2.1.1 Hurwitz data of special loci

Let $\mathcal {M}_{g,[m]}[G]$ be the moduli stack of curves of genus g with m marked points endowed with a faithful G-action, or Hurwitz stack, whose S-sections for a $\mathbb {Q}$ -scheme S are defined as follows: the objects of $\mathcal {M}_{g,[m]}[G](S)$ are triplets $(C,D,\iota )$ where

$$\begin{align*}\begin{cases} C\text{ is a smooth projective curve of genus } g \text{ over }S,\\ \iota \colon G\rightarrow \operatorname{Aut} _S C\text{ an injective homomorphism,}\\ D\text{ an tale Cartier divisor of degree } m \\ \quad \text{ stabilized by the } G\text{-action induced by } \iota; \end{cases} \end{align*}$$

see [Reference Collas and Maugeais4] § 2.1 as well as for the rest of this section. The G-special locus $\mathcal {M}_{g,[m]}(G)$ of $\mathcal {M}_{g,[m]}$ is obtained as the image of $\mathcal {M}_{g,[m]}[G]$ in $\mathcal {M}_{g,[m]}$ under the forgetful functor defined by

$$\begin{align*}\begin{array}{cccc} \mathcal{M}_{g,[m]}[G](S) & \longrightarrow & \mathcal{M}_{g,[m]}(S) \\ (C,D,\iota) & \longmapsto & (C,D). \end{array} \end{align*}$$

In particular, the S-sections of $\mathcal {M}_{g,[m]}(G)$ are curves over S whose geometric fibers admits a faithful G-action. The stack $\mathcal {M}_{g,[m]}[G]$ having a canonical right-action of $\operatorname {Aut} G$ via $\iota $ , we can form the quotient stack $\mathcal {M}_{g,[m]}[G]/\operatorname {Aut} G$ that we denote by $\mathcal {M}_{g,[m]}(G)^{\nu }$ since, apart from a few exceptional casesFootnote ² (see [Reference Magaard, Shaska, Shpectorov and Völklein19] Theorem 5.1 and Section 4 for an account with $g\geq 2$ and also Remark 2.1 2.1.1), it identifies with the normalization of $\mathcal {M}_{g,[m]}(G)$ by the proof of [Reference Romagny29] Proposition 3.4.1. All the stacks $\mathcal {M}_{g,[m]}[G]$ , $\mathcal {M}_{g,[m]}(G)$ and $\mathcal {M}_{g,[m]}(G)^{\nu }$ are Deligne-Mumford stacks over $ \operatorname {Spec} \mathbb {Q}$ – with $\mathcal {M}_{g,[m]}[G]$ and $\mathcal {M}_{g,[m]}(G)^{\nu }$ moreover smooth over $ \operatorname {Spec} \mathbb {Q}$ .

From now on, we assume that $G\simeq \mathbb {Z}/n\mathbb {Z}$ is cyclic, so that following [Reference Collas and Maugeais4], we can investigate the subloci $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}$ of $\mathcal {M}_{g,[m]}(G)$ of S-curves whose G-action ramification data correspond to certain Hurwitz data $\underline {kr}=(k,r)$ modulo the diagonal $(\mathbb {Z}/n\mathbb {Z})^{\times }$ -action, which are abstractly defined as follows:

• • The part k corresponds to an N-tuple in $(\mathbb {Z}/n\mathbb {Z})^{N}$ , where N is the degree of the branch divisor, whose terms sum to $0$ , and which is taken up to permutation. Each component of k corresponds to a generator of one of the G-isotropy groups.

• • The second part r is an element of $\mathbb {N}^n$ , whose i-th component, in the case of a quotient map $\psi \colon C\rightarrow C/G$ , corresponds to

  $$\begin{align*}r(i)= \operatorname{Card} \{ y\in D/G \mid br(y)= i\ \mod n\}, \end{align*}$$
  where $br(y)$ is the branching order at y – that is, the ramification index of any point in the fiber $\psi ^{-1}(y)$ .

Note that the $(\mathbb {Z}/n\mathbb {Z})^{\times }$ -quotient in $\underline {kr}$ should be seen as the $(\operatorname {Aut} G)$ -quotient previously introduced. We refer to ibid. Definitions 3.5 and 3.9, and Example 3.11 for further details.

The construction of abstract Hurwitz data from G-curves defines a map

$$\begin{align*}\underline{kr}\colon \mathcal{M}_{g,[m]}[G]_{N} \longrightarrow ((\mathbb{Z}/n\mathbb{Z})^{N}/\mathfrak{S}_{N}\times \mathbb{N}^n)/(\mathbb{Z}/n\mathbb{Z})^{\times}, \end{align*}$$

where $\mathcal {M}_{g,[m]}[G]_{N}$ denotes the substack of $\mathcal {M}_{g,[m]}[G]$ of curves whose branch divisor is of degree N, which is locally constant – see [Reference Collas and Maugeais4] Lemma 3.13. For a fixed value of $\underline {kr}$ , one thus obtains a substack $\mathcal {M}_{g,[m]}[G]_{\underline {kr}}$ of $\mathcal {M}_{g,[m]}[G]$ of G-curves with abstract Hurwitz data $\underline {kr}$ so that one can define the following:

Also, since the action of $\operatorname {Aut} G$ stabilizes $\mathcal {M}_{g,[m]}[G]_{\underline {kr}}$ by definition, we have substacks $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }$ of $\mathcal {M}_{g,[m]}(G)^{\nu }$ . The stacks $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }$ and $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}$ are defined over $\mathbb {Q}$ by construction and are geometrically irreducible by Proposition 3.12 and Theorem 4.3 of [Reference Collas and Maugeais4].

One particular case of interest is when the ramification divisor is contained in the marked divisor D. In this case, we can recover r by the data of D and k. Indeed, we have

$$\begin{align*}\begin{cases} r(i)= \operatorname{Card} \{j \mid k(j)=i\}/ \operatorname{gcd}(i,n) \text{ for } i\neq 0\\ r(0)= \deg D - \sum_{i\in \mathbb{Z}/n\mathbb{Z}\setminus \{0\}} \operatorname{Card} \{j \mid k(j)=i\}. \end{cases} \end{align*}$$

Similarly to the moduli stacks of curves, the stacks $\mathcal {M}_{g,[m]}(G)$ , resp. $\mathcal {M}_{g,[m]}(G)^{\nu } $ , are not necessarily proper. We denote by $\overline {\mathcal {M}}_{g,[m]}(G)_{\underline {kr}}$ the G-stable compactification of the G-special locus $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}$ , and by $\overline {\mathcal {M}}_{g,[m]}(G)^{\nu }_{\underline {kr}}$ the G-stable compactification of $\mathcal {M}_{g,[m]}(G)^{\nu }_{\underline {kr}}$ . These are obtained from the original stacks by adding stable curves endowed with a stable G-action. We refer to [Reference Ekedahl6] and [Reference Bertin and Romagny2] § 4 and 6 for details.

2.1.2 The universal G-monodromy representation

We now consider $\mathcal {C}_{g,[m]}(G)_{\underline {kr}}$ the universal G-curve of genus g with m punctures and abstract Hurwitz data $\underline {kr}$ . We denote by $\mathcal {M}_{g,[m]+1}$ the stack of smooth projective curves with a degree m divisor and an additional marked point. We have an identification $\mathcal {C}_{g,[m]}(G)_{\underline {kr}}\simeq \mathcal {M}_{g,[m]}(G)_{\underline {kr}}\times _{\mathcal {M}_{g,[m]}} \mathcal {M}_{g,[m]+1}$ . The S-sections of $\mathcal {C}_{g,[m]}(G)_{\underline {kr}}$ are the elements of $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}(S)$ with the additional data of a section outside the marked points D; similarly, the universal punctured curve over $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }$ is given by the stack $\mathcal {C}_{g,[m]}(G)_{\underline {kr}}^{\nu }=\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }\times _{\mathcal {M}_{g,[m]}} \mathcal {M}_{g,[m]+1}$ .

One obtains the $\ell $ -universal G-monodromy representation of étale fundamental groups.

Theorem 2.2. Let g, $m \in \mathbb {N}$ such that $2g-2+m>0$ , G a finite cyclic group and $\underline {kr}$ a Hurwitz data with respect to g, m and G. Then there is an exact sequence

(2.1)

where X denotes a geometric curve of type $(g,m)$ . The $\ell $ -universal G-monodromy representation is the induced monodromy map

(2.2) $$ \begin{align} \Phi^{\ell}_{g,m}(G)_{\underline{kr}}\colon \pi_1(\mathcal{M}_{g,[m]}(G)_{\underline{kr}}) \longrightarrow \operatorname{Out} \pi_1^\ell(X), \end{align} $$

which is universal in the following sense: for any curve C over a connected $\mathbb {Q}$ -scheme S in $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}(S)$ and $\overline {\mathbb {Q}}$ -point $\overline {s}$ of S, the natural representation $\pi _1(S)\rightarrow \operatorname {Out} \pi ^{\ell }_1(C_{\overline {s}})$ factors through $\Phi ^{\ell }_{g,m}(G)_{\underline {kr}}$ . A similar result holds for $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }$ .

Note that the exact sequence above is independent of the choice of the geometric curve X of type $(g,m)$ . In the case $m\geq 1$ , one identifies $\pi _1^\ell (X)\simeq \widehat {F}^{\ell }_{2g+m-1}$ with the pro- $\ell $ completion of a free group; for general hyperbolic type $(g,m)$ , the group $\pi _1^\ell (X)$ is the pro- $\ell $ completion of a surface group.

Proof. The exactness of Equation (2.1) follows a classical argument on the geometric parts of the étale fundamental groups. Considering $\mathcal {C}_{g,[m]}(G)_{\underline {kr}}$ as the stack of m-marked curves with G-symmetries and an additional marked point, then erasing this additional point, provide a Birman point-erasing exact sequence between orbifold fundamental groups:

whose exactness is preserved after profinite completion: The profinite completion is always right-exact; the left-exactness follows from [Reference Anderson1] Proposition 4 and above by residual finitness of $\pi ^{top}(X)$ and the centerfreeness of the profinite completion of surface groups; see ibid. Proposition 8 and Proposition 18.

A similar argument provides the result for $\mathcal {M}_{g,[m]}(G)^{\nu }_{\underline {kr}}$ with ad hoc substitutions.

For a curve C over S as in Theorem 2.2, the $\ell $ -monodromy representation of C

$$\begin{align*}\varphi^{\ell}_C\colon \pi_1(S)\rightarrow \operatorname{Out} \pi_1^{\ell} (C_{\overline{s}}) \end{align*}$$

is obtained from the relative homotopy exact sequence as usual. Notice that the $\mathbb {Q}$ -scheme S also sits in a classical arithmetic-geometric homotopy exact sequence, so that $\pi _1(S)$ is naturally equipped with a projection map $p_S\colon \pi _1(S)\rightarrow G_{\mathbb {Q}}$ . We recall that, similarly, we have a canonical homomorphism $p\colon \pi _1(\mathcal {M}_{g,[m]}(G)_{\underline {kr}})\to G_{\mathbb {Q}}$ .

