Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields
Name
1191267038-MIT.pdf
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703.47 KB
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Adobe PDF
Checksum (MD5)
2dcaf4ce9e0cbe8ba3c68b4251bc1ab2
Author(s)
Kadets, Borys.
Advisor(s)
Bjorn Poonen.
Date Issued
2020
Publisher
Massachusetts Institute of Technology
Abstract
This thesis consists of three independent parts. The first part studies arboreal representations of Galois groups - an arithmetic dynamics analogue of Tate modules - and proves some large image results, in particular confirming a conjecture of Odoni. Given a field K, a separable polynomial [mathematical expression], and an element [mathematical expression], the full backward orbit [mathematical expression] has a natural action of the Galois group [mathematical expression]. For a fixed [mathematical expression] with [mathematical expression] and for most choices of t, the orbit [mathematical expression] has the structure of complete rooted [mathematical expression]. The Galois action on [mathematical expression] thus defines a homomorphism [mathematical expression]. The map [mathematical expression] is the arboreal representation attached to f and t.
In analogy with Serre's open image theorem, one expects [mathematical expression] to hold for most f, t, but until very recently for most degrees d not a single example of a degree d polynomial [mathematical expression] with surjective [mathematical expression],t was known. Among other results, we construct such examples in all sufficiently large even degrees. The second part concerns monodromy of hyperplane section of curves. Given a geometrically integral proper curve [mathematical expression], consider the generic hyperplane [mathematical expression]. The intersection [mathematical expression] is the spectrum of a finite separable field extension [mathematical expression] of degree [mathematical expression]. The Galois group [mathematical expression] is known as the sectional monodromy group of X. When char K = 0, the group [mathematical expression] equals [mathematical expression] for all curves X.
This result has numerous applications in algebraic geometry, in particular to the degree-genus problem. However, when char K > 0, the sectional monodromy groups can be smaller. We classify all nonstrange nondegenerate curves [mathematical expression], for [mathematical expression] such that [mathematical expression]. Using similar methods we also completely classify Galois group of generic trinomials, a problem studied previously by Abhyankar, Cohen, Smith, and Uchida. In part three of the thesis we derive bounds for the number of [mathematical expression]-points on simple abelian varieties over finite fields; these improve upon the Weil bounds. For example, when q = 3, 4 the Weil bound gives [ .. ] for all abelian varieties A. We prove that [mathematical expression], [mathematical expression] hold for all but finitely many simple abelian varieties A (with an explicit list of exceptions).
Description
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 93-97).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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