Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Andrea Ferraguti
Affiliation:

University of Cambridge/MPIM
Date:

Wed, 2019-03-27 14:30 - 15:30 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] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/3444

[3] http://www.mpim-bonn.mpg.de/node/246