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\title{Uniformizability of elementary Anderson $t$-motives of dimension $2$}
\author{David Niedbala Giraudin}
\address{Independent researcher, Meaux, France}
\thanks{ORCID 0009-0009-1526-1178}
\date{October 30, 2026}
\begin{document}
\begin{abstract}
Let $M(A)$, $A\in M_2(\C)$, be the elementary Anderson $t$-motive of dimension $2$ and rank $4$ defined by
$Te=\theta e+A\tau e+\tau^2e$. Using the affine equation of Ehbauer, Grishkov and Logachev, we give an explicit
sufficient condition for $M(A)$ to be uniformizable, in terms of the valuations of the coefficients of that
equation. It holds as soon as no cancellation occurs among the four summands of one of these coefficients; in
particular $h^1(M(A))=4$ in the simple case where no cancellation occurs in any coefficient. This proves their
Conjecture~3.7: $h^1(M(A))=4$ whenever $(\ord a_{11},\ord a_{12},\ord a_{21},\ord a_{22})$ lies
outside an explicit union of nine hyperplanes.
\end{abstract}
\maketitle

\section{The affine equation}
Let $q=p^s$, let $\ord$ be the valuation of $\C$ with $\ord\theta=-1$, and let $A=(a_{jk})\in M_2(\C)$. The
$t$-motive $M(A)$ is defined by $Te_*=\theta e_*+A\tau e_*+\tau^2e_*$, $e_*=(e_1,e_2)^t$; it is pure of
dimension $2$ and rank $4$ \cite{EGL}. Assume $a_{21}\ne0$. By \cite[(3.0), (3.1), Theorem 3.3]{EGL},
$h^1(M(A))$ is the $\F_q[T]$-rank of the module of small solutions $X=\sum_{i\ge0}x_iT^i$, $x_i\in\C$,
$x_i\to0$, of the system ($i\ge0$, $x_{-1}=x_{-2}=0$)
\begin{equation}\label{eq:aff}
\sum_{k=0}^{4}a_k\,x_i^{q^k}+b_{14}\,x_{i-1}^{q^4}+b_{13}\,x_{i-1}^{q^3}+b_{12}\,x_{i-1}^{q^2}+b_{24}\,x_{i-2}^{q^4}=0,
\end{equation}
with
\begin{gather*}
a_4=\frac{\theta^{q^3+q^2}}{a_{21}^{q^2}},\quad
a_3=\frac{a_{11}^{q^2}\theta^{q^2}}{a_{21}^{q^2}}+\frac{a_{22}^{q}\theta^{q^2}}{a_{21}^{q}},\quad
a_2=\frac{\theta^{q}}{a_{21}}+\frac{\theta^{q^2}}{a_{21}^{q^2}}+\frac{a_{11}^{q}a_{22}^{q}}{a_{21}^{q}}-a_{12}^{q},\\
a_1=\frac{a_{11}}{a_{21}}+\frac{a_{22}^{q}}{a_{21}^{q}},\quad a_0=\frac1{a_{21}},\quad
b_{14}=-\frac{\theta^{q^3}+\theta^{q^2}}{a_{21}^{q^2}},\quad
b_{13}=-\frac{a_{11}^{q^2}}{a_{21}^{q^2}}-\frac{a_{22}^{q}}{a_{21}^{q}},\\
b_{12}=-\frac1{a_{21}}-\frac1{a_{21}^{q^2}},\quad b_{24}=\frac1{a_{21}^{q^2}}.
\end{gather*}
Put $\alpha_k=\ord a_k$, $\beta_{jk}=\ord b_{jk}$ (with $\ord0=+\infty$) and $P_k=(q^k,\alpha_k)$. One has the
exact relations
\begin{equation}\label{eq:id}
a_3=-\theta^{q^2}b_{13},\qquad \beta_{14}=\alpha_4+q^2,\qquad \beta_{24}=\alpha_4+q^3+q^2 .
\end{equation}
Since $a_0a_4\ne0$, the solutions of the equation $i=0$ form a $4$-dimensional $\F_q$-vector space $S_0$, and the
valuations of its nonzero elements are the slopes, with sign changed, of the lower convex hull $\Pi$ of the points
$P_k$, an edge from abscissa $q^j$ to $q^l$ carrying $q^l-q^j$ of them. For every $c$, the elements of $S_0$ of
valuation $>c$ form, with $0$, an $\F_q$-subspace, of dimension $l$ if the edges of slope $<-c$ end at abscissa
$q^l$.

