Duality of Anderson T-motives

TL;DR

This paper introduces a duality concept for Anderson T-motives, establishing algebraic and analytic dualities and revealing a correspondence between pure T-motives and certain lattices in complex space.

Contribution

It defines duality for T-motives and proves the equivalence of algebraic and analytic dualities, also establishing a correspondence between pure T-motives and specific lattices.

Findings

Algebraic duality implies analytic duality for T-motives.

There is a one-to-one correspondence between pure T-motives and certain lattices in complex space.

The transposed Siegel matrix of a T-motive's lattice is the Siegel matrix of its dual.

Abstract

Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic duality (Theorem 5). Explicitly, this means that the lattice of the dual of $M$ is the dual of the lattice of $M$, i.e. the transposed of a Siegel matrix of $M$ is a Siegel matrix of the dual of $M$. 2. Let $n=r-1$. There is a 1 -- 1 correspondence between pure T-motives (all they are uniformizable), and lattices of rank $r$ in $C^n$ having dual (Corollary 8.4).