\documentclass[CRMATH,Unicode,biblatex,XML]{cedram} \TopicFR{Théorie des nombres} \TopicEN{Number theory} \addbibresource{CRMATH_Lucas_20221138.bib} \usepackage{mathrsfs} % Abréviations ensembles \newcommand{\Z} {\mathbb{Z}} \newcommand{\N} {\mathbb{N}} \newcommand{\F} {\mathbb{F}} \newcommand{\G} {\mathbb{G}} \newcommand{\att}{K\!\left[\!\left[\frac{1}{t}\right]\!\right]} \newcommand{\btt}{K\!\left(\!\left(\frac{1}{t}\right)\!\right)} \let\dfrac\frac \let\tfrac\frac \makeatletter \DeclareFontFamily{U}{mathx}{\hyphenchar\font45} \DeclareFontShape{U}{mathx}{m}{n}{ <5> <6> <7> <8> <9> <10> <10.95> <12> <14.4> <17.28> <20.74> <24.88> mathx10 }{} \DeclareSymbolFont{mathx}{U}{mathx}{m}{n} \DeclareFontSubstitution{U}{mathx}{m}{n} \DeclareMathAccent{\widecheck} {0}{mathx}{"71} \makeatother \DeclareMathOperator{\charrm}{char} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \let\save@mathaccent\mathaccent \newcommand*\if@single[3]{% \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% \ifdim\ht0=\ht2 #3\else #2\fi } %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} %If there's a superscript following the bar, then no negative kern may follow the bar; %an additional {} makes sure that the superscript is high enough in this case: \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} %Use a separate algorithm for single symbols: \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} \newcommand*\wide@bar@[3]{% \begingroup \def\mathaccent##1##2{% %Enable nesting of accents: \let\mathaccent\save@mathaccent %If there's more than a single symbol, use the first character instead (see below): \if#32 \let\macc@nucleus\first@char \fi %Determine the italic correction: \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}% \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}% \dimen@\wd\tw@ \advance\dimen@-\wd\z@ %Now \dimen@ is the italic correction of the symbol. \divide\dimen@ 3 \@tempdima\wd\tw@ \advance\@tempdima-\scriptspace %Now \@tempdima is the width of the symbol. \divide\@tempdima 10 \advance\dimen@-\@tempdima %Now \dimen@ = (italic correction / 3) - (Breite / 10) \ifdim\dimen@>\z@ \dimen@0pt\fi %The bar will be shortened in the case \dimen@<0 ! 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Anderson defined $t$-modules and the notion of purity. Related to $t$-modules, C. Namoijam and M. A. Papanikolas defined the notion of almost strict purity in~\cite[Remark~4.5]{derivative} (concept already mentioned by G.Anderson in~\cite[Section~2.2]{Anderson} and D.Goss in~\cite[Remark~5.5]{Goss}), then they proved that an almost strictly pure $t$-module is pure. We are interested here in the reciprocal, and with the help of the work of A. Maurischat in~\cite{maurischat} we show that these two notions are not equivalent by presenting a counter-example (see Theorem~\ref{4}). \section{Purity} Let $K$ a perfect field containing $\F_q$. We let $\tau:K\to K$ denote the $q$-th power Frobenius map and $K\{\tau\}$ be the ring of twisted polynomials in $\tau$ over $K$, subject to the relation, $\tau a =a^q\tau$ for any $a\in K$. We further consider the skew Laurent series ring over $K$ in $\sigma:=\tau^{-1}$ \[ K(\{\sigma\}):=\left\{\sum\limits_{i=k}^{\infty} a_i\sigma^i \;\middle|\; k\in \Z, a_i\in K\right\}.\] Consider $\ell:\F_q[t]\to K$ a homomorphism of $\F_q$-algebras and denote $\sigma=\tau^{-1}$. A $t$-module $(E, \varphi)$ over $K$ of dimension $d$ is by definition an $\F_q$-vector space scheme $E$ over $K$ isomorphic to $ \G_a^d$ together with a homomorphism of $\F_q$-algebras $\varphi : \F_q[t] \to \operatorname{End}_{\operatorname{grp},\F_q}(E)$ into the ring of $\F_q$-vector space scheme endomorphisms of $E$, such that for all $a \in \F_q[t]$, the endomorphism $d\varphi_a$ on $\operatorname{Lie}(E)$ induced by $\varphi_a$ fulfills the condition that $d\varphi_a - \ell(a)$ is nilpotent. We will fix in the following $(E,\phi)$ a $t$-module on $K$ of dimension $d$ as well as a coordinate system $\kappa$, i.e. an isomorphism of schemes in $\F_q$-vector spaces $\kappa:E\simeq \G_a^d$ defined on $K$. With respect to this coordinate system, we can represent $\phi_t$ by a matrix $D\in M_d(K\{\tau\}).$ Let $\operatorname{pr}_i:\G_a^d\to \G_a$ ($1\leq i\leq d)$ be the projection to the $i$-th component of $\G_a^d$, and let $\kappa_i=\kappa\circ \operatorname{pr}_i$. Let $\check{\kappa}_j:\G_a\to E$ be defined by $\check{\kappa}_j=\kappa^{-1}\circ \operatorname{inj}_j$ where $\operatorname{inj}_j:\G_a\to \G_a^d$ is the natural injection into the $j-$th component. We say that $E$ is almost strictly pure if there is some integer $s\geq1$ such that \[ D^s=A_0+A_1\tau+\cdots +A_r\tau^r\] with $A_r\in GL_d(K)$. The $t$-motive $M(E)$ of $E$ is the free $K\{\tau\}$-module of rank $d$ with base $\{\kappa_1,\ldots ,\kappa_d\}$ with a $t$-action on this base defined by \[ t.\begin{pmatrix} \kappa_1 \\ \vdots \\ \kappa_d\end{pmatrix}=D\begin{pmatrix} \kappa_1 \\ \vdots \\ \kappa_d\end{pmatrix}. \] We define in a similar way the dual $t$-motive $\mathscr{M}$ as the free $K\{\tau\}$-module of rank $d$ of basis $\{\check{\kappa}_1,\ldots ,\check{\kappa}_d\}$ whose $t$-action (on the right) on this basis is defined by \[ \begin{pmatrix} \check{\kappa}_1 & \cdots & \check{\kappa}_d\end{pmatrix}\cdots t=\begin{pmatrix} \check{\kappa}_1 & \cdots & \check{\kappa}_d\end{pmatrix}D. \] We say that $E$ is abelian if $M(E)$ is a finitely generated $K[t]$-module. In this case, we define $w(M)$ the weight of $M$ by \[ w(M)=\frac{d}{\operatorname{rk}(E)} \] where $\operatorname{rk}(E)$ is the rank of $M$ as a $K[t]$-module (that is finite because $E$ is abelian). We moreover consider: \begin{itemize} \item The ring of formal power series in $\frac{1}{t}$ with coefficients in $K$ denoted by $\att$. \item The field of Laurent series in $\frac{1}{t}$ with coefficients in $K$ denoted by $\btt$ (that is the field of fractions of $\att$). \end{itemize} The $t$-motive $M(E)$ and the $t$-module $E$ are called pure if there exists a $\att$-lattice $\Lambda$ in $\btt \otimes_{K[t]}M$ as well as positive integers $u,v\in \N$ such that \[ t^u\Lambda=\tau^v\Lambda. \] We will use the following result, proved by A. Maurischat in~\cite[Theorem~6.6, Theorem~7.2]{maurischat}, characterizing the fact of being abelian and being pure using Newton polygons. \begin{theoreme}[Maurischat]\label{3} The $t$-module $E$ is abelian if and only if the Newton polygon $N_{\lambda_d}$ of the last invariant factor $\lambda_d$ of the matrix $D$ has only positive slopes. In this case, $E$ is pure if and only if $N_{\lambda_d}$ has exactly one edge. Then we have that the weight of $M$ equals the reciprocal of the slope of the edge. \end{theoreme} Here we recall that the invariant factors of a matrix $D\in M_d(K\{\tau\})$ are obtained by diagonalizing the matrix $tI_d-D\in M_d(K(\{\sigma\})[t])$ by performing elementary operations on the rows and columns in $K(\{\sigma\})[t]$ that are the following (denote by $L_i$ (resp $C_i$) the $i$-th row (resp the $i$-th column)): \begin{itemize} \item add to the $i$-th row $L_i$ the $j$-th row $L_j$ multiplied on the left by $a\in K(\{\sigma\})[t]$ denote by $L_i\to L_i+a.L_j$ (resp add to $i$-th column $C_i$ the $j$-th column $C_j$ multiplied on the right by $a\in K(\{\sigma\})[t]$ denote by $C_i\to C_i+C_j\cdot a),$ \item multiply on the left $i$-th the row $L_i$ by an element $u$ of $K(\{\sigma\})^\ast$ denoted by $L_i\to uL_i$ (resp multiply on