\documentclass[12pt]{amsart} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amscd} \headheight=8pt \topmargin=0pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \sloppy \newtheorem{claim}{Claim} \newtheorem{cor}{Corollary} \newtheorem{duffy}{Definition} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \begin{document} \title{Linear systems attached to cyclic inertia} \author{Marco A. Garuti} \address{\hskip-\parindent Marco A. Garuti\\ Dipartimento di Matematica Pura ed Applicata\\ Universit\`a degli Studi di Padova\\ Via Belzoni 7, 35131\\ Padova - ITALY} \email{mgaruti@galileo.math.unipd.it} \thanks{Research at MSRI is supported in part by NSF grant DMS-9701755.} \begin{abstract} We construct inductively an equivariant compactification of the algebraic group ${\mathbb W}_n$ of Witt vectors of finite length over a field of characteristic $p>0$. We obtain smooth projective rational varieties $\overline{\mathbb W}_n$, defined over $\mathbf F_p$; the boundary is a divisor whose reduced subscheme has normal crossings. The Artin-Schreier-Witt isogeny $F-1:{\mathbb W}_n\to {\mathbb W}_n$ extends to a finite cyclic cover ${\mathbf\Psi}_n:\overline{\mathbb W}_n\to \overline{\mathbb W}_n$ of degree $p^n$ ramified at the boundary. This is used to give an extrinsic description of the local behavior of a separable cover of curves in char. $p$ at a wildly ramified point whose inertia group is cyclic. In an appendix, we give an elementary computation of the conductor of such a covering, which can otherwise be determined using class field theory. \end{abstract} \maketitle % keys: act, adjust, AS, bound, % bord, c_n, condor, h-term, H=H % LP1, LP2, new, res, rough, S_n, % sort, tele, WAS, xi Let $f:C\to D$ be a finite separable cover of (germs of) curves over a perfect field $k$ of characteristic $p>0$. Suppose that $f$ is Galois of group $G$ and let $y_0\in C$ be a ramification point at which the residue field extension $k(y_0)/k(x_0)$ is separable, where $x_0=f(y_0)$. Recall (\cite{CL}, chap. IV), that the higher ramification groups $G_i$ (in lower numbering) are defined as $$ G_i=\left\{g\in G|\, v_{y_0}\left(g(t)-t\right)\geq i+1\right\} $$ where $t$ is a local parameter at $y_0$; they do not depend on the choice of $t$. The inertia group $G_0$ is an extension of a $p$-group by a cyclic group of order prime to $p$ and, for $i>0$, the $G_i$ are $p$-groups; they are trivial for large values of $i$. The degree (at $y_0$) of the different $\mathfrak{D}_{C/D}=Ann\,(\Omega_{C/D}^1)$ of the extension $C/D$ can be expressed as $$ \deg\mathfrak{D}_{C/D,y_0} =\sum_{i=0}^\infty\left(|G_i|-1\right) $$ Using Herbrand's $\varphi$ and $\psi$ functions, one can introduce (\emph{loc.cit.}, \S 3) higher ramification groups in upper numbering $G^{\varphi(r)}=G_r$, indexed over $\mathbf{R}_{\geq0}$. Hence, if $e$ is the largest integer such that $G_e$ is nontrivial, the real number $$ m=\varphi(e) $$ is the largest for which $G^m$ is nontrivial. When $G$ is abelian, the classical Hasse-Arf theorem (\emph{loc.cit.}) guarantees that this number is in fact an integer. We shall write $m(f)$ and $e(f)$ to emphasize that these integers are related to the cover $f:C\to D$. The conductor (at $y_0$) of the extension $C/D$ is defined as $$ \mathrm{cond}\,(C/D,y_0)=m(f)+1. $$ Geometrically, the conductor can be characterized as follows (\cite{gacc}, chap. VI, 2.12 and \cite{clac}, 3.6): $m+1=\deg_{y_0}\mathfrak{m}$, where $\mathfrak{m}$ is the smallest modulus such that the cover $C/D$ is isomorphic to a pull-back from an isogeny of the generalized jacobian $J_{\mathfrak{m}}$ of $D$. \bigskip We shall be concerned with the situation where $G_0$ is cyclic of order $p^n$. By Artin-Schreier-Witt theory, there are \'etale neighborhoods $V$ of $y_0$ and $U$ of $x_0$ and a map $\underline u$, from $U^\prime=U-\{x_0\}$ to the group ${\mathbb W}_n$ of Witt vectors of length $n$, such that $V^\prime=V-\{y_0\}$ is the fibred product $$ \CD V^\prime @>>> {\mathbb W}_n \\ @V{f|_{V^\prime}}VV @VV{F-1}V \\ U^\prime @>{\underline u}>> {\mathbb W}_n \endCD $$ The maps $\underline u$ and $\underline u+F\underline w-\underline w$, define isomorphic covers for any Witt-vector valued function $\underline w$. \bigskip We will make the following assumptions: $f$ itself is cyclic of degree $p^n$, totally ramified at $y_0$; furthermore, $U=D$ and $V=C$ and $\underline u$ is defined over $k$. Working locally for the \'etale topology, this is no great loss of generality. \bigskip We are going to show that, for every positive integer $n$, there is a smooth projective rational variety ${\overline{\mathbb W}}_n$ over the prime field $\mathbf{F}_p$, equipped with a tautological line bundle $\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$ such that: \begin{enumerate} \item ${\overline{\mathbb W}}_n$ contains ${{\mathbb W}}_n$ as an affine open dense subvariety; the boundary $B_n={\overline{\mathbb W}}_n \setminus{{\mathbb W}}_n$ is a divisor such that $\mathcal{L}(B_n)= \mathcal{O}_{{\overline{\mathbb W}}_n}(1)$ and $(B_n)_{\mathrm{red}}$ has normal crossings. \item The isogeny $F-1$ extends to a separable cyclic cover $\mathbf\Psi_n: \overline{\mathbb W}_n\to\overline{\mathbb W}_n$ branched along $B_n$. \end{enumerate} \noindent Our main result is then: \begin{thm} Let $f:C_n\to D$ be a separable cyclic cover of degree $p^n$ of curves over a perfect field $k$, totally branched above $x_0\in D(k)$. Let $\underline u= (u_0,\dots,u_{n-1})$ be a Witt vector of rational functions on $D$, with poles of order $\nu_i$, prime to $p$, at $x_0$, such that $k(C_n)/k(D)$ is defined by $F(\underline{Y})-\underline Y =\underline u$. Then $\underline u$ extends to a morphism $\underline u: D\to \overline{\mathbb W}_n\times k$ such that $C_n$ is isomorphic to the normalization of the fibred product $D\times_{\overline{\mathbb W}_n}\overline{\mathbb W}_n$ and $$ \mathrm{cond}\,(C_n/D,y_0)-1 =\deg\underline u^\ast \mathcal{O}_{{\overline{\mathbb W}}_n}(1)= \max_{0\leq i\leq n-1}\{p^{n-i-1}\nu_i\}. $$ \end{thm} \bigskip This work was completed while visiting the M.S.R.I. (october-december 1999) during the program on {\it Galois Groups and Fundamental Groups.} I would like to thank the participants for many stimulating conversations and in particular Ahmed Abbes and Mohamed Sa\"\i di for pointing out to me the references \cite{bry} and \cite{schmid} respectively. \bigskip \bigskip We begin with the following \begin{duffy} For every positive integer $n$, define inductively a variety ${\overline{\mathbb W}}_n$ over the prime field $\mathbf{F}_p$, equipped with a tautological line bundle $\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$, by letting $$\begin{array}{rl} \left({\overline{\mathbb W}}_1, \mathcal{O}_{{\overline{\mathbb W}}_1}(1)\right) & = \left(\mathbf{P}^1, \mathcal{O}_{\mathbf{P}^1}(1)\right)\\ \left({\overline{\mathbb W}}_{n+1}, \mathcal{O}_{{\overline{\mathbb W}}_{n+1}}(1)\right) & = \left(\mathbf{P}\left( \mathcal{O}_{{\overline{\mathbb W}}_n} \oplus\mathcal{O}_{{\overline{\mathbb W}}_n}(p)\right), \mathcal{O}_{\mathbf P}(1)\right) \end{array} $$ (as usual, $\mathbf P(\mathcal E)$ denotes the projective bundle associated to a locally free module $\mathcal E$ and $\mathcal{O}_{\mathbf P}(1)$ its tautological line bundle cf.\cite{EGA2}, 4.1.1; for any $m\in\mathbf Z$, put $\mathcal{O}(m)=\mathcal{O}(1)^{\otimes m}$). \end{duffy} By definition, there is a natural projection \begin{equation} \label{res} \mathbf{r}: {\overline{\mathbb W}}_{n+1}\longrightarrow{\overline{\mathbb W}}_n \end{equation} which is a fibration by projective lines; it follows immediately that ${\overline{\mathbb W}}_n$ is a smooth projective rational variety of dimension $n$. Note that $\mathcal{O}_{{\overline{\mathbb W}}_{n}}(1)$ is not ample (for $n\geq2$); it is only ample relative to $\mathbf r$. On the other hand, ${\overline{\mathbb W}}_{n+1}$ is the projective closure of the vector bundle $\mathbf V(\mathcal{O}_{{\overline{\mathbb W}}_{n}}(p))$ over ${\overline{\mathbb W}}_n$ (cf.~\cite{EGA2}, 8.4) and there are two canonical sections of $\mathbf r$. The zero section, whose divisor we will denote $Z_{n+1}$, arises from the exact sequence $$ 0\longrightarrow \mathcal{O}_{{\overline{\mathbb W}}_n} \longrightarrow \mathcal{O}_{{\overline{\mathbb W}}_n} \oplus\mathcal{O}_{{\overline{\mathbb W}}_n}(p) \longrightarrow \mathcal{O}_{{\overline{\mathbb W}}_n}(p) \longrightarrow 0 $$ which gives a global section of $\mathcal{O}_{{\overline{\mathbb W}}_n}\oplus \mathcal{O}_{{\overline{\mathbb W}}_n}(p)= \mathbf r_\ast \mathcal{O}_{{\overline{\mathbb W}}_{n+1}}(1)$, whence a global section of $\mathcal{O}_{{\overline{\mathbb W}}_{n+1}}(1)$ (the latter restricts to $\mathcal{O}_{\mathbf P^1}(1)$ on the fibres of $\mathbf r$, so its higher direct images vanish and $H^i(\overline{\mathbb W}_n,\mathbf r_\ast \mathcal{O}_{{\overline{\mathbb W}}_{n+1}}(1)) =H^i(\overline{\mathbb W}_{n+1}, \mathcal{O}_{{\overline{\mathbb W}}_{n+1}}(1))$ for all $i$). The infinity section is obtained by the same construction composed with the isomorphism $$ \mathbf{P}\left(\mathcal{O}_{{\overline{\mathbb W}}_n} \oplus\mathcal{O}_{{\overline{\mathbb W}}_n}(p)\right) \simeq \mathbf{P}\left(\mathcal{O}_{{\overline{\mathbb W}}_n} (-p)\oplus\mathcal{O}_{{\overline{\mathbb W}}_n}\right). $$ We denote $\Sigma_{n+1}$ the corresponding divisor in $H^0(\overline{\mathbb W}_{n+1}, \mathcal{O}_{{\overline{\mathbb W}}_{n+1}}(1)\otimes {\mathbf r}^\ast\mathcal{O}_{{\overline{\mathbb W}}_n}(-p))$. \bigskip As suggested by the notation, ${\overline{\mathbb W}}_n$ is a compactification of the Witt group scheme ${{\mathbb W}}_n$; more precisely: \begin{prop} There is a natural system of open immersions $j_n:{{\mathbb W}}_n\subset{\overline{\mathbb W}}_n$, compatible w.r.t. the projections $\mathbf r$ and such that $\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$ trivializes over $j_n({{\mathbb W}}_n)$. The closed subscheme $B_n={\overline{\mathbb W}}_n\setminus j_n({{\mathbb W}}_n)$ is a divisor linearly equivalent to $\Sigma_n+p\ \mathbf r^\ast B_{n-1}$ (and hence to $Z_n$) and $(B_n)_{\mathrm{red}}$ has normal crossings. \end{prop} Proof: Indeed, we have a canonical embedding $j_1:\Bbb G_{\mathrm a}\cong\mathbf A^1\subset\mathbf P^1$; by induction, assume $\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$ trivializes over the affine open subset $j_n({\mathbb W}_n)=U_n$: then, by definition (cf.