Uniformizability of elementary Anderson $t$-motives of dimension $2$
Let $M(A)$, $A\in M_2(\mathbb{C}_\infty)$, 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: $h^1(M(A))=4$…
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