Computability of rational points on curves over function fields in characteristic p
Author(s)Hewett, Campbell L.(Campbell Lucas)
Massachusetts Institute of Technology. Department of Mathematics.
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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 (). 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 (). 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 (). Lastly, we use a height bound proved by Szpiro () 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 ().
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020Cataloged from the official PDF of thesis.Includes bibliographical references (pages 103-105).
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology