中央研究院數學研究所

Number Theory Seminar

Galois actions on iterated monodromy groups.

2025/11/17 (Mon.) 10:00~11:00

Date : 2025/11/17 (Mon.) 10:00~11:00
Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
Speaker : Ophelia Adams (University of Rochester)
Abstract : We discuss and compare two kinds of Galois representations arising in dynamics as analogues of the Tate module: arboreal Galois representations and the outer Galois action on profinite iterated monodromy group (pfIMG). While less is known about the latter at present, the dynamics naturally equips them with a rich "self-similar" or fractal structure which encodes a great deal of information. For example, these self-similar groups cannot have proper self-similar subgroups unless the original dynamics are quite exceptional. In joint work with Trevor Hyde, we made use of self-similarity to calculated the pfIMGs of unicritical polynomials, along with the outer Galois action, in order to determine the associated constant field extension. Remarkably, the groups and actions can be classified almost entirely by combinatorial information, the critical portrait, with just one arithmetic parameter intervening. I will outline some of these results, and comment on future directions.