\cite{EGA2}, 3.1), $\mathbf{r}^{-1}(U_n)=\mathbf P_{U_n}^1$. Clearly, $\mathcal{O}_{{\overline{\mathbb W}}_{n+1}}(1)$ trivializes over the complement $U_{n+1}$ of the infinity section, and, identifying ${\mathbf r}^{-1}(0)\cap U_{n+1}$ with $\mathbf A^1$, we can choose an isomorphism $j_{n+1}:{\mathbb W}_{n+1}\cong U_{n+1}$ fitting in the diagram $$ \CD \mathbf A^1 @ >>> U_{n+1} @>{{\mathbf r}|_{U_{n+1}}}>> U_n \\ @Aj_1AA @AAA @AAj_n A \\ \Bbb G_{\mathrm a} @>{V^n}>>{\mathbb W}_{n+1} @>{\mathbf r}>> {\mathbb W}_n \endCD $$ where the bottom one is the exact sequence of group schemes defining ${\mathbb W}_{n+1}$. \noindent The second statement follows easily, as $j_{n+1}({\mathbb W}_{n+1})$ has been defined as the intersection of $\mathbf{r}^{-1}(U_n)$, which is the complement of the vertical divisor $\mathbf r^\ast(pB_n)$, with the complement of the horizontal infinity divisor $\Sigma_{n+1}$. \begin{cor} \label{bord} For all $1\leq i\leq n$, let $B_{n,i}=({\mathbf r}^{n-i})^\ast(\Sigma_i)$ be the pull back of the infinity divisor on $\overline{\mathbb W}_{i}$ via the iteration of the projection (\ref{res}). The boundary divisor is then $$ B_n=\sum_{i=1}^np^{n-i} B_{n,i} $$ \end{cor} From now on, we identify ${{\mathbb W}}_n$ with its image in ${\overline{\mathbb W}}_n$. Choose a system of variables $Y_0,Y_1,\dots,Y_n,\dots$ such that ${{\mathbb W}}_n=\mathrm{Spec}\ \mathbf F_p[Y_0,\dots,Y_{n-1}]$ and the projection ${\mathbf r}$ corresponds to the inclusion $\mathbf F_p[Y_0,\dots,Y_{n-1}]\subset \mathbf F_p[Y_0,\dots,Y_{n}]$. On the ring of polynomials $$ {\mathsf H}=\mathbf F_p[T,Y_0,\dots,Y_n,\dots] $$ we define a grading by $\deg T=1$ and $\deg Y_i=p^i$ (recall \cite{AC} that with this choice for the degrees, the polynomials defining the sum, product etc. of Witt vectors are isobaric). As usual, we denote by ${\mathsf H}_d$ the $d$-th graded piece, consisting of homogeneous elements of degree $d$. \begin{lemma} \label{H=Gamma} The tautological line bundle $\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$ is generated by global sections; there is a canonical isomorphism $H^0(\overline{\mathbb W}_n, \mathcal{O}_{{\overline{\mathbb W}}_n}(1)) ={\mathsf H}_{p^{n-1}}$ under which $Z_n=(Y_{n-1})_0$ and $B_n=(T^{p^{n-1}})_0$. \end{lemma} Proof: The assertions being clear for $n=1$, the proof is by induction. We need a classical explicit description of the projective bundle $\overline{\mathbb W}_{n+1}$. \noindent By inductive assumption, the complete linear series $|pZ_n|$ defines a morphism $$ \phi_{|pZ_n|}: \overline{\mathbb W}_{n}\longrightarrow \mathbf P^m=\mathbf PH^0(\overline{\mathbb W}_n, \mathcal{O}_{{\overline{\mathbb W}}_n}(p)). $$ The vector space decomposition ${\mathsf H}_{p^n}=H^0(\overline{\mathbb W}_n, \mathcal{O}_{{\overline{\mathbb W}}_n}(p)) \oplus\langle Y_n\rangle$ induces a rational map $$ \pi:\mathbf P^{m+1}= \mathbf P{\mathsf H}_{p^n} \dasharrow\mathbf P^m $$ defined outside $[0:\dots:0:1]$. The indeterminacy is solved on the blow-up of this point, $\tilde\mathbf P^{m+1}= \mathbf P(\mathcal O_{\mathbf P^m}\oplus \mathcal O_{\mathbf P^m}(1))$, and $\overline{\mathbb W}_{n+1}$ is canonically isomorphic to the fibred product: $$ \CD \overline{\mathbb W}_{n+1} @>>> \tilde\mathbf P^{m+1} \\ @V{\mathbf r}VV @VV{\tilde\pi}V \\ \overline{\mathbb W}_{n} @>\phi_{|pZ_n|}>> \mathbf P^m \endCD $$ To complete the proof of the lemma it suffices to remark that the composite morphism $\overline{\mathbb W}_{n+1}\to\tilde\mathbf P^{m+1} \to\mathbf P^{m+1}$ is given by the complete linear system $|Z_{n+1}|$; its image is the projective cone over $\phi_{|pZ_n|}(\overline{\mathbb W}_{n})$. \bigskip \begin{prop} The action of ${{\mathbb W}}_n$ on itself extends to an action of ${{\mathbb W}}_n$ on $\overline{\mathbb W}_n$ and the line bundle $\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$ is stable under this action. In particular, $\mathbf Z/p^n\mathbf Z={{\mathbb W}}_n(\mathbf F_p)$ acts on $\overline{\mathbb W}_n$ and the the Artin-Schreier-Witt isogeny $(F-1):{\mathbb W}_{n} \to {{\mathbb W}}_n$ extends to a cyclic cover $$ \mathbf\Psi_n: {\overline{\mathbb W}}_{n} \longrightarrow{\overline{\mathbb W}}_{n} $$ of degree $p^n$, defined over $\mathbf F_p$, commuting with the projection $\mathbf r$. It is branched along the boundary divisor $B_n$ and the inertia subgroup of the infinity section $\Sigma_{n}$ has order $p$. The induced map $\mathbf\Psi_n^\ast$ on the group of $k$-cycles modulo rational equivalence is multiplication by $p^{n-k}$. \end{prop} Proof: Clearly, $\Bbb G_{\mathrm a}$ acts on $\mathbf P^1$ and the isogeny $\wp:\Bbb G_{\mathrm a}\to\Bbb G_{\mathrm a}$ extends to a cyclic cover of degree $p$ of $\mathbf P^1$ branched at $\infty$; we proceed by induction. \noindent The group ${\mathbb W}_{n+1}$ acts on itself by translations; recall (e.g. \cite{AC}) that the sum of two Witt vectors $\underline{X},\underline{Y}$ is expressed by polynomials $S_i(\underline{X};\underline{Y})\in\mathbf{Z} \left[X_0,\dots,X_i,Y_0,\dots,Y_i\right]$ characterized by the relations $$ \Phi_n(S_0,\dots,S_n)= \Phi_n(X_0,\dots,X_n)+\Phi_n(Y_0,\dots,X_n) $$ If the variables $X_i$ and $Y_i$ are given the weight $p^i$, then $S_i$ is isobaric of weight $p^i$. Moreover: \begin{equation} \label{S_n} S_i(\underline{X};\underline{Y})= X_i+Y_i+c_i(\underline{X};\underline{Y}), \qquad c_i\in\mathbf{Z}\left[X_0, \dots,X_{i-1};Y_0,\dots,Y_{i-1}\right] \end{equation} Therefore, given an $\mathbf F_p$-algebra $A$, for any $\underline a\in{\mathbb W}_{n+1}(A)$, the polynomial $$ T^{p^n}S_n(\frac{Y_0}{T},\dots,\frac{Y_n}{T^{p^n}}; \underline a)=Y_n+a_nT^{p^n} +T^{p^n}c_n(\frac{Y_0}{T}, \dots,\frac{Y_{n-1}}{T^{p^{n-1}}};\underline a) $$ is a homogeneous element in ${\mathsf H}_{p^n}\otimes A$. \noindent As $\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$ is stable under the action, ${\mathbb