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 $\mathfrak {p}$-adic Galois representation is $\mathfrak {p}$-adically open in the motivic Galois group for any prime $\mathfrak {p}$. 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.