Calculation of $h^1$ of some Anderson t-motives

TL;DR

This paper computes the first cohomology dimension of certain Anderson t-motives of dimension 2 and rank 4, using explicit methods, and explores possible values of this invariant.

Contribution

It introduces explicit calculation methods for $h^1$ of Anderson t-motives and extends the understanding of its possible values, including open questions about intermediate cases.

Findings

Calculated $h^1$ for specific matrix-defined t-motives.

Established an upper bound of 4 for $h^1$ in these cases.

Identified open questions about the existence of motives with $h^1=2$ or $3$.

Abstract

We consider Anderson t-motives $M$ of dimension 2 and rank 4 defined by some simple explicit equations parameterized by $2\times2$ matrices. We use methods of explicit calculation of $h^1(M)$ -- the dimension of their cohomology group $H^1(M)$ ( = the dimension of the lattice of their dual t-motive $M'$) developed in our earlier paper. We calculate $h^1(M)$ for $M$ defined by all matrices having 0 on the diagonal, and by some other matrices. These methods permit to make analogous calculations for most (probably all) t-motives. $h^1$ of all Anderson t-motives $M$ under consideration satisfy the inequality $h^1(M)\le4$, while in all known examples we have $h^1(M)=0,1,4$. Do exist $M$ of this type having $h^1=2,3$? We do not know, this is a subject of further research.