Galois theory of iterated endomorphisms

Details Galois theory of iterated endomorphisms Galois theory of iterated endomorphisms

2007-06-15

arxiv.org/abs/0706.2384v4

Proc. Lond. Math. Soc 100(3) (2010), 763-794

Mathematics

Number Theory

33 pages; The appendix has been updated, several examples have been redone, and a number of typos corrected. The paper has bee

Scientific paper

10.1112/plms/pdp051

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-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 possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes $\p$ in the ring of integers of $F$ such that the order of $(\alpha \bmod{\p})$ is prime to $\ell$. We compute this density in the general case for several classes of $A$, including elliptic curves and one-dimensional tori. For example, if $F$ is a number field, $A/F$ is an elliptic curve with surjective 2-adic representation and $\alpha \in A(F)$ with $\alpha \not\in 2A(F(A[4]))$, then the density of $\mathfrak{p}$ with ($\alpha \bmod{\p}$) having odd order is 11/21.

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