Lemma 2.3. For C a curve over a connected $\mathbb {Q}$ -scheme S represented by an S-point on $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}$ , resp. on $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }$ , we have the inclusion

$$\begin{align*}\mathbb{Q}^{\ell}_{g,[m]}(G)_{\underline{kr}}\subset \mathbb{Q}^{\ell}_C,\text{ resp. } \mathbb{Q}^{\ell}_{g,[m]}(G)_{\underline{kr}}^\nu\subset \mathbb{Q}^{\ell}_C. \end{align*}$$

The $\ell $ -monodromy fixed field $\mathbb {Q}^{\ell }_{g,[m]}(G)_{\underline {kr}}$ is furthermore obtained as the intersection of all the $\mathbb {Q}^{\ell }_C$ for such $C/S$ where S varies in the category of connected $\mathbb {Q}$ -schemes.

The field $\mathbb {Q}^{\ell }_{g,[m]}(G)_{\underline {kr}}$ can also be obtained as $\mathbb {Q}^{\ell }_{C_0}$ where $C_0= \mathcal {C}_{g,[m]}(G)_{\underline {kr}} \times _{\mathcal {M}_{g,[m]}(G)_{\underline {kr}}}\mathcal {M}_{g,[m]+m'}(G)_{\underline {kr}}$ is a curve over $S=\mathcal {M}_{g,[m]+m'}(G)_{\underline {kr}}$ with $m'$ large enough for S to be a scheme.

Proof. By the universality of the map $\Phi ^{\ell }_{g,[m]}(G)_{\underline {kr}}$ , we have a commutative diagram

where C denotes a hyperbolic curve of type $(g,m)$ , and where $\varphi ^{\ell }_C$ appears as the composition $\pi _1(S)\rightarrow \pi _1(\mathcal {M}_{g,[m]}(G)_{\underline {kr}}) \rightarrow \operatorname {Out} \pi _1^\ell (C)$ . The compatibility with the projections to $G_{\mathbb {Q}}$ ensures that we have $p_S(\operatorname {Ker} \varphi ^{\ell }_C) \subset p(\operatorname {Ker} \Phi ^{\ell }_{g,[m]}(G)_{\underline {kr}})$ and thus the inclusion. To prove the last point, by commutativity of the diagram, it suffices to show the existence of a curve C in $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}(S)$ such that the induced map $\pi _1(S)\rightarrow \pi _1(\mathcal {M}_{g,[m]}(G)_{\underline {kr}})$ is surjective. This is done by taking $C_0= \mathcal {C}_{g,[m]}(G)_{\underline {kr}} \times _{\mathcal {M}_{g,[m]}(G)_{\underline {kr}}}\mathcal {M}_{g,[m]+m'}(G)_{\underline {kr}}$ over $S=\mathcal {M}_{g,[m]+m'}(G)_{\underline {kr}}$ with $m'$ large enough for S to be a scheme.

The case of $\mathbb {Q}^{\ell }_{g,[m]}(G)_{\underline {kr}}^\nu $ is similar after replacing $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}$ by $\mathcal {M}_{g,[m]}(G)^{\nu }_{\underline {kr}}$ .

2.1.3 The case of Hurwitz data of étale type

Let us now relate the general situation to the one where the divisor of marked points D contains the ramification divisor R of the G-action, a property that we recall can be seen directly on the abstract Hurwitz data.

By base change to an algebraically closed field and reading of the $\underline {kr}$ data, one notices that the divisor $R\cup D$ is finite étale over S for a curve $C/S$ as before.

Proof. As everything is locally constant on the base, it is enough to treat the case where S is the spectrum of an algebraically closed field. By definition of $\underline {kr}$ , the degree $\deg R=N$ of the ramification divisor is the length of k. Furthermore, since the degree of $R\cap D$ is given by $\sum \nolimits _{i=1}^{n-1} \operatorname {gcd}(i,n)\cdot \underline {r}(i)$ , we have the formula

$$\begin{align*}\deg R\cup D=m+N-\sum\limits_{i=1}^{n-1} \operatorname{gcd}(i,n)\cdot r(i), \end{align*}$$

which is entirely determined by m, $\underline {kr}$ and $G=\mathbb {Z}/n\mathbb {Z}$ .

For an abstract Hurwitz data $\underline {kr}$ , we introduce $\underline {kr}^{et}$ as the minimal associated Hurwitz data such that the ramified points are contained in the marked divisor – thata is, minimal in the sense that the new marked divisor is the smallest one containing D and R – and which is thus defined by

$$\begin{align*}\begin{cases} r^{et}(0)=r(0) &\\ r^{et}(i)=\operatorname{Card} \{j\in \{1,\dots, N\} \mid k(j)=i\}, & i\geq 1. \end{cases}\end{align*}$$

Proposition 2.5. There is a natural map of stacks

$$\begin{align*}\mathcal{M}_{g,[m]}(G)^{\nu}_{\underline{kr}}\longrightarrow \mathcal{M}_{g,[m+s]}(G)_{\underline{kr}^{et}}^{\nu}, \end{align*}$$

where $\underline {r}^{et}$ and $s=\deg R-\deg R\cap D$ can be explicitly determined as above.

Theorem 2.6. We have the following inclusion of $\ell $ -monodromy fixed fields:

(2.3) $$ \begin{align} \mathbb{Q}^{\ell}_{g,[m]}(G)^\nu_{\underline{kr}}\subset \mathbb{Q}^{\ell}_{g,[m+s]}(G)^\nu_{\underline{kr}^{et}}. \end{align} $$

Proof. Let $\sigma \in p(\operatorname {Ker} \Phi ^{\ell }_{g,[m+s]}(G)_{\underline {kr}^{et}})\subset G_{\mathbb {Q}}$ . By Lemma 2.3, there is a connected $\mathbb {Q}$ -scheme S and a curve $(C,D)$ over S represented by an S-point on $\mathcal {M}_{g,[m+s]}(G)_{\underline {kr}^{et}}$ such that $\sigma $ has a lift $\tau $ in the kernel of the map

where X denotes a hyperbolic curve of type $(g,m)$ . The divisor D admits a decomposition $D=D^{un}\cup D^{ram}$ where $D^{un}$ is given by the unramified marked points and $D^{ram}$ by the ramified marked points. By definition of the component $\underline {r}^{et}$ of $\underline {kr}^{et}$ , the divisor $D^{ram}$ corresponds to all the ramified points. The divisor $D^{ram}$ splits into a disjoint union of geometrically irreducible divisors over a finite étale extension $S'=S_K$ of S where K is defined by the property that $G_K$ stabilizes each geometric component of $D^{ram}$ . In particular, $\pi _1(S')$ contains the subgroup $\{ \alpha \in \pi _1(S) \mid p_S(\alpha )\in G_K\}$ , which contains $\tau $ by construction. We can thus assume that $S=S'$ .

By removing some chosen orbits of ramified points in $D^{ram}$ according to the data given by $\underline {r}$ , we can form a divisor $D'=D^{un}\cup D^{ram'}$ such that $(C,D')$ gives an S-point of $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}$ . Hence, it is sufficient to show that $\sigma $ is the image of an element of $\pi _1(S)$ that acts trivially on the pro- $\ell $ -fundamental group of a geometric fiber $C_{\overline {s}}\setminus D^{\prime }_{\overline {s}}$ of $C\setminus D'$ . This now comes from the fact that the outer actions of $\pi _1(S)$ on $\pi _1^{\ell } (C_{\overline {s}}\setminus D_{\overline {s}})$ and $\pi _1^{\ell }(C_{\overline {s}}\setminus D^{\prime }_{\overline {s}})$ are compatible with the canonical surjection $\pi _1^{\ell } (C_{\overline {s}}\setminus D_{\overline {s}})\rightarrow \pi _1^{\ell }(C_{\overline {s}}\setminus D^{\prime }_{\overline {s}})$ .

2.2.1 First monodromy fixed fields comparisons

The comparison via the forgetful functor $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu } \rightarrow \mathcal {M}_{g,[m]}(G)_{\underline {kr}} \rightarrow \mathcal {M}_{g,[m]}$ is now straightforward.

Proof. Let C be a curve over $\overline {\mathbb {Q}}$ represented on $\mathcal {M}_{g,[m]}(G)^{\nu }_{\underline {kr}}$ . First, see that the sequence of maps

$$\begin{align*}\pi_1(\mathcal{M}_{g,[m]}(G)^{\nu}_{\underline{kr}}) \rightarrow \pi_1(\mathcal{M}_{g,[m]}(G)_{\underline{kr}}) \rightarrow \pi_1(\mathcal{M}_{g,[m]}) \rightarrow \operatorname{Out} \pi_1^{\ell}(C) \end{align*}$$

induces a sequence

$$\begin{align*}\operatorname{Ker} \Phi^{\ell}_{g,[m]}(G)_{\underline{kr}}^\nu \rightarrow \operatorname{Ker} \Phi^{\ell}_{g,[m]}(G)_{\underline{kr}} \rightarrow \operatorname{Ker} \Phi^{\ell}_{g,[m]}, \end{align*}$$

where the second map is obtained by considering the following commutative diagram with exact rows

and the first map is obtained in a similar way.

By applying the canonical projections to $G_{\mathbb {Q}}$ , and Lemma 2.7 for $\mathbb {Q}^{\ell }_{g,m}=\mathbb {Q}^{\ell }_{g,[m]}$ , one obtains the desired sequence of inclusions.

Corollary 2.9. With the notations of Theorem 2.6 we have

$$\begin{align*}\mathbb{Q}^{\ell}_{0,3} \subset \mathbb{Q}_{g,[m]}^{\ell}(G)_{\underline{kr}}\subset \mathbb{Q}_{g,[m]}^{\ell}(G)_{\underline{kr}}^\nu \subset \mathbb{Q}_{g,[m+s]}^{\ell}(G)_{{\underline{kr}}^{et}}^\nu. \end{align*}$$

Remark 2.10. In Proposition 2.8, there is no difficulty to move to the weight version of Oda’s conjecture, and we get, for all $(g,m)$ of hyperbolic type, any compatible Hurwitz data $\underline {kr}$ , and all weight w

$$\begin{align*}\mathbb{Q}_{g,m}^{\ell}(w) \subset \mathbb{Q}_{g,[m]}^{\ell}(G)_{\underline{kr}}(w)\subset \mathbb{Q}_{g,[m]}^{\ell}(G)_{\underline{kr}}^\nu(w). \end{align*}$$

In contrast, see Remark 2.13 2.

2.2.2 Quotient vs G-special loci fixed fields

The quotient map $\delta \colon \mathcal {M}_{g,[m]}[G]_{\underline {kr}} \to \mathcal {M}_{g',[m']}$ defined by $(C,D,\iota ) \mapsto (C/\iota (G),D/\iota (G))$ allows the comparison of $\ell $ -monodromy fixed fields. We first remark that the map $\delta $ is well defined at the level of the stack $\mathcal {M}_{g,[m]}(G)^{\nu }_{\underline {kr}}$ , since $\delta $ is equivariant under the action of $\operatorname {Aut} G$ .