W}_{n}(A)$ acts $A$-linearly on $H^0(\overline{\mathbb W}_{n,A}, \mathcal{O}_{{\overline{\mathbb W}}_n}(p))$: we extend it to an action of ${\mathbb W}_{n+1}(A)$ on ${\mathsf H}_{p^n}\otimes A$ by \begin{equation} \label{act} \underline a\cdot Y_n=Y_n+a_nT^{p^n} +T^{p^n}c_n(\frac{Y_0}{T}, \dots,\frac{Y_{n-1}}{T^{p^{n-1}}};\underline a) \end{equation} The point $[0:\dots:0:1]\in\mathbf P_A^{m+1} =\mathbf P{\mathsf H}_{p^n}\otimes A$ is fixed and one easily checks that the action extends on the blow-up $\tilde\mathbf P^{m+1}\subset \mathbf P^{m+1}\times\mathbf P^m$; in particular, the inertia subgroup of the exceptional divisor for the action of $\mathbf Z/p^{n+1}\mathbf Z={\mathbb W}_{n+1}(\mathbf F_p)$ has order $p$. As $\overline{\mathbb W}_{n+1}=\tilde\mathbf P^{m+1} \times_{\mathbf P^{m}}\overline{\mathbb W}_n$, we have the obtained the desired action; moreover, the $\mathbf Z/p^{n+1}\mathbf Z$-inertia of the infinity section $\Sigma_{n+1}$ is of order $p$. \noindent Let $\pi:P\to\overline{\mathbb W}_{n+1}$ be the cyclic cover of degree $p^n$ given by the fibred product $$ \CD P @>{\pi}>> \overline{\mathbb W}_{n+1} \\ @V{q}VV @VV{\mathbf r}V \\ \overline{\mathbb W}_{n} @>{\mathbf\Psi_n}>> \overline{\mathbb W}_n \endCD $$ As $\mathbf\Psi_n^\ast$ is multiplication by $p$, we have $P=\mathbf{P}\left(\mathcal{O}_{{\overline{\mathbb W}}_n} \oplus\mathcal{O}_{{\overline{\mathbb W}}_n}(p^2)\right)$. We have a sheaf of graded $\mathcal{O}_{{\overline{\mathbb W}}_n}$-algebras $$ \mathcal S=\bigoplus_{d\geq0} \mathcal{O}_{{\overline{\mathbb W}}_n}(pd) $$ and, with the notations as in \cite{EGA2} \S2, $\overline{\mathbb W}_{n+1}=\mathrm{Proj}\ \mathcal S[Y_n]$ and $P=\mathrm{Proj}\ \mathcal S^{(p)}[X_n]$. Define a morphism of graded $\mathcal{O}_{{\overline{\mathbb W}}_n}$-algebras: $$ \varphi^\#:\mathcal S^{(p)}[X_n] \longrightarrow \left(\mathcal S[Y_n]\right)^{(p)} $$ which is the identity on $\mathcal S^{(p)}$ and sends $X_n$ to $$ T^{p^{n+1}}S_n(\frac{Y_0^p}{T^p},\dots, \frac{Y_n^p}{T^{p^{n+1}}}; -(\frac{Y_0}{T},\dots,\frac{Y_n}{T^{p^n}})) =Y_n^p-Y_nT^{p^n(p-1)} +T^{p^{n+1}}c_n(\frac{Y_0^p}{T^p}, \dots;-\frac{Y_0}{T},\dots). $$ This defines a $\overline{\mathbb W}_n$-morphism $\varphi: G(\varphi^\#)\to P$ from an open subset of $\mathrm{Proj}\ (\mathcal S[Y_n])^{(p)}=\overline{\mathbb W}_{n+1}$ (\textit{loc.cit}, prop. 3.1.8). The only homogeneous prime ideal containing $\mathcal S^{(p)}$ and $Y_n$ is the irrelevant ideal, hence $G(\varphi^\#)=\overline{\mathbb W}_{n+1}$. As $\varphi$ is equivariant under the action of $\mathbf Z/p\mathbf Z$ given by equation (\ref{act}) (via the $n$-th iteration of the Verschiebung), the composite morphism $$ \CD \mathbf\Psi_{n+1}:\overline{\mathbb W}_{n+1} @>{\varphi}>> P @>{\pi}>> \overline{\mathbb W}_{n+1} \endCD $$ extends the Artin-Schreier-Witt isogeny and is equivariant under the $\mathbf Z/p^{n+1}\mathbf Z$-action defined above (it suffices to check it over the dense open subset ${\mathbb W}_{n+1}$) and satisfies $$ \mathbf \Psi_n\circ\mathbf r =\mathbf r\circ\mathbf \Psi_{n+1}. $$ To prove the last statement, recall that, via $\mathbf r^\ast$, the cohomology group $A^\ast(\overline{\mathbb W}_{n+1})$ is a free $A^\ast(\overline{\mathbb W}_{n})$-module generated by the classes $1$ and $\xi=[Z_{n+1}]$. As $\mathbf\Psi$ commutes with $\mathbf r$, it suffices to prove that $\mathbf \Psi_{n+1}^\ast\xi=p\,\xi$. Write: \begin{equation} \label{xi} \mathbf\Psi_{n+1}^\ast\xi= a\xi+\mathbf \beta \qquad a\in\mathbf Z,\,\beta \in{\mathbf r}^\ast A^1(\overline{\mathbb W}_{n}). \end{equation} Intersecting with a fibre of ${\mathbf r}$, we deduce $a=p$. On the other hand, let $\eta={\mathbf r}^\ast[Z_n]$; since $c_t(\mathcal{O}_{{\overline{\mathbb W}}_n} \oplus\mathcal{O}_{{\overline{\mathbb W}}_n}(p)) =c_t(\mathcal{O}_{{\overline{\mathbb W}}_n}(p)) =1+p\,[Z_n]t$, we have $$ \xi^2=p\,\xi\eta. $$ As $\mathbf\Psi_{n+1}^\ast\eta=p\,\eta$, taking self intersection on both sides of (\ref{xi}) we get: $$ p^2\xi^2+p^2\beta\eta= p(p\xi+\beta)p\eta= \mathbf\Psi_{n+1}^\ast(p\xi\eta) =\mathbf\Psi_{n+1}^\ast(\xi^2) =p^2\xi^2+2p\xi\beta+\beta^2 $$ whence $(2p\beta)\,\xi+(\beta^2-p^2\beta\eta)=0$. Since $1,\xi$ form a $A^\ast(\overline{\mathbb W}_{n})$-basis, this implies $\beta=0$. \begin{cor} With the notation as in Corollary~\ref{bord}, the inertia subgroup at the $i$-th component $B_{n,i}$ of the boundary divisor $B_n$ on $\overline{\mathbb W}_{n}$ is cyclic of order $p^{n-i}$. \end{cor} Let now $f_n:C_n\to D$ be a cyclic cover as in Theorem 1 and $\underline u: D-\{x_0\}\to {\mathbb W}_{n}$ the corresponding Witt representative; clearly, $\underline u$ extends to $$ \underline u: D\longrightarrow \overline{\mathbb W}_{n}. $$ \begin{lemma} With the notation as above, $\deg\underline u^\ast \mathcal{O}_{{\overline{\mathbb W}}_n}(1)= \max_{0\leq i\leq n-1}\{p^{n-i-1}\nu_i\}$. \end{lemma} Proof: It suffices to consider the composite map (cf. lemma~\ref{H=Gamma}) $$ \CD D@>{\underline u}>> {\overline{\mathbb W}}_n @>{\phi_{|Z_n|}}>> \mathbf P{\mathsf H}_{p^{n-1}}. \endCD $$ In a neighborhood of $x_0$, it is defined by $[s^{M}:s^{M}u_0^{p^{n-1}}:\dots,s^{M}u_{n-1}]$, where $s$ is a local parameter at $x_0$ and $$ \begin{array}{rl} M & =\max\,\{i_0\nu_0+\dots+i_{n-1}\nu_{n-1}|\, 0\leq i_h\leq p^{n-1-h}\, \mathrm{and}\, \sum_{h=0}^{n-1} p^hi_h=p^{n-1}\}\\ & = \max_{0\leq i\leq n-1}\{p^{n-i-1}\nu_i\}. \end{array} $$ \begin{prop} \label{bound} Let $f_n:C_n\to D$ be as in Theorem 1 and let $M(f_n)=\deg\underline{u}^\ast\mathcal{O}_{{\overline{\mathbb W}}_n}(1)$. If $m(f_n)+1$ is the degree of the conductor, then $$ m(f_n)\leq M(f_n). $$ \end{prop} \noindent Proof: We show that $\mathfrak{u}=(1+M(f_n)).\infty$ is a modulus, in the sense of~\cite{gacc}, chap. III, for the rational map $\underline{u}:D-\{\infty\}\to\Bbb{W}_n$, with the smallest possible degree. By the universal property of the generalized jacobians, $\underline{u}$ factors through $J_{\mathfrak{u}}$ and, by the minimality of the conductor, $\mathfrak{m}\leq\mathfrak{u}$. We have to show that $(\underline{u},\alpha)_\infty=\underline{0}\in{\mathbb W}_n(k)$ for any rational function $\alpha$ on $D$ such that $\alpha\equiv1 \bmod \mathfrak{u}$, where $(\underline{u},-)_\infty$ denotes the local symbol associated to $\underline{u}$: it is the "Residuenvektor" defined in~\cite{witt}, \S 2, whose $j$-th phantom component is $$ \Phi_j\left((\underline{u},\alpha)_\infty\right) =\mathrm{Res}\,\left(\Phi_j(u_0,\dots,u_j)\frac{d\alpha}\alpha\right) $$ Let $s$ be a local parameter on $D$ at $\infty$; if $A$ is any complete discrete valuation ring of characteristic zero with residue field $k$, we can lift $\alpha\in k[[s]]$ and $u_0,\dots,u_{n-1}\in k((s))$ to formal power series $\tilde{\alpha}\in A[[s]]$ and $\tilde{u}_0,\dots,\tilde{u}_{n-1}\in A((s))$, with the $\tilde{u}_i$ having a pole at $s=0$ of the same order as $u_i$: by definition, $(\underline{u},\alpha)$ is the reduction modulo $pA$ of $(\underline{\tilde u},\tilde\alpha)$. \noindent Recall that $u_i$ has a pole of order $\nu_i$ and that $M=\max\,\{p^{n-i-1}\nu_i:\,0\leq i\leq n-1\}$. It is then clear that $M$ is the smallest integer $\rho$ for which $$ \begin{array}{rl} \Phi_j\left((\underline{\tilde u},\tilde \alpha)\right) & = \mathrm{Res}\,\left(\Phi_j(\tilde{u}_0,\dots,\tilde{u}_j) \frac{d\tilde\alpha}{\tilde\alpha}\right) \\ & = \mathrm{Res}\,\left(\left(\tilde{u}_0^{p^j}+\dots+p^j\tilde{u}_j\right) \frac{d\tilde\alpha}{\tilde\alpha}\right) \end{array} $$ vanishes for any $0\leq j\leq n-1$ and for any $\tilde\alpha$ such that $1-\tilde\alpha$ has a zero of order $\rho+1$ at $s=0$. \bigskip To complete the proof of the Theorem, we need to show that the bound provided by Proposition~\ref{bound} is attained. When $k$ is finite, this was done by H.L. Schmid, \cite{schmid}, \S3: if $\chi:G_{k(D)}^{ab}\to \mathbf Z/p^n\mathbf Z$ is the character corresponding to the cover $f_n$, then the residue vector is related to the reciprocity map by the formula $$ \chi(\mathrm{rec}(\alpha))= \mathrm{Tr}\left((\underline u,\alpha)_\infty\right) \qquad \forall\alpha\in k(D)^\times, $$ where $\mathrm{Tr}:{\mathbb W}_n(k)\to{\mathbb W}_n(\mathbf F_p)$ is the trace map. The computation of the conductor follows at once from its classical description in terms of norms of units (e.g. \cite{CL}, chap. XV, \S2). For an arbitrary field $k$, this classical approach no longer holds, but it is still possible to prove the desired equality using higher class field theory: see \cite{bry}, \S2. In the appendix we give an elementary proof of this formula; the reader who is willing to venture through it will certainly appreciate the difference between the clean and conceptual approach via class field theory and the following tedious but elementary computations. \bigskip \bigskip \bigskip \appendix \noindent {\bf Appendix: Elementary computation of the conductor of a cyclic extension.} \bigskip \bigskip Our goal is to show that the inequality in Proposition~\ref{bound} is indeed an equality. Again, our proof is by induction on $n$, the case $n=1$ being classical (e.g.~\cite{clac}, 4.4). We shall therefore assume that the Theorem is true for cyclic extensions of degree $p^l$, for $l\leq n$ and prove that it is true for $l=n+1$ by showing that, with the notations of Proposition~\ref{bound}, $m(f_{n+1})=M(f_{n+1})$. To this end, we need to compute the conductor of $C_{n+1}/D$ or, which amounts to the same, that of $C_{n+1}/C_n$. The latter is a cyclic extension of degree $p$ and hence it is described by an Artin-Schreier equation $y^p-y=z$, for a suitable rational function $z$ on $C_n$ which is not, however, in standard form i.e. $z$ has a pole of order divisible by $p$, and we cannot read off the conductor immediately. \bigskip Some work is needed in order to bring the Artin-Schreier equation describing the cover $C_{n+1}/C_n$ in standard form. The key to all our computations is the following, elementary \begin{lemma} \label{adjust} (Adjustment lemma). Let $f:Y\to X$ be a finite separable cover of curves of degree $p^n$, totally ramified at $y_0$. Let $t$ (resp. $s$) be local parameters at $y_0$ (resp. $x_0=f(y_0)$). Then, in a neighborhood of $y_0$ $$ s=t^{p^n}\left(\alpha_1^p+t^{\mu(f)}\alpha_0\right) $$ for some $\alpha_0,\alpha_1\in\mathcal{O}_{Y,y_0}^\times$, with $$ \mu(f)=\deg\mathfrak{D}_{Y/X,y_0}-p^n+1 $$ \end{lemma} \noindent Proof: Let $s=\alpha t^{p^n}$ for some unit $\alpha\in\mathcal{O}_{Y,y_0}^\times$ and expand $\alpha=\sum_{r\geq0}a_rt^r$ with $a_r\in k$ and $a_0\in k^\times$. If $\mu$ is the smallest positive integer prime to $p$ such that $a_\mu\neq0$, then $$ \deg\mathfrak{D}_{Y/X,y_0}=v_{y_0}\left(\frac{ds}{dt}\right) =p^n+v_{y_0}\left(\frac{d\alpha}{dt}\right)=p^n+\mu-1 $$ We can therefore collect: $\alpha=\sum_{0\leq r<\frac\mu p}a_{pr}t^{pr}+t^\mu\sum_{r\geq\mu}a_rt^r =\alpha_1^p+t^\mu\alpha_0$. \begin{cor} With