Rational points and lines on cubic hypersurfaces

by Christian Bernert
Doctoral thesis
Date of Examination:2023-06-26
Date of issue:2023-10-02
Advisor:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Damaris Schindler
Persistent Address: http://dx.doi.org/10.53846/goediss-10116

 

 

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Abstract

English

We study the solubility of cubic diophantine equations. In the first chapter, we discuss the convergence of the singular series of a cubic form, which is a central object in the study of the solutions of the associated cubic equation. We establish new bounds for the relevant exponential sums (Gauß sums) and use these to establish the absolute convergence of the singular series in the critical case of 10 variables. In the second chapter, which is joint work with Leonhard Hochfilzer, we use the Hardy-Littlewood Circle Method to prove the existence of solutions to homogeneous cubic equations in at least 14 variables over imaginary quadratic number fields. As an application, we deduce the existence of a rational line on a cubic hypersurface in at least 33 variables, improving on previous work by Wooley. In the final chapter, we study inhomogeneous cubic equations and establish the existence of solutions in new cases. Moreover, we provide a method to bound the smallest solution of such an equation, which also has interesting applications in the homogeneous case.
Keywords: Exponential sums; Analytic number theory; Hardy-Littlewood Circle Method; Cubic forms; Rational points; Diophantine equations; Number fields
 
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