Therefore, we have a map $\delta \colon \mathcal {M}_{g,[m]}(G)^{\nu }_{\underline {kr}}\rightarrow \mathcal {M}_{g',[m']}$ that fits in a commutative square

where the map on the left is induced by the quotient in the same way. For a curve $\overline {X}$ over $\overline {\mathbb {Q}}$ represented on $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }$ , let us denote $\overline {Y}$ the quotient proper curve, and X, Y their open counterparts. This leads to a commutative diagram with exact rows

which in turn provides an $\ell $ -monodromy representation

$$\begin{align*}\Phi_{g',[m']}^{\ell}(\delta G)_{\underline{kr}}^\nu\colon\pi_1(\mathcal{M}_{g,[m]}(G)_{\underline{kr}}^{\nu}) \rightarrow \operatorname{Out} \pi_1^{\ell}(Y) \end{align*}$$

in the quotient curve, so that one obtains

(2.4) $$ \begin{align} p(\operatorname{Ker} \Phi_{g',[m']}^{\ell}(\delta G)^\nu_{\underline{kr}})\subset p(\operatorname{Ker} \Phi_{g',[m']}^{\ell})\text{ or equivalently } \mathbb{Q}^{\ell}_{g',[m']}\subset \mathbb{Q}^{\ell}_{g',[m']}(\delta G)^\nu_{\underline{kr}} \end{align} $$

where $\mathbb {Q}^{\ell }_{g',[m']}(\delta G)_{\underline {kr}}^\nu $ denotes the fixed field of the subgroup $p(\operatorname {Ker} \Phi _{g',[m']}^{\ell }(\delta G)_{\underline {kr}}^\nu )$ as usual. Lemma 2.7 then gives $\mathbb {Q}^{\ell }_{g',m'}\subset \mathbb {Q}^{\ell }_{g',m'}(\delta G)_{\underline {kr}}^\nu $ .

In the rest of this section, we finally establish that $\mathbb {Q}^{\ell }_{g',[m']}(\delta G)_{\underline {kr}}^\nu =\mathbb {Q}^{\ell }_{g,m}(G)_{\underline {kr}}^\nu $ in the case where $X\rightarrow Y$ is a finite étaleFootnote ³ geometric cover and where $G\simeq \mathbb {Z}/\ell ^n\mathbb {Z}$ . The finite étale condition guarantees that the inclusion $\iota \colon \pi _1(X)\rightarrow \pi _1(Y)$ induces an inclusion at the pro- $\ell $ completion level $\iota ^{\ell }\colon \pi _1^{\ell }(X)\rightarrow \pi _1^{\ell }(Y)$ .

Denoting by $\operatorname {Aut} \pi _1^{\ell } (Y)^{X}$ the subgroup of the automorphisms of $\pi _1^{\ell }(Y)$ that stabilizes $\pi _1^{\ell }(X)$ , we thus obtain a big commutative diagram

By tracking the conjugation action of $\pi _1(\mathcal {C}_{g,[m]}(\mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^\nu )$ on $\pi _1^{\ell }(X)$ on the first square of the back face, we see that the dashed arrow $\pi _1(\mathcal {C}_{g,[m]}(\mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^\nu ) \rightarrow \operatorname {Aut} \pi _1^{\ell }(X)$ factors by $\operatorname {Aut} \pi _1^{\ell } (Y)^{X}$ through its conjugation action on $\pi _1^{\ell }(Y)$ and the restriction map.

Theorem 2.11. For $(g,m)$ of hyperbolic type, and $\underline {kr}$ an abstract Hurwitz data of étale type associated to $\mathbb {Z}/\ell ^n\mathbb {Z}$ with quotient topological data $(g',m')$ , we have the following inclusions of $\ell $ -monodromy fixed fields:

$$\begin{align*}\mathbb{Q}^{\ell}_{g',[m']} \subset \mathbb{Q}^{\ell}_{g,[m]}(\mathbb{Z}/\ell^n\mathbb{Z})_{\underline{kr}}^\nu. \end{align*}$$

Proof. Since $\mathbb {Q}^{\ell }_{g',[m']}\subset \mathbb {Q}^{\ell }_{g',[m']}(\delta G)_{\underline {kr}}^\nu $ , by Equation (2.4), it suffices to show the equality $\mathbb {Q}^{\ell }_{g',[m']}(\delta \mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^\nu =\mathbb {Q}^{\ell }_{g,[m]}(\mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^\nu $ . We do so by introducing some intermediate fields as can be seen in Diagram (2.5).

We first have a map

$$\begin{align*}\Psi\colon \pi_1^{\ell}(\mathcal{C}_{g,[m]}(\mathbb{Z}/\ell^n\mathbb{Z})_{\underline{kr}}^\nu)\longrightarrow \operatorname{Aut} \pi_1^{\ell} (X) \times \operatorname{Aut} \pi_1^{\ell} (Y)^{X} \end{align*}$$

such that $\Phi _{g',[m']}^{\ell }(\delta \mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^{\nu }$ and $\Phi _{g,[m]}^{\ell }(\mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^{\nu }$ are obtained by composing $\Psi $ with the projections and quotients by the inner automorphisms. One checks directly that $\operatorname {Inn} \iota ^{\ell } \pi _1^{\ell }(X)$ is a normal subgroup of $\operatorname {Aut} \pi _1^{\ell } (Y)^{X}$ . We thus have a quotient map

$$\begin{align*}\operatorname{Aut} \pi_1^{\ell} (X) \times \operatorname{Aut} \pi_1^{\ell} (Y)^{X} \longrightarrow \operatorname{Out} \pi_1^{\ell} (X) \times \operatorname{Aut} \pi_1^{\ell} (Y)^{X}/\operatorname{Inn} \iota^{\ell}( \pi_1^{\ell} (X)), \end{align*}$$

which by composition with $\Psi $ results in a map

$$\begin{align*}S^{\ell}\colon\pi_1(\mathcal{M}_{g,[m]}(\mathbb{Z}/\ell^n\mathbb{Z})^{\nu}_{\underline{kr}})\longrightarrow \operatorname{Out} \pi_1^{\ell} (X) \times \operatorname{Aut} \pi_1^{\ell} (Y)^{X}/\operatorname{Inn} \iota^{\ell}( \pi_1^{\ell} (X)). \end{align*}$$

Considering the quotient map $p_{Y}\colon \operatorname {Aut} \pi _1^{\ell } (Y)^{X}/\operatorname {Inn} \iota ^{\ell }( \pi _1^{\ell } (X)) \rightarrow \operatorname {Out} \pi _1^{\ell } (Y)$ and the canonical projections $p_i$ , $i=1,2$ , of the product $\operatorname {Out} \pi _1^{\ell } (X) \times \operatorname {Aut} \pi _1^{\ell } (Y)^{X}/\operatorname {Inn} \iota ^{\ell }( \pi _1^{\ell } (X))$ , one observes that by construction,

$$\begin{align*}\Phi_{g,[m]}^{\ell}(\mathbb{Z}/\ell^n\mathbb{Z})_{\underline{kr}}^\nu=p_1 \circ S^{\ell} \text{ and } \Phi_{g',[m']}^{\ell}(\delta \mathbb{Z}/\ell^n\mathbb{Z})_{\underline{kr}}^\nu=p_{Y} \circ p_2 \circ S^{\ell}. \end{align*}$$

By setting $\mathbb {Q}_S^{\ell }$ to be the fixed field of $p(\operatorname {Ker} S^{\ell })$ and $\mathbb {Q}_{S_Y}^{\ell }$ to be the fixed field of $p(\operatorname {Ker} p_2\circ S^{\ell })$ , we obtain the following diagram of inclusions of $\ell $ -monodromy fixed fields:

(2.5)

It remains to show some equalities. First, $\mathbb {Q}_{S_Y}^{\ell }=\mathbb {Q}_{S}^{\ell }=\mathbb {Q}_{g,[m]}^{\ell }(\mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^\nu $ since by the inclusion $\iota ^{\ell }$ , we have that $p_2$ restricted to the image of $S^{\ell }$ is injective, and by slimness of $\pi _1^{\ell }(X)$ (see Section 3.1.2 for a definition), we have that $p_1$ restricted to the image of $S^{\ell }$ is also injective.

For the remaining equality $\mathbb {Q}_{S_Y}^{\ell }=\mathbb {Q}_{g',[m']}^{\ell }(\delta \mathbb {Z}/\ell ^n\mathbb {Z})_{\underline {kr}}^\nu $ , we consider the stack inertia injection $G\subset \mathcal {I}_{\mathcal {M},\overline {x}}\hookrightarrow \pi _1(\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu })$ as in [Reference Noohi26], where $\overline {x}\in \mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu }( \overline {K})$ corresponds to the curve $\overline {X}$ , and where the injectivity follows from ibid. Theorem 6.2 with the arguments of Remark 4.4 of [Reference Collas and Maugeais5]. The injection $G\hookrightarrow \pi _1(\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu })$ can be shown to be independent of the choice of point $\overline {x}$ and maps, through our construction, $G=\mathbb {Z}/\ell ^n\mathbb {Z}$ isomorphically to the quotient $\operatorname {Inn} \pi _1^{\ell }(Y)/\operatorname {Inn} \iota ^{\ell }( \pi _1^{\ell } (X)$ . Let $\sigma \in p(\operatorname {Ker} p_{Y}\circ p_2\circ S^{\ell })$ , which lifts to $\tau \in \pi _1(\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu })$ by definition which in turn maps to $h\in \operatorname {Inn} \pi _1^{\ell }(Y)/\operatorname {Inn} \iota ^{\ell }( \pi _1^{\ell } (X))\simeq G$ . The element $h^{-1}\tau \in \pi _1(\mathcal {M}_{g,[m]}(G)_{\underline {kr}}^{\nu })$ is in $\operatorname {Ker} p_2\circ S^{\ell }$ and verifies $p(h^{-1}\tau )=p(\tau )$ . Thus, we have proven that $p(\operatorname {Ker} \Phi ^{\ell }_{g',[m']}(\delta \mathbb {Z}/\ell ^n\mathbb {Z})^\nu ) \subset p(\operatorname {Ker} S_Y^{\ell })$ , and the reverse inclusion is given by Diagram 2.5.

By Theorem A of [Reference Nakamura23], Theorem 4.3 of [Reference Matsumoto20] and Theorem 3.6 [Reference Takao31], there is an inclusion $\mathbb {Q}^{\ell }_{0,3}\subset \mathbb {Q}^{\ell }_{g,m}$ for all $(g,m)$ of hyperbolic type. Thus, we can complete the diagram Diagram (2.5) of field inclusions as follows.

Corollary 2.12. For $(g,m)$ of hyperbolic type and $\underline {kr}$ compatible Hurwitz data, we have a diagram of inclusions of fields

Establishing the G-special loci Oda’s conjecture in the case of $G=\mathbb {Z}/\ell \mathbb {Z}$ – that is, that $\mathbb {Q}^{\ell }_{g,m}(\mathbb {Z}/\ell \mathbb {Z})_{\underline {kr}}$ is independent of the topological and Hurwitz data and indeed equal to $\mathbb {Q}^{\ell }_{0,3}$ – is thus reduced to establishing the last inclusion $\mathbb {Q}^{\ell }_{g,[m]}(\mathbb {Z}/\ell \mathbb {Z})_{\underline {kr}}^\nu \subset \mathbb {Q}^{\ell }_{0,3}$ . We proceed to do so in the rest of this paper by developing for G-special loci a refinement of Ihara-Nakamura’s degeneration method used in their original proof of the containment $\mathbb {Q}^{\ell }_{g,m}\subset \mathbb {Q}^{\ell }_{0,3}$ in [Reference Ihara and Nakamura17].

3.1.1 Tangential base points on curves

We follow the elementary definition of tangential base point of the survey [Reference Nakamura24] Section I; that is, for X connected smooth curve over a field K, a K-tangential base point v on X is a morphism $v\colon \operatorname {Spec} K((t))\rightarrow X$ (see ibid. Definition 1.1).