\section{Two elementary lemmas}
For $g\in\Q$ put $W(g)=\max_k (g-\alpha_k)/q^k$, the maximum over the $k$ with $a_k\ne0$.

\begin{lemma}\label{lem:W}
Let $c\in\C^{\times}$ with $\ord c=g$. The equation $\sum_k a_kx^{q^k}=-c$ has solutions, all of valuation
$\le W(g)$, and some of valuation exactly $W(g)$. The function $W$ is strictly increasing.
\end{lemma}

\begin{proof}
If $\ord x>W(g)$, every term $a_kx^{q^k}$ has valuation $>\alpha_k+q^kW(g)\ge g$, which is impossible. The
Newton polygon of $c+\sum_ka_kx^{q^k}$ has its first edge from $(0,g)$ to the point $P_k$ minimizing
$(\alpha_k-g)/q^k$; its roots have valuation $W(g)$, and $\C$ is algebraically closed.
\end{proof}

Given $x_{i-1},x_{i-2}$ with valuations $v,v'$, put $T_{14}=\beta_{14}+q^4v$, $T_{13}=\beta_{13}+q^3v$,
$T_{12}=\beta_{12}+q^2v$, $T_{24}=\beta_{24}+q^4v'$, and $\Phi(v)=W(\beta_{12}+q^2v)$.

\begin{lemma}\label{lem:drift}
$W(T_{14})\ge v+q^{-2}$, $W(T_{13})\ge v+q^{-1}$ (if $b_{13}\ne0$), $W(T_{24})\ge v'+(q+1)q^{-2}$, and
$\Phi(v)\ge v+(\beta_{12}-\alpha_2)q^{-2}$ (if $a_2\ne0$).
\end{lemma}

\begin{proof}
Bound $W$ below by the term $k=4$, $k=3$, $k=4$, $k=2$ respectively and use \eqref{eq:id}.
\end{proof}

\section{The criterion}
For $k\in\{0,1,3,4\}$ with $a_k\ne0$ put $f_k=(\beta_{12}-\alpha_k)/(q^k-q^2)$, and let
\[
I=\{v\in\mathbb{R}:\Phi(v)\le v\}.
\]
A direct computation shows that $v\in I$ if and only if $\alpha_2\ge\beta_{12}$ and
$\max(f_3,f_4)\le v\le\min(f_0,f_1)$ (indices with $a_k=0$ omitted); geometrically, the line through
$(q^2,\beta_{12})$ of slope $-v$ lies on or below all the points $P_k$.

\begin{theorem}\label{thm:main}
Assume $a_{21}\neq0$.
\begin{enumerate}
\item If $I=\emptyset$, then $h^1(M(A))=4$. This holds in particular if $\ord a_2<\ord b_{12}$.
\item If $I\neq\emptyset$ and $a_1\ne0$, $f_1<f_0$, then $h^1(M(A))\ge1$.
\end{enumerate}
\end{theorem}

\begin{proof}
Let $x_0\in S_0$, $v_0=\ord x_0$, and define $x_1,x_2,\dots$ inductively: if the tail of \eqref{eq:aff} vanishes,
take $x_i=0$; otherwise take for $x_i$ a solution of maximal valuation (Lemma \ref{lem:W}). Since the valuation of
the tail is at least $\min(T_{14},T_{13},T_{12},T_{24})$, Lemma \ref{lem:W} gives, with $v_j=\ord x_j$,
\begin{equation}\label{eq:rec}
v_i\ \ge\ \min\bigl(W(T_{14}),W(T_{13}),\Phi(v_{i-1}),W(T_{24})\bigr),
\end{equation}
the terms with a zero coefficient being omitted (for $i=1$ the term $T_{24}$ is absent).

