{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/126927"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/126927","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Kadets, Borys."],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Bjorn Poonen."],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020","date_published":"2020","updated_at":"2026-07-22T22:22:31Z","subjects":["Mathematics."],"languages":["eng"],"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."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/126927","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Bjorn Poonen."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Department of Mathematics","Math"]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Department of Mathematics."]},{"key":"dc:creator","label":"Author","values":["Kadets, Borys."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-09-03T16:41:08Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-09-03T16:41:08Z"]},{"key":"dc:date.issued","label":"Date","values":["2020"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctoral"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1721.1/126927"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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)."]},{"key":"dc:description.abstract","label":"Abstract","values":["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)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields"]}]}],"canonical_facts":{"dc:contributor.advisor":["Bjorn Poonen."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics","Math"],"dc:contributor.other":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:creator":["Kadets, Borys."],"dc:date.accessioned":["2020-09-03T16:41:08Z"],"dc:date.available":["2020-09-03T16:41:08Z"],"dc:date.issued":["2020"],"dc: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)."],"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.","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)."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/126927"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"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."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral"]},"updated_at":"2026-07-22T22:22:31Z"}