the notations above, let $g$ be a rational function on $X$. Then $f^\ast(g)=g_1^p+g_0$ for suitable rational functions $g_i$ on $Y$ with $v_{y_0}(g_0)=p^nv_{x_0}(g)+\mu(f)$. \end{cor} \noindent Proof: Expanding $g$ as a power series in $s$, it is sufficient to prove it when $g(s)=as^r$ is a monomial of degree prime to $p$: $$ as^r=at^{p^nr}\left(\alpha_1^p+t^\mu\alpha_0\right)^r =at^{p^nr}\left(\alpha_1^{pr}+t^\mu\alpha_0^\prime\right) =(a^\frac1pt^{p^{n-1}}\alpha_1^r)^p+at^{p^nr+\mu}\alpha_0^\prime $$ \bigskip When the cover is Galois, it is easy to relate the number $\mu(f)$ to the conductor of the extension and how it varies in towers: \begin{prop} \label{condor} With the notation as in Lemma~\ref{adjust}, suppose moreover that $f$ is Galois of group $G$ and conductor $m(f)+1$. \begin{enumerate} \item If $e(f)$ is the largest integer such that $G_e\neq1$, then $$ \mu(f)=p^nm(f)-e(f) $$ \item Let $g:Z\to Y\to X$ be a second separable Galois cover, totally ramified above $x_0$ whose group $\tilde G$ is an extension of $G$ by $H=\mathbf{Z}/p\mathbf{Z}$. Let $e(g)$ be the largest integer such that $\tilde G_e\neq1$ and suppose that $\tilde G_{e(g)}\cap H\neq1$. If $m(g)+1$ denotes the conductor of $g$, then $$ e(g)=p^nm(g)-\mu(f) $$ \end{enumerate} \end{prop} \noindent Proof: (1). Denote, as usual, by $g_i=|G_i|$; as $f$ is totally ramified, $G_0=G$. By\cite{CL}, chap. IV, proposition 12, $m=\frac1{g_0}\sum_{i=0}^eg_i-1$, hence $$ \begin{array}{rl} \mu(f) & =\deg\mathfrak{D}_{Y/X,y_0}-p^n+1 \\ & = \sum_{i=0}^e(g_i-1)-p^n+1 \\ & = \sum_{i=0}^eg_i-e-p^n \\ & = p^n\left(\frac1{p^n}\sum_{i=0}^eg_i-1\right)-e. \end{array} $$ (2). Denote by $\varphi$ the Herbrand function of the cover $g$ and by $\tilde g_i=|\tilde G_i|$. The hypotheses imply that $\tilde g_i=pg_i$ for $0\leq i\leq e(f)$, hence $$ \begin{array}{rl} m(g)=\varphi(e(g)) & =\frac1{p^{n+1}}\left(\tilde g_1+\dots+\tilde g_{e(f)}+ p\,[e(g)-e(f)]\right) \\ & = \frac1p\varphi(e(f))+\frac1{p^n}[(e(g)-e(f)] \\ & = m(f)+\frac1{p^n}[e(g)-e(f)] . \end{array} $$ \begin{cor} \label{tele} Let $f_n:C_n\to D$ be a separable cyclic cover of degree $p^n$, totally ramified above $\infty$. For any $0\leq i\leq n$, let $f_i:C_i\to D$ be the quotient cover of degree $p^i$ and by $g_i:C_n\to C_i$ the subcover of degree $p^{n-i}$. Put $m_i=m(f_i)$ (resp. $e_i=e(f_i)$, resp. $\mu_i=\mu(f_i)$). Then: $$ e_n=\sum_{i=1}^np^{i-1}(m_i-m_{i-1}); \qquad \mu_n=\sum_{i=1}^n(p^i-p^{i-1})m_i; \qquad \mu(g_i)=\mu_n-p^{n-i}\mu_i. $$ \end{cor} \noindent Proof: From Proposition~\ref{condor}.2, we deduce the inductive relations $$ e_n-e_{n-1}=p^{n-1}(m_n-m_{n-1});\qquad \mu_n-\mu_{n-1}=(p^n-p^{n-1})m_n$$ from which the first two formulas in the claim follow at once. As for the third, it is a consequence of the transitivity of the different of a composite extension: $$ \mu(g_i)=\deg\mathfrak{D}_{C_n/C_i}-p^{n-i}+1 =\deg\mathfrak{D}_{C_n/D}-p^n+1 -p^{n-i}\left(\deg\mathfrak{D}_{C_i/D}-p^i+1\right). $$ \bigskip We are now ready to resume the discussion interrupted before Lemma~\ref{adjust}. We are given a separable cyclic cover $f_{n+1}:C_{n+1}\to D$ of degree $p^{n+1}$, as in the theorem, and we want to compute its conductor, to show that the inequality provided by Proposition~\ref{bound} is indeed an equality. Equivalently, we can compute the conductor of the degree $p$ subextension $C_{n+1}/C_n$. With the notations as in Proposition~\ref{condor}, let $\tilde G=\mathbf{Z}/p^{n+1}\mathbf{Z}$ be the group of $C_{n+1}/D$ and $H=\mathbf{Z}/p\mathbf{Z}$ that of $C_{n+1}/C_n$: by~\cite{CL}, IV, prop. 2, $H_i=H\cap\tilde G_i$ and it is clear that $H=H_{e_{n+1}}=\tilde G_{e_{n+1}}$ is the last nontrivial subgroup for both extensions. \bigskip We begin by making the Artin-Schreier equation defining the cover $C_{n+1}/C_n$ more explicit. Away from $\infty$, the cover $C_{n+1}/D$ is described by the Witt vector equation \begin{equation} \label{WAS} F(\underline{Y})-\underline{Y}=(u_0,\dots,u_n) \end{equation} where the left hand side should be understood as a difference of two vectors, while the equation defining $C_n/D$ is the image of (\ref{WAS}) under the restriction homomorphism $\mathbf r:\Bbb{W}_{n+1}\to\Bbb{W}_n$. Therefore, the $n$-th component of (\ref{WAS}) is $$ Y_n^p-Y_n+c_n\left(F(\underline{Y}),-\underline{Y}\right)=u_n $$ where the polynomial $c_n(\underline{X},\underline{Y})$ is the one defined by equation (\ref{S_n}). If $\underline{y}=(y_0,\dots,y_{n-1})$ denotes any solution of the Artin-Schreier-Witt equation defining $C_n/D$, the cover $C_{n+1}/C_n$ is described by the Artin-Schreier equation \begin{equation} \label{AS} Y_n^p-Y_n=u_n-c_n\left(\underline{y}^p,-\underline{y}\right). \end{equation} As remarked, this equation is not in standard form: the datum on the right hand side has, in general, poles of order divisible by $p$. We can get rid of them by adding terms of the type $\wp(h)=h^p-h$ for suitable rational functions $h$ on $C_n$; the result will be a rational function with a pole at $\infty$ of order $e_{n+1}$. By Proposition~\ref{condor}.2, $e_{n+1}=p^nm_{n+1}-\mu_n$. To complete the proof of the theorem, we should then prove the following: \begin{claim} The right hand side of equation (\ref{AS}) is congruent $\bmod\,\wp\left(k(C_n)\right)$ to a function with a pole at $\infty$ of order $p^nM-\mu_n$, where $M=\max\,\{p^{n-i}\nu_i:\,0\leq i\leq n\}$. \end{claim} It is easy to handle the term $u_n$ in (\ref{AS}) with