(1) The function $\Phi(v)-v$ is convex and piecewise linear, with slope $q^2-1>0$ for $v\gg0$ and slope
$q^{-2}-1<0$ for $v\ll0$; if $I=\emptyset$ it is positive, hence $\ge c$ for some $c>0$. By \eqref{eq:rec} and
Lemma \ref{lem:drift}, $v_i\ge\min(v_{i-1},v_{i-2})+\min(c,q^{-2})$, so $v_i\to+\infty$ and $X=\sum x_iT^i$ is a
small solution. If $\alpha_2<\beta_{12}$, then $\Phi(v)-v\ge(\beta_{12}-\alpha_2)q^{-2}>0$ for all $v$ by Lemma
\ref{lem:drift}, so $I=\emptyset$. Applying this to an $\F_q$-basis $x_0^{(1)},\dots,x_0^{(4)}$ of $S_0$ gives small
solutions $X^{(1)},\dots,X^{(4)}$. They are $\F_q[T]$-linearly independent: in a relation $\sum c_jX^{(j)}=0$ with
$c_j\in\F_q[T]$ not all zero we may assume, $T$ being injective on $\C[[T]]$, that some $c_j(0)\ne0$, and the
constant term gives $\sum c_j(0)x_0^{(j)}=0$. Hence $h^1(M(A))\ge4$, and $h^1(M(A))\le4$ always.

(2) Let $v_+=\max I=\min(f_0,f_1)=f_1$. The line $\ell$ through $(q^2,\beta_{12})$ of slope $-v_+$ passes through
$P_1$, strictly below $P_0$ (as $f_1<f_0$), and on or below the other points; hence $P_0P_1$ is the first edge of
$\Pi$, of slope $<-v_+$, and $S_0$ contains an element $x_0\ne0$ with $v_0>v_+$. The function $\Phi(v)-v$ is convex,
$\le0$ on $I$ and $>0$ to the right of $v_+$, so it is nondecreasing on $[v_+,\infty)$ and
$\Phi(v)-v\ge c_0:=\Phi(v_0)-v_0>0$ for $v\ge v_0$. By induction, \eqref{eq:rec} and Lemma \ref{lem:drift} give
$v_i\ge v_0$ and $v_i\ge\min(v_{i-1},v_{i-2})+\min(c_0,q^{-2})$, so $X$ is a small solution and $h^1\ge1$.
\end{proof}

\begin{corollary}[simple case]\label{cor:simple}
Assume $a_{21}\ne0$. If the valuation of $a_2$ equals the minimum of the valuations of its four summands (in
particular in the simple case of \cite{L26b}, where the valuation of each of the nine coefficients equals the
minimum of the valuations of its summands), then $h^1(M(A))=4$.
\end{corollary}

\begin{proof}
The first two summands of $a_2$ have valuations $-q-\ord a_{21}$ and $-q^2-q^2\ord a_{21}$, smaller than the
valuations $-\ord a_{21}$ and $-q^2\ord a_{21}$ of the two summands of $b_{12}$. Hence
$\alpha_2<\min(-\ord a_{21},-q^2\ord a_{21})\le\beta_{12}$, and Theorem \ref{thm:main}(1) applies.
\end{proof}

\begin{remark}
If at every step the minimum of $T_{14},T_{13},T_{12},T_{24}$ is attained by a single term, the inequalities above
become equalities, and one finds that the starts $x_0$ with $\ord x_0\le\max I$ stay of valuation $\le\max I$;
the value of $h^1$ is then $4$, $1$ or $0$ according as $I=\emptyset$, $f_1<f_0$, or neither. Values $2$ and $3$
can therefore only occur through cancellations in the tail.
\end{remark}

\section{Conjecture 3.7 of Ehbauer, Grishkov and Logachev}
Write $(u_{11},u_{12},u_{21},u_{22})=(\ord a_{11},\ord a_{12},\ord a_{21},\ord a_{22})\in(\Q\cup\{+\infty\})^4$.
In \cite[Conjecture 3.7]{EGL} (see also \cite[Conjecture 4.2.3]{L26}) it is conjectured that $h^1(M(A))=4$ whenever
this point lies outside a countable union of linear subspaces of dimension $\le3$.

