Computability of rational points on curves over function fields in characteristic p

The 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]).

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2020

Author dc:creator

• Hewett, Campbell L.(Campbell Lucas)

Advisor dc:contributor.advisor

• Bjorn Poonen.

• Mathematics.

Statement 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.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/126924

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/126924