# 1-t-motifs

## TL;DR

This paper establishes a canonical isomorphism between the rational points on an abelian t-module and an extension module of t-motifs, generalizing previous results and drawing analogies with Deligne's 1-motifs.

## Contribution

It introduces a new canonical isomorphism linking rational points on abelian t-modules with extension modules of t-motifs, extending prior work and clarifying the analogy with Deligne's 1-motifs.

## Key findings

- Module of rational points is isomorphic to an extension module of t-motifs.
- Extension module aligns with Pink-Hodge structures for uniformizable t-modules.
- The theory parallels Deligne's 1-motifs, enriching the understanding of t-motifs.

## Abstract

We show that the module of rational points on an abelian t-module E is canonically isomorphic with the module Ext^1(M_E, K[t]) of extensions of the trivial t-motif K[t] by the t-motif M_E associated with E. This generalizes prior results of Anderson and Thakur, Papanikolas and Ramachandran, and Woo. In case E is uniformizable then we show that this extension module is canonically isomorphic with the corresponding extension module of Pink-Hodge structures. This situation is formally very similar to Deligne's theory of 1-motifs and we have tried to build up the theory in a way that makes this analogy as clear as possible.