103 research outputs found
Galois theory of iterated endomorphisms
Given an abelian algebraic group over a global field , , and a prime , the set of all preimages of under some
iterate of generates an extension of 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 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 such that the order of (\alpha \bmod{\p}) is prime to .
We compute this density in the general case for several classes of ,
including elliptic curves and one-dimensional tori. For example, if is a
number field, is an elliptic curve with surjective 2-adic representation
and with , then the density of
with (\alpha \bmod{\p}) having odd order is 11/21.Comment: 33 pages; The appendix has been updated, several examples have been
redone, and a number of typos corrected. The paper has been accepted for
publication in Proceedings of the London Mathematical Societ
The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions
Let be a
newform with squarefree level that does not have complex multiplication.
For a prime , define to be the angle for which
. Let be a closed
subinterval, and let be the
Sato-Tate measure of . Assuming that the symmetric power -functions of
satisfy certain analytic properties (all of which follow from Langlands
functoriality and the Generalized Riemann Hypothesis), we prove that if is
sufficiently large, then with an implied constant of . By letting be a short interval
centered at and counting the primes using a smooth cutoff, we
compute a lower bound for the density of positive integers for which
. In particular, if is the Ramanujan tau function, then under
the aforementioned hypotheses, we prove that
We also discuss the connection between the density of positive integers for
which and the number of representations of by certain
positive-definite, integer-valued quadratic forms.Comment: 29 pages. Significant revisions, including improvements in Theorems
1.2, 1.3, and 1.5 and a more detailed account of the contour integration, are
included. Acknowledgements are update
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