the adjustment lemma: $u_n$ is a rational function on $D$ with a pole of order $\nu_n$ at $\infty$, hence we can find rational functions $u_n^\prime,\,u_n^{\prime\prime}$ on $C_n$ such that: \begin{equation} \label{new} u_n={u_n^\prime}^p+u_n^{\prime\prime} \qquad \mathrm{with}\ v_n(u_n^{\prime\prime})=-p^{n}\nu_n+\mu_n \end{equation} We can then rewrite $u_n=u_n^{\prime\prime}+u_n^\prime+\wp(u_n^\prime)$. When $M=\nu_n$, the function $u_n^{\prime\prime}$ has the predicted valuation and in any event the contribution of $u_n^\prime$ is negligible, as $v_n(u_n^\prime)=-p^{n-1}\nu_n\geq-p^{n-1}M$, hence \begin{equation} \label{rough} \begin{array}{rl} -p^nM+\mu_n+p^{n-1}\nu_n &\leq -(p^n-p^{n-1})M+\sum_{i=1}^n(p^i-p^{i-1})m_i\\ & < -m_n(p^{n+1}-2p^n+1) <0 \end{array} \end{equation} as $m_ii$, then by the induction hypothesis we can write $$ c_h=-X_i^{p^{h-i}-1} \left(Y_i+c_i\right) +R_{i,h} $$ hence the highest power of $X_i$ in (\ref{h-term}) appears in the term $c_h^{p^{n-h}}$ and its contribution in $c_n$ is of degree at most $(p^{h-i}-1)p^{n-h}$. \noindent For $h=i$, the variable $X_i$ does not appear in $c_i$ hence the term containing the highest power of $X_i$ in (\ref{h-term}) is $$ -{p^{n-i}\choose1}X_i^{p^{n-i}-1}Y_i-{p^{n-i}\choose p^{n-i}-1}{p^{n-i}-1\choose p^{n-i}-1}X_i^{p^{n-i}-1}c_i =-p^{n-i}X_i^{p^{n-i}-1} \left(Y_i+c_i\right). $$ \bigskip Our task is now to compute the valuation at $\infty$ of a rational function on $C_n$, congruent to $c_n(\underline{y}^p,-\underline{y}) \bmod\,\wp(k(C_n))$ and having a pole of order prime to $p$, where $\underline{y}=(y_0,\dots,y_{n-1})$ is a vector of solutions for the Artin-Schreier-Witt equation defining $C_n/D$. Lest the reader should get lost or depressed (or both) by the following computations, let us state immediately that we are going to show that, for any $i\leq n-1$, only the monomials $$ y_i^{p(p^{n-i}-1)+1} $$ are relevant. More precisely, if $j\leq n-1$ is the (necessarily unique) integer for which $$ m_n=p^{n-j-1}\nu_j=\max\,\{p^{n-i-1}\nu_i:\,0\leq i\leq n-1\} $$ then $y_j^{p(p^{n-j}-1)+1}$ is congruent $\bmod\,\wp\left(k(C_n)\right)$ to a function with a pole at $\infty$ of order $p^{2n-j}\nu_j-\mu_n$ and that $c_n(\underline{y}^p,-\underline{y})-y_j^{p(p^{n-j}-1)+1}$ is congruent to a function with a pole of order strictly smaller. \bigskip The approach is in two steps: first, we obtain an estimate of the valuation of $c_n(\underline{y}^p,-\underline{y})$ and then we apply the adjustment lemma to the functions $y_i$ to get rid of the terms in $\wp(k(C_n))$. In both steps, the problem to sort out the relevant monomials is reduced to a linear optimization. The proof is only a tedious exercise in linear programming, and the reader would loose very little, should he indulge in the temptation to skip the rest of this section. \bigskip When the variables $X_i$ and $Y_i$ are given the weight $p^i$, the polynomial $c_n(\underline{X},\underline{Y})$ is isobaric of weight $p^n$; evaluating at $\underline{X}=\underline{y}^p$ and $\underline{Y}=-\underline{y}$, our task is to compute the valuation of a sum of monomials $\prod_{i=0}^{n-1}y_i^{pa_i+b_i}$. Put $v_n(y_i)=w_i$; regarding $a_i$ and $b_i$ ad variables, the question can be rephrased as an optimization problem for a linear function under a linear constraint: \begin{equation} \label{LP1} \mathrm{Minimize:} \sum_{i=0}^{n-1}w_i(pa_i+b_i)\qquad \mathrm{under:}\, \sum_{i=0}^{n-1}p^i(a_i+b_i)=p^n \end{equation} with $a_i\in[0,p^{n-i}-1]$ and $b_i\in[0,p^{n-i}]$. The range of the $a_i$'s is restricted by Lemma~\ref{c_n}; we do not apply it to the $b_i$'s because of the irregular behavior in characteristic $2$ of the polynomials $I_i(\underline Y)$ expressing the opposite of a Witt vector. Actually, it is more convenient to rephrase the problem slightly: putting $\alpha_i=pa_i+b_i$ and taking $\alpha_i,b_i$ as variables, it becomes: \begin{equation} \label{LP2} \mathrm{Minimize:}\ \sum_{i=0}^{n-1}w_i\alpha_i\qquad \mathrm{under:}\, \sum_{i=0}^{n-1}p^i\alpha_i+\sum_{i=0}^{n-1}(p^{i+1}-p^i)b_i=p^{n+1} \end{equation} with $\alpha_i\in[0,p^{n-i+1}-p+1]$ and $b_i\in[0,p^{n-i}]$. Extremal values of a linear function on a convex polytope are attained at the vertices; as $b_i=p^{n-i}$ forces $a_i=0$, hence $\alpha_i=p^{n-i}$, best admissible solutions are for $\alpha_i=p^{n-i+1}-p+1$, $b_i=1$ and and $\alpha_l=b_l=0$ for $l\neq i$. We have thus: $$ v_n(c_n(\underline{y}^p,-\underline{y})) \geq\min\left\{(p^{n-i+1}-p+1)w_i\,|\quad0\leq i\leq n-1\right\} $$ and this in turn implies our first estimate: \begin{lemma} \label{sort} \begin{enumerate} \item $v_n(c_n(\underline{y}^p,-\underline{y}))\geq-(p^{n+1}-p+1)m_n$. \item $v_{n+1}(y_n)\geq-p^nm_{n+1}$, with an equality if and only if $\nu_n=m_{n+1}$. \end{enumerate} \end{lemma} \noindent Proof: Induction on $n$: the case $n=0$ being clear, assume (1) and (2) hold for $c_i$ and $y_i$ for all $i\leq n-1$. Then, since $m_n\geq p^{n-i-1}m_{i+1}$, we have $$ \begin{array}{rl} v_n(y_i^{p(p^{n-i}-1)+1}) & = p^{n-i-1}v_{i+1}(y_i^{p(p^{n-i}-1)+1}) \\ & \geq -p^{n-i-1}(p^{n+1}-p^{i+1}+p^i)m_{i+1} \\ & \geq -(p^{n+1}-p^{i+1}+p^i)m_n \end{array} $$ Finally, as $y_n$ is a solution of the equation $y_n^p-y_n=u_n-c_n$ and clearly $v_n(u_n)=-p^n\nu_n\geq-p^nm_{n+1}$, with equality holding if and only if $\nu_i=m_{i+1}$, (2) follows at once. \bigskip To compute the valuation of $c_n(\underline{y}^p,-\underline{y})\bmod\wp(k(C_n))$, we