Anderson t-modules with thin t-adic Galois representations
Abstract
Pink has given a qualitative answer to the Mumford-Tate conjecture for Drinfeld modules in the 90s. He showed that the image of the v-adic Galois representation is v-adically open in the motivic Galois group for any prime v. In contrast to this result, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate-modules are "far from" having t-adically open image in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group which is in accordance to the Mumford-Tate conjecture. For the proof, we explicitly determine the motivic Galois group as well as the Galois representation for these t-modules.
Keywords
Cite
@article{arxiv.1907.05144,
title = {Anderson t-modules with thin t-adic Galois representations},
author = {Andreas Maurischat},
journal= {arXiv preprint arXiv:1907.05144},
year = {2022}
}
Comments
10 pages; v1->v2: Added a definition of density of the image and of a thin image. Now 12 pages, v2->v3: Added Remark 1.2 and Theorem 6.6 + minor changes