The key feature of such a choice of a K-rational tangential base point is, via the field of Puiseux series $\overline {K}\{\{t\}\}$ , to provide at once a geometric base point for the étale fundamental group of X and a section to the related homotopy exact sequence:

(3.1)

In other words, one obtains a specific $G_K$ -action $\varphi _{\vec {v}}$ on $\pi _1(X_{\overline {K}},v)$ given by conjugation which lifts the canonical outer Galois action

(3.2)

and can be chosen to reflect some good arithmetic properties of X. More explicitly, the $G_K$ action $\varphi _v$ is given, via the function fields of X, by the action on the coefficients of the formal series in $\overline {K}\{\{t\}\}$ ; see also Equation (3.3) below.

For $X=\mathbb {P}_{\mathbb {Q}}^1\setminus \{ 0,1, \infty \}$ , let us denote the set of fiber functor associated, as in 2 above, to Deligne-Ihara’s original $\mathbb {Q}$ -tangential base points by

$$\begin{align*}\mathbb{B}=\{\vec{01}, \vec{0\infty}, \vec{10}, \vec{1\infty}, \vec{\infty 1}, \vec{\infty 0}\}, \end{align*}$$

where, for example, $\vec {01}\colon \operatorname {Spec} \mathbb {Q}((t))\to \mathbb {P}_{\mathbb {Q}}^1\setminus \{ 0,1, \infty \}$ and $\vec {0\infty }\colon \operatorname {Spec} \mathbb {Q}((-t))\to \mathbb {P}_{\mathbb {Q}}^1\setminus \{ 0,1, \infty \}$ , and refer to the Appendix of [Reference Ihara16] for further details on the associated $G_{\mathbb {Q}}$ -action. For our study, the main property of these tangential base points is that

(3.4) $$ \begin{align} \operatorname{Ker} \varphi^{\ell}_{\vec{ij}}= \operatorname{Ker} \varphi^{\ell}_{\mathbb{P}^1\setminus \{ 0,1, \infty\}}\quad \text{for every } \vec{ij}\in \mathbb{B}. \end{align} $$

While even the simplest rational scaling of the parameter (see, for example, $\vec {01}$ vs $\vec {0\infty }$ above or [Reference Tsunogai33] Section 1.5), changes the tangential Galois action, we have the following Galois invariance property.

Proof. Let $x\in \overline {X}$ be a closed K-rational point. It suffices to show that if t and $t'$ are both uniformizers at x (i.e., we have $\widehat {{\mathcal O}}_{\overline {X},x}\simeq K[[t]] \simeq K[[t']]$ and $t'=t(1+tF)$ in $K[[t]]$ with $F\in K[[t]]$ ), then the isomorphism $\delta _{t',t}\colon \overline {K}\{\{t'\}\} \rightarrow \overline {K}\{\{t\}\}$ is $G_K$ -equivariant. But as $\delta _{t',t}$ is defined by ${t'}^{\frac {1}{N}} \mapsto t^{\frac {1}{N}}(1+tF)^{\frac {1}{N}}$ for $N\geq 0$ , this comes from the fact that $(1+tF)^{\frac {1}{N}}=G_N$ with $G_N\in K[[t]]$ by the series expansion of $(1+tF)^{\frac {1}{N}}$ .

Indeed, let $v_t$ (resp. $v_t'$ ) be the tangential base points given by t (resp. $t'$ ) and denote by $\varphi _{v_t}$ (resp. $\varphi _{v_t'}$ ) the associated tangential $G_K$ -action. Let $\sigma \in G_K$ and consider a meromorphic function $f=\sum \limits _{k} a_k {t'}^{\frac {k}{N}}\in \mathcal {M}_{v_{t'}}\subset K\{\{t'\}\}$ . Then we have

$$ \begin{align*} \sigma^{-1}_{v_t}\circ \delta_{t',t} \circ \sigma_{v_{t'}}(f) &= \sigma^{-1}_{v_t} \circ \delta_{t',t}( \sum\limits_{k} \sigma(a_k) {t'}^{\frac{k}{N}}) \\ &= \sigma^{-1}_{v_t}( \sum\limits_{k} \sigma(a_k) {t}^{\frac{k}{N}}G_N) \\ &=\sum\limits_{k}a_k {t}^{\frac{k}{N}}G_N; \end{align*} $$

that is,

$$\begin{align*}\sigma^{-1}_{v_t}\circ \delta_{t',t} \circ \sigma_{v_{t'}}(f) = \delta_{t',t}(f), \end{align*}$$

which shows that $\sigma _{v_t}^{-1}\circ \delta _{t',t}\circ \sigma _{v_{t'}}= \delta _{t',t}$ and thus $\varphi _{v_t'}=\varphi _{v_t}\circ \delta _{t',t}$ as intended.

3.1.2 Monodromy fixed fields and tangential base points

We recall that a profinite group is said to be slim if any of its open subgroup has trivial centralizer. Examples of slim groups include the absolute Galois group of rational numbers and the pro- $\ell $ fundamental group of hyperbolic curves; see [Reference Mochizuki and Tamagawa21] Proposition 1.4.

We record the following inclusions between the $\ell $ -monodromy fixed fields of the various tangential and non-tangential Galois actions in the case of étale coverings.

Proof. The homotopy exact sequence for X and Y and the covering map $\psi $ gives the diagram

that is commutative by definition of $\psi (v)$ and the étaleness of $\psi $ . One thus recovers, via $\pi _1(Y,\vec {\psi (v)}) \to \operatorname {Aut} \pi _1^{\ell }(Y_{\overline {K}},\vec {\psi (v)})$ whose image stabilizes $\pi _1^\ell (X_{\overline {K}},\vec {v})$ , the monodromy action $\varphi ^{\ell }_{\vec {v}}$ as the composition

$$\begin{align*}G_K\rightarrow \operatorname{Aut} \pi_1^{\ell}(Y_{\overline{K}},\vec{\psi(v)})^{X_{\overline{K}}} \rightarrow \operatorname{Aut} \pi_1^{\ell}(X_{\overline{K}},\vec{v}), \end{align*}$$

which leads to the inclusion given in 1. In the case of slimness, the right restriction map is injective, which yields the equality.

The remaining inclusions are obtained by adding the following commutative diagram:

The inclusions of 2 are thus direct by the diagram and the definitions of the maps involved. To see the remaining inclusion, we remark that by slimness $\operatorname {Ker} \varphi ^{\ell }_X= \operatorname {Ker} d_X\circ \varphi ^{\ell }_{\vec {\psi (v)}}$ , and the inclusion follows as $\varphi ^{\ell }_Y= d_Y\circ d_X\circ \varphi ^{\ell }_{\vec {\psi (v)}}$ .

3.2.1 Tangential base points and Galois actions comparisons

The set of curves $\{C_r \mid r=1,\dots , \ell -2\}$ admits an $\mathfrak {S}_3$ -action that is compatible with the $\mathbb {Z}/\ell \mathbb {Z}$ -action and, in particular, with the $\mathfrak {S}_3$ -action on $\mathbb {P}^1_K\setminus \{0,1,\infty \}$ through the quotient map; see [Reference Seyama30] Corollary 2.5. This allows us to define, for every r, the tangential base points on $C_r'=C_r\setminus \{P_{r,0}, P_{r,1}, P_{r,\infty }\} $ at the punctures by doing so at $P_{r,1}$ .

Indeed, for $\sigma \in \mathfrak {S}_3$ , we have $\sigma (P_{r,1})=P_{\sigma (r),\sigma (1)}$ so that for every $r\in \{1,\dots , \ell -1\}$ and every $P\in \{P_{r,0}, P_{r,1}, P_{r,\infty }\}$ , there is an element $\sigma \in \mathfrak {S}_3$ such that P is the image of $P_{r,1}$ for some r. Now, the smooth affine open $U=C_r\setminus \{P_{r,0}, P_{r,\infty }\}$ is given by $U= \operatorname {Spec} K[x,y, \frac {1}{y}]$ where $x^{\ell }=y^r(y-1)$ . Looking at the equation, we see that x is a uniformizer at $P_{r,1}$ , and we have $\psi (x)=x^{\ell }=y^r(y-1)$ where $\psi $ is the quotient map to $\mathbb {P}^1_K$ .

As stated before, by using the $\mathfrak {S}_3$ -action on the previous subset of Matsumoto-Seyama curves, we obtain tangential base points $T^r_{ij}$ for $i,j\in \{0,1,\infty \}$ , whose set of associated fiber functors on the categories of finite étale covers $\mathrm {Et}(C_r')$ we denote by

$$\begin{align*}\mathbb{B}^r=\{\overrightarrow{T^r_{ij}} \mid i,j\in \{0,1,\infty\}\},\text{ for } r\in \{1,\dots, \ell-2\}. \end{align*}$$

These tangential base points induce the same tangential $G_K$ -actions on the fundamental group of $\mathbb {P}^1_{\overline {K}} \setminus \{0,1,\infty \}$ given by Deligne-Ihara as in Lemma 3.4.

In what follows, r will be omitted from notations when clear from context.

3.2.2 The fixed field of the Matsumoto-Seyama curves

For $r=0$ , we consider the covering of $\mathbb {P}^1_K$ given by

$$\begin{align*}C_0\colon x=y^{\ell} \text{ with usual }\mathbb{Z}/\ell\mathbb{Z}\text{-action having two ramified points } 0 \text{ and }\infty \end{align*}$$

with abstract Hurwitz data $\underline {k}=(1,-1)$ . The $\ell +2$ -marking is given by the two ramified points and by the unramified points $P_1,\dots , P_{\ell }$ of the fiber at $1$ .

We further set

$$\begin{align*}\mathbb{B}^0=\{\overrightarrow{T^0_{0\infty}}, \overrightarrow{T^0_{\infty0}}\} \text{ and } C_0'=C_0\setminus \{0,\infty, P_1,\dots, P_{\ell}\}, \end{align*}$$

where the fiber functors $\overrightarrow {T^0_{0\infty }}$ and $\overrightarrow {T^0_{\infty 0}}$ are induced by the tangential base points associated to the parameter x and $\frac {-1}{x}$ , and which are direct lifts of the Deligne tangential base points $\overrightarrow {0\infty }$ and $\overrightarrow {\infty 0}$ .

We finish this section by showing that the $\ell $ -monodromy fixed field of the Matsumoto-Seyama curves is $\mathbb {Q}^{\ell }_{0,3}$ .

Proof. For $r\in \{0,\dots , \ell -2\}$ given, it follows from Lemma 3.3 that $\operatorname {Ker} \varphi ^{\ell }_{\overrightarrow {T_{0\infty }}}= \operatorname {Ker} \varphi ^{\ell }_{\vec {01}}$ as $\pi _1(\mathbb {P}^1_K\setminus \{0,1,\infty \}, \vec {0\infty })$ is slim. From the same lemma, we also get the inclusions

$$\begin{align*}\operatorname{Ker} \varphi^{\ell}_{\overrightarrow{T_{0\infty}}} \subset \operatorname{Ker} \varphi^{\ell}_{C_r'} \subset \operatorname{Ker} \varphi^{\ell}_{\mathbb{P}^1\setminus\{0,1,\infty\}}. \end{align*}$$

Since the two outmost terms are equal as in Equation (3.4), it follows that $\operatorname {Ker} \varphi ^{\ell }_{C_r'}=\operatorname {Ker} \varphi ^{\ell }_{\mathbb {P}^1\setminus \{0,1,\infty \}}$ , thus the desired equality.