have to apply the adjustment lemma twice, first on $C_i$ and then on $C_n$. Fix $i\leq n-1$ and denote by $v_i$ the discrete valuation of $\mathcal{O}_{C_i,\infty}$. By definition, $y_i$ is the solution (on $C_{i+1}$) of the Artin-Schreier equation $y_i^p-y_i=u_i-c_i(\underline{y}^p,-\underline{y})$; applying the adjustment lemma to $u_i$ and the induction hypothesis to $c_i$, we can find rational functions $g_i$ and $\tilde u_i$ on $C_i$ such that $$ u_i-c_i(\underline{y}^p,-\underline{y})=\tilde u_i+g_i^p-g_i $$ with $v_i(\tilde u_i)=p^{i}m_{i+1}-\mu_{i}$ and, by the lemma above, $v_i(g_i^p)\geq-p^nm_{n+1}$, equality holding iff $\nu_i=m_{i+1}$. On $C_{i+1}$ we have thus $y_i=\tilde y_i+g_i$ with $$ v_{i+1}(\tilde y_i)=-p^im_{i+1}+\mu_i; \qquad v_{i+1}(y_i)=v_{i+1}(g_i)\geq p^im_{i+1} $$ with an equality iff $\nu_i=m_{i+1}$. We now apply the adjustment lemma on $C_n$ both to $\tilde y_i$ and $g_i$: we can write $$ \tilde y_i=\tilde\gamma_i^p+\tilde\eta_i \qquad g_i={\gamma_i^\prime}^p+\eta_i^\prime $$ with $v_n(\tilde\gamma_i)=p^{-1}v_n(y_i)=-p^{n-2}m_{i+1}+p^{n-i-1}\mu_{i+1}$ and, applying Corollary~\ref{tele}, $$ \begin{array}{rl} v_n(\tilde\eta_i) & =v_n(\tilde y_i)+\mu(C_n/C_{i+1}) \\ & = -p^{n-i-1}(p^im_{i+1}-\mu_i)+ \mu_n-p^{n-i-1}\mu_{i+1} \\ & = -p^{n-1}m_{i+1}-p^{n-i-1}(p^{i+1}-p^i)m_{i+1}+\mu_n \\ & = -p^nm_{i+1}+\mu_n. \end{array} $$ As $g_i$ comes from $C_i$, we have $v_n(\gamma_i^\prime)=p^{-1}v_n(g_i)\geq-p^{n-2}m_{i+1}$, with equality iff $\nu_i=m_{i+1}$ and $$ \begin{array}{rl} v_n(\eta_i^\prime) & =v_n(g_i)+\mu(C_n/C_i) \\ & = p^{n-i}v_i(g_i)+\mu_n-p^{n-i}\mu_i \\ & \geq -p^{n-1}m_{i+1}+\mu_n-p^{n-i}\mu_i \end{array} $$ Trivially, $v_n(\gamma_i^\prime)\lneq v_n(\tilde\gamma_i)$; the same rough estimates as in equation (\ref{rough}) yield $v_n(\tilde\eta_i)\lneq v_n(\eta_i^\prime)$. Putting $\gamma_i=\tilde\gamma_i+\gamma_i^\prime$ and $\eta_i=\tilde\eta_i+\eta_i^\prime$, we can write $$ y_i=\gamma_i^p+\eta_i \qquad \textrm{with:} \quad v_n(\eta_i)=-p^nm_{i+1}+\mu_n; \ v_n(\gamma_i)\geq-p^{n-2}m_{i+1} $$ again the last inequality being an equality iff $\nu_i=m_{i+1}$. Substituting in $\prod_{i=0}^{n-1}y_i^{\alpha_i}$ we get: $$ \prod_{i=0}^{n-1}(\gamma_i^p+\eta_i)^{\alpha_i} =\prod_{i=0}^{n-1}\gamma_i^{p\alpha_i}+ \sum_{i=0}^{n-1}\left(\alpha_i\gamma_i^{p\alpha_i-1}\eta_i \prod_{h\neq i}\gamma_h^{p\alpha_h}\right)+ \dots $$ The first term is a $p$-power and, by Lemma~\ref{sort}, its valuation is bounded from below by $-(p^{n+1}-p+1)m_n$; the same estimates as those of formula (\ref{rough}) show that its $p$-th root has a pole of order strictly smaller than $p^{n+1}m_n-\mu_n$ and we can get rid of it $\bmod\wp(k(C_n))$. \bigskip To prove the claim, we should therefore compute, for all $i,h\leq n-1$, the valuation of the monomials $\gamma_i^{p\alpha_i-1}\eta_i \prod_{h\neq i}\gamma_h^{p\alpha_h}$. Our optimization problem (\ref{LP2}) becomes then: $$ \mathrm{Minimize:}\ v_n(\eta_i)+w_i\alpha_i+ \sum_{h\neq i}w_h\alpha_h\qquad \mathrm{under:}\, \sum_{l=0}^{n-1}p^l\alpha_l+\sum_{l=0}^{n-1}(p^{l+1}-p^l)b_l=p^{n+1} $$ with $\alpha_l\in[0,p^{n-l+1}-p+1]$ and $b_l\in[0,p^{n-l}]$. \bigskip \noindent \textbf{Proof of Claim \ref{old}.} All we have to do is check the extremal values of the function above, for all $i$. For $\alpha_i=p^{n-i+1}-p+1$ and $b_i=1$ we get $$ v_n(\gamma_i^{p^{n-i+2}-p^2}\eta_i)\geq -(p^{n-i+2}-p^2)p^{n-2}m_{i+1}-p^nm_{i+1}+\mu_n=-p^{2n-i}m_{i+1}+\mu_n $$ with an equality iff $\nu_i=m_{i+1}$. Hence, if $j\leq n-1$ is the (necessarily unique) integer for which $$ m_n=p^{n-j-1}\nu_j=\max\,\{p^{n-i-1}\nu_i:\,0\leq i\leq n-1\} $$ then $y_j^{p(p^{n-j}-1)+1}$ is congruent $\bmod\,\wp\left(k(C_n)\right)$ to a function with a pole at $\infty$ of order $p^{2n-j}\nu_j-\mu_n$. If $i\neq j$ then $p^{n-i-1}\nu_i\lneq m_n$ and the contribution of $y_i$ is strictly smaller. \bigskip \noindent Finally, consider $c_n(\underline{y}^p,-\underline{y})-y_j^{p(p^{n-j}-1)+1}$: repeating the argument above, to show that it is congruent $\bmod\wp(k(C_n))$ to a function with a pole of order strictly smaller than $p^{2n-j}\nu_j-\mu_n$, it suffices to show that the term containing the highest power of $y_j$, namely $$ y_j^{p(p^{n-j}-1)}c_j(\underline{y}^p,-\underline{y}) $$ (cf. Lemma \ref{c_n}) satisfies this property. As above, after a double application of the adjustment lemma on $C_n$, we can write $$ c_j(\underline{y}^p,-\underline{y})=\theta_j^p+\xi_j $$ with $v_n(\theta_j^p)=p^{n-j}v_j(c_j)\geq-p^{n-j}(p^{j+1}-p+1)m_j$ (cf. Lemma \ref{sort}) and $$ v_n(\xi_j)=-p^{n-j}(p^{j+1}m_j-\mu_j)+\mu_n-p^{n-j}\mu_j =-p^{n+1}m_j+\mu_n. $$ Therefore, $\bmod\wp(k(C_n))$ we can get rid of $y_j^{p(p^{n-j}-1)}\theta_j^p$ and, by Lemma \ref{sort}.2, $$ \begin{array}{rl} v_n(y_j^{p(p^{n-j}-1)}\xi_j) & = p(p^{n-j}-1)p^{n-j-1}v_{j+1}(y_j) -p^{n+1}m_j+\mu_n\\ & = -(p^{n+1}-p^{j+1})p^{n-j-1}\nu_j -p^{n+1}m_j+\mu_n\\ & = -p^{2n-j}\nu_j+p^j(\nu_j-pm_j)+\mu_n \end{array} $$ and the assumption on $j$ implies that $\nu_j-pm_j$ is strictly positive. \bigskip \begin{thebibliography}{Marco 1999} \bibitem[AC IX]{AC} \textsc{N. Bourbaki}, \textit{Alg\`ebre Commutative, Chap IX}, Masson, Paris (1983). \bibitem[E.G.A. II]{EGA2} \textsc{A. Grothendieck et J. Dieudonn\'e}, \textit{El\'ements de G\'eom\'etrie Alg\'ebrique, Chap II}, Publications Math\'ematiques de l'I.H.E.S. \textbf{8} (1961). \bibitem{bry} \textsc{J.-L. 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