For $u_{21}<\infty$ put $s_1=-q-u_{21}$, $s_2=-q^2-q^2u_{21}$, $s_3=q(u_{11}+u_{22}-u_{21})$, $s_4=qu_{12}$, with
$s_3=+\infty$ if $u_{11}=+\infty$ or $u_{22}=+\infty$, and $s_4=+\infty$ if $u_{12}=+\infty$; these are the valuations
of the four summands of $a_2$.

\begin{corollary}\label{cor:conj}
Let $H\subset(\Q\cup\{+\infty\})^4$ be the set of points with $u_{21}<\infty$ at which two of the finite numbers
$s_1,\dots,s_4$ are equal. It is the union of the six sets defined by
\begin{gather*}
u_{21}=-\tfrac{q}{q+1},\qquad u_{11}+u_{22}-\tfrac{q-1}{q}u_{21}+1=0,\qquad u_{12}+\tfrac1q u_{21}+1=0,\\
u_{11}+u_{22}+(q-1)u_{21}+q=0,\qquad u_{12}+q\,u_{21}+q=0,\qquad u_{11}+u_{22}-u_{21}-u_{12}=0,
\end{gather*}
each equation being imposed with all the coordinates it involves finite and the other coordinates arbitrary in
$\Q\cup\{+\infty\}$. Let $H'$ be the image of $H$ under $u_{11}\leftrightarrow u_{22}$, $u_{12}\leftrightarrow u_{21}$. This exchange maps the set given by $s_1=s_4$ onto the set given by $s_2=s_4$ and fixes the set given by $s_3=s_4$, so $H\cup H'$ is the union of nine such sets.
If $(u_{11},u_{12},u_{21},u_{22})\notin H\cup H'$, then $M(A)$ is uniformizable: $h^1(M(A))=h_1(M(A))=4$.
\end{corollary}

\begin{proof}
Suppose first $u_{21}<\infty$. The equalities $s_j=s_k$ between finite $s_j,s_k$ are the six equations
defining $H$, and $s_1,s_2$ are always finite. Outside $H$ the minimum of the $s_j$ is attained once, so no cancellation can occur in $a_2$, and Corollary \ref{cor:simple} gives
$h^1=4$. If $u_{21}=\infty$ and $u_{12}<\infty$, the basis $(e_2,e_1)$ gives
$M(A)=M(JAJ)$ with $J=\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$, which exchanges $a_{11}\leftrightarrow a_{22}$
and $a_{12}\leftrightarrow a_{21}$; apply the first case. If $a_{12}=a_{21}=0$, $M(A)$ is the direct sum of two Drinfeld
modules of rank $2$, which are uniformizable. Finally $h^1=4$ if and only if $h_1=4$ by Anderson's theorem
\cite{EGL}.
\end{proof}

\begin{remark}
For antidiagonal $A$ ($u_{11}=u_{22}=+\infty$), $H\cup H'$ meets the plane of $(u_{12},u_{21})$ in the four lines
$u_{21}=-\tfrac{q}{q+1}$, $u_{12}=-\tfrac{q}{q+1}$, $u_{21}=-q\,u_{12}-q$ and $u_{21}=-u_{12}/q-1$. The first, third and
fourth contain the rays $R_1,R_2,R_3$ of \cite[Section 6]{EGL}, on which the non-uniformizable $M(A)$ of
\cite[Theorem 5.1]{EGL} lie. On $H\cup H'$ uniformizability may fail only if the
leading terms of the tied summands of $a_2$ cancel to the point that $I\ne\emptyset$. The uniformizable locus is
open in families \cite{BH}.
\end{remark}

\section{Numerical checks}
\emph{Theorem 5.1 of \cite{EGL}.} For antidiagonal $A$, the value of $h^1$ obtained by iterating \eqref{eq:rec} with
equality coincides with \cite[Theorem 5.1]{EGL} at all $38\,640$ points of a grid in
$(\ord a_{21},\ord a_2)$, for $q=2$ and $q=3$.

