Galois theory of iterated endomorphisms

Galois Theory of Iterated Endomorphisms

Given an abelian algebraic group A over a global field F , α ∈ A(F ), and a prime `, the set of all preimages of α under some iterate of [`] generates an extension of F that contains all `-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as p...

Iterated Endomorphisms of Abelian Algebraic Groups

Given an abelian algebraic group A over a global field K, α ∈ A(K), and a prime l, the set of all preimages of α under some iterate of [l] has a natural tree structure. Using this data, we construct an “arboreal” Galois representation ω whose image combines that of the usual l-adic representation and the Galois group of a certain Kummer-type extension. For several classes of A, we give a simple...

Fixed-point Free Endomorphisms and Hopf Galois Structures

Let L|K be a Galois extension of fields with finite Galois group G. Greither and Pareigis [GP87] showed that there is a bijection between Hopf Galois structures on L|K and regular subgroups of Perm(G) normalized by G, and Byott [By96] translated the problem into that of finding equivalence classes of embeddings of G in the holomorph of groups N of the same cardinality as G. In [CCo06] we showed...

Galois Theory

Galois Theory

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...