3.3.1 Gluing curves with G-actions

While the gluing, or clutching, of marked points for stable curves can be found in details in [Reference Knudsen18], the similar gluing for curves with G-action requires an additional constraint as follows.

Consider two curves $C_r$ and $C_{r'}$ with $r,r'\in \{1,\dots , \ell -2\}.$ The gluing of both curves at the points $P_{r,1}$ and $P_{r',1}$ can be constructed as the union

$$\begin{align*}C_{r,r'}^{1,1}=C_r \times \{P_{r',1}\} \cup C_{r'}\times \{ P_{r,1}\} \text{ in the fiber product } C_r\times_{ \operatorname{Spec} K} C_{r'}. \end{align*}$$

The result of the gluing is a curve $X^0$ of genus $\ell -1$ with $2$ irreducible components and $4$ marked points given by $\{P_{r,0}, P_{r,\infty }, P_{r',0}, P_{r',\infty } \}$ , that is equipped with a $\mathbb {Z}/\ell \mathbb {Z}$ -action by pullback of the action on the product.

For $X^0$ to be a G-stable curve, the G-actions must be chosen such that Hurwitz data at the points $P_{r,1}$ and $P_{r',1}$ have opposite characters (see [Reference Bertin and Romagny2] Section 4.1), which is easily done by choosing that $G=\mathbb {Z}/\ell \mathbb {Z}$ acts by $x\mapsto \overline {\zeta _{\ell }}x$ on $C_{r'}$ and by $x\mapsto \zeta _{\ell }x$ on $C_r$ . The same construction can be made by gluing together any two ramified points $P_{r,i}$ and $P_{r',j}$ into a curve $C_{r,r'}^{i,j}$ , where $i,j\in \{0,1,\infty \}$ denotes which points are glued.

Note that the $\mathfrak {S}_3$ -action on the curves $(C_r)_{r\in \{1,\dots , \ell -2\}}$ extends naturally to a $\mathfrak {S}_3\times \mathfrak {S}_3$ -action on the fiber products $(C_r\times _{ \operatorname {Spec} K} C_{r'})_{r,r'\in \{1,\dots , \ell -2\}}$ of such curves. One checks that this action stabilizes the closed subsets $(C_{r,r'}^{i,j})_{r,r'\in \{1,\dots \ell -2\},~i,j\in \{0,1,\infty \}}$ globally; that is, for $\sigma , \tau \in \mathfrak {S}_3\times \mathfrak {S}_3$ , we have $(\sigma ,\tau )\cdot C_{r,r'}^{i,j}=C_{\sigma (r), \tau (r')}^{\sigma (i), \tau (j)}$ . It results that the affine neighborhood of $C_{r,r'}^{i,j}$ with the $4$ marked points removed is, for some r, always isomorphic to

$$\begin{align*}C_{r,r'}^{1,1}\setminus \{P_{r,0}, P_{r,\infty}, P_{r',0}, P_{r',\infty} \} = \operatorname{Spec} K[x,y,x',y'][\frac{1}{y},\frac{1}{y'}]/(xx'), \end{align*}$$

which serves as a model for the construction of the $U_\mu $ s as in Section 3.4.1.

3.3.2 Diagrams of Matsumoto-Seyama curves

We will build our $\mathbb {Z}/\ell \mathbb {Z}$ -stable $C_r$ -diagrams from the two types of Matsumoto-Seyama curves $C_r$ of Section 3.2. Recall that the genus $0$ curves have two distinguished rational sections given by the ramified points, and that the genus $(\ell -1)/2$ ones have three.

The isomorphisms of 1 come with choices of variables $x_{\lambda },y_{\lambda }$ and choices of tangential base points $T^{\lambda }_{ij}\colon \operatorname {Spec} K((t)) \rightarrow X^0_{\lambda }$ with the properties of the ones defined in Section 3.2. The corresponding set of fiber functors will be denoted by $\mathbb {B}^r_{\lambda }$ . We will omit $\lambda $ and r from the notations when it is clear from context.

The following three kinds of $\mathbb {Z}/\ell \mathbb {Z}$ -stable $C_r$ -diagrams will be used as basic building blocks for the special fiber of our $1$ -parameter deformation families.

• • Seyama curve (Figure 1a): a curve of genus $g=(\ell -1)/2$ with $\nu =3$ ramified points and $\underline {k}$ free;

  Figure 1

  Elementary building blocks of $\mathbb {Z}/\ell \mathbb {Z}$ -stable $C_r$ -diagrams.

  

• • A $\mathbb {Z}/\ell \mathbb {Z}$ -curve of genus $0$ (Figure 1b): a curve of genus $g=0$ with $\nu =2$ ramified points, $\ell $ unramified points and $\underline {k}=(1,-1)$ ;

• • A $2$ -Seyama curve (Figure 1c): a curve of genus $g=\ell $ with $\nu =2$ ramified points and $\underline {k}=(1,-1)$ . These curves are obtained by gluing two Seyama curves twice. The correspond data of the $\mathbb {Z}/\ell \mathbb {Z}$ -stable $C_r$ -diagram is $\Lambda =\{1,2\}$ where $X^0_{1}$ , $X^0_{2}$ are Matsumoto-Seyama curves isomorphic to $C_r$ with $r\in \{1,\dots , \ell -2\}$ and Hurwitz data $\underline {k}=(1,a,-(1+a))$ with ramified points $\{\nu _1,\nu _2,\nu _3\}$ and $\underline {k'}=(-1,-a,1+a)$ with ramified points $\{\nu ^{\prime }_1,\nu ^{\prime }_2,\nu ^{\prime }_3\}$ . We have $M=\{1,2\}$ , and the pairs $P^0_{1}$ , $P^0_{2}$ are, respectively, $(\nu _1,\nu ^{\prime }_1)$ and $(\nu _2,\nu ^{\prime }_2)$ . The remaining set of distinguished sections is $\{\nu _3,\nu ^{\prime }_3\}= \{Q^0_v\}_{v\in N}$ .

In Figure 1 above, the bold points represent ramified points under the G-action, the lined markings represent the unramified points ( $\ell =3$ here), and the dashed lines represent the glued points. The hook at the end of the genus $g=(\ell -1)/{2}$ curves is to differentiate them from the genus $0$ ones, which are represented by straight lines.

Recall that we denote by $\overline {\mathcal {M}}_{g,[m]}(G)^{\nu }$ the stable compactification of $\mathcal {M}_{g,[m]}(G)^\nu $ , and accordingly, $\overline {\mathcal {M}}_{g,[m]}(G)_{\underline {kr}}$ the closure of $\mathcal {M}_{g,[m]}(G)_{\underline {kr}}$ in $\overline {\mathcal {M}}_{g,[m]}(G)^{\nu }$ .

Let us first recall that, by [Reference Collas and Maugeais4] Proposition 3.7, the locus $\mathcal {M}_{g,[m]}(\mathbb {Z}/\ell \mathbb {Z})_{\underline {kr}}$ is nonempty as soon as g can be obtained by the Hurwitz formula

$$\begin{align*}g=(N-2)\frac{\ell-1}{2}+g'\ell \text{ with } g'\geq 0 \text{ and } N \geq 0, N\neq 1, \end{align*}$$

where N is the number of ramified points in the cover, and $N\neq 1$ by the nullity assumption modulo $\mathbb {Z}/\ell \mathbb {Z}$ as given in ibid. Définition 3.5. The m marked points are distributed freely in G-orbits. This is a particular instance of the Frobenius coin problem, and it is thus known that all $g\geq (\frac {\ell -1}{2})(\frac {\ell -3}{2})$ are attainable with $N-2\geq 0$ , as well as one element of each pair $(k, \ell \frac {\ell -1}{2}-\ell -\frac {\ell -1}{2}-k)$ for $k\in \{ 0,\dots , \frac {\ell -1}{2}\frac {\ell -3}{2}-1\}$ . When $g\in \{ 0,\dots , \frac {\ell -1}{2}\frac {\ell -3}{2}-1\}$ is attainable only by the choice $N=0$ , we say that g is an unramified case. For example, this is the case for $g=1$ by considering the translation action by a choice of order $\ell $ point on an elliptic curve.

Proof. First, suppose g is not unramified. Then by gluing along the dotted lines as in Figure 2a, we obtain the desired $\mathbb {Z}/\ell \mathbb {Z}$ -stable $C_r$ -diagram $X^0$ as follows. The first part is made by gluing p copies of $\mathbb {Z}/\ell \mathbb {Z}$ -curves of genus $0$ , which contributes to the $p\ell $ unramified marked points that are permuted by $\mathbb {Z}/\ell \mathbb {Z}$ to $1$ ramified point with Hurwitz data $\underline {k}=(1)$ and does not contribute to the genus. The second portion is composed of $N-2$ Seyama curves of genus $(\ell -1)/2$ glued in a chain, which contributes to $(N-2)(\ell -1)/{2}$ to the genus and to $N-2$ to the ramified points with free Hurwitz data. The last part is made by gluing $g'$ copies of $2$ -Seyama curves. It contributes to $g'\ell $ to the genus and to $1$ ramified point with imposed Hurwitz data.

Figure 2

The $\mathbb {Z}/\ell \mathbb {Z}$ -stable curve $X^0$ .

To achieve the unramified g, we remove the middle section made of Seyama curves in the previous construction and glue the remaining parts on the added dotted line as in Figure 2b. One can easily check in the same way that it gives a desired curve.

3.4.1 The three families

Consider the following kind of families $W^0_{\lambda }$ , $U^0_{\mu }$ , and $V^0_{v}$ of affine open of $X^0$ .

1. 1. The family $(W^0_{\lambda })_{\lambda \in \Lambda }$ , resp. $(W^0_{\lambda })_{\lambda '\in \Lambda '}$ , given for each $\lambda \in \Lambda $ , resp. $\lambda '\in \Lambda '$ , by the open complement in $X^0_{\lambda }$ of the three ramified points, resp. of the two ramified points, and represented as below:

   

2. 2. The family $(U^0_{\mu })_{\mu \in M}$ , that we will specify as three subfamilies $U^0_{\mu ,0,0}$ , $U^0_{\mu ,0,1}$ , and $U^0_{\mu ,1,1}$ , which for $\mu \in M$ are defined such that $P^0_{\mu }$ consists of a pair of distinguished sections over $X^0_{\lambda }$ and $X^0_{\lambda '}$ with $\lambda ,\lambda '\in \Lambda \sqcup \Lambda '$ , and are, respectively, given as below (see also Figure 3):

   $$ \begin{align*} U^0_{\mu,0,0}&= \operatorname{Spec} K[y_\lambda,x_\lambda,y_{\lambda'},x_{\lambda'}][\frac{1}{y_\lambda}, \frac{1}{y_{\lambda'}}]/(T^{\lambda}_{ij} T_{kl}^{\lambda'}), & \\ U^0_{\mu,0,1}&= \operatorname{Spec} K[x_\lambda,y_\lambda,y_{\lambda'}, \frac{1}{y_\lambda}]/(T^{\lambda}_{ij}T^{\lambda'}_{kl}),\\ U^0_{\mu,1,1}&= \operatorname{Spec} K[y_\lambda,y_{\lambda'}]/(T^{\lambda}_{ij}T^{\lambda'}_{kl}), \end{align*} $$
   Figure 3

   The three subfamilies of $U^0_{\mu }$ .