\emph{The example of \cite{GL21}.} For $q=2$ and $A=\bigl(\begin{smallmatrix}\theta&\theta^6\\ \theta^{-2}&0\end{smallmatrix}\bigr)$ one has
$s_2=s_4=-12$ (and $u_{22}=+\infty$): the point lies on $H$, and $\theta^{12}$ cancels in $a_2=\theta^4$, so that $\alpha_2=-4\ge-8=\beta_{12}$.
This agrees with $h^1(M(A))=0$ and $h^1(M(A^t))=1$ \cite[Theorems 4.1 and 4.7]{GL21}.

\emph{The example of \cite{L26b}.} For $q=2$ and $A=\bigl(\begin{smallmatrix}\theta^3&\theta^{-2}\\ \theta^{-1}&1\end{smallmatrix}\bigr)$
the nine coefficients are those of \cite{L26b}; the two summands $\theta^8$ of $a_2$ cancel ($s_2=s_3=-8$), and
$a_2=\theta^3+\theta^{-4}$. Here $\beta_{12}=-4$ and $(f_0,f_1,f_3,f_4)=(1,0,4,1)$, so $\max(f_3,f_4)>\min(f_0,f_1)$
and $I=\emptyset$ whatever the valuation of $a_2$. Since $a_1,a_3,b_{12},b_{13}$ have no ties among their summands,
$h^1(M(A))=4$ for every $A$ whose entries have the valuations $\bigl(\begin{smallmatrix}-3&2\\1&0\end{smallmatrix}\bigr)$,
in accordance with \cite{L26b}. The lower bounds \eqref{eq:rec} for the start of valuation $-1/2$ are
$-1/2,0,1/2,1,\dots$, as in \cite{L26b}.

\emph{Exact computations.} For $40\,000$ matrices $A$ with entries in $\F_q[\theta,\theta^{-1}]$ ($q=2,3$), chosen so
that cancellations in $a_2$ are frequent, the nine coefficients were computed exactly. In every case $I=\emptyset$
holds for $A$ if and only if it holds for $A^t$, in accordance with $h^1(M(A))=4\Leftrightarrow h^1(M(A^t))=4$.

The scripts used for these checks are not part of this deposit.

\subsection*{Acknowledgements}
The author thanks D. Logachev for explaining the computation of $h^1$ through affine equations. The computations
and part of the writing were carried out with the assistance of Claude (Anthropic).

\begin{thebibliography}{L26b}
\bibitem[BH]{BH} G. B\"ockle, U. Hartl, \emph{Uniformizable families of $t$-motives}, Trans. Amer. Math. Soc.
\textbf{359} (2007), no.~8, 3933--3972; arXiv:math/0411262.
\bibitem[EGL]{EGL} S. Ehbauer, A. Grishkov, D. Logachev, \emph{Calculation of $h^1$ of some Anderson
$t$-motives}, J. Algebra Appl. \textbf{21} (2022), no.~1, 2250017; arXiv:2006.00316.
\bibitem[GL21]{GL21} A. Grishkov, D. Logachev, \emph{$h^1\neq h_1$ for Anderson $t$-motives}, J. Number Theory
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\bibitem[L26a]{L26} D. Logachev, \emph{Research plan}, complete version, 2026, private communication.
\bibitem[L26b]{L26b} D. Logachev, \emph{How to calculate $h^1$ of $t$-motives: an example}, 2026, private communication.
\end{thebibliography}
\end{document}