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dc.contributor.advisorBjorn Poonen.en_US
dc.contributor.authorKadets, Borys.en_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Mathematics.en_US
dc.date.accessioned2020-09-03T16:41:08Z
dc.date.available2020-09-03T16:41:08Z
dc.date.copyright2020en_US
dc.date.issued2020en_US
dc.identifier.urihttps://hdl.handle.net/1721.1/126927
dc.descriptionThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020en_US
dc.descriptionCataloged from the official PDF of thesis.en_US
dc.descriptionIncludes bibliographical references (pages 93-97).en_US
dc.description.abstractThis 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.abstractIn 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.abstractThis 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.statementofresponsibilityby Borys Kadets.en_US
dc.format.extent97 pagesen_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsMIT 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.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectMathematics.en_US
dc.titleArboreal representations, sectional monodromy groups, and abelian varieties over finite fieldsen_US
dc.typeThesisen_US
dc.description.degreePh. D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.identifier.oclc1191267038en_US
dc.description.collectionPh.D. Massachusetts Institute of Technology, Department of Mathematicsen_US
dspace.imported2020-09-03T16:41:08Zen_US
mit.thesis.degreeDoctoralen_US
mit.thesis.departmentMathen_US


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