the right the $i$-th column $C_i$ by $u$ denoted by $C_i\to C_i u)$, \item exchanging two lines $L_i$ and $L_j$ (resp two columns $C_i$ and $C_j$) denote by $L_i\leftrightarrow L_i$ (resp $C_i\leftrightarrow C_j$). \end{itemize} Contrary to the commutative case the invariant factors are only unique up to similarity. \begin{exam*} In~\cite{maurischat}, Maurischat defined the $t$-module given by the matrix \[ M:=\begin{pmatrix} \theta & 0 \\ 1& \theta \end{pmatrix}+\begin{pmatrix} 0 & 0 \\1 &0 \end{pmatrix}\cdot \tau+\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\cdot \tau^2+\begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}\cdot \tau^3\in M_2(K\{\tau\}). \] By diagonalizing the matrix $tI_2-M$ we get the matrix \[ \begin{pmatrix}1 &0 \\ 0 & \lambda_2\end{pmatrix} \] where \[ \lambda_2= \left(-\sigma^{-3}+(\theta+\theta^{q^2})\sigma^{-2}+\theta^{q^3+1}\right)-\left(2\sigma^{-2}+\theta^{q^{-3}}+\theta\right)\cdot t+t^2\in K(\{\sigma\}). \] If $\charrm (K)=2$ then we represent the Newton polygon of $\lambda_2$ in Figure~\ref{Fig1}. It has only one edge of slope $\dfrac{3}{2}$ hence by Theorem~\ref{3} the $t$-module is abelian and pure of weight $\dfrac{2}{3}.$ \end{exam*} The authors of~\cite{derivative} showed in the same paper the next result. \begin{theoreme}[Namoijam, Papanikolas]\label{pspur} With the previous notation, an almost strictly pure $t$-module is pure of weight $\frac{s}{r}$. \end{theoreme} We now turn our interest to the reciprocal of the above result, and we answer negatively. \begin{theoreme}\label{4} For any integer $d\geq 2$ there exists a pure but not almost strictly pure $t$-module of dimension $d$. \end{theoreme} \begin{proof} We first consider the case $d=2$. Let us note $\theta=\ell(t)$. Consider the $t$-module given by the matrix \[ D_2=\begin{pmatrix} \theta & 0 \\ 1&\theta\end{pmatrix}+\begin{pmatrix}1 & 1 \\ \theta& \theta\end{pmatrix}\cdot \tau\in M_d(K\{\tau\}). \] Let us diagonalize the matrix $tI_2-D_2$ (the diagonalization is being taken over $K(\{\sigma\}))$: \begin{gather*} \begin{pmatrix} t-\theta-\tau & -\tau \\ -1-\theta\tau & t-\theta-\theta\tau \end{pmatrix} \xrightarrow{L_1\leftrightarrow L_2} \begin{pmatrix} -1-\theta\tau& t-\theta-\theta\tau \\ t-\theta-\tau&-\tau \end{pmatrix} \xrightarrow{C_1\to C_1\gamma} \begin{pmatrix} 1 &t-\theta-\theta\tau \\ (t-\theta-\tau)\gamma & -\tau \end{pmatrix} \\ \begin{pmatrix} 1 &t-\theta-\theta\tau \\ (t-\theta-\tau)\gamma & -\tau \end{pmatrix} \xrightarrow{L_2\to L_2-(t-\theta-\tau)\gamma L_1} \begin{pmatrix} 1& t-\theta-\theta\tau \\ 0&-\tau -(t-\theta-\tau)\gamma(t-\theta -\theta\tau) \end{pmatrix} \\ \begin{pmatrix} 1& t-\theta-\theta\tau \\ 0&-\tau -(t-\theta-\tau)\gamma(t-\theta -\theta\tau) \end{pmatrix} \xrightarrow{C_2\to C_2+C_1(-t+\theta+\theta\tau)} \begin{pmatrix} 1&0 \\0&\lambda'\end{pmatrix} \xrightarrow{L_2\to -\gamma^{-1}L_2} \begin{pmatrix} 1 & 0 \\ 0 & \lambda_2\end{pmatrix} \end{gather*} where \[ \begin{aligned}\lambda_2&=-\gamma^{-1}\lambda' \\ &=-\gamma^{-1}\left(-\tau-(t-\theta-\tau)\gamma(t-\theta-\theta\tau)\right)\\ &=t^2+t\cdot (-\theta -\theta\tau -\gamma^{-1}\theta\gamma-\gamma^{-1}\tau\gamma)+\gamma^{-1}\tau+\gamma^{-1}\theta\gamma\theta+\gamma^{-1}\theta\gamma\theta\tau+\gamma^{-1}\tau\gamma\theta+\gamma^{-1}\tau\gamma\theta\tau\end{aligned} \] and $\gamma=(-1-\theta\tau)^{-1}$. \goodbreak We represent the Newton polygon of $\lambda_2$ in Figure~\ref{Fig2}. It has only one edge of positive slope equal to $1$, hence according to Theorem~\ref{3} this $t$-module is pure of weight 1. An immediate induction shows that for $n\geq 2$, the leading coefficient of $D_2^n$ is given by the matrix \[ \prod\limits_{k=1}^{n-1}(1+\theta^{q^k})\cdot A \] where \[ A=\begin{pmatrix}1 & 1 \\ \theta& \theta\end{pmatrix} \] whose determinant is zero, so this $t$-module is not almost strictly pure. \end{proof} Now we consider the general case $d>2$. We put $m:=d-2>0$ and consider the $t$-module given by the matrix \[ D_{2+m}=\begin{pmatrix} D_2 & \empty &\empty &\empty\\ \empty &\theta+\tau &\empty &\empty \\ \empty& \empty & \ddots &\empty \\ \empty & \empty & \empty & \theta+\tau \end{pmatrix}\in M_{2+m}(K\{\tau\}). \] This $t$-module is the direct sum of pure $t$-modules of weight $1$ (the $t$-module associated to $D_2$ and $d-2$ copies of the Carlitz module), so we can prove it is a pure $t$-module of weight $1$, but here we give a proof using Maurischat's algorithm. For $n\geq 1$, the leading coefficient of the matrix $D_{2+m}^n$ is the matrix \[ \begin{pmatrix} D_2^n &\empty \\ \empty & I_m\end{pmatrix} \] whose determinant is zero, so this $t$-module is not almost strictly pure. Consider $(J_0)$ the algorithm that diagonalize as previously the matrix $tI_2-D_2$. Applying $(J_0)$ and exchanging row and columns, we get the matrix: \[ tI_{2+m}-D_{2+m}\to S=\begin{pmatrix} 1 & \empty &\empty &\empty &\empty \\ \empty &t-\theta-\tau &\empty &\empty &\empty \\ \empty& \empty & \ddots &\empty &\empty \\ \empty & \empty & \empty & t-\theta-\tau & \empty \\ \empty & \empty & \empty & \empty & \lambda_2 \\ \end{pmatrix}\in M_{2+m}(K\{\tau\}[t]). \] Consider the euclidean division of $\lambda_2$ by $t-\theta-\tau$: \[ \lambda_2=q(t-\theta-\tau)+r, \ r\neq0 \text{ and } \operatorname{deg}_t(r)=0. \] Let \[ S'=\begin{pmatrix} t-\theta-\tau & \empty \\ \empty & \lambda_2\end{pmatrix}\in M_2(K\{\tau\}[t]). \] We apply the following operations to the matrix $S'$ (and denote by $(J_1)$ this algorithm): \[ \begin{aligned}L_{2}&\to L_{2}-qL_{1} \\ C_{2}&\to C_{2}+C_{1} \\ L_{2}&\to r^{-1}L_{2}\\ L_{1}&\to L_{1}-(t-\theta-\tau)L_{2}\\ C_{2}&\to C_2 -C_{1}r^{-1}\lambda_2 \\ L_1&\leftarrow\mkern -8mu\rightarrow L_2 \\ L_2&\to -L_2. \end{aligned} \] We get the matrix: \[ \begin{pmatrix} \lambda_2 & \empty \\ \empty & (t-\theta-\tau)r^{-1}\lambda_2 \end{pmatrix}. \] By successively applying the algorithm $(J_1)$ to the matrices $S'$ which appear from the matrix $S$, we obtain the matrix \[ S\to \begin{pmatrix} 1 & \empty &\empty &\empty &\empty \\ \empty &\lambda_2 &\empty &\empty &\empty \\ \empty& \empty & (t-\theta-\tau)r^{-1}\lambda_2&\empty &\empty \\ \empty & \empty & \empty & \ddots& \empty \\ \empty & \empty & \empty & \empty & (t-\theta-\tau)r^{-1}\lambda_2 \end{pmatrix}. \] As $\lambda_2$ and $t-\theta-\tau$ have Newton polygons consisting of only one edge of slope 1, the Newton polygon of the last coefficient of the last matrix has only one edge of slope $1$. It follows that the Newton polygon of the last invariant factor of $D_{m+2}$ has only one edge of slope 1. Hence $D_{m+2}$ is also a $t$-module which is pure of weight 1 but not almost strictly pure for all $m\geq 0$. \begin{figure}[htb] \centering \begin{minipage}[t]{0.45\linewidth}\centering \includegraphics{Figures/Figure1.pdf} \caption{\centering Newton polygon of the Anderson module $M$ constructed by Maurischat when $\charrm (K)=2$ .}\label{Fig1} \end{minipage} \hfill \begin{minipage}[t]{0.45\linewidth}\centering \includegraphics{Figures/Figure2.pdf} \caption{Newton polygon of the Anderson module $D_2$ in Theorem~~\ref{4}.}\label{Fig2} \end{minipage} \end{figure} \section*{Aknowledgements} This result is part of my master-Thesis at University of Caen under the supervision of Floric Tavares Ribeiro and Tuan Ngo Dac that I would like to thank. \section*{Declaration of interests} The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations. \printbibliography \end{document}