   

   where $T^{\lambda }_{ij}$ denotes the image of $T^{\lambda }_{ij}\colon \operatorname {Spec} K((t)) \rightarrow X^0_{\lambda }$ and the couple $(\lambda ,\lambda ')$ is related to $\mu \in M$ as in Definition 3.1 and below.

3. 3. The family $(V^0_{v})_{v\in N}$ given for each $v\in N$ by taking the component $X^0_{\lambda }$ that supports the section $Q^0_v$ and removing all the other distinguished sections, to obtain

   (3.6) $$ \begin{align} V^0_v= \operatorname{Spec} K[y_\lambda,x_\lambda,\frac{1}{1-y_\lambda}],\text{ resp. }V^0_v= \operatorname{Spec} K[y_\lambda]. \end{align} $$

   for $\lambda \in \Lambda $ , resp. $\lambda \in \Lambda '$ .

We thus obtain an affine cover of the $\mathbb {Z}/\ell \mathbb {Z}$ -stable curve $X^0$

$$\begin{align*}X^0= \bigcup_{\lambda\in \Lambda\sqcup \Lambda'} W^0_{\lambda} \bigcup_{\mu \in M} U^0_{\mu} \bigcup_{v\in N} V^0_v, \end{align*}$$

where each open is $\mathbb {Z}/\ell \mathbb {Z}$ -stable by construction, and such that:

1. 1. For $\mu \in M$ such that $P^0_{\mu }$ contains a distinguished section of $X^0_{\lambda }$ and $X^0_{\lambda '}$ we have $W^0_{\lambda }$ and $W^0_{\lambda '}$ as open subsets of $U^0_{\mu }$ and $W^0_{\lambda }\cap W^0_{\lambda '}=\varnothing $ .

2. 2. For $v\in N$ such that $Q^0_v$ is a distinguished section of $X^0_{\lambda }$ , we have $W^0_{\lambda }$ as an open subset of $V^0_v$ .

3. 3. The intersection of $U^0_{\mu }$ or $V^0_v$ with any other member of the affine cover is either empty, $W^0_{\lambda }$ or $W^0_{\lambda }\sqcup W^0_{\lambda '}$ .

These properties ensure, in the next section, the possibility of patching local formal schemes over the affine cover that we just defined.

3.4.2 Patching formal schemes in algebraic family

Consider the affine formal scheme $\mathcal {S}= \operatorname {Spf} K[[q]]$ with ideal of definition $\mathfrak {q}=(q)$ equipped with the G-action $q\mapsto \zeta _{\ell }q$ by our choice of isomorphism $G\simeq \mu _{\ell }$ . In order to construct a formal scheme $\mathfrak {X}$ with base $X^0$ over $\mathcal {S}$ with a compatible G-action, we shall define affine formal $\mathcal {S}$ -schemes $\mathcal {W}_{\lambda }$ , $\mathcal {U}_{\mu }$ and $\mathcal {V}_v$ with bases $W^0_{\lambda }$ , $U^0_{\mu }$ and $V^0_v$ with ideal of definition the pullbacks of $\mathfrak {q}$ denoted $\mathfrak {q}$ again.

For $\lambda ,v$ , we set

$$ \begin{align*} \mathcal{W}_{\lambda}= \operatorname{Spf} \Gamma(W_{\lambda}^0,{\mathcal O}_{X^0})[[q]]\text{ and }\mathcal{V}_v= \operatorname{Spf} \Gamma(V_{v}^0,{\mathcal O}_{X^0})[[q]], \end{align*} $$

where the K-algebras of sections $\Gamma (-,{\mathcal O}_{X^0})$ are given by one of the explicit K-algebra of the affine schemes of Section 3.4.1 items 1. and 3. above.

Whenever $\lambda $ and v are such that $W^0_{\lambda }$ is an open subset of $V^0_v$ , the open immersion $j_{v/\lambda }\colon \mathcal {W}_{\lambda }\rightarrow \mathcal {V}_v$ over it is obtained without effort. For instance, let us assume $W^0_{\lambda }= \operatorname {Spec} K[y_\lambda ,x_\lambda ,\frac {1}{y_\lambda },\frac {1}{1-y_\lambda }]$ and $V^0_v= \operatorname {Spec} K[y_\lambda ,x_\lambda ,\frac {1}{1-y_\lambda }]$ . By [Reference Grothendieck8] Proposition 10.1.4, it suffices to check that the map $K[y_\lambda ,x_\lambda ,\frac {1}{1-y_\lambda }][\frac {1}{y_\lambda }][[q]]^{\wedge } \rightarrow \Gamma (\mathcal {W}_{\lambda })$ , where $\wedge $ denotes the $\mathfrak {q}$ -adic completion, is an isomorphism. But this is clear by construction. Note that $j_{v/\lambda }$ is an $\mathcal {S}$ -morphism.

Let us consider the case of $\mathcal {U}_\mu $ , whose base $U^0_\mu $ is obtained via 3 subfamilies $U^0_{\mu ,0,0}$ , $U^0_{\mu ,0,1}$ , and $U^0_{\mu ,1,1}$ as in Section 3.4.1 2.

Proposition 3.11. For $\mu \in M$ such that $U^0_{\mu }$ is of the form $U^0_{\mu ,0,0}$ . Let us define

$$ \begin{align*} \begin{aligned} \textstyle \mathcal{U}_{\mu,0,0}= \operatorname{Spf} K[T,T',X,X'][\frac{1}{1-T},\frac{1}{1-T'}]&[[q]]/(T_{r}T^{\prime}_{r'}-q)\\ &\text{ with }\begin{cases} \textstyle X^{\ell}=T^r(1-T), &\textstyle T_{r}=\zeta_{2\ell}X\\ \textstyle {X'}^{\ell}={T'}^{r'}(1-T'), &\textstyle T^{\prime}_{r'}=\zeta_{2\ell}{X'}.\\ \end{cases} \end{aligned} \end{align*} $$

Then we can identify $\mathcal {U}_{\mu ,0,0}\ \mod \mathfrak {q}$ with $U^0_{\mu ,0,0}$ by $(T,T')\mapsto (y_\lambda ,y_{\lambda '})$ with the choices $(X,X')\mapsto (x_\lambda ,x_{\lambda '})$ . Furthermore, for $\lambda \in \Lambda $ such that $W^0_{\lambda }= \operatorname {Spec} K[y_\lambda ,x_\lambda ,\frac {1}{y_\lambda },\frac {1}{1-y_\lambda }]$ , the scheme $W^0_{\lambda }$ is an open subset of $U^0_{\mu ,0,0}$ given by inverting $y_\lambda $ , so that $\Gamma (\mathcal {U}_{\mu })[\frac {1}{T}]^{\wedge }\rightarrow \Gamma (\mathcal {W}_{\lambda })$ , given by $T\rightarrow y_\lambda $ , is an isomorphism, which induces an open immersion $j_{\mu /\lambda }\colon \mathcal {W}_{\lambda } \rightarrow \mathcal {U}_{\mu ,0,0}.$

Proof. By assumption, $T_{r}T^{\prime }_{r'}=q$ so that for $N\geq 1$ , we have

$$\begin{align*}\Gamma(\mathcal{U}_{\mu})[\frac{1}{T}]/\mathfrak{q}^N= K[T,T',X,X'][\frac{1}{1-T},\frac{1}{1-T},\frac{1}{T'}]/(T_{r}T^{\prime}_{r}-q)^N. \end{align*}$$

As T and $1-T$ are invertible, so is $X^{\ell }=T^r(T-1)$ , and thus so is X and $T_{r}=\zeta _{2\ell }X$ . It follows that $(T_{r}T^{\prime }_{r'})^N=0$ if and only if ${T^{\prime }_{r'}}^N=0$ . Now as ${T^{\prime }_{r'}}^{\ell N}= -{T'}^{r'N} (1-T')^{N}$ , we have ${(1-T')}^N=0$ , which gives that ${T'}^{-1}$ can be written as $\sum \nolimits _{k=0}^{N-1} {(1-T')}^k$ . To recover $T'$ and $X'$ first as ${T'}^{\ell }_{r'}=-{T'}^{r'}(T'-1)$ , we have ${T'}^{\ell }_{r'}=P(T')$ with P invertible for the composition in $K[[T'-1]]$ . So there is $F\in K[[T'-1]]$ such that $F({T'}^{\ell }_{r'})=T'$ . As ${T'}^{\ell }_{r'}$ is nilpotent of order N, we can truncate F to get a polynomial $\widetilde {F}$ that verifies the equality $\widetilde {F}({T'}^{\ell }_{r'})=T'$ in $\Gamma (\mathcal {U}_{\mu })[\frac {1}{T}]/\mathfrak {q}^N$ . Thus, as $T^{\prime }_{r'}=\frac {q}{\zeta _{2\ell } T}$ , we have

$$\begin{align*}\Gamma(\mathcal{U}_{\mu})[\frac{1}{T}]/\mathfrak{q}^N=K[X,T,\frac{1}{T},\frac{1}{1-T}][q]/(q^N) \end{align*}$$

and the desired isomorphism by passing to the $\mathfrak {q}$ -adic completion. It is clear that this isomorphism is compatible with the $\mathbb {Z}/\ell \mathbb {Z}$ -action on both sides.

The other open immersions are proven in the same way.

One thus obtains a proper formal regular $\mathcal {S}$ -scheme $\mathfrak {X}$ with a collection of sections $(\mathcal {Q}_v)_{v\in N}$ with base space $X^0$ by gluing along the affine formal schemes $\mathcal {U}_{\mu },\mathcal {V}_v$ and $\mathcal {W}_{\lambda }$ .

The formal scheme $\mathfrak {X}$ has the property that, for each $\mu $ , $\lambda $ or v, we have $\mathcal {S}$ -isomorphisms

$$\begin{align*}\varphi_{\mu}\colon\mathfrak{X}_{|U^0_{\mu}}\rightarrow \mathcal{U}_{\mu},~\varphi_{\lambda}\colon \mathfrak{X}_{|W^0_{\lambda}}\rightarrow \mathcal{W}_{\lambda}, \varphi_v\colon \mathfrak{X}_{|V^0_v}\rightarrow \mathcal{V}_v \end{align*}$$

extending the identity maps of $U^0_{\mu }$ , $W^0_{\lambda }$ and $V^0_v$ , respectively, such that

1. 1. for each $v\in N$ , $\mathcal {Q}_v$ is induced from the canonical section $\mathcal {S}\rightarrow \mathfrak {X}_{|V^0_v}$ that lift the section $Q_v$ of $V^0_v$ ,

2. 2. the isomorphisms $\varphi _{\lambda }$ , $\varphi _{\lambda }$ and $\varphi _v$ are compatible with the open immersions $j_{v/\lambda }$ and $j_{\mu /\lambda }$ .

A direct application of Grothendieck’s existence theorem [Reference Grothendieck9] 5.4 as in [Reference Ihara and Nakamura17] 2.4.1 and 3.1 provides the algebraization of the formal scheme $\mathfrak {X}$ into a scheme X over $ \operatorname {Spec} K[[q]]$ , whose generic fiber $X_{\eta }$ is a smooth geometrically irreducible genus g curve with m marked points and a $\mathbb {Z}/\ell \mathbb {Z}$ -action, coming by pullback of the one on X, with Hurwitz data $\underline {kr}$ , and whose special fiber is $X^0$ .

