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The inverse problem for arboreal Galois representations of index two

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Speaker: 
Andrea Ferraguti
Affiliation: 
University of Cambridge/MPIM
Date: 
Wed, 2019-03-27 14:30 - 15:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

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.

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