# Reading roadmap: \'{E}tale covers with a prescribed exterior sheet The argument constructs semistable-model cycles, uses graph characters to obtain cyclic covers supported above the chosen disks, and takes a stabilizer quotient of their Galois composite to obtain an exterior sheet and disk degrees divisible by p. Source SHA-256: `22909d746ababe3e7547271619ddf54da33f386a954f5b7cfab13c251f32ebb8`. Indexed 25,388 characters. Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates. ## Internal claim index ### Theorem 1.1 — `thm:exterior` Source lines 67–80. For the fixed disks and exterior above, there is a smooth proper connected curve $Y/\bar k$ and a finite \'etale surjective morphism $g:Y\to X$ such that: [label=\textup{(\roman*)}] \item a connected component of $g^{-1}(U)$ maps isomorphically to $U$; \item for every $i$, every connected component of $g^{-1}(V_i)$ maps to $V_i$ with degree divisible by $p$. Here the inverse images and the degrees of their components refer to the induced finite \'etale analytic maps. The disks are the prescribed $V_i$; they are not shrunk in the conclusion. *Extractive statement; proof gist not independently reconstructed.* Proof opening: Take the covers of Lemma~\ref{lem:graph-character} for all $i$, their conjugates under $\Gal(Z/X)$, and a connected component of their fibre product over $Z$. Call the resulting curve $T$. The intersection of their kernels in $\pi_1(Z)$ is normal in $\pi_1(X)$, so $T\to X$ is connected, finite \'etale and Galois. Put \[ G=\Gal(T/X),\qquad P=\Gal(T/Z). \] The group $P$ is elementary abelian of exponent $p$, since it embeds in the product of the cyclic deck groups. All conjugate covers split over $U_Z$: each deck transformation of $Z/X$ preserves $U_Z$. Their composite $P$-cover therefore splits over $U_Z$ as well. Let $H\subset G$ be the stabilizer of a connected component of the inverse imag Candidate references: `lem:graph-character` (candidate_proof_dependency); `lem:components` (candidate_proof_dependency); `eq:nontrivial` (candidate_proof_dependency); `eq:disjoint` (candidate_proof_dependency). ### Lemma 2.1 — `lem:components` Source lines 162–169. Let $f:C\to X$ be a connected finite \'etale Galois cover with group $G$, and let $D$ be either $U$ or one of the $V_i$. Every connected component of $f^{-1}(D)$ is finite \'etale and surjective over $D$. The group $G$ acts transitively on these components. If $B$ is the stabilizer of a component, its degree over $D$ is $|B|$. *Extractive statement; proof gist not independently reconstructed.* Proof opening: The spaces in question are locally path connected, and their connected components are open and closed. For the compact domain $U$ this also follows from a finite affinoid cover and the finite component decompositions of affinoids. An open-and-closed restriction of a finite \'etale map is finite \'etale: on affinoids it is given by an idempotent summand of the finite algebra. Its rank is locally constant, hence constant and positive on the connected base $D$. Choose a classical point of $D$. Its geometric fibre is a free transitive $G$-set. Every component meets that fibre, which proves transitivity on components. Two points of the fibre lying in the same component are related by an element p ### Proposition 2.2 — `prop:cycles` Source lines 185–190. There is a connected finite \'etale Galois cover $Z\to X$ whose stable skeleton contains, for each $i$, an embedded circle lying entirely above $V_i$. *Extractive statement; proof gist not independently reconstructed.* Proof opening: Positive-genus paths. All the algebraic data descend to a finite extension of $k$. Choose a split semistable model of $X$ with, for each $i$, two distinct smooth closed special-fibre points whose tubes lie inside the fixed $V_i$. To arrange this without altering $V_i$, insert a rational-radius type-two point strictly inside $V_i$ as a vertex of a semistable refinement. Choose two smooth residue directions at this vertex away from the direction toward the exterior; their tubes lie in $V_i$. After a finite extension their centres are rational. The finite set of these type-two points can be inserted simultaneously, and the correspondence between vertex sets and models gives the required model \ Candidate references: `lem:components` (candidate_proof_dependency); `fig:path-cycle` (candidate_proof_dependency); `external:BakerPayneRabinoff` (external_reference); `external:MochizukiTsujimura` (external_reference); `external:MilneAV` (external_reference); `external:SGA1` (external_reference). ### Lemma 3.1 — `lem:graph-character` Source lines 353–358. For every $i$ there is a connected cyclic finite \'etale cover of $Z$ of degree $p$ which is trivial over all of $U_Z$ and whose restriction to at least one component above $V_i$ is connected. *Extractive statement; proof gist not independently reconstructed.