TDX
author
Vrioni, Brikena
authoremail
brikena.vrioni@yahoo.com
authoremailshow
false
director
Lario Loyo, Joan Carles
2022-01-26T10:26:45Z
2022-01-26T10:26:45Z
2021-10-29
http://hdl.handle.net/10803/673261
An algebraic variety defined over a field is said to have Diophantine stability for an extension of this field if the variety does not acquire new points in the extension. Diophantine stability has a growing interest due to recent conjectures of Mazur and Rubin linked to the well-known Lang conjectures, generalizing the celebrated Faltings theorem on rational points on curves of genus grater or equal than 2. Their framework is characteristic zero, and we shall focus on the analogous and related questions in positive characteristic. More precisely, the aim of the thesis is to initiate the study of Diophantine stability for curves and surfaces defined over finite fields. First we prove the finiteness of the finite field extensions where an algebraic variety can exhibit Diophantine stability (DS) in terms of its Betti numbers (the genus in the case of curves, the Hodge diamond in the case of surfaces, etc.) Then, we analyze the existence of curves with Diophantine stability. More precisely, for curves of genus g<=3 we give the complete list of (isomorphism classes of) DS-curves, and we also provide data on the candidate Weil polynomials for DS-curves of genus g=4 and 5. For curves of large genus, we exhibit certain families of DS-curves: Deligne-Lusztig curves, Carlitz curves, .... Finally, we also aim to make a contribution on surfaces defined over finite fields with Diophantine stability. From the classification of surfaces of Enriques-Munford-Bombieri we derive partial results and a census of DS-surfaces.
eng
Curves and surfaces over nite elds
Diophantine stability
A census for curves and surfaces with diophantine stability over finite fields
info:eu-repo/semantics/doctoralThesis
URL
http://www.tdx.cat/bitstream/10803/673261/1/TBV1de1.pdf
File
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application/pdf
TBV1de1.pdf