Local-global problems in diophantine geometry

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)

Abstract

This thesis is dedicated to the study of rational solutions to Diophantine equations, or equivalently rational points on varieties. In particular, we consider families of equations and ask how frequently rational solutions exist to equations in these families. We also study the frequency of examples of local-global principles, such as the Hasse principle and weak approximation, for equations in these families. We develop an asymptotic formula, in Chapter 3, for the number of biquadratic fields K of bounded discriminant for which the equation N_{K/Q}(x) = c fails to satisfy the Hasse principle for each c in Q*. In Chapter 4, we study the frequency of Hasse principle failures in a particular family of rational surfaces. Families of varieties whose base is a hypersurface of low degree is the subject of Chapter 5. In Chapter 6, we describe the frequency with which conics of the form aX^2 +bY^2 +cZ^2 = 0 have non-trivial rational points. Finally, in the last chapter we study weak approximation for certain quadric surface bundles over P^2.
Date of Award23 Jun 2021
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorTim D Browning (Supervisor) & Andrew R Booker (Supervisor)

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