The sets of divisors $D=((X^0_{\lambda })_{\lambda \in \Lambda \sqcup \Lambda '}, (Q_v)_{v\in N})$ and $\mathcal {D}=((X^0_{\lambda })_{\lambda \in \Lambda \sqcup \Lambda '}, (\mathcal {Q}_v)_{v\in N})$ are regular with normal crossings on X and $\mathfrak {X}$ , respectively, in the sense of [Reference Grothendieck and Murre10] Section 1.8.3; see [Reference Ihara and Nakamura17] 3.2 for details.

3.4.3 Local-global tracking tangential base points

Another important output of our construction, that will be of interest in the next section, is that we can explicitly track our tangential base points in the different formal completions of $\mathfrak {X}$ along chosen closed subsets of the special fiber.

Consider the completion $\mathfrak {X}_{\mu }$ of $\mathfrak {X}$ along $P^0_{\mu }$ . By construction, $\mathfrak {X}_{| U^0_{\mu }}= \operatorname {Spf} A/(T_{r,s}T^{\prime }_{r',s'} -q)$ for a ring A given in the construction of $U^0_{\mu }$ and $P^0_{\mu }$ corresponds to the ideal $(T_{r,s},T^{\prime }_{r',s'})$ , so that

$$\begin{align*}\mathfrak{X}_{\mu} = \operatorname{Spf} K[[T_{ij},T^{\prime}_{kl}]] \text{ with } T_{ij}T^{\prime}_{kl}=q \text{ as usual}. \end{align*}$$

Let $T_1,T_2$ be two indeterminacies. We have a commutative diagram

where $K[[T_1,T_2]]$ has ideal of definition $(T_1T_2)$ , and where the top horizontal map is an isomorphism. The formal scheme $\mathfrak {X}_{\mu }$ comes with a divisor $\mathcal {D}_{\mu }$ given by the pullback of $\mathcal {D}$ which has two components corresponding to $X^0_{\lambda }$ and $X^0_{\lambda '}$ where $\lambda (\mu )=(\lambda ,\lambda ')$ . They are defined by $T_1=0$ and $T_2=0$ , respectively, so that $\mathcal {D}_{\mu }$ is a set of divisors with regular normal crossing on $\mathfrak {X}_{\mu }$ .

We shall also consider the completion $\mathfrak {X}_{\lambda }$ of $\mathfrak {X}$ along $X^0_{\lambda }$ . It is also equipped with a divisor $\mathcal {D}_{\lambda }$ as the pullback of $\mathcal {D}$ to $\mathfrak {X}_{\lambda }$ which consists of the union of two divisors:

1. 1. $\mathcal {D}^0_{\lambda }$ given by $X^0_{\lambda }$

2. 2. $\mathcal {D}^{\prime }_{\lambda }$ given by the distinguished sections of $X^0_{\lambda }$ .

It is again a set of divisors with regular normal crossings.

By arguing as in the proof of Proposition 3.11, one further obtain the following compatibility result between tangential base points and formal completions.

Proposition 3.13. Let $\mu \in M$ and $\lambda \in \lambda (\mu )$ . Then we have the following commutative diagram in the category of formal schemes:

One remarks that the map $\iota _{\lambda }\colon \operatorname {Spec} K[[T]] \rightarrow \operatorname {Spf} K[[T_1,T_2]]$ which is given by the quotient by $T_2$ factors through the restriction to the special fiber $ \operatorname {Spec} K[[T_1,T_2]]/(T_1T_2)$ .

4.1.1 Tangential base points: Fiber functors and Galois actions

We start by defining fiber functors on $\operatorname {Rev} ^{\mathcal {D}}(\mathfrak {X})$ locally by fixing $\mu \in M$ and considering $\mathfrak {X}_{\mu }$ . Recall that we have a commutative diagram

given by the map $q\mapsto T_1T_2$ . Both maps s and $\mu $ define fiber functors, $\vec {\mu }$ for $\operatorname {Rev} ^{\mathcal {D}_{\mu }}(\mathfrak {X}_{\mu }) $ and $\vec {s}$ for $\operatorname {Rev} ^{S^0}(\mathcal {S})$ ; see [Reference Ihara and Nakamura17] 3.3.1 and 3.3.2.

To be explicit, consider a compatible choice of indeterminates $\{T_1^{\frac {1}{N}},T_2^{\frac {1}{N}}\}_{N\in \mathbb {N}}$ and $\{q^{\frac {1}{N}}\}_{N\in \mathbb {N}}$ to form the fields $K\{\{T_1,T_2\}\}$ and $K\{\{q\}\}$ . Then for $\mathfrak {B}= \operatorname {Spf} \mathcal {B} \in \operatorname {Rev} ^{\mathcal {D}}(\mathfrak {X}_{\mu })$ , resp. $\mathfrak {A}= \operatorname {Spf} \mathcal {A} \in \operatorname {Rev} ^{S^0}(\mathcal {S})$ , the values of the fiber functors are given by

$$ \begin{align*} \vec{\mu}(\mathfrak{B})=\operatorname{Hom} _{K[[T_1,T_2]]}(\mathcal{B}, \overline{K}\{\{T_1,T_2\}\}),\text{ resp. } \vec{s}(\mathfrak{A})= \operatorname{Hom} _{K[[q]]}( \mathcal{A}, \overline{K}\{\{q\}\}). \end{align*} $$

By choosing geometric points such that $q^{\frac {1}{N}} \mapsto (T_1T_2)^{\frac {1}{N}}$ , one obtains two compatible homotopy exact sequences

(4.1)

where the geometric parts $\widehat {\mathbb {Z}}(1)\times \widehat {\mathbb {Z}}(1)$ and $\widehat {\mathbb {Z}}(1)$ are equipped with the Galois actions coming from the sections defined by the choices of tangential base points $\mu $ and s. We refer to [Reference Ihara and Nakamura17] 3.3.1–3.3.4 for details.

We will now track explicitly the fiber functors defined by $\vec {\mu }$ on $\operatorname {Rev} ^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda })$ and $\operatorname {Rev} ^{\mathcal {D}_{\lambda '}}(\mathfrak {X}_{\lambda '})$ for $(\lambda , \lambda ')=\lambda (P^0_{\mu })$ and compare them to the one given by the tangential base points $T^{\lambda }_{ij}$ of Definition 3.1. First of all, remark that the map $\mathfrak {X}_{\mu } \rightarrow \mathfrak {X}_{\lambda }$ pulls back the divisor $\mathcal {D}_{\lambda }$ to $\mathcal {D}_{\mu }$ so that it induces a base change functor

$$\begin{align*}\operatorname{Rev} ^{\mathcal{D}_{\lambda}}(\mathfrak{X}_{\lambda}) \longrightarrow \operatorname{Rev} ^{\mathcal{D}_{\mu}}(\mathfrak{X}_{\mu}), \end{align*}$$

and thus, we have a fiber functor on $\operatorname {Rev} ^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda })$ that is given by composition with $\vec {\mu }$ , which we also denote by $\vec {\mu }$ . In particular, this comes with a map on the étale fundamental groups

$$\begin{align*}p_{\mu/\lambda}\colon \pi_1^{\mathcal{D}_{\mu}}(\mathfrak{X}_{\mu}, \vec{\mu})\longrightarrow \pi_1^{\mathcal{D}_{\lambda}}(\mathfrak{X}_{\lambda}, \vec{\mu}). \end{align*}$$

In the same way, the morphism $f_{\lambda }\colon \mathfrak {X}_{\lambda }\rightarrow \mathcal {S}$ defines a map

$$\begin{align*}p_{\lambda/\mathcal{S}}\colon \pi_1^{\mathcal{D}_{\lambda}}(\mathfrak{X}_{\lambda}, \vec{\mu}) \longrightarrow \pi_1^{S^0}(\mathcal{S}, \vec{s}) \end{align*}$$

by the fact that the pullback of $S^0$ is the divisor $\mathcal {D}_{\lambda }^0\cup \mathcal {D}_{\lambda }^{\prime \prime }$ where $\mathcal {D}_{\lambda }^{\prime \prime }$ is given by $\mathcal {D}_{\lambda }^{\prime }$ restricted to $X^0_{\lambda }$ . As the map $\mathfrak {X}_{\mu } \rightarrow \mathfrak {X}_{\lambda }$ is a map of $\mathcal {S}$ -schemes, we have the commutativity condition

$$\begin{align*}p_{\lambda/\mathcal{S}} \circ p_{\mu /\lambda} = p_{\mu/ \mathcal{S}} \end{align*}$$

and compatibility with the previous homotopy exact sequences of Equation (4.1).

4.1.2 Étale fundamental group comparisons and Galois-compatible actions

By Theorem 4.3.2 of [Reference Grothendieck and Murre10] the restriction map to $X^0_{\lambda }$ gives a categorical equivalence

$$\begin{align*}\operatorname{Rev} ^{\mathcal{D}_{\lambda}^{\prime}}(\mathfrak{X}_{\lambda}) \simeq \operatorname{Rev} ^{D_{\lambda}}(X^0_{\lambda}), \end{align*}$$

and the last one is canonically equivalent to $\operatorname {Rev} ( W_0^{\lambda })$ .

Proposition 4.1. The isomorphisms $\operatorname {Rev} ^{\mathcal {D}^{\prime }_{\lambda }}(\mathfrak {X}_{\lambda }) \simeq \operatorname {Rev} ^{\mathcal {D}_{\lambda }}(X^0_{\lambda }) \simeq \operatorname {Rev} (W^0_{\lambda })$ transform the fiber functor $\vec {\mu }$ in $\overrightarrow {T_{ij}}$ and thus yield a Galois compatible isomorphism

$$\begin{align*}\pi_1^{\mathcal{D}^{\prime}_{\lambda}}(\mathfrak{X}_{\lambda}, \vec{\mu}) \simeq \pi_1(W^0_{\lambda}, \overrightarrow{T}_{ij}). \end{align*}$$

Proof. By Proposition 3.13, the following diagram commutes:

where we recall the map $\iota _{\lambda }\colon \operatorname {Spec} K[[T]] \rightarrow \operatorname {Spf} K[[T_1,T_2]]$ is given by the quotient by $T_2$ .

It thus suffices to check that the fiber functors on $\operatorname {Rev}^{\mathcal {D}_{\lambda }^{\prime }}(\mathfrak {X}_{\lambda })$ given by $\mu $ and $\mu \circ \iota _{\lambda }$ are canonically equivalent and that they are also equivalent to the one given by composition of the pullback to the special fiber and $\overrightarrow {T}_{ij}$ .

