Non-injectivity of the lattice map for non-mixed Anderson t-motives, and a result towards its surjectivity

TL;DR

This paper investigates the properties of the lattice map for non-mixed Anderson t-motives, demonstrating its non-injectivity and exploring conditions related to its potential surjectivity, with implications for understanding t-motive uniformizability.

Contribution

It provides explicit constructions showing non-injectivity of the lattice map and explores the surjectivity by identifying lattices of non-pure t-motives.

Findings

Lattice map is not injective for non-mixed t-motives.

Some lattices of non-pure t-motives are not in the image of the lattice map.

Initial calculations on the uniformizability of certain t-motives.

Abstract

Let $M$ be an uniformizable Anderson t-motive and $L(M)$ its lattice. First, we prove by an explicit construction that for the non-mixed $M$ the lattice map $M\mapsto L(M)$ is not injective. Second, we show that some lattices which do not belong to the set $L(M)$ of pure $M$, are lattices of non-pure $M$. This is a result towards surjectivity of the lattice map. The t-motives used in the proofs are non-pure t-motives of dimension 2, rank 3. Finally, we start calculations in order to answer a question whether all these t-motives are uniformizable, or not.