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dc.contributor.advisorBjorn Poonen.en_US
dc.contributor.authorHewett, Campbell L.(Campbell Lucas)en_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Mathematics.en_US
dc.date.accessioned2020-09-03T16:40:53Z
dc.date.available2020-09-03T16:40:53Z
dc.date.copyright2020en_US
dc.date.issued2020en_US
dc.identifier.urihttps://hdl.handle.net/1721.1/126924
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 103-105).en_US
dc.description.abstractThe motivating problem of this thesis is that of explicitly computing the K-rational points of a regular nonsmooth curve X over a αnitely generated αeld K of characteristic p. We start with an in-depth study of such curves in general and the tools exclusive to characteristic p geometry needed to compute their K-points. We describe a combined going-down and going-up approach to compute X(K) that generalizes and makes effective the proof of finiteness of X(K) given by Voloch ([39]). We break the problem up into three separate cases according to the absolute genus of X. In the absolute genus 0 case, we give an algorithm to compute X(K) that is an effective version of a method given by Jeong ([16]). We also implement a special case of this algorithm in Sage and apply it to example curves. In the absolute genus 1 case, we give an algorithm to compute X(K) that works when we make extra assumptions about X, and we make some remarks in the case where those assumptions are removed. In the absolute genus at least 2 case, we give an unconditional algorithm to compute X(K). Some tools and algorithms we provide along the way do not directly involve regular nonsmooth curves and are interesting in their own right. We describe ways to effectively descend curves with respect to transcendental or purely inseparable field extensions. We explore the methods of p-descent on elliptic curves in characteristic p and provide explicit equations defining Z/pZ- and [ mu]p-torsors over them. We prove an effective de Franchis-Severi theorem for characteristic p that generalizes the one given by Baker, et al. over number fields ([3]). Lastly, we use a height bound proved by Szpiro ([34]) to give an algorithm to compute Y (K) for any smooth nonisotrivial curve Y over K followed by an algorithm to compute Y (K¹/p[infinity]), which was proved to be finite by Kim ([17]).en_US
dc.description.statementofresponsibilityby Campbell L. Hewett.en_US
dc.format.extent105 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.titleComputability of rational points on curves over function fields in characteristic pen_US
dc.typeThesisen_US
dc.description.degreePh. D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.identifier.oclc1191266689en_US
dc.description.collectionPh.D. Massachusetts Institute of Technology, Department of Mathematicsen_US
dspace.imported2020-09-03T16:40:53Zen_US
mit.thesis.degreeDoctoralen_US
mit.thesis.departmentMathen_US


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