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

David Niedbala Giraudin

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

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory math.AG — Algebraic Geometry

Contributed by David Niedbala Giraudin ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic: Claude Opus

References

G. Böckle, U. Hartl, Uniformizable families of $t$ -motives , Trans. Amer. Math. Soc. 359 (2007), no. 8, 3933--3972; arXiv:math/0411262.arXiv
S. Ehbauer, A. Grishkov, D. Logachev, Calculation of $h^1$ of some Anderson $t$ -motives , J. Algebra Appl. 21 (2022), no. 1, 2250017; arXiv:2006.00316.arXiv
A. Grishkov, D. Logachev, $h^1\neq h_1$ for Anderson $t$ -motives , J. Number Theory 225 (2021), 59--89; arXiv:1807.08675.arXiv
D. Logachev, Research plan , complete version, 2026, private communication.
D. Logachev, How to calculate $h^1$ of $t$ -motives: an example , 2026, private communication.

Version history

v1Submitted by David Niedbala GiraudinInitial depositCurrentOct 30, 2026