1-$t$-Motifs

Abstract

We show that the module of rational points on an abelian $t$\dash module $E$ is canonically isomorphic with the module ${\mathrm{Ext}}^1(M_E,K[t])$ 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 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.