dc.contributor.advisor | Bjorn Poonen. | en_US |
dc.contributor.author | Kadets, Borys. | en_US |
dc.contributor.other | Massachusetts Institute of Technology. Department of Mathematics. | en_US |
dc.date.accessioned | 2020-09-03T16:41:08Z | |
dc.date.available | 2020-09-03T16:41:08Z | |
dc.date.copyright | 2020 | en_US |
dc.date.issued | 2020 | en_US |
dc.identifier.uri | https://hdl.handle.net/1721.1/126927 | |
dc.description | Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020 | en_US |
dc.description | Cataloged from the official PDF of thesis. | en_US |
dc.description | Includes bibliographical references (pages 93-97). | en_US |
dc.description.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. | en_US |
dc.description.abstract | 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. | en_US |
dc.description.abstract | 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). | en_US |
dc.description.statementofresponsibility | by Borys Kadets. | en_US |
dc.format.extent | 97 pages | en_US |
dc.language.iso | eng | en_US |
dc.publisher | Massachusetts Institute of Technology | en_US |
dc.rights | MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided. | en_US |
dc.rights.uri | http://dspace.mit.edu/handle/1721.1/7582 | en_US |
dc.subject | Mathematics. | en_US |
dc.title | Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields | en_US |
dc.type | Thesis | en_US |
dc.description.degree | Ph. D. | en_US |
dc.contributor.department | Massachusetts Institute of Technology. Department of Mathematics | en_US |
dc.identifier.oclc | 1191267038 | en_US |
dc.description.collection | Ph.D. Massachusetts Institute of Technology, Department of Mathematics | en_US |
dspace.imported | 2020-09-03T16:41:08Z | en_US |
mit.thesis.degree | Doctoral | en_US |
mit.thesis.department | Math | en_US |