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 p \mathfrak {p} -adic Galois representation is p \mathfrak {p} -adically open in the motivic Galois group for any prime p \mathfrak {p} . In contrast to this result, we provide a family of uniformizable Anderson t t -modules for which the Galois representations of their t t -adic Tate-modules are “far from” having t 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 t -modules.