Arboreal Galois representations are central objects in modern arithmetic dynamics, that share mysterious analogies with Galois representations attached to the Tate module of abelian varieties. They are constructed by iterating a rational map of degree d>1 on P
1 and then looking at the action of the absolute Galois group of the base field on a certain infinite, regular d-ary tree. In a pioneering paper, Stoll proved a criterion for the arboreal representation attached to a quadratic polynomial to be surjective in terms of the arithmetic of the post-critical orbit, and then produced infinitely many examples over the rationals. In this talk, I wil recall the basic properties and conjectures about these objects, and then I will show how to obtain necessary and sufficient conditions on the post-critical orbit of such polynomials in order for the associated arboreal representation to have index two image in the automorphism group of the appropriate tree. Using this, I will be able to produce an infinite class of examples, which yield, in particular, instances of the infinite inverse Galois problem over the rationals. This can be seen as a first, systematic step towards the full inverse problem for arboreal representations. Joint work with D. Casazza and C. Pagano.Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/246