David Niedbala Giraudin

math.NT — Number Theory

Uniformizability of elementary Anderson $t$-motives of dimension $2$

Contributed by David Niedbala Giraudin

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: $h^1(M(A))=4$ whenever $(\mathrm{ord}\,a_{11},\mathrm{ord}\,a_{12},\mathrm{ord}\,a_{21},\mathrm{ord}\,a_{22})$ lies outside an explicit union of nine hyperplanes.