* Proof opening: Choose an embedded circle $C_i\subset\Gamma$ lying above $V_i$, and select one of its edges. We will construct a cyclic graph cover by gluing sheets only across a cut in this edge. To make its pullback split over all of $U_Z$, the cut must avoid not just $U_Z$ but its image under the retraction. Choose a point $\eta_i$ strictly inside the selected edge at an irrational position for the normalized valuation. Vertex positions and edge lengths are rational, so such a point exists. It is of type three and satisfies \[ \tau^{-1}(\eta_i)=\{\eta_i\}. \] Indeed side disks attach only at type-two points, whose coordinates on this edge have valuation in $\mathbb Q$. Since $\eta_i$ lies above $V_i$, it Candidate references: `external:DeJongCoverings` (external_reference); `external:GAGA` (external_reference); `external:TemkinAnalytic` (external_reference); `external:SGA1` (external_reference). ## External inputs and references ### [1] MochizukiTsujimura — `external:MochizukiTsujimura` Source lines 505–508. Mochizuki, Shinichi and Tsujimura, Shota. Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for {$p$}-adic Hyperbolic Curves. 2024. RIMS Preprint 1974. Revision of March 30, 2024. \href{https://www.kurims.kyoto-u.ac.jp/~stsuji/Resolution Candidate references: `section:1` (external_reference); `section:2` (external_reference); `prop:cycles` (external_reference). ### [2] TamagawaRNS — `external:TamagawaRNS` Source lines 508–511. Tamagawa, Akio. Resolution of Nonsingularities of Families of Curves. Publications of the Research Institute for Mathematical Sciences. 40. 1291--1336. 2004. Candidate references: `section:1` (external_reference). ### [3] LepageRNS — `external:LepageRNS` Source lines 511–514. Lepage, Emmanuel. Resolution of Nonsingularities for {Mumford} Curves. Publications of the Research Institute for Mathematical Sciences. 49. 861--891. 2013. Candidate references: `section:1` (external_reference). ### [4] SGA1 — `external:SGA1` Source lines 514–517. Rev{\^e}tements {\'e}tales et groupe fondamental ({SGA}~1). 3. 2003. Recomposed and annotated edition of the 1971 original. \url{https://arxiv.org/abs/math/0206203v2}. Candidate references: `section:1` (external_reference); `prop:cycles` (external_reference); `lem:graph-character` (external_reference). ### [5] GAGA — `external:GAGA` Source lines 517–520. Poineau, J{\'e}r{\^o}me. Raccord sur les espaces de {Berkovich}. Algebra \& Number Theory. 4. 297--334. 2010. Appendix A, Theorem A.1. \url{https://msp.org/ant/2010/4-3/ant-v4-n3-p03-s.pdf}. Candidate references: `section:1` (external_reference); `lem:graph-character` (external_reference). ### [6] TemkinAnalytic — `external:TemkinAnalytic` Source lines 520–523. Temkin, Michael. Introduction to {Berkovich} analytic spaces. 2010. Lecture notes, Jussieu summer school. Section 5.1, \url{https://berkovich-2010.imj-prg.fr/lecture-notes/Temkin-lecture-notes.pdf}. Candidate references: `lem:graph-character` (external_reference). ### [7] MilneAV — `external:MilneAV` Source lines 523–526. Milne, James S.. Abelian Varieties. 2022. Author version, January 2, 2022, of the chapter in Arithmetic Geometry (1986), pp. 103--150. Theorem 8.2 and Remark 8.4. \url{https://www.jmilne.org/math/xnotes/AVs.pdf}. Candidate references: `prop:cycles` (external_reference). ### [8] DeJongCoverings — `external:DeJongCoverings` Source lines 526–529. de Jong, A. J.. \'{E}tale fundamental groups of non-{Archimedean} analytic spaces. Compositio Mathematica. 97. 89--118. 1995. \url{https://www.numdam.org/item/CM_1995__97_1-2_89_0/}. Candidate references: `section:1` (external_reference); `lem:graph-character` (external_reference). ### [9] BakerPayneRabinoff — `external:BakerPayneRabinoff` Source lines 529–531. Baker, Matthew and Payne, Sam and Rabinoff, Joseph. On the structure of nonarchimedean analytic curves. Tropical and Non-Archimedean Geometry. 605. 93--121. 2013. Preprint version: arXiv:1404.0279v1, \url{https://arxiv.org/abs/1404.0279v1}. Candidate references: `section:1` (external_reference); `section:2` (external_reference); `prop:cycles` (external_reference). ## Coverage and limits {"mode": "native_latex_index", "claims": 4, "sections": 4, "external_references": 9, "source_bibliography_entries": 9, "unresolved_bibliography": 0, "unresolved": 0} - Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer. - Numbers reflect simple source counters; labels and line anchors are authoritative. - Extractive statements and proof openings are marked; they are not reconstructed proof gists. - Macros, included files and custom counters require an expanded source or annotated fallback.