Let $\mathfrak {B}\in \operatorname {Rev} ^{\mathcal {D}_{\lambda }^{\prime }}(\mathfrak {X}_{\lambda })$ and consider $A\in \operatorname {Rev} ^{D_{\lambda }}(X^0_{\lambda })$ obtained from $\mathfrak {B}$ by base change to the special fiber. The pullback of $\mathfrak {B}$ to $\mathfrak {X}_{\mu }$ is $ \operatorname {Spf} \mathcal {B} \in \operatorname {Rev} ^{(T_1=0)}( \mathfrak {X}_{\mu })$ with $\mathcal {B}$ a direct sum of subalgebras of $\overline {K}[[T_1^{\frac {1}{N}}, T_2]]$ for some $N\geq 1$ . Then we have

$$ \begin{align*} \vec{\mu}(\mathfrak{B})&= \operatorname{Hom} _{K[[T_1,T_2]]}(\mathcal{B}, \overline{K}\{\{T_1,T_2\}\})\\ &=\operatorname{Hom} _{K[[T_1,T_2]]} ( \mathcal{B}, \overline{K}\{\{T_1\}\} [[T_2]]) \\ &= \operatorname{Hom} _{K[[T]]}(\mathcal{B}/T_2, \overline{K}\{\{T\}\}) \\ \vec{\mu}(\mathfrak{B})&= \overrightarrow{\mu\circ \iota_{\lambda}}(\mathfrak{B}) =\overrightarrow{T_{ij}}(A).\\[-34pt] \end{align*} $$

We can now compare $\pi _1^{\mathcal {D}_{\lambda }^{\prime }}(\mathfrak {X}_{\lambda }, \vec {\mu })$ and $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu })$ by Grothendieck-Murre theory since $\mathcal {D}_{\lambda }$ and $\mathcal {D}_{\lambda }^{\prime }$ , as defined in Section 3.4.3, are two divisors that differ by the special fiber; see [Reference Grothendieck and Murre10] Corollary 5.1.11.

Proposition 4.3. We have an exact sequence

where $\alpha = p_{\mu /\lambda }\circ j_{\mu }\circ j_{\lambda }$ and where $\beta $ comes from the canonical projection induced by the inclusion $\operatorname {Rev} ^{\mathcal {D}_{\lambda }^{\prime }}(\mathfrak {X}_{\lambda })\subset \operatorname {Rev} ^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda })$ .

Proof. By [Reference Grothendieck and Murre10] Theorem 7.3.1, we have the exactness of the sequence

The injectivity of $\alpha $ can be deduced from the injectivity of $p_{\lambda /\mathcal {S}} \circ \alpha = j_{\mathcal {S}}$ .

4.2.1 Galois actions and inertia

Following [Reference Ihara and Nakamura17] 3.4.7, we define geometric parts of the fundamental groups $\pi _1^{\mathcal {D}_{\lambda }} (\mathfrak {X}_{\lambda }, \vec {\mu })$ as the kernels of such projections to $G_K$ .

4.2.2 A trivial Galois action condition

We can now state the basic result that determines when an element of $G_K$ acts trivially on $\pi _1^{\mathcal {D}_{\lambda }}({\mathfrak {X}_{\lambda }}_{\overline {K}}, \vec {\mu }).$

Proof. The decomposition of $\pi _1^{\mathcal {D}_{\lambda }}({\mathfrak {X}_{\lambda }}_{\overline {K}}, \vec {\mu })$ given by 4.2.1 of the previous result passes to the pro- $\ell $ -completion, which gives

$$\begin{align*}\pi_1^{\ell,\mathcal{D}_{\lambda}}({\mathfrak{X}_{\lambda}}_{\overline{K}}, \vec{\mu})\simeq \widehat{\mathbb{Z}_{\ell}}(1) \times \pi_1^{\ell,D_{\lambda}}({X^0_{\lambda}}_{\overline{K}}, \overrightarrow{T_{ij}}). \end{align*}$$

As the $G_K$ -action preserves the product, the implication is straightforward. For the reciprocal, let $\sigma \in G_K$ that acts trivially on $\pi _1^{\ell ,\mathcal {D}_{\lambda }}({\mathfrak {X}_{\lambda }}_{\overline {K}}, \vec {\mu })$ . Let us choose a representation $(y_1,\dots , y_{2g}, x_1,\dots x_n \mid \prod \nolimits _i [y_i,y_{i+1}] x_1\cdots x_{n} )$ of $\pi _1^{\ell ,D_{\lambda }}({X^0_{\lambda }}_{\overline {K}}, \overrightarrow {T_{ij}})$ in the usual way, where $x_1$ denotes the loop around the closed point image of $T_{ij}$ in $X^0_{\lambda }$ . We have $\sigma (x_1)=x_1^{\chi _{\ell }(\sigma )}=x_1$ by assumption. But $\sigma $ also acts by $\chi _{\ell }(\sigma )$ on the first factor $\widehat {\mathbb {Z}_{\ell }}(1)$ , so the action of $\sigma $ on $\pi _1^{\ell ,\mathcal {D}_{\lambda }}({\mathfrak {X}_{\lambda }}_{\overline {K}}, \vec {\mu })$ is trivial.

4.2.3 From fundamental groups to groupoids

In order to conclude, we first we need to explain how to move from fundamental groups to fundamental groupoids. This is essentially formal and comes down to the fact that the set of étale paths are principal homogeneous spaces under the translation actions of the fundamental groups. As such, the technical details will mostly be avoided.

Let $M_{\lambda }=\{\mu \in M \mid \lambda \in \lambda (\mu )\}$ and fix $\lambda \in \Lambda \sqcup \Lambda '$ . Let $\mu _1,\mu _2 \in M_{\lambda }$ . The set of étale paths between the fiber functors $\vec {\mu _1}$ and $\vec {\mu _2}$ of the category $\operatorname {Rev} ^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda })$ is the profinite set $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1},\vec {\mu _2})$ of ismorphisms between these two functors. The fundamental groups $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1})$ and $\pi _1^{\mathcal {D}_{\lambda }}( \mathfrak {X}_{\lambda }, \vec {\mu _2})$ act by left and right translation canonically on $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1},\vec {\mu _2})$ , and these actions are simply transitive. By construction, $\vec {\mu _1}$ and $\vec {\mu _2}$ are turned into the fiber functor $\vec {s}$ of $\operatorname {Rev} ^{S^0}(\mathcal {S})$ through the base change by the map $f_{\lambda }\colon \mathfrak {X}_{\lambda } \rightarrow \mathcal {S}$ so that we have a map

$$ \begin{align*} p_{\lambda/\mathcal{S}}\colon \pi_1^{\mathcal{D}_{\lambda}}(\mathfrak{X}_{\lambda}, \vec{\mu_1}, \vec{\mu_2}) \longrightarrow \pi_1^{S^0}(\mathcal{S}, \vec{s}). \end{align*} $$

By composition, we get a canonical map $p_{\lambda }=p_{\mathcal {S}}\circ p_{\lambda /\mathcal {S}}\colon \pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1}, \vec {\mu _2})\rightarrow G_K$ .

The maps $p_{\lambda }$ (for varying $\mu \in M_{\lambda }$ ) induce a groupoid homomorphism from $\Pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, (\vec {\mu })_{\mu \in M_{\lambda }})$ to $G_K$ . This groupoid compatibility ensures that the canonical actions of the groups $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1})$ and $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _2})$ on $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1},\vec {\mu _2})$ induce by restriction simply transitive actions from their geometric part to the geometric part of the latter.

This construction can be made when considering $\vec {\mu _1}$ and $\vec {\mu _2}$ as fiber functors with respect to the category of étale covers of $\mathfrak {X}_\lambda $ tamely ramified over $\mathcal {D}_\lambda '$ instead of of $\mathcal {D}_{\lambda }$ . As in Proposition 4.3, we have a natural map

$$\begin{align*}\beta_{\mu_1,\mu_2}\colon \pi_1^{\mathcal{D}_{\lambda}}(\mathfrak{X}_{\lambda}, \vec{\mu_1}, \vec{\mu_2}) \longrightarrow \pi_1^{\mathcal{D}_{\lambda}^{\prime}}(\mathfrak{X}_{\lambda}, \vec{\mu_1}, \vec{\mu_2})\end{align*}$$

which is compatible with the canonical actions on both sides with regard to the maps $\beta _{\mu _1}$ and $\beta _{\mu _2}.$ In particular, the map $\beta _{\mu _1,\mu _2}$ is surjective and also induces a bijection from $\overline {p_{\lambda /\mathcal {S}}}^{-1}(\{1\})$ to $\pi _1^{\mathcal {D}_{\lambda }^{\prime }}({\mathfrak {X}_{\lambda }}_{\overline {K}}, \vec {\mu _1}, \vec {\mu _2})$ as in Proposition 4.5. Moreover, the base change functor to the special fiber induces again a canonical bijection

$$ \begin{align*}\pi_1^{\mathcal{D}_{\lambda}^{\prime}}(\mathfrak{X}_{\lambda}, \vec{\mu_1}, \vec{\mu_2}) \simeq \pi_1^{D_{\lambda}}(X^{0}_{\lambda}, \overrightarrow{T_{ij}}, \vec{T_{kl}}).\end{align*} $$

Definition 4.3. We define an action of $G_K$ on $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1},\vec {\mu _2})$ in the following way. For $\gamma \in \pi _1^{\mathcal {D}_{\lambda }}({\mathfrak {X}_{\lambda }}_{\overline {K}}, \vec {\mu _1},\vec {\mu _2})$ and $\sigma \in G_K$ , let

$$\begin{align*}\sigma\cdot \gamma= s_{\lambda/\mu_1}(\sigma) \circ \gamma \circ s_{\lambda/\mu_2}(\sigma)^{-1},\end{align*}$$

where $s_{\lambda /\mu }= p_{\mu /\lambda } \circ s_{\mu }$ for $\mu \in M_{\lambda }$ .

By the compatibility with $p_{\lambda }$ , this action induces an action of $G_K$ on the geometric part of $\pi _1^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda }, \vec {\mu _1},\vec {\mu _2})$ . This action is compatible with the bijection $p_{\lambda /\mathcal {S}}^{-1}(\{1\})\simeq \pi _1^{D_{\lambda }}({X^{0}_{\lambda }}_{\overline {K}}, \overrightarrow {T_{ij}}, \vec {T_{kl}})$ , and we recover the $G_K$ -action induced by our choice of tangential base points on the right-hand side.

We can now state the groupoid analog of Proposition 4.6 and establish the main result of this section.

Consider the formal scheme $\mathfrak {X}$ . The maps $\mathfrak {X}_{\lambda }\rightarrow \mathfrak {X}$ for $\lambda \in \Lambda \sqcup \Lambda '$ , which send $\mathcal {D}$ to $\mathcal {D}_{\lambda }$ by pullback, induce base change functors $\operatorname {Rev} ^{\mathcal {D}}(\mathfrak {X}) \rightarrow \operatorname {Rev} ^{\mathcal {D}_{\lambda }}(\mathfrak {X}_{\lambda })$ . Hence, for $\mu \in M$ , we have fiber functors $\vec {\mu }$ for $\operatorname {Rev} ^{\mathcal {D}}(\mathfrak {X})$ and a fundamental groupoid $\Pi _1^{\mathcal {D}}(\mathfrak {X}, (\vec {\mu })_{\mu \in M})$ which comes with a geometric part $\Pi _1^{\mathcal {D}}(\mathfrak {X}_{\overline {K}}, (\vec {\mu })_{\mu \in M})$ equipped with a Galois action. For every $\lambda \in \Lambda \sqcup \Lambda '$ and $\mu _1,\mu _2\in M_{\lambda }$ , the induced canonical maps

$$\begin{align*}p_{\lambda/\mathfrak{X},\mu_1,\mu_2}\colon \pi_1^{\mathcal{D}_{\lambda}}({\mathfrak{X}_{\lambda}}, \vec{\mu_1},\vec{\mu_2}) \longrightarrow \pi_1^{\mathcal{D}}(\mathfrak{X}, \vec{\mu_1}, \vec{\mu_2})\end{align*}$$

are compatible with taking geometric parts and Galois actions on both sides.