Lattice map for Anderson T-motives: first approach

TL;DR

This paper investigates the lattice map for Anderson t-motives, proving it is locally an isomorphism near a specific t-motive, using monodromy groups and an explicit approximation method.

Contribution

It provides the first local result indicating the lattice map may be close to one-to-one near a particular t-motive, using monodromy group analysis and Hensel lemma techniques.

Findings

Lattice map is an isomorphism in a neighborhood of a specific t-motive.

The size of the neighborhood depends on monodromy group elements.

Explicit solutions via successive approximations are used to analyze isomorphisms.

Abstract

There exists a lattice map from the set of pure uniformizable Anderson t-motives to the set of lattices. It is not known what is the image and the fibers of this map. We prove a local result that sheds the first light to this problem and suggests that maybe this map is close to 1 -- 1. Namely, let $M(0)$ be a t-motive of dimension $n$ and rank $r=2n$ \ --- \ the $n$-th power of the Carlitz module of rank 2, and let $M$ be a t-motive which is in some sense "close" to $M(0)$. We consider the lattice map $M \mapsto L(M)$, where $L(M)$ is a lattice in $C^n$. We show that the lattice map is an isomorphism in a "neighborhood" of $M(0)$. Namely, we compare the action of monodromy groups: (a) from the set of equations defining t-motives to the set of t-motives themselves, and (b) from the set of Siegel matrices to the set of lattices. The result of the present paper gives that the size of a neighborhood, where we have an isomorphism, depends on an element of the monodromy group. We do not know whether there exists a universal neighborhood. Method of the proof: explicit solution of an equation describing an isomorphism between two t-motives by a method of successive approximations using a version of the Hensel lemma.