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A Finite Index Conjecture for Iterated Galois Groups

Introductory Workshop: Diophantine Geometry February 06, 2023 - February 10, 2023

February 06, 2023 (11:00 AM PST - 12:00 PM PST)
Speaker(s): Thomas Tucker (University of Rochester)
Location: MSRI: Simons Auditorium, Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC

A Finite Index Conjecture For Iterated Galois Groups


Let f be a rational function of degree greater than 1 defined over a number field k and let x in k.  Iterating f, we obtain
a sequence of field extensions K_n generated by the inverse images of x under f^n. Passing to the inverse limit one obtains an iterated Galois group G_x.  There is a natural "generic" Galois group G attached to f in which G_x sits. Conjecturally, G_x will have finite index in G unless x is "special" in some fashion.  While in its earliest form, this conjecture (originally made in a special case by Boston and Jones) was made in analogy with the Serre finite index theorem, it seems to have a very different flavor.  We will discuss various results towards this conjecture, the diophantine techniques used in these results, a more general higher-dimensional conjecture, and a very simple conjecture on the irreducibility of iterates of polynomials that is completely unresolved.

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A Finite Index Conjecture For Iterated Galois Groups

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