{"id":{"repo_id":"unm","oai_identifier":"oai:digitalrepository.unm.edu:math_etds-1027"},"canonical_url":"https://search.dev.ndltd.org/etd/unm/oai:digitalrepository.unm.edu:math_etds-1027","repository":{"repo_id":"unm","name":"University of New Mexico","base_url":"https://digitalrepository.unm.edu/do/oai/"},"display":{"title":"Lifts of Frobenius on Arithmetic Jet Spaces of Schemes","abstract":"Lifts of Frobenius on formal schemes X over the p-adic completion of the maximal unramified extension of the p-adic integers may be viewed as arithmetic analogues of vector fields on manifolds. In particular, vector fields on the tangent bundle of a manifold, appearing for instance in Hamiltonian mechanics, have as arithmetic analogues lifts of Frobenius on arithmetic jet spaces J^1(X) of schemes. In this thesis, we first consider the projective space P^m and prove that lifts of Frobenius do not exist on its arithmetic jet spaces J^n(P^m_R) for n, m >= 1. Exhibiting a contrast in the case n=m=1 between the arithmetic and geometric frameworks, we show on the other hand that the space of vector fields on the tangent bundle T(P^1_k) lifting vector fields on P^1_k, where k is an algebraically closed field, has dimension 6 over k. Nevertheless, \"normalized\" vector fields, which play a role in Hamiltonian mechanics, do not exist on T(P^1_k). We proceed to prove a stronger result for the case n=m=1, that there are no effective Cartier divisors on J^2(P^1) that are finite-to-one over J^1(P^1), and discover that an analogous result holds in geometry. As a final result, we prove the nonexistence of lifts of Frobenius on the first jet space of any smooth quadric hypersurface in projective space.","abstract_html":"Lifts of Frobenius on formal schemes X over the p-adic completion of the maximal unramified extension of the p-adic integers may be viewed as arithmetic analogues of vector fields on manifolds. In particular, vector fields on the tangent bundle of a manifold, appearing for instance in Hamiltonian mechanics, have as arithmetic analogues lifts of Frobenius on arithmetic jet spaces J^1(X) of schemes. In this thesis, we first consider the projective space P^m and prove that lifts of Frobenius do not exist on its arithmetic jet spaces J^n(P^m_R) for n, m >= 1. Exhibiting a contrast in the case n=m=1 between the arithmetic and geometric frameworks, we show on the other hand that the space of vector fields on the tangent bundle T(P^1_k) lifting vector fields on P^1_k, where k is an algebraically closed field, has dimension 6 over k. Nevertheless, "normalized" vector fields, which play a role in Hamiltonian mechanics, do not exist on T(P^1_k). We proceed to prove a stronger result for the case n=m=1, that there are no effective Cartier divisors on J^2(P^1) that are finite-to-one over J^1(P^1), and discover that an analogous result holds in geometry. As a final result, we prove the nonexistence of lifts of Frobenius on the first jet space of any smooth quadric hypersurface in projective space.","abstract_has_math":false,"creators":["Medina, Erik"],"institution":null,"degree_name":"Mathematics","degree_level":"Doctoral","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["Buium, Alexandr","Alexandru Buium","Janet Vassilev","Michael Nakamaye","James Borger"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-06-09T07:00:00Z","date_published":"2016-06-09T07:00:00Z","updated_at":"2026-07-24T05:27:37Z","subjects":["lifts of Frobenius","arithmetic geometry","vector fields"],"languages":["English"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalrepository.unm.edu/math_etds/28"],"render_values":[{"text":"https://digitalrepository.unm.edu/math_etds/28","href":"https://digitalrepository.unm.edu/math_etds/28","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1928/32319","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Buium, Alexandr","Alexandru Buium","Janet Vassilev","Michael Nakamaye","James Borger"]},{"key":"dc:creator","label":"Author","values":["Medina, Erik"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics & Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral","Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["lifts of Frobenius","arithmetic geometry","vector fields"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/1928/32319","https://digitalrepository.unm.edu/math_etds/28"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Lifts of Frobenius on formal schemes X over the p-adic completion of the maximal unramified extension of the p-adic integers may be viewed as arithmetic analogues of vector fields on manifolds. In particular, vector fields on the tangent bundle of a manifold, appearing for instance in Hamiltonian mechanics, have as arithmetic analogues lifts of Frobenius on arithmetic jet spaces J^1(X) of schemes. In this thesis, we first consider the projective space P^m and prove that lifts of Frobenius do not exist on its arithmetic jet spaces J^n(P^m_R) for n, m >= 1. Exhibiting a contrast in the case n=m=1 between the arithmetic and geometric frameworks, we show on the other hand that the space of vector fields on the tangent bundle T(P^1_k) lifting vector fields on P^1_k, where k is an algebraically closed field, has dimension 6 over k. Nevertheless, \"normalized\" vector fields, which play a role in Hamiltonian mechanics, do not exist on T(P^1_k). We proceed to prove a stronger result for the case n=m=1, that there are no effective Cartier divisors on J^2(P^1) that are finite-to-one over J^1(P^1), and discover that an analogous result holds in geometry. As a final result, we prove the nonexistence of lifts of Frobenius on the first jet space of any smooth quadric hypersurface in projective space."]},{"key":"dc:title","label":"Title","values":["Lifts of Frobenius on Arithmetic Jet Spaces of Schemes"]}]}],"canonical_facts":{"dc:contributor":["Buium, Alexandr","Alexandru Buium","Janet Vassilev","Michael Nakamaye","James Borger"],"dc:creator":["Medina, Erik"],"dc:description.abstract":["Lifts of Frobenius on formal schemes X over the p-adic completion of the maximal unramified extension of the p-adic integers may be viewed as arithmetic analogues of vector fields on manifolds. In particular, vector fields on the tangent bundle of a manifold, appearing for instance in Hamiltonian mechanics, have as arithmetic analogues lifts of Frobenius on arithmetic jet spaces J^1(X) of schemes. In this thesis, we first consider the projective space P^m and prove that lifts of Frobenius do not exist on its arithmetic jet spaces J^n(P^m_R) for n, m >= 1. Exhibiting a contrast in the case n=m=1 between the arithmetic and geometric frameworks, we show on the other hand that the space of vector fields on the tangent bundle T(P^1_k) lifting vector fields on P^1_k, where k is an algebraically closed field, has dimension 6 over k. Nevertheless, \"normalized\" vector fields, which play a role in Hamiltonian mechanics, do not exist on T(P^1_k). We proceed to prove a stronger result for the case n=m=1, that there are no effective Cartier divisors on J^2(P^1) that are finite-to-one over J^1(P^1), and discover that an analogous result holds in geometry. As a final result, we prove the nonexistence of lifts of Frobenius on the first jet space of any smooth quadric hypersurface in projective space."],"dc:identifier":["http://hdl.handle.net/1928/32319","https://digitalrepository.unm.edu/math_etds/28"],"dc:language":["English"],"dc:subject":["lifts of Frobenius","arithmetic geometry","vector fields"],"dc:title":["Lifts of Frobenius on Arithmetic Jet Spaces of Schemes"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Doctoral","Dissertation"],"thesis:degree_name":["Mathematics"]},"updated_at":"2026-07-24T05:27:37Z"}