]_{v_{1}})^{\mathfrak{C}^{\prime}_{i}}},$

that verifies
$\mathcal{C}(\omega_{\mathcal{T}_{n}}(v_{1}))=\omega_{\mathcal{T}_{n-1}}(v_{1})$
( which exists by Proposition 6.1) as the differential conductor at $v_{1}$.
We then extend $\mathcal{T}_{n-1}(e_{1})$ to an
additive-$a^{\prime}_{n}$-tree, $\mathcal{T}_{n-1}(e_{2})$ to a minimum tree,
and each $\mathcal{T}_{n-1}(e_{i})$ where $i\neq 1,2$ to an
additive-$(p-1)$-tree, all with depth $p\delta_{\mathcal{T}_{n-1}}(v_{1})$,
and so that they are compatible with the differential conductor
$\omega_{\mathcal{T}_{n}}(v_{1})$. That makes the current sum of conductors to
be $\mathfrak{C}(v_{1}):=p\iota_{n-1}-2p+a^{\prime}_{n}+1$ which is strictly
smaller than $p\iota_{n-1}-p+1$ and not congruent to $1$ modulo $p$. Let us
now consider the trunk $e_{0}$ of $\mathcal{T}_{n-1}$. A simple calculation
shows that one can break down the edge $e_{0}$ into two rational ones, called
$e_{0,1}$ and $e_{0,2}$, so that
$(\mathfrak{C}(v_{1})-1)\epsilon_{e_{0,2}}+(\iota_{n}-1)\epsilon_{e_{0,1}}=p\delta_{\mathcal{T}_{n-1}}$.
We then assign to $t(e_{0,1})$ on $\mathcal{T}_{n}$ the exact differential

$\omega_{\mathcal{T}_{n}}(t(e_{0,1}))=\frac{-C_{0}dx}{(\iota_{n}-1)x^{\mathfrak{C}(v_{1})}(x-a)^{p(u_{n}+1)}},$

where
$a:=\big{(}-C_{0}/((\iota_{n}-1)\mathfrak{C}(v_{1}))\big{)}^{1/(p(u_{n}+1))}$.
A straightforward computation shows that the choice of $a$ makes $v_{0}$,
$v_{1}$, and $t(e_{0,1})$ compatible. We then glue a leaf of conductor
$p(u_{n}+1)$ and monodromy group $\mathbb{Z}/p^{n-1}$ to $t(e_{0,1})$.
Finally, we set the remaining degeneration data at $e_{0,1}$, $e_{0,2}$, and
$v_{0}$ in the obvious manner, completing the proof. ∎

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